Manfred Jürgen Matschke Gerrit Brösel Business Valuation Functions, Methods, Principles utb 5520 Eine Arbeitsgemeinschaft der Verlage Böhlau Verlag · Wien · Köln · Weimar Verlag Barbara Budrich · Opladen · Toronto facultas · Wien Wilhelm Fink · Paderborn Narr Francke Attempto Verlag / expert verlag · Tübingen Haupt Verlag · Bern Verlag Julius Klinkhardt · Bad Heilbrunn Mohr Siebeck · Tübingen Ernst Reinhardt Verlag · München Ferdinand Schöningh · Paderborn transcript Verlag · Bielefeld Eugen Ulmer Verlag · Stuttgart UVK Verlag · München Vandenhoeck & Ruprecht · Göttingen Waxmann · Münster · New York wbv Publikation · Bielefeld Wochenschau Verlag · Frankfurt am Main 45520_Matschke_Griffleiste_SL5.indd 1 45520_Matschke_Griffleiste_SL5.indd 1 16.03.2021 16: 20: 26 16.03.2021 16: 20: 26 Prof. Dr. Gerrit Brösel, a former student of Matschke, is a successful author of textbooks. He has worked for one of the Big Four international auditing firms and the Chamber of Industry and Commerce as a publicly appointed expert on business valuation. Prof. Dr. Manfred Jürgen Matschke is one of the original exponents of the functional business valuation theory presented in this book that has grown incrementally in popularity in Germany over the last 50 years. 45520_Matschke_Griffleiste_SL5.indd 2 45520_Matschke_Griffleiste_SL5.indd 2 16.03.2021 16: 20: 27 16.03.2021 16: 20: 27 Manfred Jürgen Matschke / Gerrit Brösel Business Valuation Functions, Methods, Principles UVK Verlag · München 45520_Matschke_Griffleiste_SL5.indd 3 45520_Matschke_Griffleiste_SL5.indd 3 16.03.2021 16: 20: 27 16.03.2021 16: 20: 27 Umschlagmotiv: iStockphoto CHUNYIP WONG Bibliografische Information der Deutschen Nationalbibliothek. Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http: / / dnb.dnb.de abrufbar. © UVK Verlag 2021 - ein Unternehmen der Narr Francke Attempto Verlag GmbH + Co. KG Dischingerweg 5 · D-72070 Tübingen Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Internet: www.narr.de eMail: info@narr.de Einbandgestaltung: Atelier Reichert, Stuttgart CPI books GmbH, Leck utb-Nr. 5520 ISBN 978-3-8252-5520-6 (Print) ISBN 978-3-8385-5520-1 (ePDF) www.fsc.org MIX Papier aus verantwortungsvollen Quellen FSC ® C083411 ® 45520_Matschke_Griffleiste_SL5.indd 4 45520_Matschke_Griffleiste_SL5.indd 4 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 Preface Valuation methods originating in the Anglo-Saxon world (especially those methods from the USA) are considered state of the art when it comes to business valuation. These valuation methods are spread through numerous English-language articles and books and are applied in practice almost without exception. However, these marketvalue-oriented valuation methods were established through a theoretical model based on neoclassical finance theory. Moreover, what we now recognize as unrealistic premises are inconsistently combined. The result is for some kind of market value because that is what everyone seems to ask for. In practice especially, when money is to be earned, the methods used are most often those in vogue. However, buyers or sellers can find the purchase price they pay or are paid based on those valuation methods disregarded their decision value, and consequently, their good faith in fashionable valuation methods can lead to their incurring accompanied by an economic loss. Moreover, those buyers and sellers neither know the appropriate decision value nor are they aware of how it is derived.The alternative valuation theory, functional business valuation theory, arose in Germany in the 1970s and has been developing ever since. Its central aspect is that any valuation conducted is purpose-driven. While several studies dealing with these ideas can now be found in the English-language literature, the ideas of functional business valuation have not yet been made available to an international audience in a compact form. This book, which has already been published in German, Polish, and Russian, now offers an audience of English-speaking valuation professionals a theoretically sound alternative to the fashionable market-value-oriented form of valuation. The first chapter of the book presents the basics of functional business valuation theory and the three main functions of business valuation in the following three chapters. This book certainly does not ignore methods for determining market values but tests their suitability in real-world situations. Those methods are assigned to the relevant main function of business valuation to show the context in which they can be useful in practice. Chapter 5 then summarizes the purpose-oriented principles of business valuation. We would like to thank Mrs. A LICJA S OMMERMEIER , Mr. P ATRICK T HIELMANN , and Mr. C HRISTOPH H EIDLER for their support in the preparation of the manuscript, especially with regard to translation. Furthermore, we would also like to thank the publisher for their excellent cooperation. M ANFRED J ÜRGEN M ATSCHKE G ERRIT B RÖSEL 45520_Matschke_Griffleiste_SL5.indd 5 45520_Matschke_Griffleiste_SL5.indd 5 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 45520_Matschke_Griffleiste_SL5.indd 6 45520_Matschke_Griffleiste_SL5.indd 6 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 Content Overview Content Overview Page Preface V Contents IX List of Figures XV List of Symbols XXI 1 Fundamentals of Business Valuation 1 1.1 Basic Terminology 3 1.2 Concepts of Business Valuation 11 1.3 Occasions of Business Valuation 37 1.4 Matrix of Functional Business Valuation and Overview of Methods of Business Valuation 46 1.5 Selected Control Questions 52 2 Decision Function and Decision Value 53 2.1 Basics 55 2.2 Determination of Multi-Dimensional Decision Values 60 2.3 Determination of One-Dimensional Decision Values in Non-Dominated, Disjoint Conflict Situations of the Acquisition/ Sale Type 70 2.4 Selected Problems of Decision Value Determination 159 2.5 Selected Control Questions 210 3 Mediation Function and Arbitration Value 213 3.1 Basics 215 3.2 Value Determination in Non-Dominated Conflict Situations 218 3.3 Value Determination in Dominated Conflict Situations 255 3.4 Selected Control Questions 257 4 Argumentation Function and Argumentation Value 259 4.1 Basics 261 4.2 Value Determination 270 4.3 Selected Control Questions 325 5 Principles of Business Valuation 329 5.1 Principles of Business Valuation as a Norm System 331 5.2 Principles of Functional Business Valuation 338 5.3 Selected Control Questions 344 References 345 About the Authors 365 Index 367 45520_Matschke_Griffleiste_SL5.indd 7 45520_Matschke_Griffleiste_SL5.indd 7 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 45520_Matschke_Griffleiste_SL5.indd 8 45520_Matschke_Griffleiste_SL5.indd 8 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 Contents Contents Page Preface V Content Overview VII List of Figures XV List of Symbols XXI 1 Fundamentals of Business Valuation 1 1.1 Basic Terminology 3 1.2 Concepts of Business Valuation 11 1.2.1 Functional Business Valuation Theory via the German School 11 1.2.2 Neoclassical Business Valuation Theory via the Anglo-Saxon School 16 1.3 Occasions of Business Valuation 37 1.3.1 Requirement to Systematize Occasions of Business Valuation 37 1.3.2 Systematization of Occasions of Business Valuation According to the Main Functions 39 1.3.2.1 Conflict Situations of the Acquisition/ Sale Type and of the Merger/ Demerger Type 39 1.3.2.2 Non-dominated and Dominated Conflict Situations 41 1.3.2.3 Disjoint and Joint Conflict Situations 41 1.3.2.4 One-Dimensional and Multi-Dimensional Conflict Situations 42 1.4 Matrix of Functional Business Valuation and Overview of Methods of Business Valuation 46 1.4.1 Matrix of Functional Business Valuation 46 1.4.2 Overview of Methods of Business Valuation 48 1.5 Selected Control Questions 52 2 Decision Function and Decision Value 53 2.1 Basics 55 2.2 Determination of Multi-Dimensional Decision Values 60 2.2.1 Utility Values as Basics for Decision Value Determination 60 2.2.1.1 The Term “Utility Value” 60 45520_Matschke_Griffleiste_SL5.indd 9 45520_Matschke_Griffleiste_SL5.indd 9 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 X Contents 2.2.1.2 Target Plan and Decision Field as Parameters of the Utility Value 60 2.2.2 A General Model for the Determination of a Multi-Dimensional Decision Value 65 2.2.2.1 Determination of Decision Values as a Two-Step Calculation 65 2.2.2.2 Determination of the Base Program 65 2.2.2.3 Determination of the Valuation Program 66 2.2.2.4 Multi-Dimensional Decision Value 67 2.2.2.5 Set of the Acceptable Conflict Resolutions 69 2.2.2.6 Set of Agreement Resolutions 69 2.3 Determination of One-Dimensional Decision Values in Non-Dominated, Disjoint Conflict Situations of the Acquisition/ Sale Type 70 2.3.1 Examination steps on the Matrix of Functional Business Valuation 70 2.3.1.1 Overview 70 2.3.1.2 The Steps in Detail 71 2.3.1.2.1 First Step 71 2.3.1.2.2 Second Step 75 2.3.1.2.3 Third step 77 2.3.2 Characterizing the Conflict Situation 79 2.3.3 Valuation Methods 82 2.3.3.1 Basic Models of Marginal Price Determination 82 2.3.3.1.1 Basic Model of Decision Value Determination 82 2.3.3.1.1.1 Determination of the Base Program 82 2.3.3.1.1.2 Determination of the Valuation Program 83 2.3.3.1.1.3 Numerical Example 84 2.3.3.1.2 Basic Model of the Present Value Calculation 94 2.3.3.1.2.1 Structural Equality of the Price Limit Calculation and the Present Value Calculation 94 2.3.3.1.2.2 Extended Interpretation of the Term “Comparison Object” based on the Present Value Calculation 96 2.3.3.2 The State Marginal Price Model - a General Model 97 2.3.3.2.1 Basics 97 2.3.3.2.2 The Model from the Presumptive Buyer’s Perspective 100 2.3.3.2.2.1 Presentation 100 2.3.3.2.2.2 Numerical Example 102 45520_Matschke_Griffleiste_SL5.indd 10 45520_Matschke_Griffleiste_SL5.indd 10 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 Contents Contents XI 2.3.3.2.3 The Model from the Presumptive Seller’s Perspective 107 2.3.3.2.3.1 Presentation 107 2.3.3.2.3.2 Numerical Example 109 2.3.3.2.4 Consideration of Uncertainty 114 2.3.3.2.5 Critical Evaluation 120 2.3.3.3 Future Performance Value Method - a Partial Model 121 2.3.3.3.1 Presentation 121 2.3.3.3.2 Relation between the Total Model and the Partial Model 125 2.3.3.3.2.1 Derivation of the Future Performance Value Method 125 2.3.3.3.2.2 Numerical Example 134 2.3.3.3.3 Consideration of Uncertainty 139 2.3.3.3.4 Critical Evaluation 142 2.3.3.4 Approximate Decomposed Business Valuation - a Heuristic Model 145 2.3.3.4.1 Basics 145 2.3.3.4.2 Heurististic Planning Method of Approximate Decomposition under Consideration of Uncertainty 145 2.3.3.4.3 Combination between Business Valuation and Approximate Decomposition under Uncertainty 154 2.3.3.4.4 Critical Evaluation 157 2.4 Selected Problems of Decision Value Determination 159 2.4.1 Valuation of Small and Medium-Sized Enterprises 159 2.4.1.1 Valuation-relevant Idiosyncrasies 159 2.4.1.2 State Marginal Price Model in Light of Valuation-Relevant Idiosyncrasies 160 2.4.2 Effects on the Decision Value Through Modifications of the Decision Field 162 2.4.3 Determination of the Decision Value in Conflict Situations of the Merger Type and Demerger Type 166 2.4.3.1 Conflict Situations of the Merger Type 166 2.4.3.1.1 Presentation 166 2.4.3.1.2 Numerical Example 172 2.4.3.2 Conflict Situation of the Demerger Type 180 2.4.3.2.1 Presentation 180 2.4.3.2.2 Numerical Example 187 2.4.4 Joint Conflict Situations 199 2.4.4.1 Preliminary Remarks 199 2.4.4.2 Exemplary Representation of the Joint Situation of the Acquisition/ Acquisition Type 200 2.5 Selected Control Questions 210 45520_Matschke_Griffleiste_SL5.indd 11 45520_Matschke_Griffleiste_SL5.indd 11 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 XII Contents 3 Mediation Function and Arbitration Value 213 3.1 Basics 215 3.2 Value Determination in Non-Dominated Conflict Situations 218 3.2.1 Further Investigation Steps within the Matrix of Functional Business Valuation 218 3.2.1.1 Overview 218 3.2.1.2 The Steps in Detail 218 3.2.1.2.1 First Step 218 3.2.1.2.2 Second Step 222 3.2.1.2.3 Third Step 224 3.2.2 Selected Valuation Methods 225 3.2.2.1 Preliminary Remarks 225 3.2.2.2 Combined Valuation Methods 229 3.2.2.2.1 Mean Value Method 229 3.2.2.2.2 Goodwill Rent Methods 231 3.2.2.3 Overview of typical method-specific characteristics and their arbitration-theoretic interpretation 241 3.2.3 Selected Problems at the Determination of the Arbitration Value 245 3.2.3.1 Determination of the Arbitration Value at an IPO 245 3.2.3.2 Arbitration Value Determination at Auctions of Mergers & Acquisitions 252 3.3 Value Determination in Dominated Conflict Situations 255 3.4 Selected Control Questions 257 4 Argumentation Function and Argumentation Value 259 4.1 Basics 261 4.2 Value Determination 270 4.2.1 Steps of Determination within the Matrix of Functional Business Valuation 270 4.2.1.1 Overview 270 4.2.1.2 The Steps in Detail 270 4.2.1.2.1 First Step 270 4.2.1.2.2 Second Step 271 4.2.1.2.3 Third Step 280 45520_Matschke_Griffleiste_SL5.indd 12 45520_Matschke_Griffleiste_SL5.indd 12 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 Contents Contents XIII 4.2.2 Selected Valuation Methods 282 4.2.2.1 Comparison Methods 282 4.2.2.1.1 Comparison Methods based on Single Valuation 282 4.2.2.1.1.1 Stock-and-Debt Method 282 4.2.2.1.1.2 Similar Public Company Approach 284 4.2.2.1.1.3 Initial Public Offering Approach 289 4.2.2.1.2 Comparison Methods based on Overall Valuation 289 4.2.2.1.2.1 Recent Acquisitions Approach 289 4.2.2.1.2.2 Market Multiples Approach 290 4.2.2.2 Finance-theoretic Methods 292 4.2.2.2.1 Capital Market-theoretic Methods (DCF Methods) 292 4.2.2.2.1.1 Basics 292 4.2.2.2.1.2 Weighted Avarage Cost of Capital Approach 302 4.2.2.2.1.3 Adjusted Present Value Approach 308 4.2.2.2.1.4 Net Method (Flow to Equity Approach) 311 4.2.2.2.1.5 Summary Overview 313 4.2.2.2.2 Methods of Strategic Valuation 319 4.3 Selected Control Questions 325 5 Principles of Business Valuation 329 5.1 Principles of Business Valuation as a Norm System 331 5.1.1 Characteristics 331 5.1.2 Purposes 333 5.1.3 Sources 336 5.2 Principles of Functional Business Valuation 338 5.3 Selected Control Questions 344 References 345 About the Authors 365 Index 367 45520_Matschke_Griffleiste_SL5.indd 13 45520_Matschke_Griffleiste_SL5.indd 13 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 45520_Matschke_Griffleiste_SL5.indd 14 45520_Matschke_Griffleiste_SL5.indd 14 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 List of Figures Chapter 1 Page Figure 1.1: Contents of the economic subjective value concept 5 Figure 1.2: Determination of utility values 6 Figure 1.3: Price as an exchange interaction 7 Figure 1.4: Standard value 8 Figure 1.5: Value types within the functional business valuation (main functions) 13 Figure 1.6: Starting point of the A RROW -D EBREU example 18 Figure 1.7: Market simulation of the starting point 18 Figure 1.8: Payments of financial instruments based on states 19 Figure 1.9: Representation of arbitrage based on specific traded securities 20 Figure 1.10: Representation of arbitrage based on financial instruments 20 Figure 1.11: Optimal market portfolio and capital market line 25 Figure 1.12: Review of actual market prices 29 Figure 1.13: Incompatible assumptions of the DCF elements 35 Figure 1.14: Classification of occasions of business valuation 38 Figure 1.15: Demerger types with changed ownership interests 40 Figure 1.16: Classification of joint/ affiliated conflict situations 42 Figure 1.17: Classification of valuation situations for the main functions 45 Figure 1.18: Matrix of functional business valuation 47 Figure 1.19: Selected methods of business valuation 48 Figure 1.20: Major differences between theory of finance and investment theory 50 Figure 1.21: Methods of the market-oriented value determination 51 Chapter 2 Figure 2.1: Presentation of an agreement situation in a conflict situation of the acquisition/ sale type with price as the only conflictresolution-relevant fact 57 Figure 2.2: Multi-dimensional conflict situation of the acquisition/ sale type 58 Figure 2.3: Parameters and steps for the determination of the utility value for an alternative 64 Figure 2.4: Matrix of functional business valuation 71 Figure 2.5: Systematization of planning processes under uncertainty 76 Figure 2.6: Conflict cube of the acquisition/ sale type for a non-dominated, disjoint, and one-dimensional conflict situation 79 Figure 2.7: Overview of the used symbols in the basic model representation 82 Figure 2.8: Decision situation of the buyer 85 Figure 2.9: Set A of alternatives of the buyer 86 Figure 2.10: Alternative set B (P) of the buyer 87 Figure 2.11: Set of non-dominated alternatives B (P) of the seller 88 45520_Matschke_Griffleiste_SL5.indd 15 45520_Matschke_Griffleiste_SL5.indd 15 16.03.2021 16: 20: 28 16.03.2021 16: 20: 28 XVI List of Figures Figure 2.12: Relative target contributions of the investment objects 88 Figure 2.13: Base program of the buyer 89 Figure 2.14: Valuation program of the buyer (determination with the tabular method) 90 Figure 2.15: Decision value of the buyer 91 Figure 2.16: Decision situation of the seller 92 Figure 2.17: Base program of the seller 92 Figure 2.18: Valuation program of the seller (determination with the tabular method) 93 Figure 2.19: Decision value of the seller 93 Figure 2.20: Data of the example from the buyer’s perspective 103 Figure 2.21: Comprehensive financial plan of the buyer’s base program 104 Figure 2.22: Comprehensive financial plan of the buyer’s valuation program 105 Figure 2.23: Determination of the buyer’s comparison object 106 Figure 2.24: Determination of the decision value from the buyer’s perspective based on the internal rate of return of the comparison object 107 Figure 2.25: Data of the example from the seller’s perspective 110 Figure 2.26: Comprehensive financing plan of the seller’s base program 111 Figure 2.27: Comprehensive financial plan of the seller’s valuation program 112 Figure 2.28: Determination of the seller’s comparison object 113 Figure 2.29: Determination of the decision value from the seller’s perspective based on the internal rate of return of the comparison object 114 Figure 2.30: Pessimistic data of the example from the buyer’s perspective 116 Figure 2.31: Comprehensive financial plan of the buyer’s base program from a pessimistic perspective 116 Figure 2.32: Comprehensive financial plan of the buyer’s valuation program from a pessimistic perspective 117 Figure 2.33: Optimistic data of the example from the buyer’s perspective 117 Figure 2.34: Comprehensive financial plan of the buyer’s base program from an optimistic perspective 118 Figure 2.35: Comprehensive financial plan of the buyer’s valuation program from an optimistic perspective 119 Figure 2.36: Future performance value with growth rate w 125 Figure 2.37: Upper and lower limit of the decision value P max 135 Figure 2.38: Components of the complex calculation formula for the buyer 136 Figure 2.39: Upper and lower limit of the decision value P min 137 Figure 2.40: Components of the complex calculation formula for the seller 138 Figure 2.41: Estimated frequency distribution of the future performance value 140 Figure 2.42: Frequency function of the future performance value 141 Figure 2.43: Risk profile of the future performance value 142 Figure 2.44: Basics and steps of the approximative decomposition 146 Figure 2.45: Decision about feedback or termination 151 Figure 2.46: Overview of the approximative decomposition 153 Figure 2.47: Approximate decomposed valuation 156 Figure 2.48: Exemplary data from the buyer’s perspective - variation A 162 45520_Matschke_Griffleiste_SL5.indd 16 45520_Matschke_Griffleiste_SL5.indd 16 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 List of Figures XVII Figure 2.49: Comprehensive financial plan of the buyer’s base program - variation A 163 Figure 2.50: Comprehensive financial plan of the buyer’s valuation program - variation A 163 Figure 2.51: Exemplary data from the seller’s perspective - variation B 164 Figure 2.52: Comprehensive financial plan of the seller’s base program - variation B 165 Figure 2.53: Comprehensive financial plan of the seller’s valuation program - variation B 165 Figure 2.54: Data for the determination of the pre-merger program 173 Figure 2.55: Comprehensive financial plan of the pre-merger program 173 Figure 2.56: Data for the determination of the merger program 174 Figure 2.57: Comprehensive financial plan of the merger program (case a) 175 Figure 2.58: Comprehensive financial plan of the merger program (case b1) 177 Figure 2.59: Comprehensive financial plan of the valuation program (case b1) 178 Figure 2.60: Comprehensive financial plan of a merger program (case b2) 179 Figure 2.61: Comprehensive financial plan of the valuation program (case b2) 180 Figure 2.62: Special case of a structural change of ownership at a non-ratio-preserving demerger into two businesses (split-up) 188 Figure 2.63: Data for the determination of the pre-demerger program 189 Figure 2.64: Comprehensive financial plan of the pre-demerger program 190 Figure 2.65: Share of X in the distribution of the pre-demerger program 190 Figure 2.66: Data of business Ü 1 for the determination of the demerger program 191 Figure 2.67: Data of business Ü 2 for the determination of the demerger program 191 Figure 2.68: Comprehensive financial plan of business Ü 1 in the demerger program (case a) 192 Figure 2.69: Comprehensive financial plan of business Ü 2 in the demerger program (case a) 193 Figure 2.70: Comprehensive financial plan of the valuation program of shareholder X (case a) 194 Figure 2.71: Comprehensive financial plan of business Ü 1 in the demerger program (case b) 195 Figure 2.72: Comprehensive financial plan of business Ü 2 in the demerger program (case b) 196 Figure 2.73: Comprehensive financial plan of the valuation program of shareholder X (case b) 198 Figure 2.74: Conflict cube for a non-dominated, joint, and one-dimensional conflict situation 200 Figure 2.75: Data of the joint conflict situation of the acquisition/ acquisition type 201 Figure 2.76: Joint marginal price of business U A 203 Figure 2.77: Joint marginal price of business U B 207 45520_Matschke_Griffleiste_SL5.indd 17 45520_Matschke_Griffleiste_SL5.indd 17 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 XVIII List of Figures Figure 2.78: Decision values in a joint conflict situation of the acquisition/ acquisition type 208 Figure 2.79: Example for the state marginal price model 211 Figure 2.80: Example of joint conflict situations 212 Chapter 3 Figure 3.1: Arbitration area at freedom of choice 216 Figure 3.2: Set of all conflict resolutions S and the agreement area E 219 Figure 3.3: Dominance of agreements 220 Figure 3.4: Relations of dominance 221 Figure 3.5: Explanation of the rule of absolutely equal division and the rule of relatively equal division by means of a numerical example 223 Figure 3.6: Conflict cube for a non-dominated, disjoint, one-dimensional conflict situation 226 Figure 3.7: Constellations for the application of the traditional combination methods to determine the arbitration value 228 Figure 3.8: Example for the mean value method 230 Figure 3.9: Example for goodwill rent method I (majority opinion) 233 Figure 3.10: Example for goodwill annuity method I (minority opinion) 234 Figure 3.11: Example for goodwill rent method II (majority opinion) 235 Figure 3.12: Example for the method of profit layering (F RITZ method) 236 Figure 3.13: Determination of the adjusted operating profit with the EVA method 238 Figure 3.14: Determination of the invested capital with the EVA method 239 Figure 3.15: Factors of the combined valuation methods 242 Figure 3.16: Initial situation of the determination of the arbitration value 243 Figure 3.17: Exemplary arbitration values and advance benefit distributions 244 Figure 3.18: Exemplary arbitration values and no advance benefit distributions 244 Figure 3.19: Issuing processes 246 Figure 3.20: Methods of issuing price determination 247 Figure 3.21: Formal definition of the bookbuilding process 248 Figure 3.22: Variants of auction processes 250 Figure 3.23: Formal definition of the Dutch price tender process 251 Figure 3.24: Formal definition of the American price tender process 251 Figure 3.25: Process of an M&A auction 253 Figure 3.26: Forms of M&A auctions 254 Chapter 4 Figure 4.1: Use of argumentation values in business valuation 263 Figure 4.2: Distribution of the valuation-relevant profit contribution considering an internal delimitation of responsibilities 264 45520_Matschke_Griffleiste_SL5.indd 18 45520_Matschke_Griffleiste_SL5.indd 18 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 List of Figures XIX Figure 4.3: Negotiation as a mere sequence of price offers 265 Figure 4.4: Main features of argumentation values 267 Figure 4.5: Argumentation factors of business valuation 271 Figure 4.6: Present value annuity factor, depending on the interest rate i and the observation period n 274 Figure 4.7: Present value annuity factor of a (negatively) growing annuity at i = 5 % 276 Figure 4.8: Present value annuity factor of a (negatively) growing annuity at i = 10 % 276 Figure 4.9: Argumentation values, according to the stock-and-debt method 284 Figure 4.10: Argumentation values, according to the similar public company approach (a single comparable business) 288 Figure 4.11: Argumentation values, according to the similar public company approach (several comparable businesses) 289 Figure 4.12: EBIT and sales multiples for the business value (as of September 2004) 291 Figure 4.13: Variants of DCF methods 293 Figure 4.14: Pragmatic CAPM-based determination of the cost of equity 296 Figure 4.15: Indirect determination of the free cash flow 298 Figure 4.16a: Comparison of different free cash flow definitions 299 Figure 4.16b: Comparison of different free cash flow definitions 300 Figure 4.17: Relationship between different cash flow terms 301 Figure 4.18: Input data of the APV examples (predefined debt capital) 310 Figure 4.19: Overview of the DCF methods 314 Figure 4.20: Price movements in the binomial tree over n periods 321 Figure 4.21: Exercise example für the APV approach 327 Chapter 5 Figure 5.1: Purposes for principles of functional business valuation 333 Figure 5.2: (Basic) principles of functional business valuation 339 45520_Matschke_Griffleiste_SL5.indd 19 45520_Matschke_Griffleiste_SL5.indd 19 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 45520_Matschke_Griffleiste_SL5.indd 20 45520_Matschke_Griffleiste_SL5.indd 20 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 List of Symbols Chapter 1 a, b price of goods X and Y expressed in monetary units beta factor of an unencumbered enterprise j beta factor of the indebted enterprise j β j ∆r M beta factor as a measure of the systematic risk of an investment j compared with the market portfolio M expected distributions of an investment j at time t + 1 expected distributions of a fraction of the equity of entity j at time t market risk premium for systematic market risk ∆r j FT j i risk premium for an investment j pure financing title (A RROW -D EBREU financing title) with a deposit of one monetary unit in the state s j risk-free capital market interest rate expected market value of an investment j at time t + 1 expected market value of an investment j at time 0 market value of an investment j at time t market value of a fraction in the equity of company j at time 0 price of a unit of risk (related to the standard deviation) n price of a unit of risk (related to variance) utility of the risk-free monetary unit at time 0 utility of 1+i risk-free monetary units at the moment number of net single financial instruments p i ρ j A RROW -D EBREU price of securities portfolio equilibrium price of security i price of a security i price of a pure financing instrument j; discount factor for a payment in state s j correlation coefficient between investment j and market portfolio Mexpected one-period returns on investment j return required by a company's equity investors j (cost of equity) return required by a company's lenders j (cost of debt) r M return required by a company's investors j (weighted average cost of capital) expected return of a risky market portfolio expected return of the risky optimal market portfolio β j FK=0 β j FK>0 D j,t +1 * D j,t *EK K j,t +1 * K j,0 K j,t K j,0 EK λ * λ N([1] 0 ) N([1+ i] 1 ) P * p i * ρ j,M r j * r j,EK * r j,FK * r j,GK * r M * 45520_Matschke_Griffleiste_SL5.indd 21 45520_Matschke_Griffleiste_SL5.indd 21 16.03.2021 16: 20: 29 16.03.2021 16: 20: 29 XXII List of Symbols return required by equity investors of an unlevered firm j; cost of equity of an unlevered firm j σ j,M return required by equity investors of an indebted firm j; cost of equity of an indebted firm j standard deviation of the one-period return capital investment j covariance between the uncertain single-period returns on investment j and the market portfolio M standard deviation of the (optimal) market portfolio s j (s 1 *, …, s n *) variance of the (optimal) market portfolio environmental status j characteristics of facts relevant to conflict resolution S 1 , ..., S n , on which the conflict parties have agreed in an agreement price as a quantitative ratio of exchangeable goods X and Y S 1 , …, S n facts relevant to conflict resolution market value of a company j at time 0 market value of the equity of a company j at time 0; market value of all equity securities of a company j at time 0 market value of the debt capital of a company j at time 0; market value of all debt instruments of a company j at time 0 WP i x security i quantity of good X expected cash flow per all capital investors of enterprise j at time t expected cash flow for all equity investors of a company j at time t y z ij expected cash flow per all investors of borrowed capital of enterprise j at time t quantity of good Y conditional payment of a security i in the environmental state s j expected payment of an investment j at time t + 1 Chapter 2 expected payment of capital investment j at time t + 2 A K A V advantage of a buyer advantage of a seller minimum shareholding (marginal shareholding) in the new company created by the merger minimum share (marginal share) of shareholder h in the demerger company Ü f after the demerger A a i a ct a opt set of all alternatives a i alternative i of the set A optimal alternative; base program AEW AK r j,EK *FK=0 r j,EK *FK>0 σ j σ M σ M 2 tg α, tg β W j,0 W j,0 EK W j,0 FK X j,t * X j,t *EK X j,t *FK Z j,t +1 * Z j,t+2 * α min α min h Üf A 0 neu ABW U ABW VO 45520_Matschke_Griffleiste_SL5.indd 22 45520_Matschke_Griffleiste_SL5.indd 22 16.03.2021 16: 20: 30 16.03.2021 16: 20: 30 List of Symbols XXIII β β h participant's share in the company Ü without merger fraction of shareholder h in the company UG without demerger B * valuation program; subset of the alternatives B(s 1 , ..., s n ) open to the decision subject after agreement on the conflict resolution (s 1 , ..., s n ), for which the utility N(b opt (s 1 , ..., s n )) is just equal to or minimally greater than the utility N(a opt ) of the base program (not sign-constrained) autonomous deposit surpluses of the investment and financing objects from the buyer's perspective at time t (not sign-constrained) autonomous deposit surpluses of the investment and financing objects from the seller's perspective at time t B (s 1 , …, s n ) set of all possible actions b j when conflict resolution is agreed upon (s 1 , ..., s n ) alternative j in case of agreement on conflict resolution (s 1 , ..., s n ) optimal alternative in case of agreement on conflict resolution (s 1 , ..., s n ) BEW present value of a growing pension net present value from the buyer's perspective net present value from the seller's perspective (non-negative) net present value for an investment or financing object included in the buyer's valuation program j (non-negative) net present value for an investment or financing object included in the seller's valuation program j net present value difference due to restructuring from the base to the valuation program from the buyer's perspective net present value difference due to restructuring from the base to the valuation program from the seller's perspective dual variable for the restriction of the protection of the extraction current d t E e ij dual variables for the liquidity constraints in t = 0, ..., T agreement set as the intersection of those quantities that comprise the conflict resolutions that are reasonable from the perspective of each conflict party preference-relevant consequence of an alternative i in the environmental state s j ; result constellation e bt ED EM current investment deposit EN width of the withdrawal flow for consumption purposes maximum width of the withdrawal flow of the base program from the buyer's perspective maximum width of the withdrawal flow of the base program from the seller's perspective width of the withdrawal flow of the valuation program from the buyer's perspective β h b Kt b Vt B alt B neu b j (s 1 , …, s n ) b opt (s 1 , …, s n ) BW 0 C K C V C Kj Be C Vj Be ΔKW K Be−Ba ΔKW V Be−Ba δ EMW U EN K Ba max EN V Ba max EN K Be 45520_Matschke_Griffleiste_SL5.indd 23 45520_Matschke_Griffleiste_SL5.indd 23 16.03.2021 16: 20: 30 16.03.2021 16: 20: 30 volume of the cash withdrawal stream of the seller's valuation program maximum possible withdrawals of the company F created after the merger (merger program); maximum benefit of all conflicting parties from the company F after a merger maximum width of the extraction flow based on the optimistic input data variant maximum width of the extraction flow based on the pessimistic input data variant maximum amount of cash withdrawal based on a realistic version of the baseline data maximum width of the withdrawal flow from the company created by the demerger Ü f maximum width of the withdrawal flow from the company UG without splitting maximum amount of withdrawal flow from enterprise A depending on the price of enterprise B maximum amount of withdrawal flow from enterprise B depending on the price of enterprise A maximum width of the withdrawal flow from the buyer's perspective; benefit of the base program from the buyer's perspective maximum width of the withdrawal flow from the seller's perspective; benefit of the base program from the seller's perspective maximum possible withdrawals of the contributing company Ü of the valuation subject (pre-merger program); maximum benefit of the valuation subject from the company Ü without agreement on merger EW f(a i , z j ); f: A × Z → K (objectified) capitalized earnings value (as arbitration or as argumentation value) outcome function; assignment of an outcome constellation e ij to an alternative i and an environmental state z j (not sign-constrained) cash inflows of the investment and financing objects from the buyer's perspective at time t GA (unsigned) cash inflows of the investment and financing objects from the seller's perspective at time t future cash inflows of the company to be valued U from the buyer's perspective future cash inflows of the company to be valued U from the seller's perspective H ij (r) H ijt (r) H ijv (r) result level r per alternative i and per environmental state z j result level r per alternative i and per environmental state z j at time tresult level r per alternative i and per environmental state z j as well as per result type v EN V Be EN F max EN opt max EN pess max EN real max EN Üf max EN UG max EN UA max (P UB ) EN UB max (P UA ) EN K max EN V max EN Ü g Kjt g Vjt g UKt g UVt XXIV List of Symbols 45520_Matschke_Griffleiste_SL5.indd 24 45520_Matschke_Griffleiste_SL5.indd 24 16.03.2021 16: 20: 30 16.03.2021 16: 20: 30 H ijtv (r) result level r per alternative i and per environmental state z j as well as per result type v at time t i i K (uniform subjective) calculation interest rate in determining the future performance value ZEW (period-specific subjective) calculation interest rate in determining the future performance value ZEW calculation interest rate from the buyer's perspective i V calculation interest from the seller's perspective period-specific internal rates of return (internal interest rates) of the base program period-specific internal rates of return (internal interest rates) of the base program from the buyer's perspective period-specific endogenous marginal interest rate feet of the base program from the seller's perspective I b period-specific internal rates of return (internal interest rates) of the valuation program (period-specific endogenous) marginal interest rate feet of the valuation program from the buyer's perspective period-specific endogenous marginal interest rate feet of the valuation program from the seller's perspective investment objects available for the valuation subject with b ∈ {1, ..., B} IF K KKA internal financing of the company set of all possible preference-relevant consequences or result constellations e ij amount of investment capital available to the valuation subject at valuation time t = 0 K 1 , ... , K 9 KU ∅ alternative combinations of the non-price conflict-resolution-relevant facts empty set LW µ expected value MU utility of a monetary unit expected at time 0 utility of 1 + i at time 1 expected monetary units benefit of an alternative i N(a opt ) N(b j (s 1 , …, s n )) N(b opt (s 1 , …, s n )) success/ benefit of the base program success/ utilization of an alternative b j when conflict resolution is agreed upon (s 1 , ..., s n ) success/ benefit of an alternative b j when conflict resolution is agreed upon (s 1 , ..., s n ) N b utility value assigned to the investment object I b by the decision subject N Ba N Be utility of the base program utility of the valuation program i τ i t Ba i K τ Ba i V τ Ba i t Be i K τ Be i V τ Be N 1 ⎡⎣ ⎤⎦ 0 ( ) N 1 + i ⎡⎣ ⎤⎦ 1 ( ) N(a i ) List of Symbols XXV 45520_Matschke_Griffleiste_SL5.indd 25 45520_Matschke_Griffleiste_SL5.indd 25 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 n ij partial benefit of a result constellation e ij overall utility of the base program from the the buyer's perspective N U N VO value in use of the company from the perspective of the valuation subject overall utility of the base program from the seller's perspective benefit of the comparison object P P b P max agreement price price to be paid for investment object I b at valuation time t = 0, investment amount per unit of investment object maximum price payable from the buyer's perspective P min minimum asking price from the seller's perspective maximum price payable on the basis of the optimistic input data variant maximum price payable on the basis of the pessimistic input data variant maximum payable price based on the realistic input data variant P U P VO price of the company still to be negotiated U price of the comparison object maximum payable price for the company U A in dependence for the price of the company U B maximum payable price for the company U B in dependence for the price of the company U A compounding factor 1 + i period-specific discount factors applicable to the buyer's base program period-specific discount factors applicable to the seller's base program period-specific discount factors applicable to the buyer's valuation program r K r V period-specific discount factors applicable to the seller's valuation program internal interest rate of the comparative object from the buyer's perspective internal interest rate of the comparative object from the seller's perspective internal rate of return (interest rate) of the comparative object of the buyer r VO s s 1 , …, s n internal rate of return (interest rate) of the comparative object of the seller internal interest rate of the comparative object standard deviation characteristics of the facts relevant for conflict resolution (s 1 , …, s n ) a conflict resolution; possible settlement solution N K (a opt ) N V (a opt ) P max opt P max pess P max real P max UA (P UB ) P max UB (P UA ) q t ρ Kt Ba ρ Vt Ba ρ Kt Be ρ Vt Be r VO K r VO V XXVI List of Symbols 45520_Matschke_Griffleiste_SL5.indd 26 45520_Matschke_Griffleiste_SL5.indd 26 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 S 1 , …, S n S facts relevant to conflict resolution reliability index set of all conflict resolutions {(s 1 , ..., s n )} S zK S zV S z t set of reasonable conflict resolutions from the buyer's perspective set of reasonable conflict resolutions from the seller's perspective set of reasonable conflict resolutions from the perspective of one party time, time index τ (auxiliary) time index U u j UG company to be valued dual variables for the capacity constraints with j = 1, ..., J Ü VG objekt advantageousness of an object VO W w w Kt multi-dimensional decision value; set of all conflict resolutions (s 1 , ..., s n ) for which the utility N(b opt (s 1 , ..., s n )) is equal to or minimally greater than the utility N(a opt ) of the base program constant growth rate of an annuity time structure factor for the withdrawals from the buyer's perspective w Vt time structure factor for the withdrawals from the seller's perspective Number of investment or financing object to be realized from buyer's perspective capacity constraints per investment or financing object from the buyer's perspective set of all environmental states z j z b z j z U ZE number of investment objects I b that can be acquired by the valuation subject with 0 ≤ z b ≤ z bmax (in case of arbitrary divisibility) or z b ∈ {0, 1, 2, ..., z bmax }(in case of integer) state of the environment j variable characterizing the acquisition/ sale of the company consistent subjective future success ZE t ZE K ZE V period-specific subjective future success constant (optimal) future success (cash surplus per period) from the buyer's perspective future success from the buyer's perspective future success from the seller's perspective ZE U ZE VO ZEW, ZEW U future success of the company being valued future success of the comparison object (subjective) future performance value (as decision value/ border price) of the enterprise U future performance value from the buyer's perspective future performance value from the seller's perspective SI Objekt x Kj x Kj max Z ZE K * ZEW K ZEW V List of Symbols XXVII 45520_Matschke_Griffleiste_SL5.indd 27 45520_Matschke_Griffleiste_SL5.indd 27 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 future performance value of company U from the buyer's perspective based on the period-specific discount factors of the base program future performance value of company U from the buyer's perspective based on the period-specific discount factors of the valuation program future performance value of company U from the seller's perspective based on the period-specific discount factors of the base program Chapter 3 future performance value of company U from the seller's perspective based on the period-specific discount factors of the valuation program AW method-specific weighting factor of traditional combinatorial methods arbitration value bBA BK cd E agreement set modified agreement set in a dominated conflict situation subset of efficient, non-dominated conflict resolutions from the agreement set EEVA EW subset of inefficient, dominated conflict resolutions from the agreement set constant future earnings surplus (for capitalized earnings value) economic value added earned value f gGR goodwill amortization rate goodwill annuity; excess profit derivative goodwill i * i ** k original goodwill capitalization/ calculation interest rate (for capitalized earnings value) goodwill annuity interest rate; capitalization/ calculation interest rate for discounting the goodwill annuity (excess profit) (in the profit-shifting method/ procedure II of goodwill annuities) cost of capital as weighted average of cost of equity and cost of debt, decision value of buyer MVA NG NOA market value added normal profit net operating assets NOPaT net operating profit after taxes decision value of the buyer; maximum payable price decision value of the seller; minimum price to be demanded ZEW U K (ρ Kt Ba ) ZEW U K (ρ Kt Be ) ZEW U V (ρ Vt Ba ) ZEW U V (ρ Vt Be ) a E ' ˆ E E GW deri GW orig P max P min XXVIII List of Symbols 45520_Matschke_Griffleiste_SL5.indd 28 45520_Matschke_Griffleiste_SL5.indd 28 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 SW return on invested capital NOA (defined individually for each company as required for operation) in a period t substance value TUW v number of years to consider a goodwill annuity GR company value discount factor; reciprocal of the compounding factor q t Chapter 4 AA BO AK BO number of shares issued in the valuation object (average) price of the shares of the valuation object AK VU BG BO BG VU price of the share of the comparable company selected reference value of the valuation object selected reference value at the comparable company continuing value, terminal value, residual value, value of all performance indicators discounted to the end of the detailed planning period E T EK success variable at time T market value of equity market value of the equity of an indebted company according to the adjusted present value approach market value of the equity of an indebted company according to the free cash flow approach EM 0 FA market value of the equity of an indebted company according to the flow-to-equity approach market value of the equity of an indebted company according to the total cash flow approach success multiplier at time 0 fungibility discount FK FCF FTE GK market value of debt capital free cash flow; cash flow available to providers of equity and debt capital flow to equity; inflow to equity providers (after income taxes) market value of total capital of the company GK e GK f GW i, i rf market value of the total capital of an unleveraged (only self-financed) company market value of the total capital of an indebted (i.e., also or only debt-financed) company risk-free capital market interest rate; borrowing rate k k e risk-adequate capitalization rate cost of capital of an unleveraged, only self-financed company k f M VU MK VU OW cost of capital of an indebted, equityand debt-financed company multiplier applicable at the comparable company market capitalization of the comparable company PZ r NOA t v t CV τ FCF EK APV EK FCF EK FTE EK TCF List of Symbols XXIX 45520_Matschke_Griffleiste_SL5.indd 29 45520_Matschke_Griffleiste_SL5.indd 29 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 return demanded by the equity investors of an unleveraged company sTCF TS return demanded by the equity providers of an indebted company profit tax rate total cash flow tax shield UW UW APV UW FCF company value company value according to the adjusted present value approach company value according to the flow to equity approach company value according to the free cash flow approach UW TCF W 0 W BO W T company value according to the total cash flow approach discounted target selling price of the company at time 0 value of the valuation object (according to the method of corrected stock market value) target sales price of the company at time T X Z 0 cash flow available to all providers of capital, i.e., equity and debt providers capital injection by the venture capitalist at time 0 Chapter 5: ZE r e r f UW FTE XXX List of Symbols 45520_Matschke_Griffleiste_SL5.indd 30 45520_Matschke_Griffleiste_SL5.indd 30 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 Chapter 1: Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 1 45520_Matschke_Griffleiste_SL5.indd 1 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 Overview The first chapter contains an Introduction to the Topic of Business Valuation. Section 1.1 provides the essential definitions relating to business valuation. The functional business valuation concept of the German School and the neoclassical business valuation concept of the Anglo-Saxon school are presented in Section 1.2. The following discussion focuses on the functional conception; a theory contrasted with the neoclassical concept, developing its perspective depending on the conditions of the real world. Section 1.3 offers a systematization of business valuation events, according to the main functions. Finally, Section 1.4 goes on to study the matrix of functional business valuation and its substantial procedures in more detail, thereby providing the basis for the following chapters. The first chapter concludes the introduction with selected control questions (Section 1.5). Learning objectives After studying this chapter, you should 1. understand the meanings of the terms: valuation subject, valuation object, value, and further specific terms used as a part of the business valuation process; 2. know which general value theories can be found in literature; 3. be able to name the differences between the prevailing concepts of business valuation in German and Anglo-Saxon literature; 4. know the fallacies of the neoclassical valuation theory; 5. be able to differentiate between main and minor functions; 6. be able to systematize the events of main functions; 7. know the purpose of a matrix of functional business valuation theory; and 8. be able to name and systematize the business valuation methods. Kapitel 1: Einführung 2 2 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 2 45520_Matschke_Griffleiste_SL5.indd 2 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 Chapter 1 1.1 Basic Terminology Valuation means the allocation of a value by a valuation subject, in most cases in the form of a monetary value, to a valuation object (S IEBEN / L ÖCHERBACH / M ATSCHKE 1974, p. 840, M ATSCHKE 2017a). The valuation subject is the person for whom the valuation is conducted. A valuation subject can be a single natural or legal entity or a group. The terms decision subject, and presumptive buyer, and presumptive seller in targeted acquisition or sale situations are used below. Since the main functions of business valuation concentrate on interpersonal conflicts, the opposing parties in the negotiation representing the valuation subjects are called conflicting parties. If a valuation for one of the conflicting parties made by a third party, the client (i.e., the conflicting party) is the valuation subject but an external expert, such as an appraiser, is not. If the client for a valuation report is not one of the conflicting parties, but for instance a court, the parties to the conflict are the valuation subjects for whom the court has to reconcile the interests of disputes through a court order or court ruling. However, the terms company or business in the context of business valuation describe the valuation object, that is, the object that is valuated. Hence, the company is to be the object of valuation in that the valuation is determined by a value calculation. An object of valuation can be the company as a whole but also definable parts of the company. That is not a contradiction because the term definable parts of the company is used to describe complex divisions of a company (e.g., individual facilities, or divisions), and sometimes even shares in a company, for example, in the form of a block of shares or shares of a limited liability company that can be characterized as similar to an entire business. The term definable is thus not limited to the spatial delimitation of part of a business, but also applies occasionally in the sense of delimitating an abstract share of an entire business (B ALLWIESER / H ACHMEISTER 2016, p. 6). The term as a whole means that the valuation object constitutes a complex and unique conglomeration of tangible and intangible assets (production factors). The value of this conglomeration of assets stems from the utility provision for the valuation subject. It arises from the best possible efficient combination of the production factors. Successful entrepreneurial action causes the whole to be more valuable than the sum of its parts, resulting in value-increasing effects (positive effects of synergy, positive economies of scope, or original goodwill). These advantages of a combination are lost if the whole is demerged into its individual parts. Nevertheless, it could be that value-increasing effects do not appear until they are triggered by a change to the encountered whole valuation object, because previous, barely successful actions have not created positive effects from a combination or may even have had negative effects: Therefore, a company valuation has to be preceded by comprehensive due diligence, which is meant to discover value-increasing potential, referring to the whole company or its definable parts, from the view of the valuation subject. While one valuation subject (e.g., a buyer) is able to realize the value-increasing potential relating to the further opportunities, another (e.g., a seller) cannot recognize them due to a lack of financial means and abilities. An increase in value can be achieved if the negative effects of combination are curtailed. Such a company analysis is indispensable not only for the buyer to recognize the emerging opportunities of prospective successful entrepreneurial action or to assess the risks in relation to a purchase, but also 1.1 Basic Terminology 3 45520_Matschke_Griffleiste_SL5.indd 3 45520_Matschke_Griffleiste_SL5.indd 3 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 for the sellers of a company so that they do not have to sell their own company for less than it is worth. The development of value-increasing potential necessitates extensive, complex, and lengthy legal, organizational, industrial, or financial restructuring of the company and therefore is only recognizable with regard to an analysis and valuation of a company as a whole and should become discernible in terms of its advantages and disadvantages and also well as opportunities and risks with regard to the strategic plan of the respective valuation subject. Consequently, the valuation of a business needs to be embedded into the plan of the valuation subject. The business value is subjective, as it depends on sound planning and thus on the future performance of the business. The awareness of the subjectivity of a value is a long-established finding from economic literature (see G OSSEN 1854, M ENGER 1871 such as H ERBENER / R APP 2016). The value of an asset depends on a target and a preference system and on the decision field of the valuation subject derived from its individual marginal utility and that value is therefore subjective. Every student of economics is probably aware of the aphorism that the first beer provides a greater benefit than the tenth (H ERING 2000a, p. 435). The decreasing value is to be strictly distinguished from the constant price. After consuming a certain amount of beer, the guest does not ask for beer anymore, even if the market value stays constant. The subjective value of beer has fallen below the objective price of a beer. This means that the net present value of drinking is negative and continuing to drink (not only as an economic factor) would prove detrimental (H ERING 2004b, p. 108). That is why the economic value concept is understood as a subject-object-object-relationship (M ATSCHKE 1972, p. 147). The value represents the utility which the valuation subject (during a certain period and at a certain location) expects from the valuation object with regard to other available comparable objects. Hence, this is not only a question of value judgment, but one of rationally explainable determination, which results from the economic fundamentals of 1. the infinity of human needs, and in opposition 2. the scarcity of available goods. Hence, a value can be determined by everyone who has the relevant information. This means that the valuation object has a tangible value only relative to a valuation subject. Therefore, it cannot have a value per se, but only a value for someone. In literature (S IEBEN / L ÖCHERBACH / M ATSCHKE 1974) five different meanings of the subjective term value are elicited, which arise in accordance with certain objectives of value determination (see Figure 1.1): 1. Value in the sense of utility value (success, utility, value in use), that is, of the level of target performance for a subject; 2. value in the sense of decision value, that is, the concession limit of a determined (decision-)subject; 3. value in the sense of argumentation value, that is, a variable or size consulted by the subject to justify and to support the subject’s negotiation position; 4. value in the sense of a differently interpreted exchange value, that is, valid terms of trade of goods between a number of subjects; and 5. value in the sense of standard value, that is, information derived from norms/ standards of reality for a recipient interested in that reality. Kapitel 1: Einführung 4 4 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 4 45520_Matschke_Griffleiste_SL5.indd 4 16.03.2021 16: 20: 31 16.03.2021 16: 20: 31 Chapter 1 The utility value (S IEBEN / L ÖCHERBACH / M ATSCHKE 1974, p. 841) is the result of the valuation of decision alternatives (actions or objects) and it indicates the degree of target performance for each alternative, in the sense of fulfilling needs for a decision subject. To determine utility values, information about the target plan, derived from the target system and about the decision field of the valuation subject is necessary. The target system of a subject includes the objective and formal target. It is the context for deriving a target plan, which contains the facts and features (definition of result) that interest the subject. It also provides information about preferences and the utility function. While the target plan is an expression of willingness of the conflicting party, the decision field gives information about the individual options relating to the valuation subject. It outlines the totality of all available action options and restrictions for the valuation subject that must be taken into account. Hence, the decision field is an “expression of ability” of the respective subject (S IEBEN / S CHILDBACH 1994, p. 15). For the respective decision alternative the amount of the desired facts of all known states is predicted, according to the definition of the result and dependent on time. Hence, the consequences relevant to a judgment on each decision alternative mean utility values must ultimately be assessed, which involves having a preference system that includes preferences regarding time, risk, amount, and manner. The necessary information regarding both the target plan and the decision field to determine the utility value is presented graphically in Figure 1.2 (cf. S IEBEN / L ÖCHERBACH / M ATSCHKE 1974, p. 842). Inhalte des ökonomischen subjektiven Wertbegriffs Nutzwert Entscheidungswert Argumentationswert Tauschwert Mengenverhältnis von Gütern Marktpreis Komplexer Einigungswert Schiedsspruchwert Normwert Figure 1.1: Contents of the economic subjective value concept Utility value Decision value Argumentation value Proportion of assets Market price Complex agreed value Arbitration value Exchange value Contents of the economic subjective value concept Standard value 1.1 Basic Terminology 5 45520_Matschke_Griffleiste_SL5.indd 5 45520_Matschke_Griffleiste_SL5.indd 5 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 A decision value (M ATSCHKE 1969, M ATSCHKE 1972, M ATSCHKE 1975) constitutes the concession limit. The concession limit is the limit of a party’s willingness to grant concessions in a specific conflict situation; like the utility value, it is dependent on the current target plan and the current decision field of the valuation subject. The decision subjects involved in a negotiation or conflict situation and seeking agreement hope for a higher degree of target performance (utility value) than would arise without the realization of the action. Therefore, the utility value that could be attained without the agreement must at least be achieved following an agreement. The determination of the decision value and the utility value are closely related to each other. Argumentation values (M ATSCHKE 1976 and F OLLERT / H ERBENER / O LBRICH / R APP 2018) are variables taken into account in negotiations to justify and support a party’s position. The party seeks a favorable result by using those values as a negotiation tool. An exchange value (S IEBEN / L ÖCHERBACH / M ATSCHKE 1974, p. 840) is the result of a concrete negotiation or conflict process between a number of subjects, who communicate directly or indirectly (e.g., using intermediary services) and equalize supply and demand of (scarce) resources between the exchange partners. These exchange values can be defined in many ways, for example, as already shown in Figure 1.1 (p. 5): 1. as quantity ratio of assets, for example the assets X and Y can be exchanged according to the ratio: x : y = 4 : 6, that is, for four units of X, six units of Y are necessary. The exchange value is dependent on the asset that is taken as the numeraire and can be derived by (see Figure 1.3; German: Mengeneinheit ME = quantity unit): 1 1 This book follows European number and decimalization conventions. Definition of result Environmental constellations Alternatives of decision Valuation-relevant consequences Utility values Preference system Information about the target plan Information about the decision field Figure 1.2: Determination of utility values 1 unit X = 6 4 units Y = 1,5 ⋅ ME Y ⎡⎣ ⎤⎦ = tg α or 1 unit Y = 4 6 units X = 0, 6 ⋅ ME X ⎡⎣ ⎤⎦ = tg β . Kapitel 1: Einführung 6 6 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 6 45520_Matschke_Griffleiste_SL5.indd 6 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 Chapter 1 2. as the market price of an asset determined in monetary units, for example, if a monetary units (German: Geldeinheit GE = monetary unit) have to be paid for an unit of X, and b monetary units (GE) for an unit (ME) of Y. Referring to the example, the following equation must hold true: All prices a and b with a : b = 3 : 2 would also meet the requirements of the example. For 1.5 units of Y, the same monetary amount has to be paid as for one unit of X. 3. as a complex agreed value in a conflict situation, that is, as a combination of forms of so-called conflict-resolution-relevant facts S 1 , …, S n , on which the conflicting parties have reached an agreement. The combination in legal terms, is a contract with a specified content according to the conflict-resolution-relevant facts between the involved conflicting parties. mP 0,08 0,09 sP* 6 4 4 2 2 Menge y von Gut Y Menge x von Gut X Austauschgerade: y = -6/ 4·x + 6 β α Figure 1.3: Price as an exchange interaction Set y of asset Y Exchange line y = -6/ 4 · x + 6 Set x of asset X 6 4 ⋅ ME Y ⎡⎣ ⎤⎦ ME X ⎡⎣ ⎤⎦ ⋅ b ⋅ GE ⎡⎣ ⎤⎦ ME Y ⎡⎣ ⎤⎦ = a ⋅ GE ⎡⎣ ⎤⎦ ME X ⎡⎣ ⎤⎦ or 6 4 ⋅ b ⋅ GE ⎡⎣ ⎤⎦ ME X ⎡⎣ ⎤⎦ = a ⋅ GE ⎡⎣ ⎤⎦ ME X ⎡⎣ ⎤⎦ . If a = 3⋅ GE ⎡⎣ ⎤⎦, then b = 2 ⋅ GE ⎡⎣ ⎤⎦ has to be: a = 6 4 ⋅ b = 3 = 6 4 ⋅ 2 = 3 and b = 4 6 ⋅ a = 4 6 ⋅ 3 = 2. (s 1 * , …, s n * ) (s 1 * , …, s n * ) 1.1 Basic Terminology 7 45520_Matschke_Griffleiste_SL5.indd 7 45520_Matschke_Griffleiste_SL5.indd 7 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 4. as an arbitration, a mediation, or a negotiation value (M ATSCHKE 1969, M ATSCHKE 1971, M ATSCHKE 1979), which is defined as an exchange value in the sense of a proportion of assets, a market price or a complex agreed value, which is proposed by an impartial third party in a conflict situation. The exchange value can also be denoted as price. Owing to it having no value per se, it makes no sense to derive one’s own decision values of assets directly from prices of comparable assets, since these prices include interactions between the value judgments of the respective sellers and/ or buyers (in the sense of their decision values) and is also dependent on their bargaining power and negotiating skills in the specific market situation. Unfortunately, this point is often forgotten or overlooked. The standard value (S IEBEN / L ÖCHERBACH / M ATSCHKE 1974, p. 841) represents a value that transfers information about real facts. In this case, encoded information about the facts is derived from norms or conventions and is given to a recipient, who is interested in the standardized information according to the standard-setter. Finally, the recipient of the information has to decode the standard value to get an idea of the real facts (see Figure 1.4). Consequently, three individuals are involved: The information sender (valuator, standard-user I), who represents the reality respective to the norms/ standards of the standard-setter, and the information recipient (standard-user II), who has to know the norms/ standards for collecting information in the sense of gaining purpose-driven knowledge. These individuals can be identical, for example, when the subject is a control calculation, where according to the regulations of the decision subject, parameters are determined and later re-used to verify the decision. Often the individuals are different; for example, in accounting the annual financial statement of the reporting company is prepared based on codified norms (for example, US-GAAP) for the addressees (e.g., creditors, shareholders). Abzubildender realer Sachverhalt Normwert Normanwender I Regeln zur Kodierung Kodierung (Abbildung des Sachverhalts, Bewertung) Vorstellung über den realen Sachverhalt Normanwender II Regeln zur Dekodierung Dekodierung (Interpretation der Bewertung) Normgeber Figure 1.4: Standard value Standard-setter Rules for coding Rules for decoding Standard-user I Standard-user II Standard value Described real facts Imagination of the real facts Coding (description of facts, valuation) Decoding (interpretation of the valuation) Kapitel 1: Einführung 8 8 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 8 45520_Matschke_Griffleiste_SL5.indd 8 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 Chapter 1 The significant relationships between the interpretations of the economic concept of value can be presented as follows: The determination of utility values is a requirement of the determination of decision values because the decision value constitutes the concession limit, which does not minimize the degree of target performance (utility value) that can be achieved without an agreement at this point, if there is agreement in the conflict situation. Argumentation values often enter the negotiation process as apparent decision values. Purpose-driven argumentation values assume knowledge of one’s own decision value and a presumption on the opposing decision value. Exchange values only exist if there is an existing decision or negotiation field, for example, when the decision value of the supplier is below the decision value of the customer. For instance, decision values vary because of different target systems, decision fields, and expectations of future events. Standard values can be based on decision values if the standard-setter regards the interpretation of the standard value as appropriate, or if the interpretation of the targets is considered adequate from an economic perspective. The clear differentiation of the valuation objects, their isolation from the real world, that is, the determination of system limits, results from the specific remit. To support general theoretical reasoning, it is sufficient to consider the valuation object as a business in the sense of a unit of results/ successes. For this purpose, a delimitation must be made insofar as results (i.e., measurable facts), that the valuation subject is interested in (in the sense of “striving” or “avoiding”), or successes (in the sense of subjectively judged results) (utility)can be assigned to the valuation object (S IEBEN 1968). The company contributes to the target performance of the valuation subject and provides a prospective benefit (present value of future expected cash flows). One example of such objects that can be classified under results/ successes is longterm economic organizations, which can be social, technical, organizational, financial, or legal units. In addition, the use of human resources and materials as well as other forms of energy and knowledge causes purpose-driven actions that aim to satisfy human needs and are in danger of failing (business as an economic unit). This view generally dominates the statements of the relevant literature because the terms of business often remain undetermined. The interesting facts are often the financial indicators from the owners’ perspective, that is, the options to withdraw funds/ capital. In this context, the process of valuation of an individual share is similar to the process of valuating the company as a whole (cf. below O LBRICH 2000, p. 460). In this case, the utility that the share will provide to the subject in the future is relevant. This utility results from the expected proceeds from dividends and those from liquidation by selling the stock. With regard to only generating success, there are differences between the company as a whole and the individual shares: According to the business as a whole concept or its definable parts, in the case of abstract shares, the owner can generally actively influence the business policy and, for instance, spur value-increasing changes, which maximizes his utility. The owner can continue the business unchanged or change its operations and/ or can liquidate it as a whole or in parts, depending on the way in which he or she increases the utility value. At the level of individual shares, or a relatively small volume of them, the shareholder cannot significantly influence the business policy of the company but only have the right to decide between a continuation (buy/ holding of the share) and the related recognition of dividends or liquidation (sale of the share) and the corresponding recognition of the liquidation proceeds. On first examinati- 1.1 Basic Terminology 9 45520_Matschke_Griffleiste_SL5.indd 9 45520_Matschke_Griffleiste_SL5.indd 9 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 on, it would be an exaggeration to regard the valuation of an individual share as a business valuation; but by simply regarding the company and the individual shares as future cash flows that have to be valuated, the valuation methods do not vary from each other. “As a matter of course, the shareholders and the sellers of a business will demand a monetary amount that exceeds the decision value of their share as much as possible, but never falls below it. It is the mirror-image of such investors that buy stock at the stock exchange and for whom the price of the share is below their decision value. The price of the share is consequently between the decision values of a buyer and a seller; the different estimation and appraisal of buyers and sellers regarding the valuation of a share render stock exchange trading possible” (O LBRICH 2000, p. 460). Such decision values become effective at the point when stock is traded in form of specified limits of prospective buyers and sellers. On the stock market, for instance, which comes very close to the idealized conditions of the capital market theory, it is notable how important it is to distinguish the price and value of an asset, whether in the form of a business as a whole or an individual share (M ÜNSTERMANN 1966, p. 11). Only in the context of the interpretation of value in relation to the exchange value an equation between value and price is allowed, and then only for marginal customers and suppliers who create the market equilibrium. According to business valuations, valuations of shares, and other valuations, we must remember that from a rational perspective transactions for exchange value/ price only take place if from the perspective of a presumptive buyer the exchange value/ price is not higher, and from the perspective of a presumptive seller, is not lower than the decision value that each party attributes to the asset. The market price will be judged acceptable or unacceptable according to the relevant decision value and therefore a potential business will be judged profitable or not profitable. This conclusion cannot be made only from the amount of the market value relating to an asset. Kapitel 1: Einführung 10 10 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 10 45520_Matschke_Griffleiste_SL5.indd 10 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 Chapter 1 1.2 Concepts of Business Valuation 1.2.1 Functional Business Valuation Theory via the German School With the paradigm shift to the theory of functional business valuation, controversial views of objective (e.g., M ORAL 1920, L EITNER 1926, M ELLEROWICZ 1952) and subjective (e.g., B USSE VON C OLBE 1957, M ÜNSTERMANN 1966) theories were resolved in the German business valuation literature. A major aspect of this prevailing opinion in the theoretical literature since the mid-1970s is its purpose-driven (subjective) company value (M ATSCHKE 1969, 1971, 1972, 1975, 1979, 2017b and 2017c, M ATSCHKE / B RÖSEL 2008, 2011, 2013a, 2013b and 2018, H ERING 1999, 2000a, 2000b and 2014, M OXTER 1983, O L- BRICH 1999, R EICHERTER 2000, W ITT 2006, K LINGELHÖFER 2006, B YSIKIEWICZ 2008, X. M ATSCHKE 2008, T OLL 2011, K ARAMI 2014, Q UILL 2016 and 2020, Z IMMERMANN 2016, W ASMUTH 2018, F OLLERT 2020, W ALOCHNIK 2021). The functional form of business valuation emphasizes the necessity of an analysis of objectives and the dependence of the company value on objectives (purpose and function). The value of a company is determined by the relevant expectations and plans and also the options available to a specific interested party at the time of valuation, including an explicit consideration of the remits of business valuation. A business not only has a specific value for every valuation subject but can also have different values for different remits. Any valuation conducted is purpose-driven; there is no one and only business value or any single method to calculate such a value. The functional business valuation theory is based upon the fundamentals of subjectivity, future orientation, overall valuation, and dependence of purpose. The determination of company values requires the valuation object to be embedded in the expectations and plans of the party interested in the valuation, in accordance with the principle of subjectivity (cf. K REUTZ 1909, p. 31, B ERLINER 1913, p. 12, S CHMALEN- BACH 1917/ 18, p. 4, S IEBEN 1969b). Accordingly, the value of a company is determined by the aims of the valuation subject, by the options available to it, the limitation on action in financial and real economic terms, and by the planned utility of the valuation subject for the company. The value of the company stands out due to the focus on its target system, decision field, and activities that are determined factors affected by the decision subject. In addition, the result of the principle of subjectivity is that the individual positive and negative synergy effects during the value determination, which are expected by the valuation subject, have to be considered (T OLL / K INTZEL 2019). Different plans and the potential for synergy as well as utilization options and utilization restrictions lead companies to have an individual value for each valuation subject. The identical expectations of different valuation subjects do not lead to an identical value of the company, for example, if the alternative options of capital appropriation or capital procurement of the subjects differ. Although subjective values are determined within the framework of functional business valuation theory, they can be examined intersubjectively through the consideration of the pursued objective. This means that the values determined are traceably correct, as long as they refer to a logical, consistent, complete, and rationally explainable approach, and neglect to address value judgments. 1.2 Concepts of Business Valuation 11 45520_Matschke_Griffleiste_SL5.indd 11 45520_Matschke_Griffleiste_SL5.indd 11 16.03.2021 16: 20: 32 16.03.2021 16: 20: 32 In the Anglo-Saxon literature on business valuation, the reference to the subject is rather unfamiliar and hence only recognized in a rudimentary manner by comparatively few authors (e.g., F ISHBURN 1964, p. 2, E CCLES / L ANES / W ILSON 2000). The principle of future orientation indicates that relating only the utility to the valuation of the company is relevant for the valuation subject. For an existing matter, the historic and currently achieved utility is insignificant, because a merchant does not pay for the past (S CHMALENBACH 1917/ 18, p. 11). The past successes of a company can serve only as a possible indicator of the future results of utility. Finally, a future orientation causes a problem of uncertainty because the exact prospective utility of the company and all prospective alternatives and the consequences of action are not known by the valuation subject at the valuation date. This issue is aggravated by the fact that particularly with regard to the valuation of young and small and medium-sized companies that a trend extrapolation based on the results of past performance can be used only in a very restrictive manner or cannot be used at all as a method in forecasting. According to the principle of overall valuation, it is not the total of single values of assets of the company that is relevant for a valuation but the the company itself as an economic entity in the context of the conflict situation. If there is an isolated (stand-alone) valuation of the economic single values, there is a risk of disregarding both the positive and negative synergy effects in the valuated company (T OLL / K INTZEL 2019), because the total of single values does not have to be identical to the total value of the valuation subject (e.g., A ULER 1926/ 1927, p. 42, M ÜNSTER- MANN 1966, p. 18). In exceptional cases, the principle of overall valuation is dominated by the principle of subjectivity with its components: the goodwill (expression of the target system) and the ability (expression of decision-making scope) of the valuation subject. When the subject finally desires to dismantle the company into its single components and to sell each one, or is unable to achieve value-increasing effects higher than the total of the disposal value of the single components of the company, a single unit valuation under the consideration of the principle of subjectivity and future orientation is necessary. Consider the following example: The presumptive seller S determines a subjective value of 100 monetary units (GE; German: Geldeinheiten) for the company C in consideration of the principle of overall valuation in the case of continuation of the business. A single unit valuation by S would result in only 80 GE for C due to the realization of positive effects of synergy. The presumptive buyer B determines a value of 120 GE for C in the case of an immediate liquidation of the company, because B has different disposal options owing to its decision-making scope than does S. But B is not able to trigger the positive effects of synergy, and that is why a value of 90 GE would accrue to C from the perspective of B in the case of an overall valuation. If B acts rationally, he or she will valuate in this case according to the principle of single unit valuation. Hence, there is room for negotiation between 100 GE and 120 GE between S and B. The concept of the functional (i.e., objective-oriented) business valuation distinguishes between main and minor functions that can have a value. The central criterion for the main functions is the focus on interpersonal conflicts regarding a dispute of conditions surrounding a change of ownership. The main functions refer to those valuations targeting a change of ownership of the business or its definable parts to be valuated (M ATSCHKE 1979, p. 17). Changes of ownership can be subsumed under situations where Kapitel 1: Einführung 12 12 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 12 45520_Matschke_Griffleiste_SL5.indd 12 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 Chapter 1 conditions about a change of ownership are considered or have already occurred. Events that lead to a change in ownership include, in addition to events where the owner changes (e.g., an acquisition or sale), also the events in which the shareholders are unchanged, but their share in ownership changes as a result of the conflict, according to the valuation object(s) (e.g., merger, demerger). Below, we outline the value types of the main functions of functional business valuation (see Figure 1.5). A detailed description of these functions and the examination of the resulting characteristics follows in Chapters 2 (decision function), 3 (mediation function) and 4 (argumentation function). At this point, a connection between business value and remit of business valuation will be established. There is a far-reaching consensus regarding the three main functions: involving decision, mediation, and argumentation (M ATSCHKE 2008, H ERING / T OLL / G ERBAULET 2019): 1. The result of business valuation according to the decision function is called the decision value of the company. The term decision function focuses on the purpose of business value calculation and provides a basis for rational decisions for a specific decision subject (a person interested in the valuation, e.g., a buyer or seller) in a very specific decision and conflict situation (e.g., acquisition or sale) and with regard to this project. When the target system and the decision-making scope are determined, the decision value indicates the circumstances in which the realization of a definable, planned activity does not yet reduce the level of the utility value that could be achieved without this activity. It refers to all conditions relevant for conflict resolution between the parties (so-called conflict-resolution-relevant facts) and states what may still be acceptable in case of an agreement. The examined decision Entscheidungsfunktion Entscheidungswert des Unternehmens Der Entscheidungswert gibt die Grenzeinigungsbedingungen einer Konfliktpartei in der zugrunde gelegten Konfliktsituation an. Vermittlungsfunktion Arbitriumwert des Unternehmens Der Arbitriumwert ist ein vom unparteiischen Gutachter vorgeschlagener Einigungswert, der für die Konfliktparteien zumutbar ist und die Interessen der beteiligten Konfliktparteien angemessen wahrt. Argumentationsfunktion Argumentationswert des Unternehmens Der Argumentationswert ist ein Instrument zur Beeinflussung des Verhandlungspartners, um für den damit Argumentierenden eine möglichst günstige Übereinkunft zu erzielen. Figure 1.5: Value types within the functional business valuation (main functions) Decision function Mediation function Argumentation function Decision value of the company Arbitration value of the company Argumentation value of the company The decision value shows the marginal agreement conditions of a conflicting party in the underlying conflict situation. The arbitration value is an agreed value suggested by an independent appraiser, which is reasonable for the conflicting parties and protects the interests of all involved. The argumentation value is an instrument used to influence the opposing party so as to achieve the most favorable conflict resolution for the party deploying the instrument. 1.2 Concepts of Business Valuation 13 45520_Matschke_Griffleiste_SL5.indd 13 45520_Matschke_Griffleiste_SL5.indd 13 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 value is the basic value for all main functions, and hence constitutes the limit of concession willingness of a party in a specific conflict situation, and should therefore not become known to the other party. 2. The arbitration value in contrast is the result of a business valuation within the scope of the mediation function and it is meant to facilitate or affect an agreement between the buyer and the seller regarding the conditions for the ownership change of the business to be valuated. It is the value suggested by an independent appraiser, as the appraiser as a mediator considers a possible resolution of the conflict. The arbitration value constitutes a compromise that is reasonable for the participating parties and ultimately adequately protects their interests. Therefore, it is necessary for an expert to determine the decision values of the conflicting parties. 3. The argumentation value is the result of a business valuation utilizing the argumentation function. It is an instrument used to influence the opposing party so as to achieve the most favorable conflict resolution for the party deploying the instrument. Hence, the argumentation value is a prejudiced value. This value cannot reasonably be determined without each party knowing their own decision value and without their making assumptions about the decision value of the opposing party, because only the relevant decision values allow a party to decide what negotiation results are compatible with a rational action and should be targeted with a reasonable argumentation value. While the mediation function focuses on all conflicting parties, the decision function and argumentation function concentrate on one conflicting party. In this context the results of the decision function constitute confidential self-information (inside orientation during the negotiation process) and the results of the argumentation function constitute information directed at the opponent party (outside orientation during the negotiation process). While the main functions presented are event-oriented and refer to changes of ownership of the valuated company considered or already actioned, the minor functions focus on events, in which companies are not primarily valuated for the purpose of decisions intending to cause a change of ownership of the valuated company (M ATSCHKE 1979, p. 17). For some years an increasing significance of the so-called company value, including the difficulty of valuating a company, has been noticed in the branches of economic science. Hence, the terms value-oriented management or value-oriented payment/ remuneration are examples of the steadily growing relevance of business valuation. The term minor function should not be understood to indicate a less significant function but to refer to a historically grounded explanation for the valuation objectives, that are not focused on a change of ownership of the valuated company, and which regards the major functions as basic functions, where the values can acquire major significance including in the context of minor functions, as in the examples provided valuation purposes are determined that have to be classified as belonging to the minor functions. A current catalogue of minor functions mentioned in literature (B RÖSEL 2006) includes: 1. Crediting-support function (business valuations that are carried out by credit institutions as valuation subjects according to their credit investigation and loan securitization); 2. Control function (value-oriented business management); Kapitel 1: Einführung 14 14 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 14 45520_Matschke_Griffleiste_SL5.indd 14 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 Chapter 1 3. Information function (i.e., the conventionalized information transfer about the business value: business valuations that have to be carried out in the scope of accounting, e.g., to determine the business values according to norms like IFRS 3 and IAS 36), 4. Motivation function (i.e., business valuations that encourage work-focused behavior and determine its manner, direction, duration, and intensity and also ultimately influence the performance of the (motivated) person in a positive way: “value-oriented payment/ remuneration”), 5. Tax assessment function (i.e., the determination of tax bases) and 6. Contract design function (i.e., the analysis of corporate law and contractual design problems, which refer to the value assessment when the conflict sets in, for example, according to compensation clauses, and influence the behavior of the shareholders, S IEBEN / L UTZ 1983). 1.2 Concepts of Business Valuation 15 45520_Matschke_Griffleiste_SL5.indd 15 45520_Matschke_Griffleiste_SL5.indd 15 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 1.2.2 Neoclassical Business Valuation Theory via the Anglo-Saxon School Functional business valuation (H ERING / O LBRICH / S TEINRÜCKE 2006, K LINGELHÖFER 2009, O LBRICH / B RÖSEL / H ASSLINGER 2009, M ATSCHKE / B RÖSEL / M ATSCHKE 2010, K LINGELHÖ- FER / K URZ 2011, B RÖSEL / M ATSCHKE / O LBRICH 2012, L ERM / R OLLBERG / K URZ 2012, H ERING / T OLL / K IRILOVA 2014a, 2014b, 2015a and 2015b, H ERING / T OLL 2015) is individualistic in its main functions, that is, it is directed toward the specific goals, plans, expectations, and options for action of the valuation subjects in incomplete markets and it is conflictoriented, that is, directed at a possible interpersonal conflict in connection with changes of ownership between a few decision subjects and with several conflict-relevant facts. Functional business valuation considers real existing conditions. In contrast, the neoclassical company valuation theory of the Anglo-Saxon school (the so-called market-value-oriented valuation) establishes - contrary to the classical and long-forgotten Anglo- Saxon valuation doctrine (D EAN 1951, H IRSHLEIFER 1958, W EINGARTNER 1963) - an idealized model world based on the neoclassical finance theory (R APP 2015, R APP / H ASSLIN- GER / O LBRICH 2018). It is oriented toward the anonymous, exchange-organized perfect and complete capital market and toward the (equity, debt) capital providers acting in it, that is, it is supraindividually-oriented. This neoclassical valuation concept rejects the difference between value and price. Although it is not often recognized, the market-value-oriented valuation in its literal sense is not about determining the value of a company as a whole, but about determining the market value of the acting objects on a complete capital market with perfect competition, in which the subjective values of the market players match the resulting objective price. On an exchange-style capital market the acting objects are basically single shares of equity and debt capital of a listed company that are securitized in traded securities. Hence, the market value of a company derives from the total of the market values of the acting objects. The terminological source of market-value-oriented valuation is the value concept in the sense of an exchange value. The market value of equity of an exchange-organized joint-stock company is equivalent to its stock market value, which is the product of the number of shares and the price per share. At this point, another real problem occurs: The current price of shares can be interpreted as an instantly achievable marginal expense/ marginal income on the market, where this limit value was determined on the basis of the current trade volume. The valuation of an asset with such a limit value determines the value of the total only accidentally, if the limit value (referring to the trade volume) corresponds to the average value (referring to the total). If there is no high free floating of shares, the issue arises that the owners have a position of power and the share price does not represent a date for them. Every standard quotation of the share price shows that the subjective values of the market participants, which are expressed by their limitations, are different from the ones implicitly assumed by the determination of the market price. This means that the capitalmarket-oriented valuation has completely lost and forgotten the simplest references to the conditions of real capital markets. With regard to the the assumed form of the market by the representatives of the market-oriented valuation (G ORDON 1959, S HARPE 1964, B LACK / S CHOLES 1973, M ERTON 1973, C OX / R OSS / R UBINSTEIN 1979, D E A NGELO 1981, R APPAPORT 1981 and 1998, K APLAN Kapitel 1: Einführung 16 16 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 16 45520_Matschke_Griffleiste_SL5.indd 16 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 Chapter 1 1986, D AMODARAN 2006, 2012, 2015 and 2018, J ENNERGREN 2013, K OLLER / G OEDHART / W ESSELS 2020); it is assumed that homogenous (equal) goods are traded at the same time (i.e., on the same market) at the same price. The degree of knowledge of all market participants is the same: the conclusions from the available information coincide. The individual market participant has no market power in this market. The price is given, the market participant’s actions cannot influence the price; the price is fixed and is not customizable. Value and price coincide theoretically under these ideal market conditions. This equilibrium theory is based on the work of A RROW (1964) and D EBREU (1959) and enables the valuation of uncertain cash flows so that the same decision value - that for reasons of arbitrage has to become the market price - applies for all market participants, regardless of their individual risk propensity. As a precondition for this situation, the restrictive and greatly idealistic conditions for an arbitrage-free valuation, particularly a perfect and complete market and perfect competition, must be established in this market (D EBREU 1959, A RROW 1964): 1. A perfect market means all market participants are aware of the financial returns (cash flows) of all traded securities and the returns are equal for all market participants in terms of those returns’ amount and temporal structure. All market participants can buy or sell each cash flow (i.e., the traded securities) on an unlimited scale without transaction costs for the same price. 2. A complete market means all possible environmental constellations can be reproduced by traded securities (cash flows) with the help of linear combinations so that a random cash flow (traded security), which has to be valuated, can be reproduced on the market by traded securities. The traded securities generate a so-called state space. This characteristic is also referred to as spanning. 3. Perfect competition means that no market participant has market power and is able to influence the prices of the traded securities. Therefore, everyone is either a price taker or quantity adjuster because new cash flows do not change the current market prices (the so-called quality of competitiveness). Under these unrealistic premises, every random cash flow can be valuated with the (A RROW -D EBREU ) price p* of the portfolio, which is composed of the traded securities (cash flows) on the market that are required for reproduction. Indeed, no buyer would be willing to pay more than this price p* (the value for reproduction of the cash flow), because one could generate an identical cash flow (in terms of the valuation subject) for p* on the market. Finally, no seller would be satisfied with less than the liquidation proceeds p*, due to the complete returns of the valuation object on this market being saleable at p*. The value for reproduction would also be the limit price of the buyer and the liquidation proceeds p* the limit price of the seller. Both correspond in this (illusory) world, so that the resulting price (i.e., the market price) can also only represent the amount p* since there is no room for negotiation available, but nor is any required. It is a general setting of the market-oriented valuation, which we will now explicate using a simple numerical example (quoted from M ATSCHKE / H ERING / K LINGELHÖFER 2002, p. 14). In this assumption of a model world two linearly independent traded securities WP 1 and WP 2 (German: Wertpapiere WP = traded securities) are examined that anticipate payments in two possible future states s 1 and s 2. The current prices for these traded securities on the capital market are and . A traded security WP 3 ought to be valup 1 * p 2 * 1.2 Concepts of Business Valuation 17 45520_Matschke_Griffleiste_SL5.indd 17 45520_Matschke_Griffleiste_SL5.indd 17 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 ated, hence, the the following question must be answered: Which equilibrium price must be applied to this traded security? Figure 1.6 below provides a numerical representation of the situation: The traded security WP 3 can be replicated from the linear independent securities WP 1 and WP 2 : Figure 1.7 below provides numerical information on the reproduction of the market. This finding means that the equilibrium price for the traded security WP 3 is = 70. If it were not like this, there would be a possibility of risk-free profits (arbitrage profits): Every price deviating from p* could be used for profitable businesses. If p < p*, it would be profitable to buy the cash flow for p and to sell it (divided into its single components) for p*. For p > p*, conversely, the misvaluated cash flow for p* could be reproduced from each other deposit and sold as a unit for a higher price p. Because the market swiftly recognizes such arbitrage profits, the price eventually adjusts itself to p = p*. This ensures that the market price p* coincides with the equal decision value for all market participants; hence, it is also the market value. With the help of state-dependent prices (A RROW -D EBREU prices) for a valuation of standardized cash flows (“pure securities”), the market value of a cash flow is formally determined as (net) income value in the case of certainty. This argumentation expands the simple valuation principle of the complete capital market in conditions of certainty to the case in conditions of uncertainty. p 3 * Starting point: WP i 1 50 State s 1 50 State s 2 100 2 Traded security for valuation: 3 Figure 1.6: Starting point of the A RROW -D EBREU example 165 300 100 150 100 Price p i * p 3 * = ? 6 9 ⋅ WP 1 + 2 9 ⋅ WP 2 = 1 ⋅ WP 3 or 6 ⋅ WP 1 + 2 ⋅ WP 2 = 9 ⋅ WP 3 Starting point: WP i Quantity State s 1 State s 2 123 Figure 1.7: Market simulation of the starting point 62 300 330 9 630 300 600 600 300 900 900 Price p i * p 3 * Kapitel 1: Einführung 18 18 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 18 45520_Matschke_Griffleiste_SL5.indd 18 16.03.2021 16: 20: 33 16.03.2021 16: 20: 33 Chapter 1 The financial instruments FT j (A RROW -D EBREU Financial Instruments; German: Finanzierungstitel = FT) lead only in one single state (s j ) to a payment of the amount of 1 and in all other states to no payments (see Figure 1.8). The state-dependent payments Z ij from specific traded securities WP i can also be illustrated as linear combinations of n pure securities: If, with the exception of the payments z ij in the states s j , the market prices p i * of the traded securities WP i are also known, the prices ρ j of the pure securities (and thus of the state-dependent claims) can be derived: As far as the matrix of payments z ij of traded securities is regular in this equation (i.e., 1. The number of examined traded securities WP i and states s j is equal, thus m = n; 2. No traded security can be illustrated as a linear combination of all others; 3. Price inconsistencies of the different traded securities are eliminated.), the vector of the prices ρ j of the pure securities can be determined: FT j State s 1 State s j State s n 12 10 01 … ú · . 0: … 0 ú · . 0: : j: : : 0 ú · . 0 : : ú · . ú · . ú · . 010: ú · . 0 ú · . ú · . ú · . 1 : 0: 0 n Figure 1.8: Payments of financial instruments based on states 0 0 … 0 … 0 1 WP i → = 1 0 ! 0 ! 0 0 0 1 " # # # # 0 " 0 # # # # " 1 " # # # # 0 " 0 # # # # " 1 0 0 0 ! 0 ! 0 1 ⎛ ⎝ ⎜⎜⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟⎟⎟ ⋅ z i1 # # z ij # # z in ⎛ ⎝ ⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟⎟⎟⎟ = z i1 , …, z ij , …, z in ( ) T . p 1 * : p i * : p m * ⎛ ⎝ ⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟ = z 11 … z 1 j … z 1n : : : z i1 … z ij … z in : : : z m1 z mj z mn ⎛ ⎝ ⎜⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟⎟ ⋅ ρ 1 : ρ j : ρ n ⎛ ⎝ ⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟ 1.2 Concepts of Business Valuation 19 45520_Matschke_Griffleiste_SL5.indd 19 45520_Matschke_Griffleiste_SL5.indd 19 16.03.2021 16: 20: 34 16.03.2021 16: 20: 34 From the numerical example follows: The equilibrium prices of a traded security WP i can be interpreted as a summarized valuation of the state-dependent demands, thus, in the numerical example: With an interpretation of states as points in time, the prices ρ j of the state-dependent claims can be interpreted as discount factors. If the derived equilibrium price = 70 cannot be reached for the new traded security WP 3 , but for example, p 3 = 60, it would be profitable to buy the cash flow of the traded security WP 3 for p 3 = 60 and at the same time to sell it (divided into its single components, namely as traded security WP 1 and traded security WP 2 ) for a balanced total income of + = = 70. This is illustrated in Figure 1.9. The connections explained above also mean an equivalent representation based on financial instruments (pure securities) is of course also possible (see Figure 1.10): ρ 1 # ρ j # ρ n ⎛ ⎝ ⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟ = z 11 ! z 1j ! z 1n # # # z i1 ! z ij ! z in # # # z n1 ! z nj ! z nn ⎛ ⎝ ⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟ − 1 ⋅ p 1 * # p i * # p n * ⎛ ⎝ ⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟ . ρ 1 ρ 2 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = 50 100 300 150 ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ − 1 ⋅ 50 165 ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ = − 1 150 1 225 1 75 − 1 450 ⎛ ⎝ ⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟ ⋅ 50 165 ⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟ = 0, 4 0,3 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ . p 1 * = 50 ⋅ 0, 4 + 100 ⋅ 0,3 = 20 + 30 = 50 p 2 * = 300 ⋅ 0, 4 + 150 ⋅ 0,3 = 120 + 45 = 165. p 3 * p 1 * p 2 * p 3 * WP i Action Quantity State s 1 State s 2 12 Sell Sell 6/ 9 2/ 9 33,33 36,67 -33,33 -66,67 -66,67 -33,33 3 Balance Figure 1.9: Representation of arbitrage based on specific traded securities Buy 1 -60 10 100 0 100 0 Price p i * WP i Action Quantity State s 1 State s 2 FT 1 FT 2 Sell Sell 100 100 40 30 -100 0 0 -100 WP 3 Balance Figure 1.10: Representation of arbitrage based on financial instruments Buy 1 -60 10 100 0 100 0 Price p i * Kapitel 1: Einführung 20 20 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 20 45520_Matschke_Griffleiste_SL5.indd 20 16.03.2021 16: 20: 34 16.03.2021 16: 20: 34 Chapter 1 An execution of the above described actions therefore leads in both cases in t = 0 to a positive balance (i.e., to a payment) to the amount of 10, without bringing about a payout in one of the states s 1 or s 2 . If the actions were actually operated frequently at will, a theoretically infinite arbitrage profit could be achieved. To sum up, the following position can be established with regard to the general background of the market-oriented valuation (cf. M ATSCHKE / H ERING / K LINGELHÖFER 2002, p. 17): The principle of the arbitrage-free valuation excels owing to the theoretical accomplishment of the uncertainty problem in an idealized world of thought. With regard to the decision-oriented business valuation, its set of assumptions proves to be too restrictive to permit recommending an application. In addition, the requirements related to a complete market are too high to be approximately feasible in reality. A particular issue in this regard appears to be the unmanageable number of (mainly unknown) states, which expands every payoff to a vector of almost infinite dimensions that can only be filled with numerals to a minor fraction. Under real-life conditions, it can therefore be assumed that value and (market) price of a cash flow, a stock, or a whole enterprise do not need to be and are not identical; otherwise there would hardly be any reason to undertake transactions in the market. Value enhancement can only result if the value of the realized operation (e.g., a company acquisition) is above its purchase price. However, as soon as the presence of favorable transactions means a distinction must be made between subjective value and objective market price, there is no longer a need to strive for any defined maximizing of market value, but rather a maximizing of the absolute difference between (subjective) decision value and (objective) market price. For example, the prospective buyer of a company that following a planned takeover exhibits a certain value to the buyer, will want to minimize the purchase price to be paid. In contrast, the seller or proprietor (as a presumptive seller) of a company, will naturally seek a high (market) price: The seller only achieves an actual advantage in the sale if the subjective value is lower than that price. While the (market) price is negotiable, the decision value is defined as the acceptable marginal price from the individual objectives and options of the buyer. Accordingly, in an acquisition/ sale situation, the objective market value maximization blurs this elementary essential difference between value and price and therefore loses its point in real decision-making situations, in which the three above-mentioned tight assumptions are not fulfilled and where value and price fall apart. However, it should be stressed that this does not imply ceteris paribus that proprietors might not be interested in rising stock market prices. Alongsides this general assumption of equilibrium on the capital market, two special different concepts are relevant within the framework of the market-value-oriented, capital-market-theoretical valuation methods: the Capital Asset Pricing Model (CAPM) (cf. M ARKOWITZ 1952, S HARPE 1963, L INTNER 1965, M OSSIN 1966) and the M ODIGLIANI - M ILLER approach (1958 and 1963). In fact, as will be described below, these concepts are based on different (but equally unrealistic) model premises; however, these approaches are unconditionally interlinked for the most part within the framework of the market-value-oriented valuation methods, despite their verifiable incompatibility (e.g., M UL- LINS 1982, H ERING 2014, p. 297, S CHILDBACH 2021). With regard to the adequate target rate required for theoretical capital market valuation methods, the CAPM is widely used. The CAPM is based on the neoclassical thin- 1.2 Concepts of Business Valuation 21 45520_Matschke_Griffleiste_SL5.indd 21 45520_Matschke_Griffleiste_SL5.indd 21 16.03.2021 16: 20: 34 16.03.2021 16: 20: 34 king of equilibrium and is notably used to explain the pricing on the capital market under restrictive and greatly idealized assumptions described below (cf. H ERING 2017, p. 297): 1. Perfect capital market: (a) Market access can be free, unlimited, and lacking transaction costs; (b) there are no information asymmetries; (c) the reaction rate is infinite; (d) all market participants can invest any amount of money for a given rate of interest i; (e) homogenous goods are traded, and the market participants assess them to be of equal quality, because no time, functional, or other preferences are provided. (f) Taxes remain unconsidered. 2. Homogenous expectations: According to the expectation values and standard deviations of all traded securities and the existing covariances, it is generally presumed risk-averse market participants have the same homogenous objective and right expectations. 3. Single period examination: The model (generally) has a planning horizon of a single period, thus only the payment of the point in time t = 1 is uncertain. Models expanded to a multi-period examination do exist, albeit operating under very strict assumptions (cf. F AMA 1977), however the failure of the multi-period-CAPM has already been established (cf. R ÖDER / M ÜLLER 2001, K OCHERLAKOTA 1996). In light of everyday economic life and real economic decision problems, these assumptions must surely be regarded as highly unrealistic (cf. F AMA / F RENCH 1992, H ERING 2017, p. 303). According to the CAPM, only risky future cash flows bonded to the commodity are valuated, “Generally speaking, the market value of a traded security can be defined as the price for which this cash flow can be bought” (M ANDL / R ABEL 1997, p. 18). Hence, the market value is determined as the price, that is, exchange value in terms of a monetary value per unit of the commodity. The basis of valuation is the future-oriented and uninfluenced (by capital market participants) cash flows of commodities that are derived from the cash flows of the company involved, which can be influenced by the management and also the investment returns of the actual and potential investors. Their expected returns and requirements, according to the discount rate used for valuation in the scope of theoretical capital market methods, are based on considerations with foundational principles in the CAPM. The CAPM generally includes all capital investment opportunities. Transferred to the real world, this means that not only the tradable amounts, but also properties, buildings, gold, works of art, or similar goods. The market participants influence their own acquisition and selling activities indirectly through the management who are subject to market controls. A management team that can improve the valuation-relevant cash flow (directly or indirectly) in comparison to that of other companies is rewarded by heightened demand for the traded securities of the business and the enhanced market value of the owners who reward themselves by increasing the market value of their capital. A management team that cannot improve the valuation-relevant cash payment flow in comparison to that of the firm’s peers, must consider the intensified sales efforts of the investors and can be punished by a reduced market value. The interest of the management in an increasing market value for the benefit of actual and potential owners, that is, a so-called shareholder-value-orientation, can be intensified by incentives such as share options for managers driving profit as a result of an increasing market value. This process plays a vital role for agency-theoreti- Kapitel 1: Einführung 22 22 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 22 45520_Matschke_Griffleiste_SL5.indd 22 16.03.2021 16: 20: 34 16.03.2021 16: 20: 34 Chapter 1 cal concerns, although that topic will not be discussed any further here (see V INCENTI 2002, 2004, p. 321). The homogenous objectives, to be achieved by the capital market participants according to the assumptions of the CAPM, are exclusively in the form of expected monetary payments. Payoffs are valuation-relevant if they are expected at the end of the single reporting period. The traded securities are representative of these expected payments, hence, market assets are the expected payments and not the security. Actual payments (in the same currency) at the same time are physically equivalent. In contrast, expected payments are not homogenous assets per se, but that is true only in the case of an equal risk, referring to the same quantities. Given the general property of a complete market that homogenous assets have the same price, it follows that risk-equivalent expected cash flows are valued equally. Therefore, expected payments of the same amount with a varying degree of risk content have to be valuated differently. According to the CAPM, the market participant (as outlined in the assumptions) is risk averse. This means that expected payments with a higher risk will be valuated comparatively lower on the market. In other words, capital market participants who are willing to acquire expected payments with a higher risk want that risk assumption to be compensated by a premium in the form of a higher interest rate than a risk-free capital investment offers. Risk-free expected payments are homogenous assets regardless of whether they result from a decision to raise or to deliver capital. Finally, from the perspective of the market participants involved, the delivery of capital of one participant includes the raising of capital on the part of the other. In a complete capital market, riskfree capital can be raised and invested at the same interest rate i. Including interest rates, payments of today and tomorrow are estimated as equivalent by the capital market participants, thus for a monetary unit of today it follows: This means the present benefit of a risk-free monetary unit is equivalent to the benefit of 1 + i risk-free monetary units of tomorrow . Consequently, i as the interest rate is the compensation for waiving the benefit of a present monetary unit. Under the CAPM, only the so-called systemic market risk is remunerated in the form of a market risk premium ∆r M , which is measured as the difference between the expected return of the risky market portfolio and the risk-free capital market interest rate: All commodities are subjected to systematic risk (e.g., economic or political risk). The special unsystematic risk of a risky capital investment, under the conditions of the N 1 ⎡⎣ ⎤⎦ 0 ( ) = N 1 + i ⎡⎣ ⎤⎦ 1 ( ) or 1 0 ~ 1 + i ⎡⎣ ⎤⎦ 1 . N 1 ⎡⎣ ⎤⎦ 0 ( ) N 1+i ⎡⎣ ⎤⎦ 1 ( ) Δr M = r M * − i with Δ r M = market risk premium, r M * = expected return on investment of an optimal market portfolio with inherent risks, i = risk-free capital market interest. 1.2 Concepts of Business Valuation 23 45520_Matschke_Griffleiste_SL5.indd 23 45520_Matschke_Griffleiste_SL5.indd 23 16.03.2021 16: 20: 34 16.03.2021 16: 20: 34 CAPM, may be eliminated at no cost through diversification by the market participants setting up portfolios and is therefore not remunerated on the market. All market participants may generally assemble the optimal market portfolio and are thus completely risk diversified. The optimal market portfolio is the risky portfolio with the highest price per risk unit. If, with regard to the general considerations, a market price is given, it is applied to the market price that from the demand perspective is an expression of the slightest, barely applicable, willingness to pay. From the supply perspective, the market price represents the highest, barely fulfilled, payment claim. The variable λ * is then determined as illustrated in Figure 1.11 by the angle α between the abscissa and the smallest line from the capital investment point to a portfolio point and simultaneously the angle β between the longest line from the capital investment point to a portfolio point and the perpendicular from this portfolio point to the abscissa. Additionally, it is: tg β = 1/ tg α. If the risk is measured with the standard deviation, the following expression for the price λ * of a risk unit results: λ * and λ are determined as the surplus yield per risk unit. This process is in accordance with the assumption of risk-averse capital market participants. Such risk-averse investors want a high profit to compensate for accepting risk. If, however, the reference value for the risk is instead the variance of the market portfolio, the following price formula λ for a risk unit is determined as: The portfolio, which is realized by the capital market participants, is showcased in Figure 1.11. The capital market participants compile their individual portfolio, in line with their individual risk profiles; that might equate to a combination of a market portfolio and a risk-free capital market investment or only the market portfolio. These portfolios from individual investors follow the capital market line, which consists of two sectors: The first sector (between the capital market investment point and the optimal market portfolio) represents what is termed the mixing line, whereby the share x M of the market portfolio lies in the area of 0 ≤ x M ≤ 1 and only equity capital is utilized by the market participants (leverage ratio: FK/ EK = 0, where the market value of debt capital FK (German: Fremdkapital = FK) and the market value of equity EK (German: Eigenkapital = EK)); in the second sector (on the right of the optimal market portfolio, with the coordinates in Figure 1.11 as following: (r M ; σ M ) = (0,150236; 0,091018)) the only investment is made in the market portfolio (x M = 1), where for the realization additional debt capital has to be taken up at the risk-free capital market interest rate i = 0,08 (FK/ EK > 0). λ * = r M * − i σ M . σ Μ 2 λ = r M * −1 σ M 2 with λ * = λ ⋅ σ M . Kapitel 1: Einführung 24 24 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 24 45520_Matschke_Griffleiste_SL5.indd 24 16.03.2021 16: 20: 35 16.03.2021 16: 20: 35 Chapter 1 The price λ * per risk unit of the non-diversifiable systemic risk is outlined in Figure 1.11 in the slope of the capital market line (The angle α is the angle between the capital market line and the abscissa. Due to 1/ tg α = tg β, λ * as the tangent of the angle between the line of the capital investment point to a portfolio and the perpendicular of this portfolio point to the abscissa in Figure 1.11. The angle β is not to be confused with the beta factor.): that is, with a view to the example: λ * = (0,150236 - 0,08)/ 0,091018 = 0,771675 = 1/ [0,091018/ (0,150236 - 0,08)] = 1/ 1,295882 = 0,771675 per unit of the standard deviation of the market portfolio. If the variance is taken as a risk measure, this results in: or in values for the example λ = 0,771675/ 0,091018 = 8,478293 per unit of the variance of the market portfolio. 0,08 0,1 0,12 0,14 0,16 0,18 0,2 Expectation value of return on investment 0 0,05 0,1 0,15 0,2 0,25 Standard deviation inefficient portfolio of the capital investment with inherent risks capital market line (at a equity financed total portfolio) efficiency line of the capital investment with inherent risks optimal market portfolio with inherent risks capital market investment point (risk-free capital market investment) capital market line (at partly leveraged total portfolio) Figure 1.11: Optimal market portfolio and capital market line λ * = r M * − i σ M = 1 tg α = 1 σ M r M * − i , λ = r M * − i σ M 2 = λ * σ M 1.2 Concepts of Business Valuation 25 45520_Matschke_Griffleiste_SL5.indd 25 45520_Matschke_Griffleiste_SL5.indd 25 16.03.2021 16: 20: 35 16.03.2021 16: 20: 35 The capital market is assumed to be efficient, thus, all equity providers and all debt capital providers of a business have the same required rate of return. However, since liability risks are involved, each investor group will have a different required rate of returns: The claim of equity investors exceeds that of debt capital providers. The prevailing information efficiency in the capital market leads to a consensus of expectations of the capital market participants. In the CAPM, a certain commodity is valuated with respect to its relationship to the market risk. This means that the returns expected (and demanded) of a capital investment j increase with the growing risk associated with the analyzed capital investment j in comparison to the market portfolio. The relationship between the risk of the market portfolio M and the risk of a certain capital investment j can be determined with the covariance σ j,M so that relying on λ as an instrument of value for a unit of systemic risk is appropriate, because with regard to the the market portfolio the risk premium is due: Generally speaking, the price λ is paid on the market for a risk unit, in such a way that the whole risk premium ∆r j for a capital investment j is . The expected return of a risky capital investment j can be illustrated as the sum of a risk-free capital market interest rate i and the demanded (and paid) risk premium of ∆r j = λ · σ j,M for the examined capital market investment j: The systemic risk of a capital investment j can also be presented as a multiple of the market risk; hence, it is then paid with the same multiple of the market risk premium. That multiple can be determined by the so-called beta factor β j of the examined capital investment j. The beta factor β j is a criterion for the systemic risk of a capital investment j in comparison to the market portfolio M and can be measured as the ratio of the covariance σ j,M between the uncertain single period return of the capital investment j and the market portfolio M (as numerator) and the variance of the market portfolio (as denominator), that is, owing to its reference to the variance of the market portfolio, the beta factor β j determines a standardized risk measure. Consequently, the paid risk premium ∆r j can be written as: and for the expected return : while the beta factor β j is defined as: (r M * − i) λ ⋅ σ M ,M = λ ⋅ σ M 2 = r M * − i. Δr j = λ ⋅ σ j,M r j * r j * = i + λ ⋅ σ j,M = i + r M * − i σ M 2 ⋅ σ j,M . σ M 2 Δr j = r M * − i ( ) ⋅ β j r j * r j * = i + r M * − i ( ) ⋅ β j , β j = σ j,M σ M 2 . Kapitel 1: Einführung 26 26 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 26 45520_Matschke_Griffleiste_SL5.indd 26 16.03.2021 16: 20: 35 16.03.2021 16: 20: 35 Chapter 1 The covariance σ j,M can be determined with the correlation coefficient ρ j,M between the capital investment j and the market portfolio M as well as the standard deviations σ j and σ M , so that for the beta factor the following relations formally result: and the expected return The summarized results show that the required internal rate of discount for a valuation of an object corresponds to the expected (and demanded) return . In equilibrium this corresponds to the sum of the risk-free interest rate i and the object-specific risk premium ∆r j , the latter as the product of the price payable on the market for a unit of the non-diversified systemic risk λ (measured here on the basis of the variance of the market portfolio ) and the covariance (between the uncertain single period return of the capital investment j and the market portfolio M). Converted, the internal rate of discount ultimately equals the sum of the risk-free interest rate i and the product of the market risk premium ∆r M and the so-called object-specific beta factor β j ), which is measured as a difference between the expected return of the risky market portfolios and the risk-free capital market interest rate i (as an expression of the payable systemic market risk). Again, the beta factor as a criterion for the systemic risk of an investment j in comparison to the market portfolio M is the ratio of the covariance (between the valuation object j and the market portfolio M) and the variance of the market portfolio : The market value is determined under the conditions of the CAPM as an equilibrium price. This equilibrium price is certainly not the result of different estimates of the market participants, as in reality, but represents the general consensus of estimations according to the assumptions. The CAPM is, as mentioned above, modeled as a single period model, so that it results for the market value of a capital investment j at a time point t + 1 with the expected payments and the expected return as the internal rate of discount: The expected payments (cash flows) in t + 1 correspond to the expected pay-outs as well as to the expected market value in t + 1. If stationary conditions are assumed for the future, the single period model can be utilized in an intertemporal, that is, multi-period context: β j = σ j,M σ M 2 = ρ j,M ⋅ σ j ⋅ σ M σ M 2 = ρ j,M ⋅ σ j σ M r j * = i + r M * − i ( ) ⋅ ρ j,M ⋅ σ j σ M . r j * σ Μ 2 σ j, Μ r Μ * σ j, Μ σ Μ 2 r j * = i + Δr j = i + λ ⋅ σ j,M = i + (r M * − i) ⋅ σ j,M σ M 2 = i + (r M * − i) ⋅ β j with β j = σ j,M σ M 2 . r j * K j,t = Z j,t +1 * 1 + r j * = D j,t + 1 * + K j,t + 1 * 1 + r j * = D j,t + 1 * 1 + r j * + K j,t + 1 * 1 + r j * . D j,t + 1 * Κ j,t + 1 * 1.2 Concepts of Business Valuation 27 45520_Matschke_Griffleiste_SL5.indd 27 45520_Matschke_Griffleiste_SL5.indd 27 16.03.2021 16: 20: 35 16.03.2021 16: 20: 35 or, if t = 0 is chosen as the starting point and t = 2 as the end point, or, if t = 0 is chosen as the starting point and t = T as the end point, or, if or generally, due to , at constant expectations over time as a return or dividend model Strictly speaking K j,0 is an expected market price. But since all market participants have that expectation with regard to the examined capital investment j, K j,0 is the price, that has to be determined on the market in t = 0 owing to the assumptions, because all marginal prices correspond to each other. There is just one existing problem according to this valuation concept: The question arises as to why people constantly act, that is, buy and sell traded securities (capital investments), despite having no individual additional utility (in terms of a positive net present value). After all, every market participant has the same interests and expectations regarding similar commodities because they have the same level of information. The marginal prices of all market participants in reference to the same capital investment are absolutely identical. But to do business at a marginal price means to do no business at all, that is, to expect no advantage from doing business. Despite this problem, which actually severely questions the market-oriented valuation regarding real tasks, the market approach itself is seldom challenged. Instead of thinking, one simply applies (for criticism see T OLL / L EONHARDT 2019). Discrepancies between the market price obtained in this way and the market price actually observed are K j,t = Z j,t +1 * 1 + r j * = D j,t + 1 * 1 + r j * + K j,t + 1 * 1 + r j * = D j,t + 1 * 1 + r j * + Z j,t + 2 * 1 + r j * 1 + r j * = D j,t + 1 * 1 + r j * + Z j,t + 2 * 1 + r j * ( ) 2 K j,0 = Z j,1 * 1 + r j * = D j,1 * 1 + r j * + K j,1 * 1 + r j * = D j,1 * 1 + r j * + Z j,2 * 1 + r j * 1 + r j * = D j,1 * 1 + r j * + Z j,2 * 1 + r j * ( ) 2 K j,0 = D j,1 * 1 + r j * + D j,2 * + K j,2 * 1 + r j * ( ) 2 = D j,1 * 1 + r j * + D j,2 * 1 + r j * ( ) 2 + K j,2 * 1 + r j * ( ) 2 = D j,t * 1 + r j * ( ) t t = 1 2 ∑ + K j,2 * 1 + r j * ( ) 2 K j,0 = D j,t * 1 + r j * ( ) t t = 1 T ∑ + K j,T * 1 + r j * ( ) T T → ∞, K j,0 = D j,t * 1+ r j * ( ) t t = 1 ∞ ∑ lim T→∞ (1+ r) T − 1 r ⋅ (1 + r) T ⎡ ⎣⎢ ⎤ ⎦⎥ = lim T →∞ 1 r − 1 r ⋅ (1 + r) T ⎡ ⎣⎢ ⎤ ⎦⎥ = 1 r , D j,t * = D j * K j,0 = D j * r j * . Kapitel 1: Einführung 28 28 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 28 45520_Matschke_Griffleiste_SL5.indd 28 16.03.2021 16: 20: 36 16.03.2021 16: 20: 36 Chapter 1 considered to be insignificant differences between theory and practice or even as infallible indicators for risk-free arbitrage profits from so-called undervaluation, that is, for cases in which the market price calculated according to this approach is higher than the actual market price, or as a signal for withdrawal from an investment if the calculated market price falls below the observed market price (overvaluation; O LBRICH / R APP / V E- NITZ 2016, R APP / O LBRICH / V ENITZ 2017 and 2018). The model-related determined market price is consequently described with new, embellished terminology: The term market value is suddenly no longer mentioned, and the concept is referred to as either the theoretically right value, intrinsic value, true value, or fair value of a capital investment, from which the current extrinsic value as the market value (market price) deviates. If the claim of the market-oriented valuation to determine not any but the exchange value is abandoned, it means at the same time that a value defined on the basis of a particular theory is immunized against falsification. The theoretically right value becomes fiction. It is an argumentation value, which is valid until someone says, “but the emperor market value isn’t wearing any clothes.” If it is not the market value but something else that is determined, the underlying theory is apparently proven not to be the right one because the assumptions do not correspond to reality. If the theory is not right, then the determined value is not the theoretically right value and is inadequate to inform a judgment of actual market prices, since it remains open whether it has the essential characteristics of an individual marginal price for such a judgment. No rational recommendation can be derived from a comparison between the actual market value (price) of a commodity and a value (e.g., like the calculated market value), which is not an individual marginal price at the same time (cf. Figure 1.12). The figure above presents two separate situations. The buyers in the example above depicting an undervaluated situation could achieve a comparable success to the commodity for 95 GE with their best alternative. They would act irrationally if they bought the commodity for the actual market price of 100 GE, because instead of a sup- Buying situation Amount Reference Actual market price Calculated market value 100 105 So-called undervaluated situation! Buy! Real individual marginal price Selling situation Actual market price 95 Rational behavior: Do not buy commodity/ share! Realize alternatives! Amount 100 Recommendation Calculated market value Real individual marginal price Figure 1.12: Review of actual market prices 95 105 So-called overvaluated situation! Sell! Rational behavior: Do not sell commodity/ share! 1.2 Concepts of Business Valuation 29 45520_Matschke_Griffleiste_SL5.indd 29 45520_Matschke_Griffleiste_SL5.indd 29 16.03.2021 16: 20: 36 16.03.2021 16: 20: 36 posed profit of 5 GE, as in the comparison between actual and calculated market value, they would in fact have a disadvantage of 5 GE (the result of a comparison between marginal price and market price). Sellers, who want to sell their commodity in the hypothesized overvaluated situation (lower section), have to receive compensation of at least 105 GE, to achieve a comparable success, as obtained from the best alternative of selling the commodity. But if they really sold for the actual market price, the discrepancy with the calculated market value would mean they only receive 100 GE, and could therefore not achieve the best alternative from which their marginal price is derived, and they would have no advantage, but a disadvantage of 5 GE (comparison of market price and limit price). To avoid such problems, they have to invest in the commodity once again. In a world without transaction costs, this excessive desire for action is compatible with rational behavior. Otherwise, it is better to follow the recommended course of action according to the marginal price and not to sell. Ignoring the differences of the assumptions, the financial valuation methods try to theoretically integrate the ideal world of the CAPM (in which the approach focuses on the determination of the market value of a unit of the commodity “capital investment j”) under the premises of the M ODIGLIANI -M ILLER approach. M ODIGLIANI / M ILLER established in 1958 that a capital structure of a company has no influence on the market value of this company under certain restrictive conditions. Hence, under the premises of the M ODIGLIANI -M ILLER theorem, it is valid to have a market value of an unleveraged company that corresponds to the market value of a comparable levered company (of the same risk). The key assumptions, which include the thesis of irrelevance of the debtequity ratio, are (M ODIGLIANI / M ILLER 1958, p. 265): 1. Both a perfect and complete market and perfect competition are a given. 2. Equity and debt are treated the same way for tax purposes. 3. Private debt and raising of credits of companies follow the same terms and conditions. 4. Investors judge private debt and contribution to the indebted company indifferently. 5. The interest rate on borrowings is independent of the capital structure. There is an abstraction of insolvency costs and illiquidity risks. This step into the premises of the M ODIGLIANI -M ILLER theorem is in accordance with a change of perspective. The views are no longer focused on the unit of the commodity of a “capital investment j”, but are now highly concentrated on the whole company j, that is, all units of this “capital investment j”. The market value W j,0 of the company j at the valuation time t = 0 is analyzed. The step from the market value of equity K j,0 to the company market value W j,0 can easily be understood if it is elaborated on the example of the unlevered firm, because then it holds true that the equity of the capital investment j represents the equity capital of the company j and can be identified as a traded security (share). The enterprise value W j,0 as the market value of the overall or total equity capital is computed as follows: W j,0 EK W j,0 = n 0 ⋅ K j,0 = n 0 ⋅ D j,t * 1 + r j * ( ) t = n 0 ⋅ D j,t * 1 + r j * ( ) t t = 1 ∞ ∑ t = 1 ∞ ∑ = X j,t * 1 + r j * ( ) t t = 1 ∞ ∑ Kapitel 1: Einführung 30 30 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 30 45520_Matschke_Griffleiste_SL5.indd 30 16.03.2021 16: 20: 36 16.03.2021 16: 20: 36 Chapter 1 and as a perpetuity Referred to an unlevered firm, it follows: and for a perpetuity with n 0 being the number of the available equity capital shares in t = 0 and with as an expression of the available future cashflow of the company for all investors, hence it applies under the conditions: However, if the company is levered, under the premises of the M ODIGLIANI -M ILLER theorem the market value of a company W j,0 equals the sum of the market value of the equity on the one hand and the market value of debt on the other: with or as a perpetuity with is the market value of all shares and is the market value of debt of the company j, for example, the value of the company’s publicly issued bonds. represents the (overall) cost of capital of the company, which are calculated from the arithmetical mean of the required rate of return of equity providers (cost of equity) and that of debt capital providers (cost of debt), (known as the weighted average cost of capital or WACC approach). represents the so-called free (gross) cash flow, which is entirely at the investors’ disposal. Under the premises of the M ODIGLIANI -M ILLER theorem, it is assumed that the cost of capital of a company with a specific risk category is completely independent of the realized debt-equity ratio (in terms of a relation of the market value of all loan titles to the market value of all shares of a company), because otherwise the financing would have an influence on the market value of the company . The result is that the cost of equity of an unlevered business corresponds to the W j,0 = n 0 ⋅ D j * r j * = X j * r j * . W j,0 = W j,0 EK = n 0 ⋅ K j,0 EK = n 0 ⋅ D j,t *EK 1 + r j,EK * ( ) t t = 1 ∞ ∑ = n 0 ⋅ D j,t *EK 1 + r j,EK * ( ) t t = 1 ∞ ∑ = X j,t *EK 1 + r j,EK * ( ) t t = 1 ∞ ∑ W j,0 = W j,0 EK = n 0 ⋅ D j *EK r j,EK * = X j *EK r j,EK * X j,t * X j,t * = X j *EK und X j * = X j *EK . W j,0 = W j,0 EK + W j,0 FK = X j,t *EK 1 + r j,EK * ( ) t t = 1 ∞ ∑ + X j,t *FK 1 + r j,FK * ( ) t t = 1 ∞ ∑ = X j,t * 1 + r j,GK * ( ) t t = 1 ∞ ∑ X j,t * = X j,t *EK + X j,t *FK W j,0 = W j,0 EK + W j,0 FK = X j *EK r j,EK * + X j *FK r j,FK * = X j * r j,GK * r j,GK * = r j,EK * ⋅ W j,0 EK W j,0 + r j,FK * ⋅ W j,0 FK W j,0 . W j,0 EK W j,0 FK r j,GK * r j,EK * r j,FK * X j * r j,GK * W j,0 1.2 Concepts of Business Valuation 31 45520_Matschke_Griffleiste_SL5.indd 31 45520_Matschke_Griffleiste_SL5.indd 31 16.03.2021 16: 20: 36 16.03.2021 16: 20: 36 cost of capital of the firm: In contrast, the cost of equity of the levered firm is dependent on the debt-equity ratio and can be derived from the weighted (overall) cost of capital by rearranging as follows: or or after consideration of or or or due to If the cost of debt is independent of the debt-equity ratio, under the premises of the M ODIGLIANI -M ILLER theorem, the cost of equity of the levered company results from the cost of equity of the unlevered company [= (total) cost of capital] plus a risk premium which is linearly dependent on the debt-equity ratio. Under the premises of the M ODIGLIANI -M ILLER theorem, the cost of debt , which is independent of the debtequity ratio, is limited to the amount of the risk-free capital market interest rate i. The M ODIGLIANI -M ILLER approach also makes no assumptions about the risk attitude of the economic subject (also known as a preference-free valuation). The relation above is often used for the estimation of the cost of equity in a capitalmarket-theory-oriented business valuation. However, as mentioned in the context of the CAPM approach, the assumptions around the validity of the irrelevance theory of the debt-equity ratio do not hold up in everyday economic life. In 1963, M ODIGLIANI / M IL- LER (1963) revised their views from 1958 and replaced their initial assumptions of equal tax treatment for equity and debt capital with the more realistic assumption that owing to the possible tax deductibility of the cost of debt, financing with borrowed capital provides advantages over financing with equity capital. This first loosening of the unrealistic assumptions by M ODIGLIANI and M ILLER leads to the elimination of the irrelevance principle. Therefore, an uncritical transfer of the M ODIGLIANI -M ILLER approach, which r j,GK * = r j,EK *FK=0 . r j,GK * = r j,EK *FK=0 = r j,EK *FK>0 ⋅ W j,0 EK W j,0 + r j,FK * ⋅ W j,0 FK W j,0 r j,GK * ⋅ W j,0 = r j,EK *FK=0 ⋅ W j,0 = r j,EK *FK>0 ⋅ W j,0 EK + r j,FK * ⋅ W j,0 FK W j,0 = W j,0 EK + W j,0 FK r j,EK *FK>0 ⋅ W j,0 EK = r j,EK *FK=0 ⋅ W j,0 EK + W j,0 FK ( ) − r j,FK * ⋅ W j,0 FK r j,EK *FK>0 = r j,EK *FK=0 ⋅ W j,0 EK + W j,0 FK ( ) W j,0 EK − r j,FK * ⋅ W j,0 FK W j,0 EK r j,EK *FK>0 = r j,EK *FK=0 + r j,EK *FK=0 − r j,FK * ( ) ⋅ W j,0 FK W j,0 EK r j,EK *FK=0 = r j,GK * r j,EK *FK>0 = r j,GK * + r j,GK * − r j,FK * ( ) ⋅ W j,0 FK W j,0 EK . r j,FK * r j,FK * Kapitel 1: Einführung 32 32 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 32 45520_Matschke_Griffleiste_SL5.indd 32 16.03.2021 16: 20: 37 16.03.2021 16: 20: 37 Chapter 1 became a centerpiece of capital-market-theoretical valuation methods, to real situations in which decisions have to be prepared and ultimately are made, should be out of the question. The consideration set out above gives expression to the positive leverage effect of debt financing, but does not explicitly include the risk that arises with the increasing debt (leverage risk). The leverage risk can be determined by implementation of the CAPM with the help of the variance or standard deviation of the expected return on equity. In accordance with the general leverage formula, it follows that: with i as risk-free capital market interest rate. For the variance of the return on equity that is required in the situation of debt financing, it is then determined: or [due to and with the result with Y as random variable and p as probability of occurrence] or or For the standard deviation, accordingly it follows: r j,EK *FK>0 = r j,EK *FK=0 + r j,EK *FK=0 − i ( ) ⋅ W j,0 FK W j,0 EK σ 2 r j,EK *FK>0 ( ) = σ 2 r j,EK *FK=0 Y $ ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ a % & ' ( ' − i ⋅ W j,0 FK W j,0 EK b %&( ⎛ ⎝ ⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟ Var a ⋅ Y + b ( ) = p ⋅ a ⋅ Y + b ( ) − E a ⋅ Y + b ( ) ⎡⎣ ⎤⎦ ∑ 2 b - E(b) = 0, Var a ⋅ Y + b ( ) = p ⋅ a ⋅ Y − E a ⋅ Y ( ) ⎡⎣ ⎤⎦ ∑ 2 = p ⋅ a ⋅ Y − E Y ( ) ( ) ⎡⎣ ⎤⎦ ∑ 2 = a 2 ⋅ Var Y ( ) σ 2 r j,EK *FK>0 ( ) = σ 2 r j,EK *FK=0 Y $ ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ a % & ' ( ' − i ⋅ W j,0 FK W j,0 EK b %&( ⎛ ⎝ ⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟ σ 2 r j,EK *FK>0 ( ) = σ 2 r j,EK *FK=0 ( ) Y % & ' ( ' · 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ a 2 % & ' ( ' 2 − σ 2 i ⋅ W j,0 FK W j,0 EK ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = 0 % & ' ( ' σ 2 r j,EK *FK>0 ( ) = σ 2 r j,EK *FK=0 ( ) · 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ 2 . 1.2 Concepts of Business Valuation 33 45520_Matschke_Griffleiste_SL5.indd 33 45520_Matschke_Griffleiste_SL5.indd 33 16.03.2021 16: 20: 37 16.03.2021 16: 20: 37 or due to Pursuant to this relation, it is evident that the risk for the equity capital in case of an unlevered company with = 0 is the lowest and can be measured by using the variance or the standard deviation of the (overall) cost of capital and that furthermore the risk that is measured by using the standard deviation for equity capital increases proportionally to the debt-equity ratio, hence there is a linear relation between the equity capital risk and the debt-equity risk. Owing to this linear relation between the equity risk and the level of debt, the expected return on equity (from the perspective of the investors) and hence the cost of equity has to grow linearly with the leverage ratio under the conditions of the CAPM. Considering the already explained general relation: it follows that: as cost of equity of the unlevered company and as cost of equity of the levered company. If the derived relation between equity risk and debt is taken into consideration, the cost of equity of the levered company can alternatively be written as: Since the correlation between general market development and development of the firm is independent of a financing decision, the following holds true: σ r j,EK *FK>0 ( ) = σ r j,EK *FK=0 ( ) ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ r j,EK *FK = 0 = r j,GK * σ r j,EK *FK>0 ( ) = σ r j,GK * ( ) ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ . W j,0 FK σ 2 (r j,GK * ) σ(r j,GK * ) r j * = i + (r M * − i) ⋅ ρ j,M · σ j σ M = i + (r M * − i) ⋅ σ j,M σ M 2 ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ r j *FK = 0 = i + (r M * − i) ⋅ ρ j,M FK = 0 · σ j r j *FK = 0 ( ) σ M = i + (r M * − i) ⋅ β j FK = 0 r j *FK >0 = i + (r M * − i) ⋅ ρ j,M FK > 0 ⋅ σ j r j *FK > 0 ( ) σ M = i + (r M * − i) ⋅ β j FK > 0 σ r j,EK *FK >0 ( ) = σ r j,EK *FK = 0 ( ) ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ r j,EK *FK >0 = i + (r M * − i) ⋅ ρ j,M FK > 0 · σ r j,EK *FK = 0 ( ) ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ σ M = i + (r M * − i) ⋅ β j FK > 0 r j,EK *FK > 0 = i + (r M * − i) ⋅ β j FK = 0 ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ . Kapitel 1: Einführung 34 34 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 34 45520_Matschke_Griffleiste_SL5.indd 34 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 Chapter 1 This implies cost of equity of the levered firm from the CAPM: thus, the inclusion of the leverage risk does not change anything in the original context: Both under the premises of the CAPM and the M ODIGLIANI -M ILLER theorem, the cost of equity of the levered company increases proportionally with the debt-equity ratio. With regard to the beta factor, the following relation is determined: The premises of the CAPM and the M ODIGLIANI -M ILLER theorem ultimately build the theoretical basis for a variety of competing discounted cash flow approaches (DCF approaches). With the (standard) CAPM and M ODIGLIANI -M ILLER approach, it is the special charm of “putting assumptions together that don’t belong together.” While the planning horizon of the standard CAPM consists of two dates or just one period, the general M ODIGLIANI -M ILLER approach works with perpetuities. The M ODIGLIANI -M ILLER theory, based on the loosened assumptions introduced in the later 1963 study, considers taxes and the tax deductibilty of debt, whereas the standard form of the CAPM neglects to address taxes. In addition, the subjects in the CAPM are risk averse, but under the premises of the M ODIGLIANI -M ILLER theorem they are not bound to any preferences. In Figure 1.13 (cf. H ERING 2000a, p. 445), once again the main incompatible assumptions of these models are outlined. With regard to the DCF approaches, the wide variety of approaches, argumentations, and methods makes the situation similar to the mixed methods approach combining asset and income based approaches that was more common in the past. This becomes clear - aside from the different DCF approaches - if the concrete proposals of the introduced methodical concepts are considered directly. This, however, does not apply to every single parameter since they are assessed and determined differently, despite allegedly sharing the same theoretical basis (C OLEMAN 2014, O LBRICH / Q UILL / R APP 2015). Numerous different definitions of the so-called free cash flow (cf. G ÜNTHER 1997, p. 112) and the value parameters from the CAPM give plenty of latitude for valuation fantasies, especially if privatedly owned, non-listed companies are analyzed. However, all these approaches have a common standard: The determination of the one and only true market value of the company. This standard cannot be fulfilled by all approaches at once, on the contrary it must be understood as merely a platitude referring to the market-oriented valuation. The following also has to be said: In relation to the market-orienρ j,M FK=0 = ρ j,M FK > 0 = ρ j,M . r j,EK *FK >0 = i + r M * − i ( ) ⋅ ρ j,M ⋅ σ r j,EK *FK = 0 ( ) σ M ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ , β j FK >0 = β j FK = 0 ⋅ 1 + W j,0 FK W j,0 EK ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ = β j FK = 0 + β j FK = 0 ⋅ W j,0 FK W j,0 EK . CAPM Planning horizon Taxes 1 No Infinite No CAPM M ODIGLIANI / M ILLER Planning horizon Taxes Preference-free valuation Figure 1.13: Incompatible assumptions of the DCF elements 1 No Infinite Yes No Yes 1.2 Concepts of Business Valuation 35 45520_Matschke_Griffleiste_SL5.indd 35 45520_Matschke_Griffleiste_SL5.indd 35 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 ted business valuation (W ILLIAMS 1938, D EAN 1954, G ORDON 1959, C OASE 1981, D AM- ODARAN 2015, K OLLER / G OEDHART / W ESSELS 2020), there is much calculation and little thinking. Bothersome calculations do not make the question of the purpose of calculations superfluous, but rather prompt that question and also offer an answer to it. While this question is consequently being avoided, an answer is essential in order to reliably estimate both the usefulness and the applicability of the approaches at hand. The determined market values are generally not values that can offer any decision support. The major use of those values can be seen in the determination of argumentation values, and that is why they are addressed in greater detail in the fourth chapter. A (random) market value has to be determined, because everyone desires it, or at least that seems to be the case. The assessment of the market value becomes a rule of the game to establish the correct behavior. To play by the rules is certainly essential; especially if you want to make money, in practice, you will likely have to follow the current vogue (B RÖSEL / T OLL / Z IMMERMANN 2011a, 2011b, 2012). Kapitel 1: Einführung 36 36 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 36 45520_Matschke_Griffleiste_SL5.indd 36 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 Chapter 1 1.3 Occasions of Business Valuation 1.3.1 Requirement to Systematize Occasions of Business Valuation There is a range of occasions and reasons for, and causes of, business valuation. For a long time it was established practice to list and to briefly describe possible occasions for business valuation, like the acquisition of a company, the merger of several companies, dispossession for legal reasons, investment in a company as an contribution in kind (a non-cash contribution), compensation when minority shareholders were removed, the exclusion of an unsuitable partner, etc. Generally, this listing practice did not advance the explanations authors offered, because subsequent discussion was mostly limited to an examination of purchasing and selling activity relating to a (whole) company. However, a model-theoretical analysis of business valuation functions is required to reveal the similarities and differences in relation to such business valuation events. The examination of business valuation problems in relation to the respective valuation purpose has to be based on a precisely defined starting point so that the adequacy of the proposed proceedings can be examined intersubjectively. Finally, the business valuation calculation - as with any other value calculation - must be purpose-oriented and therefore does not have general validity. The calculation purpose can be sensibly materialized only with respect to the occasion of the calculation and the result of the calculation must in turn be judged in the context of the calculation purpose and the calculation cause. A fundamental distinction of occasions was already determined when they were categorized as main and minor functions. The quite different events mentioned as an example share the common ground of affecting the ownership of the company. These are situations that focus on a solution of interpersonal conflicts regarding the terms of a change being considered in the ownership of a company or one that has already occurred. These events can be classified as main functions. In addition, there are also occasions that, as explained above, are not related to considerations of a change of ownership (minor functions). The main functions, which are examined below, concern interpersonal conflict situations, that is, disputes about the terms under which a change in ownership of a company shall occur. The functional business valuation is thus no equilibrium theory but a theory that accepts the real world is imperfect. The observed events are consequently decision dependent and interpersonally conflicting. To avoid being exposed to such a highly complex world at least in theory, M ATSCHKE early on proposed a systematization of the occasions underlying the main functions (M ATSCHKE 1975, p. 30, M ATSCHKE 1979, p. 30, M ANDL / R ABEL 1997, p. 14). This framework helps to separate similar form distinguishable cases and thus serves to support the model-theoretical analysis and the derivation of adequate valuation models. The events of the main functions can be classified into the following categories: 1.3 Occasions of Business Valuation 37 45520_Matschke_Griffleiste_SL5.indd 37 45520_Matschke_Griffleiste_SL5.indd 37 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 1. with regard to the type of property change in conflict situations of the acquisition/ sale type and the merger/ demerger type, 2. with regard to the degree of relationship in joint (affiliated) and disjointed (unaffiliated) conflict situations, 3. with regard to the degree of complexity in oneand multi-dimensional conflict situations, and 4. with regard to the degree of dominance in dominated and non-dominated conflict situations. Figure 1.14 illustrates this classification of possible occasions of business valuation according to the main functions that will be discussed further below. Because the characteristics mentioned should be accounted for in combination, a broad theoretical foundation for objective and situation-specific business valuation models results; accordingly, every valuation occasion can be thoroughly analyzed. Anlässe zur Unternehmensbewertung Anlässe mit Änderung der Eigentumsverhältnisse des Unternehmens Konfliktsituationen vom Typ Kauf/ Verkauf und vom Typ Fusion/ Spaltung Jungierte und disjungierte Konfliktsituationen Ein- und mehrdimensionale Konfliktsituationen Dominierte und nicht dominierte Konfliktsituationen Anlässe ohne Änderung der Eigentumsverhältnisse des Unternehmens Figure 1.14: Classification of occasions of business valuation Occasions of business valuation Occasions with a change of ownership situation Occasions without a change of ownership situation Conflict situations of the acquisition/ sale type and of the merger/ demerger type Joint and disjoint conflict situations One-dimensional and multi-dimensional conflict situations Dominated and non-dominated conflict situations Kapitel 1: Einführung 38 38 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 38 45520_Matschke_Griffleiste_SL5.indd 38 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 Chapter 1 1.3.2 Systematization of Occasions of Business Valuation According to the Main Functions 1.3.2.1 Conflict Situations of the Acquisition/ Sale Type and of the Merger/ Demerger Type In a conflict situation of the acquisition/ sale type, the ownership situation of the company to be valued is changed such that one conflicting party (seller) surrenders its ownership of the company in favor of the other conflicting party (buyer) and receives from the buyer compensation (a price, in the broader sense) in exchange. Examples of such conflict situations of the acquisition/ sale type are the acquisition (M ATSCHKE 1975, H ERING / T OLL / K IRILOVA 2014a and 2015a) and sale (M ATSCHKE 1975, H ERING / T OLL / K IRILO- VA 2014b and 2015b, H ERING / T OLL 2015) of a whole company or the interests thereof (e.g., subsidiaries, affiliates, or permanent establishments). In addition, expropriations of private companies and privatization of public companies as well as the exclusion of socalled annoying partners or a compulsory expulsion or withdrawal of minority shareholders can be classified in this category. The amount of the monetary compensation (cash price), which is usually provided by the buyer, takes center stage in these types of transactions. However, merger type conflict situations do not result in a change of ownership, but in changed ownership relations between the same shareholders when the conflict is resolved (M ATSCHKE 1975, Y AGIL 1987, R EICHERTER 2000, H ERING 2004a, T OLL / H ERING 2017). In such a situation several companies to be valuated are combined and the ownership relations are changed in that the owners of the companies to be merged acquire direct or indirect ownership of the new economic entity resulting from the merger. Examples of such conflict situations are mergers of companies with different constitutions of shareholders and business start-ups, which bring the whole company as a contribution in kind. Additionally, the significant case of losing ownership of the valuated company in a legal sense, but acquiring ownership of the new company through receiving shares in it is classified in this category because the former shareholders of the company still participate in the risks and opportunities associated with their former company, albeit rather indirectly. In addition, the admission of a new shareholder to an established company that does not adversely affect the financial commitment of the former shareholders can be classified under this conflict type (H ERING / T OLL / K IRILOVA 2016). The distribution of ownership shares and thus the distribution of the future successes of the companies to be merged among the conflict parties is at the centre of the interpersonal conflict to be resolved in cases of a conflict situation of the merger type. The term demerger can be understood as essentially the real division of a company or the spin-off of parts of the previous company to the previous owners. An outsourcing with a hand-over to a third party is an example of a conflict situation of the acquisition/ sale type. An interpersonal conflict situation of the demerger type (M ANDL / R ABEL 1997, p. 14, B YSIKIEWICZ / M ATSCHKE / B RÖSEL 2005, B YSIKIEWICZ 2008, T OLL 2018) occurs if the arrangements of ownership of the businesses that arose from demergers differ from tho- 1.3 Occasions of Business Valuation 39 45520_Matschke_Griffleiste_SL5.indd 39 45520_Matschke_Griffleiste_SL5.indd 39 16.03.2021 16: 20: 38 16.03.2021 16: 20: 38 se in place before the demergers (change of structure of ownerships). A change of structure of ownership often has the following features: 1. The previous owners are involved in the new business, with changed conditions of shares (change of structure of ownership in the case of a ratio-maintaining or not ratio-maintaining demerger due to changed ownership interests). A special form of this feature is the separation of ownership. In this variation, a complete separation of the shareholders occurs, that is, some receive something, while others receive something else (see Figure 1.15). 2. Even if the previous owners are involved in the new businesses under the same conditions as in the former businesses, a change of ownership structure can occur because the cumulative income stream of the developing businesses with regard to their amount and/ or structure deviates from the demerging business (change of structure of ownerships in the case of a ratio-maintaining demerger due to changed successes in future). Both variations, which can be combined, have the same following effect: The former shareholders will soon be involved in different ways in the opportunities and risks presented by the newly created businesses. A. Allgemeiner Fall der Eigentumsstrukturänderung bei einer nicht-verhältniswahrenden Spaltung aufgrund veränderter Beteiligungsquoten B. Sonderfall „Eigentumstrennung“ Unternehmen vor der Spaltung mit den Eignern A (x %) und B (y %) Neues Unternehmen mit den Eignern A (xI %) und B (yI %) Neues Unternehmen mit den Eignern A (xII %) und B (yII %) Neues Unternehmen des Eigners A (x I = 100 %) [B (yI = 0 %)] Neues Unternehmen des Eigners B (yII = 100 %) [A (xII = 0 %)] Unternehmen vor der Spaltung mit den Eignern A (x %) und B (y %) Figure 1.15: Demerger types with changed ownership interests А. General part of structural ownership changes at a demerger with a not maintaining ratio due to changed stakes B. Special case “separation of ownership” Companies before the demerger with shareholders А (x %) and B (y %) Companies before the demerger with shareholders А (x %) and B (y %) New company with shareholders А (x I %) and B (y I %) New company of shareholder А (x I = 100 %) [B (y I = 0 %)] New company with shareholders А (x II %) and B (y II %) New company of shareholder B (y II = 100 %) [А (x II = 0 %)] Kapitel 1: Einführung 40 40 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 40 45520_Matschke_Griffleiste_SL5.indd 40 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 Chapter 1 1.3.2.2 Non-dominated and Dominated Conflict Situations The distinction between dominated and non-dominated conflict situations describes the balance of power between the conflicting parties with regard to the change of ownership in the company to be valuated. The issue is if such a change of ownership can be accomplished by one party, that is, whether that change of ownership is forced through by one party. These could be conflict situations of the acquisition/ sale type as well as of the merger/ demerger type, depending on whether a cash settlement or compensation in shares, especially stock, is sought (cf. M ATSCHKE 1979, p. 39). A non-dominated conflict situation (M ATSCHKE 1979, p. 31) exists if no single participating party to the conflict can enforce a change of ownership without collaboration and against the declared will of the other conflict party, in the company to be valuated. In a non-dominated conflict situation a change of ownership by negotiation occurs only if there is a settlement proposal acceptable to all parties. The situation demands a solution that is advantageous to all parties. Examples of non-dominated conflict situations can include sales or acquisitions of firms and the merger or demerger of several companies with free entrepreneurial responsibility and decision-making power. Conversely, in the case of a dominated conflict situation (M ATSCHKE 1979, p. 33), one of the participating parties to the conflict has the power to enforce a change of ownership against the declared will of the other party. Such a unilaterally enforceable change of ownership (e.g., minority shareholders being squeezed out) usually has to follow legal rules and the dominated party can appeal to the law to examine the conditions under which the change of ownership is executed. The legal legitimation permits the parties to execute a change of ownership directly, including the opportunity for other parties to appeal to the courts against the conditions of change of ownership, or indirectly, that is, instigated by one party by means of the courts, which might authorize one sided changes of ownership of the company under specific conditions. The dominated party cannot influence the change of ownership, but only the conditions of its realization. Hence, dominated conflict situations are those in which primary norms of court decisions and legislation or from existing contracts are highly relevant and must ultimately be respected in the conflict situation (S IEBEN 1966, S IEBEN 1969a, O LBRICH 1982, B USSE VON C OLBE 1992, B ÖCKING 2003). 1.3.2.3 Disjoint and Joint Conflict Situations The existing literature implicitly assumes with only a few exceptions (M ATSCHKE 1975, p. 336, B RÖSEL 2002, p. 98, H ERING 2014, p. 166) that the decision subject valuates a company according to a single conflict situation of the acquisition/ sale or merger/ demerger type, which has no relation to other conflict situations of the acquisition/ sale or merger/ demerger type. Such conflict situations are usually called unaffiliated or disjoint conflict situations. If however, as is usually the case in practice (M ATSCHKE 1975, p. 34), it is assumed that the conflicting parties want to buy/ sell and/ or merge/ demerge several companies, an isolated company valuation related solely to one conflict situation is not adequate to address the problem because it disregards the interdependencies between the conflict situations. It is important to recognize that agreements in other conflict situations of the 1.3 Occasions of Business Valuation 41 45520_Matschke_Griffleiste_SL5.indd 41 45520_Matschke_Griffleiste_SL5.indd 41 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 acquisition/ sale or merger/ demerger type influence the decision field of the decision subject and hence the level to achieve after an agreement on acquisition/ sale or on merger/ demerger of the company, which has to be maintained to determine the decision value. In such joint or affiliated conflict situations the decision value of the company in a specific conflict situation can only be properly determined with reference to possible agreements in other conflict situations. In this case, the decision value is a conditional variable. Referring to one decision subject and assuming that it appears only in one other conflict situation of the acquisition/ sale or the merger/ demerger type, the following joint conflict situations can be distinguished (see Figure 1.16) because symmetry means it is not important which type characterizes the “1. conflict situation” and which type the “2. conflict situation”: • joint conflict situations of the acquisition-acquisition, sale-sale, merger-merger, or demerger-demerger types; and • joint conflict situations of the sale-acquisition, merger-acquisition, merger-sale, merger-demerger, demerger-acquisition, or demerger-sale types. 1.3.2.4 One-Dimensional and Multi-Dimensional Conflict Situations Acquisitions and sales and mergers or demergers of companies are very complex conflict situations, even if the only conflicts are between the parties; but even more so if conflict emerges among the parties. With respect to the relevant number of terms of agreement in these situations, a (theoretical) distinction between one-dimensional and multi-dimensional conflict situations can be made. In reality, an agreement between the parties depends on many factors, among which the (cash) price for the company in the case of acquisitions and sales as well as the distribution of ownership shares in companies after a merger or in the companies after a demerger are crucial; nevertheless they are not the only conditions determining an agreement between the parties and the group as a whole are referred to as conflict-resolution-relevant facts. The presence of several problem-solving facts in negotiations makes it logical to describe company valuation situations as multi-dimensional conflict situations. In contrast, it is generally assumed in theory that company valuation situations represent a one-dimensional conflict situation, namely of the acquisition/ sale type (for the few exceptions based on implicit assumptions, see e.g., M ATSCHKE 1975, p. 55 and p. 356, H IN- TZE 1992a, H INTZE 1992b, M ATSCHKE 1993a, O LBRICH 1999, p. 177, R EICHERTER 2000, 1. Conflict situation type 2. Conflict situation type Acquisition Acquisition acquisition-acquisition Sale acquisition-sale Merger acquisition-merger Demerger acquisition-demerger Sale Merger Demerger Figure 1.16: Classification of joint/ affiliated conflict situations sale-acquisition merger-acquisition sale-sale merger-sale demerger-acquisition demerger-sale sale-merger merger-merger sale-demerger merger-demerger demerger-merger demerger-demerger Kapitel 1: Einführung 42 42 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 42 45520_Matschke_Griffleiste_SL5.indd 42 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 Chapter 1 p. 232, B RÖSEL 2002, p. 143, H ERING 2014, p. 171). This means that normally the dependence of the business value on the conflict situation is examined in only a rudimental fashion, because in the context of the conflict situations of the acquisition/ sale type the (cash) price is examined as the sole factor relevant to conflict resolution. Hence, the firm value is most often regarded as the current cash equivalent for the company. It is a restriction according to the examination of the real conflict situation because the (cash) price, especially if payable at once, for the transfer of the rights of ownership in the case of conflict situations of the acquisition/ sale type or the distribution of both shares of ownership and rights to vote in the case of conflict situations of the merger/ demerger type constitute only one area of concern among others, where the parties have to find a solution if they are to reach an agreement. The common understanding on the particular takeover according to civil law (asset, stock acquisition), when the valuation object is a stock corporation, is another area of concern in purchase/ sales situations. A further significant problem of negotiation is the common understanding on the delimitation of the company to be acquired/ sold (company volume), and thus the exclusion of the parts of the company that are not being taken over. Further areas of negotiation can address the future composition of the company management and a possible further participation of the seller, furthermore the understanding between buyer and seller on business policy interests beyond the activity areas of the company to be acquired/ sold, the agreement/ non-agreement on competition restraints whether temporal or regional, the increased flexibility of purchase price components owing to their dependence on future events. Therefore, a (fixed) basic purchase price can be supplemented by one or more (variable) payments that are dependent on achieving target figures agreed within a defined period of time or at a certain point after the company sale. Corresponding contractual agreements are also referred to as “earnout-clauses” (B EHRINGER 2004, T OLL / R OLINCK 2017). Scholars have recently discussed issues related to the arrangements for both the financial and temporal responsibility of the seller in the case of environmental contamination (e.g., of brownfield sites) (T ILLMANN 1998). In connection with the takeover of an unprofitable company capable of restructuring, the measures still to be taken by the seller as well as their financial contribution toward further necessary measures for the company’s reconstruction planned by the purchaser may be relevant to conflict resolution. Additional meaningful conditions for conflict resolution are arrangements on liability issues and assurances with regard to completeness of information, exemptions from previous or existing liabilities and much more. Even if it was just about the level of remuneration in a negotiation, it should still be noted that the payment of remuneration is customizable in various ways, for example, through the allocation of cash or installments payable immediately, temporary or life annuities due in the future, the life/ term (i.e., the run time) of an annuity, and through tax rates. Therefore, each party involved in the conflict would be well advised to consider tax-related consequences and estimate potential effects on remuneration arrangements on the rational conduct of negotiations. However, even in the event of an agreement on the price as a single fact, which is relevant for the conflict resolution during negotiation and with the help of a certain traditional company valuation method and via agreement on parameters significant to this valuation (e.g., future success, adequate target rate, the form and extent of consideration 1.3 Occasions of Business Valuation 43 45520_Matschke_Griffleiste_SL5.indd 43 45520_Matschke_Griffleiste_SL5.indd 43 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 of risk and inflation, and consideration of an asset value), both purchasers and sellers preparing for a negotiation would be well advised to clarify which negotiation margin may result with regard to these parameters, and which extensions of them are just about acceptable so that the particular marginal price is not violated. The example facts mentioned relevant to conflict resolution can be classified into two categories (M ATSCHKE 1975, p. 57): 1. those directly modifying the decision field (original facts relevant to conflict resolution, such as the legal form of takeover according to civil law, level and form of remuneration, extent of the company at issue, agreement on business policy interests beyond this company, regulations on temporal and regional competition restraints, regulations regarding financial and temporal responsibility with regard to brownfields and to redevelopment measures), and 2. those serving to substantiate extensions of the original facts relevant to conflict resolution or to derive substantiated extensions of the original facts relevant to conflict resolution (derivative facts relevant to conflict resolution such as parameters of a company valuation, on the basis of which the parties argue). Therefore, these example facts only indirectly affect the decision field. If parameters directly affect the decision-making area, as in the case of an agreement in interpersonal conflict situations, they are to be assigned to the original facts relevant to conflict resolution. For a change of ownership of a company that is occurring, it is thus necessary for the parties in conflict to agree on those parameters. Therefore, the original facts complement or supplement each other. Agreements in those situations lead to changes to the decision fields (quantity of action alternatives) of the parties involved in the conflict. A company seller, for example, is giving up the company as a means of achieving a target and in return receives payments, which open up new options for action. The purchaser acquires the company and with it new options for action at the same time, but due to the price to be paid, has to postpone alternative actions that might otherwise be available. There is a close relationship between the amount of the price to be paid for the company and the change of the decision fields after an agreement. The higher the price to be paid, the more action alternatives are no longer available to the buyer and the more options for action become feasible for the seller in a monetary economy. However, such a close relation between changes of decision-making areas and the extension of facts relevant to the conflict resolution or the variables of an agreement does not necessarily need to be the case in all facts relevant to conflict resolution. This particularly concerns the derivative facts relevant to conflict resolution that only indirectly modify the decision field of the valuation subject. Instead, they serve to derive or substantiate certain extensions of original facts relevant to conflict resolution that change the decision fields. For this reason, those same derivative facts are in a meansend-relationship with the original facts relevant to conflict resolution. That relationship includes, for example, the variables valuation methods, level and point of the influx of cash flows, and level of capitalization rate, which only indirectly modify the decision field of a conflict party and instead facilitates deriving the original fact level of purchase price. Kapitel 1: Einführung 44 44 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 44 45520_Matschke_Griffleiste_SL5.indd 44 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 Chapter 1 With regard to a positive negotiation result, it is not necessary that the parties ultimately agree on all derivative facts relevant to conflict resolution. The substitution of an original fact (most likely the price level) through various derivative facts (e.g., valuation methods, level and point of the influx of cash flow, and level of capitalization rate) may just serve to approximate the viewpoints of the parties, in order that they might subsequently agree on a particular price. The vital role of both derivative and original facts relevant to conflict resolution in the negotiation process is discussed in greater detail in Chapter 4. The major characteristics of the valuation situations within the framework of the main functions are summarized in Figure 1.17 (following O LBRICH 1999, p. 13). In order to present the conflict situations that underly the following statements transparently, we adopt the so-called conflict cube (R EICHERTER 2000, p. 122) where appropriate. Typ des Kaufs/ Verkaufs Typ der Fusion/ Spaltung Art der Eigentumsänderung Ausprägungen der Bewertungssituationen der Hauptfunktionen Grad der Dominanz Grad der Verbundenheit Grad der Komplexität nicht dominiert dominiert disjungiert (unverbunden) jungiert (verbunden) mehrdimensional eindimensional Acquisition/ sale type Merger/ demerger type One-dimensional Multidimensional Non-dominated Dominated Disjoint (unaffiliated) Joint (affiliated) Degree of relationship Degree of dominance Type of property change Degree of complexity Forms of valuation situations for the main functions Figure 1.17: Classification of valuation situations for the main functions 1.3 Occasions of Business Valuation 45 45520_Matschke_Griffleiste_SL5.indd 45 45520_Matschke_Griffleiste_SL5.indd 45 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 1.4 Matrix of Functional Business Valuation and Overview of Methods of Business Valuation 1.4.1 Matrix of Functional Business Valuation The following Chapters 2, 3 and 4 will explain the main functions of business valuation. The first part of each chapter offers an analysis of the characteristics of the main functions and their corresponding standards (and types) of value before the value determination is presented. We use the matrix of functional business valuation to illustrate this valuation process. Valuation methods from which corresponding values can be generated are presented for each main function selected. At this point, the matrix of functional business valuation and an overview of the main methods of business valuation are presented. Regardless of the firm value that has to be determined with regard to the intended purpose, the process of business valuation in the broader sense can usually be divided into three steps (H ERING 2014, p. 4): Step 1: Determination of relevant data. Step 2: Transformation of the determined data into the firm value (business valuation in the narrower sense). Step 3: Use of the determined business value. Only if these steps lead from the function of business valuation, can rules of procedure be coherently derived to facilitate valuation (M ATSCHKE 1981, p. 115). If the variables business valuation functions and valuation steps are marked in a coordinate system, the matrix of a functional business valuation can be determined (M ATSCHKE / B RÖSEL 2013a, p. 120). This matrix is illustrated in Figure 1.18, which offers a transparent representation of the valuation process and supports the elaboration of each valuation step according to the assignment of business valuation. However, it should be noted that these steps cannot be clearly differentiated in reality, however, they do not necessarily follow only one direction and the direction(s) taken will subsequently be determined in an example of a decision value calculation. Kapitel 1: Einführung 46 46 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 46 45520_Matschke_Griffleiste_SL5.indd 46 16.03.2021 16: 20: 39 16.03.2021 16: 20: 39 Chapter 1 During a negotiation over a company, negotiators should bear in mind that they can know neither all possible conflict-resolution-relevant facts nor the possible features according to the final relevant conditions of qualification at the beginning of the negotiation. It is therefore first necessary to discover the volume of possible conflict resolutions during the negotiations. In order to ensure negotiation assumptions are rational and accurately reflect the influence on the conflict resolution process of the valuation object, hypotheses about the relevant resolution conditions and their possible extensions are required. Subsequently, corresponding decision values must be determined (M ATSCHKE 1993a, p. 11). A presumptive buyer can thus also gain new insights into the company during the negotiation process and, especially in drawn-out negotiations, changes can occur in their decision field that are independent of the conflict so that the decision value becomes a dynamic limit (and in view of an uncertain reality even a dynamic range), which can shift several times during the negotiations and in different directions (B RÖSEL / B URCHERT 2004, p. 351). The valuation process has to run repeatedly through steps 1 (field A of the matrix) and 2 (field B of the matrix). Hauptfunktionen Nebenfunktionen Entscheidungsfunktion Vermittlungsfunktion Argumentationsfunktion Nebenfunktion 1 Jeweils weitere Nebenfunktion Ermittlung der relevanten Daten Transformation der relevanten Daten in einen Wert Verwendung des ermittelten Wertes A B C D G J M E F H K N I L O Bewertung i. e. S. )* ' + ' Bewertung i. w. S. ) * '''' + '''' Valuation in the broader sense Valuation in the narrower sense Main functions Minor functions Decision function Mediation function Argumentation function Minor function 1 Respective further minor function Determination of relevant data Transformation of relevant data in a value Use of the determined value Figure 1.18: Matrix of functional business valuation 1.4 Matrix of Functional Business Valuation 47 45520_Matschke_Griffleiste_SL5.indd 47 45520_Matschke_Griffleiste_SL5.indd 47 16.03.2021 16: 20: 40 16.03.2021 16: 20: 40 1.4.2 Overview of Methods of Business Valuation For terms like future performance value, income value, net asset value, combination value, and the calculation methods involved, it remains unknown for which task the determined value is relevant. Finally, methods of business valuation refer to the determination of the standard of value and not to the purpose of the business valuation. However, regarding the respective valuation methods both purpose and task have a pivotal role because the goals, which result from the respective valuation purpose, determine the methods used for business valuation. It is necessary to verify if the respective method and its basic conditions fulfill the valuation purpose and if possible improvements to the business value, delivered by employing a more suitable method, could have been neglected (M ATSCHKE 2017c). According to the matrix of business valuation, the purpose-related valuation methods have to be classified under the second category, the transformation of the determined data in the company value (business valuation in the narrower sense). Over the years, a multitude of different valuation methods came into use in the theoretical and empirical literature of business valuation. While the valuation methods in the Anglo-Saxon literature are categorized into the asset approach (also referred to as the cost approach), the income approach, and the market approach, the methods considered here are distinguished according to their approach into single and overall (total) valuation methods and into combined (mixed) valuation methods (see Figure 1.19). Рис. 19: Матрица функциональной оценки предприятий Methoden der Unternehmensbewertung Einzelbewertungsverfahren Substanzwertverfahren Rekonstruktionswert Liquidationswert Verfahren der börsenkursgestützten Bewertung Methode des börsennotierten Vergleichsunternehmens Methode des Börsengangs Kombinierte Bewertungsverfahren Mittelwertverfahren Verfahren der Goodwillrenten Verfahren der Geschäftswertabschreibung Gesamtbewertungsverfahren Gesamtbewertungsorientierte Vergleichsverfahren Verfahren der kürzlichen Akquisition Multiplikatormethoden Finanzwirtschaftliche Verfahren Finanzierungstheoretische Verfahren Kapitalmarkttheoretische Verfahren „Adjusted Present Value”-Verfahren (APV-Verfahren) „Weighted Average Cost of Capital”-Verfahren (WACC-Verfahren) „Equity”-Verfahren Verfahren der „strategischen Bewertung” Bewertung auf Basis der Optionspreistheorie Investitionstheoretische Verfahren Zustands-Grenzpreismodell (ZGPM)/ Zustands-Grenzquotenmodell Zukunftserfolgswertverfahren/ Ertragswertverfahren Approximativ dekomponierte Bewertung Substanzwert als Auszahlungsersparniswert Methods of business valuation Single valuation methods Overall valuation methods Methods of the net asset value Recent acquisition approach Reconstruction value Value of liquidation Methods of stock-marketaided valuation Similar public company approach Initial public offering (IPO) approach Overall valuation methods of comparison Financial methods Market multiple methods Approximate value decomposition APV method WACC method Equity method Valuation based on the option price theory State marginal price/ quote model German income approach Method of average value Method of goodwill income Method of goodwill amortization Net asset value as a payment saving Strategic valuation Figure 1.19: Selected methods of business valuation Investment theory Capital market theory Finance theory Combined valuation methods Kapitel 1: Einführung 48 48 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 48 45520_Matschke_Griffleiste_SL5.indd 48 16.03.2021 16: 20: 40 16.03.2021 16: 20: 40 Chapter 1 1. The single valuation methods hold a business value is to be determined via a) the realized prices of shares in a company b) the sum of the individually determined values of single business components (material and immaterial production factors). Case 1a) defines the aforementioned issue that realized prices do not have any direct reference to values, which remain to be determined. Since the businesses or their definable parts, for example, complex divisions of the company, need to be disassembled into their parts, there is a risk according to case 1b) that positive and negative economies of scope are not taken into account. 2. As for the combined valuation methods, business values are determined by a mix of overall and single valuation methods. They can be classified into a) an average value method, b) the method of goodwill annuities, and c) the method of goodwill amortization. 3. In contrast, the businesses or definable parts that are valuated with the overall valuation methods can be categorized into a) the overall-valuation-oriented comparison methods and b) the financial methods, and are regarded as respective valuation units. The financial valuation methods have a vital role in theory and practice. Depending on whether the methods are based on the findings of financial or investment theory, they have to be strictly distinguished into finance-theoretical or investment-theoretical valuation methods (H ERING / V INCENTI 2004, p. 343). The scope of the theory of finance is about the explanation of market processes and market results under ideal requirements. Central aspects of these finance-theoretical explanatory models are the hypothetical, objective, market equilibrium price and the perspective of an aggregated perfect and complete market, whose market participants are characterized by homogenous decision fields. Since these valuation models neglect to address the individual circumstances of the valuation subject, which are based on the theory of finance, in respect to the individual relations of the valuation subject, they generally cannot be used as decision models. In contrast, investment theory is oriented toward economically assessing the profitability of cashflows to support individual decisions made under real, incomplete conditions (O LBRICH / R APP / F OLLERT 2020). The major profitability ratio of the investment theory is the net present value, which ought to provide decision support to the solution of real economic problems. These problems will be analyzed under the broadest possible consideration of the goals and the decision field, that is, restrictions, opportunities for action, and expectations of the valuation subject. Valuation methods based on investment theory usually consider the individual situations of the valuation subject in order to enable a more appropriate calculation of the firm value for specific and individual purposes. The main differences between finance theory and investment theory are outlined in Figure 1.20. They clarify that a determination of the decision value cannot be carried out aided by finance-theoretical methods. However, since these approaches are widely applied in business valuation, they will be discussed in greater detail in Chapter 4. 1.4 Matrix of Functional Business Valuation 49 45520_Matschke_Griffleiste_SL5.indd 49 45520_Matschke_Griffleiste_SL5.indd 49 16.03.2021 16: 20: 40 16.03.2021 16: 20: 40 Section 1.2.2 above offered a basic explanation of the market-oriented neoclassical valuation, relating how the represented CAPM and M ODIGLIANI -M ILLER model was based on finance theory and focused on the theoretical basis of the discounted cash flow approaches (DCF approaches), that is, capital-market-theoretical approaches used for the determination of market values. The DCF methods comprise the APV approach (APV = “Adjusted Present Value”), the WACC approach (WACC = “Weighted Average Cost of Capital”), and the equity approach. While the equity approach already determines the equity value, both the APV approach and the WACC approach are techniques in which the market value of debt (borrowed capital) FK (German: Fremdkapital = FK) is to be substracted from the determined overall value of the business V in order to determine the market value of equity EK (German: Eigenkapital = EK): EK = V - FK. These approaches are referred to as gross methods, whereas the equity approach represents a net method of valuation. There are further procedures in addition to these theoretical methods of market value determination, however, their theoretical orientation leaves much to be desired. They are called methods of comparison in literature, and derive business values in terms of potential market prices and especially for non-listed companies on sector-based multiples (sales, profit, and cash flow) of comparable reference companies (known as the market multiples approach) from both the estimated and realized market prices of these reference companies. This latter approach can be distinguished in singleand overalloriented comparison methods. The single-valuation-oriented comparison methods, that is, methods of market-oriented value determination, seek to establish a connection between the price of a share and the value of the company through the construct of a market value (O LBRICH 2000, p. 459). These popular approaches in the Anglo-Saxon world are designed to derive business values from stock market prices (known as the similar public company approach) or from issue prices (known as the initial public offering approach) of similar companies. The overall-valuation-oriented comparison methods are, like the multiples method, approaches that try to derive business values from realized transaction prices [sic.] of (entire) reference companies (also known as the recent acquisitions approach). Corporate finance Main objective Theory of finance Exploration of market processes and market results Investment theory Decision support for the solution of real economic problems Conditions Purpose Perspective Figure 1.20: Major differences between theory of finance and investment theory Idealized conditions Determination of a hypothetical objective market equilibrium price Real incomplete conditions Economic valuation of the profitability of cash flows Aggregated overall market Conditions of a specific valuation subject Kapitel 1: Einführung 50 50 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 50 45520_Matschke_Griffleiste_SL5.indd 50 16.03.2021 16: 20: 41 16.03.2021 16: 20: 41 Chapter 1 Figure 1.21 structures the approaches to the market-oriented value determination illustrated in the overview of selected methods (Figure 1.19). Methoden der marktorientierten Wertermittlung DCF-Verfahren Gesamtkapitalbewertungsansatz (Entity-Approach): Bruttoverfahren WACC-Verfahren APV-Verfahren Eigenkapitalbewertungsansatz (Equity-Approach): Nettoverfahren Equity-Verfahren Vergleichsverfahren Ableitung potentieller Marktpreise auf Basis von Multiplikatoren Multiplikatorverfahren Ableitung potentieller Marktpreise aus (geschätzten oder realisierten) Preisen vergleichbarer Unternehmen Verfahren der kürzlichen Akquisition Verfahren des börsennotierten Vergleichsunternehmens Methode des Börsengangs WACC method APV method Equity method Market multiples method Recent acquisition approach Similar public company approach Initial public offering (IPO) approach Derivation of potential market prices from (estimated or realized) prices of companies (Comparative company approach) Derivation of potential market prices based on multiples Equity valuation approach (Equity Approach): net method Overall capital valuation approach (Entity Approach): gross method DCF methods Methods of comparison Methods of market-oriented value determination Overall valuation methods Single valuation methods Figure 1.21: Methods of the market-oriented value determination 1.4 Matrix of Functional Business Valuation 51 45520_Matschke_Griffleiste_SL5.indd 51 45520_Matschke_Griffleiste_SL5.indd 51 16.03.2021 16: 20: 41 16.03.2021 16: 20: 41 1.5 Selected Control Questions Exercise 1 (30 points) - Valuation Object and Valuation Subject a) What are valuation objects in business valuation? How are they defined and how are they classified? (5 points) b) Prototypical valuation objects are the company as a whole and definable parts of it. Explain why this is not a contradiction in terms. (10 points) c) List the differences and common grounds of business valuation as a whole and the valuation of single shares. (10 points) d) You are instructed to determine the value of a business. Discuss the term valuation subject. (5 points) Exercise 2 (20 points) - The Concept of Value a) What do you understood by the economic term subjective value? What are the historical roots of subjective value theory? (5 points) b) Outline the content of different interpretations of the term economic value. (15 points) Exercise 3 (40 points) - Concepts of Business Valuation a) Write a brief but structured essay on: Critical comparison of the neoclassical (market-value-oriented) business valuation concept of the Anglo-Saxon school and the modern functional business valuation concept of the German school. (30 points) b) Briefly discuss both the standard CAPM and the M ODIGLIANI -M ILLER approach, and critically compare their assumptions. (10 points) Exercise 4 (9 points) - Main Functions List the main functions of business valuation according to functional theory. Exercise 5 (15 points) - Minor Functions a) How can main functions and minor functions be distinguished? (3 points) b) Briefly present four possible minor functions of business valuation. (12 points) Exercise 6 (15 points) - Systematization of Valuation Occasions a) Briefly describe why potentially different occasions of valuation should be systematized. (3 points) b) Structure the occasions of business valuation within the main functions according to four aspects and explain the different types of conflict situations arising by noting their general features and then provide an example of each type. (12 points) Kapitel 1: Einführung 52 52 1 Fundamentals of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 52 45520_Matschke_Griffleiste_SL5.indd 52 16.03.2021 16: 20: 41 16.03.2021 16: 20: 41 Chapter 2: Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 53 45520_Matschke_Griffleiste_SL5.indd 53 16.03.2021 16: 20: 41 16.03.2021 16: 20: 41 Overview The second chapter deals with the decision function, which is the first main function of functional business valuation (M ATSCHKE / B RÖSEL 2013a, p. 131). It determines the decision value of the business. This value represents the limit of the willingness to make concessions for a party in a certain conflict situation. Additionally, it forms the basis for the derivation of the arbitration and argumentation values from further main functions. Therefore, the decision function is described as the basic function of the functional business valuation. The current Section 2.1 provides the basics of the decision function. The overall focus is on the representation of the attributes of the decision value. The chapter will show how multi-dimensional decision values are determined (Section 2.2). A central aspect of this part is a general model for the determination of decision values according to M ATSCHKE . The following Section 2.3 deals with the calculation methods of one-dimensional decision values in a non-dominated, disjoint conflict situation of the type acquisition/ sale. In Section 2.4 special aspects relating to the determination of decision values are discussed. Finally, selected control questions will be posed in Section 2.5 in order to deepen the acquired knowledge and to apply the new skill set. Learning objectives After studying this chapter, you should be able to 1. define the term decision value and to list its attributes; 2. outline the significance of utility values, targets (aims/ objectives), and decision fields for the determination value; 3. present the general model of the decision value according to M ATSCHKE ; 4. apply and critically review the models of investment theory for the determination of decision values. Kapitel 1: Einführung 54 54 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 54 45520_Matschke_Griffleiste_SL5.indd 54 16.03.2021 16: 20: 41 16.03.2021 16: 20: 41 Chapter 2 2.1 Basics The decision value of a company is the result of the business valuation in the context of the decision function (see primarily M ATSCHKE 1972 and 1975, H ERING 1999 and 2014). The term does not refer to the valuation methods, but to the purpose of the business valuation calculation. Within the decision function, individual foundations for rational decisions are provided to (or are identified by) a certain decision subject - the person interested in the valuation - in a certain decision and conflict situation, which concentrate precisely on the specific subject, situation, and intention at the decision point. Generally speaking, a decision value relating to a given target system and decision field indicates to the decision subject the exact conditions under which the execution of a certain action does not yet reduce the level of target achievement (utility value and success), that could be obtained without this action. Hence, the decision value is based upon the principle of investment theory, emphasizing the target and decision field (M ATSCHKE 1993b, p. 182, H ERING 2014, p. 23, C OENENBERG 1992, p. 107). The subject of the process of negotiation and agreement between the parties cannot focus on utility values, but only on relevant facts of conflict resolution, which also change the achievable utility values of the parties through causing a change of decision fields. Assuming a rational way of acting, the decision subject in a non-dominated conflict situation confirms an agreement only if the achievable degree of fulfilling the goals (utility value) after the agreement is not lower than without it. To choose between various conflict resolutions, the decision subject has to develop an idea in which way different aspects and relevant facts for a conflict resolution after the agreement could change the degree of accomplishing the goals. In particular, for a rational negotiation, each negotiating party must precisely clarify these aspects and facts relevant to achieve a conflict resolution, which leads to the same degree of fulfilling the goals of an agreement that could be achieved without it (in the case of non-agreement). The decision value determines which conflict-resolution-relevant facts may still be acceptable for a decision subject. It is quite possible that there are many combinations of facts relevant to conflict resolution; in which case, the quantity of such critical combinations reflects the decision value. The decision value states the marginal agreement conditions of the considered decision subject in the underlying decision situation, that is, it includes the (absolute) limit of concession willingness. As the concession limit, the decision value is highly sensitive information that should remain confidential so as not to weaken a party’s negotiation position. Finally, the decision value is determined by four features (M ATSCHKE 1972, p. 147 and 1975, p. 26): 1. It is a critical parameter (characterized by a limit value or a concession limit). 2. It is determined for certain provided actions (characterized by relations with regard to specific actions). 3. It is related to a certain decision subject and its target system (characterized by both a subject relation and a target relation). 4. It only holds true for a concrete decision field (existing during the decision period) of the decision subject and the implicit alternatives - according to the considered action (characterized by a decision field relation). 2.1 Basics 55 45520_Matschke_Griffleiste_SL5.indd 55 45520_Matschke_Griffleiste_SL5.indd 55 16.03.2021 16: 20: 42 16.03.2021 16: 20: 42 If there is an agreement on the decision value of a party, that party does not improve its outcome compared to situation of a “non-agreement.” The decision subject is thus indifferent regarding the conflict situations “agreement at marginal conditions” and “non-agreement.” This indifference results because the utility value (success) as the expression of the achievable target satisfaction is the same in the case of an “agreement at marginal conditions” and in the case of “non-agreement.” In conflict situations of the acquisition/ sale of a company type, the possible price of a company plays a special and (usually also) a dominating role so that the determination of the decision value often concentrates exclusively on the determination of an agreed price limit that is rationally acceptable. The only dispute in this negotiation situation, which is addressed almost exclusively in literature, is about the price. Because of this strong model simplification of the actual conflict situation, the decision value becomes a critical price for the respective negotiation party: the upper price limit (marginal price) in the view of the presumptive buyer and the lowest price limit (marginal price) in the view of the presumptive seller. In other words: In the view of the presumptive buyer, the decision value (its upper price limit) is exactly the price it can accept without an economic detriment after the acquisition (M ATSCHKE 1969, p. 59). In the view of the presumptive seller, it is the lowest price limit that it must achieve without suffering an economic detriment from the sale. Each party knows only its own marginal price, possibly not as an exact price point but as an interval size. Finally, it must be considered that owing to the prevailing uncertainty existing in reality, it is not possible to determine ex-ante a unique value about future facts, which are relevant to a decision. While speaking of the determination of a value for decision support, the term decision support can be interpreted as a range of possible values - even if the level of the price is considered as a conflict-resolution-relevant fact (H ERING 2014, p. 9) - but that interpretation is still neglected in the figure below (see Figure 2.1). If the upper price limit P max of the presumptive buyer exceeds the lowest price limit P min of the presumptive seller, in other words, P max > P min applies, an area of agreement exists with regard to the amount of price P. A transaction that is advantageous for both parties, that is, an acquisition/ sale, is possible if the parties recognize that situation and agree a price that satisfies the condition P max ≥ P ≥ P min . Ideally, that price does not coincide with one of the price limits. It is more like a medium (although not necessarily an average) price instead (see Figure 2.1). Kapitel 1: Einführung 56 56 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 56 45520_Matschke_Griffleiste_SL5.indd 56 16.03.2021 16: 20: 42 16.03.2021 16: 20: 42 Chapter 2 In this case, the buyer (K; German: Käufer) economizes the advantage of A K = P max - P compared to the decision value. To the extent of this differential, they can make further investments, that lead to an additional success (e.g., to consume something), not achievable in a situation without an agreement. Such an agreement is also advantageous for the seller (V; German: Verkäufer) since that party does not get less than it demands: the advantage is determined by A V = P - P min . The seller is also able to afford more than in a situation without the agreement on the price P. Hence, both parties benefit from the agreement on P. In this simple conflict situation, the entire distributable advantage can be measured by the difference (mostly unknown to the parties) between the marginal prices: A = P max - P min . This is the whole achievable welfare gain due to the agreement. But the advantage of the one can only increase at the expense of the other. It is referred to one party if the negotiated price corresponds to the decision value of one of the parties: it is determined either by P = P max or by P = P min . The area of agreement in the conflict situation shrinks to a single point, where the decision values of the parties coincide, this is determined by P max = P min . Even in such a situation, an agreement on P max = P = P min is still possible, but no party could improve. In such a negotiation situation in which only the price might be disputed, a bazaarlike confrontation of the parties arises (M ATSCHKE 2003, p. 12). The multi-dimensional disjoint (or disjunct) conflict situation of the type acquisition/ sale shown in Figure 2.2 should describe the reality far better than the just addressed one-dimensional conflict situation. Pmin Pmax Einigungsbereich gesamter verteilbarer Vorteil V = Pmax - Pmin P Vorteil des Verkäufers VV = P - Pmin Vorteil des Käufers VK= Pmax - P Figure 2.1: Presentation of an agreement situation in a conflict situation of the acquisition/ sale type with price as the only conflict-resolution-relevant fact Advantage of the seller A V = P - P min Advantage of the buyer A K = P max - P Area of an agreement Total distributable advantage A = P max - P min 2.1 Basics 57 45520_Matschke_Griffleiste_SL5.indd 57 45520_Matschke_Griffleiste_SL5.indd 57 16.03.2021 16: 20: 42 16.03.2021 16: 20: 42 To present this situation graphically, all non-price conflict-resolution-relevant facts regarding different combinations were nominally combined on the abscissa. The price limits of the conflicting parties are interpreted as conditional variables. Depending on the non-price components, the buyer might offer more or less and the seller would have to demand more or less. There are two potential agreement areas in the example, namelythe combinations (K; German: Kombinationen) K 3 , K 4 and K 5 , and also the combinations K 7 and K 8 of the non-price facts because in these cases the upper price limit of the buyer is higher than the lowest price limit of the seller. Such a multi-dimensional situation requires creativity from both sides to discover the potential areas of agreement. It is not certain whether this will be possible at all since only in theory is such a clear picture of a negotiation situation provided. The parties involved in the negotiation process do not notice that. Viewers of Figure 2.2 have a clear advantage. They can even recognize if they interpret the dotted line as the expected price of agreement, that this price is certainly compatible with K 3 , K 4 , K 5 , and K 7 , but not with K 8 , and that the whole achievable advantage is divided very unequally as the difference of the marginal prices: at K 3 only in favor of the buyer, at K 5 only in favor of the seller and at K 7 the seller receives as much as at K 5 , whereas the buyer receives more than at K 5 . It would be desirable for the parties to find and to agree on the combination K 4 of the non-price conflict-resolution-relevant facts in the situation because this solution is pareto optimal compared to others. It is uncertain if this could be achieved. Finally, it should be noted that the decision value is not only the result according to the decision function, but also represents the main value (base value) of the functional business valuation theory, because it serves as the basis of the mediation (or arbitration) and the argumentation function (see Chapters 3 and 4 for details). Figure 2.2: Multi-dimensional conflict situation of the acquisition/ sale type Conditional price limits of the parties Combinations of non-price facts Lowest price limit of the seller Upper price limit of the buyer Potential agreement area Potential agreement area Kapitel 1: Einführung 58 58 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 58 45520_Matschke_Griffleiste_SL5.indd 58 16.03.2021 16: 20: 43 16.03.2021 16: 20: 43 Chapter 2 After this representation of the decision value, the focus is on the problem of its determination, regarding not the most simple, but the most complex situation, namely the determination of a multi-dimensional decision value. The methodology developed in this context can generally be deployed in all other cases (M ATSCHKE 1975, p. 387). However, there are certainly methods for less complex situations that are far easier to handle. Therefore, they should be preferred in such circumstances. 2.1 Basics 59 45520_Matschke_Griffleiste_SL5.indd 59 45520_Matschke_Griffleiste_SL5.indd 59 16.03.2021 16: 20: 43 16.03.2021 16: 20: 43 2.2 Determination of Multi-Dimensional Decision Values 2.2.1 Utility Values as Basics for Decision Value Determination 2.2.1.1 The Term “Utility Value” The decision value of a business should specify which agreements according with the conflict-resolution-relevant facts one party could just about agree to. The party can only agree to an agreement if no lower target level is to be expected after an acquisition/ sale or a merger/ demerger of the business than if no agreement is reached on the acquisition/ sale or a merger/ demerger of the business. The determination of the decision value requires that a party (valuation subject or decision subject) can decide what degree of target performance could be achieved without acquisition/ sale or merger/ demerger of the business, and which degree of target performance could be expected after an acquisition/ sale or a merger/ demerger of the business, depending on the various extensions of the conflict-resolution-relevant variables. Consequently, the party must be capable of classifying the action program without acquisition/ sale or merger/ demerger of the business and also the action program including acquisition/ sale or merger/ demerger of the business. The latter depends on the extensions of the conflict-resolution-relevant variables in categories like success, utility, degree of target performance, or utility value (S IE- BEN / L ÖCHERBACH / M ATSCHKE 1974, p. 841). Such a categorization is also defined as the determination of success. It is always subjective, but can be understood by a third party under certain circumstances and therefore it is generally intersubjectively verifiable. 2.2.1.2 Target Plan and Decision Field as Parameters of the Utility Value Such an intersubjective verification is possible if a third party (e.g., a valuation expert) possesses the following information: • the premeditation of the decision subject (information about the target plan) and • the possibilities of the decision subject (information about the decision field). The target plan is an idea of the decision subject. On the one hand, it contains information about the facts and characteristics (target or result definitions) the decision subject is interested in, on the other hand, it represents information about the intensity of desire (preferences). The definition specifies which facts the decision subject is interested in, whether that be in striving for or avoiding them. For the purpose of simplification, often only one single interesting fact is taken as a basis (homogenous definition, simple objective). In the context of business valuation, this is often a financial excess, which is still defined in a variety of ways, as is outlined below. However, it is more realistic that several facts are simultaneously relevant to a decision. This is called a heterogeneous definition, and in such a situation, the problem solution is far more complex and difficult to handle. In particular, this applies when the acquisition is strategically motivated and when not directly measurable monetary facts are replaced by estimations Kapitel 1: Einführung 60 60 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 60 45520_Matschke_Griffleiste_SL5.indd 60 16.03.2021 16: 20: 43 16.03.2021 16: 20: 43 Chapter 2 of a general nature about the contribution of the valuated company for the realization of a business strategy (S IEBEN 1969b, p. 71, H AFNER 1989). To ensure an intersubjectively verifiable determination of success, the definition of the result must represent an identification of the interesting facts from the body of all facts that characterize alternative options of the decision subject. The definition must set out the interesting facts, a requirement that encompasses the necessity for the definition of the result specifying what can be understood as an interesting fact (content description), how the interesting facts might be measured (measurement regulations) when they can be measured (temporal validity period), and where they could be measured (objective and geographical validity area). The decision field reflects different options (alternatives) of the decision subject and several circumstances (states), but it also represents options depending on the environment of specific facts (preference-relevant consequences of the valuation, results, result constellations, results matrix). If A is the set of each alternative a i with i ∈ {1, ..., m}, Z the set of all states z j with j ∈ {1, ..., n), K the set of all possible consequences relevant to a preference or result constellation e ij , the result function f: A × Z → K will assign a result constellation e ij ∈ K for each combination (a i , z j ): e ij = f(a i , z j ). Each determination of the utility value refers to a certain option, which can be characterized by the classified, predicted interesting facts of a specific type, specific amount, specific time, and specific uncertainty. Accordingly, the consequences for a valuation can be further described through four different characteristics: 1. type (or form), 2. amount (or height), 3. time, and 4. uncertainty. These chiefly subjective characteristics are called preferences. Preferences express subjective likes with regard to these characteristics and make it possible to combine the identified results with the indicator of preference (utility value) while choosing an alternative that can differ in height, time, form, and the degree of uncertainty in the general case of a heterogeneous result definition.Determining a homogenous result definition means it is sufficient to name three preferences: preference for amount, time, and certainty. In the case of a heterogeneous result definition, the preference of type (form) should be itemized too. These preferences can be preferences of a lower order if they are formulated on a separate basis, or preferences of a higher-order if more characteristics are subject to a combined analysis. Again, the preference of type is only necessary in a heterogeneous target definition. It expresses a relative advantageousness and the results are given only due to different variety characteristics for the decision-maker (S IEBEN / S CHILDBACH 1994, p. 26). Preferences of type can be formulated in a cardinal form as linear and non-linear target weightings or in ordinal form as a lexicographical system. At the same time, scoring models or methods transforming credit targets into monetary terms (e.g., pricing-outmethods or analysis of willingness to pay) ultimately serve to represent the preference of type. With the preference of amount, the decision subject determines the order of different extensions of an interesting fact. The preference of amount indicates the way the decision-maker judges the results only due to their characteristics of amount according to their advantageousness - ceteris paribus how only variations of this preference change 2.2 Determination of Multi-Dimensional Decision Values 61 45520_Matschke_Griffleiste_SL5.indd 61 45520_Matschke_Griffleiste_SL5.indd 61 16.03.2021 16: 20: 43 16.03.2021 16: 20: 43 the degree of desirability for the decision-maker (S IEBEN / S CHILDBACH 1994, p. 25). Those preferences can be concretized by extreme requirements (targets of maximization or minimization), as requirements that outline the direction toward a “better” judgment (targets of amelioration or improvement), or as requirements, which determine a desirable level in the sense of exceeding or falling short of an ambition level (targets of satisfaction), or an exact achievable level (point targets) (Z ELEWSKI 2008, p. 13). In situations of uncertainty (e.g., risk or a game situation) different alternative result constellations are to be expected, depending on external environmental conditions (e.g., states or actions of the opponent). In such situations a preference of certainty is essential. The preference of certainty reflects the subjective attitude of the decision-maker according to a certain fact, which is a consequence of the choice involved in an action and outlines a quantity of different possible results (result combinations). The preference of certainty also describes a relative advantageousness, which is selected only due to the uncertainty of its occurrence for the decision-maker out of the comparable quantities of possible results (S IEBEN / S CHILDBACH 1994, p. 26). Hence, the preference of certainty can be concretized as the expectation-value principle (risk neutrality or the B AYES rule) or as the (µ, σ) principle for the determination of risk-seeking and risk-aversion if the probabilities of occurrence are known. Otherwise, the minimax principle (the W ALD rule) could determine the level of risk-aversion and the maximax principle that of risk appetite. Further rules available include the H URWICZ rule, which focuses on the optimismpessimism-index of a preference; the rule of the slightest sign of regret (the S AVAGE - N IEHANS rule); and the rule of insufficient reason (the L APLACE rule). A temporal preference is necessary if the results are expected at different operating times. The temporal preference outlines the relative advantageousness of the results occurring at different times only according to their occurring time for the decision-maker (S IEBEN / S CHILDBACH 1994, p. 27). That temporal preference can be expressed by determining a desirable temporal structure of the desired facts, as an increasing, falling, constant, or other determined income. But the temporal preference can also be determined by constant, increasing, falling, or unstable time-dependent weighting factors. The underestimation of future payments is often determined by a multiplication of the expected future payments with the help of geometric falling interest-based weighting factors (discount factors) in the economic sector. This underestimation is already expressed in the relationship of equivalence, from which the one-period discount-factor [1/ (1 + i)] can be derived. This factor determines that (1 + i) risk-free monetary units at the end of the period (at the time t = 1) are estimated as high as a single risk-free monetary unit at the beginning of this period (at the time t = 0). If this subjective one-period weighting remains steady (constant) over time, the following relations of equivalence can be concluded: N 1 ⎡⎣ ⎤⎦ 0 ( ) = N 1 + i ⎡⎣ ⎤⎦ 1 ( ) or 1 0 ~ 1 + i ⎡⎣ ⎤⎦ 1 Kapitel 1: Einführung 62 62 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 62 45520_Matschke_Griffleiste_SL5.indd 62 16.03.2021 16: 20: 44 16.03.2021 16: 20: 44 Chapter 2 Figure 2.3 presents and schematizes the parameters of the utility value and all-important determination steps. In terms of the elements of the matrix, it is important to note that these can be scalars, vectors, or matrices. If the decision subject is only interested in one single result type and if the temporal validity area of the homogenous definition is limited to one single point in time, the result constellation e ij represents the result height H ij (r) for every alternative a i and every states z j , while r ∈{1, …, R} illustrates the mentioned amount class, that is, the elements of the matrix are scalar quantities. At a homogenous result definition with a temporal validity area, which includes several points in time t ∈ {l, …, T}, the result constellations e ij are (row) vectors. Their elements H ijt (r) determine that alternative a i and state z j at the point in time t correspond to an interesting fact of a specific level that can be allotted to the amount class r. For each alternative a i , each state z j , and each point in time t, r can be different. (Column) vectors are result constellations e ij , if a heterogeneous result definition has a temporal validity area of only one single point in time. Consequently, the elements H ijv (r) are of the expected height for each interesting result type E v with v ∈ {l, …, V}, for each alternative a i and each state z j . Hence, the amount class r can differ for each alternative, state, and result type. At a heterogeneous result definition with a temporal validity area, which encompasses several points in time t ∈{l, …, T}, the result constellations are e ij matrices. Their elements H ijtv (r) determine the expected result height of the different interesting result types E v at the points in time t for alternative a i and state z j , while the amount class r can differ for each alternative, state, point in time, and result type. 1 0 ~ 1+ i ⎡⎣ ⎤⎦ 1 1 0 ~ 1+i ( ) 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 2 1 0 ~ 1+i ( ) 3 ⎡⎣⎢ ⎤⎦⎥ 3 … 1 0 ~ 1+i ( ) t ⎡ ⎣⎢ ⎤ ⎦⎥ t . 2.2 Determination of Multi-Dimensional Decision Values 63 45520_Matschke_Griffleiste_SL5.indd 63 45520_Matschke_Griffleiste_SL5.indd 63 16.03.2021 16: 20: 44 16.03.2021 16: 20: 44 Zielplaninformationen Entscheidungsfeldinformationen Präferenzen Ergebnisdefinition Alternativen Umweltzustände Ergebnismatrix (Ergebniskonstellationen) z1 z2 z3 … zn a1 a2 … am e11 e12 e13 … e1n e21 e22 e23 … e2n … em1 em2 em3 … emn Entscheidungsmatrix (Teilnutzen) z1 z2 z3 … zn a1 a2 … am n11 n12 n13 … n1n n21 n22 n23 … n2n … nm1 nm2 nm3 … nmn Höhen-, Zeit-, Artenpräferenz N(ai) a1 a2 … am N(a1) N(a2) … N(am) Sicherheitspräferenz ai Entscheidungsvektor (Nutzwerte) Figure 2.3: Parameters and steps for the determination of the utility value for an alternative Target plan information Decisions field information Preferences Result definitions Alternatives Environmental conditions Decision matrix (Part worth utility) Decision vector (Utility values) z 1 z 2 z 3 … z n n 11 n 21 n 12 n 22 n 13 n 23 … … n 1n n 2n n m1 n m2 … n m3 … n mn z 1 z 2 z 3 … z n e 11 e 21 e 12 e 22 e 13 e 23 … … e 1n e 2n e m1 e m2 … e m3 … e mn Result matrix (Result constellations) a i N(a i ) a 1 a 2 N(a 1 ) N(a 2 ) … a m … N(a m ) a i a 1 a 2 … a m a i a 1 a 2 … a m Certainty preference Amount, time, and type preference Kapitel 1: Einführung 64 64 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 64 45520_Matschke_Griffleiste_SL5.indd 64 16.03.2021 16: 20: 45 16.03.2021 16: 20: 45 Chapter 2 2.2.2 A General Model for the Determination of a Multi-Dimensional Decision Value 2.2.2.1 Determination of Decision Values as a Two-Step Calculation At this point, it is important to introduce a general model for the determination of the decision value according to M ATSCHKE (1975, p. 387) from which all other decision value determination methods can be derived. This general model does not require determination of the targets and decision fields of the conflicting parties nor of the number and type of conflict-resolution-relevant facts. Its field of application is by no means limited to business valuation problems, and instead it applies to any number of decisiondependent and interpersonal conflict situations without compulsory character. The decision value is a concession limit, which is always essential in cases when the decision subject can only carry out an expected action if it communicates with one or several other decision subjects about the conditions of the action. For instance, if the decision subject agrees on a certain constellation of the conflict-resolution-relevant facts. The conflict parties involved use this agreement to enhance their situation (an increase in utility) compared to that available in a non-agreement scenario. In the case of rational behavior, the expected utility in the event of a non-agreement therefore becomes a standard of comparison for each possible constellation of the conflict-resolutionrelevant facts. Hence, the decision value is calculated in two steps: • The first step comprises the determination of the comparative measure in the sense of the achievable level of utility for the conflicting party without agreement. This step is called determination of the base program. • The second step comprises the establishment of the conflict-resolution-relevant facts that are to be declined, preferred, or judged indifferently from the viewpoint of a conflicting party because, in the case of an agreement, a lower, higher or equal level of utility is achievable from the conflicting party’s perspective. An issue of special interest in a negotiation is the set of conflict-resolution-relevant facts that after an agreement result in the same level of utility as without an agreement or - in the case of discontinued relationships - the lowest possible higher level of utility compared to that available in a non-agreement scenario. During the negotiation, those facts form the limit of concession willingness, or the decision value. The second step is called determination of the valuation program as far as it results in the decision value. 2.2.2.2 Determination of the Base Program Without an agreement on acquisition, sale, or merger the decision subject can choose between action option . Every alternative matches a certain utility value N(a i ) to the decision subject with regard to the expected result constellations and the preferences. Assuming rational behavior, the decision subject will choose the alternative with the highest utility value. For the optimal alternative a opt , it is computed: , which represents the alternative with the hig- A = {a 1 , …, a i , …, a k } a i ∈ A N(a opt ) = max{N(a i ) | a i ∈ A } 2.2 Determination of Multi-Dimensional Decision Values 65 45520_Matschke_Griffleiste_SL5.indd 65 45520_Matschke_Griffleiste_SL5.indd 65 16.03.2021 16: 20: 45 16.03.2021 16: 20: 45 hest utility. The optimal alternative a opt with the utility value N(a opt ) is defined as the base program. The success (utility value) of the base program is at least to be achieved again at the point rational behavior of the decision subject after an agreement. It becomes the standard of comparison after each agreement. The comparison is done on the utility level. If the considered decision subject is a presumptive buyer, the valuated business is not a part of the base program (best action alternative without the acquisition of the company). If the valuation is carried out from the presumptive seller’s perspective, the valuated business is still a part of the base program (best action alternative without the sale of the business). The same procedure applies to the case of a conflict situation of the type merger, starting from not a merged businesses and according to the conflict situation of the type demerger from a not (yet) demerged businesses within the determination of the base program. 2.2.2.3 Determination of the Valuation Program Any agreement between the conflict parties is based on agreement on the original conflict-resolution-relevant facts S 1 , …, S n , which have the concrete attributes s 1 , …, s n . Each mutually exclusive combination of the extensions of the conflict-resolution-relevant facts represents a possible solution for the agreement in the considered conflict situation, which is specified by the conflict-resolution-relevant facts S 1 , …, S n . The set of all possible conflict results is: or The set S can be interpreted as the set of all possible contracts. After the agreement on a certain conflict resolution , that is, on a concrete contract content, the decision cubject can choose between the set of actions , to which again a certain success is allocated with regard to expected result constellations and preferences. Assuming rational behavior, the decision subject decides for the action from the set of all available action possibilities having the highest utility after an agreement on the conflict resolution The utility value of the optimal alternative - referring to the considered conflict resolution - is computed with the equation or a function f of the conflict resolution In this manner a certain utility value can be clearly allocated to a conflict resolution This utility value is equal to the utility value of the best alternative option which (s 1 , …, s n ) ∈ S 1 × … × S n S : = S 1 × … × S n S = {(s 1 , …, s n ) | (s 1 , …, s n ) ∈ S 1 , …, S n }. (s 1 , …, s n ) ∈ S (s 1 , …, s n ) ∈ S B (s 1 , …, s n ) = {b 1 , …, b j , …, b p } b j (s 1 , …, s n ) N(b j (s 1 , …, s n )) b j (s 1 , …, s n ) B (s 1 , …, s n ) (s 1 , …, s n ). N(b j (s 1 , …, s n )) b opt (s 1 , …, s n ) N(b opt (s 1 , …, s n )) = max{N(b j (s 1 , …, s n ))| b j (s 1 , … , s n ) ∈ B (s 1 , … , s n )} N(b opt (s 1 , …, s n )) : = f(s 1 , …, s n ), (s 1 , …, s n ). f(s 1 , …, s n ) (s 1 , …, s n ). b opt (s 1 , …, s n ), Kapitel 1: Einführung 66 66 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 66 45520_Matschke_Griffleiste_SL5.indd 66 16.03.2021 16: 20: 45 16.03.2021 16: 20: 45 Chapter 2 could be grasped by the decision subject after an agreement on the conflict resolution Whether a conflict resolution is acceptable from the decision subject’s perspective, that is, if the solution of the agreement is useful, depends on the scale (or level) of the utility value compared to the utility value of the base program: • If it is assumed that , the conflict resolution is not acceptable for the decision subject. • If, however, it is assumed that , the conflict resolution is acceptable for the decision subject. In view of the conflict resolution of the non-dominated alternative options which lead to the equality of the utility value with the utility value of the base program or in case of a discontinuous function where it is minimally higher than the success of the base program, the valuation program of the decision subject is defined by: The valuation program is at the same time a partial set of all outstanding alternative options for the decision subject depending on the conflict resolutions For the buyer, the business to be valuated is a part of the valuation program, whereas, for the seller, it is no longer part of the valuation program. In the case of a conflict situation of the type merger, the business after the merger is a part of the valuation program (the merger program). According to the conflict situation of the type demerger, the business in question is included in the valuation program in its demerged form (shift of ownership, change of ownership) or with at least one newly formed business (separation of ownership). 2.2.2.4 Multi-Dimensional Decision Value From the perspective of the decision subject, the decision value of the business to be be valuated is thus the set of all conflict resolutions for which the utility value equals or marginally exceeds the utility value of the base program: (s 1 , …, s n ). (s 1 , …, s n ) N(b opt (s 1 , …, s n )) N(a opt ) N(b opt (s 1 , …, s n )) < N(a opt ) (s 1 , …, s n ) N(b opt (s 1 , …, s n )) ≥ N(a opt ) (s 1 , …, s n ) (s 1 , …, s n ) b opt (s 1 , …, s n ), N(b opt (s 1 , …, s n )) N(a opt ) N(b opt (s 1 , …, s n )) = f(s 1 , …, s n ), B * B *: = {b opt (s 1 , …, s n ) | N(b opt (s 1 , …, s n )) = min{N(b opt (s' 1 , …, s' n ))| N(b opt (s' 1 , …, s' n )) ≥ N(a opt ) as well as (s' 1 , …, s' n ) ∈ S , b opt (s' 1 , …, s' n ) ∈ B (s' 1 , …, s' n ) and a opt ∈ A }}. B * B * ⊂ ∪ ( s 1 , .., s n ) ∈ S B (s 1 , … , s n ) S. W (s 1 , …, s n ) N(b opt (s 1 , …, s n )) N(a opt ) 2.2 Determination of Multi-Dimensional Decision Values 67 45520_Matschke_Griffleiste_SL5.indd 67 45520_Matschke_Griffleiste_SL5.indd 67 16.03.2021 16: 20: 45 16.03.2021 16: 20: 45 The decision value is a partial set of the (total) set of all conflict resolutions, namely Taking into consideration the decision value the valuation program can also be defined as: The determination of a multi-dimensional decision value as a partial set of all possible conflict resolutions requires a model of conflict situations in the sense of a description of conflict-resolution-relevant facts S 1 , …, S n and their concrete attributes and the knowledge of the function as well as the knowledge of the utility value of the base program. Due to the dependence on the target plan (result definition and preferences) and the decision field and as well as the generally possible changes over time, the decision value has generally to be regarded as time-dependent. Since the concept of the multi-dimensional decision value requires a model of conflict situation, the decision subject should formulate hypotheses about conflict-resolution-relevant facts and their possible attributes to define positive and negative negotiation objectives and to actively influence the negotiation process. Thos hypotheses permit ideas about acceptable conflict resolutions (that is, the decision value) to be developed; however those ideas are conditional statements and have to be corrected if in the negotiation the underlying hypotheses prove not to be robust. In other words, the decision value is only usable to the extent that the underlying model of the conflict situation reflects reality. To summarize, the old adage garbage in, garbage out applies. Even if the idea of a decision subject who knows precisely the decision value and the set of the acceptable conflict resolutions from their perspective at the beginning of a negotiation process was abstracted from the time dependency, that idea would certainly be not realistic. To summarize, negotiation processes are and will remain discovery processes, that is, processes of a common problem structuring and a creative problem solution by the negotiation partners. Neither should it be assumed that it is possible to know ex-ante which concrete assumptions are required in the negotiations according to the conflict-resolution-relevant facts. It cannot therefore be assumed that the set of all possible conflict resolutions can be known in advance. Accordingly, the knowledge of the decision value as a set of the just about acceptable conflict resolutions in the negotiation process is changing, because the decision subject has to choose if new solutions are acceptable, just about acceptable, or unacceptable. The determination of the decision va- W : ={(s 1 , …, s n )|N(b opt (s 1 , …, s n )) = min{N(b opt (s' 1 , …, s' n ))| N(b opt (s' 1 , …, s' n )) ≥ N(a opt ) as well as (s' 1 , …, s' n ) ∈ S , b opt (s' 1 , …, s' n ) ∈ B (s' 1 , …, s' n ) and a opt ∈ A }}. W S W ⊂ S. W , B * : ={b opt (s 1 , …, s n )|(s 1 , …, s n ) ∈ W }. W S s 1 , …, s n N(b opt (s 1 , …, s n )) = f(s 1 , …, s n ) N(a opt ) ( A B (s 1 , …, s n )) W Kapitel 1: Einführung 68 68 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 68 45520_Matschke_Griffleiste_SL5.indd 68 16.03.2021 16: 20: 46 16.03.2021 16: 20: 46 Chapter 2 lue and the set of acceptable conflict resolutions are thus realistically not to be understood as comprising a completed process before the actual beginning of the negotiation. 2.2.2.5 Set of the Acceptable Conflict Resolutions The set of the acceptable conflict resolutions might be more important to a rational negotiation process than the decision value. This is mainly because that quality determines not only the limit of concession willingness, but also indicates to the decision subject which conflict resolutions should be pursued. From the perspective of the decision subject, the acceptable conflict resolutions are defined as the set while it can be assumed that the decision value is a partial set of the set of the acceptable conflict situations, that is, From this definition, it can be concluded that the utility value of an acceptable conflict resolution is at least equal to the utility value of the base program The party can agree with an acceptable conflict resolution, because such an agreement does not contradict rational behavior. However, this does not mean that the party is indifferent to all acceptable conflict resolutions. An indifference applies only in relation to such acceptable conflict resolutions that belong to the decision value and only if their utility value corresponds with the utility value of the base program. Conflict resolutions, which in the case of an agreement lead to a higher utility value than that of the base program, are preferred. During the negotiation, the party will make an effort to secure an agreement on a conflict resolution that is significantly beneficial. Nonetheless, that conflict resolution must at the same time remain reasonable for the other party. If it is, the resolution can be added to the set of all possible conflict resolutions. 2.2.2.6 Set of Agreement Resolutions From the perspective of q participating decision subjects, the possible agreement set can be determined as the intersection of the sets, which contain the acceptable conflict solutions from the perspective of each single conflict party: An agreement between the conflict parties in a non-dominated conflict situation, assuming rational behavior of all conflict parties, can only be expected when the agreement set is not an empty set, that is, it is assumed that It is common in negotiations that not all negotiation parameters remain unknown until the end of the process. Frequently, the negotiation partners work out partial solutions, which still depend on an overall agreement, but essentially can be considered as (variable) milestones in the ongoing negotiation. Such a step-by-step negotiation, which results in causes a reduction of the set of acceptable conflict resolutions and hence of the agreement area. Accordingly, the initial complexity of the negotiation problem is considerably reduced. S z : ={(s 1 , …, s n ) | f(s 1 , …, s n ) ≥ f(s 1 ' , … , s n ' ) as well as (s' 1 , …, s' n ) ∈ W and (s 1 , … , s n ) ∈ S }, W S z W ⊂ S z . f(s 1 , …, s n ) f(s 1 , …, s n ) ≥ N(a opt ). E E : = S z1 ∩ S z 2 ∩ … ∩ S zq . E ≠ ∅. 2.2 Determination of Multi-Dimensional Decision Values 69 45520_Matschke_Griffleiste_SL5.indd 69 45520_Matschke_Griffleiste_SL5.indd 69 16.03.2021 16: 20: 46 16.03.2021 16: 20: 46 2.3 Determination of One-Dimensional Decision Values in Non-Dominated, Disjoint Conflict Situations of the Acquisition/ Sale Type 2.3.1 Examination steps on the Matrix of Functional Business Valuation 2.3.1.1 Overview With regard to the matrix of functional business valuation, the following valuation steps arise within the decision-making function (valuation in the broader sense: B RÖSEL / D ECHANT 2003, p. 135): Step 1 (field A of the matrix): Delimitation and quantification of the relevant future performance. Step 2 (field B of the matrix): Transformation of determined future performances of the decision value of the business (valuation in the strict sense). Step 3 (field C of the matrix): Use a) to weigh (subjective) decision value and (objective) price according to the decision function, b) as base value for determination of arbitration values, according to the mediation/ arbitration function and/ or c) as base value for determination of argumentation values, according to the argumentation function. This matrix, which is shown in Figure 2.4, enables a transparent presentation of the valuation process and supports the explanation of the individual valuation steps depending on the task followed by business valuation. It should be noted that the steps cannot be differentiated so clearly in reality and that the steps are not always taken in one direction. Those factors are illustrated in the following example of decision value determination. Kapitel 1: Einführung 70 70 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 70 45520_Matschke_Griffleiste_SL5.indd 70 16.03.2021 16: 20: 47 16.03.2021 16: 20: 47 Chapter 2 2.3.1.2 The Steps in Detail 2.3.1.2.1 First Step For the valuation subject, all future performances evoked by the valuation object are of key importance in the valuation. The valuation object is a future utility for the valuation subject and therefore contributes to its target fulfillment. The determination of these performances, relevant to the valuation subject from the valuation object, usually tends not to be focused on in valuation theory. Delimitation and quantification of the utility, caused by the businesses to be valuated (step 1 and field A of the matrix) are instead incumbent upon the specialists of the respective sectors. The generation of future performances, the mutual dependencies to be taken into account, the determination of appreciation potentials, and the estimation of the changes of future performances during the period to be examined are commonly neglected in the literature. However, in the following step 2, the quality of the decision value of the business, based on theoretical investment methods in certain models, is determined by both the quality of information and the delimited and quantified future performances. Those need to be provided with regard to the valuation in the strict sense. It is noteworthy that the difficulty of the actual decision value determination - especially for startup companies - is the problem of estimating future performances (step 1) and not - as is often mistakenly believed - the matter of valuation methodology (step 2) (H ERING / O LBRICH 2002, p. 156). The decision value is a relative value, which arises from a subject-object-object relation (M ATSCHKE 1972, p. 147, S IEBEN 1988, p. 87). In consideration of the target and preference system, the presumptive buyer as the (decision/ valuation) subject expects a Hauptfunktionen Nebenfunktionen Entscheidungsfunktion Vermittlungsfunktion Argumentationsfunktion Nebenfunktion 1 Jeweils weitere Nebenfunktion Ermittlung der relevanten Daten Transformation der relevanten Daten in einen Wert Verwendung des ermittelten Wertes A B C D G J M E F H K N I L O Bewertung i. e. S. "# $ % $ Bewertung i. w. S. " # $$$$ % $$$$ Valuation in the broader sense Valuation in the strict sense Main functions Minor functions Decision function Mediation function Argumentation function Minor function 1 Respective further minor function Determination of relevant data Transformation of relevant data in a value Use of the determined value Figure 2.4: Matrix of functional business valuation 2.3 Determination of One-Dimensional Decision Values 71 45520_Matschke_Griffleiste_SL5.indd 71 45520_Matschke_Griffleiste_SL5.indd 71 16.03.2021 16: 20: 47 16.03.2021 16: 20: 47 certain utility from the (decision/ valuation) object. The decision value here results from the comparison of the valuation object with the alternatively available objects, according to the decision area, with regard to the level of target fulfillment. Which future performances the valuation subject is ultimately interested in (and definitely not interested in) therefore depends on its target plan. Which future performances the valuation subject may realize, is conversely primarily determined by its decision area. Step 1 therefore comprises an analysis of the targets and the decision area of the valuation subject. The target plans of the valuation subjects depict their systems of values that are relevant to the business valuation. Within the framework of the business valuation theory, it is mainly assumed (H ERING 2014, p. 25) that the interest of the valuation subjects is primarily focused on financial advantages or financial benefits, that is, valuation subjects strive for an influx, which may occur and be measured in the form of payments to the owner(s) (withdrawal or payouts) as well as payment savings by the owner(s). Therefore, the assessment is based on cash flows, not on accounting measures like income or sales. In the case of profit retention, such performance measures do not directly contribute to the satisfaction of needs, because retained profits are not available for consumption purposes in the period in question. The unadjusted consideration of performance metrics leads to double-counting because retained profits and the future excess profits thereof are recognized. With the correct fixing of imputed interest on capital commitment caused by retention, the assessment can be alternatively oriented to performance flows (L ÜCKE 1955). Since the application of profitability measures extensive auxiliary calculations, usually cash flows are expediently chosen as operands (H ERING 2017, p. 248). With the “advantage flow” (expected future cash flow) resulting from the business, the opportunity to satisfy the consumption needs is given to the owner(s). To operationalize the commonly assumed simple financial objective with asset and income maximization, two different cashflow-oriented variants of wealth maximization can be distinguished on the imperfect capital market (H ERING 2017, p. 20): • In a asset maximization approach, the objective under the restriction of a fixed withdrawal rate is to maximize a payout, according to the consumption preference. The amount of weighted payouts corresponds to the target function. The weighting factor to be specified for each point in time thus mirrors the subjective appreciation of a payout in relation to other payout time points. Special cases in asset maximization are the future value method and the present value method. • In an income maximization approach and under the restriction of fixed payouts at specific times, the aim is to discover the investment and financing program that maximizes the range of a “withdrawal flow”. The relation of the withdrawal amounts to be determined is already prescribed. The crucial factors in the choice between these objectives are the individual preferences of the valuation subject. If a company (valuation object) is supposed to be acquired by another firm, the management of the acquiring business has to conform to the individual (withdrawal and consumption) preferences of the enterprise owners (valuation subject and measuring plane of the target fulfillment). While the demonstration of the essential consumption utility function of an individual is already difficult, the problems increase with a multitude of valuation subjects. The choice of the “correct” objective evades valuation theory because here value judgments have to be made on subjective Kapitel 1: Einführung 72 72 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 72 45520_Matschke_Griffleiste_SL5.indd 72 16.03.2021 16: 20: 47 16.03.2021 16: 20: 47 Chapter 2 preferences. If using appropriately formulated restrictions ensures that the withdrawals do not lead to a loss of corporate integrity, both alternatives can be suitable objectives. However, the valuation subjects are required to specify as accurately as possible the objectives relevant to them prior to the valuation process. Alongside the objectives of the valuation and decision subject, its individual decision field also determines the respective firm value. It is characterized by both corporate finance and real economic action alternatives and limitations (H ERING 2014, p. 27). The real economic sphere of activity arises inter alia from the current endowment with goods and personnel as well as the overall possibilities of acquiring or selling further goods and of hiring or dismissing employees. Corporate finance encompasses available funds, the posibilities of investment and borrowing plus credit rationing, among other aspects. It should also be noted that with an increased leverage ratio (debt ratio), creditors will generally demand higher debit interest. Due to the imperfection of the capital market, the corporate financial sphere of activity is furthermore characterized by the following valuation-relevant limitations (B URCHERT / H ERING / H OFFJAN 1998, p. 247): 1. Debit interest (borrowing rate) and credit interest differ from each other and capital is scarce. In the application of partial models, this results in a significant problem which is attributable to the dilemma of costs on a value basis or the theory of the marginal cost pricing: The investment theoretically-correct control or adequate target (interest) rates that are necessary for the valuation with partial models, such as the discounted cash flow method, will only be defined if the associated general model is resolved. 2. There is a requirement for permanent solvency (liquidity). 3. In addition, interdependencies, requirements for integers (whole numbers) and exclusion requirements for selection problems may influence corporate financial (but also the real economic) sphere of action. In reality, the decision field in the valuation is characterized by its openness. In such an open decision area (H ERING 2014, p. 30) at the valuation date, neither all the possible actions nor the limitations are known, nor can the payment consequences of the known possible actions be predicted accurately. The future success, and therefore the decision value, are influenced inter alia by the uncertain changes of the interest and salary levels, over time occurring yet unknown financing and investment opportunities as well as the uncertain payment consequences of the business to be valuated and other investment measures. The present uncertainty is characterized by the following features (H ERING 2017, p. 12): 1. The expectations (of success) are polyvalent. 2. Not all decision variables and constraints (side or second-order conditions) are known. 3. The planning period is open . At this point, it is clear why at the beginning of this chapter it was stated that an “optimal” answer for a valuation problem is not defined ex-ante in the case of uncertainty. The answer to the problem under uncertainty at best can take place heuristically whereby with suitable procedures it is to search for “satisfactory” or “good” but not for clearly “correct” optimal answers. It will be necessary to artificially and gradually narrow or close the open decision field by plausible assumptions. In addition, the determi- 2.3 Determination of One-Dimensional Decision Values 73 45520_Matschke_Griffleiste_SL5.indd 73 45520_Matschke_Griffleiste_SL5.indd 73 16.03.2021 16: 20: 47 16.03.2021 16: 20: 47 nation of the planning period should be pragmatic through a reasonable choice of planning horizon (R OLLBERG 1999, p. 106, H ERING 2017, p. 13). The decisive future performances of the businesses to be valuated, however, can only be approximately quantified because of the uncertainty in real life. For the determination of the decision value, it is therefore useful, if the performance expectations are narrowed to a certain range by undertaking a diligent analysis of the probability of certain success factors occurring. Conducting profound estimations of the distribution of these performance factors within this range makes an additional reduction of the polyvalence possible. However, a narrowing to a factual monovalence should not be sought. The basis of the gain of future performances for both presumptive buyers and presumptive sellers is the determination of appreciation potentials within the framework of a holistic business analysis. On the one hand, opportunities and threats and also strengths and weaknesses are identified; a SWOT analysis offers a means to find the “right” course of action. On the other hand, the tactical practicability of the planned strategies is investigated. In this context, it is about finding feasible positive and negative synergies for the business to be valuated as a whole, which arise in the interaction of the particular production factors of the business and the interaction with the elements of the remaining investment and financing programs of the valuation subject. Appreciation potentials can arise if negative synergies can be reduced, if different valuation subjects have not realized possible synergies, or especially if with regard to different decision fields, they cannot open up for themselves owing to lack of skills and means. Since the potentially necessary restructuring measures - that may be vital in organizational, legal, and financial or real economic terms - are usually long-term in nature, they must be embedded in the long-term planning of the valuation subject. That clearly illustrates the dependence of the decision values on both the future and on planning. Moreover, possible effects on mergers, which might arise from different corporate cultures, play a key role since a culture shock in terms of a collision of corporate values and corporate identity is possible. Hence, these potential impacts should be monitored closely (O LBRICH 1999). The characteristics of many sectors and markets, like being fast-paced, featuring highly intensive competition, and a rapid technological change, intensify the already present problems of determination of future performances. Backward looking mathematical-statistical forecasting methods prove to be useless. If useful results are supposed to be obtained by the valuation, the future performances resulting from the business are to be appropriately differentiated and quantified. But which future performances with regard to valuation have to be precisely considered? In essence, future performances for the valuation subject, according to total return principle (M OXTER 1983, p. 75) are the sum of all advantages that will flow from the business as a whole to the subject in the future: Those advantages include financial as well as non-financial elements. Originating from the individual target system of the valuation subject, the reality is that it would be necessary to identify all interesting facts and to determine their emphasis. Due to limited possibilities of quantification, the valuation of the non-financial advantages, such as an increase in prestige, proves particularly difficult. If in the sense of complexity reduction, it can be assumed within the target analysis that the main interest of the valuation subject is the cashflow-oriented wealth Kapitel 1: Einführung 74 74 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 74 45520_Matschke_Griffleiste_SL5.indd 74 16.03.2021 16: 20: 47 16.03.2021 16: 20: 47 Chapter 2 maximization, then only financial advantages (benefits) and the corresponding monetary consequences can (and must) be regularly taken into account for the investment-theoretically substantiated determination of the decision value. It should also be pointed out that this simplified assumption must not lead under any circumstances to neglecting non-financial advantages or even not considering them at all. Therefore, the decisionmaker should not only monitor the decision value that only comprises financial advantages, but also consider non-financial benefits when it comes to pricing because generally future performances consist of the entire advantage expectations. In this light, only payment variables function as operands for the determination of the decision value (H ERING 2014, p. 32). Deposits and withdrawals (payouts) are intersubjectively verifiable because they are subject neither to accounting policies nor to periodization considerations. The limitation to cash inflows and outflows as the relevant financial values prevents the risk of a double count, also with regard to the described L ÜCKE theorem. As payment variables, payment surpluses, and payment savings come into question. Accounting-oriented performance metrics can only have an impact on the valuation if they affect the level of actual payments, such as profit-related tax payments. The business in question is being interpreted as an uncertain future stream of cash flows in the context of valuation. The relevant cash flows from the firm at hand might also arise from constant or discontinuous deposits and withdrawals, which are related to possible buying or selling decisions. Illustrated with the example of payments for personnel costs, it is not only about payments for the staff of the business to be valuated, but also about those payments arising from acquisition within the business, petrhaps already owned by the valuation subject (e.g., in the event of the acquisition of a foreign business for employees to be hired in Germany, who are entrusted with international personnel management). 2.3.1.2.2 Second Step The information collected in this way on dispersions, ranges, and interdependencies of the future performances in the sense of payment surpluses (positive cash flows) set the starting point for the decision value determination in the strict sense. At this point this information must be transformed into a value capable of being used as a decisionmaking tool. This transformation of qualitative and quantitative information on future performances, determined from solid estimations, into a value that fulfills its underlying function - here the decision function - is the main task of the valuation (step 2 and field B of the matrix). The only methods suitable are those aimed at valuating the advantageousness of cash flows under real (and therefore imperfect) conditions and that also take account of both the targets and the decision field of the decision subject. Hence, only the recourse on the investment-theoretic methods is justified, such as the state marginal price model, the future performance value method, or the approximate decomposed valuation, because only within those methods will the target system and the decision field of the valuation subject be sufficiently factored in. However, it is important to find the right balance between theoretical rigor and practical applicability (H ERING 2014, p. 5). 2.3 Determination of One-Dimensional Decision Values 75 45520_Matschke_Griffleiste_SL5.indd 75 45520_Matschke_Griffleiste_SL5.indd 75 16.03.2021 16: 20: 48 16.03.2021 16: 20: 48 The performance estimations to be considered within the context of this transformation are characterized under uncertainty by polyvalent expectations. The given performance expectations are the basis of the valuation. They are (by applying subjective complexity reduction, for example, the discontinuous investigation of almost continuously arising cash flows) narrowed down to subjective ranges and assigned to probabilities of occurrence, ideally determined by substantiated estimations. Methods regarding the polyvalence of future expectations of the valuation subject, which eventually try to solve the present defect target setting valuation issue (A DAM 1996, p. 19) in a heuristic approach, can be divided into uncertainty solidifying or uncertainty revealing (or disclosing) methods (H ERING 2017, p. 269) (cf. also Figure 2.5). In the planning processes solidifying uncertainty, the aforementioned is compressed either on the level of input data or of the target value. The consideration of the uncertainty issue on the level of input data can occur by the use of risk premia for cash flows and/ or interest rates. Alternatively, parameters considered uncertain are narrowed down to actual monovalence. This “correction” of performance measures or interest rates subsequently allows a valuation with deterministic models. The adjustment of performance metrics occurs, for instance, with the so-called certainty equivalent method. Here, the aggregated data at hand are transformed into the socalled certainty equivalents, either as a point value, in ranges, or as subjective probability distributions. Based on the B ERNOULLI principle (1738) and with the required knowledge of the risk preferences of the valuation subject, the uncertain future performance flows are transformed into a certain payment flow, which is considered equivalent by the valuation subject. In other words, the certainty equivalent describes a safe performance, which is worth just as much to the valuation subject as the estimated uncertain performance range. If the identification of the decision value should not become merely an intuitive process, the risk-utility function of the valuation subject or, in the case of Planungsverfahren unter Unsicherheit Unsicherheit verdichtende Planungsverfahren Komprimierung auf Ebene der Eingangsdaten Anpassung der Erfolgsgrößen Anpassung der Zinssätze Komprimierung auf Ebene des Zielwertes Unsicherheit aufdeckende Planungsverfahren Planning processes solidifying uncertainty Adjustment of performance variables Compression on the level of input data Planning processes under uncertainty Planning processes revealing uncertainty Compression on the level of the target value Adjustment of interest rates Figure 2.5: Systematization of planning processes under uncertainty Kapitel 1: Einführung 76 76 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 76 45520_Matschke_Griffleiste_SL5.indd 76 16.03.2021 16: 20: 48 16.03.2021 16: 20: 48 Chapter 2 several subjects, of all valuation subjects has to be considered. This, of course, implies considerable practical difficulties. In fact, you will not be able to recognize them in practice (B ALLWIESER 1981, p. 101). The adjustment of interest rates takes place in the subjective risk premium method and the Capital Asset Pricing Model (CAPM). Here, for the determination of the decision value, the expected values of the cash flows, and more or less randomly risk-adjusted interest rates are used. The level of the selected premium or discount for considering the risk is not rationally justifiable (H ERING 2017, p. 292). The disadvantages of both methods, where the uncertainty is compressed on the level of input data, finally lie in the fact that planning data are randomly corrected, parameter ranges are not considered, and the dynamics of the states over time are not represented so that the significance of the detected (point) value is instead low. The compression of the uncertainty on the level of the target value, however, occurs in such a manner, that the information on the ranges and the distribution of the polyvalent input values of the valuation issue are explicitly used to determine a consistent point value as the recommendation for action. These methods, for example, include stochastic optimization and fuzzy linear optimization. The stochastic optimization method interprets the individual input data of the valuation issue as a random variable with known probability distributions to ultimately condense the polyvalent expectations to a point value (R OLLBERG 1999, p. 107, H ERING 2017, p. 315). All versions of uncertainty solidifying methods attempt in a loss of information situation to compress the polyvalent expectations of the complex valuation issue into one point value to artificially “adjust” the uncertainty. Eventually, a monovalent decision value with rather low significance is delivered to the valuation subject. In contrast, the result of the uncertainty revealing (or disclosing) planning methods is a decision value that is provided to the valuation subject as a range or a distribution. Since the determined decision value should serve as a decision support to the valuation subject and the following third step demands a transparent information base with regard to the “consideration of (subjective decision) value and (objective) price”, the uncertainty of the valuation issue in step two of the decision value determination should not be concentrated, thereby reducing information, but instead be disclosed to the fullest extent. Owing to the inexpediency of those methods, the usage of uncertainty revealing methods is close at hand. These methods create the necessary transparency with regard to the subjectively suspected consequences of a decision and therefore serve as a decision criterion in a vivid and comprehensible way (H ERING 2017, p. 274). Against this background, uncertainty revealing methods will be presented in this chapter, including sensitivity analysis and risk analysis, and will be applied in the context of the decision value determination. 2.3.1.2.3 Third step From now on, the application of the determined decision value depends on the pursued function (purpose) of business valuation. Within the framework of the decision function, the phase of negotiation and decision ends with the non-formalizable weighing between the objective price as the major object of negotiation and the subjective decision value as the basis for negotiation (step 3 as well as field C of the matrix), in which the individual risk appetite of the decision-maker is integrated (H ERING 2014, 2.3 Determination of One-Dimensional Decision Values 77 45520_Matschke_Griffleiste_SL5.indd 77 45520_Matschke_Griffleiste_SL5.indd 77 16.03.2021 16: 20: 48 16.03.2021 16: 20: 48 p. 44). With the decision value as the result of an investment-theoretically supported valuation, quantitative information regarding the business in question is made available to the decision-maker. Therefore, the information available should be considered the starting point for negotiations between the relevant parties (K USSMAUL 1996, p. 266). Informed decisions on the advantageousness of a change of ownership of a business also demand an additional analysis of the qualitative aspects. Hence, a decision requires consideration of both quantitative and qualitative aspects. The determined decision value, however, constitutes the most important but not the single economic criterion. With regard to the non-financial targets of the valuation subject, it is conceivable that a presumptive acquirer accepts a higher payment than the (marginal) price limit under strictly financial aspects would suggest (M OXTER 1983, p. 75). Kapitel 1: Einführung 78 78 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 78 45520_Matschke_Griffleiste_SL5.indd 78 16.03.2021 16: 20: 49 16.03.2021 16: 20: 49 Chapter 2 2.3.2 Characterizing the Conflict Situation After the demonstration of the basic model for decision value determination by M ATSCHKE in Section 2.2, more specific methods for the decision value determination will be subsequently presented in Section 2.3 with regard to the matrix of functional business valuation. This relates to step 2 of decision value determination. To this end, the following conflict situation needs to be characterized. Here, a non-dominated, disjoint, one-dimensional conflict situation of the acquisition/ sale type (cf. Figure 2.6) is assumed, in which only the level of the (cash) price is resolution-relevant to the business at the valuation date: • Non-dominated means that no party may bring about the change of ownership by unilateral actions. The change of ownership will only occur in case of a voluntary agreement that is advantageous for both parties. • Disjoint means that the decision subjects “buyer” and “seller” have not simultaneously entered into negotiations with third parties on the acquisition or sale as well as merger or demerger of another business. • One-dimensional means that the only controversial issue in the negotiation, upon which the agreement between the parties depends, is the level of the (cash) price to be paid to the seller by the buyer. • It is implied that the valuation date, the decision date of acquisition/ sale and in case of an agreement, also the payment date for the price coincide. In the following, this date is set as t = 0. Figure 2.6: Conflict cube of the acquisition/ sale type for a non-dominated, disjoint, and one-dimensional conflict situation Degree of dominance Non-dominated Dominated Disjoint Degree of attachment Joint Multi-dimensional One-dimensional Degree of complexity 2.3 Determination of One-Dimensional Decision Values 79 45520_Matschke_Griffleiste_SL5.indd 79 45520_Matschke_Griffleiste_SL5.indd 79 16.03.2021 16: 20: 49 16.03.2021 16: 20: 49 To simplify, it is assumed that the business to be valuated is continued to be managed in isolation by the buyer so that so-called positive synergy effects (combined effects) do not exist, because an economic integration in a larger group of enterprises is either not intended by or not possible for the buyer. In the same way, it is implied for the seller that the business to be valuated is not being released out of a larger group so that consequently no negative repercussions (negative synergy effects) on other activities of the seller have to be taken into account. The last two assumptions lead to the principle of isolated valuation (stand-alone principle) that the valuation becomes practical. However, it is noteworthy that the applicability of this principle arises from the selected constellation and that potentially occurring positive and negative combined effects should generally be considered too. Further feasible investment activities by the buyer or seller (investment objects) are admitted but under the assumption of the stand-alone principle, that is, there is neither mutual influence nor influence on the business to be valuated, enabling the utility value of the business and/ or the utility values of such investment objects to be added (assumption of the utility additivity). Consequently, further investments are essentially randomly combinable among each other and with the business to be valuated (assumption of the random combinability). These further investments of each party ought to be randomly divisible (assumption of random divisibility) but are sometimes omitted from this assumption. More importantly, the investments could be different for each party (principle of subjectivity). Nonetheless, this does not necessarily exclude the possibility of their actual equality (parity). With regard to the target plan (result definitions, preferences) no ascertainments and therefore no restrictions will initially be necessary. The utility value determination will not be addressed any further here. Instead, it is presumed that the particular utility values regarding both the business and the further investment objects are known to the parties (assumption of knowledge of the utility values). Regarding the explanations concerning the utility values, this assumption demands further explanations. The explanations of the result function showed that the result constellations, apart from environmental impact, depend on the alternatives at hand. The term alternative is used in a decision-theoretic sense as an excluding combination of action parameters of the valuation subject. Result constellations, and therefore ultimately decision values, can principally not be attributed to specific (real) investment objects or certain institutions like businesses, but only to certain combinations of action parameters of the valuation subject. Therefore, these combinations are planned and comprise the investment objects, the business alone, or a combination thereof. If the cash flow of a (real) investment object or in business valuation theory of the payment flow of a business is nevertheless spoken of, this not only implies a result definition of the valuation subject fixed upon payments but also presupposes certain payments prompting measures by the decision subject and therefore a certain combination of action parameters referring to the particular (real) investments or the respective firm. In the case of rational behavior of the decision subject, the cash flow of the business must not be the result of any of the possible combinations of action parameters of the decision subject concerning the business but must instead be the optimal combination of these action parameters from the perspective of the decision subject. Kapitel 1: Einführung 80 80 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 80 45520_Matschke_Griffleiste_SL5.indd 80 16.03.2021 16: 20: 49 16.03.2021 16: 20: 49 Chapter 2 It must consequently be considered that (real) investment objects and the business to be valuated as organizational, economic, and legal units (business-related term) always represent a set of (investment and business-related) alternatives. Both result constellations and utility values can be attributed to the alternatives. If rational behavior is assumed, the mentioned utility value refers strictly to the particular (investment and business-related) optimum alternative (assumption of optimal planning from the perspective of the decision subject). While this assumption of optimal planning flows from the principle of rational behavior, it is also necessary to ensure a distinct allocation of utility values to the objects (investments or business as valuation objects). The assumptions of utility additivity and the random combinability, and random divisibility for their part, imply that the optimal planning, with regard to all action activities of the decision subject, results from the investmentand business-related partial optimal planning. Moreover, it is assumed that the decision subject is endowed with a certain fund of financial resources for investment purposes (investment capital) at the valuation date. At first, it is usually presumed that the funds only contain equity capital. The following presentation of the basic model omits the possibility of raising further capital (especially debt capital). The liquidity condition is solely explained with regard to the decision date t = 0. In terms of the following periods, solvency is constantly assumed. The capital endowment of the decision subject for investment purposes and the guarantee (or warranty) of solvency allows a very simple presentation of the basic approach in the determination of the decision value as the price limit (price cap, price ceiling, or upper price limit for the presumptive buyer; bottom price or lowest price limit for the presumptive seller) without arising serious limitations regarding the insights just gained. What should be demonstrated can be explained on this basis. 2.3 Determination of One-Dimensional Decision Values 81 45520_Matschke_Griffleiste_SL5.indd 81 45520_Matschke_Griffleiste_SL5.indd 81 16.03.2021 16: 20: 49 16.03.2021 16: 20: 49 2.3.3 Valuation Methods 2.3.3.1 Basic Models of Marginal Price Determination 2.3.3.1.1 Basic Model of Decision Value Determination 2.3.3.1.1.1 Determination of the Base Program The following symbols will be used below (cf. Figure 2.7): The underlying assumptions at the valuation date t = 0 mean the only topics subject to negotiation are how many units of the investment objects I b should be acquired if the business U is not acquired/ sold (determination of the base program) and how many units of the investment objects I b should be acquired if the business to be valuated is bought at the maximum price (price ceiling) P Umax (decision value of the buyer)/ is sold at the lowest price (bottom price) P Umin (decision value of the seller) (determination of the valuation program). The action parameters to be considered in the basic model are: 1. Acquisition of the investment objects I b , which can take on the values 0 ≤ z b ≤ z bmax (for b ∈ {1, …, B}) with random divisibility, as well as. 2. Acquisition/ sale of the business U, which can have the values z U ∈ {0, 1} from the perspective of the buyer/ seller. From the perspective of the buyer, z U = 0 means that the business U should not be acquired and z U = 1, means that it should. From the perspective of the seller, z U = 1 means that the business should not be sold and z U = 0, means that it should. U Business to be valuated N U I b Utility value of the business from the perspective of the valuation subject Investment objects available for the valuation subject with b ∈ {1, …, B} z b P b N b z U Number of the acquirable investment objects I b for the valuation subject with 0 ≤ z b ≤ z bmax (with random divisibility) or z b ∈ {0, 1, 2, …, z bmax } (as integers) The payable price for the investment object I b at the valuation date t = 0, Investment amount per unit of the investment object Utility value, attributed to the investment object I b by the decision subject Variable for the characterization of the acquisition/ sale of the business KN Ba N Be P U Capital endowment for the valuation subject at the valuation date t = 0 Utility of the base program from the perspective of the valuation subject Utility of the valuation program from the perspective of the valuation subject The price to be negotiated for the business U Figure 2.7: Overview of the used symbols in the basic model representation Kapitel 1: Einführung 82 82 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 82 45520_Matschke_Griffleiste_SL5.indd 82 16.03.2021 16: 20: 50 16.03.2021 16: 20: 50 Chapter 2 Based on the assumptions of the model, the following target function for the determination of the base program can be formulated: From the perspective of the valuation subject, the utility is to be maximized, which means nothing other than the assumption of rational behavior. Here, the following constraints have to be considered which secure the financial balance at the valuation date t = 0, that is, the capital requirement must not be higher than the investment capital K and the B constraints which indicate the limitations with regard to the amount of the investment objects I b to be acquired at the valuation date as well as the constraint if the base program is to be determined for the buyer, or the constraint if the base program is to be determined for the seller. Due to the constraints (4.1) and (4.2), this approach can be simplified with regard to both the buyer and the seller (see target function). However, this option is waived here to maintain the consistent presentation of the determination of the base program, with respect to both parties. The solution of this approach ultimately leads to the base program, whose utility value N Ba becomes the minimum requirement in the determination of the decision value. 2.3.3.1.1.2 Determination of the Valuation Program The approach (target function) for determination of the valuation program is: from the perspective of the buyer from the perspective of the seller, whereby the following constraints have to be considered in order to ensure that the valuation subject is not placed worse in the case of acquisition/ sale of the business for the decision value P Umax / P Umax than without acquisition/ sale, further or (1) z b b ∑ ⋅ N b +z U ⋅ N U → max! (2) z b b ∑ ⋅ P b ≤ K, (3) 0 ≤ z b ≤ z bmax for b ∈ 1, …, B { }, (4.1) z U = 0, (4.2) z U = 1, (1.1) P U → max! (1.2) P U → min! (2) z b b ∑ ⋅ N b + z U ⋅ N U ≥ N Ba , (3.1) z b b ∑ ⋅ P b ≤ K− P U (3.2) z b b ∑ ⋅ P b ≤ K+ P U , 2.3 Determination of One-Dimensional Decision Values 83 45520_Matschke_Griffleiste_SL5.indd 83 45520_Matschke_Griffleiste_SL5.indd 83 16.03.2021 16: 20: 50 16.03.2021 16: 20: 50 to secure the financial equilibrium at the valuation date in case of an acquisition (3.1) or sale (3.2), as well as the B constraints which limit the acquisition of the investment objects I b to the corresponding currently available amount, and finally the constraint or if the valuation program is to be determined for the buyer (5.1) or for the seller (5.2). The constraint (3.1) expresses that the acquisition of the business P U > 0 limits the scope of the buyer for the realization of other investment objects (K - P U ). Conversely, it applies to the seller as can be seen from the constraint (3.2) that the sale of the business leads to an increase of capital endowment K. While the buyer has to waive investments, which would have been realized without the acquisition of the business, the seller is given the opportunity to make investments. Those investments would not be possible without the sale of the business owing to the limited capital endowment. The acquisition/ sale of the company, however, not only has financial impacts but also impacts on the utility, both of which are considered by constraint (2). Hence, the buyer/ seller gains/ loses the attributable utility value N U through the acquisition/ sale of the business. Accordingly, the buyer can waive the purchase of other investment objects, while the seller has to carry out new investments to compensate for the reduction of the utility value arising from the sale of the company. This mathematical approach can also be simplified due to the conditions (5.1) and (5.2). But yet again, this is omitted here, because at this point the important task is to present the structure of the task and not the issues around solution techniques. 2.3.3.1.1.3 Numerical Example a. Valuation from the Buyer’s Perspective (1) General Valuation Method The following decision situation with regard to the investment objects I b in question (for now assumed as not divisible) and the business U to be valuated should apply for the decision subject (buyer) (GE = monetary units; German: Geldeinheiten; NE = utility units; German: Nutzeneinheiten): (3) 0 ≤ z b ≤ z bmax for b ∈ 1, …, B { }, (5.1) z U = 1 (5.2) z U = 0, Kapitel 1: Einführung 84 84 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 84 45520_Matschke_Griffleiste_SL5.indd 84 16.03.2021 16: 20: 51 16.03.2021 16: 20: 51 Chapter 2 The available capital endowment K of the buyer at the valuation date is K= 5.000 GE. The mathematical approach for the determination of the base program from the perspective of the buyer is - considering the possible simplification due to z U = 0 - in this example: The assumption of the indivisibility of the investment objects I b has been introduced to carry out the following determination of the decision value as the price cap, also in analogy to the general model of a multi-dimensional decision value, and to recapitulate this model, with regard to a more simple conflict situation and to thus make this model more comprehensible. In this given decision situation, only eight alternatives a i (with i ∈ {1, 2, …, 8}) exist for the buyer, in the sense of excluding combinations of the action parameters acquisition of the investment objects I b , which could take on the values z b ∈ {0, 1} for b ∈ {1, 2, 3}. Accordingly, z b = 0 means that the investment object I b is not purchased, and z b = 1 that it should be. In the example of these eight mutually exclusive combinations of action parameters (alternatives), the buyer cannot implement the alternatives a 6 , a 7 and a 8 (cf. Figure 2.9), because their capital requirement exceeds the initially available capital endowment of K = 5000 GE. Since no financing options exist, the buyer is not able to realize those alternatives. The decision field (without acquisition) A = {a i | Σz b · P b ≤ K} is represented in the example only with the alternatives a 1 to a 5 : A = {a 1 , a 2 , a 3 , a 4 , a 5 }. In Figure 2.9 the alternatives within the decision field of the buyer are bold and underlined: Investment objects I b Price P b per unit for the investment objects Utility value N b per unit for the investment objects Maximum possible amount z bmax of the investment objects Investment object I 1 Investment object I 2 2.000 GE 3.000 GE 6.000 NE 6.000 NE 1 1 Investment object I 3 Business U Figure 2.8: Decision situation of the buyer 5.000 GE ? 6.000 NE 6.000 NE 1 (1) z 1 ⋅ 6.000 + z 2 ⋅ 6.000 + z 3 ⋅ 6.000 → max! under the constraints (2) z 1 ⋅ 2.000 + z 2 ⋅ 3.000 + z 3 ⋅ 5.000 ≤ 5.000 (3) z b ∈ 0, 1 { } for b ∈ 1, 2, 3 { } . 2.3 Determination of One-Dimensional Decision Values 85 45520_Matschke_Griffleiste_SL5.indd 85 45520_Matschke_Griffleiste_SL5.indd 85 16.03.2021 16: 20: 51 16.03.2021 16: 20: 51 Alternative a 1 is the refraining alternative, a 2 to a 5 are various performance alternatives. The combinations a 6 to a 8 are not financeable; they thus do not constitute possible actions. The optimal alternative a opt in the example is alternative a 5 , which contains the acquisition of the investment objects I 1 and I 2 . This optimal alternative represents the base program of the buyer: a opt = {a i | max{N(a i ) | a i ∈ A }. The utility value N(a opt ) of this optimal alternative is the utility value N Ba of the base program: N Ba = N(a opt ) = {max{N(a i ) | a i ∈ A }. It will be explained later that under the conditions of the base model the optimal alternative a opt can also be directly defined without explicitly determining the set A of all alternatives of the buyer. The mathematical approach for calculating the decision value under the conditions of the numerical example is: Here, the solution can also be found without further mathematical programming, solely on the basis of combinatorics, which is done subsequently to apply the procedure of determining a multi-dimensional utility value on the present one-dimensional conflict situation. Figure 2.10 illustrates all alternatives B (P) = {b j (P)} available to the buyer, according to the determined conditions after an agreement of a certain price P, where even negative prices are possible. In such a case, the buyer receives money if the business is taken over and continued: Alternative a i Utility value N(a i ) Capital requirement of the alternative Σ z b · P b a 1 = (z 1 =0, z 2 =0, z 3 =0) a 2 = (z 1 =1, z 2 =0, z 3 =0) 0 NE 6.000 NE 0 GE 2.000 GE a 3 = (z 1 =0, z 2 =1, z 3 =0) a 4 = (z 1 =0, z 2 =0, z 3 =1) 6.000 NE 6.000 NE a 5 = (z 1 =1, z 2 =1, z 3 =0) a 6 = (z 1 =1, z 2 =0, z 3 =1) 12.000 NE 12.000 NE 3.000 GE 5.000 GE 5.000 GE 7.000 GE > K a 7 = (z 1 =0, z 2 =1, z 3 =1) a 8 = (z 1 =1, z 2 =1, z 3 =1) 12.000 NE 18.000 NE Figure 2.9: Set A of alternatives of the buyer 8.000 GE > K 10.000 GE > K (1) P U → max! under the constraints (2) z 1 ⋅ 6.000 + z 2 ⋅ 6.000 + z 3 ⋅ 6.000 + z U ⋅ 6.000 ≥ 12.000 (3) z 1 ⋅ 2.000 + z 2 ⋅ 3.000 + z 3 ⋅ 5.000 ≤ 5.000 − P U (4) z b ∈ 0, 1 { } for b ∈ 1, 2, 3 { } (5) z U = 1. Kapitel 1: Einführung 86 86 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 86 45520_Matschke_Griffleiste_SL5.indd 86 16.03.2021 16: 20: 52 16.03.2021 16: 20: 52 Chapter 2 Figure 2.10 shows that due to the integer (whole number) condition the set B (P) = {b j (P)} contains only one element, except for the price P = 0 GE. The set B (P = 0) has two elements, while the alternative b 5 = (z 1 = 0, z 2 = 0, z 3 = 1, z U = 1) is dominated and worse than the alternative b 4 = (z 1 = 1, z 2 = 1, z 3 = 0, z U = 1) and consequently does not come in to question for P = 0; that is, the utility value of the best alternative N(b opt (P = 0)) = N(b 4 ) = 18.000 NE (in terms of the conflict solution) is to be classified to price P = 0. The alternative b 1 is only achievable if the buyer receives the amount of 5.000 GE from the seller, that is, at the representation of the set B (P) of the action possibilities after an acquisition, according to the level of the negotiated price P, it is abstracted from the restriction that the buyer has to regularly pay a price P > 0. Moreover this dominance relation between the alternatives according to the same price (here P = 0 GE), there still are more dominance relations between the represented alternatives illustrated in Figure 2.10. The utility value of N(b j (P)) = 18.000 NE is achieved at the prices P ∈ {0 GE, -2.000 GE, -3.000 GE}. In this case, the conflict solution P = 0 is dominating the other conflict solutions (due to P U → max.). Therefore, only the price P = 0 can be attributed to the utility value N(b j (P)) = 18.000 NE. With respect to the utility value N(b j (P)) = 12.000 NE, it follows that it is achieved at prices P ∈ {2.000 GE, 3.000 GE} with the alternatives {b 6 = (z 1 = 0, z 2 =1, z 3 = 0, z U = 1), b 7 = (z 1 = 1, z 2 = 0, z 3 = 0, z U = 1)}. The latter of which dominates the other. If such dominance relations are considered, the dominated alternatives are neglected. It results in the following set of the non-dominated alternatives of the buyer as well as the one-to-one classification of a certain utility value to a conflict solution (here a specific price P). This is represented in Figure 2.11. Price P Alternative set B (P) = {b j (P)} Utility value N(b j (P)) -5.000 -3.000 B (P = -5.000) = {b 1 = (z 1 = 1, z 2 = 1, z 3 = 1, z U = 1)} B (P = -3.000) = {b 2 = (z 1 = 0, z 2 = 1, z 3 = 1, z U = 1)} 24.000 NE 18.000 NE -2.000 0 B (P = -2.000) = {b 3 = (z 1 = 1, z 2 = 0, z 3 = 1, z U = 1)} B (P = 0) = {b 4 = (z 1 = 1, z 2 = 1, z 3 = 0, z U = 1); 2.000 b 5 = (z 1 = 0, z 2 = 0, z 3 = 1, z U = 1)} B (P = 2.000) = {b 6 = (z 1 = 0, z 2 = 1, z 3 = 0, z U = 1)} 18.000 NE 18.000 NE 12.000 NE 12.000 NE 3.000 5.000 B (P = 3.000) = {b 7 = (z 1 = 1, z 2 = 0, z 3 = 0, z U = 1)} B (P = 5.000) = {b 8 = (z 1 = 0, z 2 = 0, z 3 = 0, z U = 1)} Figure 2.10: Alternative set B (P) of the buyer 12.000 NE 6.000 NE 2.3 Determination of One-Dimensional Decision Values 87 45520_Matschke_Griffleiste_SL5.indd 87 45520_Matschke_Griffleiste_SL5.indd 87 16.03.2021 16: 20: 52 16.03.2021 16: 20: 52 The alternative b 7 = (z 1 = 1, z 2 = 0, z 3 = 0, z U = 1) leads to exactly the same utility value as the base program. Hence, it represents the valuation program: B * = {b 7 }, which is a one-element set in the example. The set of the feasible conflict solutions S z results from all prices, which lead to a utility value N(b j (P)) ≥ N Ba = 12.000 NE: S z = {P | P ≤ 3.000 GE}. Then, the decision value is W = {P = 3.000 GE}. (2) Specific Valuation Method: Tabular Method Under the conditions of the basic model, both the base program and the valuation program can be determined without the aid of the set A and B (P) = {b j (P)} and without the linear programming approach. This will now be examined in greater detail. The optimal alternative a opt , that is, the base program, is directly determined under the conditions of the basic model since the underlying situation can be classified as a situation with only one known relative scarce factor (situation with a known limiting factor). The relatively scarce factor is represented by the capital endowment K. Therefore, K should be optimally allocated to the most promising investment objects I b . The rule for using of a known scarce factor is: Use the scarce factor in order of relative target contributions of its different possible uses. Start with the use that has the highest relative target contribution! The relative target contribution is the target contribution with the respective possible use per operational unit of the scarce factor. The condition for the application of this general rule is the divisibility of both the scarce factor and the possible uses which come into question. This condition is fulfilled in the base model. In Figure 2.12 the relative target contributions of the investment objects I b are presented. The utilization of the scarce factor (capital endowment K) by the investment objects is expressed by their respective price P b . Price P Alternative set B (P) = {b j (P)} Utility value N(b j (P)) -5.000 0 B (P = -5.000) = {b 1 = (z 1 = 1, z 2 = 1, z 3 = 1, z U = 1)} B (P = 0) = {b 4 = (z 1 = 1, z 2 = 1, z 3 = 0, z U = 1)} 24.000 NE 18.000 NE 3.000 5.000 B (P = 3.000) = {b 7 = (z 1 = 1, z 2 = 0, z 3 = 0, z U = 1)} B (P = 5.000) = {b 8 = (z 1 = 0, z 2 = 0, z 3 = 0, z U = 1)} Figure 2.11: Set of non-dominated alternatives B (P) of the seller 12.000 NE 6.000 NE Investment objects I b Price P b per unit of the investment objects Utility value N b per unit of the investment objects Relative target contribution N b / P b Investment object I 1 Investment object I 2 2.000 GE 3.000 GE 6.000 NE 6.000 NE 3 NE/ GE 2 NE/ GE Investment object I 3 Figure 2.12: Relative target contributions of the investment objects 5.000 GE 6.000 NE 1,2 NE/ GE Kapitel 1: Einführung 88 88 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 88 45520_Matschke_Griffleiste_SL5.indd 88 16.03.2021 16: 20: 53 16.03.2021 16: 20: 53 Chapter 2 The above rule indicates the scarce factor should initially be allocated in favor of investment object I 1 and afterward - if still available - in favor of investment object I 2 and finally - if still available - in favor of the investment object I 3 . In the example, the scarce factor is K = 5.000 GE. Accordingly, it consequently has to be used for the acquisition of investment objects I 1 (with P 1 = 2.000 GE and a relative target contribution of N 1 / P 1 = 3) and afterward for the acquisition of investment object I 2 (with P 2 = 3.000 GE and N 2 / P 2 = 2). Then it is completely depleted so that investment object I 3 does not play a role here. This procedure is based on the so-called tabular valuation method (cf. Figure 2.13): The utility value of the base program N Ba = 12.000 NE becomes the minimum requirement of the valuation program, which comprises the business to be valuated. The question to be answered is: How much of financial means could the buyer provide at most for the acquisition of the business at the price ceiling, without the utility value shrinking under the level of the base program? The answer to that question can be found via the rule of using a scarce factor; unfortunately, not in the situation of including investment objects in the base program, but in a situation of displacing investment objects from the base program. This can be interpreted as a renouncement of a previously planned usage of the scarce factor “capital endowment” and releasing capital for the new purpose of use such as the possible acquisition of a company. By including the investment objects in the base program, the optimal inclusion rule schedules a procedure according to the order of relative target contributions, while it starts with the inclusion of the investment object, which indicates the highest relative target contribution. Henceforth, the optimal displacement rule requires an orientation on the relative target contribution, namely in the sense of renouncing the planned use of the scarce factor in the base program. This factor has the lowest relative target contribution. Accordingly, the same procedure is put into effect regarding further displacements. The displacement of investment objects in the base program of the buyer is finally accomplished, when the utility value N Be of the valuation program that contains the business to be valuated is as high as the utility value N Ba of the base program. If the described condition N Be = N Ba is met, the utility value N VO of the displaced investment objects, which represents the so-called comparison object, corresponds to the utility value N U of the business to be valuated as the valuation object: N U = N VO . Again, it follows that by dis- Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) Investment object I 1 Investment object I 2 -2.000 GE -3.000 GE 6.000 NE 6.000 NE Capital endowment K Utility value N Ba Figure 2.13: Base program of the buyer 5.000 GE 12.000 NE 2.3 Determination of One-Dimensional Decision Values 89 45520_Matschke_Griffleiste_SL5.indd 89 45520_Matschke_Griffleiste_SL5.indd 89 16.03.2021 16: 20: 53 16.03.2021 16: 20: 53 placing the capital endowment, the “released” capital can be used for the acquisition of the business. Following the optimal displacement rule, the investment objects of the base program displaced are those whose utility value includes the highest capital commitment. One could also say that the highest capital amount for the acquisition of a business is released, according to the utility value N U of the business. The buyer could pay this exact amount for the business; it corresponds to the decision value of the buyer as an upper price limit (ceiling price). In other words: In the basic model, the decision value P max of the buyer results from the price P VO of the investment objects (comparison object), equally successful, and displaced from the base program. The determination of the upper price limit P max (decision value of the buyer) contains the transfer of the price P VO of the displaced investment objects of the business to be valuated (valuation object): “The price of the comparison object becomes the value of the valuation object.” This also holds true from the perspective of the seller (see discussion below). Hence, the decision value as a price limit can generally be regarded as a relation between the valuation subject, the valuation object, and the comparison object. Figure 2.14 illustrates the essential steps of the valuation program according to the tabular valuation method: The decision value of the buyer is equal to the price of the investment objects that have equal utility and were displaced from the base program. In the numerical example, that is equal to the price P 2 of the investment object I 2 , an amount that can be determined with tabular valuation by the subtraction of the quantities of the base program from those from the valuation program (cf. Figure 2.15). Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) 1. Step: Inclusion of the valuation object Investment object I 1 -2.000 GE 6.000 NE Investment object I 2 Capital endowment K Business U Utility value N Ba + N U -3.000 GE 5.000 GE 6.000 NE 6.000 NE 18.000 NE 2. Step: Displacement of the investment object I 2 Investment object I 1 Capital endowment K Business U -2.000 GE 6.000 NE 5.000 GE 6.000 NE Utility value N Be = N Ba Figure 2.14: Valuation program of the buyer (determination with the tabular method) 12.000 NE Kapitel 1: Einführung 90 90 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 90 45520_Matschke_Griffleiste_SL5.indd 90 16.03.2021 16: 20: 53 16.03.2021 16: 20: 53 Chapter 2 With the aid of this tabular valuation, the characterization of the decision value as a subject-object-object-relation becomes apparent. The current relation (ratio) between the target contribution (utility) and the capital investment (price) of the comparison object (VO; German: Vergleichsobjekt): The comparison object VO can be composed of several displaced investment objects. In the example, the comparison object is the investment object I 2 , which was displaced from the base program. b. Valuation from the Seller’s perspective The determination of the decision value as lowest price limit (bottom price) of the seller is described with this numerical example (cf. Figure 2.16): Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) Valuation program Investment object I 1 -2.000 GE 6.000 NE Capital endowment K Business U ./ . Base program Investment object I 1 5.000 GE 6.000 NE -2.000 GE 6.000 NE Investment object I 2 Capital endowment K = Decision value P max Figure 2.15: Decision value of the buyer -3.000 GE 5.000 GE 6.000 NE 3.000 GE P max = N U K N VO K P VO K = P VO K due to N U K = N VO K , in the exampel: P max = 6.000 NE 6.000 NE 3.000 GE = 3.000 GE. 2.3 Determination of One-Dimensional Decision Values 91 45520_Matschke_Griffleiste_SL5.indd 91 45520_Matschke_Griffleiste_SL5.indd 91 16.03.2021 16: 20: 53 16.03.2021 16: 20: 53 The investment objects are infinitely divisable. The numerical approach for the determination of the base program from the seller’s perspective is as follows: This approach constitutes the maximization of the utility value, taking into consideration liquidity conditions at time t = 0 as well as the limitation according to the available investment objects I b . The business is included in the base program of the seller. This is represented by z U = 1. Because of this, the programming approach could again be simplified, but this will not happen here, because the process is examined with respect to the tabular valuation method. The investment objects are included according to the relative target contributions. The (initial) capital endowment of the seller K = 1.000 GE should be used to acquire the investment object I 1 . The base program of the seller reads as follows (cf. Figure 2.17): The steps leading to the valuation program of the seller are stated in Figure 2.18 pursant to the tabular valuation method: Investment objects I b Price P b Utility value N b Quantity z bmax Relative target contribution N b / P b Investment object I 1 Investment object I 2 1.000 GE 2.000 GE 5.000 NE 6.000 NE 1 1 5 NE/ GE 3 NE/ GE Investment object I 3 Business U Capital endowment K Figure 2.16: Decision situation of the seller 4.000 GE ? 8.000 NE 5.000 NE 1.000 GE 1 2 NE/ GE (1) z 1 ⋅ 5.000 + z 2 ⋅ 6.000 + z 3 ⋅ 8.000 + z U ⋅ 5.000 → max! under the constraints (2) z 1 ⋅ 1.000 + z 2 ⋅ 2.000 + z 3 ⋅ 4.000 ≤ 1.000 (3) 0 ≤ z b ≤ 1 for b ∈ 1, 2, 3 { } (4) z U = 1. Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) Investment object I 1 Capital endowment K -1.000 GE 1.000 GE 5.000 NE Business U Utility value N Ba Figure 2.17: Base program of the seller 5.000 NE 10.000 NE Kapitel 1: Einführung 92 92 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 92 45520_Matschke_Griffleiste_SL5.indd 92 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 Chapter 2 The decision value of the seller is equal to the price P 2 of the investment object I 2 in the example. It can be determined by the tabular valuation method by subtracting the quantities of the valuation program from those of the base program (cf. Figure 2.19): In the example, the seller should at least require a price of P min = 1.667 GE for the company. If he receives that sum, he could acquire the investment object I 2 for which the utility value is as high as the utility value of the valuation object, resulting in compensation. If the seller receives more, he could achieve a utility value that exceeds the value of the base program (N Ba ). The comparison object of the seller represents the investment objects, which have an equal utility value and in which he has to invest after a Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) 1. Step: Exclusion of the valuation object Investment object I 1 -1.000 GE 5.000 NE Captial Endowment K Utility value N Ba - N U 2. Step: Inclusion of the investment object I 2 Investment object I 1 1.000 GE 5.000 NE -1.000 GE 5.000 NE Capital endowment K Investment object I 2 Utility value N Be = N Ba Figure 2.18: Valuation program of the seller (determination with the tabular method) 1.000 GE -1.667 GE 5.000 NE 10.000 NE Capital requirement (-) and capital requirement coverage (+) at time t = 0 (column 1) Utility values (column 2) Base program Investment object I 1 -1.000 GE 5.000 NE Capital endowment K Business U ./ . Valuation program Investment object I 1 1.000 GE 5.000 NE -1.000 GE 5.000 NE Capital endowment K Investment object I 2 = Decision value P min Figure 2.19: Decision value of the seller 1.000 GE -1.667 GE 5.000 NE 1.667 GE 2.3 Determination of One-Dimensional Decision Values 93 45520_Matschke_Griffleiste_SL5.indd 93 45520_Matschke_Griffleiste_SL5.indd 93 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 sale of the valuation object at the decision value P min . These are the best investment objects that have not yet been utilized. The tabular valuation illustrates the subject-object-object-relation for the seller. In other words, the decision value indicates a transfer from the relation of the comparison object VO between the target contribution and the capital investment (price) to the valuation object: Additionally, it is assumed that the comparison object consists of several reinvestment opportunities. 2.3.3.1.2 Basic Model of the Present Value Calculation 2.3.3.1.2.1 Structural Equality of the Price Limit Calculation and the Present Value Calculation In the examples focusing on the determination of the decision value as upper price limit P max of the buyer or as lowest price limit P min of the seller, it is generally apparent that the price limit calculation implies a transfer from the relation of the comparison object between the target contribution and investment (price) to the valuation object. The formal structure of the price limit calculation is defined as follows: • for the buyer: and • for the seller: However, it should be pointed out once again that the quantities used in these calculations are interpreted from the perspective of the respective valuation subject (buyer or seller). The comparison object is the respective optimal alternative investment of each party. For the buyer, that means the investment objects that have equal utility value and have been displaced from the base program. For the seller, that refers to investment objects with equal utility value included in the valuation program instead of the business. P min = N U V N VO V P VO V = P VO V due to N U V = N VO V , in the example: P min = 5.000 NE 5.000 NE 1.667 GE = 1.667 GE. P max = N U K N VO K P VO K = P VO K due to N U K = N VO K P min = N U V N VO V P VO V = P VO V due to N U V = N VO V . Kapitel 1: Einführung 94 94 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 94 45520_Matschke_Griffleiste_SL5.indd 94 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 Chapter 2 These price limit calculations can be directly transferred into a structure of a present value calculation (as a perpetuity). The present value is the current value of all future payments (caused after the valuation date). First, they are discounted, then their individual, period-specific present values are added up. The structural equality of the price limit calculation and the present value calculation results from the interpretation of the respective utility amount N as the positive cash flow (future performance ZE; German: ZE = Zukunftserfolg) and the respective ratio between target contribution and capital investment (price) as the subjective scarcity price of the capital r VO . The scarcity price depends on the comparison objects of the respective valuation subject: • for the buyer: and • for the seller: In the present value calculation, the relative target contribution results from the optimal alternative investment and hence should be utilized as the interest rate i by the valuation subject. If this is considered, the following present value calculations (as a perpetuity) will result for the determination of the price limits (with i as interest rate from the perspective of the buyer i K or the seller i V ): • for the buyer K: and • for the seller V: These formulas represent the basic model for a purely financial business valuation. ZE K and ZE V represent the valuation-relevant cash flows (future performances) of the business from the perspective of the buyer K and the seller V. Furthermore, i K and i V represent the comparison objects of the buyer K and the seller V, that is, their optimal alternative investment. 1/ i K and 1/ i V represent the business capitalization formula. They are the result of an infinite series of discount factors Σ(1+i K ) -t and Σ(1+i V ) -t (for t → ∞). However, this requires that the future performances ZE K and ZE V are expected on the basis of a constant perpetuity (perpetual annuity immediate). P max = ZE U K r VO K = P VO K due to ZE U K = ZE VO K and r VO K = ZE VO K P VO K P min = ZE U V U r VO V = P VO V due to ZE U V U = ZE VO V and r VO V = ZE VO V P VO V . P max = ZE K i K = ZE K ⋅ 1 i K P min = ZE V i V = ZE V ⋅ 1 i V . 2.3 Determination of One-Dimensional Decision Values 95 45520_Matschke_Griffleiste_SL5.indd 95 45520_Matschke_Griffleiste_SL5.indd 95 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 2.3.3.1.2.2 Extended Interpretation of the Term “Comparison Object” based on the Present Value Calculation Based on the present value calculation, both the comparison object and the business to be valuated (valuation object) are illustrated as a cashflow. Hence, an investment is a specific stream of cash flows that is usually typified by initial payouts (negative cash flows) and later by deposits (positive cash flows). In the simplest case of a (normal) investment, the initial payout a S0 is the price of the investment object at time t = 0 followed by a series of deposits e St corresponding to the (projected) life (or term) T of that investment with t = 1, …, T: (- a S0 , + e S1 , …, + e ST ). The comparison object of the buyer is categorized in the price limit calculation as the respective investment objects that are excluded from the base program. In other words, the buyer refrains from some investment objects in favor of the business. Under the conditions of the present value calculation the investments displaced from the base program (comparison object) are represented by the following cash flow: -(-a VO0 , + e VO1 , …, +e VOT ) with (+e VO1 , …, +e VOT ) = (+e U1 , …, +e UT ), while the index VO defines the comparison object and the index U defines the business. The comparison object of the buyer is formally computed by the payment flow -(-a VO0 , +e VO1 , …, +e VOT ) = (+a VO0 , -e VO1 , …, -e VOT ), which starts with an initial positive payment in the amount of +a VO0 and continues with payouts (negative excess deposits) in the amount of (-e VO1 , …, -e VOT ). This cash flow is a classic example of financing (funding measure). However, this also generally implies that under the assumptions of the present value calculation a comparison object of the buyer is not only comprised of displaced investments but also of additional financing opportunities that have not yet been incorporated in the base program. Since the displacement of investment objects from the base program and the additional financing opportunities in the base program are not mutually exclusive, the comparison object can subsequently be derived from a combination of these two alternatives. The reflections on the generalization of the term “comparison object” under the assumptions of the present value concept can also be transferred to the valuation situation of the seller. In such a price limit determination, the comparison object of the seller is classified as the respective investment object with equal utility that is included in the valuation program instead of the business. Under the conditions of the present value calculation, the investments (comparison objects), which can be included in the valuation program, are illustrated by this payment flow: (-a VO0 , +e VO1 , …, +e VOT ) with (+e VO1 , …, +e VOT ) = (+e U1 , …, +e UT ). The index VO represents the comparison object from the perspective of the seller and the index U indicates the business. Such a cash flow from the comparison object also results when you regard a payment flow of a (displaced) financing opportunitiy: - (+a VO0 , -e VO1 , …, -e VOT ) = (-a VO0 , +e VO1 , …, +e VOT ). This is a classic example of a series of cash flows from a (normal) investment. Under the assumptions of the present value calculation, a comparison object for the seller not only encompasses the (additional) inclusion of investment objects but also those from the displaced financing possibilities (debt relief opportunities) of the base program. Because the inclusion of the investment object into the valuation program and Kapitel 1: Einführung 96 96 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 96 45520_Matschke_Griffleiste_SL5.indd 96 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 Chapter 2 the displacement of the existing financing possibilities are not mutually exclusive, the comparison object of the seller can also be derived from a combination thereof. 2.3.3.2 The State Marginal Price Model - a General Model 2.3.3.2.1 Basics Subsequently, the investment-theoretic models of business valuation will be presented in greater detail. They are all based on the previously discussed basic model for the determination of multi-dimensional decision values. These models comprise the general “state marginal price model”, the partial “future performance value method”, and the heuristic model of the “approximate decomposed valuation”. All these models serve to valuate cash flows economically and to support decision-making under real conditions and with respect to the targets and the decision field of the decision subject. First, these methods will be critically reviewed according to the following six model requirements (B RÖSEL 2002, p. 85). Moreover the requirements in real decision situations, the characteristics of the decision value will also be addressed: 1. Subject, target system, and action relation: The model is to enable a business valuation explicitly respecting the principles of overall valuation, of future relation, and of subjectivity in the respective conflict situation (a feature of the action relation), while the targets of the presumptive buyer must be considered (a feature of the subject and target relation). 2. Decision field relation and determination of the marginal (limit) value: The determined value of the company in the model should represent the limit of negotiation willingness (a feature of the marginal value) and it is valid exclusively for the concrete decision field of the valuation subject and thus the derived alternatives (a feature of decision field relation). 3. Possibility of connecting with methods revealing uncertainty: So that the determinated business value serves as a basis for decision-making in an illustrative form, the effects of uncertainty in real life are transparently revealed by combining suitable methods with this model. 4. Reasonable effort for information acquisition and information processing: The effort for information acquisition and processing of essential information should remain within economically acceptable levels. 5. Computability/ Calculability: The solvability of the model is provided/ ascertained. 6. Providing individual business decision support: Decision-making competence regarding the changes of ownership can be integrated differently within the business being acquired or sold. Hence, the valuation model should provide individual and necessary requested/ demanded centralized and/ or decentralized decision support. Based on the basic model of the decision value according to M ATSCHKE (1975, p. 387), H ERING (1999) determines a general state marginal price model (in German: Zustands-Grenzpreismodell (ZGPM)) for the valuation of a stream of cash flows to calculate the marginal price of a business in two simple steps (H ERING 2000b, p. 363, O L- BRICH 2001, p. 1329, B RÖSEL 2002, p. 91, H ERING 2002, p. 74, P FAFF / P FEIFFER / G ATHGE 2002). Both the base program (1. step) and the valuation program (2. step) are based on multi-period, simultaneous planning approaches of W EINGARTNER (1963) and H AX 2.3 Determination of One-Dimensional Decision Values 97 45520_Matschke_Griffleiste_SL5.indd 97 45520_Matschke_Griffleiste_SL5.indd 97 16.03.2021 16: 20: 54 16.03.2021 16: 20: 54 (1964). They are consequently determined by means of linear optimization. However, it must be taken into consideration: Due to the limited human information acquisition and processing capabilities, a general model, which reflects all operational contexts, does not and will likely never exist (R OLLBERG 2001, p. 4). The deterministic variation of this model, namely the “time marginal price model”, enables the valuation of (quasi-) certain cash flows. Generally speaking, if the points in time are interpreted as states, the model transforms (back) into a general model of equal structure. It can be used for the valuation of uncertain payment streams and also considers a system of restrictions. Before a formal presentation and a transparent exemplary application of the state marginal price model is provided, the model is described verbally. The perspectives of both the buyer and the seller will be discussed to clarify the connection to the general model according to M ATSCHKE and the representations of the mathematical (linear) approaches explained below. In the first step, the investment and financing program is calculated as the base program. This investment and financing program maximizes the target function on the assumption that there is no change of ownership. To determine this base program, an adequate linear optimization approach must be formulated and ultimately resolved. This approach focuses on the question of which maximum utility level can be achieved by the decision subject without a settlement of the conflict situation. After valuators (valuers, appraisers) have determined the framework conditions (e.g., the planning horizon and the valuation, the decision, and the acquisition date), they have to determine the target function as well as the action alternatives. It is also vital that the restrictions are formulated as constraints. The target function for the determination of the base program depends on the target system of the valuation subject. Following the methodology of the state marginal price model, the valuation subject has to decide between the two variants; either asset maximization or income maximization. Once the appraiser has decided upon an expedient objective, the action alternatives and the action limits of the decision field must be elicited to formulate the constraints. Hence, the investment and financing objects have to be included in the model as variables with determined capacity limits as well as non-negativity and integer (whole number) conditions: Bear in mind that the valuation object is not a part of the investment program of the presumptive buyer, but it is an integral part of the investment program of the seller. The constraints also comprise the financing/ borrowing possibilities, the (unlimited) cash management, and the potentially available interest-bearing investments. Specific payments such as those from current business activity and from existing loan obligations are also considered: Eventually, the process arrives at the consideration of a (fixed) (payment/ accounting) balance. This payment balance is independent of the objects to be valuated and could be positive, negative, or zero. The returns from investment and financial objects and from the balance of the “predisposed” payments should always be sufficient to allow for payouts (distributions) to the owner(s). In other words, the financial equilibrium in the sense of constant solvency (ability to pay) must exist at all times by maintaining liquidity restrictions. It is also noteworthy that the liquidity of the valuation subject is not permanent, but is only guaranteed at the beginning and end of a period. This is because the model focuses on a point in time rather than on a period of time. Kapitel 1: Einführung 98 98 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 98 45520_Matschke_Griffleiste_SL5.indd 98 16.03.2021 16: 20: 55 16.03.2021 16: 20: 55 Chapter 2 The simplex algorithm based on G AUSSIAN elimination indicates the formulated optimization problem can be solved. The result is the base program, which leads to the maximum target function value by using the inherent investment and financing opportunities. The acquisition (from the perspective of the presumptive buyer) and the sale (from the perspective of the presumptive seller) of the business to be valuated is economically reasonable only if the respective valuation subject achieves at least the same target value in the valuation program as in the base program. In the second step, only in the case of an acquisition situation is the valuation object included in the investment and financing program of the presumptive buyer. The ceiling price is determined as the decision value of the presumptive buyer under the condition that at least the same target function contribution is achieved as in the base program. The result of this step is the valuation program, which must include the valuation object. Again, the target function and the constraints for the optimization problem must be formulated. Ultimately, the valuation subject (buyer) pays the price, which is assumed to be the only conflict-resolution-relevant fact. Remember, the aim is to find an upper price limit as the decision value from the perspective of the presumptive buyer. This value then corresponds to the ceiling price, which could be accepted by the valuation subject for the stream of cash flows from the business as long as it does not impair the situation compared to the first step of the determined base program. Accordingly, the target function must now be formalized. With the acquisition of the business and its integration in the investment and financing program; from the perspective of the valuation subject, as a minimum, the target function value from the base program must be regained. Henceforth, this condition is included in the mixed whole-number linear optimization approach. Taking into consideration the remaining constraints of the original decision field, the model uses the simplex algorithm to compute the decision value and the valuation program in the second step. The valuation object is finally included in the transformed optimal investment and financing program of the presumptive buyer. Conversely, the valuation object is eliminated from the investment program in the second step in the case of a sale situation. The bottom price is then determined as the decision value of the presumptive seller, assuming that the target function contribution of the base program is also achieved. The result of this step is the valuation program. It does not contain the valuation object because, in the case of a sale, the valuation subject (seller) receives the price paid by the buyer. The aim is to find the lowest price limit as the decision value from the perspective of the presumptive seller. This value corresponds to the lowest price that is still acceptable to the seller. With the sale of the business and its withdrawal of the investment and financing program, the target function value from the base program at least must be regained. However, in contrast to the perspective of the buyer, the valuation object must not be included in the modified optimal investment and financing program from the perspective of the seller. 2.3 Determination of One-Dimensional Decision Values 99 45520_Matschke_Griffleiste_SL5.indd 99 45520_Matschke_Griffleiste_SL5.indd 99 16.03.2021 16: 20: 55 16.03.2021 16: 20: 55 2.3.3.2.2 The Model from the Presumptive Buyer’s Perspective 2.3.3.2.2.1 Presentation The next issue is the determination of the base program and the valuation program for a presumptive buyer with the state marginal price model in a one-dimensional, disjoint conflict situation. The price serves as the only conflict-resolution-relevant fact. The objective is purely financial, namely the maximization of withdrawals or payouts (H ERING 2014, p. 49). While the model in Section 2.3.3.1 abstracted from the time dimension, now a multi-period model is considered (M ATSCHKE / B RÖSEL / M ATSCHKE 2010). The planning horizon is finite and extends over T periods. The buyer can make investment and financial decisions at all times. Additionally, it is assumed that there there is precisely one investment and financing opportunity that is not limited to a specific amount; other opportunities, however, are only available to a certain extent. Therefore, capacity constraints per investment or financing object must also be noted. Even the consideration of an instance of cash management could be represented by the payment sequence (-1, 1). The (positive and negative) payments g Kjt per unit of the investment and financing object j, realized at time t, are known and independent of their type of combination in an investment and financing program. This means that the hypothesis of linearity hold true and that there are no synergies between the objects due to a specific constellation. At all times of the planning horizon, a payment balance of any amount that is decision-independent is expected. These decision-irrelevant payments b Kt may be positive, negative, or zero. The buyer is looking for a broad stream of withdrawals EN that serves as the target function for consumption purposes. Nonetheless, the desired withdrawals follow a predefined structure as an expression of the time preference of the buyer. The actual withdrawals at time t are determined by the realized size of the withdrawal stream EN, which results from the base program, multiplied with the so-called temporal structural factor w Kt . Formally speaking, it is w Kt · EN for t = 0, 1, …, T. Regarding the desired temporal structure, there are no restrictive assumptions so that any kind of withdrawal (payout), for instance, constant, increasing, decreasing, or irregular can be modeled within the predefined structure. By using the weighting factor w KT at the end of the planning horizon T, a sufficient amount of terminal assets can be ascertained. In practice, the allows the business to continue beyond the planning horizon T. If for example, w KT = a + 1/ i was given, this would mean that at time T a withdrawal in the amount of w KT · EN = a · EN + EN/ i was possible. While a · EN represents the amount of consumption at time T, EN/ i is equal to the present value of a perpetuity. With both EN and i constant, it leads to a perpetual annuity stream EN in the future that is discounted with the interest rate i. Since investment and financing decisions are not only made at the beginning of the planning horizon, but possibly also at each following point in time t > 0, the ability to pay at all times must be upheld by restrictions in later points in time. Then, the following mathematical model for the determination of the base program from the buyer’s perspective can be designed: x Kj max Kapitel 1: Einführung 100 100 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 100 45520_Matschke_Griffleiste_SL5.indd 100 16.03.2021 16: 20: 55 16.03.2021 16: 20: 55 Chapter 2 Target function: Restrictions: (1) Liquidity restrictions (safeguarding of the ability to pay at all times): The sum of the cash flows from realized investment and financing objects and from decisionindependent payouts ≥ withdrawals • at t = 0: Already at t = 0, an amount of may be withdrawn. b K0 can be interpreted as initially available capital endowment. • at t = 1, 2, …, T: The structure of the desired withdrawals in the future is w K1 : w K2 : … : w KT-1 : w KT . b Kt can be interpreted as equity capital increases planned for the future, but also as autonomous future payment obligations. Still, b Kt = 0 is also permitted. (2) Capacity limits: The number of realized investment and financing objects ≤ the upper capacity limit for j =1, 2, …, J: (3) Non-negativity conditions: The result of this approach is the base program of the buyer with maximum size of the expected withdrawal stream . The expected withdrawals have the amount of at specific times t. The investments and financings to be realized form the buyer’s base program. A rationally acting buyer could pay every price P ≤ P max for the business, from which the buyer gains at least the same utility after the purchase as from the base program. The utility of the base program for the buyer is represented by the vector of expected withdrawals. They are only determined by because of the predefined structure factors For the determination of the valuation program, it is therefore sufficient to presuppose that the size of the withdrawal stream of the valuation program is at least as high as The following approach leads to the determination of the valuation program and the decision value of a buyer if the future cash flows from the business at t are summarized as the payment vector g UK = (0; g UK1 ; g UK2 ; …; g UKT ). Note that at t = 0 the negotiable price P would be additionally due. EN K Ba → max! - g Kj0 j=1 J ∑ ⋅ x Kj + w K 0 ⋅ EN K Ba ≤ b K 0 w K 0 ⋅ EN K Ba - g Kjt j=1 J ∑ ⋅ x Kj + w Kt ⋅ EN K Ba ≤ b Kt x Kj ≤ x Kj max x Kj ≥ 0 EN K Ba ≥ 0. EN K Ba max w Kt ⋅ EN K Ba max (w K 0 ⋅ EN K Ba max ; w K1 ⋅ EN K Ba max ; w K 2 ⋅ EN K Ba max ; …; w K T ⋅ EN K Ba max ) EN K Ba max w Kt . EN K Be EN K Ba max . 2.3 Determination of One-Dimensional Decision Values 101 45520_Matschke_Griffleiste_SL5.indd 101 45520_Matschke_Griffleiste_SL5.indd 101 16.03.2021 16: 20: 55 16.03.2021 16: 20: 55 Target function: P → max! Restrictions: (1) Liquidity restrictions (safeguarding of ability to pay at all times): The sum of cash flows (income) from investment and financing objects and from decision-independent payments as well as cash flows from the business ≥ withdrawals • at t = 0: • at t = 1, 2, …, T: (2) Compliance with withdrawal stream of the base program: (3) Capacity limits: The number of realized investment and financing objects ≤ the upper capacity limit for j =1, 2, …, J: (4) Non-negativity conditions: Hence, the case of the buyer being subsidized by the seller (negative purchase price) is explicitly excluded in this model. On the one hand, the optimal solution of this approach leads to the ceiling price P max , that is, the decision value from the buyer’s perspective. On the other hand, the respective investment and financing program is revealed that buyers should realize if they would have to pay a price equal to their decision value. This program is the valuation program of the buyer. 2.3.3.2.2.2 Numerical Example The process for determining the decision value using the state marginal price model from the perspective of a presumptive buyer is now illustrated by an example (B RÖSEL / M ATSCHKE 2004, M ATSCHKE / B RÖSEL / M ATSCHKE 2010) with a multi-period planning period (T = 4) under the assumption of (quasi-)certain expectations. On incomplete markets, certainty means that the decision subject waives the consideration of multi-valued expectations and only expects a specific data constellation (F RANKE / H AX 2009, p. 148 and 245). The quasi-certainty is influenced by the individual expectations of the valuation subject and therefore strictly subjective. Hence, for the valuation subject the decision field is then completed. It is assumed in an uncertain environment that all action alternatives including their payment conditions can be anticipated. However, errors are not ruled out by definition. Due to the absence of complete foresight of the decision subject, (subjective) certainty should be regarded in the context of a definite planning horizon. The valuation subject already owns a small enterprise KU (German: Kleines Unternehmen) during the valuation date t = 0 that is also the decision and acquisition date. - g Kj0 j=1 J ∑ ⋅ x Kj + P+ w K 0 ⋅ EN K Be ≤ b K 0 - g Kjt j=1 J ∑ ⋅ x Kj + w Kt ⋅ EN K Be ≤ b Kt + g UKt EN K Ba max EN K Be ≥ EN K Ba max x Kj ≤ x K j max x Kj ≥ 0 P ≥ 0. Kapitel 1: Einführung 102 102 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 102 45520_Matschke_Griffleiste_SL5.indd 102 16.03.2021 16: 20: 55 16.03.2021 16: 20: 55 Chapter 2 The valuation subject manages KU as general manager and achieves a perpetuity from internal financing (IF) in the amount of 30 GE. At t = 0, the valuation subject has the opportunity to invest AK. The series of cash flows from this investment including its price payable is given (-100 GE, +30 GE, +40 GE, +50 GE, +55 GE). At t = 0, the valuation subject owns equity assets (EM) from (personal) family resources in the amount of 10 GE. It is further assumed that the primary (main) bank of the general manager makes available a bullet loan ED in the amount of 50 GE at t = 0 at an annual interest rate of 8 % p. a. with a total term of four periods. Additional financial funds are available as operating loans (KA t ) in unlimited amounts at a short-term borrowing (interest) rate of 10 % p. a. Financial investments (GA t ) may be made in the main bank of the general manager in any amounts at an interest rate of 5 % p. a. The valuation subject generally aims to achieve an uniform income stream (income maximization) to safeguard its existence. At T = 4, we obtain (H ERING 2014, p. 50) so that the desired time structure reads: w K0 : w K1 : w K2 : w K3 : w K4 = 1 : 1 : 1 : 1 : 21. This means that the last payout should not only comprise the regular (annual) payment, but also the present value of the perpetuity at an interest rate of 5 %. This is necessary to guarantee the income EN after the (definite) planning period, because for t > 4 the estimated interest rate of i = 5 % p. a. is taken into account in the example. The valuation subject may acquire another business U (German: Unternehmen) starting at t = 0. For this business the payment stream (0 GE, 60 GE, 40 GE, 20 GE, 20 GE) is estimated for the planning horizon. Additionally, a perpetuity of 20 GE is expected from it starting in t = 5. The aim is to determine the maximum payable price P max for the business U. The table below summarizes the data from the example. In order not to omit vertical interdependencies between the selected planning period and the periods beyond the planning horizon, factor 21 was taken into account. This relates to both the perpetual cash surplus from internal financing and the perpetuity from the business U to be valuated starting at time t = 5. The amounts of 630 GE (= 30 GE · 21) at IF and 420 GE (= 20 GE · 21) at U taken into account in t = 4 accordingly include the payment in t = 4 and the perpetual annuity expected from t = 5 onwards at a calculation interest rate of 5%. In other words, at time t = 4 payments comprise the periodic amount and the perpetuity. The payments expected after time t > T = 4 are therefore taken into account, again at an estimated interest rate of i = 5 % p. a. (see Figure 2.20). w T ⋅ EN = EN + EN i ⇒ w T = 1 + 1 i = 1 + 1 0,05 = 21, t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U 01 -100 30 50 -4 -1 1,05 -1 1 -1,1 1 10 30 30 P? 60 234 Limit 40 50 -4 -4 55 1 -54 1 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ 30 30 1 630 1 40 20 420 1 Grenze Figure 2.20: Data of the example from the buyer’s perspective 1 1 1 1 1 2.3 Determination of One-Dimensional Decision Values 103 45520_Matschke_Griffleiste_SL5.indd 103 45520_Matschke_Griffleiste_SL5.indd 103 16.03.2021 16: 20: 56 16.03.2021 16: 20: 56 To determine the base program, the existing data must be used to formulate a linear optimization approach that can be solved using the simplex algorithm: From the base program originates a maximum uniform income stream of size EN max = 32,6176 GE. The future value amounts to 652,3520 GE at an interest rate of 5 % p. a. This represents the result of the perpetuity using EN max . The investment AK is to be realized. To this end, internal financing IF, equity assets EM and the bullet loan ED as well as the operating loans KA at t = 0 and t = 1 are used. At t = 3, single period financial investments GA are made. The payment balance at times t = 1, 2, 3 equals 0 so that the liquidity restriction is maintained. At t = 4, a cash flow of 652,3520 GE results after the deduction of the withdrawal in the amount of EN max . The comprehensive financial plan of the base program is presented below in Figure 2.21: If company U is included in the valuation program, the size of the uniform income stream of the base program must at least be reached again. To determine the valuation program, the linear approach to be formulated must again be solved using the simplex algorithm: EN → max! 100 ⋅ AK - 50 ⋅ ED + 1 ⋅ GA 0 - 1 ⋅ KA 0 + 1 ⋅ EN ≤ 40 -30 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 - 1 ⋅ KA 1 + 1 ⋅ EN ≤ 30 -40 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 - 1 ⋅ KA 2 + 1 ⋅ EN ≤ 30 -50 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 - 1 ⋅ KA 3 + 1 ⋅ EN ≤ 30 -55 ⋅ AK + 54 ⋅ ED - 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN ≤ 630 GA 0 , GA 1 , GA 2 , GA 3 , KA 0 , KA 1 , KA 2 , KA 3 , EN ≥ 0 AK, ED ≤ 1. Personal equity assets EM Internal financing IF t = 0 t = 1 10 30 30 Investment AK Bullet loan ED Operating loan KA Financial investments GA -100 42,7680 30 -3,4214 49,8496 30,8736 t = 2 t = 3 30 30 t = 4 630 40 -3,4214 50 -3,4214 -43,9610 55 -46,1894 KA-, GA-paybacks Withdrawal EN Payment balance Debt from KA -32,6176 -54,8346 -32,6176 0 49,8496 0 30,8736 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.21: Comprehensive financial plan of the buyer’s base program -33,9610 -32,6176 -32,6176 0 0 46,1591 -32,6176 652,3520 43,9610 652,3520 Kapitel 1: Einführung 104 104 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 104 45520_Matschke_Griffleiste_SL5.indd 104 16.03.2021 16: 20: 56 16.03.2021 16: 20: 56 Chapter 2 The determined marginal price P max for company U is 391,4550 GE. At t = 0, the valuation subject invests in company U as well as - already in the base program - in object AK. In addition to internal financing IF, personal equity assets EM, the bullet loan ED, and the short-term operating loans KA are used in all planning periods. The comprehensive financial plan of the valuation program is represented in the following Figure 2.22. The process below shows that the decision value can be determined as the maximum payable price from the buyer’s perspective by subtracting the data of the valuation program from those of the base program. One can recognize which modifications have to be made to calculate the valuation program from the base program. The differences between those two determine the “comparison object” within the business valuation theory (see Figure 2.23). For any time t > 0, the stream of cash flows of the comparison object corresponds to the payment stream of the company to be valuated so that there is “profit equality” between valuation and comparison object. It is the mirror-image of the business to be valuated because the buyer waives the payment stream of the comparison object if the valuation object is acquired. The funds of the comparison object released at t = 0 are the upper price limit for the company to be valuated. If the payment streams of P → max! 100 ⋅ AK - 50 ⋅ ED + 1 ⋅ GA 0 - 1 ⋅ KA 0 + 1 ⋅ EN + P ≤ 40 -30 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 - 1 ⋅ KA 1 + 1 ⋅ EN ≤ 90 -40 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 - 1 ⋅ KA 2 + 1 ⋅ EN ≤ 70 -50 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 - 1 ⋅ KA 3 + 1 ⋅ EN ≤ 50 -55 ⋅ AK + 54 ⋅ ED - 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN ≤ 1.050 EN ≥ 32,6176 GA 0 , GA 1 , GA 2 , GA 3 , KA 0 , KA 1 , KA 2 , KA 3 , P ≥ 0 AK, ED ≤ 1. t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 30 30 30 630 Business U Investment AK Bullet loan ED Operating loan KA -100 60 30 50 434,0726 -4 394,0975 40 40 20 50 -4 360,1248 -4 332,7549 420 55 -54 Financial investments GA KA-paybacks Withdrawal EN Payment balance -477,4799 -32,6176 391,4550 -32,6176 0 Debt from KA Deposits from GA Present Value of perpetuity EN/ 0,05 Figure 2.22: Comprehensive financial plan of the buyer’s valuation program 434,0726 394,0975 -433,5073 -396,1373 -32,6175 0 -32,6176 0 -366,0304 -32,6176 652,3520 360,1248 332,7549 652,3520 2.3 Determination of One-Dimensional Decision Values 105 45520_Matschke_Griffleiste_SL5.indd 105 45520_Matschke_Griffleiste_SL5.indd 105 16.03.2021 16: 20: 56 16.03.2021 16: 20: 56 the valuation and comparison object are added, payment balances are equalized for any t > 0 since the profitability is the same. However, at t = 0 a payment balance exists, namely in the amount of the decision value. The comparison object in the example are additional operating loans at t = 0, t = 1, t = 2 and t = 3 as well as not executed financial investments at t = 2 and t = 3. The expected future cash flows from the additional third-party funds and from suppressed investments correspond to the cash flows of the company. Therefore, the payment balance of the comparison object resulting at the time t = 0 reflects the ceiling price. From this t = 0 t = 1 t = 2 t = 3 t = 4 Valuation program of the buyer Personal equity assets EM 10 Internal financing IF Business U Investment AK Bullet loan ED 30 30 60 -100 50 30 -4 30 40 30 20 40 -4 50 -4 630 420 55 -54 Operating loan KA Financial investments GA KA-paybacks Withdrawal EN 434,0726 394,0975 -32,6176 -477,4799 -32,6176 Payment balance ./ . Base program of the buyer Personal equity assets EM 391,4550 0 10 360,1248 332,7549 -433,5073 -32,6175 -396,1373 -32,6176 -366,0304 -32,6176 0 0 652,3520 Internal financing IF Investment AK Bullet loan ED Operating loan KA 30 -100 30 30 42,7680 49,8496 -3,4214 30,8736 Financial investments GA KA-, GA-paybacks Withdrawal EN Payment balance -54,8346 -32,6176 0 -32,6176 0 30 40 30 50 -3,4214 -3,4214 630 55 -46,1894 -33,9610 -43,9610 -32,6176 0 -32,6176 0 46,1591 -32,6176 652,3520 = Comparison object (differences between both programs; modifications of the programs) ∆ Personal assets EM ∆ Internal financing IF 0 0 0 0 ∆ Investment AK ∆ Bullet loan ED ∆ Operating loan KA ∆ Financial investments GA 0 7,2320 0 -0,5786 384,2230 0 363,2239 0 0 0 0 0 0 0 0 -0,5786 0 -0,5786 360,1248 0 332,7549 43,9610 0 -7,8106 0 0 ∆ KA-, GA-paybacks ∆ Withdrawal EN = Payment balance of the modifications (comparison object) Business U 0 0 -422,6453 0 391,4550 -60 60 Decision value P max Figure 2.23: Determination of the buyer’s comparison object 391,4550 0 -399,5463 0 -396,1373 0 -40 40 -20 20 -412,1894 0 -420 420 0 0 0 Kapitel 1: Einführung 106 106 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 106 45520_Matschke_Griffleiste_SL5.indd 106 16.03.2021 16: 20: 57 16.03.2021 16: 20: 57 Chapter 2 “price” of the comparison object the buyer’s decision value P max of 391,4550 GE can be derived. The internal rate of return of the comparison object from the perspective of the buyer is r K = 0,098364 (= 9,8364 %). The decision-oriented interpretation of the term “comparison object” has nothing to do with a “comparable” business as the term is often erroneously understood in the literature. Accordingly, when determining the decision value the issue is not to find a business comparable to the business to be valuated. The “comparison object” represents all measures of the transformation of the base program into the valuation program instead. If the expected payments from the business are discounted at the internal rate of return of the “comparison object”, the result is a future performance value (or present value) that is equal to the upper price limit, that is, the decision value (see Figure 2.24). 2.3.3.2.3 The Model from the Presumptive Seller’s Perspective 2.3.3.2.3.1 Presentation The mathematical approach (H ERING 2014, p. 74) for the determination of the base program from the seller’s perspective varies formally from the one of the buyer’s only in the aspect that the expected payments g UV = (0; g UV1 ; g UV2 ; …; g UVT ) of the business to be valuated are included. They could also be regarded as part of autonomous payments b Vt , but do not have to be modeled separately: Target function: Restrictions: (1) Liquidity restrictions (safeguarding of ability to pay at all times): The sum of the cash flows from investment and financing objects and from decision-independent payments (including those from the valuation object) ≥ withdrawals • at t = 0: Already at t = 0, an amount of may be withdrawn. b V0 can be interpreted as the initially available capital endowment of the seller. t 1 2 3 4 Business U r K 0,098364 60 40 20 420 (1 + r K ) - t (Periodic) Present value Future performance value/ Present value Figure 2.24: Determination of the decision value from the buyer’s perspective based on the internal rate of return of the comparison object 1 0,910445 54,6267 391,4550 0,828910 33,1564 0,754677 15,0935 0,687091 288,5784 EN V Ba → max! - g Vj0 j=1 J ∑ ⋅ x Vj + w V 0 ⋅ EN V Ba ≤ b V 0 w V 0 ⋅ EN V Ba 2.3 Determination of One-Dimensional Decision Values 107 45520_Matschke_Griffleiste_SL5.indd 107 45520_Matschke_Griffleiste_SL5.indd 107 16.03.2021 16: 20: 57 16.03.2021 16: 20: 57 • at t = 1, 2, …, T: The structure of the desired withdrawals in the future is represented by w V1 : w V2 : … : w VT-1 : w VT . b Vt includes not only payments of the valuation object, but also possible future equity capital increases or autonomous future payment obligations. (2) Capacity limits (restrictions of quantity of the investment and financing objects): The number of the investment and financing objects ≤ upper capacity limit for j = 1, 2, …, J: (3) Non-negativity conditions: The result of this approach is the base program of the presumptive seller with the expected withdrawal stream with the maximum size of The realizable withdrawals at different points in time t are and follow the desired temporal structure. Then, the investment and financing objects represent the base program of the seller including the enterprise. A rational seller is willing to sell the company to be valued for a price P if, after the sale at this price, he again achieves at least the same (high) utility as in the base program, that is, P must be at least equal to his minimum requirement. Hence, the aim is to determine the lowest price that fulfills this condition. The utility of the base program is expressed by the following vector of expected withdrawals without the sale of the company Due to the defined and desired temporal structure, it is necessary for the determination of the base program that the size of the withdrawal stream from the valuation program is at least as high as After that, the following approach for the determination of the valuation program and of the decision value from the perspective of the seller can be established if the expected payments of the company at time t are defined as g UV = (0; g UV1 ; g UV2 ; …; g UVT ). Note that at t = 0 the still to be negotiated price P for the business must also be considered: Target function: P → min! Restrictions: (1) Liquidity restrictions (safeguarding of ability to pay at all times): The sum of the cash flows from investment and financing objects and from decision-independent payments as well as from the price for the business to be valuated (at t = 0) ≥ withdrawals - g Vjt j=1 J ∑ ⋅ x Vj + w V t ⋅ EN V Ba ≤ b V t x Vj ≤ x Vj max x Vj ≥ 0 EN V Ba ≥ 0. EN V Ba max . w V t ⋅ EN V Ba max (w V 0 ⋅ EN V Ba max ; w V1 ⋅ EN V Ba max ; w V 2 ⋅ EN V Ba max ; …; w V T ⋅ EN V Ba max ). EN V Be EN V Ba max . Kapitel 1: Einführung 108 108 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 108 45520_Matschke_Griffleiste_SL5.indd 108 16.03.2021 16: 20: 57 16.03.2021 16: 20: 57 Chapter 2 • at t = 0: • at t = 1, 2, …, T: (2) Compliance with withdrawal stream of the base program: (3) Capacity limits (restrictions of quantity of the investment and financing objects): The number of the realized investment and financing objects ≤ upper capacity limit for j = 1, 2, …, J: (4) Non-negativity conditions: The decision value P min and the structure of the valuation program from the perspective of the seller perspective are represented in the optimum solution. 2.3.3.2.3.2 Numerical Example The process of the determination of the decision value from the seller’s perspective can also to be explained with the aid of a simple intuitive example (B RÖSEL / M ATSCHKE 2003, p. 2241) with a multi-period planning horizon (T = 4) assuming (quasi-)certain expectations. The valuation subject, now the presumptive seller, already owns a small enterprise (KU; German: Kleines Unternehmen) and a medium enterprise (MU; German: Mittleres Unternehmen) at the valuation date. Both companies operate independently of each other. The seller is also the general manager (GF; German: Geschäftsführer) of both companies and receives a perpetuity from internal financing (IF; German: Innenfinanzierung) in the amount of 30 GE at each time t. The high workload involved in being general manager of both enterprises has persuaded the seller to dispose of enterprise KU, which has the following payment sequence of (0 GE, 12 GE, 11 GE, 12 GE, 10 GE) and from t = 5 onwards the perpetuity of 10 GE. At t = 0, the valuation subject has the opportunity to make an investment AK in the remaining company MU. The payment stream of this investment is (-100 GE, +30 GE, +40 GE, +50 GE, + 55 GE), including its initial investment. At the time of decision, the valuation subject owns personal equity assets (EM) from family resources in the amount of 10 GE. At t = 0, it is also assumed that the primary bank of the general manager makes available a bullet loan ED in the amount of 50 GE at an annual interest rate of 8 % with a total term of four periods (years). Additional financial funds are available as operating loans in unlimited amounts at a short-term borrowing (interest) rate of 10 % p. a. (KA t ). Moreover, financial investments (GA t ) may be made in the main bank of the general manager in any amounts at an interest rate of 5 % p. a. The table below summarizes the data of the example (see Figure 2.25). - g Vj0 j=1 J ∑ ⋅ x Vj - P+ w V 0 ⋅ EN V Be ≤ b V 0 - g Vjt j=1 J ∑ ⋅ x Vj + w Vt ⋅ EN V Be ≤ b Vt - g UVt EN V Ba max EN V Be ≥ EN V Ba max x Vj ≤ x Vj max x Vj ≥ 0 P ≥ 0. 2.3 Determination of One-Dimensional Decision Values 109 45520_Matschke_Griffleiste_SL5.indd 109 45520_Matschke_Griffleiste_SL5.indd 109 16.03.2021 16: 20: 58 16.03.2021 16: 20: 58 To safeguard its existence, the valuation subject aims to achieve a uniform income stream, which anticipates the identical withdrawals EN at t = 1, 2, 3, 4. Moreover the (normal) payout EN, at T = 4 the present value of the perpetuity, that is, EN is discounted at a rate of 5 % p. a., is to be obtained. The perpetuity is necessary to guarantee the income EN after the planning horizon. The desired time structure reads: w V0 : w V1 : w V2 : w V3 : w V4 = 1 : 1 : 1 : 1 : 21. The aim is to determine the price P min that has at least to be demanded for the enterprise KU. To compute the base program, the existing data are used to formulate a linear optimization approach that can be solved using the simplex algorithm: The comprehensive financing plan of the base program of the seller is shown below (see Figure 2.26): t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF thereof 0 -100 50 -1 1 10 30 MU 30 KU P? 1234 30 40 -4 -4 50 55 -4 -54 1,05 -1 1,05 -1 1,05 -1 1,05 -1,1 1 -1,1 1 -1,1 1 -1,1 30 30 30 630 18 19 12 11 18 420 12 210 limit Grenze Figure 2.25: Data of the example from the seller’s perspective 11 11 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 1 1 11 11 11 EN → max! 100 ⋅ AK - 50 ⋅ ED + 1 ⋅ GA 0 - 1 ⋅ KA 0 + 1 ⋅ EN ≤ 40 -30 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 - 1 ⋅ KA 1 + 1 ⋅ EN ≤ 30 -40 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 - 1 ⋅ KA 2 + 1 ⋅ EN ≤ 30 -50 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 - 1 ⋅ KA 3 + 1 ⋅ EN ≤ 30 -55 ⋅ AK + 54 ⋅ ED - 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN ≤ 630 GA 0 , GA 1 , GA 2 , GA 3 , KA 0 , KA 1 , KA 2 , KA 3 , EN ≥ 0 AK, ED ≤ 1. Kapitel 1: Einführung 110 110 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 110 45520_Matschke_Griffleiste_SL5.indd 110 16.03.2021 16: 20: 58 16.03.2021 16: 20: 58 Chapter 2 As one can see from the base program, a uniform income stream 32,6176 GE is attained. At the end of the planning period (t = 4), the payment balance amounts to 652,3520 GE. The very same income stream is reached if is discounted at 5 % p. a. This is the present value of the perpetuity. With regard to the base program, the investment AK should be made in the business MU. For this purpose, the internal financing IF as well as the personal equity assets EM are used in full and the bullet loan ED is used at 85.5360 %. While single-period working capital loans KA are taken out at t = 0 and t = 1, a single-period financial investment GA is made at t = 3. The size of the uniform income stream of the base program must at least be reached again by the valuation program, when the enterprise KU is sold. For the determination of the valuation program, the simplex algorithm is used again: The lowest price P min that needs to be demanded for the enterprise KU is 196,2159 GE. This price represents the decision value from the perspective of the seller. The valuation object KU is no longer part of the valuation program. Instead, it is structured as follows: The valuation subject, who disposes of the perpetuity of 32,6176 GE, invests in the object AK at t = 0 and receives (at least) the limit price for KU. Moreover, the seller uses personal equity assets EM and internal financing IF, the latter now limited to MU. At all times t, money can be invested with an interest rate of 5 %. The comprehensive t = 0 t = 1 t = 2 t = 3 t = 4 Personal assets EM Internal financing IF 10 30 30 30 30 630 Investment AK Bullet loan ED Operating loan KA Financial investments GA -100 42,7680 30 -3,4214 49,8496 30,8736 40 -3,4214 50 -3,4214 -43,9610 55 -46,1894 KA-, GA-paybacks Withdrawal EN Payment balance Debt from KA -32,6176 -54,8346 -32,6176 0 49,8496 0 30,8736 Deposits from GA Present value of the perpetuity EN/ 0,05 Figure 2.26: Comprehensive financing plan of the seller’s base program -33,9610 -32,6176 -32,6176 0 0 46,1591 -32,6176 652,3520 43,9610 652,3520 EN V Ba max = EN V Ba max P → min! 100 ⋅ AK - 50 ⋅ ED + 1 ⋅ GA 0 - 1 ⋅ KA 0 + 1 ⋅ EN - P ≤ 40 -30 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 - 1 ⋅ KA 1 + 1 ⋅ EN ≤ 18 -40 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 - 1 ⋅ KA 2 + 1 ⋅ EN ≤ 19 -50 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 - 1 ⋅ KA 3 + 1 ⋅ EN ≤ 18 -55 ⋅ AK + 54 ⋅ ED - 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN ≤ 420 EN ≥ 32,6176 GA 0 , GA 1 , GA 2 , GA 3 , KA 0 , KA 1 , KA 2 , KA 3 , P ≥ 0 AK, ED ≤ 1. 2.3 Determination of One-Dimensional Decision Values 111 45520_Matschke_Griffleiste_SL5.indd 111 45520_Matschke_Griffleiste_SL5.indd 111 16.03.2021 16: 20: 59 16.03.2021 16: 20: 59 financial plan of the valuation program is outlined in the following table (see Figure 2.27). Subsequently, it will be possible to outline which measures constitute the “comparison object” from the perspective of the seller This is, again, achieved by subtracting the values of the valuation program from those of the base program (see Figure 2.28). t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 30 30 30 630 Business KU Investment AK Bullet loan ED Operating loan KA -100 -12 30 -11 40 -12 50 -210 55 Financial investments GA GA-paybacks Withdrawal EN Payment balance -103,5983 -124,1607 108,7782 -32,6176 -196,2159 -32,6176 0 Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.27: Comprehensive financial plan of the seller’s valuation program 103,5983 124,1607 -156,7511 130,3687 -199,9711 164,5887 -32,6176 0 -32,6176 0 209,9696 -32,6176 652,3520 156,7511 199,9711 652,3520 Kapitel 1: Einführung 112 112 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 112 45520_Matschke_Griffleiste_SL5.indd 112 16.03.2021 16: 20: 59 16.03.2021 16: 20: 59 Chapter 2 The comparison object comprises the bullet loan ED not to be included at t = 0 anymore, the displaced operating loans KA at t = 0 and t = 1, and additional financial investments GA at t = 0, t = 1, t = 2, and t = 3. The expected future payments from displaced loans and newly included investments correspond in their amounts to the future cash flows of the company KU that are waived due to its sale. In other words, profit equality between valuation and comparison object is given once again since the payments of valuation and comparison object balance out at t > 0. Only at time t = 0, a difference remains, namely the decision value. The “price” of the comparison object defines the decision value P min of the seller of the business KU in the amount of t = 0 t = 1 t = 2 t = 3 t = 4 Base program of the seller Personal equity assets EM 10 Internal financing IF Investment AK Bullet loan ED Operating loan KA 30 -100 30 30 42,7680 49,8496 -3,4214 30,8736 30 40 30 50 -3,4214 -3,4214 630 55 -46,1894 Financial investments GA KA-, GA-paybacks Withdrawal EN Payment balance -54,8346 -32,6176 0 -32,6176 0 ./ . Valuation program of the seller Personal equity assets EM Internal financing IF 10 30 30 -33,9610 -43,9610 -32,6176 0 -32,6176 0 46,1591 -32,6176 652,3520 30 30 630 Business KU Investment AK Bullet loan ED Operating loan KA -100 -12 30 Financial investments GA GA-paybacks Withdrawal EN Payment balance -103,5983 -124,1607 108,7782 -32,6176 -196,2159 -32,6176 0 -11 40 -12 50 -210 55 -156,7511 130,3687 -199,9711 164,5887 -32,6176 0 -32,6176 0 209,9696 -32,6176 652,3520 = Comparison object (differences between both programs; modifications of the programms) ∆ Personal assets EM ∆ Internal financing IF 0 0 0 0 ∆ Investment AK ∆ Bullet loan ED ∆ Operating loan KA ∆ Financial investments GA 0 42,7680 0 -3,4214 49,8496 103,5983 30,8736 124,1607 0 0 0 0 0 0 0 -3,4214 0 -3,4214 0 156,7511 0 156,0101 0 -46,1894 0 0 ∆ KA-, GA-paybacks Withdrawal EN = Payment balance of the modifications (comparison object) Business KU 0 0 -163,6128 0 196,2159 -12 12 Decision Value P min Figure 2.28: Determination of the seller’s comparison object 196,2159 0 -164,3297 0 -164,5887 0 -11 11 -12 12 -163,8106 0 -210 210 0 0 0 2.3 Determination of One-Dimensional Decision Values 113 45520_Matschke_Griffleiste_SL5.indd 113 45520_Matschke_Griffleiste_SL5.indd 113 16.03.2021 16: 20: 59 16.03.2021 16: 20: 59 196,2159 GE. The internal rate of return of the comparison object from the seller’s perspective is r V = 0,061920 (= 6,1920 %). If the expected cash flows from the business are discounted at that rate, the future performance value (present value) results equate to the lowest price (bottom price) that must at least be demanded (see Figure 2.29). 2.3.3.2.4 Consideration of Uncertainty The future expectations of the valuation subject under uncertainty are characterized by polyvalence. As explained in Section 2.3.1.2.2, polyvalent expectations can be considered either by using methods revealing uncertainty or by methods consolidating uncertainty. Since it is the aim of the valuation within the context of the decision function to provide sound bases for decision-making for the valuation subject, the methods revealing uncertainty should be preferred compared to the methods consolidating uncertainty. The valuation of (quasi-)certain payment streams was exemplarily demonstrated, using the deterministic variant of the state marginal price model. As already outlined, the initial model is transferred into a structurally equal general state marginal price model if points in time are generalized as states. Under consideration of a system of restrictions, the state marginal price model can then be used for the valuation of (randomly structured) uncertain cash flows. However, regarding the total analysis, it will now be shown that the sensitivity analysis can easily be combined with the deterministic variant of the state marginal price model (B RÖSEL 2002, p. 124). The sensitivity analysis, which is used to examine the sensitivity of valuation results (or generally: planning results) by changing input data (valuation parameters), can be distinguished into two types (H ERING 2017, p. 322): 1. Regarding the question of the limits in which the input data of the model can vary without changing the structure of the optimum solution, it is generally spoken about the sensitivity analysis of the first type. Answering this question, the analysis provides information about the stability of the optimum solution. However, the aim is to determine the critical value for uncertain input parameters of the planning problem. The simplest case of a sensitivity analysis of the first type is given when the fluctuation range of a single coefficient (parameter) is to be determined. All the other coefficients remain unaltered. This is also known as the ceteris paribus assumption. Regarding the total model at hand, as long as the (tableau) coefficients of the base variables are not influenced by this analysis, such isolated fluctuation ranges can be t 0 1 2 3 4 Business KU r V 0,061920 12 11 12 210 (1 + r V ) - t (Periodic) Present value Future performance value/ Present value Figure 2.29: Determination of the decision value from the seller’s perspective based on the internal rate of return of the comparison object 1 0,941691 11,3003 196,2159 0,886782 9,7546 0,835074 10,0209 0,786382 165,1401 Kapitel 1: Einführung 114 114 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 114 45520_Matschke_Griffleiste_SL5.indd 114 16.03.2021 16: 20: 59 16.03.2021 16: 20: 59 Chapter 2 easily computed. However, if the (tableau) coefficients of the base variables are the center of attention, the determination of isolated fluctuation ranges becomes increasingly complex. By the same token, if the sensitivity analysis of the first type is multi-parametric, the computation causes great difficulties and quickly becomes incomprehensible. Considering that real valuation problems usually have more than one uncertain coefficient, the sensitivity analysis of the first type is an inadequate solution procedure for the current problem. 2. The sensitivity analysis of the second type aims to answer the question of which new optimum solution results from the modification of one or more coefficients. The analysis aims to identify the range of variation of the optimum solution. It also examines how the alternative data constellation affects the structure of the optimum solution. Generally speaking, there are two different possibilities: The linear optimization approach resulting from the modified database can be solved again using the simplex algorithm; or the optimum solution can be determined from the existing optimum tableau. The sensitivity analysis of the second type is now illustrated in the total model using a plain example. Due to the polyvalent structure of the expectations, the determination of the decision value becomes increasingly complex. Therefore, it is reasonable for practical reasons to estimate the future performances, to limit them to a realistic, a pessimistic, and an optimistic variant, and to compute their respective decision values. Such an approach corresponds to a straightforward sensitivity analysis of the second type, because the influence of modified input data on the problem solution is shown. As a starting point of the examination, the numerical example of Section 2.3.3.2.2.2 is selected. It represents a buyer’s perspective in a non-dominated, disjoint, one-dimensional conflict situation of the type acquisition/ sale. The data illustrated in Figure 2.20 showcase a realistic version. After the solution of the linear optimization approach, a uniform income stream of the size = 32,6176 GE was determined in the base program; also see the comprehensive financial plan of the buyer’s base program in Figure 2.21. The size of this uniform income stream must at least be reattained after the inclusion of the business U if the price for U does not exceed 391,4550 GE. Accordingly, the marginal price for the business U is = 391,4550 GE for the realistic variant. For the calculations, refer to Section 2.3.3.2.2.2. See also especially the illustrated comprehensive financial plan of the valuation program in Figure 2.22. It is now assumed that two additional constellations of input data - a pessimistic and an optimistic variant - can be presented by sound estimates. Furthermore, neither a solely positive nor a solely negative development of the performances of the business U should be assumed. All input data must apply consistently. In other words, the payment streams of all objects of the model have to be determined considering uniform assumptions. If there is a positive correlation between the successes of the investment AK and those of the business U, it is not reasonable to use the payment stream of the investment AK for the determination of the decision value according to the realistic version, assuming decreasing cash flows of the business U. EN real max P max real 2.3 Determination of One-Dimensional Decision Values 115 45520_Matschke_Griffleiste_SL5.indd 115 45520_Matschke_Griffleiste_SL5.indd 115 16.03.2021 16: 21: 00 16.03.2021 16: 21: 00 In the pessimistic variant, the payment stream (0 GE, 60 GE, 35 GE, 15 GE, 19 GE) from t = 0 to t = 4 and a perpetuity in the amount of 19 GE starting at t = 5 are expected by the valuation subject for the valuation object “Business U”. The payment stream of the possible investments AK is including the price to be paid (- 100 GE, 25 GE, 30 GE, 40 GE, 50 GE). The cash flows from internal financing (IF) of the small business KU are 30 GE expected at t = 0 and 20 GE at any time t > 0. At the decision point, the valuation subject owns personal equity assets (EM) in the amount of 10 GE. Other than that, the decision field remains unaltered. The data illustrating the pessimistic variant for the determination of the highest affordable price (price ceiling) are summarized in Figure 2.30. To compute the ceiling price for the business U in the pessimistic scenario, the base program has be to modified and the resulting maximum target function value has to be determined. Finally, from the pessimistic base program reveals that a uniform income stream of the size = 21,8681 GE can be achieved (see Figure 2.31). After including the business U in the valuation program, the size of the uniform income stream must be at least permanently attainable. P max pess t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U 01 -100 25 50 -4 -1 1,05 -1 1 -1,1 1 10 30 20 P? 60 234 limit 30 40 -4 -4 50 1 -54 1 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ 20 20 1 420 1 35 15 399 1 Grenze Figure 2.30: Pessimistic data of the example from the buyer’s perspective 1 1 1 1 1 P max pess EN pess max EN pess max Personal equity assets EM Internal financing IF t = 0 t = 1 10 30 20 Investment AK Bullet loan ED Operating loan KA Financial investments GA -100 43,6501 25 -3,4920 38,218 22,3999 t = 2 t = 3 20 20 t = 4 420 30 -3,4920 40 -3,4920 -34,6399 50 -47,1421 0,873001319 KA-, GA-paybacks Withdrawal EN Payment balance Debt from KA -21,8681 -42,0398 -21,8681 0 38,2180 0 22,3999 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.31: Comprehensive financial plan of the buyer’s base program from a pessimistic perspective -24,6399 -21,8681 -21,8681 0 0 36,3719 -21,8681 437,3617 34,6399 437,3617 116 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 116 45520_Matschke_Griffleiste_SL5.indd 116 16.03.2021 16: 21: 00 16.03.2021 16: 21: 00 Chapter 2 After the determination of the valuation program, the marginal price of the business U can be determined for the pessimistic scenario of input data with = 368,8487 GE (see Figure 2.32). In the optimistic variant, the valuation subject ultimately expects a stream of cash flows (0 GE, 62 GE, 45 GE, 25 GE, 21 GE) during the periods t = 0 to t = 4. Moreover, a perpetuity starting at t = 5 in the amount of 21 GE is expected from valuation object, the business U. The payment stream for the additional possible investment AK, including its price, is (-100 GE, 35 GE, 45 GE, 55 GE, 60 GE). The valuation subject expects cash flows of 30 GE at t = 0 and of 40 GE at any time t > 0 from internal financing (IF) of the small enterprise KU. Again, the rest of the decision field remains unchanged. This means that at the decision date, the valuation subject owns personal equity assets (EM) in the amount of 10 GE and still has the option at t = 0 to obtain a bullet loan ED of 50 GE at an annual interest rate of 8 % p. a. with a total term of four periods. Moreover, financial funds as operating loans can be borrowed in unlimited amounts at a short-term borrowing (interest) rate of 10 % p. a. (KA t ). Moreover, the valuation subject may make financial investments (GA t ) in any amounts at an interest rate of 5 % p. a. Figure 2.33 summarizes the data of this variant for the determination of the maximum payable price of P max pess t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 20 20 20 420 Business U Investment AK Bullet loan ED Operating loan KA -100 60 25 50 400,7168 -4 361,6566 35 30 15 40 -4 338,6903 -4 323,4274 399 50 -54 Financial investments GA KA-paybacks Withdrawal EN Payment balance -440,7885 -21,8681 368,8487 -21,8681 0 Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.32: Comprehensive financial plan of the buyer’s valuation program from a pessimistic perspective 400,7168 361,6566 -397,8222 -372,5593 -21,8681 0 -21,8681 0 -355,7702 -21,8681 437,3617 338,6903 323,4274 437,3617 P max opt . t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U 01 -100 35 50 -4 -1 1,05 -1 1 -1,1 1 10 30 40 P? 62 234 Limit 45 55 -4 -4 60 1 -54 1 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ 40 40 1 840 1 45 25 441 1 Grenze Figure 2.33: Optimistic data of the example from the buyer’s perspective 1 1 1 1 1 2.3 Determination of One-Dimensional Decision Values 117 45520_Matschke_Griffleiste_SL5.indd 117 45520_Matschke_Griffleiste_SL5.indd 117 16.03.2021 16: 21: 00 16.03.2021 16: 21: 00 If this data constellation is used for the determination of the decision value of the optimistic scenario, a payment stream in the amount of = 42,9463 GE results. All necessarily realized investments and financings are shown in Figure 2.34. Without even achieving agreement in the conflict situation, the realizable success (success of the base program) must at least be attainable from the valuation program, too. Utilizing the linear optimization approach, which is essential for the determination of the valuation program, the “optimistic” decision value = 415,4942 GE can finally be computed. Both the valuation program and the decision value are outlined in Figure 2.35. P max opt EN opt max Personal equity assets EM Internal financing IF t = 0 t = 1 10 30 40 Investment AK Bullet loan ED Operating loan KA Financial investments GA -100 45,3476 35 -3,6278 57,5987 34,9327 t = 2 t = 3 40 40 t = 4 840 45 -3,6278 55 -3,6278 -48,4259 60 -48,9754 KA-, GA-paybacks Withdrawal EN Payment balance Debt from KA -42,9463 -63,3586 -42,9463 0 57,5987 0 34,9327 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.34: Comprehensive financial plan of the buyer’s base program from an optimistic perspective -38,4259 -42,9463 -42,9463 0 0 50,8472 -42,9463 858,9255 48,4259 858,9255 P max opt 118 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 118 45520_Matschke_Griffleiste_SL5.indd 118 16.03.2021 16: 21: 01 16.03.2021 16: 21: 01 Chapter 2 After performing a plain sensitivity analysis of the second type, the decision-maker can be provided with three different decision values pursuant to a different scenario = 391,4550 GE, = 368,8487 GE, and = 415,4942 GE. It is the task of the decision-maker to choose between the price and the range of the decision values represented by these values. Additionally, the valuation subject hast to consider both the individual risk propensity (risk-aversion, risk neutrality, or risk-seeking) and qualitative aspects (H ERING 2014, p. 43). If a multitude of further data sets is available in a manageable total model, the it is possible to compute, statistically evaluate, and document the optimum solution for each data set. The determined frequency distributions contain valuable quantitative information for decision-making under uncertainty. Furthermore, there is an option to carry out a scenario analysis for each of these three mutually exclusive basic situations (scenarios). It can be assumed that the input data of each scenario have fluctuation ranges (H ERING 2017, p. 359). Considering the multitude of alternatively possible scenarios, the analysis should focus on realistic, most likely, and hence reasonable data situations. The possible range containing the decision value, which is based on the so-called realistic input data scenario, is superior to a monovalent decision value, especially when polyvalent expectations are explicitly considered. It is noteworthy that the application of the sensitivity analysis of the second type only provides a range of results that demonstrate subjectively possible data situations. However, it does not make decisions for the decision-makers. Additionally, it was assumed in the example that polyvalent expectations are only given concerning the payment consequences of known objects. In reality, uncertainty also exists regarding the (real economic) restrictions. Nevertheless, incorporating real economic (non-financial) restrictions in the total model can lead to significant complexity, even under the assumption of (quasi-)certainty. If the decision-maker requires further detailed information, like frequency distributions, in addition to the presented possible range of the decision value, an increased computational effort is likely and a multitude of available input data becomes inevitable. t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 40 40 40 840 Business U Investment AK Bullet loan ED Operating loan KA -100 62 35 50 468,4405 -4 425,2309 45 45 25 55 -4 384,7002 -4 350,1165 441 60 -54 Financial investments GA KA-paybacks Withdrawal EN Payment balance -515,2846 -42,9463 415,4942 -42,9463 0 Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.35: Comprehensive financial plan of the buyer’s valuation program from an optimistic perspective 468,4405 425,2309 -467,7539 -423,1702 -42,9463 0 -42,9463 0 -385,1282 -42,9463 858,9255 384,7002 350,1165 858,9255 P max real P max pess P max opt 2.3 Determination of One-Dimensional Decision Values 119 45520_Matschke_Griffleiste_SL5.indd 119 45520_Matschke_Griffleiste_SL5.indd 119 16.03.2021 16: 21: 01 16.03.2021 16: 21: 01 Therefore, if uncertainty is considered, a reduction of complexity and computational effort is strongly advised. 2.3.3.2.5 Critical Evaluation After the examination of the general state marginal price model as the basic total model for the determination of decision values within the context of business valuation, this model is now critically evaluated according to the six model conditions (specifications) formulated in Section 2.3.3.2.1 (H ERING 2000a, p. 440, B RÖSEL 2002, p. 129). 1. Subject, and target system, and action relation: On an incomplete market, the decision value of a business can be computed as a marginal price in the represented general state marginal price model. This happens under the explicit consideration of the principles of overall valuation, of future actions, and of subjectivity. The value determined with this two-step cashflow-oriented total model is characterized by the principle of subjectivity according to its emphasis on the target system, decision field, and the action alternatives. With the aid of the corresponding formulation of the target function under the consideration of the valuation subject’s target system, there is a possible choice between the targets of asset and income maximization. Hence, the total model fulfills the first model condition. 2. Decision field relation and determination of the marginal (limit) value: In the described multi-period model the possibilities for investment and financing objects as well as the financial object interdependencies of the decision field are determined simultaneously and instead realistically. The guarantee of liquidity is assured by liquidity restrictions at each time t. Theoretically, the integration of further linear restrictions into the model is possible. The limit of negotiation willingness of the valuation subject in a specific conflict situation is determined by the model. Hence, the second model condition is also fulfilled. 3. Possibility of connecting with methods revealing uncertainty: Within the state marginal price model, the decision value represents a monovalent amount under the assumption of (quasi-)certainty. However, real uncertainty requires an interpretation of the points in time as states or the connection of adequate uncertainty revealing methods with the deterministic version of this total model. According to the examples of the sensitivity analysis of the second type, the model can be combined by using corresponding methods to provide significant quantitative information in form of possible ranges or frequency distributions of the decision value for the decision-makers. Since possibilities to show the consequences of uncertainty by applying the total model are given, it also fulfills the third model condition. 4. Reasonable effort for information acquisition and information processing: To realistically illustrate and to consider the interdependencies and the consequences of the marginal price in the total model, an enormous number of restrictions would have to be formulated. For the computation of the decision value, the simultaneous model has to be filled with the corresponding data. According to the data acquisition both all disposed objects and all investment and financing objects that might influence each other, the planning period, or its payment consequences as well as all investment and financing objects, which have already been realized, must be integrated into the model. That is why the total model suffers from major weaknesses 120 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 120 45520_Matschke_Griffleiste_SL5.indd 120 16.03.2021 16: 21: 01 16.03.2021 16: 21: 01 Chapter 2 in practice, especially within major companies according to the efforts for information acquisition and information processing. Furthermore, the polyvalent expectations increase the required data size even further. Hence, due to the not justifiable effort, the fourth model condition can only be fulfilled in plain models and instead only in small or medium-sized enterprises. This issue will be later addressed again in an excursus below. 5. Computability/ calculability: If all required data and information for the solution of the formulated model are collected and processed, the determined total models are of enormous complexity, especially in major enterprises, even under the assumption of (quasi-)certainty. The computer-aided solution of this optimization problem can present extra difficulties that are caused by numerous constraints, especially regarding integers. Furthermore, if an open decision field is assumed, the computational effort rises, too. As long as the essential computing capacities for the solution of the problem are not available, the fifth model condition can only be fulfilled in more straightforward total models. 6. Providing individual business decision support: Total models are generally designed for businesses with centralized decision-making authorities, where the management has the only decision-making competence. However, in companies with a more decentralized decision-making organization, the state marginal price model usually cannot fulfill the sixth model condition due to its complex form. Moreover, it has to be considered that the determination of the total model asserts that all decisions about alternative investment and financing possibilities are incorporated at the planning and valuation date (L EUTHIER 1988, p. 198). On the contrary, such decisions will have to be made continuously in reality, and this would require an iterative solution of a complex total model. 2.3.3.3 Future Performance Value Method - a Partial Model 2.3.3.3.1 Presentation Considering the necessity of a complexity reduction in business valuation, the determination of the decision value is usually accomplished by utilizing a partial model called the future performance value method. However, using the future performance value (which is in essence a present value), it cannot be concluded from the term “performance” (e.g., profit) that a key parameter of accounting is given. Instead, to avoid misunderstandings, it is advisable to clearly state the specific sizes/ variables and their respective assumptions pertaining to the model. The terms “future performance value” and “future performance value method” have recently also been associated with the so-called equity approach of the Discounted Cash Flow (DCF) method. While there are many similarities with the future performance value - especially regarding the application of the present value concept -there are conceptual differences too: While the future performance value aims for the determination of the decision value, the DCF equity market value instead focuses on the determination of an argumentation value (or possibly an arbitration value). That is why a blending of different contents by using the same terms is not useful, because this almost inevitably results in misunderstandings. Therefore, it is essential not to classify the DCF methods 2.3 Determination of One-Dimensional Decision Values 121 45520_Matschke_Griffleiste_SL5.indd 121 45520_Matschke_Griffleiste_SL5.indd 121 16.03.2021 16: 21: 01 16.03.2021 16: 21: 01 as future performance value methods. Only with the future performance method sound decision values can be determined, including the findings from investment theory. If the future successes of the business, that is, the cash flows of the business for its owners (under consideration of the subjective alternative rate of return or the endogenous marginal interest rates of the base/ valuation program) are considered, the result is henceforth referred to as future performance value (ZEW; German: Zukunftserfolgswert). A memorable explanation of this value is provided by H ERING (2014, p. 39): The interpretation of the future performance value for the buyer K as decision value or marginal price at a complete capital market results if the term of net present value is used. The acquisition of the enterprise at price P is an advantageous investment if the net present value is not negative from the buyer’s perspective Hence, the future performance value is the critical price and the upper price limit that a buyer can barely accept without suffering economic disadvantages from the acquisition. Conversely, the seller V demands For future performance value of the seller also represents the critical price, namely the lowest price limit. The marginal price defines the respective critical initial payment (deposit vs. payout) from both perspectives. The area of agreement interval for negotiations is limited by the respective decision values: In the area the business transaction at the price P is advantageous for both buyer and seller with a non-negative net present value. This conjecture holds true on the complete market even if several, periodspecific different interest rates are determined, instead of only one constant interest rate i. Accordingly, for the computation of the future performance value, each period t has its own interest rate i t . The future performance value formally represents a variant of the present value calculation, which is based on 1. the principle of overall valuation and 2. on the principle of future orientation. In the most simple case of estimating a future performance ZE (German: Zukunftserfolg) as a perpetuity, the determination of the future performance value ZEW (present value of the perpetuity) is carried out by the mere application of the capitalization rate (without a growth rate): In case of a time-limited business, the corresponding formulas are used, depending on whether ZE is deemed constant or irregular: Liquidation proceeds might also have to be considered at the end of the planning horizon, but this issue will not be further discussed here. ZEW K C K C K = - P + ZEW K ≥ 0 P ≤ ZEW K . C V = P - ZEW V ≥ 0 P ≥ ZEW V . ZEW V ≤ P ≤ ZEW K , ZEW = ZE i . ZEW = ZE t 1+ i ( ) t t = 1 T ∑ or - at ZE t = ZE = const. - ZEW = ZE ⋅ 1 + i ( ) T − 1 i ⋅ 1 + i ( ) T . 122 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 122 45520_Matschke_Griffleiste_SL5.indd 122 16.03.2021 16: 21: 02 16.03.2021 16: 21: 02 Chapter 2 If the interest rates are time-dependent, the following formula results: If the estimations of the discounted future performances ZE t follow a phase model, phase 1 has distinct estimations from t = 1 to t = T and phase 2 represents a single constant estimation from T to ∞. The formula with constant (time-invariant) interest rates is determined as follows: The term can be interpreted as present value of an estimated (target) selling price (terminal value) at the end of the first phase at time t = T. If it is assumed that the key parameters rise, this can also be incorporated into the formula. At first, the present value of an increasing series of future performances ZE t will be considered. They grow at a geometric rate w from any time t for n periods. The formula for the growing annuity reads: The parenthesized expression represents the sum of the first n terms, also known as a geometric series S (geometric progression). Note that it is implicitly assumed that i ≠ w. The series is derived as follows: ZEW = ZE t 1+ i τ ( ) τ=1 t ∏ t = 1 T ∑ . ZEW = ZE t 1+ i ( ) t t = 1 T ∑ + ZE T + 1 i ⋅ 1 1 + i ( ) T . ZE T+1 i BW 0 = ZE t ⋅ 1 (1+ i) t + ZE t ⋅ (1 + w) ⋅ 1 (1 + i) t + 1 + ZE t ⋅ (1 + w) 2 ⋅ 1 (1 + i) t + 2 + … + ZE t ⋅ (1 + w) n ⋅ 1 (1 + i) t + n BW 0 = ZE t ⋅ 1 (1 + i) t + (1 + w) ⋅ 1 (1 + i) t + 1 + (1 + w) 2 ⋅ 1 (1 + i) t + 2 + … + (1 + w) n ⋅ 1 (1 + i) t + n ⎡ ⎣⎢ ⎤ ⎦⎥ BW 0 = ZE t ⋅ 1 (1 + i) t ⋅ 1 + (1 + w) ⋅ 1 (1 + i) 1 + (1 + w) 2 ⋅ 1 (1 + i) 2 + … + (1 + w) n ⋅ 1 (1 + i) n ⎡ ⎣⎢ ⎤ ⎦⎥ BW 0 = ZE t ⋅ 1 (1 + i) t ⋅ 1 + (1 + w) (1 + i) 1 + (1 + w) 2 (1 + i) 2 + … + (1 + w) n (1 + i) n ⎡ ⎣⎢ ⎤ ⎦⎥ . S = 1+ (1+ w) (1 + i) 1 + (1 + w) 2 (1 + i) 2 + … + (1 + w) n (1 + i) n 1 + w 1 + i ⋅ S = (1 + w) (1 + i) 1 + (1 + w) 2 (1 + i) 2 + … + (1 + w) n (1 + i) n + (1 + w) n+1 (1 + i) n+1 S − 1+ w 1 + i ⋅ S = 1 − (1 + w) n+1 (1 + i) n+1 S ⋅ 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ = 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 2.3 Determination of One-Dimensional Decision Values 123 45520_Matschke_Griffleiste_SL5.indd 123 45520_Matschke_Griffleiste_SL5.indd 123 16.03.2021 16: 21: 02 16.03.2021 16: 21: 02 If this sum expression is inserted in the formula for the present value of a growing annuity ZE t from time t for n periods at the geometric rate w, it results: Again, this formula is only applicable if the growth rate w and the interest rate i are not equal, that is, it results: Under the assumption i > w, the present value of an infinitely growing annuity can be derived by searching for a convergence (limit of a sequence): If this convergence (limit of a sequence) is incorporated into the model and if a phase model (see p. 123) is assumed, the future performance value ZEW can be computed with the following formula: The expression 1/ (i - w) connects the capitalization rate with growth. If the growth rate w (accidentally) equals the inflation rate g, the difference (i - w) could be interpreted as the real interest rate. However, this simplification was the starting point for various misunderstandings regarding the consideration of a so-called inflation deduction from the (nominal) interest rate (M ATSCHKE 1979, p. 224-230, and 1986, B ALLWIESER 1981 and 1988) and for rather superfluous discussions about using nominal or real values in business valuation. This is because nominal and real values will always lead to the same results, at least as long as the growth rate w corresponds (exactly) to the inflation rate. In the following example (cf. Figure 2.36) it is assumed that the future performance is ZE 1 = 200 GE in the first period and the growth rate of the performance is w = 0,05 in the first decade. Note, however, that from the eleventh period on, the future performance equals the amount of the tenth period as a constant perpetuity. Finally, a future per- S ⋅ 1+ i − 1 + w ( ) 1 + i ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 S ⋅ i − w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ = 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 S = 1− 1+ w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ ⋅ 1 + i i − w ⎛ ⎝⎜ ⎞ ⎠⎟ . BW 0 = ZE t ⋅ 1 (1+ i) t ⋅ 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⋅ (1 + i) i − w ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ = ZE t ⋅ 1 (1 + i) t − 1 ⋅ 1 i − w ⋅ 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ w ≠ i. BW 0 = lim n→∞ and w<i ZE t ⋅ 1 (1 + i) t − 1 ⋅ 1 i − w ⋅ 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ = ZE t (1 + i) t − 1 ⋅ 1 i − w due to lim n →∞ and w<i 1 i − w ⋅ 1 − 1 + w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ = 1 i − w . ZEW= ZE t 1+ i ( ) t t = 1 T ∑ + ZE T+1 (1 + i) (T + 1) − 1 ⋅ 1 i − w = ZE t 1 + i ( ) t t = 1 T ∑ + ZE T+1 (1 + i) T ⋅ 1 i − w . 124 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 124 45520_Matschke_Griffleiste_SL5.indd 124 16.03.2021 16: 21: 03 16.03.2021 16: 21: 03 Chapter 2 formance value in the amount of ZEW = 2.684,17 GE results. For checking purposes, the future performance value was determined under consideration of the given formula for the present value of a finitely growing annuity immediate. If growth is assumed constantly throughout time, the future performance value can be calculated as the present value of a growing perpetuity: 2.3.3.3.2 Relation between the Total Model and the Partial Model 2.3.3.3.2.1 Derivation of the Future Performance Value Method In comparison to the total model “state marginal price model” a business valuation loses a substantial amount of complexity through the partial model “future performance value method” (in German: Zukunftserfolgswertmethode). The application of the future performance value method, or the concept of the present value in general, was not problematized yet. The method was given and its application was assumed. This is the typical approach in both theory and practice. Now, the question is, why is this method justified? Can it be shown that the present value calculation is a reasonable approach to determine the decision value from the perspective of a buyer or seller as a future performance value if only financial targets are considered as is customarily assumed? How are the necessary interest rates i calculated? Are the interests of the valuation subjects still upheld if the future performance value method is applied? Only if the future performance value formula can be theoretically explained, its use as a method to determine the decision value is justified. Maybe, its applicability limits Growth rate w Period t 1 0,05 Interest rate i Future performance ZE t 200 Discount factor (1 + i) - t 0,909091 2345 210 220,5 0,826446 0,751315 231,525 243,10125 0,683013 0,620921 6789 255,256313 268,019128 0,564474 0,513158 281,420085 295,491089 0,466507 0,424098 10 Subtotal 11 → ∞ Future performance value ZEW (present value) 310,265643 0,385543 3.102,656432 0,385543 0,1 Present value ZE t · (1 + i) - t 181,818182 173,553719 165,664914 158,134690 150,946750 144,085534 137,536191 131,284546 125,317067 119,620837 1.487,962430 1.196,208367 2.684,170797 Test by using formula Figure 2.36: Future performance value with growth rate w 2.684,170797 ZEW = ZE 1 i − w = 200 GE 0,1− 0,05 = 200 GE 0,05 = 4.000 GE. 2.3 Determination of One-Dimensional Decision Values 125 45520_Matschke_Griffleiste_SL5.indd 125 45520_Matschke_Griffleiste_SL5.indd 125 16.03.2021 16: 21: 03 16.03.2021 16: 21: 03 can be shown so that it becomes more apparent, which assumptions are necessary in the case of its application. What happens if these assumptions are not strictly fulfilled? Is the future performance value formula still helpful? Indeed, it can be shown that even if the decision value cannot be precisely determined with the aid of the future performance method, it is still possible to define an area (range) in which it will be found. It is the duality theory of the linear optimization (D ANTZIG 1966, p. 148, H ERING 2014, p. 53, and 2017, p. 154) that enables the valuation subject to determine the payment stream induced by the business to be valuated on a decentralized basis. In case of using the correct investment-theoretic interest rate, the interests of the owners as the valuation subjects are safeguarded (H ERING 2014, p. 36). In contrast to the total model, where the utility of the base program is compared to that of the valuation program, the future performance value method provides a comparison between the valuation object and the most favorable investment and financing program. The future performance value corresponds to the present value of future performances of the valuation object in the sense of cash flows that are then discounted at the respective interest rate. The interest rates serve as a standard of comparison and result from the best alternative capital investment of the decision subject. The theoretically correct marginal cost prices, that is, the endogenous marginal interest rates of the base program, could serve as interest rates on an incomplete capital market according to the future performance value method. In an incomplete capital market, they are, hence, determined by the decision field and by the individual preferences of the valuation subjects (H ERING 2017, p. 23). As an example, if a valuation subject anticipates a sequence of future performances from the business to be valuated with gt as cash flows at time t, the future performance value emerges according to the following formula of the simplified valuation (L AUX / F RANKE 1969, p. 210). Under the assumption of at all times t, the future performance value can be equally computed using endogenous marginal interest rates of the base program or those of the valuation program The knowledge of the endogenous marginal interest rates in each period is the foundation of (decentralized) application of the partial model “state marginal price model”. However, the determination of the endogenous marginal interest rates requires the computation of the base program and the valuation program as the optimum investment and financing program. Since the interest rates can only be computed after defining an optimum solution of the total model, it can be generally spoken about the dilemma of calculatory costs or the theory of marginal cost pricing (H IRSHLEIFER 1958, p. 340, H AX 1964, H ERING 2017, p. 144). The determination of the marginal interest rates and the verification of the applicability of the future performance value method and its limits requires the utilization of a mathematical optimization model, that is, the total model. The relation between total g = (0, g 1 , g 2 , …, g t , …, g T ) i t Ba = i t Be i t Ba i t Be : ZEW = g t 1+ i τ Ba ( ) τ= 1 t ∏ t = 1 T ∑ = g t 1 + i τ Be ( ) τ= 1 t ∏ t = 1 T ∑ . 126 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 126 45520_Matschke_Griffleiste_SL5.indd 126 16.03.2021 16: 21: 04 16.03.2021 16: 21: 04 Chapter 2 and partial model is then analyzed on the basis of the future performance value method (H ERING 2017, p. 53). The key to moving from the investment-theoretic general model “state marginal price model” to the investment-theoretic partial model “future performance value method” is the duality theory of the linear optimization, because each linear optimization task (primal problem) is closely related to a dual problem that allows conclusions regarding valid contexts contained in the optimum solution (H ERING 2014, p. 53). The primal problem of determining the ceiling price P max from the perspective of the buyer is examined in the next step. The approach for the valuation program, that is, for the primal problem from the buyer’s perspective, is outlined in the following figure: The variables to be determined in the primal problem are the number of investment and financing objects x Kj , the size of the withdrawal stream from the valuation program as well as the potential price P of the valuation object. To maximize P, the optimum solution must - in the case of assumed divisibility of the investments and financings to be realized - satisfy the withdrawal restriction (2) in the form of an equation, that is, the withdrawal stream form of the valuation program equals the highest possible withdrawal stream of the base program: The corresponding dual problem is determined as follows (G ALE / K UHN / T UCKER 1951): Primal problem: P → max! (1) Liquidity restrictions : (1a) - g Kj0 j=1 J ∑ ⋅ x Kj + P + w K0 ⋅ EN K Be ≤ b K0 (for t = 0) (1b) - g Kjt j=1 J ∑ ⋅ x Kj + w Kt ⋅ EN K Be ≤ b Kt + g UKt (for t = 1, …, T) (2) Securing of the withdrawel stream : EN K Be ≥ EN K Ba max (3) Capacity restrictions : x Kj ≤ x Kj max (for j = 1, …, J) (4) Non negativity conditions : (4a) x Kj ≥ 0 (for j = 1, …, J) (4b) EN K Be ≥ 0 (4c) P ≥ 0. EN K Be EN K Be = EN K Ba max . 2.3 Determination of One-Dimensional Decision Values 127 45520_Matschke_Griffleiste_SL5.indd 127 45520_Matschke_Griffleiste_SL5.indd 127 16.03.2021 16: 21: 04 16.03.2021 16: 21: 04 The autonomous payments b Kt correspond to the right sides of the payment restrictions of the valuation program without the payments from the business to be valuated, that is, the right sides of the payment restrictions of the base program. The right sides of the restrictions of the primal problem are contained in the target function of the dual program. The variables of the dual problem to be determined are the dual variables d t (liquidity restrictions at times t = 0, …, T), u j (capacity restrictions with j = 1, …, J), and δ (safeguarding of the withdrawal stream). The dual variables have to be assessed in the optimum of the dual problem so that the sum of the right sides of the restrictions, that is, the opportunity costs K of the use of these restrictions, becomes as small as possible. The optimum solution of the dual problem is then K min . Given the conditions and in the primal problem and because of as the optimum solution of the base program, it follows that restriction (2), at the optimum of the dual problem regarding the withdrawal stream, must equal the following: Now it has to be considered that the maximum of the primal problem (with solution: P max ) equals the minimum of the dual problem (with solution: K min ). Because of K : = b Kt ⋅ d t t=0 T ∑ valuated autonomous payments & ' $ ( $ + g UKt t = 1 T ∑ ⋅ d t valuated business payments & ' $ ( $ liquidity restrictions " # $$$$ % $$$$ − δ ⋅ EN K Ba max valuated withdrawal stream & ' $ ( $ withdrawal stream restriction " # $$ % $$ + x Kj max ⋅ u j j = 1 J ∑ valuated capacity & ' $ ( $ capacity restrictions " # $ % $ → min! (2) Weighting factor restriction of the withdrawel stream : w Kt t=0 T ∑ ⋅ d t − δ ≥ 0 (2) Weighting factor restriction of the withdrawel stream : w Kt t=0 T ∑ ⋅ d t − δ ≥ 0 (3) Dual variable restrictios of the liquidity restrictions : (3a) d 0 ≥ 1 (for t = 0) (3b) d t ≥ 0 (for t = 1, … , T) (4a) Dual variable restrictions of the capacity restrictions : u j ≥ 0 (for j = 1, … , J) (4b) Dual variable restriction of the withdrawel stream security : δ ≥ 0. EN K Be = EN K Ba max EN K Be ≥ 0 EN K Ba max > 0 w Kt t=0 T ∑ ⋅ d t − δ = 0 and δ = w Kt t = 0 T ∑ ⋅ d t . 128 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 128 45520_Matschke_Griffleiste_SL5.indd 128 16.03.2021 16: 21: 05 16.03.2021 16: 21: 05 Chapter 2 this relationship the defining equation of K can be used to calculate P max . If the solution for δ is taken into account, the equation to calculate the decision value P max reads: In the optimal solution of the primal problem P = P max > 0 holds true so that the liquidity restriction (1a) of the primal problem has been strictly complied with. The theorem of complementary slackness indicates that restriction (3a) must be satisfied with its lower limit in the dual problem so that d 0 = 1 holds. The dual variable d 0 = 1 means that today’s payments are without any adjustment included in the calculation of P max . For the other dual values d t at time t = 1, …, T, the relation d t / d 0 =: is valid. represent the discount factors for the valuation program, which may be derived from the endogenous periodic marginal interest rates of the valuation program of the buyer (R OLLBERG 2001, p. 178, H ERING 2017, p. 193): This means: 1 GE at time t > 0 is then worth units at t = 0 so that future cash flows are included in the calculation of P max with their respective present value. For investment and financing objects j contained in the valuation program, it holds that restriction (1) of the dual problem is satisfied with its lower limit and that those objects do not have a negative net present value ≥ 0 at t = 0. Since represents a current monetary value, it follows from the theory of the marginal cost pricing that · d 0 = u j and - due to d 0 = 1 - consequently u j and are identical. In case of disadvantageous investment and financing objects, restriction (1) of the dual problem is not strictly satisfied (with its lower limit). It follows that restriction (4a) of the dual problem must be made to comply with its lower limit so that for disadvantageous investment and financing objects with a negative net present value, the dual variable u j is 0. If this is considered, the equation for P max can also be written as follows: P max = b Kt ⋅ d t t=0 T ∑ + g UKt t=1 T ∑ ⋅ d t + x Kj max ⋅ u j j=1 J ∑ - EN K Ba max ⋅ w Kt t=0 T ∑ ⋅ d t . ρ Kt Be ρ Kt Be i Kt Be ρ Kt Be = 1 (1+i Kτ Be ) τ =1 t ∏ . ρ Kt Be − g Kjt t=0 T ∑ ⋅ d t + u j = 0 ↔ u j = g Kjt t = 0 T ∑ ⋅ d t C Kj Be C Kj Be C Kj Be C Kj Be P max = b Kt ⋅ d t t=0 T ∑ + g UKt t = 1 T ∑ ⋅ d t + x Kj max ⋅ u j j = 1 J ∑ − EN K Ba max ⋅ w Kt t = 0 T ∑ ⋅ d t or - because of d t d 0 =: ρ Kt Be , d 0 = 1 and C Kj Be = g Kjt t = 0 T ∑ ⋅ ρ Kt Be - P max = b Kt ⋅ ρ Kt Be t = 0 T ∑ + g UKt t = 1 T ∑ ⋅ ρ Kt Be + x Kj max ⋅ C Kj Be C Kj Be > 0 ∑ − EN K Ba max ⋅ w Kt t = 0 T ∑ ⋅ ρ Kt Be . 2.3 Determination of One-Dimensional Decision Values 129 45520_Matschke_Griffleiste_SL5.indd 129 45520_Matschke_Griffleiste_SL5.indd 129 16.03.2021 16: 21: 06 16.03.2021 16: 21: 06 An adjustment leads to the following equation for the decision value P max , the socalled complex formula of valuation (L AUX / F RANKE 1969, p. 214, B RÖSEL 2002, p. 157, H ERING 2014, p. 55): The formula says that the maximum price P max can be calculated as the difference between the net present value of the valuation program (before considering a price for the business) and the net present value of the base program, which must be given up if the business is acquired. The future performance value of the business to be valuated is part of the net present value of the valuation program (before considering a price for the business) and will principally not equal the decision value P max from the perspective of a buyer. However, in the least desirable case that the negotiated price equals the decision value P max , the valuation program of the buyer then represents the optimum program after such an agreement. Rearrangements (cf. Figure 2.28) lead to the following equation for the decision value P max from the buyer’s perspective: According to this equation, the maximum payable price P max as the decision value of the buyer results from the future performance value ZEW of the business considering the net present value difference transforming the base program into the valuation program of the buyer: If there are transformations between the base and the valuation program with a positive net present value, it follows from this relation that the determined future performance value calculated with the aid of the endogenous marginal interest rates of the valuation program is lower than the decision value of the buyer in the sense of the maximum price: The future performance value based on endogenous marginal interest rates of the valuation program thus constitutes a lower limit for the buyer’s decision value. This leads to the question of whether the upper limit can also be determined for the buyer’s P max = g UKt t=1 T ∑ ⋅ ρ Kt Be future performance value of the business to be valuated & ' $ ( $ + b Kt ⋅ ρ Kt Be t = 0 T ∑ + x Kj max ⋅ C j KBe C Kj Be > 0 ∑ capital value of the remaining valuation program & ' $$$$ ( $$$$ capital value of the valuation program (before consideration of a price for the business to be valuated) " # $$$$$$$$ % $$$$$$$$ − w Kt t = 0 T ∑ ⋅ EN K Ba max ⋅ ρ Kt Be capital value of the base program & ' $$$ ( $$$ . P max = g UKt payment of the valuation object ) t=1 T ∑ ⋅ ρ Kt Be discounting factor ) future performance value of the valuation object & ' $$ ( $$ + b Kt ⋅ ρ Kt Be + x Kj max ⋅ C Kj Be C Kj Be > 0 ∑ total of the positive capital values " # $$ % $$ t = 0 T ∑ capital value of the valuation program (without valuation object) & ' $$$$ ( $$$$ − w Kt ⋅ EN K Ba max ⋅ ρ Kt Be t = 1 T ∑ capital value of the base program & ' $$$ ( $$$ change of capital value, according to the tranformation from the base to the valuation program ≥ 0 & ' $$$$$$$$$ ( $$$$$$$$$ . P max = ZEW U K (ρ Kt Be ) + ΔKW K Be-Ba with Δ KW K Be-Ba ≥ 0, in the order that: ZEW U K ( ρ Kt Be ) = P max - Δ KW K Be-Ba . ZEW U K (ρ Kt Be ) ≤ P max . 130 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 130 45520_Matschke_Griffleiste_SL5.indd 130 16.03.2021 16: 21: 08 16.03.2021 16: 21: 08 Chapter 2 decision value. This is indeed possible. The starting point of this approach is the dual problem for the determination of the buyer’s base program (H ERING 2014, p. 58). It can be shown that the net present value difference is not negative. The upper limit for the buyer’s decision value corresponds to the computed future performance value that can be determined with the so-called simplified formula of valuation (considering endogenous marginal interest rates of the base program): In short form, it reads: The future performance value based on the endogenous marginal interest rates of the base program must constitute the upper price limit for the decision value P max from the buyer’s perspective. This already results from the consideration that if otherwise, P max was higher than the future performance value, the acquisition at P max would be disadvantageous; the result would be a negative net present value. Hence, the decision value of the buyer P max must consequently lie in the following limits (H ERING 2014, p. 59): The lower limit is the future performance value based on the endogenous marginal interest rates of the valuation program, the upper limit is the future performance value based on the endogenous marginal interest rates of the base program (computed with the formula of the simplified valuation). If the marginal interest rates differ in the base and valuation program, the valuation problem can only be solved using a general model. If the endogenous marginal interest rates of the base program are known and if those of the valuation program can be estimated, the simplified valuation formula of the future performance value method can be used to narrow down the area in which the buyer’s decision value P max will lie. If the endogenous marginal interest rates of both programs are equal, transformations of the base and valuation program are carried out. In other words, the net present value difference reads At such a difference of zero, only marginal ob- ΔKW K Be−Ba ZEW U K (ρ Kt Ba ) P max = g UKt payment of the valuation object ) t=1 T ∑ ⋅ ρ Kt Be discounting factor ) future performance value of the valuation object & ' $$ ( $$ = ZEW U K ( ρ Kt Ba ). ZEW U K (ρ Kt Ba ) ≥ P max . ZEW U K (ρ Kt Be ) ≤ P max ≤ ZEW U K (ρ Kt Ba ) or g UKt t=1 T ∑ ⋅ 1 1+i K τ Be ( ) τ =1 t ∏ discounting factor " # $ % $ future performance value of the valuation object based on the endogenous marginal interest rates of the valuation program & ' $$$ ( $$$ ≤ P max ≤ g UKt t=1 T ∑ ⋅ 1 1+i K τ Ba ( ) τ =1 t ∏ discounting factor " # $ % $ future performance value of the valuation object based on the endogenous marginal interest rates of the base program & ' $$$ ( $$$ . ΔKW K Be−Ba = 0. 2.3 Determination of One-Dimensional Decision Values 131 45520_Matschke_Griffleiste_SL5.indd 131 45520_Matschke_Griffleiste_SL5.indd 131 16.03.2021 16: 21: 08 16.03.2021 16: 21: 08 jects are displaced or additionally included. In such a situation the simplified valuation formula of the future performance value can be used as a method to calculate the exact decision value in terms of the highest price (ceiling price) from the perspective of the buyer. In case of a perfect capital market, the simplified valuation formula of the future performance value method is always valid, because in a perfect capital market, marginal transactions are always carried out at the current market interest rate I, so that - if we assume for simplicity that the interest rate is constant through time - the following holds: The derivation of the equation for P max is based on the objective of withdrawal maximization. In fact, equally structured equations can be determined based on other objectives, but that option will not be explored here (H ERING 2014, p. 60). However, it is important to note that under the assumption of an incomplete capital market, the possible marginal transactions are dependent on the respective target so that the endogenous marginal interest rates are generally target-oriented. Accordingly, even if applied, those equations might still differ numerically in their endogenous marginal interest rates and hence do not necessarily have the same results. Conversely, different valuation results are likely to occur on an incomplete capital market depending on the target. The preferences of the valuation subjects ultimately determine the decision value and also the upper and lower limit of the decision value of the business (H ERING 2014, p. 65). Analogous considerations from the buyer’s perspective can be transferred to the decision value P min from the perspective of the seller. The withdrawal maximization is once again taken as the foundation (H ERING 2014, p. 74). Starting with the dual problem of the valuation program, the complex formula for the decision value P min reads: It is also worth mentioning that the business payments g UVt are still included in the autonomous payment balances b Vt . The disposition of the business in the case of its sale is considered in the formula by the subtraction of the future performance value. This formula for P min determines that the decision value, which is the lowest price (bottom price) from the perspective of the seller, can be computed as the difference between the net present value of the base program (including the business to be valuated) and the net present value of the valuation program (excluding the business to be valuated) to reflect the procedure of the tabular method. The elimination of the business causes a reduction of the net present value (cf. the expression in brackets in the equation above). The attainable price for the business must ρ Kt Be = ρ Kt Ba = (1 + i) − t . P min = w Vt ⋅ EN V Ba max ⋅ ρ Vt Be t=1 T ∑ capital value of the base program (with valuation object) & ' $$$ ( $$$ - b Vt ⋅ ρ Vt Be t=0 T ∑ + x Vj max ⋅ C Vj Be C Vj Be >0 ∑ total of the positive capital values " # $$ % $$ capital value of the valuation program (+ valuation object) " # $$$$ % $$$$ - g UVt payment of the valuation object ) t=1 T ∑ ⋅ ρ Vt Be discounting factor ) future performance value of the valuation object & ' $$ ( $$ capital value of the valuation program & ' $$$$$$$ ( $$$$$$$ ⎛ ⎝ ⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟⎟⎟⎟⎟ . 132 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 132 45520_Matschke_Griffleiste_SL5.indd 132 16.03.2021 16: 21: 10 16.03.2021 16: 21: 10 Chapter 2 at least compensate for that reduction. Consequently, such a high price has to be achieved through the sale that the sum of the price (received at t = 0) and of the net present value of the valuation program corresponds at least the amount of the net present value of the base program after the elimination of the business. Then, the lowest acceptable price represents the decision value of the seller P min : If (and only if) this is successful, a situation of indifference between the non-sale situation (illustrated by the base program) and the sale situation (illustrated by the valuation program including the price P min ) occurs. This formula can be restructured so that it structurally resembles the situation from the perspective of the buyer: In this formula, the decision value P min is defined as future performance value plus the change of the net present value caused by transformations. The difference of the net present value is zero if the restructuring only leads to an exchange of marginal transactions. Otherwise, it is negative if the transformations are going beyond these transactions. The short form of the equation is: If there are transformations with a negative net present value , with the help of the endogenous marginal interest rates of the valuation program, the determined future performance value based on the simplified valuation formula is higher than the seller’s decision value in the sense of a lowest price that must be demanded: The future performance value based on endogenous marginal interest rates of the valuation program thus constitutes an upper limit for the decision value of the seller. Analogous to those considerations from buyer’s perspective, a lower limit can also be defined for the decision value of the seller. The dual problem for the determination of the base program of the seller. This lower limit is defined as: P min + b Vt ⋅ ρ Vt Be + x Vj max ⋅ C Vj Be C Vj Be >0 ∑ - g UVt t=1 n ∑ ⋅ ρ Vt Be future performance of the valuation object & ' $ ( $ t=0 n ∑ ⎛ ⎝ ⎜⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟⎟ capital value of the valuation program & ' $$$$$$$$ ( $$$$$$$$ = w Vt ⋅ EN V Ba max ⋅ ρ Vt Be t=1 n ∑ capital value of the base program (with valuation object) & ' $$$ ( $$$ . P min = g UVt ⋅ ρ Vt Be t=1 T ∑ future performance of the valuation object & ' $ ( $ + w Vt ⋅ EN V Ba max ⋅ ρ Vt Be t = 1 T ∑ − b Vt ⋅ ρ Vt Be + x Vj max ⋅ C Vj Be C Vj Be > 0 ∑ t = 0 T ∑ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ change of the capital value, according to the transformation from the base to the valuation program ≤ 0 & ' $$$$$$$$$ ( $$$$$$$$$ . P min = ZEW U V (ρ Vt Be ) + ΔKW V Be − Ba with Δ KW V Be − Ba ≤ 0. ΔKW V Be−Ba < 0 ZEW U V (ρ Vt Be ) ≥ P min . P min ≥ g UVt payment of the valuation object ) t=1 T ∑ ⋅ ρ Vt Ba discounting factor ) future performance value of the valuation object & ' $$ ( $$ . 2.3 Determination of One-Dimensional Decision Values 133 45520_Matschke_Griffleiste_SL5.indd 133 45520_Matschke_Griffleiste_SL5.indd 133 16.03.2021 16: 21: 10 16.03.2021 16: 21: 10 The short form result is: The seller’s decision value must consequently lie in the following limits (H ERING 2014, p. 78): In summary, the (“simplified”) future performance value method is always applicable under the condition that there will be no transformations (or only such, which do not lead to a difference of the net present value between the base and the valuation program, that is, only marginal transactions). It serves as a method for the determination of the decision value in a one-dimensional, disjoint conflict situation of the type acquisition/ sale with merely financial targets. On the contrary, if it comes to positive (buyer) or negative (seller) changes of the net present value, the simplified future performance value method can still be used for the estimation of the area, which contains the respective decision value (P max or P min ). 2.3.3.3.2.2 Numerical Example In the multi-period numerical example of the determination of the decision value from the buyer’s perspective (see Section 2.3.3.2.2.2), a maximum price of 391,4550 GE was computed. The result of using the base program (cf. Figure 2.21) is that the endogenous marginal interest rates are 10 % p. a. in the first and second period, 6,39 % in the third, and 5 % p. a. in the fourth period. In the valuation program, the marginal transactions (see Figure 2.22) are exclusively represented by short-term borrowing of operating loans KA at 10 %. In the following table, the data of the example are summarized and the upper limit and lower limit of the maximum payable price from the perspective of the buyer are calculated (see Figure 2.37). ZEW U V (ρ Vt VBa ) ≤ P min . ZEW U V (ρ Vt Ba ) ≤ P min ≤ ZEW U V (ρ Vt Be ) or g UVt t=1 T ∑ ⋅ 1 1+i V τ Ba ( ) τ =1 t ∏ discounting factor " # $ % $ future performance value of the valuation object based on the endogenous marginal interest rates of the base program & ' $$$ ( $$$ ≤ P min ≤ g UVt t=1 T ∑ ⋅ 1 1+i V τ Be ( ) τ =1 t ∏ discounting factor " # $ % $ future performance value of the valuation object based on the endogenous marginal interest rates of the valuation program & ' $$$ ( $$$ . 134 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 134 45520_Matschke_Griffleiste_SL5.indd 134 16.03.2021 16: 21: 11 16.03.2021 16: 21: 11 Chapter 2 As expected, holds. The specific values read: 389,4953 GE < = 391,4550 GE < 413,8628 GE. These numerical values must not be confused with the range of the decision value that results in utilizing the sensitivity analysis (based on pessimistic, realistic, and optimistic data). Figure 2.38 below contains the synopsis of the data for the “complex” calculation formula. It directly provides the exact decision value of the buyer To clarify that the operating loans KA are the marginal transactions, their respective summarized payments are also mentioned and their net presents value are calculated, too. Time 0 1 2 3 4 Business U Endogenous marginal interest rates of the base program 60 40 20 420 Present value 1 0,1 0,909091 413,8628 54,5455 0,1 0,826446 0,0639 0,776808 33,0579 15,5362 0,05 0,739817 310,7233 Endogenous marginal interest rates of the valuation program 1 0,1 0,909091 Present value Figure 2.37: Upper and lower limit of the decision value P max 389,4953 54,5455 0,1 0,826446 0,1 0,751315 0,1 0,683013 33,0579 15,0263 286,8657 i Kt Ba Discounting factors ρ Kt Ba ZEW U K (ρ Kt Ba ) i Kt Be Discounting factors ρ Kt Be ZEW U K (ρ Kt Be ) ZEW U K (ρ Kt Be ) ≤ P max ≤ ZEW U K (ρ Kt Ba ) P max P max . 2.3 Determination of One-Dimensional Decision Values 135 45520_Matschke_Griffleiste_SL5.indd 135 45520_Matschke_Griffleiste_SL5.indd 135 16.03.2021 16: 21: 11 16.03.2021 16: 21: 11 Time 0 1 2 3 4 Unternehmung Endogene Grenzzinsfüße des Basisprogramms 60 40 20 420 itKBa Abzinsungsfaktoren rtKBa Barwerte ZEWUK( rtKBa) 1 0,1 0,90909091 408,9991 54,5454545 0,1 0,82644628 0,08 0,76522804 33,0578512 15,3045608 0,05 0,72878861 306,091215 Endogene Grenzzinsfüße des Bewertungsprogramms itKBe Abzinsungsfaktoren rtKBe 1 0,1 0,90909091 Barwerte ZEWUK( rtKBe) Right side of the liquidity restrictions of the valuation program 389,4953 54,5454545 0,1 0,82644628 0,1 0,7513148 0,1 0,68301346 33,0578512 15,026296 286,865651 (without payments of the business to be valuated) 40 30 11 0,90909091 0,90909091 40 544,904037 27,2727273 544,9040 30 30 0,82644628 0,82644628 0,7513148 0,7513148 630 0,68301346 0,68301346 24,7933884 22,539444 430,298477 Net present values of the objects in the valuation program Payments of the investments AK -100 30 11 0,90909091 0,90909091 Present value of investment AK Net present value of investment AK Bullet loan ED -100 35,4621 27,2727 50 -4 40 50 0,82644628 0,82644628 0,7513148 0,7513148 55 0,68301346 0,68301346 33,0579 37,5657 -4 -4 37,5657 -54 Present value of total loan payments ED Net present value of total loan ED 11 0,90909091 0,90909091 50 3,1699 -3,6364 Operating loan KA 434,1446 -83,3867 1 1 0,90909091 0,90909091 0,82644628 0,82644628 0,7513148 0,7513148 -3,3058 -3,0053 0,68301346 0,68301346 -36,8827 -73,3867 -63,3867 0,82644628 0,82644628 0,7513148 0,7513148 -366,1202 0,68301346 0,68301346 Present value of operating loan payments KA Net present value of operating loan KA 434,1446 0 -75,8061 32,6176 32,6176 Present value of withdrawals 32,6176 1 32,6176 0,90909091 1 32,6176 0,90909091 29,6524 -60,6502 -47,6234 32,6176 32,6176 -250,0650 684,9696 32,6176 0,82644628 32,6176 0,7513148 0,82644628 26,9567 0,7513148 24,5061 684,9696 0,68301346 0,68301346 467,8435 Net present value of base program 581,5762 389,4953 544,9040 + Net present value of investment AK + Net present value of bullet loan ED - Net present value of base program Sum = P max 35,4621 3,1699 -581,5762 391,4550 583,5360 Figure 2.38: Components of the complex calculation formula for the buyer b Kt Discounting factors ρ Kt Be Cash values b Kt ⋅ ρ Kt Be Total cash value b Kt ⋅ ∑ ρ Kt Be Discounting factors ρ Kt Be Discounting factors ρ Kt Be Discounting factors ρ Kt Be Withdrawals w Kt ⋅ EN K Ba max Discounting factors ρ Kt Be ZEW U K ( ρ Kt Be ) + b Kt ⋅ ∑ ρ Kt Be + total cash value of the remaining valuation program ⎧ ⎨⎪ ⎩⎪ 136 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 136 45520_Matschke_Griffleiste_SL5.indd 136 16.03.2021 16: 21: 12 16.03.2021 16: 21: 12 Chapter 2 In the multi-period numerical example of the decision value determination from the seller’s perspective (see Section 2.3.3.2.3.2), the lowest price that has to be demanded amounts to 196,2159 GE. From the base program (cf. Figure 2.26), it follows that the endogenous marginal interest rates are 10 % in the first and second period, 6,39 % in the third, and 5 % in fourth period. In the valuation program, only the short-term financial investments GA at 5 % represent the marginal transactions (cf. Figure 2.27). In the following table, the data of the example are summarized and the upper and lower limits for the lowest payable price (bottom price) from seller’s perspective are determined (see Figure 2.39). As expected, holds true. The numerical values of the example read: 186,8633 GE < = 196,2159 GE < 204,5395 GE. In Figure 2.40, the data for the complex calculation are consolidated and directly deliver the decision value of the seller P min . To clarify that the short-term investments GA represent the marginal transactions, the corresponding summarized payments are also presented and their respective net present values calculated. Time 0 1 2 3 4 Business KU Endogenous marginal interest rates of the base program 12 11 12 210 Present values 0,1 0,90909091 184,683346 10,9091 0,1 0,82644628 0,0639 0,77680823 9,0909 9,3217 0,05 0,73981737 155,3616 Endogenous marginal interest rates of the valuation program 0,05 0,95238095 Present values Figure 2.39: Upper and lower limit of the decision value P min 204,539467 11,4286 0,05 0,90702948 0,05 0,8638376 0,05 0,82270247 9,9773 10,3661 172,7675 i Vt Ba Discounting factors ρ Vt Ba ZEW KU V ( ρ Vt Ba ) i Kt Be Discounting factors ρ Vt Be ZEW KU V ( ρ Vt Be ) ZEW KU V (ρ Vt Ba ) ≤ P min ≤ ZEW KU V (ρ Vt Be ) P min 2.3 Determination of One-Dimensional Decision Values 137 45520_Matschke_Griffleiste_SL5.indd 137 45520_Matschke_Griffleiste_SL5.indd 137 16.03.2021 16: 21: 12 16.03.2021 16: 21: 12 Time 0 1 2 3 4 Unternehmung Endogene Grenzzinsfüße des Basisprogramms 12 11 12 210 itVBa Abzinsungsfaktoren rtVBa Barwerte ZEWUV( rtVBa) 1 0,1 0,90909091 186,8634 10,9090909 0,1 0,82644628 0,05 0,7870917 9,09090909 9,44510035 0,05 0,74961114 157,418339 Endogene Grenzzinsfüße des Bewertungsprogramms itVBe Abzinsungsfaktoren rtVBe 1 0,05 0,95238095 Barwerte ZEWUV( rtVBe) Right side of the payment restrictions of the valuation program 204,5395 11,4285714 0,05 0,90702948 0,05 0,8638376 0,05 0,82270247 9,97732426 10,3660512 172,76752 (including payments of the business KU to be valuated) b Vt 40 30 1 40 0,95238095 28,5714286 Net present values of the objects included in the valuation program Investment AK 640,0000 -100 30 30 30 0,90702948 27,2108844 0,8638376 25,915128 630 0,82270247 518,302559 40 50 55 Present value of investment AK Net present value of investment AK 1 -100 0,95238095 28,5714 53,2931 Bullet loan ED 1 0,95238095 0 0,0000 0,0000 0,90702948 36,2812 0,8638376 43,1919 0,82270247 45,2486 0,90702948 0,8638376 0,0000 0,0000 0,82270247 0,0000 Investments GA Present value of deposits -103,5983 -15,3825 1 -103,5983 0,95238095 -14,6500 Net present value of deposits 0 32,6176 1 32,6176 0,95238095 -26,3824 -35,3824 0,90702948 -23,9296 0,8638376 -30,5646 209,9696 0,82270247 172,7425 32,6176 0,90702948 32,6176 0,8638376 684,9696 0,82270247 Present value of withdrawals Net present value of base program 32,6176 684,9696 31,0644 204,5395 + Net present value of base program - Net present value of investment AK Sum = P min 684,9696 -53,2931 -640,0000 196,2159 29,5851 28,1763 563,5262 -8,3235 Figure 2.40: Components of the complex calculation formula for the seller Discounting factors ρ Vt Be Cash values b Vt ⋅ ρ Vt Be Total cash value b Vt ⋅ ∑ ρ Vt Be Discounting factors ρ Vt Be Discounting factors ρ Vt Be Withdrawals w Vt ⋅ EN V Ba max Discounting factors ρ Vt Be ZEW KU V ( ρ Vt Be ) − Total cash value b Vt ⋅ ∑ ρ Vt Be + change of capital value, according to the transformation from the base to the valuation program ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 138 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 138 45520_Matschke_Griffleiste_SL5.indd 138 16.03.2021 16: 21: 12 16.03.2021 16: 21: 12 Chapter 2 2.3.3.3.3 Consideration of Uncertainty Traditionally, uncertainty is taken into account in the determination of the future performance value by reduction of future performances and/ or by increases in the interest rate, while methods condensing uncertainty are used. However, this means the polyvalent structure of the expectations is not sufficiently clear. It is far more effective to examine different variants of future development of the business for the determination of uncertainty, estimating the consequences of business performance. Due to potentially different time horizons, appropriate steps need to be considered. It is illustrated below how the risk analysis, that is, a planning method revealing uncertainty, can be used in the context of the future performance value method. With the aid of the risk analysis which can be performed in a simulative or analytical manner, a statistic distribution for the target value is derived from the given distributions of the planning input data (H ERTZ 1964, H ERING 2017, p. 334). Due to the high number of uncertain parameters, the analytical method for the determination of the decision value usually is not feasible. Therefore, the examination is limited to simulative risk analysis. The basis of this analysis is input data from expert estimates and their respective probability distributions for their possible manifestations. The analysis might also incorporate information about stochastic dependencies between the parameters. As far as uncertain parameters and their presumedly given probability distributions are concerned, random numbers are drawn iteratively in numerous program runs with the help of the Monte-Carlo-Simulation (C OENENBERG 1970, p. 799). A frequency distribution of the target value can be determined from the decision values resulting from an adequate number of experiments. The statistical evaluation of the simulation does not provide conditional recommendations for the decision-makers, but especially the possible graphical processing of the results provides a picturesque basis of decision-making, which is based on a subjective interpretation of input data and their frequency distributions (H ERING 2017, p. 335). If several expert estimates with different input data and frequency distributions are given, it results an amount of similar frequency distributions with serveral iterations of the risk analysis. It can be made available in its entirety to the decision-maker indicating a range of information (H ERING 2014, p. 42). In the next step the application of the simulative risk analysis is illustrated in a simple example, according to the future performance value method. A presumptive buyer wants to determine the decision value for a business U in the sense of a marginal price in a non-dominated, disjoint, multi-dimensional conflict-situation of the type acquisition. The business to be valuated can be characterized by the payment sequence gt with the stochastically independent, normally distributed expectation values (80 GE, 60 GE, 50 GE) and the corresponding standard deviations (5 GE, 4 GE, 4 GE) at time t = 1, 2, 3. An anticipated uniformly distributed payment stream between 28 GE and 32 GE is expected from the fourth period as a perpetuity. In the first three periods, a symmetric triangular distribution is assumed for the endogenous interest rates i t at an expectation value of 10 % p. a. No transformations during the shift from the base to the valuation program are expected. While the interval of the endogenous interest rates can be limited for the first period to 10 % ± 0,5 % p. a. [that is, between 9,5 % and 10,5 % p. a.], for the second period to 10 % ± 0,75 % p. a. [that is, between 9,25 % and 10,75 % p. a.], and for the third period to 10 % ± 1.0 % p. a. [that is, between 9,0 % and 11,0 % p. a.], 2.3 Determination of One-Dimensional Decision Values 139 45520_Matschke_Griffleiste_SL5.indd 139 45520_Matschke_Griffleiste_SL5.indd 139 16.03.2021 16: 21: 13 16.03.2021 16: 21: 13 for all following periods the range [9,0 %; 11,0 %] is also expected. However, this range is expected to be uniformly distributed. Hence, the assumptions imply increasing information uncertainty over time. Under consideration of the “simplified” formula for the determination of the future performance value the following results can be computed after a thousand simulation steps (in the Monte-Carlo-Simulation). The range of the decision values determined in the experiments lies between ZEW Umin = 342,3664 GE and ZEW Umax = 440,0006 GE. The median is ZEW UMedian = 385,5367 GE. The arithmetic mean amounts to 385,9148 GE; the standard deviation is 18,1939. In Figure 2.41 the simulative estimated frequency function of the future performance value is illustrated as a decision value (H ERING 2014, p. 43, B RÖSEL 2002, p. 170). The representative depiction of the future performance value has a higher impact than a point estimation or a range, according to present uncertainty. Figure 2.42 represents a further instrument for the evaluation of risk analysis, showing a distribution function of the decision value (future performance value). The diagram illustrates the estimated probability on the ordinate and payment in tterms of the future performance value on the abscissa leading to a decrease of the utility value for the decision subject by acquiring the business compared to the utility value with a nonagreement. ZEW U = g 1 1+ i 1 + g 2 1 + i 1 ( ) ⋅ 1 + i 2 ( ) + g 3 1 + i 1 ( ) ⋅ 1 + i 2 ( ) ⋅ 1 + i 3 ( ) + g 4 →∞ i 4 →∞ ⋅ 1 1 + i 1 ( ) ⋅ 1 + i 2 ( ) ⋅ 1 + i 3 ( ) 340≤ZEW<345 345≤ZEW<350 350≤ZEW<355 355≤ZEW<360 360≤ZEW<365 365≤ZEW<370 370≤ZEW<375 375≤ZEW<380 380≤ZEW<385 385≤ZEW<390 390≤ZEW<395 395≤ZEW<400 400≤ZEW<405 405≤ZEW<410 410≤ZEW<415 415≤ZEW<420 420≤ZEW<425 425≤ZEW<430 430≤ZEW<435 435≤ZEW<440 440≤ZEW<445 0 20 40 60 80 100 120 Number 3 9 24 46 46 82 92 90 101 98 99 82 63 59 48 21 18 14 4 0 1 Figure 2.41: Estimated frequency distribution of the future performance value 140 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 140 45520_Matschke_Griffleiste_SL5.indd 140 16.03.2021 16: 21: 14 16.03.2021 16: 21: 14 Chapter 2 In Figure 2.42, two situations are exemplary marked. Situation 1: If the presumptive buyer acquires the business at the price P = 367,7209 GE, this price could be too high in 17,4 % of the cases, according to the results of the simulation. Situation 2: If the buyer pays the price P = 404,1088 GE for the enterprise, it must be taken into account that this price has to be regarded as too high in 82,8 % of the cases, according to the results of the simulation. From the other perspective, an opposite representation is shown in Figure 2.43 with the risk profile of the future performance value as decision value. This profile indicates those probabilities that are necessary to achieve at least the same level of target fulfillment as without any agreement during the negotiation. Both situations in Figure 2.42 are shown below. 340 350 360 370 380 390 400 410 420 430 440 450 Future performance value 0,000 0,200 0,400 0,600 0,800 1,000 Cumulative probability (367,720891/ 0,17430042) (404,108782/ 0,82760496) Figure 2.42: Frequency function of the future performance value 2.3 Determination of One-Dimensional Decision Values 141 45520_Matschke_Griffleiste_SL5.indd 141 45520_Matschke_Griffleiste_SL5.indd 141 16.03.2021 16: 21: 14 16.03.2021 16: 21: 14 The simulative risk analysis approximates a representative stable probability distribution of the target value by applying a random number generator from the probability distribution of the uncertain input data. If reliable information and estimations about the input data are given, the computer-aided risk analysis proves to be a valuable and an easily manageable instrument for decision support, according to the simple partial model. As demonstrated in the previous explanations and figures, the results of the analysis are clearly illustrated and therefore easily interpretable. The uncertainty of the valuation problem is not aggregated reducing information but fully revealed instead. Hence, financial performance consequences are transparently presented. The decision-maker has to make the not rationally verifiable choice between the uncertain decision value and a price falling into that established range (H ERING 2014, p. 44). 2.3.3.3.4 Critical Evaluation According to the critical evaluation of the total models, it was established in the case of a complex model formulation that they do not meet the requirements regarding a reasonable information acquisition and information processing effort as well as the computability of the calculations. Moreover, they renounce the general decentral decision organization. The insights gained in the total analytic examination led to a partial model being represented and analyzed with the future performance value method, which is now critically evaluated referring to the six model conditions formulated in Section 2.3.3.2.1 (B RÖSEL 2002, p. 173). 1. Subject, target system, and action relation: The valuation subject can determine the decision value of a business under consideration of the principles of overall valuation, future relation, and subjectivity with the simplified future performance value method. The duality theorem of linear optimization enables a decentralized valuation of actions for the valuation subject, according to the induced payment streams and a simultaneous pursuit of the operationalized target of the business. However, 340 350 360 370 380 390 400 410 420 430 440 450 Future performance value 0,000 0,200 0,400 0,600 0,800 1,000 1 - cumulative probability (367,720891/ 0,82569958) (404,108782/ 0,17239504) Figure 2.43: Risk profile of the future performance value 142 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 142 45520_Matschke_Griffleiste_SL5.indd 142 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 Chapter 2 the determination of the decision value with the simplified method of the future performance value is tied up to three conditions: a) The investment-theoretic interest rates are known or can be reliably estimated and have to be used in the calculation. b) By including the valuation object, no transformations have to be made the transfer from the base to the valuation program with respect to marginal objects. c) The action options are not limited by non-financial restrictions. The safeguarding of the interests of the owners requires, above all, the knowledge of the endogenous marginal interest rates and presupposes their so-called steady state (or stationary state) during the transfer from the base to the valuation program. If the decision value is determined on a complete capital market, the simplified valuation formula can be used. Due to the dilemma of the theory of marginal cost pricing, the interest rates are not defined until the solution of the total model. Furthermore, if continuity is not provided, the marginal price must be determined with the complex valuation formula. For this purpose, information from the total model is essential, especially according to the net present value change, which was caused by transformations at the transfer from the base to the valuation program. Nonetheless, an estimation of the interval is still possible with the formula of simplified valuation if (quasi-)certainty and the knowledge of the endogenous marginal interest rates are assumed. The fulfillment of the first model condition in the partial model usually requires the solution of the corresponding total model or its solution structure. 2. Decision field relation and determination of the limit value: Generally, the future performance value method only represents the scarcity prices of the capital by the calculatory interest rates. If further non-financial restrictions are determined, the corresponding scarcity prices of these non-financial restrictions must additionally be considered to define the marginal price. B RÖSEL has shown that the consideration of the non-financial interdependencies necessitates a corresponding modification of the future performance value method (B RÖSEL 2002, p. 160). If besides the financial restrictions, further conditions have to be taken into account during the determination of the appropriate limit of negotiation willingness, the solution of the corresponding total model is essential. With regard to the critical evaluation of the model conditions, this is not further discussed here. In this case, the simplified formula could be used for the calculation of the future performance value. 3. Possibility of connecting with methods revealing uncertainty: If the input data (including the calculatory interest rates) and their probability distributions concerning possible attributes used for the decision value calculation of the partial model could be reliably estimated, and if information about stochastic interdependencies between the parameters is available, the values can be approximated to a representative frequency distribution of the target value with the aid of the simulative risk analysis. The computer-aided simulative risk analysis proves to be a valuable and easily manageable instrument for decision support. The results are clearly represented and easily explicable. Additionally, the risk analysis can be accompanied by a sensitivity analysis (H ERING 2017, p. 335). Uncertainty of the valuation problem is fully covered, according to the third model condition. 2.3 Determination of One-Dimensional Decision Values 143 45520_Matschke_Griffleiste_SL5.indd 143 45520_Matschke_Griffleiste_SL5.indd 143 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 4. Reasonable information acquisition and information processing effort: For the calculation of the decision value with the future performance value method under the mentioned conditions, information about the expected future performances and scarcity prices of the capital (and non-financial restrictions if required) is essential. Hence, the amount of necessary data should be manageable. If a total model is not used, all essential information must be estimated, while information acquisition and information processing efforts should remain within reasonable limits in the sense of this model condition. 5. Computability of the calculations: If sound estimates of the mentioned input data are available, a possible computation of the calculations with the simplified formula of the future performance value method can be undertaken, neglecting possible transformations during the transfer from the base to the valuation program. Suitable software for the application of the simulative risk analysis should be used, which today could run on a home computer. The partial model fulfills the fifth model condition. 6. Providing individual business decision support: Due to the duality theorem of the linear optimization with the partial model called “future performance value method”, businesses can be valuated in isolation or in a decentralized manner (as long as no transformations from the base to the valuation program are necessary during the inclusion or elimination of the valuation object in the investment and financing program). The endogenous marginal interest rates (and the scarcity prices of the non-financial restrictions) have an incentive function. The marginal cost prices are subjective because their existence and their quantification depend on the target function and the individual decision field of the valuation subject (S CHMALENBACH 1947, A DAM 1970, p. 25, M ATSCHKE 1993c, H ERING 2014, p. 29, R OLLBERG 2001, p. 136, H ERING 2017, p. 3). If correct marginal cost prices are used, partial models offer crucial support for decentralized decision-makers in the sense of the sixth model condition. In view of decision support and a necessary complexity reduction of the valuation problem, a model is to be presented that enables a decentralized business valuation. It was already examined that the exact determination of the decision value in a partial model requires the solution of the corresponding total model. Especially, the dilemma of the marginal cost pricing theory and the transformations in the investment and financing program proved to be a problem. The partial model is redundant due to the solution of the total model. However, according to the consideration of the total model, a solution might not be attained due to the high level of complexity at hand. Hence, the problem of business valuation is a problem without a solution (A DAM 1996, p. 10). For the determination of the decision value a potential solution has be found heuristically - especially in divisional businesses. On the basis of the approximate decomposition, a new model is now presented, attempting to solve the described problems in a heuristic way. 144 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 144 45520_Matschke_Griffleiste_SL5.indd 144 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 Chapter 2 2.3.3.4 Approximate Decomposed Business Valuation - a Heuristic Model 2.3.3.4.1 Basics Due to their enormous or even unrealizable requirements, total or simultaneous models often pose an insoluble problem for decision-makers, according to information acquisition and information processing and regarding the computability of the optimization problem, especially in large companies. Additionally, it is doubtful whether the theoretically perfect simultaneous model delivers such good results in practical valuation situations in contrast to competing, but less challenging, valuation approaches that make the higher costs for the model determination and solution justifiable (L EUTHIER 1988, p. 25). In divisional businesses, a profound decentralized form of decision support is essential due to the delegation of responsibility. But the partial analysis has shown that decentralized decisions require sound estimates of shadow prices - especially scarcity prices of capital in the sense of individual, periodic interest rates. These marginal cost prices result from the coefficients of the target function of the total model. For decentralized decision support, a solution to the dilemma of the marginal cost pricing theory is vital. Given business valuation, an approximate decomposed valuation represents a theoretically valid investment approach that enables a weighting between necessary practicability and sufficient rigor. This approach is based on the model of approximate decomposition (H ERING 2014, p. 174, M ATSCHKE / H ERING / K LINGELHÖFER 2002, p. 221, B RÖSEL 2002, p. 179, H ERING 2017, p. 239), a combination of hierarchical and iterative coordination. The heuristic approach of the approximate decomposed valuation represents - like the approach of the approximate decomposition itself - a structuring rule, gradually transforming badly structured, at first unsolvable, initial problems into well structured and solvable subproblems. These subproblems point toward a satisfying solution to the initial problem (O LBRICH 1999, p. 81). The approximate decomposition under uncertainty is examined below. Following this, it is shown with the approximate decomposed valuation, how business valuation can be combined with the approximate decomposition under uncertainty. 2.3.3.4.2 Heurististic Planning Method of Approximate Decomposition under Consideration of Uncertainty The approximate decomposition approach for investment and financing planning combines total and partial planning in divisional businesses to resolve the dilemma of the marginal cost pricing theory. In contrast, the exact decomposition (D ANTZIG / W OLFE 1960) of the problem leads to the same difficulties as the complex total model and hence does not constitute an efficient solution approach. To safeguard the flexibility of the business and to consider the temporal structures within the planning processes, the synthesis of total and partial models occurs in the rolling-wave planning of the business accompanied by a sensitivity and a risk analysis revealing uncertainty (J ACOB 1967, p. 19). Preparatory measures for the realization of the approximate decomposition are a 2.3 Determination of One-Dimensional Decision Values 145 45520_Matschke_Griffleiste_SL5.indd 145 45520_Matschke_Griffleiste_SL5.indd 145 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 “hierarchization” and the “definition of all major variables”. These preparations and the implementation steps, which are illustrated in Figure 2.44 (B RÖSEL 2002, p. 180), are explained below. The iterative phases during the approximate decomposition are highlighted in the figure. Hierarchization Using the notion of the hierarchization (R OLLBERG 2001, p. 197, B RÖSEL 2002, p. 181, H ERING 2017, p. 356) permits the planning system to be decomposed into two partial subsystems, the “centralized planning authority” (headquarter) and the “decentralized planning authority” (divisions). With regard to their decision competence, the subsystems are in a so-called superordination-subordination relationship. The competences, tasks, and essential planning instruments are classified as hierarchical steps. Given the hierarchical relationship, the headquarter has the power to direct. Furthermore, it has the decision competence for potential marginal objects as well as for the largest and most important strategic objects (B LOHM / L ÜDER / S CHAEFER 2012, p. 207). On the centralized level, a small, well manageable total model should be used as the planning instrument. It generates the results for the necessary marginal cost prices in partial planning. By formulating the optimization approach as a total model, only major restrictions should be considered. The criterion for the scope of the total problem to be solved by the headquarter is the efficiency of the available information systems. The linear approach is similar to the simplified total model for the determination of the base program both from the perspective of the presumptive buyer and from the one of the presumptive seller. Due to polyvalent expectations, this linear approach can be combined with the sensitivity analysis of the second type. Since marginal cost prices can be determined by marginal objects, it is sufficient to include only variables of potential marginal objects and strategically important objects in the total model. With regard to liquidity constraints, large-scale financings, unlimited deposits, operating loans, and major real investments will primarily serve as marginal objects. The headquarter will decide which of these potential marginal objects reported by the divisions are included as variables in the total model and which will revert to the Preparation of the approximative decomposition • Hierarchization • Centralized determination of all major variables Application of the approximative decomposition 1. Determination of the decision-independent parameters and preselection 2. Centralized determination of the interest rates ranges 3. Decentralized investment calculation with net present values 4. Feedback or termination Figure 2.44: Basics and steps of the approximative decomposition 5. Investment decision of the headquarter 146 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 146 45520_Matschke_Griffleiste_SL5.indd 146 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 Chapter 2 area of competence of the divisions owing to being less significant. The criteria for the potential marginal objects and other strategically important objects, which have to be reported by the divisions, should be clearly defined, conclusive, and easily comprehensible. For the classification, features like strategic importance and financial project volume could be used. Within the divisions, net present values of the objects are calculated autarkically and the advantageousness of potential objects is assessed by using scarcity prices. The net present value method should be combined with the simulative risk analysis, especially with regard to uncertain expectations. The decision-making competence of the divisions is limited to objects that do not count as potential marginal objects or as a strategically important investment and financing objects. The divisions are obliged to report those objects beyond their authority to headquarter. Centralized determination of all major variables When it has finished the hierarchization by classifying the respective competences, tasks, and planning instruments into individual hierarchy levels, it is the job of headquarter to specify the major variables for the hierarchical planning process with feedback (H ERING 2014, p. 175, B RÖSEL 2002, p. 183, H ERING 2017, p. 355). Headquarter decides the target in the total model that also expresses the desires of the majority of the owners. Suitable targets are asset and income maximization (see Section 2.3.1.2.1). In the case of income maximization, headquarter must define the weighting factors for the desired temporal (withdrawal) structure as significant parameters. Also, in the case of asset maximization, the distributions have to be weighted to correspond with the consumption preferences of the owners (H ERING 2014, p. 60). Moreover, to define the target, headquarter must decide the frequency of the rolling-wave planning, differentiating between purpose(fulness) and the necessity of the planning process. Time requirements mean it is not expedient to repeat the steps of the approximate decomposition in too short intervals. According to the requirement for essential flexibility, the planning problem must, however, not be defined in too long intervals either. With regard to the now common quarterly reporting, it seems appropriate to initiate the process quarterly or at least annually. Furthermore, headquarter must determine an appropriate planning horizon and the length of the planning period pragmatically. While the planning horizon will likely not exceed 5 (up to a maximum of 10) years due to limited projections (predictive power), quarters, half years, or years might serve as planning periods depending on the respective (rolling-wave) planning. Determination of the decision-independent parameters and preselection At the beginning of each planning process, the decision-independent parameters must be determined (H ERING 2017, p. 356, B RÖSEL 2002, p. 184). Therefore, the divisions have to report the ranges of the non-available, fixed payment sequences (cash flows). Additionally, information about recognizable potential marginal objects and any strategically important object must be communicated to headquarter. Since the divisions do not have information about the interest rates at the beginning of the planning pro- 2.3 Determination of One-Dimensional Decision Values 147 45520_Matschke_Griffleiste_SL5.indd 147 45520_Matschke_Griffleiste_SL5.indd 147 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 cess, potential marginal objects should be defined with respect to the marginal cost prices of the capital taken from the prior period. Due to the required determination of the ranges of non-available, fixed payment sequences as well as of potential marginal objects at the start of the planning process, a heuristic preselection of disadvantageous and advantageous objects can take place by the divisions. Marginal cost pricing theory suggests limiting the range of endogenous marginal cost prices to the closed interval: i St ≥ i t ≥ i Ht : Normal (or normalized) investments, that is, those investments with only one change of sign, prove to be advantageous if their net present value based on the interest rates i St is positive. If a negative net present value occurs under consideration, the interest rates i Ht , the investment is disadvantageous. In contrast, normal financing is advantageous if the net present value is positive under consideration of the interest rates i Ht . The same is, however, disadvantageous if the net present value is negative by applying the interest rates i St (H ERING 2017, p. 234). There is also the possibility of taking the marginal cost prices from the prior period. For highly indebted (leveraged) firms the application of the (possibly already known) range of interest rate might be useful as a preselection (H ERING 2017, p. 241). The non-available, fixed payment sequences have to contain those objects that have to be realized for strategic reasons regardless of their net present value (R OLLBERG 2001, p. 175). Centralized determination of the interest rates ranges Headquarter determines the range of the endogenous marginal interest rates based on the available data (H ERING 2014, p. 176, R OLLBERG 2001, p. 200, B RÖSEL 2002, p. 185, H ERING 2017, p. 357). For this purpose, it uses the (simplified) base approach in combination with comprehensive sensitivity analyses of the second type. The decisionindependent data reported by the divisions are considered in the total model. Moreover, headquarter must decide if the present potential marginal objects can be included in the simultaneous model or if they fall back to the divisions due to their lower overall significance. While selecting the analyzed data sets of the present experiments, the interdependencies between the objects must be taken into account to identify diversification potentials contributing to risk compensation. The information processing and computing effort of the headquarter has to be kept within limits, and that is why the data constellations are not arbitrarily selectable. It suffices if the situations remain somewhat realistic and probable. It is feasible to limit the number of scenarios that are carried out during a sensitivity analysis. It is conceivable to analyze three to five mutually exclusive basic situations, which comprise a realistic (or neutral), a pessimistic, and an optimistic input data variant (H ERING 2014, p. 177‚). For a statistical evaluation of the results the respective optimum solutions of the experiments are documented in an ongoing transcript. The following data of each coefficient constellation are noteworthy: The optimum of the respective target function, the endogenous marginal interest rates, the marginal cost prices of the time restriction, the index of advantageousness of every single investment and financing object (listing of indices for objects at each solution: +1 for a realizable object, 0 for each marginal ob- 148 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 148 45520_Matschke_Griffleiste_SL5.indd 148 16.03.2021 16: 21: 15 16.03.2021 16: 21: 15 Chapter 2 ject, -1 for not realizable objects) and a reliability index (for each object the net present value of which falls below a set threshold). The purpose of the reliability index is to protect the planning business from potentially existence-threatening mistakes (H ERING 2017, p. 358). Therefore, the net present value has to be computed for each object in all experiments. The summary minutes of the scenarios provide detailed information on period-specific endogenous marginal cost prices, advantageous objects detected in the total model, and the range or distribution of the target function value. The result of this step in the approximate decomposition is the specification of the marginal cost pricing ranges and the possibly qualified assumptions regarding the distribution of the interest rates (i 1 , i 2 , ..., i n ) in a decentralized marginal cost pricing management. Decentralized investment calculation with net present values The divisions must make investment decisions for the objects of their areas of competence, according to the net present value method using the available intervals (ranges) or distributions of interest rates defined by the headquarter (H ERING 2014, p. 177, R OLL- BERG 2001, p. 200, B RÖSEL 2002, p. 187, H ERING 2017, p. 360). Due to the polyvalent expectations, the divisions should ideally apply the method of the simulative risk analysis in order not to artificially consolidate the net present value to a monovalent target value. Instead, uncertainty should be revealed and ranges or distributions of the net present values should be represented. The aim of the investment calculation provided by the divisions is to identify the generally disadvantageous or advantageous objects and to lead to a decision, according to the object, of which the net present value ranges have positive and negative values. This decision finally has to be made by the decision-makers of the divisions regarding their individual risk propensity (risk-aversion, risk neutrality, or risk-seeking). In addition to the net present value profiles, other criteria can be taken into account for this not further formalized “entrepreneurial” decision. However, it should be noted that the net present value is the single most important economic criterion. The following factors could be considered: strategic choices or reliability targets like prevention of essential risks, diversification, and flexibility. To make a decision, the division has to consider the non-financial restrictions, too. Those are neglected in the simplified base approach. The judgments about the objects, made in the previous iterative runs might have to be amended. The presence of marginal objects cannot be defined with the net present value profile, and that is why it is generally essential for the divisions to make an explicit decision for each object in the analysis. A payment balance of either cash flows or financial requirements results for each period of the planning horizon from the presumed advantageous investment and financing objects of the division. These values should be determined as (neutral) point estimates, ranges, or decision values with an (engineering) tolerance (limits of variation). As a result of this decomposition step, the advantageous or decentralized significant investment and financing objects of each division are reported to the headquarter based on given ranges or distributions of the marginal cost prices. To avoid an inflation of the 2.3 Determination of One-Dimensional Decision Values 149 45520_Matschke_Griffleiste_SL5.indd 149 45520_Matschke_Griffleiste_SL5.indd 149 16.03.2021 16: 21: 16 16.03.2021 16: 21: 16 base approach of the headquarter, further potential marginal objects should only be reported to the headquarter in exceptional circumstances (H ERING 2017, p. 243). Feedback or termination The reporting of each division is considered on a centralized level in the simulative base approach in a cumulated form. Now, the headquarter decides about the continuation (feedback) or discontinuation (termination) of the iteration: If the centralized decision authority verifies that the divisions did not significantly change their investment and financing decisions, the iterative approach is to be terminated (H ERING 2014, p. 178, B RÖSEL 2002, p. 188, H ERING 2017, p. 361). In the next step, the investment decisions of the headquarter are made. Otherwise, the marginal cost price ranges will be determined again. If they do not significantly differ from the result of the last run, the iteration has to be concluded and it has to be passed to the centralized investment decisions. The rather vaguely formulated criterion for a termination condition “not significantly” assures the finitude (or finiteness) of the approach, because headquarter must decide if further iteration steps lead to a substantially increased quality so that greater planning efforts are justified. Otherwise, there is no termination. The new marginal cost pricing intervals must be communicated to the divisions; the decentralized investment calculation with the net present values must be repeated. Figure 2.45 (B RÖSEL 2002, p. 189) illustrates the aforementioned process. The iterative process can be terminated if the decisions within the divisions or the marginal cost prices resulting from the simplified total model do not significantly change. If the approximate decomposition does not converge against steady marginal cost price vectors (H ERING 2017, p. 245), the procedure should be terminated after a certain number of coordination rounds (runs) with the best solution so far. After two or three rounds of feedback, it can be expected that the decision-makers of the divisions remain unimpressed with these - possibly insignificant from their perspective - changes to the marginal cost pricing ranges and that they persist in their decisions because they might already have carefully weighed all qualitative and quantitative arguments. The coordination requirements are already fulfilled in the case of uncertainty if feedbacksteps were carried out at all (H ERING 2014, p. 179). 150 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 150 45520_Matschke_Griffleiste_SL5.indd 150 16.03.2021 16: 21: 16 16.03.2021 16: 21: 16 Chapter 2 Investment decision of the headquarter After the termination of the iteration, the headquarter has to make conclusions about all realizable objects at t = 0, according to the remaining investment and financing objects in the total model (H ERING 2014, p. 179, R OLLBERG 2001, p. 201, B RÖSEL 2002, p. 189, H ERING 2017, p. 364). Hence, the headquarter does not only need to finally decide about all payment streams starting at t = 0, but also about those objects that are eventually not available anymore in the next run of the rolling-wave planning. For the decision, quantitative and qualitative factors have to be balanced, just as on the decentralized level. Particularly with regard to strategic investment objects, qualitative factors can be of greater importance than within the division. According to the quantitative factors, the report (record) has to be used, which was created during the last centralized determination of the marginal cost price range. In this report, the indices of advantageousness and reliability were defined in addition to the optimum of the respective target function and the corresponding marginal prices. If the single indices of advantageousness of an object are summed up, an overall index results for the respective action. By definition, such an index informs about whether an object has been disadvantageous (- 1), advantageous (+ 1), or a marginal object (0) during the experiment. For the valuation of the objects, the degree of advantageousness Prüfungsschritt: Haben die Divisionen ihre Investitions- und Finanzierungsentscheidungen geändert? JA Verfahrensschritt: Zentrale Ermittlung der Lenkpreisbandbreiten Prüfungsschritt: Haben sich die Lenkpreisbandbreiten wesentlich verändert? somit nächster Verfahrensschritt: Dezentrale Investitionsrechnung mit Kapitalwerten Rückkopplung somit nächster Verfahrensschritt: Investitionsentscheidung der Zentrale Abbruch JA NEIN NEIN Figure 2.45: Decision about feedback or termination Examination step: Have the divisions changed their investment and financing decisions ? Procedural step: Centralized determination of the range of marginal cost prices (scarcity prices) Feedback and next procedural step: Decentralized investment calculation with net present values Termination and next procedural step: Investment decision of the headquarter No Yes No Yes Examination step: Have the ranges of marginal cost prices changed significantly ? 2.3 Determination of One-Dimensional Decision Values 151 45520_Matschke_Griffleiste_SL5.indd 151 45520_Matschke_Griffleiste_SL5.indd 151 16.03.2021 16: 21: 17 16.03.2021 16: 21: 17 of an object VG Objekt (VG; German: Vorteilhaftigkeitsgrad) can be determined by the quotient (ratio) of this overall index, and the number of recorded data sets: The degree of advantageousness of an object, which mainly lies in the value range [-100 %, +100 %], reveals the probability of being advantageous or disadvantageous. From the overall index and the degree of advantageousness of an object, the following information is collected by the decision authorities of the headquarter: If the object had a positive net present value in all experiments of the last run, the degree of advantageousness is +100 %. If the object never had a positive net present value, the degree of advantageousness amounts to -100 %. Objects with a significant positive degree of advantageousness and hence overall index are advantageous in most constellations. Objects with a significant negative degree of advantageousness and overall index are consequently disadvantageous in most constellations. The reliability index of an object SI Objekt (SI; German: Sicherheitsindex), which gives information about the frequency of how often the net present value has fallen below a defined threshold at the beginning of the procedure. This can be considered in relation to the number of data sets registered in the report for improving the empirical significance: The decision should only consider objects that have never (or at least only seldom) fallen below the net present value threshold. The calculated solution is now to be examined with respect to the fulfillment of liquidity and integer constraints (R OLLBERG 2001, p. 180) as well as other restrictions that have been neglected so far. The attainment of the payment equilibrium requires not only a decision about objects with a positive degree of advantageousness but also a centralized determination of the corresponding marginal objects. For the determination of marginal objects matching to advantageous objects, correlation coefficients between the indices of advantageousness can be calculated with the aid of the already used (final) report. Suitable marginal objects are described as objects, which strongly correlate with the advantageous objects. Alternatively, the sum of the absolute differences of the corresponding advantageousness indices can be used as a decision criterion. The higher the sum, the more seldom will both objects have the same net present value in the optimum solution. Accordingly, those object combinations with a small sum of differences are instead suitable pairs. If headquarter makes the essential investment decisions, the divisions can be informed about the end and the corresponding result of the planning process. The marginal cost pricing ranges used at last have to be confirmed for the period until the next run of the planning process, according to the rolling-wave planning as the valid decentralized calculation foundation of the business (R OLLBERG 2001, p. 180). Figure 2.46 presents the relevant information flow and the process of investment and financing planning with the aid of the approximate decomposition under uncertainty (B RÖSEL 2002, p. 192). VG object = total of the object's advantageousness amount of documented data sets . SI object = amount of threshold decreases amount of documented data sets . 152 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 152 45520_Matschke_Griffleiste_SL5.indd 152 16.03.2021 16: 21: 19 16.03.2021 16: 21: 19 Chapter 2 The compromise between centralized and decentralized planning through the approximate decomposition results in an expedient solution according to marginal cost pricing theory. A simplified linear optimization approach serves as a total model, which is manageable and solvable with the help of efficient software. The objects are organized in a decentralized way within the divisions concerning the net present value criterion. The partial model uses theoretically founded scarcity prices. The procedure of ap- Lenkpreisbandbreiten [Knappheitspreise der (Liquiditäts-) Restriktionen]; Zurückweisung nicht relevanter Objekte Zentrale Planungsinstanz Divisionen Hierarchiestufe Konstante Zahlungen (saldierter Zahlungsüberschuß oder Finanzbedarf) und eventuelle weitere Restriktionen Planungsinstrumente: Totalmodell (Basismodell, linearer Optimierungsa nsatz); Sensitivitätsanalyse der zweiten Art Festlegung der Rahmenvariablen: Entnahmezielsetzung, Länge der Planungsperiode, Planungsrhythmus und -horizont sowie Kriterien für strategisch bedeutende und potentielle Grenzobjekte Entscheidung über zu realisierende Objekte des Totalmodells; Ermittlung geeigneter Grenzobjekte zur Einhaltung der Liquiditäts- und anderer Restriktionen Planungsinstrumente: Partialmodell (Kapitalwertmethode); Simulative Risikoanalyse Aufgaben: Dezentrale Investitionsrechnung; Treffen von klaren Entscheidungen für oder gegen ein Objekt Aufgaben: Ermittlung der Lenkpreisbandbreiten; Erzeugung von Ergebnisprotokollen; Entscheidung über Rückkopplung oder Abbruch Berücksichtigung der entscheidungsunabhängigen Parameter; Entscheidung über Aufn ahme gemeldeter Objekte in das zentrale Modell Potenti ell e Grenzobjekte, strategisch bedeutende Objekte und entscheidungsunabhängige Parameter Qualitative und quantitative Vorselektion; Pragmatische Ermittlung potentieller Grenzobjekte sowie Eruierung strategisch bedeutender Objekte Iteration Figure 2.46: Overview of the approximative decomposition Headquarter Determination of the structure variables: withdrawal objectve, term of planning period, planning rhythm and planning horizon as wel as criteria for strategically significant and potential marginal objects Consideration of decision-independent parameters, decision about the inclusion of reported objects into the main model Qualitative and quantitative preselection, pragmatic determination of potential marginal objects as well as examination of strategically significant objects Planning instruments: Total model (base model, linear optimization approach), sensitivity analysis of the second type Tasks: Determination of marginal price ranges, generation of (final) reports, decision about feedback or termination Planning instruments: Partial model (capital value method); simulative risk analysis Tasks: Decentralized investment calculations, concise decision for or againt an object Decision about objects of the total model to be realized, determination of suitable marginal objects for the fulfillment of liquidity restrictions and other restrictions Marginal price ranges [scarcity prices of (liquidity) restrictions], rejection of non-relevant objects Interation Potential marginal objects and decision-independent parameters Hierarchy level Divisions Constant payments (balanced excess or financial need) and potential further restrictions 2.3 Determination of One-Dimensional Decision Values 153 45520_Matschke_Griffleiste_SL5.indd 153 45520_Matschke_Griffleiste_SL5.indd 153 16.03.2021 16: 21: 20 16.03.2021 16: 21: 20 proximate decomposition represents an economic planning process, which serves as the starting point for business valuation (H ERING 2014, p. 181, H ERING 2017, p. 246 and 395). 2.3.3.4.3 Combination between Business Valuation and Approximate Decomposition under Uncertainty Subsequent to the remarks regarding the investment and financing planning with the approximate decomposition, a method for decision value determination of businesses is presented that can be integrated in the heuristic planning process. Business valuation is represented as a two-step process. The first step is the classification of the business to be valuated to the centralized or decentralized decision authority. After the clarification of the responsibility, the valuation is carried out with the investment-theoretic marginal price estimation in the second step (cf. H ERING 2014, p. 182, H ERING 2000b, p. 371, B RÖSEL 2002, p. 193). Classification of valuation objects to the relevant hierarchy level The investment and financing planning was determined by the combinations of the total and partial model on two hierarchy levels. If a business valuation is necessary, it has to be determined in the first step, which hierarchy level has the corresponding decision competence about the valuation object. This classification relies on the insights of Section 2.3.3.3.2: Hence, the application of the future performance value method for the determination of the decision value is influenced by a change of the endogenous marginal prices during the transfer from the base to the valuation program. The classification to the centralized or the respective decentralized decision authority consequently depends on the resulting risk from a valuation object at a base modification. Therefore, it is vital to take the individual decision field of the valuation subject into account. The business to be valuated is, thus, assigned to the corresponding planning hierarchy level, especially according to its financial volume. If a base modification, according to financial marginal objects, is not expected during a presumptive acquisition or a presumptive sale of the valuation object, the valuation of the mentioned business can be decentrally managed by the divisions. Hence, the “small” valuation objects are falling within the area of competence of the divisions. Their acquisition or sale does probably not affect the marginal objects of the centralized simultaneous approach (H ERING 2014, p. 182). On the contrary, if the inflow or outflow of the business to be valuated leads to a transformation of the centralized total model, according to the financial marginal prices, a valuation by the centralized decision instance is necessary. “Large” and strategically important enterprises are assigned to the centralized decision authority. Due to the failure of the future performance value method at a change of the endogenous marginal cost prices during the transfer from the base to the valuation program, the decision value of the corresponding company is to be determined with the aid of a suitable total model (valuation program). After the classification of valuation objects to the hierarchical planning authorities, an investment-theoretic marginal price estimation is carried out. Depending on whether the valuation ob- 154 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 154 45520_Matschke_Griffleiste_SL5.indd 154 16.03.2021 16: 21: 21 16.03.2021 16: 21: 21 Chapter 2 ject is allotted to the headquarter or the corresponding divisions, the following valuation process has to be distinguished: Investment-theoretic marginal price estimation of the divisions Analogously to the net present value criterion as the decentralized planning tool in the context of the approximate decomposition, the decision value of the company is determined in a decentralized process with the simplified formula of the future performance value; hence, the future performance value is calculated as a probability density function or as a range. Accordingly, the given ranges and distributions of the marginal cost prices known from the last planning run of the heuristic program planning are employed. To take the uncertainty into account, the procedures of the risk analysis revealing uncertainty or sensitivity analysis are applied. The information about the determined probability density functions or ranges of the future performance values should be graphically enhanced. This is the best prerequisite for decision makers at a decentralized level to be able to use this information in the negotiation process for the acquisition or sale of the company in question. Investment-theoretic marginal price estimation of the headquarter In contrast, the centralized determination of the decision value carried out in the simultaneous model from the investment and financing planning process, which serves as an approach for the determination of the so-called base program. The starting point of the calculations is the final report created at the last centralized determination of the marginal cost pricing ranges. This record outlines the optimum of the target function for each experiment within the sensitivity analysis. In the next step, a corresponding marginal price is determined with the aid of a valuation approach for each data constellation and its corresponding target function value. In the case of an acquisition, the valuation object has to be integrated into the linear optimization approach in such a way that it becomes a part of the valuation program. However, in the case of a planned sale, it is not included in the formulated optimization approach. The respective marginal cost prices of the valuation program must be adequately recorded. The record should enable a comparison of the corresponding marginal prices of the base and valuation program. The marginal price can represent the range or possibly even the probability density function after a satisfying amount of experiments, which can be divided into scenarios. If the recorded marginal cost prices of the base program are compared to the equally recorded corresponding values of the valuation program, the following two evaluation possibilities are outlined: 1. If the ranges or distributions of the endogenous marginal cost prices of the base and the valuation program are approximately equal, the determination of the decision value is completed. The present ranges or probability density functions of a decision value are an approximately reliable quantitative decision-making basis for the centralized decision-maker. 2. However, if the ranges or distributions of the endogenous marginal cost prices of the base and valuation program are distinguishable from each other, the business valuation can be expected to influence the decentralized investment and financing 2.3 Determination of One-Dimensional Decision Values 155 45520_Matschke_Griffleiste_SL5.indd 155 45520_Matschke_Griffleiste_SL5.indd 155 16.03.2021 16: 21: 21 16.03.2021 16: 21: 21 decisions of the divisions: A change to the marginal cost prices caused by an acquisition or sale can influence the investment and financing program of the divisions. That is why the divisions must be informed about the modified ranges of the marginal cost prices, but also why it is important to request their feedback. To protect the centralized valuation, the decentralized decision authorities must execute the step of the approximate decomposition decentralized investment calculation with net present values after which the decision-makers must report the hypothetical reaction to the presumed realization of the acquisition/ sale of the business to headquarter. This happens with regard to the decentralized investment and financing decisions of those decision-makers in the form of summed payment sequence ranges or distributions. To estimate the decision value more precisely, the incoming reports of each division are considered on the centralized level on the right-hand side of the depiction of the simulative valuation approach. Moreover, the marginal price ranges or probability density functions are iteratively determined. If the result is significantly changed marginal cost price ranges, new feedback is essential. Since the marginal cost price control overlaps with the problem of uncertainty, it can be expected that the divisions will not change their decision too frequently. Accordingly, a continuous recursion is highly unlikely. By determining the range or the probability density function, the result is a sound investment-theoretic estimate of the decision value. It offers crucial and reliable quantitative decision support for the decision-makers in negotiation situations. The described model of the approximate decomposed valuation, which is summarized below in Figure 2.47 (B RÖSEL 2002, p. 198), is critically evaluated in the next step. Hierarchieebene: Zentrale Planungsinstrumente: - Basisansatz - Bewertungsansatz Hierarchieebene: Divisionen Planungsinstrumente: - Kapitalwert - Zukunftserfolgswert - Steuerungszinsfüße - kon stante Zahlungsreihen 1. Schritt: Zuordnung der Bewertungsobjekte zur relevanten Hierarchieebene 2. Schritt: Investition stheoretisch geleitete Grenzpreisschätzung Bandbreiten/ Verteilungen Figure 2.47: Approximate decomposed valuation First step: Classification of valuation objects to the relevant hierarchy level Second step: Investment-theoretic marginal price estimation Hierarchy level: Headquarter Main planning instruments: • Base approach • Valuation approach Hierarchy level: Divisions Planning instruments: • Net present values • Future performance value Payment sequences Ranges/ Distributions Interest rates 156 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 156 45520_Matschke_Griffleiste_SL5.indd 156 16.03.2021 16: 21: 21 16.03.2021 16: 21: 21 Chapter 2 2.3.3.4.4 Critical Evaluation Heuristics do not generate an optimum solution for problems, especially if no optimum is ex-ante defined in the decision field. Hence, the critical evaluation of a heuristic is characterized by structural deficiency (B ERENS 1992, p. 18 and 24, R OLLBERG 2001, p. 153, H ERING 2017, p. 395). The judgment of the quality of business valuation with the aid of the approximate decomposition is limited to the six model requirements already formulated in Section 2.3.3.2.1 (H ERING 2014, p. 180, H ERING 2000b, p. 372, B RÖSEL 2002, p. 198, H ERING 2017, p. 246 and 395): 1. Subject, target system, and action relation: The investment and financing planning with the approximate decomposition and the approximate decomposed valuation include centralized and decentralized components. On a centralized level, the interests of the owners are explicitly preserved by the formulation of the target functions in the simultaneous models. Since the consideration of the owner’s targets in a partial model requires the solution of the corresponding total model, which is due to the dilemma of the marginal cost pricing theory, the algorithm approximates the necessary price control by the information flow between the hierarchical planning and the decision authorities. The marginal cost prices determined in the simplified total model are made available to the divisions in form of ranges or qualified assumptions about the distributions. Hence, it is assured that centralized and decentralized decision values of the business to be valuated are calculated in accordance with the operationalized target of the owners in the corresponding situation. The represented heuristics proves to be based on a profound marginal cost pricing theory and thus fulfills the first model condition. 2. Decision field relation and determination of the limit value: Having bridged approximate decomposed planning and business valuation, the presented heuristics model allows us to consider the subjective decision field of the valuation subject as far as possible by determining the uncertain decision value. The heuristic approach represents a purpose-oriented method for the determination of the limit of negotiation willingness (concession limit) and fulfills the second model condition, according to the significant restrictions determined in the total model. 3. Possibility of connecting with methods revealing uncertainty: The involvement of business valuation in the investment and financing planning with approximate decomposition represents a heuristic model that does not impose a concrete decision proposal. Instead, it enables an estimation of the ranges or probability density functions of the decision value to represent the opportunities and risks resulting from the valuation objects for the decision-makers. By transparently disclosing real uncertainty, a heuristic synthesis of the total and partial model can be accompanied by the methods revealing uncertainty, namely by a sensitivity and by a risk analysis. The inclusion of this process in the rolling-wave planning of the valuation subject additionally promotes its flexibility and fosters the consideration of temporal structures within the planning. The result of the approximate decomposed valuation under uncertainty is a reasonably estimated decision value of investment theory in the form of a range or a probability density function. Therefore, the just presented procedure also fulfills the third model condition. 2.3 Determination of One-Dimensional Decision Values 157 45520_Matschke_Griffleiste_SL5.indd 157 45520_Matschke_Griffleiste_SL5.indd 157 16.03.2021 16: 21: 22 16.03.2021 16: 21: 22 4. Reasonable effort of information acquisition and information processing: The possibility for decomposing the valuation problem provided in the model and the herewith combined delegation of decision competences on decentralized hierarchy levels offers a simplification, according to the intersection between information acquisition and information processing: The processing of information requires that information has already been collected, which is why the requirements for the planning and decision authorities are significantly reduced. The shift of decision competences to the divisions provokes an increased motivation of the decentralized planning and decision authorities. The motivation boost can lead to positive consequences for the acquisition of information as well as for the realization of made decision (L EUTHIER 1988, p. 206). However, the basic difficulties of the prognosis cannot be completely eradicated by this approach (H ERING 2014, p. 184). With respect to the mentioned advantages, it can be concluded that the information acquisition and information processing effort is kept within reasonable limits. Accordingly, the model fulfills the fourth model requirement. 5. Computability of the calculations: Simplified and manageable linear optimization approaches, which can be solved with efficient software, are used as base and valuation programs. The size of these centralized total models remains steady during the whole process. The more easily calculable partial models, namely the net present value and the future performance value method, are used decentrally. For disclosure of uncertainty, both the manageable total models and the rather simple partial models can be combined with efficient computer-aided procedures. The computability is given throughout the entire valuation process. Hence, the fifth model condition is also fulfilled. 6. Providing individual business decision support: The combination between the centralized and decentralized components enables (and requires) the support of centralized and decentralized decision authorities at the investment and financing planning as well as at the valuation itself. Depending on the importance and size of the valuation object, a hierarchical classification to the corresponding planning and decision authorities is necessary. Hence, the coordination processes, based on the approximate decomposition, can support the flexibility of the valuation subject and fulfill the sixth model condition. Despite the possible delegation of decision competences to the divisions, the heuristics demand high requirements for the market overview and the experience of the employees in the headquarter. The complete procedure is dependent on the quality of the centrally estimated marginal cost prices (H ERING 2014, p. 181). After the determination of multi-dimensional decision values in Section 2.2 and the examination of general opportunities for the determination of one-dimensional decision values in Section 2.3, Section 2.4 addresses selected problems of decision value determination. Section 2.4.1 considers why the state marginal price model is suitable for a decision value determination relating to smalland medium-sized enterprises. Section 2.4.2 explains the consequences for the decision value arising from changes in the decision field using numerical examples. In this context, the state marginal price model is used. Section 2.4.3 deals with the decision value determination regarding conflict situations of the type merger and demerger, before a range of joint conflict situations are examined in Section 2.4.4. 158 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 158 45520_Matschke_Griffleiste_SL5.indd 158 16.03.2021 16: 21: 22 16.03.2021 16: 21: 22 Chapter 2 2.4 Selected Problems of Decision Value Determination 2.4.1 Valuation of Small and Medium-Sized Enterprises 2.4.1.1 Valuation-relevant Idiosyncrasies Although it appears to be relatively easy to differentiate small and medium-sized enterprises (SMEs) from large businesses by examining numerical size, a clear distinction can only be made according to the enterprises’ qualitative features. Qualitative features have the advantage of revealing the essence (character) of businesses (M UGLER 1998, p. 19). Since the qualitative allocation does not primarily focus on accuracy, but instead on purpose (fulness) or expediency, a multitude of quantitative distinction features can be found in the economic literature. Below, four criteria are summarized that characterize the major idiosyncrasies (also particularities or peculiarities) of SMEs (B RÖSEL / M ATSCHKE 2004, p. 50). • Personal union of owner(s) and general manager(s): The owners of SMEs usually run the business themselves. The firms represent the economic livelihood of the owners, usually being their only source of income or at the least major one. Sometimes, the transition between the corporate and the private sphere is smooth, which requires close attention to the distinction of the spheres. The owner, who often has profound sector-specific experience, could lack economic knowledge despite having control over all issues and being involved in all conflict situations. • Grave incompleteness of the capital market: For the businesses in question the assets of the business owners and their families represent an important and usually strongly limited capital basis. Raising (additional) debt and equity and avoiding liquidity problems are more difficult tasks than they are for (listed) large companies. From the perspective of new institutional economics, these financial problems are based on an asymmetric information distribution that is far more common among SMEs than among large firms. These information problems are the reason for the existence of financial intermediaries and the presence of an incomplete capital market. The latter causes the scarcity of capital and the discrepancy of borrowing and interest rates (H ERING 2017, p. 139). The incomplete capital market represents a major financing restraint for SMEs that usually do not have access to the capital market. • The owners as centralized decision authority: For owners the size of their SME means it is manageable. The relations between the owners and their employees are often rather close and informal. The organizational structure is characterized by short instruction and information distances. Often, the owner is the centralized decision authority. • Significant dependence of business performance on the owner: The performance of an SME is strongly dependent on the owner’s character because the success of the SME is often based on a tight network of personal contacts of the owners and their colleagues. If the original owner leaves the SME due to a change of ownership, this can translate into a significant loss of knowledge and a lack of a close network of contacts (that is, customers and suppliers). A trend extrapolation on the basis of the 2.4 Selected Problems of Decision Value Determination 159 45520_Matschke_Griffleiste_SL5.indd 159 45520_Matschke_Griffleiste_SL5.indd 159 16.03.2021 16: 21: 22 16.03.2021 16: 21: 22 results of prior periods can only be used to a limited extent as a support for future performance. Furthermore, the uncertainty is at least partly significantly higher compared to large enterprises, because an SME is often less diversified. They also tend to be more crisis-prone due to their financing gaps and other liquidity problems. Additionally, SMEs often have fewer planning resources than large businesses. 2.4.1.2 State Marginal Price Model in Light of Valuation-Relevant Idiosyncrasies The valuation of SMEs requires a valuation model capable of working with all the qualitative idiosyncrasies described above. Whether the general state marginal price model fulfills these requirements will be now examined concerning all valuation-relevant particularities of SMEs (B RÖSEL / M ATSCHKE 2004, p. 64): • Personal union of owner(s) and general manager(s): The management provides business valuations in the case of a planned change of ownership on behalf of the owners of large businesses and it often examines the heterogeneous targets of several owners. In the case of a personal union, it is an advantage that the owner operating as general manager knows their own preferences. Hence, general management is the measuring plane of target fulfillment in SMEs. It has to reflect its target system with a corresponding formulation of the target function (the choice between the targets of asset maximization and income maximization). Since dividend payments, or distributions in general, for the owner are used as an operand in the general state marginal price model, it is considered that SMEs to be valuated represent a personal source of income for the owner. Because simultaneous models impose onerous requirements for users to have a sound methodological knowledge of linear optimization, the lack of economic expertise of the owner(s) can make it necessary to accept help from third parties (D ECHANT 1998, p. 143). According to the decision value determination, the owner/ manager should trust those external advisors/ consultants, who not only have profound knowledge of investment-theoretic methods but apply them in the sense of the classical functional valuation theory. • Grave incompleteness of capital markets: The application of partial models requires a knowledge of marginal interest rates of each period on the incomplete capital market (see below). This implies the solution of a total model. In the explained multi-period total model, the alternative investment and financing options and also the financial object interdependencies of the decision field are identified simultaneously and represent a fairly realistic simulation. Hence, a separate determination of marginal interest rates is not necessary. Additionally, the guarantee of solvency (ability to pay) is ensured at each time t by the liquidity restrictions. An integration of further linearly portrayable restrictions into the model is theoretically possible (and in practice also feasible in an SME). • The owners as the centralized decision authority: Total models are generally designed for businesses with centralized decision authorities, where the management body has the final authority and decision-making competence. For the determination of the decision value the simultaneous model has to be filled with the corresponding data. Since SMEs are manageable because of their size, and that the valuation 160 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 160 45520_Matschke_Griffleiste_SL5.indd 160 16.03.2021 16: 21: 22 16.03.2021 16: 21: 22 Chapter 2 of those businesses is not a day-to-day business in practice, the information acquisition and the information processing effort is regularly acceptable. The continuous implementation and solution of the total model are not necessary. • Significant dependence of business performance on the owner: The significant uncertainty for SMEs is based on business performance depending on the owner. That creates a necessity for a model that provides the decision-maker with important quantitative information; specifically, the result of the valuation process should be available in the form of a possible range or distribution of the decision value. The state marginal price model emphasizes the idea of state-based payment streams originating from the financing theory and therefore proves to be very useful for business valuations under uncertainty. Additionally, the risk analysis and the sensitivity analysis are adequate methods to reveal uncertainty. Those forms of analysis can be combined with the state marginal price model to transparently convey the consequences of uncertainty. Hence, it can be concluded that the decision value determination with the aid of the general state marginal price model, originating from investment theory, is particularly suited for the valuation of SMEs. 2.4 Selected Problems of Decision Value Determination 161 45520_Matschke_Griffleiste_SL5.indd 161 45520_Matschke_Griffleiste_SL5.indd 161 16.03.2021 16: 21: 22 16.03.2021 16: 21: 22 2.4.2 Effects on the Decision Value Through Modifications of the Decision Field Valuation methods that should provide market values often abstract broadly or even completely from a target system and the decision field of the valuation subject. However, the consideration of these subjective aspects is of considerable importance for the determination of the decision value and is therefore demonstrated in the following valuation examples. Each perspective of the presumptive buyer and presumptive seller features a numerical example of the consequences of modifications in the decision field on the amount of the decision value. For this purpose, the investment-theoretic total model is implemented, namely the state marginal price model because it explicitly considers the decision field at the decision value determination. Nevertheless, a non-dominated, disjoint, and one-dimensional conflict situation of the type acquisition/ sale is assumed (cf. the conflict cube in Figure 2.6), in which it is solely the size of the (cash) price that is relevant to an agreement at the valuation date. Variation A The following section shows from the perspective of a presumptive buyer how the modifications of the valuation subject’s decision field have effects on the firm value. Therefore, the data from the situations outlined in Section 2.3.3.2.2.2 are used as the initial situation. However, in contrast to the initial situation illustrated in Figure 2.20, it is assumed in the variation that the primary bank offers an additional bullet loan zED (German: zusätzliches endfälliges Darlehen) in the case of the acquisition of the business U. This loan is granted at t = 0 up to a maximum amount of 300 GE at an annual interest rate of 8 %. In the initial situation, additional funds could only be obtained at an interest rate of 10 %. That is why a positive change of the financial conditions is outlined here. A summary of the variation is shown in Figure 2.48. Since it is assumed that the additional financing options improved in the case of the acquisition, there are no changes in the base program. Hence, the realized uniform payment stream 32,6176 GE remains unaltered. That is also why the comprehensive financial plan of the base program in the variation corresponds to one of the base programs in the initial situation in Figure 2.49 (cf. Figure 2.21). t AK ED zED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U 01 -100 30 50 -4 300 -24 -1 1,05 -1 1 -1,1 1 10 30 30 P? 60 234 Limit 40 50 -4 -4 55 1 -54 1 -24 -24 -324 1 ∞ 1,05 -1 1,05 ∞ ∞ -1 1,05 ∞ ∞ -1,1 1 -1,1 ∞ ∞ 1 -1,1 ∞ 1 30 30 40 20 630 1 420 1 Grenze Figure 2.48: Exemplary data from the buyer’s perspective - variation A 1 1 1 1 1 1 EN A max = 162 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 162 45520_Matschke_Griffleiste_SL5.indd 162 16.03.2021 16: 21: 23 16.03.2021 16: 21: 23 Chapter 2 The payment stream 32,6176 GE in the variation is achieved at a marginal price for the business U of 410,4742 GE (initial situation 391,4550 GE). The valuation program contains - in addition to the valuation object U - the object AK, the additional bullet loan zED, and the single-period operating loans KA. Figure 2.50 shows the comprehensive financial plan of the valuation program of the variation. Conclusions: The positive change of the financing conditions (in the example) leads to an increase in value. The maximum price is rising. The example makes clear that equal future performances do not necessarily lead to identical marginal prices if the included decision fields are deviating from each other. The discounting of future performances, for instance with the aid of the pseudo-objecti- Personal equity assets EM Internal financing IF t = 0 t = 1 10 30 30 Investment AK Bullet loan ED Additional bullet loan zED Operating loan KA -100 42,7680 30 -3,4214 0 49,8496 0 30,8736 t = 2 t = 3 30 30 t = 4 630 40 -3,4214 50 -3,4214 0 0 55 -46,1894 0 Financial investments GA KA-, GA-paybacks Withdrawal EN A Payment balance -54,8346 -32,6176 -10 -32,6176 0 Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.49: Comprehensive financial plan of the buyer’s base program - variation A 49,8496 30,8736 -33,9610 -43,9610 -32,6176 0 -32,6176 0 46,1591 -32,6176 652,3520 43,9610 652,3520 EN A max = P max A P max = t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 30 30 30 630 Business U Investment AK Bullet loan ED Additional bullet loan zED -100 60 30 50 300 -4 -24 40 40 20 50 -4 -24 -4 -24 420 55 -54 -324 Operating loan KA Financial investments GA KA-paybacks Withdrawal EN A 153,0918 109,0186 -32,6176 -168,4010 -32,6176 Payment balance Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 410,4742 153,0918 0 109,0186 70,5381 38,2095 -119,9205 -32,6176 -77,5919 -32,6176 -42,0305 -32,6176 0 70,5381 0 38,2095 652,3520 652,3520 Figure 2.50: Comprehensive financial plan of the buyer’s valuation program - variation A 2.4 Selected Problems of Decision Value Determination 163 45520_Matschke_Griffleiste_SL5.indd 163 45520_Matschke_Griffleiste_SL5.indd 163 16.03.2021 16: 21: 23 16.03.2021 16: 21: 23 ve interest rates of the CAPM, leads to different values, sometimes even within the DCF methods. However, insufficient consideration of the decision field of the valuation subject means the process also inevitably leads to values that are not suitable for decision support in conflict situations like those discussed here. Variation B In the next step, the seller’s perspective is examined. Therefore, the example in Section 2.3.3.2.3.2 is used, which represents the initial situation. With the help of the state marginal price model, it is shown how modifications of the decision field affect the firm value. In contrast to the initial situation, the variation represents the rationing of credit supply for the valuation subject by the primary bank. The valuation subject can use personal equity assets EM, internal financing IF, and the bullet loan ED of 50 GE. However, in contrast to the initial situation in Figure 2.25, additional finance (KA t ) is no longer available in unlimited amounts at a short-term interest (borrowing) rate of 10 % p. a. (KA t ), but only at an interest rate of 11 % p. a. and up to a maximum of 30 GE. The data of the variation is presented in Figure 2.51. The valuation subject now has to partly waive the investment in object AK in the base program (only a fraction of 0,876685 = 87,6685 % is realized) and takes out operating loans KA 0 = 30 GE at t = 0 and also KA 1 = 13,3310 GE at t = 1. Moreover, financing investments of GA 2 = -13,9384 GE at t = 2 and of GA 3 = -52,1380 GE at t = 3 are made. Finally, one arrives at a withdrawal stream 32,3315 GE from the base program (initial situation: 32,6176 GE) and a payment balance of 646,6310 GE (initial situation: 652,3520 GE) at the end of the planning period. The comprehensive financial plan of the variation is represented in Figure 2.52. By rationing the financing opportunities, the marginal price (bottom price) drops to 190,2088 GE compared to that in the initial situation 196,2159 GE). Due to the elimination of the valuation object KU from the investment and financing program of the valuation subject and the corresponding payment of the purchase price (cash inflow), the possibility of an investment in AK arises in the valuation program. Additionally, the valuation program comprises internal financing IF, personal equity assets EM, and single-period financial investments GA (cf. Figure 2.53). t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF thereof 0 -100 50 -1 1 10 30 MU 30 KU P? 1234 30 40 -4 -4 50 55 -4 -54 1,05 -1 1,05 -1 1,05 -1 1,05 -1,11 1 -1,11 1 -1,11 1 -1,11 30 30 30 630 18 19 12 11 18 420 12 210 Limit Grenze Figure 2.51: Exemplary data from the seller’s perspective - variation B 11 11 ∞ ∞ ∞ ∞ 30 30 30 30 30 30 30 30 11 11 11 11 EN B max = EN max = P min B = (P min = 164 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 164 45520_Matschke_Griffleiste_SL5.indd 164 16.03.2021 16: 21: 23 16.03.2021 16: 21: 23 Chapter 2 Conclusions: The rationing of credits leads to a reduction of the decision value (in the example). The lowest price limit (bottom price) falls. The analysis shows that the marginal price changes because modified and rationed objects alter the target function value of the base program. The inclusion of the corresponding object in the valuation program is not carried out and thus does not represent a compulsory requirement for a change of the decision value. To sum up, the variation of the sale situation clarifies that equal future performances do not automatically have to lead to identical marginal prices if the considered decision fields differ from each other. t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 30 30 30 630 Investment AK Bullet loan ED Operationg loan KA Financial investments GA -87,6685 50 26,3005 -4 30 13,3310 35,0674 -4 43,8342 -4 -13,9384 -52,1380 48,2176 -54 0,876684502 KA-, GA-paybacks Withdrawal EN B Payment balance Debt from KA -32,3315 -33,3000 -32,3315 0 30 0 13,331 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.52: Comprehensive financial plan of the seller’s base program - variation B -14,7974 -32,3315 14,6353 -32,3315 0 0 54,7449 -32,3315 646,6310 13,9384 52,1380 646,6310 t = 0 t = 1 t = 2 t = 3 t = 4 Personal equity assets EM Internal financing IF 10 30 30 30 30 630 Business KU Investment AK Bullet loan ED Operating loan KA -100 -12 30 -11 40 -12 50 -210 55 Financial investments GA GA-paybacks Withdrawal EN B Payment balance -97,8773 -118,4397 102,7712 -32,3315 -190,2088 -32,3315 0 Debt from KA Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.53: Comprehensive financial plan of the seller’s valuation program - variation B 97,8773 118,4397 -151,0301 124,3617 -194,2500 158,5816 -32,3315 0 -32,3315 0 203,9625 -32,3315 646,6310 151,0301 194,2500 646,6310 2.4 Selected Problems of Decision Value Determination 165 45520_Matschke_Griffleiste_SL5.indd 165 45520_Matschke_Griffleiste_SL5.indd 165 16.03.2021 16: 21: 23 16.03.2021 16: 21: 23 2.4.3 Determination of the Decision Value in Conflict Situations of the Merger Type and Demerger Type 2.4.3.1 Conflict Situations of the Merger Type 2.4.3.1.1 Presentation H ERING emphasizes that despite the key role of mergers in practice this topic is largely neglected in business valuation theory (H ERING 2004a, p. 148). Mergers, and also cooperation, can be associated with business combination. However, mergers are a form of corporate concentration that go hand in hand with economic integration and the loss of legal independence of at least one involved business (merger by absorption or merger by new formation). Once a merger is finalized, the businesses involved often form a new economic and legal unit (a merger by formation), which is most obviously represented by a shared firm name. Hence, mergers have a much closer relationship than a business combination, in which one of the businesses loses its economic independence, but maintains its legal independence (corporate group; parent and subsidiaries). However, such a business combination may have an effect equivalent to a merger (R EICHERTER 2000, p. 45). In a conflict situation of the merger type, the distribution of the property rights (direct or indirect ownership shares) and ultimately the possible distribution of future performances from the new business (merger by formation) are central. The conflict situation of a merger is very broadly defined. Even the entry of a new shareholder or partner into an existing company is categorized in this conflict situation, as long as the existing shareholders do not reduce or cease their financial commitment in the business on behalf of the new shareholder. In all merger types, the number of involved businesses and involved conflicting parties is not limited to two. It is not only possible but rather common that several businesses with many owners merge. In the following, a non-dominated, disjoint, one-dimensional conflict situation of the merger type is analyzed (cf. the conflict cube of the type acquisition/ sale in Figure 2.6), in which only the level of ownership interest in the new company is deemed relevant at the valuation date. Hence, the respective decision value of the owners of the merged businesses is not the price - as in the previously analyzed conflict situations of the acquisition/ sale type - but the respective minimum ownership interest (critical ownership interest rate or marginal rate) in the new business and its distribution stream (merger by formation). Such a rate guarantees that the owner does not suffer disadvantages by the merger (M ATSCHKE 1975, p. 327, H ERING 2004a, p. 148). It is assumed in the present case that two opposing conflicting parties look for a merger by way of a new formation, that is, the incorporation of a new entity. The adopted perspective is that of the conflicting party as the valuation subject, which owns the business Ü (Geman: Übernahmegesellschaft). The valuation subject is opposed by a conflicting party, which brings a second business Z into the merger. The valuation from the perspective of the latter party is not the topic of the following discussion. Both businesses Ü and Z are merged and ultimately form the new business F (German: Fusionsα min 166 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 166 45520_Matschke_Griffleiste_SL5.indd 166 16.03.2021 16: 21: 24 16.03.2021 16: 21: 24 Chapter 2 gesellschaft). Afterwards, both parties will hold shares/ interest in the new economic unit F. This decision problem can be resolved by using the state marginal rate model (in German: Zustands-Grenzquotenmodell) after H ERING (2004a, p. 148, R EICHERTER 2000, p. 196). It is based on the general state marginal price model. The process of business valuation in the conflict situation of the merger type can be divided into three steps: 1. Determination of the maximum stream of withdrawals of the new business Ü of the valuation subject (pre-merger program): Which maximum utility, in the sense of can the valuation subject obtain from business Ü without the agreement on the merger in the conflict situation? This performance must be achieved again after the merger if the decision subject is not to be disadvantaged compared to the pre-merger situation. Hence, the performance of the pre-merger program (as the base program) becomes a constraint. 2. Determination of the maximum stream of withdrawals of the new business F (merger program): Which maximum utility, in the sense of can all conflicting parties generally obtain from the business F after an agreement in the current conflict situation? 3. Determination of the minimum marginal rate (and the valuation program): To what minimum extent (share/ stake) must the valuation subject be involved in the utility of the new business F, without being disadvantaged in compared to refraining from this action? In other words, without the merger, there is still a recourse to the withdrawal stream These three steps are generally discussed below. In Section 2.4.3.1.2 below, all steps are outlined and an example offered. From the perspective of the valuation subject, that is, the owner of Ü before the merger, the determination of a minimum ownership interest in the newly merged business requires the determination of the expected withdrawals that business Ü can distribute to its owners in future uncertain states s. In the first step of the merger, the optimum investment and financing program of business Ü is to be determined, which leads to the maximum utility level for the valuation subject without an agreement. This base approach leads to the pre-merger program. The model presented below is planned for a set of S future uncertain states s (or of S periods). The valuation subject, who desires income maximization, has the possibility of J different investment and financing objects. For each object j with g js as a stream of cash flows, the state-related payment sequence g j : = (g j0 , g j1 , ..., g js , ..., g jS ) is given for each state s. The decision variable x j defines how often an object j is realized, where represents the existing possible upper limit. Additionally, an autonomous payment balance b s of any amount comes up in each state s. The valuation subject looks for a possibly broad withdrawal stream EN for consumption purposes, where the distribution occurs in each state s. EN Ü EN Ü , EN F EN F , α min α min EN F EN Ü . EN Ü x j max w s Ü ⋅ EN Ü 2.4 Selected Problems of Decision Value Determination 167 45520_Matschke_Griffleiste_SL5.indd 167 45520_Matschke_Griffleiste_SL5.indd 167 16.03.2021 16: 21: 24 16.03.2021 16: 21: 24 The linear approach for the determination of the pre-merger program is defined as follows: with 1. Liquidity restrictions: 2. Capacity restrictions: 3. Non-negativity constraints: In the second step, the merger program and hence the expected withdrawals have to be determined that all owners are entitled to from the new business F (merger by formation Ü and Z). In this context, the problem might arise that the merged business (the newly formed business F) is aiming for another objective (e.g., asset maximization) than the single businesses without the merger. This could happen because the valuation subject is, for instance, no longer able to enforce the originally desired structure of the withdrawal stream controlled by the withdrawal weightings from business F, due to the share of votes or balance of power with regard to so-called domination agreements. It is assumed now that the weightings for the newly merged business F are already negotiated. In real conflict situations, these are original conflict-resolution-relevant facts that are often the results of the negotiation process. According to the negotiation result and with respect to the weighting factors a corresponding marginal rate has to be determined. That is also why generally no predetermined weightings are used in multi-dimensional conflict situations. Here and in the following example, the complexity is reduced and it is assumed that the weightings for the merged business F are already determined. Accordingly, it is possible to speak of a one-dimensional conflict situation. Below, the so-called merger approach for the determination of the merger program is presented. Formally, it only differs regarding the index F that indicates the new business. However, materially it can be assumed that the approach is formulated using significantly different data. This is due to fact that the merged business F, is (often substantially) larger than the originally regarded business. The linear approach for the determination of the merger program reads as follows: with 1. Liquidity restrictions: max. Entn; Entn : = EN Ü − g js Ü ⋅ x jÜ j = 1 J Ü ∑ + w s Ü ⋅ EN Ü ≤ b sÜ ∀ s ∈ {0, 1 2, … , S} x jÜ ≤ x jÜ max ∀ j ∈ {1, 2, … , J Ü } x jÜ ≤ 0 ∀ j ∈ {1, 2, … , J Ü } EN Ü ≥ 0. EN F w s Ü w w, w max. Entn; Entn : = EN F − g js F ⋅ x jF j = 1 J F ∑ + w s F ⋅ EN F ≤ b sF ∀ s ∈ {0, 1, 2, … , S} 168 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 168 45520_Matschke_Griffleiste_SL5.indd 168 16.03.2021 16: 21: 24 16.03.2021 16: 21: 24 Chapter 2 2. Capacity restrictions: 3. Non-negativity constraints: If the maximum possible withdrawals of the business Ü of the valuation subject and those of the new business after the merger are assessed, the marginal rate can be determined in the third step (M ATSCHKE 1975, p. 329). In the special case that the vector of withdrawals of the merged business is the multiple of the vector of withdrawals of the transferred business , the minimum ownership interest of the previous owners of Ü corresponds to the following quotient (ratio), according to its respective share β of the business to be transferred. This ratio is also called “trivial” valuation formula: However, this special case only arises if both businesses have the same objective and hence the structure of the distribution stream is equal, that is, if Such structural equality is not essential for the determination of the minimum marginal rate of a specific owner. On the contrary, the minimum rate has also to be determined in cases when the withdrawal of business Ü does not correspond to the target of the merged business F. This can happen when the weightings for F and Ü differ for on at least one condition s: Moreover, this can also occur if the target of business Ü is income maximization, whereas business F strives for asset maximization (H ERING 2004a, p. 151). H ERING (2004a, p. 152) states that to make an inevitable economic comparison of marginal rate determination in this situation, the private decision field of the shareholders must be taken into account. Only by supporting private financial rearrangements can the previous owners (shareholders) transform the payment stream of the merged business back into its original desired income structure. Depending on the target and the private decision field of the considered shareholder, the scope (volume) of business x j must be determined by the base approach (M AT- SCHKE / W ITT 2004, p. 262). The state-related cash flows of the private objects j are With the help of financial restructuring, that is, increases in or reductions of of the owner’s private businesses j, the structure of the distribution stream (such as dividends) of the newly merged business can be transformed into the structure of the original business to be transferred. Finally, the minimum required shares in the merged business are determined with the following linear optimization approach from the perspective of the owners (“general” valuation approach): x jF ≤ x jF max ∀ j ∈ {1, 2, … , J F } x jF ≤ 0 ∀ j ∈ {1, 2, … , J F } EN F ≥ 0. EN Ü max EN F max α min EN F * + *** EN Ü * + *** α min α min = β ⋅ EN Ü max EN F max . w s Ü = w s F ∀s. w s Ü ≠ w s F . g js priv . x j pos x j neg 2.4 Selected Problems of Decision Value Determination 169 45520_Matschke_Griffleiste_SL5.indd 169 45520_Matschke_Griffleiste_SL5.indd 169 16.03.2021 16: 21: 25 16.03.2021 16: 21: 25 with 1. Liquidity restrictions: 2. Restrictions according to the increase of businesses: 3. Restrictions according to the reduction of businesses: 4. Non-negativity constraints: H ERING (2004a, p. 153) points out that the present valuation approach minimizes the ownership interest for the shareholders (in a manner similar to the discussed “trivial” case) under the constraint that the incoming payment stream after the merger (including possible modifications in the private wealth sector) does not fall below its distribution before the merger in any state s. Then, the following is true: While the approach for the determination of the pre-merger program explicitly includes the target function of the shareholder, the objective of the shareholder is reflected only implicitly in the valuation approach by the resulting upper limits for increases or decreases of the businesses j. The maximum possible increase in the business volume is the difference from the upper limit, regarding the realization of the respective object and the planned implementation x j . Accordingly, it can only be reduced by the planned implementation If a lower limit for the execution of a business exists, the upper limit for a reduction of this object corresponds to the difference of the actual implementation and this lower limit. If several owners are involved in the business to be transferred, the individual minimum ownership interest rates of each owner have to be determined and added up for the determination of the required minimum marginal rate in the merged business. The resulting minimum rate of the transferred business in the merged business does not disadvantage the previous owners after the merger. However, it could still lead to conflicts among the shareholders of the transferred business if the ratios of the individual ownership interests do not correspond to the shares in the new business. To avoid such conflicts, the highest resulting marginal rate has to be determined for the whole business by min. A; A : = α g js priv j = 1 J priv ∑ ⋅ x j priv + α ⋅ w s F ⋅ EN F max ≥ w s priv ⋅ EN priv max − b s priv ∀ s ∈ 0, 1, 2, ..., S { } x j priv ≤ x j priv max ∀ j ∈ {1, 2, … , J priv } x j priv ≤ x j priv max ∀ j ∈ {1, 2, … , J priv } x j priv ≤ 0 ∀ j ∈ {1, 2, … , J priv } α ≥ 0. w s Ü ⋅ EN Ü max g js priv j = 1 J priv ∑ ⋅ x j pos − g js priv j = 1 J priv ∑ ⋅ x j neg transformation of private credits and investment ! " #### $ #### + α ⋅ w s F ⋅ EN F max distribution according to part α of the optimal withdrawal stream of the merged business F ! " ## $ ## excess deposits of the shareholders of the businiess F (after merger) ! " ######## $ ######## ≥ β ⋅ w s Ü ⋅ EN Ü max excess deposits of the shareholders of the business Ü, according to his or her part β ! " ## $ ## . x j max x j neg . 170 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 170 45520_Matschke_Griffleiste_SL5.indd 170 16.03.2021 16: 21: 26 16.03.2021 16: 21: 26 Chapter 2 extrapolating the individual marginal rates on the total capital (total assets) of the previous business (H ERING 2004a, p. 153). As already shown in the example of the conflict situation of the acquisition/ sale type in Section 2.3.3.3.2, there is a relation between the total model (in the example, the state marginal price model) and the partial model (in the example, the future performance value method). The formula for the future performance value can be derived based on the duality theory of the linear optimization, assigning a corresponding duality problem to each primal problem. Even in the case of a merger, a complex valuation formula for the determination of the marginal rate can be derived from this relation in the state marginal rate model (H ERING 2004a, p. 153). The minimum marginal rate is calculated as a ratio from the proportional net present value of the business to be valuated Ü without a merger and the net present value of the modifications in the private program as well as the net present value of the merged business F (merger by formation). While the net present value after the merger results from the distributions of the merged business F in the private program, having the endogenous discount factors the net present value without a merger can be calculated as the net present value with the discounted distributions of the business Ü, with the endogenous discount factors of the private program. The net present value of the transformations in the private program describes the resulting changes of the net present values of the private transformation businesses j at the transfer from the basis to the valuation program. The complex valuation formula reads as follows: If the net present value of the transformations in the private program amounts to zero, the complex valuation formula can be transformed into the simplified valuation formula for the marginal rate at the merger. Consequently, if no marginal object of the private program reaches its upper limit, the simplified valuation formula reads as follows (H ERING 2004a, p. 152): α min ρ s priv , ρ s priv C j priv α min = β ⋅ w s Ü ⋅ EN Ü max ⋅ ρ s priv s = 0 S ∑ part β of the valuation subject at the capital value without merger = distribution of the business Ü " # $$$ % $$$ − Δ x j ⋅ C j p r iv j = 0 J priv ∑ capital value of the tranformations in the private program " # $ % $ w s F ⋅ EN F max ⋅ ρ s priv s = 0 S ∑ capital value after the merger = capital value of the distribution of the merged business F & ' $$$ ( $$$ . α min = β ⋅ w s Ü ⋅ EN Ü max ⋅ ρ s priv s = 0 S ∑ part β of the valuation subject at the capital value without merger = distribution of the business Ü " # $$$ % $$$ w s F ⋅ EN F max ⋅ ρ s priv s = 0 S ∑ capital value after the merger = capital value of the distribution of the merged business F & ' $$$ ( $$$ 2.4 Selected Problems of Decision Value Determination 171 45520_Matschke_Griffleiste_SL5.indd 171 45520_Matschke_Griffleiste_SL5.indd 171 16.03.2021 16: 21: 27 16.03.2021 16: 21: 27 and for β = 1, that ist, for a (non-influential) sole owner of the business Ü: The complex and the simplified formulas enable differently structured distribution streams before and after the merger. They can ultimately be transformed back into the trivial valuation formula if recourse to the private program is necessary because for all states s the weightings are 2.4.3.1.2 Numerical Example In the following, the statements for the merger are examined with a plain numerical example under the assumption of (quasi-)certain expectations. The valuation subject, who has a planning horizon of four periods, owns business U as a sole proprietor at the valuation date (β = 1). This business represents one of the businesses to be transferred in the merger. A cashflow of 40 GE results from business U at t = 0 followed by a perpetuity of 30 GE (starting at t = 1). At t = 0, the valuation subject has the option to make an investment AK which the following payment sequence including its acquisition cost (-100 GE, +30 GE, +40 GE, +50 GE, +55 GE). The valuation subject can raise a bullet loan ED of 50 GE from the primary bank at an annual interest rate of 8 % p. a. with a total term of four periods. Financial investments (GA t ) may be made for economic purposes at an interest rate of 5 % p. a. Furthermore, financial funds are available in unlimited amounts at a short-term interest (borrowing) rate of 10 % p. a. (KA t ). The valuation subject strives for income maximization. The desired time structure is = 1 : 1 : 1 : 1 : 21. Again, this means that the last distribution contains both the regular distribution and the present value of the perpetuity at an interest rate of 5 %. Regarding the payment stream of business Ü, the present value of the perpetuity is considered also. The initial data are represented in Figure 2.54. α min = w s Ü ⋅ EN Ü max ⋅ ρ s priv s = 0 S ∑ part β of the valuation subject at the capital value without merger = distribution of the business Ü " # $$$ % $$$ w s F ⋅ EN F max ⋅ ρ s priv s = 0 S ∑ capital value after the merger = capital value of the distribution of the merged business F & ' $$$ ( $$$ = future performance value of the company Ü for the sole owner of Ü without merger future performance value of the company Ü for the sole owner of Ü in the merged business F after the merger . w s Ü = w s F : α min = β ⋅ w s Ü ⋅ EN Ü max ⋅ ρ s priv s = 0 S ∑ − 0 capital value of the tranformations in the private program " # $ % $ w s F ⋅ EN F max ⋅ ρ s priv s = 0 S ∑ = β ⋅ EN Ü max EN F max ⋅ w s Ü ⋅ ρ s priv s = 0 S ∑ w s F ⋅ ρ s priv s = 0 S ∑ = β ⋅ EN Ü max EN F max . Ü Ü Ü Ü Ü 0 1 2 3 4 w : w : w : w : w 172 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 172 45520_Matschke_Griffleiste_SL5.indd 172 16.03.2021 16: 21: 28 16.03.2021 16: 21: 28 Chapter 2 In the first step, the pre-merger program (base program) is to be determined. This can be done with the aid of the following linear optimization approach, which is solved with the simplex algorithm: under the constraints: A maximum uniform income stream of = 32,6176 GE results from the base program. The assets at the end of the planning period amount to 652,3520 GE. This value equals the present value of the perpetuity at an interest rate of 5 % p. a. Figure 2.55 demonstrates the comprehensive financial plan of the pre-merger program. t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 Ü 01 -100 30 50 -4 -1 1,05 -1 1 -1,1 1 40 30 234 Limit 40 50 -4 -4 55 1 -54 1 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ 30 30 630 1 Grenze Figure 2.54: Data for the determination of the pre-merger program 1 1 1 max . Entn; Entn : = EN Ü 100 ⋅ AK - 50 ⋅ ED + 1 ⋅ GA 0 - 1 ⋅ KA 0 + 1 ⋅ EN Ü ≤ 40 -30 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 - 1 ⋅ KA 1 + 1 ⋅ EN Ü ≤ 30 -40 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 - 1 ⋅ KA 2 + 1 ⋅ EN Ü ≤ 30 -50 ⋅ AK + 4 ⋅ ED - 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 - 1 ⋅ KA 3 + 1 ⋅ EN Ü ≤ 30 -55 ⋅ AK + 54 ⋅ ED - 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN Ü ≤ 630 GA t , KA t , EN Ü ≥ 0 ∀ t AK, ED ≤ 1. EN Ü max Business to be transferred Ü Investment AK t = 0 t = 1 40 -100 30 30 Bullet loan ED at 0,855356 Operating loan KA Financial investments GA KA-, GA-paybacks 42,7680 49,8496 -3,4214 30,8736 -54,8346 t = 2 t = 3 30 40 30 50 t = 4 630 55 -3,4214 -3,4214 -33,9610 -43,9610 -46,1894 46,1590 Withdrawal EN U Payment balance Debt from KA Deposits from GA -32,6176 0 -32,6176 0 49,8496 30,8736 Present value of perpetuity EN/ 0,05 Figure 2.55: Comprehensive financial plan of the pre-merger program -32,6176 0 -32,6176 0 43,9610 -32,6176 652,3520 652,3520 2.4 Selected Problems of Decision Value Determination 173 45520_Matschke_Griffleiste_SL5.indd 173 45520_Matschke_Griffleiste_SL5.indd 173 16.03.2021 16: 21: 29 16.03.2021 16: 21: 29 The valuation subject is in a conflict situation in the merger type where the ownership interest represents the single conflict-resolution-relevant fact. In this context, the valuation subject negotiates with another conflicting party, who desires to include the business Z in the merger. Again, such a merger is usually referred to as a merger by formation. Business F (a merged business) is to be founded, hence the previous businesses Ü (the valuation subject) and Z (of the negotiation party) are merged. Shortly before the negotiations, both merger parties develop their ideas of the measures to be implemented in the case of the merger (merger strategy) and the modified decision field (R EICHERTER 2000, p. 206). From the business F, the sequence of cash flows (70 GE, 80 GE, 90 GE, 90 GE, 100 GE) is expected from t = 0 to t = 4 and a perpetuity of 100 GE from t = 5 on. Additionally, an investment opportunity AK I is available where the payment sequence including the initial price now is (-100 GE, +35 GE, +45 GE, +55 GE, +60 GE). This is due so-called economies of scope. Moreover, the investment possibility AK II is available for the business F with the payment sequence (-80 GE, +20 GE, +40 GE, +70 GE) for three periods. After the merger at t = 0, the primary bank makes offers a bullet loan ED in the amount of 50 GE to business F at an annual interest rate of 8 % p. a. with a total term of four periods. Additional financial funds are available as operating loans in unlimited amounts at a short-term (borrowing) interest rate of 9 % p. a. (KA t ). Financial investments (GA t ) can still be made at an interest rate of 5 % p. a. The data for the determination of a merger program are illustrated in Figure 2.56. It is assumed that both the valuation subject and the negotiation partner desire income maximization. The following scenarios should be distinguished from one another: a) The valuation subject and the negotiation partner agree on the temporal withdrawal structure = 1 : 1 : 1 : 1 : 21, which corresponds to the desired structure of the valuation subject ( = 1 : 1 : 1 : 1 : 21). b) The valuation subject and the negotiation partner agree on a temporal withdrawal structure, which does not correspond to the desired structure of the valuation subject ( = 1 : 1 : 1 : 1 : 21). To find out whether the agreement on w has an influence on the minimum marginal rate at the merger, the following variants (subcases) are examined: t AK I AK II ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 F 01 -100 35 -80 20 50 -4 -1 1,05 -1 1 -1,09 1 70 80 234 Limit 45 55 40 70 60 1 1 -4 -4 -54 1 ∞ 1,05 -1 1,05 ∞ ∞ -1 1,05 ∞ ∞ -1,09 1 -1,09 ∞ ∞ 1 90 90 -1,09 ∞ 2.100 1 Grenze Figure 2.56: Data for the determination of the merger program 1 1 1 F a) F a) F a) F a) F a) 0 1 2 3 4 w : w : w : w : w Ü Ü Ü Ü Ü 0 1 2 3 4 w : w : w : w : w Ü Ü Ü Ü Ü 0 1 2 3 4 w : w : w : w : w 174 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 174 45520_Matschke_Griffleiste_SL5.indd 174 16.03.2021 16: 21: 29 16.03.2021 16: 21: 29 Chapter 2 b1) The valuation subject and the negotiation partner agree on the temporal structure = 0 : 1 : 1 : 1 : 21. b2) The valuation subject and the negotiation partner agree on the temporal structure = 3 : 1 : 2 : 4 : 15. The desired different temporal withdrawal structures require different linear optimization approaches and lead to different merger programs. For case a), the following approach for the determination of the merger program, the second step, must be solved: under the constraints: The resulting withdrawal stream is =100,9657 GE. At the end of the last planning period, the assets amount to 2.019,3144 GE. Again, this amount is equivalent to the present value of the perpetuity at an annual interest rate of 5 % p. a. Figure 2.57 illustrates the comprehensive financial plan of the merger program in case a). Case a) corresponds to the special case, in which the vector of withdrawals of the merged business is a multiple of the withdrawal vector of the transferred busi- F b1) F b1) F b1) F b1) F b1) 0 1 2 3 4 w : w : w : w : w F b2) F b2) F b2) F b2) F b2) 0 1 2 3 4 w : w : w : w : w max. withdrawal Entn F a) ; Entn F a) : = EN F a) 100 ⋅ AK I + 80 ⋅ AK II − 50 ⋅ ED + 1 ⋅ GA 0 − 1 ⋅ KA 0 + 1 ⋅ EN F a) ≤ 70 − 35 ⋅ AK I − 20 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 1 − 1,05 ⋅ GA 0 − 1 ⋅ KA 1 + 1,09 ⋅ KA 0 + 1 ⋅ EN F a) ≤ 80 − 45 ⋅ AK I − 40 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 2 − 1,05 ⋅ GA 1 − 1 ⋅ KA 2 + 1,09 ⋅ KA 1 + 1 ⋅ EN F a) ≤ 90 − 55 ⋅ AK I − 70 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 3 − 1,05 ⋅ GA 2 − 1 ⋅ KA 3 + 1,09 ⋅ KA 2 + 1 ⋅ EN F a) ≤ 90 − 60 ⋅ AK I + 54 ⋅ ED − 1,05 ⋅ GA 3 + 1,09 ⋅ KA 3 + 21 ⋅ EN F a) ≤ 2.100 GA t , KA t , EN F a) ≥ 0 ∀ t ED,AK I ,AK II ≤ 1. EN F a) max EN F a) max Merged business F Investment AK I t = 0 t = 1 70 -100 80 35 Investment AK II Bullet loan ED Operating loan KA Financial investments GA -80 50 20 -4 160,9657 145,4184 t = 2 t = 3 90 45 90 55 t = 4 2.100 60 40 -4 70 -4 88,4717 -13,6001 -54 KA-, GA-paybacks Withdrawal EN F a) Payment balance Debt from KA -100,9657 -175,4527 -100,9657 0 160,9657 0 145,4184 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.57: Comprehensive financial plan of the merger program (case a) -158,506 -100,9657 -96,4342 -100,9657 0 88,4717 0 14,2801 -100,9657 2.019,3144 13,6001 2.019,3144 EN F a) * + ***** 2.4 Selected Problems of Decision Value Determination 175 45520_Matschke_Griffleiste_SL5.indd 175 45520_Matschke_Griffleiste_SL5.indd 175 16.03.2021 16: 21: 30 16.03.2021 16: 21: 30 ness because both negotiation parties pursue the objective of income maximization. Also, the structure of the distribution stream of the valuation object equals to the structure of those distributions of the merged business, that is, Hence, the minimum ownership interest of the valuation subject is determined (in the third step), depending on the respective shares β of the valuation subject in the transferred business Ü. Until this point, , this quotient (ratio) is β = 1, because the valuation subject is the sole proprietor (shareholder) of the business Ü. Using the “trivial” valuation formula, the result is: Hence, the minimum required marginal rate from the perspective of the valuation subject amounts to 32,3056 %. However, this special case is no longer a given in cases b1) and b2). For while both negotiation partners pursue the same objective, income maximization, the structure of the desired distribution stream in the merged business F does not correspond to the structure of the distribution stream that the valuation subject desires. On the contrary, at least at one point in time t it follows and In case b1) the valuation subject and the negotiation partner agree on the temporal structure = 0 : 1 : 1 : 1 : 21. The following approach for the determination of the merger program (step 2 of case b1) that can be solved using the simplex algorithm, is shown below: under the constraints: A withdrawal stream in the amount of = 106,5792 GE results from the merger program, starting at t = 1. At t = 4, the total assets (payment balance) amounts to 2.131,5848 GE. Once again, this equals the present value of the perpetuity at an annual interest rate of 5 % p. a. The bullet loan ED of 61,3020 % is used in the merger program. Figure 2.58 represents the comprehensive financial plan of the merger program in case b1). EN Ü * + **** , w t Ü = w t F a) ∀ t. α min F a ) α min F a ) = β ⋅ EN Ü max EN F a ) max = 1 ⋅ 32,6176 GE 100,9657 GE = 0,323056 , 32,3056 %. w t Ü ≠ w t F b1) w t Ü ≠ w t F b2) . F b1) F b1) F b1) F b1) F b1) 0 1 2 3 4 w : w : w : w : w max. withdrawal Entn F b1) ; Entn F b1) : = EN F b1) 100 ⋅ AK I + 80 ⋅ AK II − 50 ⋅ ED + 1 ⋅ GA 0 − 1 ⋅ KA 0 ≤ 70 − 35 ⋅ AK I − 20 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 1 − 1,05 ⋅ GA 0 − 1 ⋅ KA 1 + 1,09 ⋅ KA 0 + 1 ⋅ EN F b1) ≤ 80 − 45 ⋅ AK I − 40 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 2 − 1,05 ⋅ GA 1 − 1 ⋅ KA 2 + 1,09 ⋅ KA 1 + 1 ⋅ EN F b1) ≤ 90 − 55 ⋅ AK I − 70 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 3 − 1,05 ⋅ GA 2 − 1 ⋅ KA 3 + 1,09 ⋅ KA 2 + 1 ⋅ EN F b1) ≤ 90 − 60 ⋅ AK I + 54 ⋅ ED − 1,05 ⋅ GA 3 + 1,09 ⋅ KA 3 + 21 ⋅ EN F b1) ≤ 2.100 GA t , KA t , EN F b1) ≥ 0 ∀ t ED,AK I ,AK II ≤ 1. EN F b1) max EN F b1) max 176 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 176 45520_Matschke_Griffleiste_SL5.indd 176 16.03.2021 16: 21: 31 16.03.2021 16: 21: 31 Chapter 2 Because the temporal structure of the withdrawals of the business to be transferred and that of the merged business are no longer equal, even if it is only at one point in time (t = 0), the calculation of the marginal rate is more complicated. The valuation subject has to use its private decision field to transform the new payment stream resulting from the merged business into the desired structure. Therefore, it is assumed that the valuation subject has the following (private) options at the primary bank: Single-period funds are available at an annual interest rate of 5 % p. a.; one-period operating loans are available up to an amount of 40 GE at an interest rate of 11 % p. a. For the determination of the minimum required rate in the merged business, the following optimization approach results in step 3 of case b1) under the following constraints: The solution embodied in this approach is the minimum required ownership interest (marginal rate) = 32,4843 %. Only if at least this rate is negotiated, is the valuation subject is not at a disadvantage compared to the case without a merger. The comprehensive financial plan of the valuation program illustrated in Figure 2.59 shows that the valuation subject can replicate (or reproduce) the desired temporal withdrawal structure, Merged business F Investment AK I t = 0 t = 1 70 -100 80 35 Investment AK II Bullet loan ED at 0,6130201 Operating loan KA Financial investments GA -80 30,6510 20 -2,4521 79,3490 60,5217 t = 2 t = 3 90 45 90 55 t = 4 2.100 60 40 -2,4521 70 -2,4521 -105,9687 -33,1031 KA-, GA-paybacks Withdrawal EN F b1) Payment balance Debt from KA 0 -86,4904 -106,5792 0 79,3490 0 60,5217 Deposits from GA Present value of perpetuity EN/ 0,05 Figure 2.58: Comprehensive financial plan of the merger program (case b1) -65,9687 -106,5792 -106,5792 0 0 111,2671 -106,5792 2.131,5848 105,9687 2.131,5848 GA priv KA priv α min F b1) min. A; A : = α F b1) −1⋅ GA 0 priv + 1 ⋅ KA 0 priv ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 0 priv − 1 ⋅ GA 1 priv − 1,11 ⋅ KA 0 priv + 1 ⋅ KA 1 priv + α F b1) ⋅ 1 ⋅ 106,5792 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 1 priv − 1 ⋅ GA 2 priv − 1,11 ⋅ KA 1 priv + 1 ⋅ KA 2 priv + α F b1) ⋅ 1 ⋅ 106,5792 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 2 priv − 1 ⋅ GA 3 priv − 1,11 ⋅ KA 2 priv + 1 ⋅ KA 3 priv + α F b1) ⋅ 1 ⋅ 106,5792 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 3 priv − 1,11 ⋅ KA 3 priv + α F b1) ⋅ 21 ⋅ 106,5792 ≥ 21 ⋅ 32,6176 −KA t priv ≥ − 40 ∀ t GA t priv , KA t priv ≥ 0 ∀ t α F b1) ≥ 0. α min F b1) 2.4 Selected Problems of Decision Value Determination 177 45520_Matschke_Griffleiste_SL5.indd 177 45520_Matschke_Griffleiste_SL5.indd 177 16.03.2021 16: 21: 31 16.03.2021 16: 21: 31 according to this rate under consideration of the private decision field. In the grayly highlighted fields of the table, the payment stream can be recognized that the valuation subject would have in the case of a non-merger. In other words, it showcases the withdrawals only from business Ü (see also Figure 2.55). In case b2), the valuation subject and the negotiation partner agree on the temporal structure = 3 : 1 : 2 : 4 : 15. For the determination of the merger program (step 2 of case b2), the following approach is once again solved with the simplex algorithm: under the following restrictions: It results a withdrawal stream in the amount of 96,2412 GE. Figure 2.60 presents the comprehensive financial plan of the merger program of case b2). Distribution from the merger program F Thereof 32,4843 % t = 0 t = 1 0 0 106,5792 34,6215 Private operating loan KA priv Private financial investments GA priv Paybacks KA priv , GA priv Desired withdrawal EN max 32,6176 34,2016 -32,6176 -36,2055 -32,6176 t = 2 t = 3 106,5792 34,6215 106,5792 34,6215 t = 4 2.238,1632 727,0514 35,9599 37,9116 -37,9638 -32,6176 -39,9155 -32,6176 -42,0819 -32,6176 Payment balance Debt from KA priv Deposits from GA priv Present value of perpetuity EN/ 0,05 0 32,6176 0 34,2016 Figure 2.59: Comprehensive financial plan of the valuation program (case b1) 0 35,9599 0 37,9116 652,3520 652,3520 F b2) F b2) F b2) F b2) F b2) 0 1 2 3 4 w : w : w : w : w max. withdrawal Entn F b2) ; Entn F b2) : = EN F b2) 100 ⋅ AK I + 80 ⋅ AK II − 50 ⋅ ED + 1 ⋅ GA 0 − 1 ⋅ KA 0 + 3 ⋅ EN F b2) ≤ 70 − 35 ⋅ AK I − 20 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 1 − 1,05 ⋅ GA 0 − 1 ⋅ KA 1 + 1,09 ⋅ KA 0 + 1 ⋅ EN F b2) ≤ 80 − 45 ⋅ AK I − 40 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 2 − 1,05 ⋅ GA 1 − 1 ⋅ KA 2 + 1,09 ⋅ KA 1 + 2 ⋅ EN F b2) ≤ 90 − 55 ⋅ AK I − 70 ⋅ AK II + 4 ⋅ ED + 1 ⋅ GA 3 − 1,05 ⋅ GA 2 − 1 ⋅ KA 3 + 1,09 ⋅ KA 2 + 4 ⋅ EN F b2) ≤ 90 − 60 ⋅ AK I + 54 ⋅ ED − 1,05 ⋅ GA 3 + 1,09 ⋅ KA 3 + 15 ⋅ EN F b2) ≤ 2.100 GA t , KA t , EN F b2) ≥ 0 ∀ t ED,AK I ,AK II ≤ 1. EN F b2) max 178 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 178 45520_Matschke_Griffleiste_SL5.indd 178 16.03.2021 16: 21: 33 16.03.2021 16: 21: 33 Chapter 2 For the determination of the minimal ownership interest in the merged business F, the private investment and financing program of the valuation subject has to be used due to the modified withdrawal structure. If it is assumed that it corresponds to case b1), the following optimization approach has to be solved in step 3 of case b2), under consideration of the withdrawal structure = 3 : 1 : 2 : 4 : 15: under the constraints: Due to the modified distribution structure in the merged business, a modified minimum required ownership interest results. It amounts to = 33,0077 %. The valuation subject is not at a disadvantage compared to the case without a merger if at least this rate is negotiated during the merger. In Figure 2.61, it is shown that the valuation subject can replicate the desired temporal withdrawal structure, under consideration of the private decision field (see also Figure 2.55). Merged business F Investment AK I t = 0 t = 1 70 -100 80 35 Investment AK II Bullet loan ED Operating loan KA Financial investments GA -80 50 20 -4 348,7235 345,3498 t = 2 t = 3 90 45 90 55 t = 4 2.100 60 40 -4 70 -4 397,9136 607,6905 -54 KA-, GA-paybacks Payment balance Debt from KA -288,7235 -380,1086 -96,2412 0 348,7235 0 345,3498 Deposits from GA Figure 2.60: Comprehensive financial plan of a merger program (case b2) -376,4313 -192,4823 -433,7258 -384,9647 0 397,9136 0 607,6905 -662,3825 -1.443,6175 0 Withdrawal EN F b2) α min F b1) F b2) F b2) F b2) F b2) F b2) 0 1 2 3 4 w : w : w : w : w min. A; A : = α F b2) −1⋅ GA 0 priv + 1 ⋅ KA 0 priv + α F b2) ⋅ 3 ⋅ 96, 2412 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 0 priv − 1 ⋅ GA 1 priv − 1,11 ⋅ KA 0 priv + 1 ⋅ KA 1 priv + α F b2) ⋅ 1 ⋅ 96,2412 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 1 priv − 1 ⋅ GA 2 priv − 1,11 ⋅ KA 1 priv + 1 ⋅ KA 2 priv + α F b2) ⋅ 2 ⋅ 96,2412 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 2 priv − 1 ⋅ GA 3 priv − 1,11 ⋅ KA 2 priv + 1 ⋅ KA 3 priv + α F b2) ⋅ 4 ⋅ 96,2412 ≥ 1 ⋅ 32,6176 1,05 ⋅ GA 3 priv − 1,11 ⋅ KA 3 priv + α F b2) ⋅ 15 ⋅ 96,2412 ≥ 21 ⋅ 32,6176 KA t priv ≥ − 40 GA t priv , KA t priv ≥ 0 ∀ t α F b2) ≥ 0. α min F b2) α min F b2) 2.4 Selected Problems of Decision Value Determination 179 45520_Matschke_Griffleiste_SL5.indd 179 45520_Matschke_Griffleiste_SL5.indd 179 16.03.2021 16: 21: 34 16.03.2021 16: 21: 34 This examplifies that the weightings represent original conflict-resolution-relevant facts. Different weighting factors will thus lead to different marginal rates: = 32,3056 %, = 32,4843 %, and = 33,0077 % in the scenarios. 2.4.3.2 Conflict Situation of the Demerger Type 2.4.3.2.1 Presentation Despite the great importance of business demergers in practice, economics literature does not yet offer any model-theoretic considerations on the valuation of such demergers. However, demergers represent a unique event, one in which economic goods, assets, and liabilities of a legal entity are transferred to various newly established or existing businesses without liquidation (F REITAG 1998, p. 1). Accordingly, the known valuation models are not applicable without modifications. Therefore, the conflict situations of the demerger type are now discussed in greater detail, and methods regarding the determination of the decision value for this situation are presented and explained by way of an example (B YSIKIEWICZ / M ATSCHKE / B RÖSEL 2005). The conventional understanding of a business demerger lies in its assets being divided into at least two businesses, in which the shareholders of the former single business (transferring (legal) entity) are involved (F REITAG 1998, p. 11). In a conflict situation of the demerger type in business valuation - similar to the conflict situation of the merger type - it is about distributing the property rights (direct or indirect ownership interests, shares, or stake respectively) and therefore it is ultimately about distributing the future performances of the businesses that are available after the demerger (acquiring (legal) entity). As pointed out in Section 1.3.2.1, an interpersonal conflict situation of the demerger type is present if the distribution of the ownership interests (in the sense of a distribution of the future performances) in the businesses created by the demerger differentiates from those prior to the demerger. This is the case if the former owners (shareholders) are still participating in the businesses created by the demerger but in a diffe- Distribution from the merger program case b2) Thereof 33,0077 % t = 0 t = 1 288,7235 95,3009 96,2412 31,7670 Private operating loan KA priv Private financial investments GA priv Paybacks KA priv , GA priv Desired withdrawal EN max -62,6833 -64,9668 -32,6176 65,8174 -32,6176 t = 2 t = 3 192,4823 63,5339 384,9646 127,0679 t = 4 1.443,6174 476,5045 -99,1315 -198,5383 68,2152 -32,6176 104,0880 -32,6176 208,4652 -32,6176 Payment balance Debt from KA priv Deposits from GA priv Present value of perpetuity EN/ 0,05 0 0 62,6833 64,9668 Figure 2.61: Comprehensive financial plan of the valuation program (case b2) 0 0 99,1315 198,5383 652,3520 652,3520 w s w s α min F a ) α min F b1) α min F b2) 180 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 180 45520_Matschke_Griffleiste_SL5.indd 180 16.03.2021 16: 21: 34 16.03.2021 16: 21: 34 Chapter 2 rent ratio (change of ownership structure in a ratio-changing or not ratio-preserving demerger due to changed ownership interests). This is also the case if the accumulated income stream of the new businesses with the same ownership interests differs from the original business (change of ownership structure in a ratio-preserving demerger due to changed future performances). Here, ratio-preserving is to be understood as a constant rate (or quota). Seldom, if ever, do the accumulated future performances of the new businesses comply with those of the business to be demerged so that even with constant rates, changes in the ownership structure occur. This is because the valuation subjects are still participating in the companies with the same percentage, but on another basis (be it about the amount and/ or to the structure of the future performances). Hence, even those situations are valuation-relevant, in which the shareholders of the original business (transferring legal entity) receive a proportionate share in the new businesses (acquiring entities). The following effect is similar in both valuation-relevant situations: The previous shareholders are to different extents involved in the opportunities and risks of the new businesses. The question facing the single shareholder in a relation-changing demerger is how large the share in the demerged business has to be if that shareholder is not to be disadvantaged compared to the present position with the original business. In the ratiopreserving demerger, when the rates in the new businesses have already been fixed, it can be questioned whether the valuation subject can agree on these rates without suffering a disadvantage compared to the case if the demerger did not happen. Generally speaking, the agreement in a non-dominated conflict situation is not excluded if the cumulated future performances of the acquiring entities exceed those of the transferring entity. Alternatively, they might have temporal or structural advantages. Moreover, an interpersonal conflict situation of the demerger type can occur as a special case of a ratio-changing demerger if a complete separation of the shareholders takes place after the demerger (separation of ownership). The following simple example shows that ownership separation is a special form of a ratio-changing demerger: a transferring business, in which A and B are involved in equal measure (each 50 %), is divided into the acquiring businesses U 1 and U 2 . If A participates 100 % and B 0 % in U 1 and A participates 0 % and B 100 % in U 2 , a ratio-changing demerger occurred. From the perspective of the valuation subject, the conflict-resolution-relevant facts can be either the size of the remaining share and its resulting future performances or the size of the payable or demandable compensation. If compensation was negotiated, a conflict situation of the type acquisition/ sale is given. In the next step, a non-dominated, disjoint, and one-dimensional conflict situation of the demerger type is discussed where only the size of the (not uniform) ownership interests in the newly demerged businesses is relevant (shift of ownership at a non-ratiopreserving or ratio-changing demerger) at the valuation date. The most prominent case in practice is when a business is divided into two enterprises (split-up). The business UG (original business; German: Ursprungsgesellschaft) before the demerger is divided into F adopting businesses Ü (demerged businesses; German: übernehmendes Unternehmen). Before and after the demerger, the number of shareholders is determined as H. During the ratio-changing demerger, the shares β h (share in the future performance of the business UG that the owner (or demerger party) h receives without 2.4 Selected Problems of Decision Value Determination 181 45520_Matschke_Griffleiste_SL5.indd 181 45520_Matschke_Griffleiste_SL5.indd 181 16.03.2021 16: 21: 34 16.03.2021 16: 21: 34 the demerger) of the shareholders of business UG do not correspond to the shares of the demerged business Ü f . Due to the changed share of votes (balance of power), a different distribution stream is preferred in a new demerged business U than in the original business UG. That is why the weighting factors of the withdrawals, which determine the distribution structure, must be considered during the value determination. In the following, it is assumed that the weightings for the newly demerged businesses are already negotiated. In a real conflict situation, these are original conflictresolution-relevant facts, which are similar to the conflict situation of the merger type. They are typically the results of the negotiation process. Depending on those results, a corresponding minimal marginal rate has to be determined according to the weighting factors In this and the following example, the complexity is reduced and it is assumed that the weightings for the new demerged businesses are already determined so that it can generally be spoken of as a one-dimensional conflict situation. The previously examined state marginal rate model serves as the foundation of the determination of a decision value at the demerger. The marginal rate model determines the single minimum ownership interests that the owners must receive if they do not want to weaken their original economic position by the demerger of the original business.The process for the determination of the minimum ownership interests can be divided into three steps: 1. Determination of the maximum stream of withdrawals from the original business UG without a demerger (pre-demerger program): What is the maximum utility - in the sense of a share β h at - the valuation subject h can derive from business UG without the agreement on the demerger? That level of performance must be reached again after the demerger and is computed as the sum of the shares of the performances of the demerged businesses Ü f so that the valuation subject does not suffer a disadvantage as compared to the case without the demerger. The partial performance of the valuation subject in the pre-merger program becomes a constraint for the demandable ownership interest after the demerger. 2. Determination of the maximum of possible withdrawals of the single new demerged businesses Ü f (demerger program or division program, respectively): Which maximum utility can all conflicting parties receive from the businesses Ü f ? 3. Determination of the minimal marginal rate (and the valuation program): For the determination of the valuation program, a comparison of shares regarding the withdrawals from the base and the demerger (division) program is drawn. Which share in the utility of the single newly demerged businesses Ü f , does the valuation subject h at least need without suffering a disadvantage? Given that the result of the demerger is the creation of several businesses, the conflicting parties have to agree on several ownership interests. Therefore, a demerger can generally be described as a multi-dimensional decision situation. The determination α Üf h min w s Sp w s Sp w s Sp . EN s UG max EN s UG max α min Üf h α min Üf h EN s Üf Sp 182 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 182 45520_Matschke_Griffleiste_SL5.indd 182 16.03.2021 16: 21: 35 16.03.2021 16: 21: 35 Chapter 2 of the single minimum marginal rates for a business Ü 1 can be accomplished using the following model if the ownership interests, basically subject to negotiation, in the other businesses can be defined as ceteris paribus conditions. If there is a substantive agreement during the negotiations over the shares in the demerged businesses Ü h , the situation ultimately becomes a one-dimensional conflict situation but only if the shares in the last business are left undetermined. Accordingly, it is assumed below that only the rate of an acquiring business Ü is to be defined and all other ownership interests have already been determined. Formally, this is equivalent to a one-dimensional conflict situation. The base approach illustrates such a linear optimization approach that reflects the situation of the business UG without the demerger. Analogous to the case of the merger, the following assumptions are made for the first step: The company makes plans about S uncertain future states s (or about S periods). There are J investment and financing objects available. The payment sequence of any given investment or financing object j is g j : = (g j0 , g j1 , ..., g js , ..., g js ) with g js as cash flows in the respective state (or at time) s. The structure or decision variable x j indicates how often an object j is realized. For each x j , an upper limit may exist. Prescheduled payments are to be considered with a fixed payment balance b s . They can - equating to the sizes g js - be positive, zero, or negative. The valuation subject h, which participates in business UG with the share β h , desires a large withdrawal stream EN for consumption purposes. This objective corresponds to the one of business UG, where a distribution of should occur in each state s. If the pursued objective of the valuation subject would not correspond to the one of the business UG, for instance, due to an insufficient share β h , the valuation subject would have to establish structural equality by using the subject’s private decision field. This would necessitate an intermediate step just after this first step. Only through additional private financial restructuring can the valuation subject successfully transform the payment stream of the business into the preferred income structure. The linear approach for the determination of the pre-merger program reads as follows: under the 1. Liquidity restrictions: 2. Capacity restrictions: 3. Non-negativity conditions: This base approach can be solved with the help of the simplex algorithm. The solution provides the maximum size of the withdrawal stream of the business UG. Ü h ≠ Ü 1 x j max w s UG ⋅ EN UG max max. withdrawal Entn; Entn : = EN UG − g js j=1 J ∑ ⋅ x j + w s UG ⋅ EN UG ≤ b s ∀ s ∈ {0, 1, 2, … , S} x j ≤ x j max ∀ j ∈ {1, 2, … , J} x j ≥ 0 ∀ j ∈ {1, 2, … , J} EN UG ≥ 0. EN UG max 2.4 Selected Problems of Decision Value Determination 183 45520_Matschke_Griffleiste_SL5.indd 183 45520_Matschke_Griffleiste_SL5.indd 183 16.03.2021 16: 21: 35 16.03.2021 16: 21: 35 The corresponding optimum solution is the pre-merger program, in which the decision subject h is invested in with β h . This partial withdrawal has to be at least achieved again by the decision subject after the demerger, in order not to be disadvantaged. In the second step, the determination of the demerger program, the maximum possible withdrawals of each newly demerged business Ü f are determined. The necessary (demerger) approach is solved by the simplex algorithm and has to be formulated from the perspective of the valuation subject for each newly demerged business Ü f with The valuation subject should have a financial investment in those businesses in the future. Formally, this approach differs from the base approach only regarding the index Ü f , which indicates the respective newly demerged businesses. However, materially, that is, considering the number of variables, the upper limits of variables, the weightings of withdrawals, and the cash flows, it represents a linear model varying (substantially) from the base approach, which presents each single new businesses Ü f resulting from the previous original businesses UG: under 1. Liquidity restrictions: 2. Capacity restrictions: 3. Non-negativity conditions: After the determination of the maximum possible withdrawals of the original business UG and the maximum possible withdrawal of the respective demerger businesses Ü f , the marginal rate in the business Ü f has to be determined for the decision subject in a third step. The procedure of the demerger is economically justifiable from the perspective of the decision subject only in cases when the respective shares of the distributions of the demerged businesses Ü f have the cumulative target value as the result. Generally speaking, a multi-dimensional and joint decision problem is given, because the change regarding the agreed-upon ownership interest (and hence also a change of the future performances concerning the shares of the valuation subject) of one business after the demerger implies a change of the marginal rates of the other demerged businesses: EN Üf max f ∈ 1, 2, …, F { }. max . withdrawal Entn Üf ; Entn Üf : = EN Üf − g js Üf ⋅ x j,Üf j = 1 J Üf ∑ + w s Üf ⋅ EN Üf ≤ b s,Üf ∀ s ∈ {0, 1, 2, … , S} x j,Üf ≤ x j,Üf max ∀ j ∈ {1, 2, … , J Üf } x j,Üf ≥ 0 ∀ j ∈ {1, 2, … , J Üf } EN Üf ≥ 0. EN UG max EN Üf max α Üf h min EN Üf max EN UG max α min Üf h = f α min Ün ∈ Ü1, Ü 2, ..., Üf , ..., Ün, ..., ÜF { } \ Üf { } ,h ( ) ∀ f ∈ 1, 2, ..., n, ..., F { } . 184 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 184 45520_Matschke_Griffleiste_SL5.indd 184 16.03.2021 16: 21: 36 16.03.2021 16: 21: 36 Chapter 2 The single marginal rate is dependent on the other ownership interests. Due to the present solution problem, it is recommended to solve the third step heuristically. This can be done by calculating the respective marginal rate under the systematically varying ceteris paribus condition, that is, under systematically varying ownership interests of the other demerged businesses. If those rates for the demerged businesses not considered are given, the minimum demandable ownership interest (in the sense of a marginal rate) has to be determined for the demerged business in focus. Analogous to the merger situation, two different cases should be distinguished: a) It is assumed that the desired structure of the withdrawals of all demerged businesses U is equal and corresponds to the one of the original business UG. With regard to a single demerged busine ss (here exemplary business Ü 1 ), the owner h has the following valuation approach for the determination of the minimum ownership interest at hand, where all other rates are predetermined: Below, the determination of the minimum ownership interests resulting from the demerger of the original business UG into two new businesses Ü 1 und Ü 2 is discussed (split-up). If rate in business Ü 2 is already negotiated or given, the critical marginal rate in business Ü 1 of the conflicting party h can be determined.The marginal rate in business Ü 1 is calculated as follows: In case of separation of ownership, that is, if the decision subject h is not invested in the business Ü 2 after the demerger, the critical ownership interest can finally be determined analogously to the case of the merger situation since b) During the discussion of the merger, it should have been conveyed that the independent businesses (before the merger) can have entirely different withdrawal structures Accordingly, negotiations about the withdrawal structure of the merged business can become a part of the future corporate policy. Moreover, with respect to α min Üf h w s UG = w s Üf w s UG = w s Üf α min Ü1 h ⋅ EN Ü1 max excess deposits of the shareholder h from the business Ü1 & ' $ ( $ + α Üf h ⋅ EN Üf max f = 2 F ∑ excess deposits of the shareholder h from the demerged business f = 2, ..., F & ' $$ ( $$ excess deposits of the shareholder h form all demerged businesses Üf & ' $$$$$$ ( $$$$$$ ≥ β h ⋅ EN UG max excess deposits of the shareholder h form the original business UG & ' $ ( $ α min Ü1 h = β h ⋅ EN UG max EN Ü1 max − α Üf h ⋅ EN Üf max f = 2 F ∑ EN Ü1 max . α Ü 2 h α min Ü1 h α min Ü1 h α min Ü1 h = β h ⋅ EN UG max − α Ü 2 h ⋅ EN Ü 2 max EN Ü1 max = β h ⋅ EN UG max EN Ü1 max − α Ü 2 h ⋅ EN Ü 2 max EN Ü1 max . α min Ü1 h α Ü 2 h = 0 : α min Ü1 h = β h ⋅ EN UG max EN Ü1 max . w s . 2.4 Selected Problems of Decision Value Determination 185 45520_Matschke_Griffleiste_SL5.indd 185 45520_Matschke_Griffleiste_SL5.indd 185 16.03.2021 16: 21: 37 16.03.2021 16: 21: 37 the conflict situation of the demerger type, a withdrawal structure different from that of the original business can occur, for instance, due to a changed share of votes in one (or more) demerged businesses Ü f . This means that is true at least in one state s and for one business Ü f . If different withdrawal structures exist before and after the demerger, the private decision field of the shareholders has to be used in the third step to compare the distribution stream of the business UG, without the demerger with the sum of the other payment streams that result from each single business Ü f after the demerger. The valuation subject has to transform the sum of the new distribution streams back into the original payment stream by doing additional private financial restructuring, that is, increases or decreases of the private businesses . The transformations in the private decision field comprise the corresponding cash flows in each state s. Consequently, shareholder h has to solve the linear optimization approach (valuation approach), where business Ü 1 is considered and where the other ownership interests are predetermined, which on the other hand represent systematically varying ceteris paribus conditions: under the following constraints 1. Liquidity restrictions: 2. Restrictions regardings the increase of businesses: 3. Non-negativity conditions: The previous valuation approach minimizes the marginal rate in the demerged business Ü 1 for the shareholder h, assuming the fulfillment of the constraints so that in every state s the incoming distribution stream for the owner h from the demerged business, including possible transformations in the private sector, does not differ from the distribution before the demerger: w s UG ≠ w s Üf Δ x j pos Δ x j neg x j priv g js priv min. A; A : = α Ü1 h g js priv ⋅ Δ x js priv j = 1 J priv ∑ + α h Ü1 ⋅ w s Ü1 ⋅ EN Ü1 max + α Üf h ⋅ w s Üf ⋅ EN Üf max f = 2 F ∑ constant ! " ### $ ### ≥ β h ⋅ w s UG ⋅ EN UG max ∀ s ∈ {0, 1, 2, …, S} −Δ x j priv ≥ x j priv max ∀ j ∈ {0, 1, 2, …, J priv } Δ x j priv ≥ 0 ∀ j ∈ {0, 1, 2, …, J priv } α Ü1 h ≥ 0. α h Ü1 β h ⋅ w s UG ⋅ EN UG max 186 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 186 45520_Matschke_Griffleiste_SL5.indd 186 16.03.2021 16: 21: 39 16.03.2021 16: 21: 39 Chapter 2 By using the duality theory of the linear optimization, a complex valuation formula for the determination of the marginal rate can be derived for the case of the demerger. The minimum rate results for business Ü 1 under the simplified assumption that the respective marginal rates for the other demerged businesses, here Ü 2 , are given or represent the systematically varying ceteris paribus conditions: In the special case that in all states s is true, the simplified valuation formula can finally be derived from the complex formula, because it is needless to incorporate the private decision field: or or 2.4.3.2.2 Numerical Example For the purposes of illustration, the previously mentioned approach to determine a decision value is examined with the aid of an example assuming (quasi-)certain expectations. This example refers to the case of a one-dimensional conflict situation of the demerger type with the critical shareholding as the single conflict-resolution-relevant fact. g js priv ⋅ Δx j priv j=1 J priv ∑ tranformations of the private credits and investments & ' $$ ( $$ + α Üf h ⋅ w s Üf ⋅ EN Üf max f = 1 F ∑ distributions, according to the parts α Üf h of the optimum withdrawal stream EN Üf max of the demerged businesses Üf & ' $$$ ( $$$ excess deposits of the shareholder h from all demerged businesses Üf, under consideration of the essential transformation in the private sector & ' $$$$$$ ( $$$$$$ ≥ β h ⋅ w s UG ⋅ EN UG max excess deposits of the shareholder h from the original business UG (without demerger) & ' $$ ( $$ . α min Ü1 h = β h ⋅ w s UG ⋅ EN UG max ⋅ ρ s priv s=0 S ∑ part β of the valuation subject h at the capital value of the distribution of the original business UG " # $$$$ % $$$$ - Δ x j ⋅ C j priv j=0 J priv ∑ capital value of the transformations in the private program " # $ % $ w s Ü1 ⋅ EN Ü1 max ⋅ ρ s priv s=0 S ∑ capital value of the business Ü 1 after the demerger & ' $$$ ( $$$ − α Ü 2 h ⋅ w s Ü 2 ⋅ EN Ü 2 max ⋅ ρ s priv s = 0 S ∑ part of the valuation subject h at the capital value of the distribution of business Ü 2 after the demerger " # $$$$ % $$$$ w s Ü1 ⋅ EN Ü1 max ⋅ ρ s priv s = 0 S ∑ capital value of the business Ü 1 after the demerger & ' $$$ ( $$$ . w s UG = w s Ü α min Ü1 h = β h ⋅ w s UG ⋅ EN UG max ⋅ ρ s priv s = 0 S ∑ − 0 w s Ü1 ⋅ EN Ü1 max ⋅ ρ s priv s = 0 S ∑ − α Ü 2 h ⋅ w s Ü 2 ⋅ EN Ü 2 max ⋅ ρ s priv s = 0 S ∑ w s Ü1 ⋅ EN Ü1 max ⋅ ρ s priv s = 0 S ∑ α min Ü1 h = β h ⋅ EN UG max EN Ü1 max ⋅ w s UG ⋅ ρ s priv s = 0 S ∑ w s Ü1 ⋅ ρ s priv s = 0 S ∑ − α Ü 2 h ⋅ EN Ü 2 max EN Ü1 max ⋅ w s Ü 2 ⋅ ρ s priv s = 0 S ∑ w s Ü1 ⋅ ρ s priv s = 0 S ∑ α min Ü1 h = β h ⋅ EN UG max EN Ü1 max − α Ü 2 h ⋅ EN Ü 2 max EN Ü1 max . 2.4 Selected Problems of Decision Value Determination 187 45520_Matschke_Griffleiste_SL5.indd 187 45520_Matschke_Griffleiste_SL5.indd 187 16.03.2021 16: 21: 41 16.03.2021 16: 21: 41 As a consequence, only the ownership interest in an acquiring entity needs to be assessed, because the withdrawal structures of the new business and the shareholdings for all businesses except the business under consideration have already been predetermined. This decision value is determined in the form of a marginal rate based on the simplified total model concerning the financial target of income maximization. Figure 2.62 illustrates the assumed conflict situations for the shareholders X and Y, namely in the general case of ownership change, where the original business UG is divided into Ü 1 and Ü 2 (split-up). The shares before and after the demerger do not necessarily have to be equivalent. The first step considers the determination of the pre-demerger program, in which the withdrawal options of business UG are defined without the demerger. The planning horizon extends over four periods, both the valuation date and the demerger date are at t = 0. Shareholder X, that is, the valuation subject, and shareholder Y desire for a uniform payment stream that provides for the withdrawal in the periods t = 1 to t = 3. The last distribution contains both the regular distribution and the present value of the perpetuity to guarantee the income beyond the planning period.Business UG has two investment possibilities I 1 and I 2 with the payment sequences (-100 GE, 30 GE, 40 GE, 50 GE, 55 GE) and (-110 GE, 10 GE, 80 GE, 80 GE) and the upper limits 30 and 25, respectively. For financing, bond A could be raised under the following conditions: face value (par value) 100 GE, a term of four periods (years), bond or issue price 99 % (below par), nominal interest rate 6 % p. a., repayment in equal installments at the end of the third and fourth year, upper limit: maximum 35 GE. Further financial funds KA t are available as operating loans in unlimited amounts at a short-term interest (borrowing) rate of 10 % p. a. One-period financial investments (GA t ) may be made in any amounts at an interest rate of 5 % p. a. The fixed cash flows bt amount to 133 GE at t = 0. For any time t after that, they are negative (-55 GE, -9 GE, -9 GE, -8 GE). The aim is to find a maximum periodic uniform withdrawal Unternehmen UG Anteil des Eigners X an UG: 70 % Anteil des Eigners Y an UG: 30 % Unternehmen Ü Unternehmen Ü Unternehmen Ü Unternehmen Ü 22 2 2 Anteil des Eigners X am Ü 2 : α 2 X % Anteil des Eigners Y am Ü 2 : α 2 Y % (mit 70 % : 30 % α 2 X % : α 2 Y %) ≠ Unternehmen Ü Unternehmen Ü Unternehmen Ü Unternehmen Ü 11 1 1 Anteil des Eigners X an Ü 1 : α 1 X % Anteil des Eigners Y an Ü 1 : α 1 Y % (mit 70 % : 30 % α 1 X % : α 1 Y %) ≠ Business Ü 2 Business Ü 1 The owner's part X at Ü 2 : α 2 X % The owner's part Y at Ü 2 : α 2 Y % (for 70 % : 30 % ≠ α 2 X % : α 2 Y %) Business UG The owner's part X at UG : 70 % The owner's part Y at UG : 30 % Figure 2.62: Special case of a structural change of ownership at a nonratio-preserving demerger into two businesses (split-up) The owner's part X at Ü 2 : α 2 X % The owner's part Y at Ü 2 : α 2 Y % (for 70 % : 30 % ≠ α 2 X % : α 2 Y %) EN UG w 4 UG ⋅ EN UG EN UG EN UG 188 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 188 45520_Matschke_Griffleiste_SL5.indd 188 16.03.2021 16: 21: 43 16.03.2021 16: 21: 43 Chapter 2 The initial data are represented in Figure 2.63. The minimum marginal rate (as the marginal price in the case of an acquisition/ sale situation) is not yet known. The valuation subject, that is, shareholder X, determines this value based on own ideas about the measures to be taken in the case of a demerger (demerger strategy). Hence, the data collected during the three steps of the decision value determination are data from the perspective of the valuation subject X: For the determination of the pre-demerger program (or pre-division program), the solution of the following linear optimization problem can be calculated once again with the help of the simplex algorithm: under the constraints: The end-tableau of the simplex algorithm is represented in the pre-demerger program. A uniform payment stream of the size of =118,3474 GE originates from this program. The payment balance of 2.366,9484 GE at the end of the planning horizon is equal to the present value of the perpetuity discounted at 5 % p. a. Formally, this calculation reads as follows: 118,3474 GE · 1/ 0,05 = 2.366,9484 GE. The following investments and financing rounds occur: Investments I 1 and I 2 as well as bond A reach their fixed upper limits. At t = 0 and t = 1, the respective one-period operating loan KA t is taken out. At t = 2 and t = 3, financial investments GA t are made. The comprehensive financial plan of the pre-demerger program is represented in Figure 2.64: EN UG . t I 1 I 2 GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 A b t 01 -100 30 -110 10 -1 1,05 -1 1 -1,1 1 99 -6 133 -55 234 Limit 40 50 80 80 55 30 25 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ -6 -56 -9 -9 -53 35 -8 1 Grenze Figure 2.63: Data for the determination of the pre-demerger program 1 1 1 1 max . withdrawal Ent; Ent : = EN UG 100 ⋅ I 1 +110 ⋅ I 2 + 1 ⋅ GA 0 − 1 ⋅ KA 0 − 99 ⋅ A≤ 133 − 30 ⋅ I 1 − 10 ⋅ I 2 − 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 − 1 ⋅ KA 1 + 6 ⋅ A+1 ⋅ EN UG ≤ − 55 − 40 ⋅ I 1 − 80 ⋅ I 2 − 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 − 1 ⋅ KA 2 + 6 ⋅ A+1 ⋅ EN UG ≤ − 9 − 50 ⋅ I 1 − 80 ⋅ I 2 − 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 − 1 ⋅ KA 3 + 56 ⋅ A+1 ⋅ EN UG ≤ − 9 − 55 ⋅ I 1 − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 53 ⋅ A+21 ⋅ EN UG ≤ − 8 I 1 , I 2 , GA t , KA t , A, EN UG ≥ 0 ∀ t I 1 ≤ 30 I 2 ≤ 25 A≤ 35. EN UG max EN UG max 2.4 Selected Problems of Decision Value Determination 189 45520_Matschke_Griffleiste_SL5.indd 189 45520_Matschke_Griffleiste_SL5.indd 189 16.03.2021 16: 21: 43 16.03.2021 16: 21: 43 Since the valuation subject X is invested in the business UG without demerger with a share of β h = 70 %, the following distribution stream from the pre-demerger program is illustrated in Figure 2.65. In the demerger program, the maximum withdrawals and of the demerged businesses Ü 1 and Ü 2 are determined for shareholders X and Y in the second step. Business Ü 1 can make investment I 1 due to the demerger and business Ü 2 investment I 2 . Due to positive demerger effects, the upper limits of the single investments increase because better support can be guaranteed by the demerged businesses. For the business Ü 1 , the upper limit rises to 34 for investment I 1 . For business Ü 2 , the upper limit for investment I 2 amounts to 28. The investment and financing options retain the same conditions as without a demerger. However, the financing options by the (issuance of the) bond are reduced since the business size has decreased. Accordingly, the upper limits fall for business Ü 1 to 18 and business Ü 2 to 14. The autonomous payments (cash flows) can be divided into (fixed) payments for both businesses: Business Ü 1 achieves an internal financing stream of (73 GE, -30 GE, -5 GE, -5 GE, -4 GE) during the planning horizon. Business Ü 2 obtains a payment stream of (60 GE, -25 GE, -4 GE, -4 GE, -4 GE). Finally, the current action options for the businesses Ü 1 and Ü 2 are outlined in Figure 2.66 as well as in Figure 2.67. Investment I 1 Investment I 2 t = 0 t = 1 -3.000 -2.750 900 250 Bond A Autonomous payments Operating loan KA Financial investments GA 3.465 133 -210 -55 2.152 1.600,5474 t = 2 t = 3 1.200 2.000 1.500 2.000 t = 4 1.650 -210 -9 -1.960 -9 -1.102,0505 -2.569,8055 -1.855 -8 KA-, GA-paybacks Withdrawal EN UG Payment balance Debt from KA -2.367,2000 -118,3474 0 2.152 0 1.600,5474 Deposits from GA Present value of perpetuity EN UG / 0,05 Figure 2.64: Comprehensive financial plan of the pre-demerger program -1.760,6021 -118,3474 1.157,1529 -118,3474 0 0 2.698,2958 -118,3474 2.366,9484 1.102,0505 2.569,8055 2.366,9484 Withdrawal EN UG Present value of perpetuity EN UG / 0,05 t = 0 t = 1 -118,3474 Share of X (70 %) in the withdrawal EN UG Present value of perpetuity of the share of X (70 %) EN UG / 0,05 Figure 2.65: Share of X in the distribution of the pre-demerger program -82,8432 t = 2 t = 3 -118,3474 -118,3474 t = 4 -118,3474 2.366,9484 -82,8432 -82,8432 -82,8432 1.656,8639 EN Ü1 EN Ü 2 190 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 190 45520_Matschke_Griffleiste_SL5.indd 190 16.03.2021 16: 21: 44 16.03.2021 16: 21: 44 Chapter 2 Regarding the withdrawal structure, the target of income maximization is maintained. It is assumed that the conflicting parties agree on a structure of an income stream. Below, two cases are distinguished: Case a) The conflicting parties agree on the uniform income stream for the respective businesses Ü 1 and Ü 2 , which corresponds to the structure of the business UG before the demerger: = 0 : 1 : 1 : 1 : 21. Case b) The conflicting parties agree on the income structure for the business Ü 1 = 0 : 1 : 1,3 : 1,69 : 22,197, which includes an increase of the withdrawals of 30 % in the first four years, and for business Ü 2 = 0 : 1 : 1,15 : 1,3225 : 21,5209, which corresponds to an increase of 15 % in each of the first four years and, in addition, in t = 4, an amount that, at 5 % per year, provides a perpetuity equal to the payment in t = 1 (i.e., 21,5209 = 1/ 0,05 + 1,15 3 ). Due to the presumption of a one-dimensional, disjoint conflict situation, the share of the conflicting parties in the business has to be determined. Therefore, it is assumed that the conflicting parties have agreed on the shares in business Ü 2 so that only the minimum ownership interest of the shareholders in business Ü 1 has to be determined. The respective rates can be used as decision values during the demerger negotiations. The owners X and Y divide business Ü 2 into the following parts: X receives the share of = 55 %, Y the complement of = 45 %. For case a), the following approach for business Ü 1 must be solved for the determination of the first part of the demerger program: t I 1 GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 A Ü1 b t Ü1 01 -100 30 -1 1,05 -1 1 -1,1 1 99 -6 73 -30 234 Limit 40 50 55 34 ∞ 1,05 -1 1,05 ∞ ∞ -1 1,05 ∞ ∞ -1,1 1 -1,1 ∞ ∞ 1 -6 -56 -1,1 ∞ -53 18 -5 -5 -4 1 Grenze Figure 2.66: Data of business Ü 1 for the determination of the demerger program 1 1 1 t I 2 GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 A Ü2 b t Ü2 01 -110 10 -1 1,05 -1 1 -1,1 1 99 -6 60 -25 234 Limit 80 80 28 ∞ 1,05 -1 1,05 ∞ ∞ -1 1,05 ∞ ∞ -1,1 1 -1,1 ∞ ∞ 1 -6 -56 -1,1 ∞ -53 14 -4 -4 -4 1 Grenze Figure 2.67: Data of business Ü 2 for the determination of the demerger program 1 1 1 w 0 Sp : w 1 Sp : w 2 Sp : w 3 Sp : w 4 Sp w Ü1,0 Sp : w Ü1,1 Sp : w Ü1,2 Sp : w Ü1,3 Sp : w Ü1,4 Sp w Ü 2,0 Sp : w Ü 2,1 Sp : w Ü 2,2 Sp : w Ü 2,3 Sp : w Ü 2,4 Sp α X Ü 2 α Y Ü 2 2.4 Selected Problems of Decision Value Determination 191 45520_Matschke_Griffleiste_SL5.indd 191 45520_Matschke_Griffleiste_SL5.indd 191 16.03.2021 16: 21: 45 16.03.2021 16: 21: 45 under the following constraints: From the perspective of shareholder X, an uniform income stream in the amount of = 82,7713 GE from business Ü 1 after the demerger is attainable. This withdrawal must be divided between the shareholders X and Y. The payment balance amounts to 1.655,4268 GE at the end of the planning horizon. Again, this amount represents the present value of the perpetuity discounted at an annual interest rate of 5 % p. a. By applying the (economic) capitalization formula, the calculation reads: 85,0114 GE · 1/ 0,05 = 1.655,4268 GE. As in the base program, the investment I 1 and the bond A reach the present upper limits. The comprehensive financial plan of business Ü 1 in the demerger program depicts the optimum solution in detail (cf. Figure 2.68). Now, the maximum income stream for business Ü 2 is determined to complete the demerger program. Consequently, the following optimization approach has to be solved: under the constraints: max . withdrawal Entn Ü1 ; Entn Ü1 : = EN Ü1 a ) 100 ⋅ I 1 +1 ⋅ GA 0 − 1 ⋅ KA 0 − 99 ⋅ A≤ 73 − 30 ⋅ I 1 − 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 − 1 ⋅ KA 1 + 6 ⋅ A+1 ⋅ EN Ü1 a) ≤ − 30 − 40 ⋅ I 1 − 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 − 1 ⋅ KA 2 + 6 ⋅ A+1 ⋅ EN Ü1 a) ≤ − 5 − 50 ⋅ I 1 − 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 − 1 ⋅ KA 3 + 56 ⋅ A+1 ⋅ EN Ü1 a) ≤ − 5 − 55 ⋅ I 1 − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 53 ⋅ A+21 ⋅ EN Ü1 a) ≤ − 4 I 1 , GA t , KA t , A, EN Ü1 a) ≥ 0 ∀ t I 1 ≤ 34 A≤ 18. EN Ü1 max a ) EN Ü1 max a ) Investment I 1 Bond A t = 0 t = 1 -3.400 1.782 1.020 -108 Autonomous payments Operating loan KA Financial investments GA KA-, GA-paybacks 73 1.545 -30 900,2713 -1.699,5000 t = 2 t = 3 1.360 -108 1.700 -1.008 t = 4 1.870 -954 -5 -5 -173,9302 -990,2985 -786,8554 182,6267 -4 826,1981 Withdrawal EN Ü1 Payment balance Debt from KA Deposits from GA 0 -82,7713 0 1.545 900,2713 Present value of perpetuity EN Ü1 / 0,05 Figure 2.68: Comprehensive financial plan of business Ü 1 in the demerger program (case a) -82,7713 0 -82,7713 0 173,9302 786,8554 -82,7713 1.655,4268 1.655,4268 max . withdrawal Entn Ü 2 ; Entn Ü 2 : = EN Ü2 a ) 192 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 192 45520_Matschke_Griffleiste_SL5.indd 192 16.03.2021 16: 21: 47 16.03.2021 16: 21: 47 Chapter 2 Business Ü 2 generates an uniform withdrawal stream of = 48,3789 GE if the investment I 2 is made to its maximum amount of 28. It is assumed that the shareholder (and valuation subject) X will already have negotiated a share of = 55 %. Figure 2.69 shows the comprehensive financial plan of the optimum investment and financing program for the demerged business Ü 2 . Hence, the following minimum marginal rate of the owner X in business Ü 1 results in the third step at a previously determined share in the business Ü 2 of 55 %: 110 ⋅ I 2 +1 ⋅ GA 0 − 1 ⋅ KA 0 − 99 ⋅ A≤ 60 − 10 ⋅ I 2 − 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 − 1 ⋅ KA 1 + 6 ⋅ A+1 ⋅ EN Ü2 a ) ≤ − 25 − 80 ⋅ I 2 − 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 − 1 ⋅ KA 2 + 6 ⋅ A+1 ⋅ EN Ü2 a ) ≤ − 4 − 80 ⋅ I 2 − 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 − 1 ⋅ KA 3 + 56 ⋅ A+1 ⋅ EN Ü2 a ) ≤ − 4 − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 53 ⋅ A+21 ⋅ EN Ü2 a ) ≤ − 4 I 2 , GA t , KA t , A, EN Ü2 a ) ≥ 0 ∀ t I 2 ≤ 28 A≤ 14. EN Ü2 max a ) α X Ü 2 Investment I 2 Bond A t = 0 t = 1 -3.080 1.386 280 -84 Autonomous payments Operating loan KA Financial investments GA KA-, GA-paybacks 60 1.634 -25 1.674,7789 -1.797,4000 t = 2 t = 3 2.240 -84 2.240 -784 t = 4 -742 -4 -4 -261,3644 -1.842,2567 -1.678,0538 274,4327 -4 1.761,9569 Withdrawal EN Ü2 Payment balance Debt from KA Deposits from GA 0 -48,3789 0 1.634 1.674,7789 Present value of perpetuity EN Ü2 / 0,05 Share of X (55,00 %) in the withdrawal EN Ü2 Present value of perpetuity of the share of X (55,00 %) EN Ü2 / 0,05 Figure 2.69: Comprehensive financial plan of business Ü 2 in the demerger program (case a) -26,6084 -48,3789 0 -48,3789 0 261,3644 1.678,0538 -48,3789 967,5780 -26,6084 -26,6084 967,578 -26,6084 532,1676 α min X Ü1 a ) = EN UG max EN Ü1 max a ) ⋅ β X − EN Ü 2 max EN Ü1 max a ) ⋅ α X Ü 2 α min X Ü1 a ) = 118,3474 82,7713 ⋅ 0,7 − 48,3789 82,7713 ⋅ 0,55 ↔ α min X Ü1 a ) = 0,6794. 2.4 Selected Problems of Decision Value Determination 193 45520_Matschke_Griffleiste_SL5.indd 193 45520_Matschke_Griffleiste_SL5.indd 193 16.03.2021 16: 21: 48 16.03.2021 16: 21: 48 This means that the shareholder X with 55 % in business Ü 2 has to receive at least 67,94 % of business Ü 1 to have the same advantage as would be the case without the demerger of the business UG (see Figure 2.65). The comprehensive financial plan of the valuation subject X (valuation program) in case a) is represented in Figure 2.70. For case b), in the second step the maximum possible distribution for the business Ü 1 is to be determined with the following linear optimization approach. The already negotiated withdrawal structure for business Ü 1 reads = 0 : 1 : 1,3 : 1,69 : 22,197: under the constraints: From the demerger program for business Ü 1 an income stream of the size = 75,7676 GE can be calculated, again using the simplex algorithm. Under consideration of the withdrawal structure, the distributions for the owners can be calculated for each period t. At the end of the planning period, the payment balance amounts to Withdrawal EN U1 Present value of EN Ü1 / 0,05 t = 0 t = 1 82,7713 Withdrawal EN U2 Present value of EN Ü2 / 0,05 Share of X (67,94 %) in withdrawal EN Ü1 Present value of the share of X (67,94 %) EN Ü1 / 0,05 48,3789 56,2348 t = 2 t = 3 82,7713 82,7713 t = 4 82,7713 1.655,4268 48,3789 48,3789 56,2348 56,2348 48,3789 967,578 56,2348 1.124,6963 Share of X (55,00 %) in withdrawal EN Ü2 Present value of the share of X (55,00 %) EN Ü2 / 0,05 Sum of the shares of X in withdrawals EN Ü1 and EN Ü2 Present value of the shares of X (EN Ü1 +EN Ü2 )/ 0,05 26,6084 -82,8432 Figure 2.70: Comprehensive financial plan of the valuation program of shareholder X (case a) 26,6084 26,6084 -82,8432 -82,8432 26,6084 532,1676 -82,8432 1.656,8639 w Ü1,0 Sp : w Ü1,1 Sp : w Ü1,2 Sp : w Ü1,3 Sp : w Ü1,4 Sp max . withdrawal Entn Ü1 ; Entn Ü1 : = EN Ü1 b) 100 ⋅ I 1 +1 ⋅ GA 0 − 1 ⋅ KA 0 − 99 ⋅ A≤ 73 − 30 ⋅ I 1 − 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 − 1 ⋅ KA 1 + 6 ⋅ A+1 ⋅ EN Ü1 b) ≤ − 30 − 40 ⋅ I 1 − 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 − 1 ⋅ KA 2 + 6 ⋅ A+1,3 ⋅ EN Ü1 b) ≤ − 5 − 50 ⋅ I 1 − 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 − 1 ⋅ KA 3 + 56 ⋅ A+1,69 ⋅ EN Ü1 b) ≤ − 5 − 55 ⋅ I 1 − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 53 ⋅ A+22,197 ⋅ EN Ü1 b) ≤ − 4 I t , GA t , KA t , A, EN Ü1 b) ≥ 0 ∀ t I 1 ≤ 34 A≤ 18. EN Ü1 max b) 194 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 194 45520_Matschke_Griffleiste_SL5.indd 194 16.03.2021 16: 21: 49 16.03.2021 16: 21: 49 Chapter 2 1.515,3522 GE. If discounted at 5 %, it is the foundation of the perpetuity The comprehensive financial plan of business Ü 1 in the demerger program in case b) depicts the optimum solution in Figure 2.71. According to the demerger program for business Ü 2 , a linear approach is to be formulated and solved, too. However, now the following withdrawal stream is considered = 0 : 1 : 1,15 : 1,3225 : 21,5209: under the constraints: In case b), a withdrawal stream of = 46,4261 GE results for business Ü 2 , in which the valuation subject X holds a share of = 55 %. The comprehensive financial plan, considering the corresponding withdrawal structure of the optimum investment and financing program of the newly demerged business Ü 2 , is illustrated in Figure 2.72. EN Ü1 max b) . Investment I 1 Bond A t = 0 t = 1 -3.400 1.782 1.020 -108 Autonomous payments Operating KA Financial investments GA KA-, GA-paybacks 73 1.545 -30 893,2676 -1.699,5000 t = 2 t = 3 1.360 -108 1.700 -1.008 t = 4 1.870 -954 -5 -5 -165,9077 -982,5944 -733,1558 174,2031 -4 769,8136 Withdrawal EN Ü1 Payment balance Debt from KA Deposits from GA 0 -75,7676 0 1.545 893,2676 Present value of perpetuity EN Ü1 / 0,05 Figure 2.71: Comprehensive financial plan of business Ü 1 in the demerger program (case b) -98,4979 0 -128,0473 0 165,9077 733,1558 -166,4614 1.515,3522 1.515,3522 w Ü 2,0 Sp : w Ü 2,1 Sp : w Ü 2,2 Sp : w Ü 2,3 Sp : w Ü 2,4 Sp max . withdrawal Entn Ü 2 ; Entn Ü 2 : = EN Ü2 b) 110 ⋅ I 2 +1 ⋅ GA 0 − 1 ⋅ KA 0 − 99 ⋅ A≤ 60 − 10 ⋅ I 2 − 1,05 ⋅ GA 0 + 1 ⋅ GA 1 + 1,1 ⋅ KA 0 − 1 ⋅ KA 1 + 6 ⋅ A+1 ⋅ EN Ü2 b) ≤ − 25 − 80 ⋅ I 2 − 1,05 ⋅ GA 1 + 1 ⋅ GA 2 + 1,1 ⋅ KA 1 − 1 ⋅ KA 2 + 6 ⋅ A+1,15 ⋅ EN Ü2 b) ≤ − 4 − 80 ⋅ I 2 − 1,05 ⋅ GA 2 + 1 ⋅ GA 3 + 1,1 ⋅ KA 2 − 1 ⋅ KA 3 + 56 ⋅ A+1,3225 ⋅ EN Ü2 b) ≤ − 4 − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 53 ⋅ A+21,5209 ⋅ EN Ü2 b) ≤ − 4 I 2 , GA t , KA t , A, EN Ü2 b) ≥ 0 ∀ t I 2 ≤ 28 A≤ 14. EN Ü2 max b) α X Ü 2 Kapitel 1: Einführung 195 2.4 Selected Problems of Decision Value Determination 195 45520_Matschke_Griffleiste_SL5.indd 195 45520_Matschke_Griffleiste_SL5.indd 195 16.03.2021 16: 21: 50 16.03.2021 16: 21: 50 It is obvious that the maximum withdrawals and neither reflect individually nor in a cumulated manner reflect the desired uniform withdrawal (distribution) structure = 0 : 1 : 1 : 1 : 21 of the original business UG. In order to make a corporate financial comparison between the demerged businesses and the original business, the valuation subject has to transform the distributions of the demerged businesses into the structure of the (desired) distribution stream of the business UG. This happens with the help of asset restructuring in the private decision field in the third step. For the private decision field of shareholder X, the following parameters are defined: The valuation subject X can invest liquid funds on the capital market at the interest rate of 5 % p. a. (GA priv ). Furthermore, owner X is equipped with 5 % p. a. deposits at all times that can be reduced by 2 GE. Valuation subject X also receives liquidity by taking out operating loans (KA priv ) at a short-term interest (borrowing) rate of 10 % p. a. The basis of the transformation is the expected distributions from both demerged businesses, where the distributions of Ü 1 read = 0 : 1 : 1,3 : 1,69 : 22,197 and those of Ü 2 have the structure = 0 : 1 : 1,15 : 1,3225 : 21,5209. The required minimum ownership interest in business Ü 1 has to guarantee that the withdrawal structure of business UG = 0 : 1 : 1 : 1 : 21 and the amount of the distributions from the business UG, which the decision subject X receives from their share β X = 0,7, can be replicated (reproduced) by private transactions of owner X. For this purpose, the decision subject has to solve the following optimization approach for the determination of the required marginal rate Investment I 2 Bond A t = 0 t = 1 -3.080 1.386 280 -84 Autonomous payments Operating loan KA Financial investments GA KA-, GA-paybacks 60 1.634 -25 1.673 -1.797,4000 t = 2 t = 3 2.240 -84 2.240 -784 t = 4 -742 -4 -4 -258,5013 -1.840,1087 -1.662,0279 271,4264 -4 1.745,1293 Withdrawal EN U2 Payment balance Debt from KA Deposits from GA 0 -46,4261 0 1.634 1.672,8261 Present value of perpetuity EN Ü2 / 0,05 Share of X (55,00 %) in withdrawal EN Ü2 Present value of the share of X (55,00 %) EN Ü2 / 0,05 Figure 2.72: Comprehensive financial plan of business Ü 2 in the demerger program (case b) -25,5343 -53,39 0 -61,3985 0 258,5013 1.662,0279 -70,6082 928,5211 -29,3645 -33,7692 928,5211 -38,8345 510,6859 EN Ü1 max b) EN Ü2 max b) w UG ,0 Sp : w UG ,1 Sp : w UG ,2 Sp : w UG ,3 Sp : w UG ,4 Sp w Ü1,0 Sp : w Ü1,1 Sp : w Ü1,2 Sp : w Ü1,3 Sp : w Ü1,4 Sp w Ü 2,0 Sp : w Ü 2,1 Sp : w Ü 2,2 Sp : w Ü 2,3 Sp : w Ü 2,4 Sp w UG ,0 Sp : w UG ,1 Sp : w UG ,2 Sp : w UG ,3 Sp : w UG ,4 Sp α min X Ü1 b) : 196 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 196 45520_Matschke_Griffleiste_SL5.indd 196 16.03.2021 16: 21: 52 16.03.2021 16: 21: 52 Chapter 2 under the constraints: These constraints can be simplified to: Finally, the required minimum ownership interest = 67,93455 % is only slightly lower in comparison to case a). Only if this rate is negotiated will the valuation subject not suffer a disadvantage as opposed to the case without the demerger. The comprehensive financial plan of the valuation program of shareholder X in Figure 2.73 shows that the valuation subject can replicate the desired temporal withdrawal structure and the amount of the distributions of business UG at this rate if the private decision field is considered. The example of the demerger with its equal (see case a) and different (see case b) withdrawal weightings as an expression of the distribution policy for both demerged businesses makes clear that not only the ownership interests but also the weightings are original conflict-resolution-relevant facts, which lead to different marginal rates. min. A; A : = α X Ü1 b) −1⋅ GA 0 priv + 1 ⋅ KA 0 priv + α X Ü1 b) ⋅ 0 ⋅ 75,7676 + 0,55 ⋅ 0 ⋅ 46,4261 ≥ 0 ⋅ 0,7 ⋅ 118,3474 1,05 ⋅ GA 0 priv − 1 ⋅ GA 1 priv − 1,1 ⋅ KA 0 priv + 1 ⋅ KA 1 priv + α X Ü1 b) ⋅ 1 ⋅ 75,7676 + 0,55 ⋅ 1 ⋅ 46,4261 ≥ 1 ⋅ 0,7 ⋅ 118,3474 1,05 ⋅ GA 1 priv − 1 ⋅ GA 2 priv − 1,1 ⋅ KA 1 priv + 1 ⋅ KA 2 priv + α X Ü1 b) ⋅ 1,3 ⋅ 75,7676 + 0,55 ⋅ 1,15 ⋅ 46,4261 ≥ 1 ⋅ 0,7 ⋅ 118,3474 1,05 ⋅ GA 2 priv − 1 ⋅ GA 3 priv − 1,1 ⋅ KA 2 priv + 1 ⋅ KA 3 p + α X Ü1 b) ⋅ 1,69 ⋅ 75,7676 + 0,55 ⋅ 1,3225 ⋅ 46,4261 ≥ 1 ⋅ 0,7 ⋅ 118,3474 1,05 ⋅ GA 3 priv − 1,1 ⋅ KA 3 priv + α X Ü1 b) ⋅ 22,197 ⋅ 75,7676 + 0,55 ⋅ 21,5209 ⋅ 46,4261 ≥ 21 ⋅ 0,7 ⋅ 118,3474 − GA t priv ≥ − 2 ∀ t KA t priv ≥ 0 ∀ t. −1⋅ GA 0 priv + 1 ⋅ KA 0 priv ≥ 0 1,05 ⋅ GA 0 priv − 1 ⋅ GA 1 priv − 1,1 ⋅ KA 0 priv + 1 ⋅ KA 1 priv + α X Ü1 b) ⋅ 1 ⋅ 75,7676 ≥ 57,3088 1,05 ⋅ GA 1 priv − 1 ⋅ GA 2 priv − 1,1 ⋅ KA 1 priv + 1 ⋅ KA 2 priv + α X Ü1 b) ⋅ 1,3 ⋅ 75,7676 ≥ 53, 4787 1,05 ⋅ GA 2 priv − 1 ⋅ GA 3 priv − 1,1 ⋅ KA 2 priv + 1 ⋅ KA 3 p + α X Ü1 b) ⋅ 1,69 ⋅ 75,7676 ≥ 49,0740 1,05 ⋅ GA 3 priv − 1,1 ⋅ KA 3 priv + α X Ü1 b) ⋅ 22,197 ⋅ 75,7676 ≥ 1.190,1845 − GA t priv ≥ − 2 ∀ t KA t priv ≥ 0 ∀ t. α min X Ü1 b) 2.4 Selected Problems of Decision Value Determination 197 45520_Matschke_Griffleiste_SL5.indd 197 45520_Matschke_Griffleiste_SL5.indd 197 16.03.2021 16: 21: 53 16.03.2021 16: 21: 53 To summarize, it becomes apparent that the conflict situations of the type demerger, which were presented here as one-dimensional conflict situations for didactic reasons, are generally multi-dimensional conflict situations. The focus of the negotiations is on the ownership interest of the respective owners of the demerged businesses, but also on the aims and on the structures of the distribution streams. The single ownership interest of a newly demerged business determines the marginal rates of the other newly demerged business(es). This solution defect can be solved heuristically by calculating the respective marginal rate under systematically varying ceteris paribus conditions, namely with regard to the ownership interests and under consideration of the already negotiated ownership interests of the other businesses. For the reduction of complexity, the calculations should be limited to reasonable ownership interests (or possible areas thereof). Withdrawal EN Ü1 Present value of perpetuity EN Ü1 / 0,05 t = 0 t = 1 75,7676 Withdrawal EN Ü2 Present value of perpetuity EN Ü2 / 0,05 Share of X (67,93455 %) in withdrawal EN Ü1 Present value of the share of X (67,93455 %) EN Ü1 / 0,05 46,4261 51,4724 t = 2 t = 3 98,4979 128,0473 t = 4 166,4614 1.515,3522 53,39 61,3985 66,9141 86,9883 70,6082 928,5211 113,0848 1.029,4477 Share of X (55,00 %) in withdrawal EN Ü2 Present value of the share of X (55,00 %) EN Ü2 / 0,05 Sum of the shares of X in withdrawals EN Ü1 and EN Ü2 Present value of the shares of X (EN Ü1 +EN Ü2 )/ 0,05 25,5343 77,0067 Deposits from possible reduction of private financial investments GA priv Minimum deposits from the reduction of private financial investments GA priv Private operating loan KA priv Financial investments GA priv 2 3,8365 29,3645 33,7692 96,2786 120,7575 38,8345 510,6859 151,9193 1.540,1336 -2,1 -7,1152 -45,3851 KA priv -, GA priv -paybacks Payment balance Debt from KA priv 0 -82,8432 0 0 3,8365 Deposits from GA priv Desired present value of perpetuity EN UG / 0,05 Figure 2.73: Comprehensive financial plan of the valuation program of shareholder X (case b) -4,2202 -82,8432 7,4708 -82,8432 0 0 47,6542 -82,8432 1.656,8639 7,1152 45,3851 1.656,8639 Desired withdrawal EN UG max 198 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 198 45520_Matschke_Griffleiste_SL5.indd 198 16.03.2021 16: 21: 54 16.03.2021 16: 21: 54 Chapter 2 2.4.4 Joint Conflict Situations 2.4.4.1 Preliminary Remarks In the previous analysis, only one business was valuated. A simultaneous valuation of several objects at the decision date by the valuation subject was excluded so far. Now, it is assumed that the decision subject is in contact with various negotiation partners and takes into account more than just one acquisition, sale, merger, and/ or demerger (M ATSCHKE 1975, p. 336). The corresponding decision values in these conflict situations cannot be regarded in isolation. H ERING (2014, p. 166) remarks on this issue: In this case, the concession willingness concerning a certain valuation object depends on the negotiation results of other valuation objects. The valuations are interdependent because the decision value of each object depends on the maximum target function value of the corresponding base program and this again depends on the negotiation results of the other valuation objects. In order to reasonably negotiate the prices and shares of the objects, the valuation subject needs to know the value, which will be determined throughout the negotiation process. Hence, the valuation situations are inextricably interwoven. In other words, a joint conflict situation is in place. For the decision subject, who is taking part in x joint conflict situations KS 1 , KS 2 , ..., KS n , ..., KS x - in which the subject is an acquisition, sale, merger, and/ or demerger of x businesses U 1 , U 2 , ..., U n , ..., U x -, it is essential, due to the present interdependencies, to define the decision value for business U 1 and additionally the decision values for the businesses in the remaining (x - 1) conflict situations. The decision value of business U 1 is dependent on the maximum target function value of the corresponding base program. This target function value is again influenced by the negotiated prices and distributions of the property rights in the other (x - 1) conflict situations. Equally, a relationship exists between the decision value of business U 2 and the (x - 2) remaining businesses (this time under consideration of the negotiated price or distributions of property rights for U 1 ). With regard to the prices P or the distributions of property rights VE (German: Verteilung der Eigentumsrechte) still to be negotiated, the decision values EW, which are depending on the open possibilities in the decision field, can be expressed as follows: While the prices and distributions of the property rights of all other objects are ceteris paribus conditions in a disjoint situation, the decision value of one object in the joint conflict situation - and hence the respective base program and the corresponding valuation program - is determined depending on the negotiated prices and distributions of property rights of the other objects. The presence of the solution problem makes it useful to solve the parametric optimization approach heuristically by iteratively computing the decision value under the systematically varying ceteris paribus conditions (H ERING 2014, p. 166). EW U1 EW U 2 EW U a = f P ∨ VE U n ∈ U 1 , U 2 , ..., U a , ..., U n , ..., U x { } U a { } ( ) ∀ U a ∈ U 1 , U 2 , ..., U a , ..., U n , ..., U x { } . 2.4 Selected Problems of Decision Value Determination 199 45520_Matschke_Griffleiste_SL5.indd 199 45520_Matschke_Griffleiste_SL5.indd 199 16.03.2021 16: 21: 54 16.03.2021 16: 21: 54 Generally, these are non-dominated, joint, one-dimensional conflict situations that are outlined in the corresponding conflict cube in Figure 2.74. 2.4.4.2 Exemplary Representation of the Joint Situation of the Acquisition/ Acquisition Type The next step addresses the approach to decision value determination in a joint conflict situation of the acquisition/ acquisition type based on the state price marginal price model. To do so an example with a multi-period planning horizon (T = 4) (cited from B RÖSEL 2002, p. 98) is employed. At t = 0, the valuation subject could acquire another enterprise U A and/ or U B . For the valuation object U A , the payment stream (0 GE, 60 GE, 40 GE, 20 GE, 10 GE) and additionally the perpetuity of 1 GE is expected from t = 5 onwards. According to business U B , the determined payment stream is (0 GE, 30 GE, 40 GE, 50 GE, 15 GE). From t = 5, a perpetuity of 2 GE is expected. From internal financing (IF) a perpetuity of 30 GE of already owned businesses is achieved at each time t. However, this payment stream is independent of the business to be valuated. At the decision date, the valuation subject additionally owns personal equity assets (EM) in the amount of 100 GE. Further financial funds (operating loans are available in unlimited amounts at a short-term interest (borrowing) rate of 10 % p. a. Financial investments (cash investments can be made in any amount at an interest rate of 5 % p. a. Figure 2.74: Conflict cube for a non-dominated, joint, and one-dimensional conflict situation Degree of connectedness Multi-dimensional One-dimensional Degree of dominance Non-dominated Dominated Disjoint Joint Degree of complexity KA t ) GA t ) 200 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 200 45520_Matschke_Griffleiste_SL5.indd 200 16.03.2021 16: 21: 55 16.03.2021 16: 21: 55 Chapter 2 The valuation subject desires a uniform payment stream, which anticipates the withdrawal EN in each period. In addition to the regular distribution EN, the last distribution comprises the present value of a perpetuity to guarantee the income EN beyond the planning period. In the example, the estimated interest rate of i = 5 % p. a. is taken into account for t > T. By applying the capitalization formula in t = 4 - as already shown in various examples - the factor 20 (= 1/ i) results for payments after time t > T = 4. The desired temporal structure of the withdrawals reads = 1 : 1 : 1 : 1 : 21. In Figure 2.75, the data of this example are summarized once again (the values for t = 4 result as follows: for IF 30 GE + 20 · 30 GE = 630 GE, for U A 10 GE + 20 · 1 GE = 30 GE, and for U B 15 GE + 20 · 2 GE = 55 GE). For the decision subject, it is crucial to assess not only the decision value for the business U A but also the marginal price for the business U B . It must be considered that the decision values are interdependent in joint conflict situations. The marginal price for business U A depends on the maximum target function value of the corresponding base program. This is influenced by the negotiated price for business U B . The same holds true for the relationship between the marginal price for the business and the negotiated price for business U A . For those marginal prices that still depend on the open action options in the decision field, it follows that: In the example of the determination of joint decision values , the assumed or already negotiated price of business U B has to be increased incrementally and the maximum target function value of the corresponding base program has to be determined. For the respective target function value of the base program, the corresponding marginal price for business U A has to be assessed. It should be considered in both the base and the valuation program that an integer (whole number) constraint is assumed for business U B . Below, the linear optimization approach for the determination of the base program at the price = 0 is elaborated, which can be solved again utilizing the simplex algorithm: w 0 : w 1 : w 2 : w 3 : w 4 t GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U A U B 01 -1 1,05 -1 1 -1,1 1 100 30 30 P? 60 P? 30 234 Limit 1,05 ∞ ∞ -1 1,05 -1 ∞ 1,05 ∞ -1,1 ∞ ∞ 1 -1,1 1 ∞ -1,1 ∞ 30 30 1 630 1 40 20 40 50 30 1 55 1 Grenze Figure 2.75: Data of the joint conflict situation of the acquisition/ acquisition type 1 1 1 1 P max UA P UB P UA P max UA = f (P UB ) ∧ P max UB = f (P UA ). P max UA P UB EN UA max (P UB ) P max UA (P UB ) P UB 2.4 Selected Problems of Decision Value Determination 201 45520_Matschke_Griffleiste_SL5.indd 201 45520_Matschke_Griffleiste_SL5.indd 201 16.03.2021 16: 21: 55 16.03.2021 16: 21: 55 under the following constraints: The target function value = 42,0616 GE results as the optimum solution. Hence, the corresponding approach for the determination of the valuation program at the price = 0 GE is represented as follows: under the constraints: The marginal price for business U A at the price = 0 GE finally amounts to = 133,2253 GE. In Figure 2.76, the target function values and the corresponding marginal prices (decision values) for business U A as well as the endogenous marginal interest rates of the base program and those of the valuation program are determined in dependence of the different negotiation results for the price of business U B . max. withdrawal Entn; Entn : = EN UA 0 ⋅ U B + 1 ⋅ GA 0 − 1 ⋅ KA 0 + 1 ⋅ EN UA ≤ 130 − 30 ⋅ U B + 1 ⋅ GA 1 − 1,05 ⋅ GA 0 − 1 ⋅ KA 1 + 1,1 ⋅ KA 0 + 1 ⋅ EN UA ≤ 30 − 40 ⋅ U B + 1 ⋅ GA 2 − 1,05 ⋅ GA 1 − 1 ⋅ KA 2 + 1,1 ⋅ KA 1 + 1 ⋅ EN UA ≤ 30 − 50 ⋅ U B + 1 ⋅ GA 3 − 1,05 ⋅ GA 2 − 1 ⋅ KA 3 + 1,1 ⋅ KA 2 + 1 ⋅ EN UA ≤ 30 − 55 ⋅ U B − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN UA ≤ 630 GA t , KA t , EN UA ≥ 0 ∀ t U B ∈ 0; 1 { } . EN UA max (P UB = 0 GE) P UB max. W UA ; W UA : = P max UA 0 ⋅ U B + 1 ⋅ GA 0 − 1 ⋅ KA 0 + 1 ⋅ EN UA + P UA ≤ 130 − 30 ⋅ U B + 1 ⋅ GA 1 − 1,05 ⋅ GA 0 − 1 ⋅ KA 1 + 1,1 ⋅ KA 0 + 1 ⋅ EN UA ≤ 90 − 40 ⋅ U B + 1 ⋅ GA 2 − 1,05 ⋅ GA 1 − 1 ⋅ KA 2 + 1,1 ⋅ KA 1 + 1 ⋅ EN UA ≤ 70 − 50 ⋅ U B + 1 ⋅ GA 3 − 1,05 ⋅ GA 2 − 1 ⋅ KA 3 + 1,1 ⋅ KA 2 + 1 ⋅ EN UA ≤ 50 − 55 ⋅ U B − 1,05 ⋅ GA 3 + 1,1 ⋅ KA 3 + 21 ⋅ EN UA ≤ 660 EN ≥ 42,0616 GA t , KA t ≥ 0 ∀ t U B ∈ 0; 1 { } . P max UA P UB P max UA (P UB = 0 GE) EN UA max (P UB ) P max UA (P UB ) i t Ba i t Be P UB 202 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 202 45520_Matschke_Griffleiste_SL5.indd 202 16.03.2021 16: 21: 56 16.03.2021 16: 21: 56 Chapter 2 The performance of the base program has the highest value at the price = 0 GE. The range of the withdrawal stream - as target function value of the base program for the valuation of business U A - declines by increasing price for business U B as long as this object is withdrawn from the base program at a 0 10 (5; 5; 5; 5) (5; 5; 5; 5) 42,0616 41,5854 133,2253 132,7924 (10; 5; 5; 5) (10; 5; 5; 5) 20 29,5287 29,5288 30 (5; 5; 5; 5) (5; 5; 5; 5) 41,1092 40,6554 (5; 5; 5; 5) (5; 5; 5; 5) 40,6554 40,6330 132,3595 131,9470 (10; 5; 5; 5) (10; 5; 5; 5) 131,9470 131,9081 (10; 10; 5; 5) (10; 10; 5; 5) 40 50 60 70 (5; 5; 5; 5) (5; 5; 5; 5) 40,1568 39,6806 (5; 5; 5; 5) (5; 5; 5; 5) 39,2044 38,7282 80 90 92,3353 92,3354 (5; 5; 5; 5) (5; 5; 5; 5) 38,2521 37,7759 (5; 5; 5; 5) (10; 5; 5; 5) 37,6647 37,6647 131,0817 130,4484 (10; 10; 5; 5) (10; 10; 5; 5) 129,4288 128,6023 (10; 10; 5; 5) (10; 10; 5; 5) 127,7759 126,9494 (10; 10; 5; 5) (10; 10; 5; 5) 126,7564 126,7564 (10; 10; 5; 5) (10; 10; 5; 5) 100 101,9072 101,9073 110 (10; 5; 5; 5) (10; 5; 5; 5) 37,2832 37,1882 (10; 5; 5; 5) (10; 5; 5; 5) 37,1882 36,7854 114,7722 114,7723 120 130 (10; 5; 5; 5) (10; 10; 5; 5) 36,5479 36,5479 (10; 10; 5; 5) (10; 10; 5; 5) 36,2765 35,7574 126,4411 126,3627 (10; 10; 5; 5) (10; 10; 5; 5) 126,3627 125,7272 (10; 10; 10; 5) (10; 10; 10; 5) 125,3524 125,3524 (10; 10; 10; 5) (10; 10; 10; 5) 125,1485 124,7585 (10; 10; 10; 5) (10; 10; 10; 5) 140 140,3542 140,3543 145 (10; 10; 5; 5) (10; 10; 5; 5) 35,2383 35,2200 (10; 10; 5; 5) (10; 10; 5; 5) 35,2199 34,9788 146,9940 146,9941 147 148 (10; 10; 5; 5) (10; 10; 5; 5) 34,8753 34,8753 (10; 10; 5; 5) (10; 10; 10; 5) 34,8750 34,8209 124,3685 124,3547 (10; 10; 10; 5) (10; 10; 10; 5) 124,3547 129,1997 (10; 5; 5; 5) (10; 5; 5; 5) 131,2792 131,2793 (10; 5; 5; 5) (10; 5; 5; 5) 131,2857 132,3709 (10; 5; 5; 5) (10; 5; 5; 5) 149,0931 149,0932 150 160 (10; 10; 10; 5) (5; 5; 5; 5) 34,7619 34,7619 (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 170 180 190 200 (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 133,5571 133,5571 (10; 5; 5; 5) (10; 5; 5; 5) 133,5571 133,5571 (10; 5; 5; 5) (10; 5; 5; 5) 133,5571 133,5571 (10; 5; 5; 5) (10; 5; 5; 5) 133,5571 133,5571 (10; 5; 5; 5) (10; 5; 5; 5) Figure 2.76: Joint marginal price of business U A P UB in GE i t Ba in % EN UA max (P UB ) in GE P max UA (P UB ) in GE i t Be in % EN UA max (P UB ) P UB EN UA max (P UB ) P UB 2.4 Selected Problems of Decision Value Determination 203 45520_Matschke_Griffleiste_SL5.indd 203 45520_Matschke_Griffleiste_SL5.indd 203 16.03.2021 16: 21: 57 16.03.2021 16: 21: 57 price of > 149,0931 GE. Hence, at prices > 149,0931 GE the acquisition of business U B is not reasonable - independently from the negotiation result for business U A . Up to the price of ≤ 140,3542 GE, both the performance of the base program and the corresponding decision value for business U A are shrinking depending on the negotiation result for business U B . This is because besides the business U B , business U A could also be acquired at the corresponding marginal price. The reduction of the performance of the base program depends on the endogenous marginal interest rates of the base program and the withdrawal structure. By an exemplary increase of the price for business U B from 0 GE to 10 GE, a reduction of follows. Under consideration of = (0,05; 0,05; 0,05; 0,05) the decrease can be computed as follows: If, however, the price for business U B is increased from 90 GE to 100 GE, the determination of must consider that the structure of the endogenous interest rates of the base program up to the price = 92,3353 GE with remains constant, but from the price < 92,3353 GE it changes to . At an increased price for business U B of 10 GE, the determination of the reduction requires the consideration of the price areas with different interest rates structures: Up to the price ≤ 140,3542 GE the decision value is reduced due to a decrease of the performance of the base program The reduction of the marginal price is determined according to the withdrawal structure, the change of the range of the withdrawal stream, and the increased price for business U B under P UB P UB P UB EN UA max (P UB ) P max UA (P UB ) P UB EN UA max (P UB ) (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) ΔEN UA max (P UB (0 → 10)) = (0 − 10) 1 + 1 1,05 + 1 1,05 2 + 1 1,05 3 + 21 1,05 4 = − 0,4762. ΔEN UA max (P UB (90 → 100)) P UB (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) = (0,05; 0,05; 0,05; 0,05) P UB (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) = (0,1; 0,05; 0,05; 0,05) ΔEN UA max (P UB (90 → 100)) ΔEN UA max (P UB (90 → 100)) = Δ EN UA max (P UB (90 → 92,3353)) +Δ EN UA max (P UB (92,3353 → 100)) = (90 − 92,3353) 1 + 1 1,05 + 1 1,05 2 + 1 1,05 3 + 21 1,05 4 + (92,3353 − 100) 1 + 1 1,1 + 1 1,1 ⋅ 1,05 + 1 1,1 ⋅ 1,05 2 + 21 1,1 ⋅ 1,05 3 = − 0,4927. P UB P max UA (P UB ) EN UA max (P UB ). P UB 204 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 204 45520_Matschke_Griffleiste_SL5.indd 204 16.03.2021 16: 21: 58 16.03.2021 16: 21: 58 Chapter 2 consideration of the marginal interest rates of the corresponding valuation program. At the represented price increase for business U B from 0 GE to 10 GE, the following reduction of the marginal price can be determined with = (0,1; 0,05; 0,05; 0,05) and = -0,4762 GE. It should be noted that the calculation is based on non-rounded values: Consequently, at a price increase for business U B from 90 GE to 100 GE - under consideration of the reduction of the performance of the base program = -0,4927 GE as well as the relevant endogenous interest rates of the valuation program - the resultant reduction of the marginal price is illustrated as follows: Moreover, at an exemplary price increase for business U B from 20 GE to 30 GE, it has to be considered that the structure of the endogenous interest rates of the valuation program changes. While the endogenous marginal interest rates are determined by = (0,1; 0,05; 0,05; 0,05) within the price range (20 GE ≤ ≤ 30 GE) for prices 20 GE ≤ ≤ 29,5287 GE , they amount to = (0,1; 0,1; 0,05; 0,05) in the price range of 29,5287 GE < ≤ 30 GE. Under the consideration of the reduction of the performance of the base program = -0,4538 GE at a price increase from 20 GE to 29,5287 GE and = -0,0224 GE at an price increase of over 29,5287 GE up to 30 GE and the corresponding endogenous marginal interest rates, the following value results with regard to the change of the marginal prices : P max UA (P UB ) (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) ΔEN UA max (P UB (0 → 10)) ΔP max UA (P UB (0 → 10)) = (0 − 10) + 0,4762 + 0,4762 1,1 + 0,4762 1,1 ⋅ 1,05 + 0,4762 1,1 ⋅ 1,05 2 + 21 ⋅ 0,4762 1,1 ⋅ 1,05 3 = − 0,4329. ΔEN UA max (P UB (90 → 100)) (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) = (0,1; 0,1; 0,05; 0,05) ΔP max UA (P UB (90 → 100)) = (90 − 100) + 0,4927 + 0,4927 1,1 + 0,4927 1,1 2 + 0,4927 1,1 2 ⋅ 1,05 + 21 ⋅ 0, 4927 1,1 2 ⋅ 1,05 2 = − 0,5083. (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) P UB P UB (i 1 Ba ; i 2 Ba ; i 3 Ba ; i 4 Ba ) P UB ΔEN UA max (P UB (20 → 29,5287)) P UB ΔEN UA max (P UB (29,5287 → 30)) P UB P max UA (P UB ) 2.4 Selected Problems of Decision Value Determination 205 45520_Matschke_Griffleiste_SL5.indd 205 45520_Matschke_Griffleiste_SL5.indd 205 16.03.2021 16: 21: 59 16.03.2021 16: 21: 59 Within the price interval 140,3543 GE < ≤ 149,0931 GE, business U A replaces business U B in the optimum investment and financing program of the valuation subject at the transfer from the base to the valuation program. In other words, the acquisition of business U A at the corresponding marginal price is only reasonable by waiving business U B. The decision value increases in this area at an increasing price and with a decreasing withdrawal stream For the determination of the decision value of business U B , the potential price for business U A represents the variable. Figure 2.77 shows and , according to and the respective endogenous marginal interest rates for the base program and those of the valuation program. The integer constraint has to be considered for business U A . ΔP max UA (P UB (20 → 30)) = Δ P max UA (P UB (20 → 29,5287)) + Δ P max UA (P UB (29,5287 → 30)) = (20 − 29,5287) + 0,4538 + 0,4538 1,1 + 0,4538 1,1 ⋅ 1,05 + 0,4538 1,1 ⋅ 1,05 2 + 21 ⋅ 0, 4538 1,1 ⋅ 1,05 3 + (29,5287 − 30) + 0,0224 + 0,0224 1,1 + 0,0224 1,1 2 + 0,0224 1,1 2 ⋅ 1,05 + 21 ⋅ 0,0224 1,1 2 ⋅ 1,05 2 = − 0, 4514. P UB P max UA (P UB ) P UB EN UA max (P UB ). P max UB (P UA ) P UA EN UB max (P UA ) P max UB (P UA ) P UA i t Ba i t Be 206 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 206 45520_Matschke_Griffleiste_SL5.indd 206 16.03.2021 16: 21: 59 16.03.2021 16: 21: 59 Chapter 2 The range of the withdrawal stream is reduced by an increasing price for business U A until at the price of > 133,5571 GE it is no longer included in the base program. Within the price interval 0 GE ≤ ≤ 124,3546 GE, the decision value for business U B is reduced because it can be acquired at the marginal price in addition to business U A . Conversely, the decision value increases within the interval 124,3547 GE ≤ ≤ 133,5571 GE because the valuation sub- 0 10 (5; 5; 5; 5) (5; 5; 5; 5) 41,2087 40,7325 150,3612 149,9283 (10; 5; 5; 5) (10; 5; 5; 5) 11,6176 11,6177 20 30 (5; 5; 5; 5) (5; 5; 5; 5) 40,6554 40,6554 (5; 5; 5; 5) (5; 5; 5; 5) 40,2563 39,7801 149,8583 149,8583 (10; 5; 5; 5) (10; 10; 5; 5) 149,1655 148,3391 (10; 10; 5; 5) (10; 10; 5; 5) 40 50 60 70 (5; 5; 5; 5) (5; 5; 5; 5) 39,3039 38,8277 (5; 5; 5; 5) (5; 5; 5; 5) 38,3515 37,8753 80 84,4294 84,4295 90 (5; 5; 5; 5) (5; 5; 5; 5) 37,3991 37,1882 (5; 5; 5; 5) (5; 5; 5; 5) 37,1882 36,9229 147,5126 146,6862 (10; 10; 5; 5) (10; 10; 5; 5) 145,8598 145,0333 (10; 10; 5; 5) (10; 10; 5; 5) 144,2069 143,8408 (10; 10; 5; 5) (10; 10; 5; 5) 143,8408 143,1811 (10; 10; 10; 5) (10; 10; 10; 5) 93,2309 93,2310 100 110 (5; 5; 5; 5) (10; 5; 5; 5) 36,7691 36,7691 (10; 5; 5; 5) (10; 5; 5; 5) 36,4322 35,9344 120 124,3546 124,3547 130 (10; 5; 5; 5) (10; 5; 5; 5) 35,4367 35,2200 (10; 5; 5; 5) (10; 5; 5; 5) 35,2199 34,9390 142,7985 142,7985 (10; 10; 10; 5) (10; 10; 10; 5) 142,2669 141,4816 (10; 10; 10; 5) (10; 10; 10; 5) 140,6963 140,3543 (10; 10; 10; 5) (10; 10; 10; 5) 140,3543 145,7674 (10; 10; 5; 5) (10; 10; 5; 5) 131,2792 131,2793 133,5571 133,5572 (10; 5; 5; 5) (10; 5; 5; 5) 34,8753 34,8753 (10; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 140 150 160 170 (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 146,9940 146,9941 (10; 10; 5; 5) (10; 10; 10; 5) 149,0931 149,0931 (10; 10; 10; 5) (10; 10; 10; 5) 149,0931 149,0931 (10; 10; 10; 5) (10; 10; 10; 5) 149,0931 149,0931 (10; 10; 10; 5) (10; 10; 10; 5) 180 190 200 Figure 2.77: Joint marginal price of business U B (5; 5; 5; 5) (5; 5; 5; 5) 34,7619 34,7619 (5; 5; 5; 5) 34,7619 149,0931 149,0931 (10; 10; 10; 5) (10; 10; 10; 5) 149,0931 (10; 10; 10; 5) P UA in GE i t Ba in % EN UB max (P UA ) in GE P max UB (P UA ) in GE i t Be in % EN UB max (P UA ) P UA P UA P UA P max UB (P UA ) P max UB (P UA ) P UA Kapitel 1: Einführung 207 2.4 Selected Problems of Decision Value Determination 207 45520_Matschke_Griffleiste_SL5.indd 207 45520_Matschke_Griffleiste_SL5.indd 207 16.03.2021 16: 22: 00 16.03.2021 16: 22: 00 ject can only acquire business U B at the corresponding marginal price if the option of simultaneously acquiring business U A is waived. To illustrate the joint conflict situation, the previous single figures of the example can be put together graphically in a system. Figure 2.78 depicts five different price categories that are elaborated according to their decision support (M ATSCHKE 1975, p. 343). Each price pair can be allotted to one of these price ranges. In joint conflict situations, it is incorrect to cumulatively determine the decision value for both businesses and then to arbitrarily distribute it to the objects. The calculation of a common (collective, combined, or joint) marginal price for both businesses would not represent the situation adequately. In the example, a common marginal price [P max UB (P UA ), P max UA (P UB )]-coordinate (P UB ; P UA ) 0 20 40 60 80 100 120 140 160 180 200 Decision value for business U B, according to the price for U A and for the price U B in GE 0 20 40 60 80 100 120 140 160 180 200 Decision value for business U A , according to the price for U B and the price for U A in GE (130; 143,1888) P max UA (P UB ) P max UA (P UB ) P max UB (P UA ) P max UB (P UA ) Price range, in which only the acquisition of business U B , but not of business U A is reasonable! Price range, in which only the acquisition of business U A , but not of business U B is reasonable! Price range, in which the simultaneous acquisition of business U A and of business U B is reasonable! Price range, in which neither an acquisition of business U A nor of business U B is reasonable! The line AB represents the price range, in which either the acquisition of business U A or of business U B is reasonable! A B Figure 2.78: Decision values in a joint conflict situation of the acquisition/ acquisition type (P UB ; P UA ) = 208 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 208 45520_Matschke_Griffleiste_SL5.indd 208 16.03.2021 16: 22: 00 16.03.2021 16: 22: 00 Chapter 2 = 273,1888 GE would result. If a price = 130 GE were negotiated for business U B , the decision subject would be wrong with the assumption that for business U A still the price = 273,1888 GE - 130 GE = 143,1888 GE could be paid because = 143,1888 GE is higher than the joint marginal price = 124,7585 GE (see Figure 2.76). The price combination = (130 GE; 143,1888 GE) falls into the price range, in which only an acquisition of business U B is reasonable (see Figure 2.78). Therefore, an acquisition of business U A should not take place at a price = 143,1888 GE because otherwise, the valuation subject would be worse off as compared to only acquiring business U B . In the example of the joint conflict situation of the acquisition/ acquisition type with two valuation objects, it was shown that the marginal price of one valuation object is determined with regard to the negotiated price of the other valuation object. The determination of a common marginal price of all objects and its potentially arbitrary distribution leads to an incorrect representation of the decision situation. If, however, a joint conflict situation presents itself with more than two businesses to be valuated, it is not only the graphical illustration that reaches its limits: For the reduction of complexity, the calculations should therefore be limited to important valuation objects and to suitable, and rather realistic, price ranges (H ERING 2014, p. 171). P max UA+UB P UB P UA = P max UA+UB - P UB P UA P max UA (P UB = 130) (P UB ; P UA ) P UA 2.4 Selected Problems of Decision Value Determination 209 45520_Matschke_Griffleiste_SL5.indd 209 45520_Matschke_Griffleiste_SL5.indd 209 16.03.2021 16: 22: 00 16.03.2021 16: 22: 00 2.5 Selected Control Questions Exercise 1 (15 Points) - Decision Value a) Define the term decision value and name its characteristic features. (5 points) b) Explain the general approach for the determination of the decision value (in the sense of a marginal price) from the perspective of a presumptive buyer. (10 points) Exercise 2 (15 Points) - Matrix of Functional Business Valuation Explain the valuation steps that result in the decision function, according to the matrix of functional business valuation. Exercise 3 (30 Points) - Business Valuation under Uncertainty a) What is a decision field? In what ways is it important to business valuation? (5 points) b) By which valuation-relevant restrictions is the corporate financial sphere of activity characterized? (5 points) c) Describe an open decision field. (5 points) d) Systematize a planning approach under uncertainty and critically evaluate its purpose regarding the decision value determination. (10 points) e) Explain the sensitivity analysis and outline its application according to the determination of the decision value. (5 points) Exercise 4 (40 Points) - State Price Marginal Model A valuation subject owns a small business KU at the valuation date t = 0, which simultaneously represents the decision and also the acquisition date. That business generates a perpetuity of 30 GE from internal financing (IF). At t = 0, there is also an opportunity to make an investment AK. The payment sequence of this investment is (-100 GE, 30 GE, 40 GE, 50 GE, 55 GE), including the initial cost. In addition, at t = 0, the valuation subject has personal equity assets (EM) in the amount of 10 GE. It is assumed that the primary bank of the general manager (valuation subject) offers a bullet loan ED of 50 GE at an annual interest rate of 9 % p. a. for investments with a total term of four periods (years). Further financial funds are available in unlimited amounts at a shortterm interest (borrowing) rate of 11 % p. a. (KA t ). Financial investments (GA t ) can be made at the primary bank in any amount at an interest rate of 5 % p. a. The valuation subject desires a uniform income payment stream from t = 0 to ascertain its existence (income maximization). Since the valuation subject assesses the planning horizon t = T = 4, the last distribution contains not only the regular payment EN but also the present value of a perpetuity at an interest rate of 5 % to guarantee the income EN beyond the planning period. For any time t > 4, the estimated interest rate of i = 5 % p. a. remains constant over time. 210 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 210 45520_Matschke_Griffleiste_SL5.indd 210 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 Chapter 2 For information only: An adequate approach requires that these assumptions are taken into consideration for objects whose payments extend beyond the planning horizon. At t = 0, the valuation subject has to decide whether to acquire another enterprise U or not. For this business, the payment stream (0, 60, 40, 20, 20) was estimated for the planning horizon. Additionally, a perpetuity of 20 GE is expected starting in t = 5. The objective is to find the maximum payable price P max for business U. a) Describe the consideration of payments beyond the planning horizon. Determine the desired temporal structure of the withdrawals: w 0 : w 1 : w 2 : w 3 : w 4 . (5 points) b) Complete Figure 2.79 by using the data given in the exercise. (5 points) c) Formulate the approach for the determination of the base program as well as the approach for the determination of the valuation program and the decision value. (20 points) d) Explain the special features of small and medium-sized enterprises (SME). To what extent does the state marginal price model address these particular features? (10 points) Exercise 5 (45 Points) - Investment-Theoretic Models Discuss the following statement in form of a little essay: “Totally analytical, partially analytical, and heuristic models for decision value determination: Their representation and critical evaluation.” Start with a table of contents. Exercise 6 (10 Points) - Decision Value Determination at a Merger Elucidate the process and the result of the decision value determination in a nondominated, disjoint, one-dimensional conflict situation of the merger type. Exercise 7 (20 Points) - Joint Conflict Situations A decision-maker can sell a business and/ or buy another business in the present conflict situation. Therefore, the valuation subject is in contact with various negotiation partners. The only controversial issiuse in the negotiations is the respective price. Neither the decision-maker nor either opponent can enforce an agreement. t AK ED GA 0 GA 1 GA 2 GA 3 KA 0 KA 1 KA 2 KA 3 EM IF U 01234 Limit Figure 2.79: Example for the state marginal price model 2.5 Selected Control Questions 211 45520_Matschke_Griffleiste_SL5.indd 211 45520_Matschke_Griffleiste_SL5.indd 211 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 a) Describe generally, which type of conflict situation is represented here and which aspects have to be considered during the determination of the decision value of each business. (6 points) b) The decision situation of the current decision-maker is outlined as follows (see Figure 2.80): First, describe the line of of the decision values using the numerical values. Then, indicate which independent variable the decision value depends on. These independent variables have to be added as axes labels in the diagram. (6 points) c) The diagram is divided into four price ranges marked A, B, C, and D. Determine precisely which recommendations for action result when the current negotiation situation regarding the acquisition and sale object is alternatively characterized as a conflict solution by a point in the area A or B or C or D. Note: To arrive at the answer, it is reasonable to describe in which areas an acquisition would be rational and in which areas a sale would be. (8 points) Figure 2.80: Example of joint conflict situations 0 50 100 150 200 250 0 50 100 150 200 250 Decision value of the sale object Decision value of the acquisition object 135 140 A B C D 120 212 2 Decision Function and Decision Value 45520_Matschke_Griffleiste_SL5.indd 212 45520_Matschke_Griffleiste_SL5.indd 212 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 Chapter 3: Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 213 45520_Matschke_Griffleiste_SL5.indd 213 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 Overview The third chapter discusses another main function of functional business valuation: the mediation function (arbitration function). The result of this function is the arbitration value of the firm. It is regarded as a compromise, which is essentially characterized by the feature of rationality of action of the conflicting parties and the feature of partyrelated adequacy. The introductory section of the third chapter (Section 3.1) deals with the basics of the mediation function and the features of the arbitration value. In Section 3.2 shows how arbitration values can be determined in non-dominated conflict situations. After illustrating the steps of arbitration value determination with the aid of the matrix of functional business valuation, valuation methods are represented and interpreted according to arbitration theory. These methods lead to the determination of appropriate conflict resolutions based on the Number of conflict solutions and the number of possible agreements for the conflicting parties. Subsequently, the differences are outlined when the mediation function is used in the context of dominated conflict situations (Section 3.3). The third chapter also ends with control questions for an in-depth study of the presented learning material (Section 3.4). Learning objectives After studying this chapter you should be able to 1. define the term arbitration value, characterize its features, and describe the significance of the decision value for the determination of the arbitration value; 2. explain how the number of potential conflict solutions and the number of possible agreement solutions can be found by the appraiser (mediator or arbitrator), in nondominated conflict solutions and which role dominance considerations can play; 3. list the particularities that can result during the determination of the arbitration values in dominating conflict situations; 4. explain the options to determine the arbitration value from the quantity of (both original and derivate) conflict resolutions; 5. list the combined valuation methods, elaborate their influence on the determination of business values, and discuss how these values can be interpreted, according to arbitration theory; 6. elucidate how arbitration values can be used. Kapitel 1: Einführung 214 214 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 214 45520_Matschke_Griffleiste_SL5.indd 214 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 Chapter 3 3.1 Basics The arbitration value is the result of a business valuation in the context of the mediation function (arbitration function) (M ATSCHKE 1969, 1971, and in particular 1979, K ÖNIG 1977, M OXTER 1983). The value is suggested by an appraiser (mediator, arbitrator), representing an independent third party. The appraiser mediates between the parties with diverging interests regarding the business to facilitate or reach a compromise. The arbitration value can be relevant for the conflict between the parties in various ways in case of non-dominated conflict situations (S IEBEN 1983, p. 541): 1. It could be stated as a recommendation for a conflict resolution. 2. The parties could submit to it if it is contractually agreed upon. 3. It can be the starting point for further negotiations. The arbitration value is therefore defined as an agreement value that is generally proposed by an independent third party. It states the essential conditions for a conflict resolution binding the parties. The arbitration value constitutes a compromise (M ATSCHKE 1969, p. 58) that is reasonable for the participating parties and sufficiently safeguards their interests (M ATSCHKE 1969, p. 57, M ATSCHKE 1984, p. 562). Conflict resolutions are reasonable if they are compatible with rational actions of the participating parties. Hence, the arbitration value must not violate the limits of the concession willingness of the parties (a feature of rationality of action of the conflicting parties) (M ATSCHKE 1979, p. 48). Therefore in case of freedom of choice of the conflicting parties, the precondition is the existence of an agreement or arbitration area in a non-dominated conflict situation. If only the level of the price is relevant for a conflict resolution in a non-dominated conflict situation of the acquisition/ sale type, the arbitration value (AW; German: Arbitriumwert) must lie between the higher maximum affordable price P max from the perspective of the buyer (decision value as marginal price of the buyer) and the lower minimum demandable price P min from the perspective of the seller (decision value as the marginal price of the seller). This is outlined in Figure 3.1. According to this feature, it is possible to limit the area of possible conflict resolutions of the arbitration value. Hence, the decision values of the conflicting parties also play a central role within the mediation function. 3.1 Basics 215 45520_Matschke_Griffleiste_SL5.indd 215 45520_Matschke_Griffleiste_SL5.indd 215 16.03.2021 16: 22: 01 16.03.2021 16: 22: 01 If more conflict resolutions can be deemed reasonable, the independent third party determines the arbitration value with regard to the feature of a party-related adequacy - based on the chosen equity postulate within the arbitration area. Such a conflict resolution should be chosen as the arbitration value corresponds best to the ideas of the conflicting parties with respect to an equitable agreement (M ATSCHKE 1971, p. 519). The appraiser has to refer to a conflict resolution rule that is used as a superior target function, which in turn can be regarded as a fairness postulate (S IEBEN 1983, p. 541). As interim conclusion, it can be stated: The arbitration value - as a result of mediation - is party-dependent in two respects. First, the business value must not violate the decision values of the conflicting parties, according to the mediation function (a feature of rationality of action of the conflicting parties). Second, the arbitration value selected by the appraiser has to fulfill fairness postulates to reflect an appropriate solution for the parties (a feature of party-related adequacy). However, it is not absolutely necessary that an independent appraiser determines the arbitration value. An argumentation value can theoretically represent an arbitration value too if it is, for instance, negotiated as a binding (price) with which the parties agree. If the knowledge and the consideration of individual’s decision value are assumed, a voluntary agreement of the parties on this (argumentation) value only occurs if the conflicting party believes that this value represents the aforementioned features of the arbitration value (H ERING / O LBRICH 2002, p. 153). Under the conditions of such an agreement, the arbitration value ultimately becomes the (market) price. A conflict situation, in which one of the conflicting parties might impose a change of ownership of the business to be valuated against the declared will of the other parties, is defined as a dominated conflict situation (M ATSCHKE 1979, p. 33). While dominance considerations are of minor importance at most during the determination of the decision value of the respective party, they play an important role in the determination of the arbitration value. This is because the conflicting parties have to submit to the arbitration value, for instance, according to statutory authority or contractual agreements. The arbid Zielplaninformationen e s prä sumtiven Verkäufers - E ntsc heidungsfeld informationen des präsumtiven Ve rkä ufers Zielplaninforma tione n des präs umtive n Käufers - E ntsche idun gsfeld informationen de s präs umtiven Pre is in GE E ntsche idung swert des prä sumtiven Ve rkä ufers (Preisu ntergrenz e) Ents cheidungswert des prä s umtiven Käufers (P reisoberg renze) A rbitriumbereich (V erh andlungsspie lrau m) Käufers Figure 3.1: Arbitration area at freedom of choice Information about the decision field of the presumptive seller Information about the target plan of the presumptive buyer Information about the decision field of the presumptive buyer Information about the target plan of the presumptive seller Decision value of the presumptive seller (Lower price limit) Decision value of the presumptive buyer (Upper price limit) Arbitration area (Bargaining or negotiating scope) Price (in monetary units) Kapitel 1: Einführung 216 216 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 216 45520_Matschke_Griffleiste_SL5.indd 216 16.03.2021 16: 22: 02 16.03.2021 16: 22: 02 Chapter 3 tration value is not considered as a recommendation or foundation for further negotiations, as is often the case in the non-dominated conflict situations ans instead serves as a (binding) conflict resolution that is accepted by the parties because the change of ownership is not disposible, at least not for the dominated party. Hence, the starting position for the impartial appraiser is different from the case of a non-dominated conflict situation because now it should be assumed that a change of ownership of the business to be valuated will be executed, or has already been executed, against the will of the other parties. This implies that the independent mediator has to suggest an arbitration value in cases when no agreement between equal parties would have been possible because of a non-existing overall agreement solution. In this context, a notable problem of balancing the various interests occurs for the arbitrator. If the arbitration value is consistent with rational behavior for all parties, which seems adequate because a party could impose a change of ownership against the will of the others, the question to be answered is: Which interests of which conflicting party are or should be preferred in a dominated conflict situation with no universally acceptable arbitration value for all parties? If there is no agreement area in a dominated conflict situation, the literature (M ATSCHKE 1979, p. 49, H AYN 2000, p. 1347, D RUKARCZYK / S CHÜLER 2016, p. 11) has argued for an arbitration value that is equal to the decision value of the dominated party. M EYER (2005, p. 41) states that in doing so the dominated party is not disadvantaged in comparison to the omission of the transaction. Contrarily, the dominating party could waive the transaction if their decision value is violated. Hence, it does not require any special protection. At the determination of the arbitration value, superior objectives might also have to be considered, especially legislation and case law. However, before major aspects of the determination of the arbitration value in dominated conflict situations are discussed (Section 3.3), the following Section 3.2 focuses on the arbitration value determination in non-dominated conflict situations. 3.1 Basics 217 45520_Matschke_Griffleiste_SL5.indd 217 45520_Matschke_Griffleiste_SL5.indd 217 16.03.2021 16: 22: 02 16.03.2021 16: 22: 02 3.2 Value Determination in Non-Dominated Conflict Situations 3.2.1 Further Investigation Steps within the Matrix of Functional Business Valuation 3.2.1.1 Overview If a mediator joins the negotiations on a business valuation in a non-dominated conflict situation as an independent thord party to facilitate an agreement between the conflicting parties, the impartial arbitrator should propose an arbitration value for the business that is a reasonable compromise because doing so protects the interests of the conflicting parties. According to the “matrix of functional business valuation” (cf. Figure 1.1 in Section 1.4.1, p. 47), the three following steps have to be taken so that the arbitration values in the broader sense can be determined: Step 1 (field D of the matrix): Determination of the number of acceptable conflict resolutions from the perspective of each conflicting party and the number of possible agreements (arbitration area) by the appraiser. Step 2 (field E of the matrix): Determination of the arbitration value (valuation in the narrower sense) within the determined area of agreement (arbitration area) by the appraiser as reasonable and adequate conflict resolution. Step 3 (field F of the matrix): Effective usage (application) of the arbitration value by the negotiating parties. 3.2.1.2 The Steps in Detail 3.2.1.2.1 First Step To be reasonable, the arbitration value must not violate the respective concession limits of the conflicting parties (a feature of rationality of action). In case of a non-dominated conflict situation, the appraiser has to determine the arbitration area, including the arbitration value. The determination of the set of reasonable conflict resolutions and the set of possible agreements (M ATSCHKE 1979, p. 49) by the appraiser is represented in step 1 and field D of the matrix of functional business valuation. The set of reasonable conflict resolutions from the perspective of the conflicting parties depends on their decision values. The respective decision values will usually not be disclosed to the arbitrator. Therefore, it is up to the appraiser to make reasonable and comprehensible assumptions about the decision values of the conflicting parties. Again, as outlined in the fields A and B of the matrix of functional business valuation, this step involves the same procedures as already represented in the second chapter. However, here the mediator has to take the perspective of each conflicting party. The respective assumptions about their target systems and decision fields as well as the relevant subjective future performances Kapitel 1: Einführung 218 218 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 218 45520_Matschke_Griffleiste_SL5.indd 218 16.03.2021 16: 22: 03 16.03.2021 16: 22: 03 Chapter 3 of the valuation object are transformed into the presumed decision values of the parties involved with the aid of investment-theoretic valuation methods. These decision values are defined from the perspective of the mediator. It is needless to say that the notions of the arbitrator concerning the decision values can differ from the actual (expected) decision values of the parties. However, in further theoretical considerations it is abstracted from this information problem since it is not universally solvable. In the next step, the reasonable conflict solutions of the conflicting parties have to be derived. Finally, the arbitration area is represented by the possible agreement set E as the intersection of the sets S z , which contain the acceptable conflict resolutions from the perspective of each conflicting party: E : = S z1 ∩ S z2 ∩ …∩ S zq . If the agreement set is not an empty set (that is, E ≠ ∅), it is possible for the appraiser to determine an arbitration value that is characterized by the feature of rationality of action. Due to the validity of this feature, the independent mediator should pay close attention to dominance considerations. In other words, when proposing a (multi-dimensional) arbitration value there should be no dominating alternative that would provide a party with a higher utility value. As a result, the arbitrator has to verify the conflict resolutions of the determined agreement set, according to dominant relations. By performing a dominance test of potential agreement resolutions, their amount might also be considerably reduced. The considerations about the dominance of agreement solutions are examined in the following abstract example (cf. Figure 3.2), in which a mediation between two parties (party 1 and party 2) takes place (M ATSCHKE 1979, p. 9): In Figure 3.2 presents the set of all possible conflict resolutions S and the agreement area E . The agreement area E has been divided into four subsets: Field 1: in the set of all indifferent conflict resolutions of both parties 1 and 2, Field 2: in the set of all indifferent conflict resolutions of party 2 and of all preferred conflict resolutions from the perspective of party 1, Figure 3.2: Set of all conflict resolutions S and the agreement area E E S 1 1 2 3 4 3.2 Value Determination in Non-Dominated Conflict Situations 219 45520_Matschke_Griffleiste_SL5.indd 219 45520_Matschke_Griffleiste_SL5.indd 219 16.03.2021 16: 22: 03 16.03.2021 16: 22: 03 Field 3: in the set of all indifferent conflict resolutions of party 1 and of all preferred conflict resolutions from the perspective of party 2, and Field 4: in the set of all preferred conflict resolutions of both parties 1 and 2. The conflict resolutions of field 1 and 3 represent a part of the decision value of party 1, whereas the conflict resolutions of field 1 and 2 constitute a part of the decision value of party 2. Even by this rough classification of preferences, namely into indifferent and preferred conflict resolutions, the possible solutions for a mediation proposal can be considerably reduced with respect to dominance considerations. For instance, conflict resolutions in field 1 are dominated by the conflict solutions in the fields 2, 3, and 4. While in a conflict resolution in field 1 both parties only achieve the same success as without the change of ownership, party 1 can improve in a conflict resolution in field 2, without any deterioration of party 2 regarding the conflict resolutions in field 1. The same holds for the comparison of the conflict resolutions in field 1 with those in field 3 and 4. Further segregation based on the provided information is not possible. It cannot be stated that the conflict resolutions in field 4 dominate those in field 2 or 3. For such an assessment, it is necessary to specify the expectations for the utility value of the parties, according to their preferred conflict resolutions. This is showcased in Figure 3.3. In Figure 3.3, it is assumed that the set of preferred conflict resolutions of each party is divided only into two subsets with different utility values. Considering the denotation of each field, the first digit (second digit) refers to the achieved utility value of party 1 (party 2), where 1 indicates an indifferent conflict resolution and 2 or 3 indicate a preferred conflict resolution. Conflict resolutions with a utility value at level 3 are preferred to those at level 2. The following relations of dominance are presented in Figure 3.4: Figure 3.3: Dominance of agreements Kapitel 1: Einführung 220 220 3 Mediation Function and Arbitration Value 11 11 31 22 23 21 12 13 12 E S 45520_Matschke_Griffleiste_SL5.indd 220 45520_Matschke_Griffleiste_SL5.indd 220 16.03.2021 16: 22: 04 16.03.2021 16: 22: 04 Chapter 3 The conflict resolutions in field 11 are directly dominated by the conflict resolutions in the fields 21 and 12. However, conflict resolutions in field 12 are also dominated by conflict resolutions in field 13 and 22. Additionally, conflict resolutions in field 21 are dominated by conflict resolutions in the field 22 and 31. Conflict resolutions in field 13 are dominated by those in field 23. This means that ultimately, all other fields dominate field 11. In the example, only two non-dominated conflict resolutions fields remain, field 31 and field 23. In field 31, party 1 would achieve the preference level 3, and party 2 would only achieve the indifference level 1. In the conflict resolutions in field 23, both would improve compared to the situation of indifference. In comparison to field 31, however, party 1 would only achieve the lower level of preference 2 in field 23, whereas party 2 could improve from indifference level 1 to the level of preference 3. Whether a conflict resolution in field 31 or in field 23 should be suggested can only be conclusively decided by consulting additional criteria. In this case, mere considerations about the dominance of conflict resolutions can no longer be used. The advantage of the application of the dominance criterion is because all Paretoinefficient solutions are eliminated and consequently the amount of possible conflict resolutions for the arbitration value is reduced. The final choice then only refers to Pareto-optimal conflict resolutions, in which the conflict is most clearly manifested. In the example, the transition from a conflict resolution in field 31 to a conflict resolution in field 23 illustrates the fact that a possible improvement of the position of party 2 from level 1 to the level 3 is accompanied by a corresponding deterioration of the position of party 1 from level 3 to level 2. The application of dominance considerations leads to a division of the agreement set E into two subsets: one efficient, non-dominated conflict resolutions and in the complementary subset of inefficient, dominated conflict resolutions so that the agreement set E can also be defined as the union of these subsets: A conflict resolution of the agreement set E dominates another conflict resolution if the utility value of a party j at this considered conflict resolution is higher than the utility value of this party j at another conflict resolution . For j = i as well as for all other conflicting parties j ≠ i with it is assumed that the utility values of the other parties j at the conflict resoluti- Conflict resolutions in field are dominated by conflict resolutions in field 11 12 21 and 12 13 and 22 13 21 22 Figure 3.4: Relations of dominance 23 22 and 31 23 ˆ E E ˆ E ⊂ E and E ⊂ E and E = ˆ E ∪ E . (s 1 , … , s n ) (s 1 ' , … , s n ' ), N j (b jopt (s 1 , …, s n )) (s 1 , … , s n ) N j (b jopt (s 1 ' , …, s n ' )) (s 1 ' , … , s n ' ) j ∈ {1, … , i, … , m} N j (b jopt (s 1 , …, s n )) 3.2 Value Determination in Non-Dominated Conflict Situations 221 45520_Matschke_Griffleiste_SL5.indd 221 45520_Matschke_Griffleiste_SL5.indd 221 16.03.2021 16: 22: 05 16.03.2021 16: 22: 05 on are not smaller than the utility values at the conflict resolution . The definition of dominance reads as follows: if does exist with > and > for with j ≠ i as well as and The set of the non-dominated or efficient conflict resolutions involves all conflict resolutions which are not dominated by any conflict resolution of the agreement area E : = and there is no with 3.2.1.2.2 Second Step According to the feature of party-related adequacy, the appraiser should determine the arbitration value based on a selected fairness postulate within the agreement area already defined in step 1 (step 2 as well as field E of the matrix) (M ATSCHKE 1979, p. 92). After that, the arbitration value should demonstrate a conflict resolution that best corresponds to the considerations of the negotiation partners, according to an equitable agreement. If the difference between the upper price limit (decision value of the buyer) and the lowest price limit (decision value of the seller) is regarded as a generally distributable advantage, the appraiser faces the task of an adequate distribution. For instance, this may be performed by observing the rule of absolutely equal division (also known as uniform distribution) or the rule of relatively equal division. These are easily applicable and comprehensible rules of what is kwown as just division. The rule of absolutely equal division of the distributable advantage A = P max - P min is a rule, which ensures that the arbitration value (AW; German: Arbitriumwert) fulfills the feature of party-related adequacy. The result of this rule reads as follows: A K : A V = 1 : 1 or A K = A V = 0,5 · A: AW = P max - 0,5 · A = P min + 0,5 · A AW = P max - 0,5 · (P max - P min ) = P min + 0,5 · (P max - P min ) AW = 0,5 · (P max + P min ). Therfore, the suggested arbitration value lies exactly in the middle between the (lower) decision value of the seller P min and the (higher) decision value of the buyer P max . If the appraiser considers the feature of party-related adequacy and assumes that the advantage of the parties is not absolutely but is instead relatively equal regarding their respective decision value (the rule of relatively equal division) (M ATSCHKE 1971, p. 519), the results reads: A K : A V = P max : P min or A K / P max = A V / P min : AW = P max - A · P max / (P max + P min ) = P min + A · P min / (P max + P min ). (s 1 , …, s n ) N j (b jopt (s 1 ' , …, s n ' )) (s 1 ' , …, s n ' ) (s 1 , …, s n ) ≻ (s 1 ' , …, s n ' ) i ∈ {1, ..., m} N i (b iopt (s 1 , …, s n )) N i (b iopt (s 1 ' , …, s n ' )) N j (b jopt (s 1 , …, s n )) N j (b jopt (s 1 ' , …, s n ' )) j ∈ {1, ..., m} (s 1 , …, s n ) ∈ E (s 1 ' , …, s n ' ) ∈ E. (s 1 ' , … , s n ' ), ˆ E {(s 1 , … , s n ) | (s 1 , … , s n ) ∈ E (s 1 ' , …, s n ' ) ∈ E (s 1 ' , …, s n ' ) ≻ (s 1 , …, s n )}. Kapitel 1: Einführung 222 222 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 222 45520_Matschke_Griffleiste_SL5.indd 222 16.03.2021 16: 22: 06 16.03.2021 16: 22: 06 Chapter 3 In this case, the proposed arbitration value no longer lies (exactly) in the middle of the two decision values, but is closer to the value of the seller so that the absolute amount of the buyer exceeds that amount of the seller. Both division rules are illustrated in an example (cf. Figure 3.5). The following sections clarify that the present mediative business valuation can also resort to so-called traditional combination methods (combined valuation methods) (M ATSCHKE 1998, p. 282) under certain conditions. Then, the business value (UW; German: Unternehmenswert) in the sense of an arbitration value (AW) is determined as the combination of the income value (EW; German: Ertragswert) and the net asset value (SW; German: Substanzwert). This can be illustrated by J ACOB ’ S normal form (J ACOB 1960), where the method-specific factor a represents the corresponding weighting: UW = SW + a · (EW - SW) = a · EW + (1 - a) · SW. If the term (1 - a) is again replaced by the factor b, it finally results: UW = a · EW + b · SW. If the traditional combination methods are interpreted in light of arbitration theory, the factors a and b express the method-specific desired distribution of the total advantage A = P max - P min in favor of the seller (factor a) and in favor of the buyer (factor b). However, this is only true if the (alleged) decision value of the presumptive buyer equals EW and that of the presumptive seller equals SW. In other words, P max = EW and P min = SW which, in turn, leads to UW = AW. It is also possible that other business values that have not been determined utilizing investment-theoretic substantiated valuation methods are within the arbitration area suggested by the mediator. If this is the case, they also comply with the feature of rationali- Figure 3.5: Explanation of the rule of absolutely equal division and the rule of relatively equal division by means of a numerical example It is: P max = 500 and P min = 300, so A = P max − P min = 200. Rule of absolutely equal division: AW = P max − 0,5 ⋅ A = P min + 0,5 ⋅ A = 0,5 ⋅ (P max + P min ) AW = 400 and A K = A V = 100. Rule of relatively equal division: AW = P max − A ⋅ P max P max + P min = P min + A ⋅ P min P max + P min AW = 500 − 200 ⋅ 500 800 = 300 + 200 ⋅ 300 800 AW = 500 − 125 = 300 + 75 = 375 and A K P max = A V P min = 125 500 = 75 300 = 1 4 . 3.2 Value Determination in Non-Dominated Conflict Situations 223 45520_Matschke_Griffleiste_SL5.indd 223 45520_Matschke_Griffleiste_SL5.indd 223 16.03.2021 16: 22: 07 16.03.2021 16: 22: 07 ty of action of the conflicting parties and the feature of party-related adequacy so that the conflicting parties might agree to such a method. Therefore, it is conceivable that even those methods might be deployed for the determination of the arbitration value. 3.2.1.2.3 Third Step The usage of the arbitration value is depending on the reason why the parties invoke the help of a neutral mediator (step 3 and field I of the matrix). The negotiating parties might (S IEBEN 1983, p. 541, K USSMAUL 1996, p. 266): 1. give it the status of a recommendation, for instance also in the sense of a foundation for a judgment of a court, 2. submit to the arbitration value due to prior arrangements (e.g., if the value reached during an arbitration (proceeding) between two parties resembles a final verdict), or 3. use it as a starting point for further negotiations. Kapitel 1: Einführung 224 224 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 224 45520_Matschke_Griffleiste_SL5.indd 224 16.03.2021 16: 22: 07 16.03.2021 16: 22: 07 Chapter 3 3.2.2 Selected Valuation Methods 3.2.2.1 Preliminary Remarks In this section, various methods for the determination of the arbitration value will be discussed. These approaches have in part been directly formulated for a mediative business valuation and party without such a reference. As already examined in Section 3.2.1.2.2, these are the so-called traditional combined valuation methods (combination methods). If the business value is determined as a combination of income value (sometimes referred to as earnings value) and net asset value, the result is a so-called combination value. If profits (gains) of the business are based on standardized assumptions for a likewise standardized valuation subject (generally the owner in the narrower sense), the term (objectified) income value is primarily used. The net asset value, which is also used in connection with the combination methods, is a specification of the net asset value regarding the reconstruction (old value of a net part reconstruction). The term old value of a net part reconstruction can be specified as follows: The term net relates to the fact that the type of financing compared to the gross value is also taken to consideration. The term part illustrates that unrecognizable assets are not considered. Finally, it is an old value because both the age structure and present conditions of the corporate substance are also taken into account. The term goodwill (firm value, transaction value), which occurs in the following combination methods, is used in different contexts in business administration. Subsequently, it should be used in the sense of the traditional combination methods. Here, the difference between the income value and the net asset value (or alternatively the desired business value) is defined as original goodwill. The (original) goodwill from the income value EW and the net asset value SW or from the income value EW and the searched business value UW is determined as follows by the indirect method: • according to the majority opinion and • according to the minority opinion . Besides this indirect method for determining the original goodwill as a residual, there is also a direct method, where the goodwill corresponds to the multiple of the residual income (economic value added, market value added, and other residual income methods). The residual income - here specified as a goodwill rent GR - is equal to the difference between the estimated future profit E and the interest rates on the net asset value SW (or the searched business value UW respectively). It follows • according to the majority opinion and • according to the minority opinion . The original goodwill is regarded as volatile and also depends on the competitive situation (competition risk). Therefore, E UGEN S CHMALENBACH (1937, p. 32) has already divided it into GW orig GW orig = EW − SW GW orig = EW − UW i * GR = E - i * · SW GR = E - i * · UW 3.2 Value Determination in Non-Dominated Conflict Situations 225 45520_Matschke_Griffleiste_SL5.indd 225 45520_Matschke_Griffleiste_SL5.indd 225 16.03.2021 16: 22: 07 16.03.2021 16: 22: 07 1. a part, which corresponds to the value of the unrecognized intangible assets (goodwill 1, value of internal and external organization) in the determination of the net asset value (as a part-reconstruction-value) and 2. a part that includes a return on the capital in the business to be valuated higher than the interest or discount rate (goodwill 2, excess return). After the definition of these terms, the conflict situation at hand is described. Moreover, the conditions are quoted that are necessary for the application of the traditional combined valuation methods. Finally, several of these methods are represented (Section 3.2.2.2) and interpreted with the aid of J ACOB ’s normal form (Section 3.2.2.3). The basis of further considerations is a non-dominated, disjoint, one-dimensional conflict situation of the acquisition/ sale type, in which the conflicting parties cannot achieve the desired change of ownership of the business to be valuated on their own, that is, against the will of the other party. The conflict situations only comprise two parties. Figure 3.6 shows the corresponding conflict cube. It is assumed that in case of an agreement on a certain price P, the buyer can continuously increase their expected utility at a falling price in this conflict situation, that is, decreases strictly monotonically. Moreover, it is assumed that it can also be applied vice versa for the seller. Hence, their utility increases at a rising price P. All prices between the lowest price limit of the seller P min and the upper price limit of the buyer P max constitute the agreement set and are thus deemed efficient conflict resolutions. In such conflict situations regarding a certain pri- Figure 3.6: Conflict cube for a non-dominated, disjoint, one-dimensional conflict situation Disjoint Joint One-dimensional Multi-dimensional Degree of dominance Non-dominated Dominated Degree of connectedness Degree of complexity N K (b opt K (P)) N K (b opt K (P)) N V (b opt V (P)) Kapitel 1: Einführung 226 226 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 226 45520_Matschke_Griffleiste_SL5.indd 226 16.03.2021 16: 22: 08 16.03.2021 16: 22: 08 Chapter 3 ce, there is no partial identity of interests and consequently no dominated or inefficient conflict resolutions. However, the agreement set E and the set (prices according to the feature of exclusive consideration of efficient conflict resolutions) are only equal if: Consequently, the set of inefficient conflict resolutions is empty. Hence, each price P, for which is valid, might serve as the arbitration value AW of the business in an one-dimensional conflict situation of the acquisition/ sale type. Indeed, a characteristic feature of the traditional combination methods is the lack of an examination of the price limits. Instead, the arbitration value (price offer) is a function of the income value EW as a performance measure and the net asset value SW as a benchmark for the corporate substance: EW and SW are treated as parameters without reference to a particular buyer or seller (V IEL / B REDT / R ENARD 1975, p. 23). Therefore, there is no necessary connection between the lower price limit of the seller and the net asset value as fictitious equity capital (including hidden/ secret reserves) on the one hand and the upper price limit of the buyer and the income value on the other hand. The net asset value and the lower price limit of the seller might coincide, but they also might differ substantially. The same holds tue for the income value and the upper price limit of the buyer. Hence, the usability of traditional valuation methods is limited to cases, in which the results of these methods lead to an arbitration value that is concurrently a part of the agreement set of the conflicting parties. Therefore, the impartial arbitrator has to check whether the following equation holds true: As previously mentioned, the function based on J ACOB ’s normal form (see p. 223), can be specified as the weighted arithmetic mean of the income value and the net asset value: Again, the difference (EW - SW) represents the (original) goodwill. The usability of the combination methods for the determination of the arbitration value is no longer met if the net asset value and the income value are both higher than the upper price limit or lower than the lower price limit respectively, because in this case the business value falls outside the acceptable price range . However, the latter constellation might also occur if only the income value EW or the net asset value SW lie outside the area of agreement. To sum up, the following constellations are listed below where the application of combination methods is possible, even if partially restricted: ˆ E E = ˆ E = P P min ≤ P ≤ P max { } . E P min ≤ P ≤ P max AW = P = f (EW, SW). AW = P = f (EW, SW) ∈ E = P P min ≤ P ≤ P max { } . AW = P = f (EW, SW), AW = UW = SW + a ⋅ (EW − SW) = a ⋅ EW + (1 − a ) ⋅ SW. UW = AW = SW + a ⋅ (EW − SW) P min ≤ P = AW ≤ P max 1. Case : SW < P min and EW > P max if 0 < a < 1, 2. Case : P min ≤ SW ≤ P max and EW > P max if 0 ≤ a < 1, 3. Case : SW < P min and P min ≤ EW ≤ P max if 0 < a ≤ 1, 4. Case : SW > P min and EW < P max if 0 ≤ a ≤ 1. 3.2 Value Determination in Non-Dominated Conflict Situations 227 45520_Matschke_Griffleiste_SL5.indd 227 45520_Matschke_Griffleiste_SL5.indd 227 16.03.2021 16: 22: 09 16.03.2021 16: 22: 09 Further application restrictions result from the size relations between the parameters EW and SW, from the respective price limits of the parties, and also from methodspecific particularities. Figure 3.7 illustrates the four cases graphically. The agreement area E includes the defined arbitration value. Hence, rational behavior of the conflicting parties is presumed. In case 1, the determination of the arbitration value based on the traditional combination methods is only possible if the resulting business value UW = f(SW, EW) is also lying in the interval [P min , P max ] so that neither the net asset value method nor the income value method come into question for the determination of the arbitration value. To comply the condition in case 2, factor a has to decrease depending on how much the income value exceeds the upper price limit of the buyer. As rationale for the price offer, the mediator would have to resort to the net asset value. It is clear that the income value method cannot be deployed. In case 3, factor a has to be increased depending on how much the net asset value exceeds the lower price limit of the seller. Hence, the independent arbitrator would have to justify the price offer primarily with the help of the income value, thus, the net asset value method would be inappropriate. Additionally, cases 1, 2 and 3 reveal difficulties in the application of the methods because the argumentation patterns related to those methods do not lead to any acceptable price offer for both parties in the agreement set E . Only in case 4, if neither the net asset value nor the income value exceed the price limits of the parties, are there no restrictions of application in using traditional combina- Fall 1 SW EW Pmin Pmax Fall 2 SW EW Pmin Pmax Fall 3 SW EW Pmin Pmax Fall 4 SW EW Pmin Pmax E E E E Figure 3.7: Constellations for the application of the traditional combination methods to determine the arbitration value Case 1 Case 2 Case 3 Case 4 P min ≤ P = AW ≤ P max Kapitel 1: Einführung 228 228 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 228 45520_Matschke_Griffleiste_SL5.indd 228 16.03.2021 16: 22: 10 16.03.2021 16: 22: 10 Chapter 3 tion methods to calculate the arbitration value. Moreover, solely in case 4 is it guaranteed for all values of the factor a between 0 ≤ a ≤ 1 that the arbitration value enables a price offer that is part of the agreement set: Consequently, the impartial mediator has the option to choose between all traditional methods to decide the most suitable one and to specify the feature of party-related adequacy and to propose the parties an acceptable and appropriate conflict resolution. Only in case 4, are all traditional methods alternative ways to find a resolution for the conflict between buyer and seller. Since the different methods will be described, analyzed, and compared in the following section, it is stringently required to assume the conditions of case 4. 3.2.2.2 Combined Valuation Methods 3.2.2.2.1 Mean Value Method The mean value method (M ÜNSTERMANN 1966, p. 113) can be classified as a method of business valuation, deriving the business value UW from a combination of the income value EW and the net asset value SW. According to the mean value method, the simple arithmetic mean is established from the (lower) net asset value and the (higher) income value as the business value. Its formal simplicity has resulted in the mean value method being popular in practice, although its popularity is decling. Formally, the equation reads: The above equation is also known as J ACOB ’ S normal form (J ACOB 1960), where the business value is calculated as the sum of the net asset value and the (original) goodwill: Factor a has a method-specific definition; in case of the mean value method, it is true that a = 0,5. For the exemplary determination of the business value with the mean value method, the following numeric values are assumed (cf. Figure 3.8): E = P P min ≤ P = AW = SW + a ⋅ (EW − SW) ≤ P max { } . UW = EW + SW 2 = E i * + SW 2 UW = SW + 1 2 ⋅ (EW − SW). UW = SW + GW = SW + a ⋅ (EW − SW). 3.2 Value Determination in Non-Dominated Conflict Situations 229 45520_Matschke_Griffleiste_SL5.indd 229 45520_Matschke_Griffleiste_SL5.indd 229 16.03.2021 16: 22: 10 16.03.2021 16: 22: 10 The first description of business valuation as an instrument of mediation between buyer and seller is attributed to F ELIX M ORAL . M ORAL considered that a conflict of interest that has to be balanced is motivated by different estimations of the general business risk by the parties. Therefore, predominant argumentation patterns are usually deployed. M ORAL regarded the mean value method as a method of bridging a conflict of interest between buyer and seller (M ORAL 1920, p. 131); and that conflict resulted from an underestimation of the risk of the seller and an overestimation of the risk of the buyer. Hence, th method can be seen as an substantiation of the feature of party-related adequacy. According to M ORAL , the consideration of justified claims is ensured if “the seller receives a capital gain and the buyer a corresponding income gain”. To achieve these objectives and to make the “business a desirable object of purchase for its new buyer as well as a satisfactory sale transaction for the seller, the exchange value of the business [...] lies [...] midway between the value of all assets and their capitalized income.” Accordingly, the objective of his proposal was to apply the mean value method, where the price offer is exactly half of the net asset value and the income value. The application of the mean value method is nothing more than a norm often considered just and reasonable to divide in half an achievable advantage. In other words, M ORAL promoted the application of the already discussed rule of absolute equal division, of which the overall amount is regarded by M ORAL as the difference between the net asset value and the income value. M ORAL points out arguments for the balance of interests between buyer and seller that are quite similar to the widespread perception of an adequate solution. The equal distribution of the advantage realized by buyer and seller seems adequate especially when considering that in a non-dominated conflict situation they could refuse and waive the advantage. This distribution can be convincingly explained by factoring in intangible assets like organizations and market relations not yet being compensated for. The reconciliation of these adequacy perceptions is conducted under the conditions set by M O- RAL .However, it must be emphasized that generally there is no mandatory connection between the lower price limit of the seller and the net asset value on the one hand as well as the upper price limit of the buyer and the income value on the other hand. The net asset value and the lower price limit of the seller can coincide, but they also can di- Future profit E 300.000 EUR Interest rate i * 6,0% Income value EW Net asset value SW 5.000.000 EUR 2.000.000 EUR Method-specific factor a Goodwill GW = EW - SW 0,5 3.000.000 EUR Business Value UW Figure 3.8: Example for the mean value method 3.500.000 EUR Kapitel 1: Einführung 230 230 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 230 45520_Matschke_Griffleiste_SL5.indd 230 16.03.2021 16: 22: 10 16.03.2021 16: 22: 10 Chapter 3 verge widely. This is equally assumed for the income value and the upper price limit of the buyer. If P min ≤ SW < EW ≤ P max holds, the mean value method can generally be defined as the method for determining the arbitration value based on an equal (uniform) distribution amounting to (EW - SW) ≤ (P max - P min ). However, if P min < SW and EW < P max , a division in half of the currently existing advantage (P max - P min ) is not performed, but only the smaller separated advantage (EW - SW) is divided. Additionally, the advance distribution is no longer method-specific in the sense of the differences (P max - P) for the buyer and (P - P min ) for the seller. Hence, the desired and the actual distribution of the advantages do not coincide (M ATSCHKE 1979, p. 142): The first term represents the method-based (“desired”) distribution of advantages; the second term defines the not-method-specific advance distribution of advantages. The actual (real) advantages A K and A V only coincide absolutely if the not-method-based advance distributions are incidentally equal. Only under this assumption is the mean value method based on the rule of absolute equal division of the total advantage (P max - P min ). 3.2.2.2.2 Goodwill Rent Methods According to the goodwill rent methods (M ATSCHKE 1975, p. 149), the business value is equal to the net asset value plus the so-called goodwill rents (residual income, excess profits, or excess returns). The goodwill rent is defined as the difference between the estimated future profit E and the so-called normal profit NG (German: Normalgewinn). It is an annuity. The variants discussed differ regarding • the way of calculating the goodwill rents GR as difference from the future profit E and the normal profit NG, • the assumptions about the amount and the temporal structure, and • the consideration of the goodwill rents at the value determination. Ultimately, these methods are based on the idea that for reasons of fairness, a higher future profit E (including the goodwill rent) cannot be permanently established. Conversely, the expected future profit E will suddenly or gradually decrease over time and converge to a normal profit NG due to competition risk. The gradually decreasing (linear or progressive) goodwill rents are discussed in context with a special variant. However, in the following, it is abstracted from this fact. In other words, constant goodwill rents are assumed for a certain time. For the calculation of the normal profit NG and the goodwill rent GR two approaches are discussed that differ in Terms of content: 1. The majority opinion assumes that the normal profit NG corresponds to the customary interest (normal rate of return) of the net asset value . The calculation of the normal profit based on the net asset value focuses on the fictitious use of equity capital in the amount of SW by the seller. A K = EW − SW 2 + (P max − P) and A V = EW − SW 2 + (P − P min ). i * NG = i * · SW 3.2 Value Determination in Non-Dominated Conflict Situations 231 45520_Matschke_Griffleiste_SL5.indd 231 45520_Matschke_Griffleiste_SL5.indd 231 16.03.2021 16: 22: 11 16.03.2021 16: 22: 11 2. According to the opinion held by the minority, the normal profit NG corresponds to the customary interest (normal rate of return) of the searched business value Using this method, a higher normal profit and thus a lower goodwill rent result. For instance, the Stuttgart method that will be discussed below is based on this approach. The calculation based on the searched business value focuses on the possible investment of the buyer at the acquisition of the business. In other words, UW = P (U NION E UROPÉENNE DES E XPERTS C OMPTABLES , E CONOMIQUES ET F INANCIERS 1961, p. 25). These different approaches to the concept of the normal profit can be applied in both following goodwill rent methods (goodwill rent method I and goodwill rent method II) as well as their variants. According to the goodwill rent method I (method of excess profit compensation, method of the shortened duration of goodwill rents, and excess profit purchase method), the net asset value is increased by the undiscounted goodwill rents GR for a given number of years (M ÜNSTERMANN 1966, p. 125). The time factor T is determined depending on the volatility of the excess profits (residual income). Taking the majority opinion as the basis for the calculation of the normal profit, the following formula results for the business value: Returning to the example in Section 3.2.2.2.1 (cf. Figure 3.8), Figure 3.9 outlines the resulting business value based on goodwill rent method I: i * NG = i * · UW. UW = SW + T ⋅ GR UW = SW + T ⋅ (E − NG ) UW = SW + T ⋅ (E − i * ⋅ SW) or due to EW = E i * UW = SW + T ⋅ (i * ⋅ EW − i * ⋅ SW) UW = SW + T ⋅ i * ⋅ (EW − SW). Kapitel 1: Einführung 232 232 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 232 45520_Matschke_Griffleiste_SL5.indd 232 16.03.2021 16: 22: 12 16.03.2021 16: 22: 12 Chapter 3 However, if the minority opinion is considered regarding the calculation of goodwill rents with the help of method I, the following equation comprising the searched business value on both sides of the initial equation results: If this is applied to the numerical example, the business value amounts to 2.692.308 EUR (cf. Figure 3.10). Due to the lower goodwill rent or the corresponding higher normal profit, the business value is lower than that calculated after the majority opinion in the goodwill rent method I. Future profit E 300.000 EUR Normal profit interest rate i * Number of years T 6,0%5 Income value EW Net asset value SW 5.000.000 EUR 2.000.000 EUR Normal profit NG = i * · SW Goodwill rent GR = E - i * · SW 120.000 EUR 180.000 EUR Method-specific factor a Goodwill GW = UW - SW 0,3 900.000 EUR Business value UW = SW + T · GR Figure 3.9: Example for goodwill rent method I (majority opinion) 2.900.000 EUR UW = SW + T ⋅ GR UW = SW + T ⋅ (E − NG ) UW = SW + T ⋅ (i * ⋅ EW − i * ⋅ UW) UW ⋅ (1 + T ⋅ i * ) = SW + i * ⋅ T ⋅ EW UW = 1 1 + T ⋅ i * ⋅ SW + i * ⋅ T 1 + T ⋅ i * ⋅ EW UW = 1 1 + T ⋅ i * ⋅ SW + i * ⋅ T 1 + T ⋅ i * ⋅ SW − i * ⋅ T 1 + T ⋅ i * ⋅ SW + i * ⋅ T 1 + T ⋅ i * ⋅ EW UW = SW ⋅ 1 1 + T ⋅ i * + i * ⋅ T 1 + T ⋅ i * ⎛ ⎝⎜ ⎞ ⎠⎟ + i * ⋅ T 1 + T ⋅ i * ⋅ (EW − SW) UW = SW + i * ⋅ T 1 + T ⋅ i * ⋅ (EW − SW). 3.2 Value Determination in Non-Dominated Conflict Situations 233 45520_Matschke_Griffleiste_SL5.indd 233 45520_Matschke_Griffleiste_SL5.indd 233 16.03.2021 16: 22: 12 16.03.2021 16: 22: 12 According to the goodwill rent method II (method of excess profit annuities, method of shortened duration of goodwill rents, profit excess purchase method, or U.E.C. method), the goodwill rents are discounted so that only the present value is added to the business value (M ÜNSTERMANN 1966, p. 126). The following formula results if the normal profit is calculated based on the net asset value (majority opinion): With reference to the numerical example the values are calculated as follows: 2.758.225 EUR (cf. Figure 3.11). Note that RBF (German: Rentenbarwertfaktor) denotes the annuity (present value) factor: Future profit E 300.000 EUR Normal profit interest rate i * Number of years T 6,0%5 Income value EW Net asset value SW 5.000.000 EUR 2.000.000 EUR Normal profit NG = i * · UW Goodwill rent GR = E - i * · UW 161.538 EUR 138.462 EUR Method-specific factor a Goodwill GW = UW - SW 0,230769 692.308 EUR Business value UW = SW + T · GR Figure 3.10: Example for goodwill annuity method I (minority opinion) 2.692.308 EUR UW = SW + (1 + i * ) T − 1 i * ⋅ (1 + i * ) T ⋅ GR UW = SW + (1 + i * ) T − 1 i * ⋅ (1 + i * ) T ⋅ (i * ⋅ EW − i * ⋅ SW) UW = SW + (1 + i * ) T − 1 i * ⋅ (1 + i * ) T ⋅ i * ⋅ (EW − SW) UW = SW + (1 + i * ) T − 1 (1 + i * ) T ⋅ (EW − SW) UW = SW + (1 − 1 (1 + i * ) T ) ⋅ (EW − SW) or due to v T = 1 (1 + i * ) T UW = SW + (1 − v T ) ⋅ (EW − SW). (1 + i * ) T − 1 i * ⋅ (1 + i * ) T . Kapitel 1: Einführung 234 234 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 234 45520_Matschke_Griffleiste_SL5.indd 234 16.03.2021 16: 22: 13 16.03.2021 16: 22: 13 Chapter 3 The method of profit layering (the F RITZ method) is known as a variant of the goodwill rent method II. In that method, the goodwill rent is considered indefinite. In other words, we arrive at a perpetuity. Due to the risks pertaining to a perpetuity, it is discounted at a higher interest rate i ** than the normal rate of return i * (F RITZ 1912/ 13): If this method is applied to the example and the interest rate of the goodwill rent is i ** = 0,15, the results in Figure 3.12 read as follows: Future profit E 300.000 EUR Normal profit interest rate i * Number of years T 6,0%5 Discount factor v T Income value EW 0,747258 5.000.000 EUR Net asset value SW Normal profit NG = i * · SW 2.000.000 EUR 120.000 EUR Goodwill rent GR = E - i * · SW Method-specific factor a 180.000 EUR 0,252742 Goodwill GW = UW - SW Business value UW = SW + RBF · GR 758.225 EUR 2.758.225 EUR Figure 3.11: Example for goodwill rent method II (majority opinion) UW = NG i * + E − NG i ** with i * < i ** or due to NG = i * ⋅ SW and E − NG = GR UW = SW + GR i ** UW = SW + E − i * ⋅ SW i ** UW = SW + 1 i ** ⋅ (E − i * ⋅ SW) or due to EW = E i * UW = SW + 1 i ** ⋅ (i * ⋅ EW − i * ⋅ SW) UW = SW + i * i ** ⋅ (EW − SW). 3.2 Value Determination in Non-Dominated Conflict Situations 235 45520_Matschke_Griffleiste_SL5.indd 235 45520_Matschke_Griffleiste_SL5.indd 235 16.03.2021 16: 22: 14 16.03.2021 16: 22: 14 A threeto ten-year period are often proposed as the period for a time-limited consideration of the goodwill rents (M ATSCHKE 1975, p. 152). Within this period, it is assumed that the goodwill rents are either constant (majority opinion), or are decreasing linearly or progressively (minority opinion). Decreasing goodwill rents are only discussed in context of the goodwill rent method II. For instance, the L EAKE method is characterized as a goodwill rent method II with linearly decreasing goodwill rents (L EAKE 1947, p. 38). The formulas and numerical examples presented below apply the majority opinion, that is, a constant goodwill rent used. A characteristic of the goodwill rent methods is the assumption of an excess profit (goodwill rent) that is considered for a limited time at the determination of the business value. This limited duration of the goodwill rents is justified by two separate arguments (M ATSCHKE 1979, p. 174): 1. The first argument states that there are goodwill-generating factors which imply the existence of goodwill rents. However, it is also conceded that those factors are rather volatile and might quickly start to wane. This might be expedited due to the relevant competion. Therefore, the goodwill rents will not be attainable perpetually. 2. The second argument also acknowledges the existence of goodwill-generating factors but does not rule out perpetual goodwill rents. However, the limited duration of the goodwill rent is supported because the seller can only require goodwill rents that can be attributed to thar seller and their previous activities whose effects diminish as time passes. If a goodwill rent still exists in the future, it is not primarily due to the seller, but due to the goodwill-generating and goodwill-preserving measures of the buyer. To summarize, this second argument expresses a fairness concept. Future profit E Normal profit interest rate i * Goodwill rent interest rate i** 300.000 EUR 6,0% 15,0% Normal profit NG Excess profit, goodwill rent GR Income value of the normal profit NG / i * 120.000 EUR 180.000 EUR 2.000.000 EUR + Income value of the excess profit GR / i ** = Business value UW Income value EW 1.200.000 EUR 3.200.000 EUR 5.000.000 EUR Net asset value SW Method-specific factor a 2.000.000 EUR 0,4 Goodwill GW = UW - SW Business value UW = SW + GW 1.200.000 EUR 3.200.000 EUR Figure 3.12: Example for the method of profit layering (F RITZ method) Kapitel 1: Einführung 236 236 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 236 45520_Matschke_Griffleiste_SL5.indd 236 16.03.2021 16: 22: 15 16.03.2021 16: 22: 15 Chapter 3 The market value added (MVA) method is a special variant of the method of excess profit rents. The excess profit (economic value added) is expected perpetually. The economic value added (EVA) method and the MVA method are capital market-theoretic models. The trademark-protected consulting tool was devised by the consulting company S TERN , S TEWART & C O . in New York (S TEWART 1991, S TERN / S HIELY / R OSS 2001). However, the EVA method is not an original method for business valuation. Rather, the EVA method is a concept which - interpreted as a residual income - is designed to deliver a key parameter in the sense of an excess profit for a single reporting period, and thus to make an allegedly meaningful statement regarding the increase of the business value (shareholder value). This approach is closely related to the concept of value-oriented management. The MVA as an added business value of all operating activities is the present value of the annually expected economic value added EVA t that are discounted at the discount rate k: If a constant EVA is assumed (i.e., EVA represents a perpetuity) the MVA results as follows: To derive the discount rate k, which should equal the required rate of return of both debt and equity investors, reference is made to capital market-theoretic approaches like the WACC approach discussed in chapter 4. There, the cost of capital is defined as a weighted average of the (required rate of) return on equity and debt in a capital market equilibrium. The EVA method is an operative element of the shareholder-value-oriented controlling that serves as a tool for the valuation, management, and control of businesses as well as for decentralized business units/ divisions. With the aid of the EVA method, it is attempted to assess the “value contribution” of entrepreneurial business decisions. The methodical procedure of the EVA method is sufficiently known from the operating income statement. The input data are externally-oriented, that is, the approach is open to external analysis. Therefore, the method does not provide any new economic insights. The EVA as excess profit is defined as the difference between the adjusted (periodical) operating income/ profit before interest and after taxes (NOPaT t = net operating profit after taxes) as well as the adjusted cost of capital of this period. The last cost is the product of the (necessary operating) invested capital (NOA t = net operating assets) and the discount rate k as weighted average cost of capital (WACC). Hence, the excess profit of every period t results as follows: EVA t = NOPaT t - NOA t · k. Regarding the MVA, the formula then reads: MVA= EVA t 1 + k ( ) t t = 1 ∞ ∑ . MVA= EVA k . MVA= NOPaT t 1 + k ( ) t − NOA t t = 1 ∞ ∑ 3.2 Value Determination in Non-Dominated Conflict Situations 237 45520_Matschke_Griffleiste_SL5.indd 237 45520_Matschke_Griffleiste_SL5.indd 237 16.03.2021 16: 22: 15 16.03.2021 16: 22: 15 or with EVA as a perpetuity: The MVA represents a measure of the value that is generated by a business beyond invested capital (M ANDL / R ABEL 1997, p. 380). The adjusted periodical operating profit before interest and after taxes (NOPaT t ) included in the calculation, is a results of the difference of financial income and expenses, from which only depreciations is subtracted as the only non-cash item. The reason stated is that depreciation is used for the replacement of worn assets, especially property, plant, and equipment, and hence represents true economic expenses. I addition, These depreciations have to be considered in order to easy comparability with leased assets. The first argument is only applicable if growth or at least the maintenance of assets is assumed. An argument against the second statement is that depreciation represent periodical expenses and that the structure of the payment streams from the acquisition will be different to those from leasing or rent (G ÜNTHER 1997, p. 234). The determination of the adjusted operating profit before interest and after taxes also known as operative cash flow before interest and after taxes is influenced by Anglo-Saxon accounting standards and represented in Figure 3.13 (S TEWART 1991, p. 112, G ÜNTHER 1997, p. 234). Although some adjustments to it have been made, the EVA method is learly still prone to distortions (biases) which are primarily due to accrual accounting and accounting standards in particular (G ÜNTHER 1997, p. 237). Thi situation provides several opportunities for manipulation because a positive EVA could be generated, for instance, by the extension of the useful life with correspondingly lower annual depreciation (G ÜNTHER 1997, p. 238). The (necessary operating) invested capital (NOA t = net operating assets) is calculated with the initial inventory or upon changes of more than +/ -20 % with the mean/ average value (G ÜNTHER 1997, p. 234). The determination of the invested capital, which again is based on financial values, is outlined in Figure 3.14 (G ÜNTHER 1997, p. 235). MVA= NOPaT k − NOA. (1) Operating profit before taxes and before income (“Net Operating Profit”) (2) (3) + + Increase in value adjustments on receivables Increase in the difference between the recognition of inventories with the LIFO method compared to the FIFO method a) (4) (5) (6) (7) + + Amortization of derivative goodwill Increase in the present value of research and development expenses + + Other operating income Increase in other provisions (8) (9) (10) a) In the USA, it is a disclosure requirement if a business uses the LIFO method. + - Initial cost to increase market value (e.g., exploration or market development costs) Non-tax-deductible taxes = Adjusted operating profit after taxes and before income (NOPaT) Figure 3.13: Determination of the adjusted operating profit with the EVA method Kapitel 1: Einführung 238 238 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 238 45520_Matschke_Griffleiste_SL5.indd 238 16.03.2021 16: 22: 16 16.03.2021 16: 22: 16 Chapter 3 If the following equation is used for the calculation of the interest yield on the invested capital in the sense of a return on investment or returns on invested capital, respectively (this relation is also referred to as S TEWART ’s R) (G ÜNTHER 1997, p. 234): then a positive EVA t is a manifestation of a return r NOA t on the invested capital NOA t in this period t that exceeds the discount rate k: To substantiate business decisions based on the EVA method, the parameters on the right side of the equation are categorized into three decision-making levels strategy, financing, and investment (G ÜNTHER 1997, p. 234). According to S TEWART , the following alleged supposed options of an increase in (shareholder) value can be identified (S TE- WART 1991, p. 137): 1. The business value increases with an increasing return on investment r NOA t . 2. The business value increases with a reduction of the (average) discount rate k. 3. An increase of the invested capital NOA t is only value generating if the return on investment r NOA t exceeds the (average) discount rate k. In other words, an excess return is achieved in the respective period. 4. Accordingly, a reduction of the invested capital NOA t is value generating if the return on investment r NOA t is lower than the (average) discount rate k. The EVA method is related to the common saying that value-enhancing investments are only given if the return on investment r NOA t is larger than the discount rate k. However, this is not entirely true: Even inexperienced investors will learn that it always depends on the marginal and not on the average observation! The issues with this concept are both the integration of rather unrealistic capital market-theoretic approaches and the recourse to a parameter similar to the operating result of cost accounting. This parameter is essentially derived from the data of financial accounting and then adjusted (1) Book value of the (necessary operating) fixed assets (2) (3) + - Book value of the (necessary operating) current assets Non-interest-bearing, short-term liabilities (4) (5) (6) (7) - - Marketable securities Assets under construction a) + + Value adjustments on receivables Difference between the recognition of inventories with the LIFO method compared to the FIFO method (8) (9) (10) (11) + + Cumulative amortizations of derivative goodwill Capitalized rental and leasing expenses + + Capitalized research and developement expenses Capitalized preliminary costs to increase market value (12) (13) a) They remain unconsidered since they do not serve the operative business yet. Figure 3.14: Determination of the invested capital with the EVA method + = Cumulative extraordinary losses after taxes Net Operating Assets (NOA) r NOA t = adjusted periodical operating profit invested periodical capital = NOPaT t NOA t , EVA t = (r NOA t - k) ⋅ NOA t . 3.2 Value Determination in Non-Dominated Conflict Situations 239 45520_Matschke_Griffleiste_SL5.indd 239 45520_Matschke_Griffleiste_SL5.indd 239 16.03.2021 16: 22: 17 16.03.2021 16: 22: 17 subsequently. Hence, it is neither cash-flow-oriented nor does it consider marginal interest rates. However, both observations are a precondition for a decision-oriented valuation (H ERING / V INCENTI 2004, p. 351). Other approaches focusing on the excess profit or residual income of a business are the cash flow return on investment (CFROI) approach and the corresponding concept of cash value added (CVA) approach. The single periodical CFROI and CVA approaches were developed by the B OSTON C ONSULTING G ROUP (BCG) and do not represent original business valuation methods (L EWIS / L EHMANN 1992, G ÜNTHER 1997, p. 213). These are instead methods to understand the increase of a business’s value (the so-called shareholder value) in the context of a value-oriented (business) management. Furthermore, they partly rely on capital market theory. The criticism directed toward these models is the same as that of the EVA method because strictly speaking, they lack a decision-oriented theoretical foundation. The CFROI is defined as a gross cash flow a business attains within a year relative to its invested capital. According to G ÜNTHER (1997, p. 213), the CFROI appears as the internal rate of return of the cash flow profile of a business or decentralized business unit. Although the CFROI concept, as indicated by its name, should explicitly consider cash flow based data, financially-oriented balance sheet components are often predominant in this method. The CFROI results from the gross cash flow CF br , the so-called gross investment base BIB, and the so-called economic depreciation ÖA (German: Ökonomische Abschreibung): The gross cash flow CF br (German: Brutto-Cash-flow) represents the simplified cash flow before interest, scheduled depreciation, rental and leasing expenses, but after taxes (CF br = result after taxes +/ - revisions of extraordinary items and their tax effects + scheduled depreciation + interest expense + rental and leasing expenses + adjustments of FIFO and LIFO according to inventory valuation +/ - inflation gains/ losses on net liquidity). It has been adjusted for extraordinary components and phantom profits (G ÜNTHER 1997, p. 214). The so-called gross investment base BIB (German: Bruttoinvestitionsbasis) is derived from the net value of the non-depreciable assets (monetary current assets including inventories + prepaid expenses and deferred charges + financial assets - non-interest-bearing (short-term) liabilities + properties) and the inflation-adjusted gross value of fixed assets (book value of depreciable fixed assets + accumulated depreciation + inflation adjustment) plus further corrections for an improved operative comparison (capitalized rental expenses + possibly self-created or acquired intangible assets). The economic depreciation ÖA characterize the amount at interest at the average discount rate k. This amount has to be retained each period to allow for replacement investments at the end of the useful life of the fixed assets. For simplicity, the ÖA is defined as the quotient of the BIB and the REF (German: Rentenendwertfaktor). The latter represents the annuity factor for a future value using an annuity immediate over n periods (M ATSCHKE 1993b, p. 190), where n denotes the average useful life and the discount rate k indicates the weighted average cost of capital: CFROI = CF br − ÖA ( ) BIB . Kapitel 1: Einführung 240 240 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 240 45520_Matschke_Griffleiste_SL5.indd 240 16.03.2021 16: 22: 17 16.03.2021 16: 22: 17 Chapter 3 The period-specific increase in value is ultimately represented by the CVA: CVA = (CFROI - k) · BIB = CF br - (ÖA + k · BIB). According to H ERING / V INCENTI (2004, p. 354), depreciation and interest rates constitute the lower limit of the gross cash flow that has to be earned in one period to provide a positive contribution (CVA) to the value development of the business. 3.2.2.3 Overview of typical method-specific characteristics and their arbitration-theoretic interpretation According to J ACOB ’ S (1960) proposal, the traditional combination methods can be transformed into a so-called normal form, as already shown before. The formal difference between the methods is expressed by the respective factor a. J ACOB ’ S normal form regards the business value UW as the result of the net asset value SW and the goodwill GW. The latter is the product of the method-specific factor a and the difference between the income value EW and the net asset value SW: Hence, the business value reads as follows: If J ACOB ’ S normal form is rearranged as follows the business value UW results as the weighted average from the income value EW and the net asset value SW, where the income value is multiplied with factor a and the net asset value with factor b = (1 - a). The method-specific factors a and b are summarized in Figure 3.15. T represents the duration of the goodwill rents, i* the discount rate and v t the corresponding discount factor for the period t = T, and i** the goodwill rent interest rate . ÖA = BIB REF with REF = 1+k ( ) n − 1 k . GW = a · (EW - SW). UW = SW + a ⋅ (EW − SW). UW = a ⋅ EW + (1 − a ) ⋅ SW UW = a ⋅ EW + b ⋅ SW with a + b = 1, 1 (1 + i * ) t 3.2 Value Determination in Non-Dominated Conflict Situations 241 45520_Matschke_Griffleiste_SL5.indd 241 45520_Matschke_Griffleiste_SL5.indd 241 16.03.2021 16: 22: 18 16.03.2021 16: 22: 18 The arbitration-theoretic interpretation of the traditional methods focuses on the factors a and b that show the intentional distribution of the benefit A = P max - P min in favor of the seller V (German: Verkäufer) (factor a) and in favor of the buyer K (German: Käufer) (factor b) as long as (coincidentally) P max = EW and P min = SW (M ATSCHKE 1979, p. 240). The intentional distribution of the benefit always differs from the effective distribution of the benefit if the net asset value exceeds the lower price limit (SW > P min ) and/ or the upper price limit exceeds the income value (P max > EW), respectively. In such cases, the seller only receives the amount X = SW - P min in advance and the buyer is awarded the amount Y = P max - EW. In this case, the intentional distribution norm a : b between the seller and the buyer is only applied to a part of the entire benefit: Under consideration of such an advance distribution, the benefit A V distributed to the seller V reads: The full benefit (P max - P min ) minus the advance distributions (X + Y) is placed in square brackets (i.e., residual benefit). This residual benefit has to be distributed with Method-specific factors Method Seller ’ s interest a Buyer’s interest b = (1 - a) Income Value Method Net Asset Value Method 1 0 0 1 Mean Value Method Goodwill Rent Method I (Method of excess profit compensation) (majority opinion: NG = i * · SW) 0,5 T · i * Goodwill Rent Method I (Method of excess profit compensation) (minority opinion: NG = i * · UW) Goodwill Rent Method II (Method of excess profit rents) (1 - v T ) 0,5 (1 - T · i * ) v T Profit Layering Method (F RITZ method) Figure 3.15: Factors of the combined valuation methods T · i * (1 + T · i * ) 1 (1 + T · i * ) i * i ** 1 − i * i ** EW − SW = P max − P min ( ) − X + Y ( ) . A V = X + a ⋅ P max − P min ( ) − X + Y ( ) ⎡⎣ ⎤⎦ with X = SW − P min and Y = P max − EW A V = SW − P min ( ) + a ⋅ EW − SW ( ) . Kapitel 1: Einführung 242 242 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 242 45520_Matschke_Griffleiste_SL5.indd 242 16.03.2021 16: 22: 19 16.03.2021 16: 22: 19 Chapter 3 the aid of a combined method and corresponds to the difference between the income value EW and the net asset value SW. Accordingly, the benefit in favor of the buyer A K is calculated as follows: Considering the feature of rationality of action of the conflicting parties, the arbitration value AW must lie between the price limits and can thus be expressed as the lower price limit plus the benefit of the seller V and as the upper price limit minus the benefit of the buyer K The application of the combined methods for the determination of the arbitration value AW is represented in the following situation (cf. Figure 3.16): Using a discount rate i * = 6 % p. a., the goodwill rent interest rate i ** = 15 % p. a., and the goodwill rent period T = 5 years, the following arbitration values AW, the desired benefit distribution norm a : b, and also the respective effective benefits A K and A V result (cf. Figure 3.17): A K = Y + b ⋅ P max − P min ( ) − X + Y ( ) ⎡⎣ ⎤⎦ A K = P max − EW ( ) + b ⋅ EW − SW ( ) . AW = P min + A V AW = P max − A K . Upper price limit P max of the buyer 5.200.000 EUR Lower price limit P min of the seller 1.900.000 EUR Entire benefit A = P max - P min Income value EW Net asset value SW 3.300.000 EUR 5.000.000 EUR 2.000.000 EUR Distribution in advance X in favor of the seller Distribution in advance Y in favor of the buyer 100.000 EUR 200.000 EUR Residual benefit A - (X + Y) Figure 3.16: Initial situation of the determination of the arbitration value 3.000.000 EUR 3.2 Value Determination in Non-Dominated Conflict Situations 243 45520_Matschke_Griffleiste_SL5.indd 243 45520_Matschke_Griffleiste_SL5.indd 243 16.03.2021 16: 22: 19 16.03.2021 16: 22: 19 Finally, Figure 3.18 showcases which values would result ceteris paribus if the income value corresponded to the decision value of the buyer, that is, P max = EW = 5.200.000 EUR and the decision value of the seller equaled the net asset value, that is, P min = SW = 1.900.000 EUR. Thus, there is no advance distribution. Mean Value Method Goodwill Rent Method I (Method of excess profit compensation) (majority opinion: NG = i * · SW) Goodwill Rent Method I (Method of excess profit compensation) (minority opinion: NG = i * · UW) Goodwill Rent Method II (Method of excess profit rents) FRITZ Method (Profit Layering Method) AW a 3.500.000 EUR 0,500000 2.900.000 EUR 0,300000 2.692.308 EUR 0,230769 2.758.225 EUR 0,252742 3.200.000 EUR 0,400000 b A V A K 0,500000 1.600.000 EUR 0,700000 1.000.000 EUR 1.700.000 EUR Income Value Method 2.300.000 EUR Net Asset Value Method 0,769231 792.308 EUR 0,747258 858.225 EUR 2.507.692 EUR 2.441.775 EUR 0,600000 1.300.000 EUR 2.000.000 EUR AW ab A V 5.000.000 EUR 1,000000 2.000.000 EUR 0,000000 0,000000 3.100.000 EUR 1,000000 100.000 EUR A K Figure 3.17: Exemplary arbitration values and advance benefit distributions 200.000 EUR 3.200.000 EUR Mean Value Method Goodwill Rent Method I (Method of excess profit compensation) (majority opinion: NG = i * · SW) Goodwill Rent Method I (Method of excess profit compensation) (minority opinion: NG = i * · UW) Goodwill Rent Method II (Method of excess profit rents) FRITZ Method (Profit Layering Method) AW a 3.550.000 EUR 0,500000 2.890.000 EUR 0,300000 2.661.538 EUR 0,230769 2.734.048 EUR 0,252742 3.220.000 EUR 0,400000 b A V A K 0,500000 1.650.000 EUR 0,700000 990.000 EUR 1.650.000 EUR Income Value Method 2.310.000 EUR Net Asset Value Method 0,769231 761.538 EUR 0,747258 834.048 EUR 2.538.462 EUR 2.465.952 EUR 0,600000 1.320.000 EUR 1.980.000 EUR AW ab A V 5.200.000 EUR 1,000000 1.900.000 EUR 0,000000 0,000000 3.300.000 EUR 1,000000 0 EUR A K Figure 3.18: Exemplary arbitration values and no advance benefit distributions 0 EUR 3.300.000 EUR Kapitel 1: Einführung 244 244 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 244 45520_Matschke_Griffleiste_SL5.indd 244 16.03.2021 16: 22: 20 16.03.2021 16: 22: 20 Chapter 3 3.2.3 Selected Problems at the Determination of the Arbitration Value 3.2.3.1 Determination of the Arbitration Value at an IPO The following statements are primarily based on an article by H ERING / O LBRICH (2002). There the authors show how main functions of the functional business valuation theory are connected in a conflict situation using the example of an initial public offering (IPO) involving a young firm going public via a stock market launch. Young enterprises in capital-intensive sectors, such as telecommunications, e-commerce, and biotechnology serve as a starting point. During their set-up or expansion phase, young businesses often rely on external investors, primarily due to the limited financial resources of the founders and to the influence of risk-averse creditors. Those external investors might include affluent private individuals and venture capital companies that are often referred to as angel investors, seed investors, or business angels. The involvement of such firms is usually short term. A venture capital firm sells its block of shares - generally with the aid of an issuing bank (underwriter) or a banking consortium - to institutional and private investors to divest and realize its stocks. For the founder, the IPO is an advantage because being listed on the capital market enables the firm to raise additional equity relatively easy through a capital increase (second equity offering) to finance the future growth of the business (H ERING / O LBRICH 2002, p. 148). The question concerning the value of the business or of the business shares is posed repeatedly during the IPO and within different functions (H ERING / O LBRICH 2002): 1. The presumptive seller (e.g., the venture capital company) and the presumptive buyer (e.g., institutional and/ or private investors) have to determine their marginal price (decision function). 2. The presumptive seller, the bank(s), and - as far as a corresponding influence is given by stakeholders who are organized in associations - the presumptive buyers introduce values in the process that aim to achieve a most favorable determination of the arbitration value in the sense of an issuing price or issue price respectively (argumentation function). 3. Finally, an arbitration value is assessed in cooperation with the issuing bank (underwriter). This issuing price is determined taking into account the (alleged) marginal prices of the conflicting parties, the conflict situation, the negotiation process, and the selected price discovery mechanism (mediation function). In the following paragraphs discus ithe latter function in detail and assume an agreement interval exists. In other words, the decision values of the presumptive buyers are higher than those of the presumptive seller. Now, the issue price must be determined as the arbitration value at which the venture capital firm will sell its stocks to interested private and/ or institutional investors. The question of who will determine the arbitration value and then how follows the issuing process. During an own issue event, at which the company itself or the venture capital firm strive to place the shares, the arbitration value results from direct negotiations between the venture capital company and the addressed investors. This own issue 3.2 Value Determination in Non-Dominated Conflict Situations 245 45520_Matschke_Griffleiste_SL5.indd 245 45520_Matschke_Griffleiste_SL5.indd 245 16.03.2021 16: 22: 20 16.03.2021 16: 22: 20 might not only be very bureaucratic and time-consuming, but might also involve certain risks such as an insufficient placement of stocks on the market and the liquidity risk. However, in practice the third-party issue is that predominantly used, where the shares will be placed on the stock market with the aid of one or several banks (e.g., a banking consortium or underwriting group). Consequently, the respective banks play an important role at the determination of the arbitration value. This role is dependent on the form of the cooperation agreed by the consortium, which can be distinguished between the issuance (placement) and the acquisition. Possible issuing processes are summarized in Figure 3.19. If the shares are taken over through proprietary trading, there is a takeover of the sales risk by the banking consortium. Hence, the issuing consortium or the issuing bank (underwriter) guarantees the placing of the shares to the seller. If the bank does not manage to place the stocks entirely, it has to add the remainder to its own asset base. Due to the placement risk, the mediation role of the bank has features of an intermediary that causes and promotes the active involvement in the determination of the arbitration value (H ERING / O LBRICH 2002, p. 153). In addition, there is the possibility of an issuance (placement/ placing) by an issuing consortium/ group, where the issuing risk remains with the seller. However, the firm can use the infrastructure of the banking consortium. In this case, the issuing consortium offers the stocks to potentially interested investors and does not intend to add them to their asset base; thus, the bank assumes a (rather passive) role of a broker (mediator) at the issuing price determination between seller and investors. The risk of an insufficient share placement remains with the investor and is not transferred to the bank (H ERING / O LBRICH 2002, p. 153). Furthermore, the participation of the bank is a result of the selected method of the issuing price determination because the Bank or banking consortium must carra out due diligence, regardless of the respective methods. The methods of the issuing price determination can be the fixed-price procedure, the bookbuilding (process) (price range procedure), and the auction process (cf. Figure 3.20). Emissionsverfahren Selbstemission Fremdemission Begebung Übernahme Figure 3.19: Issuing processes Issuing process Own issue Issuance Third-party issue Acquisition Kapitel 1: Einführung 246 246 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 246 45520_Matschke_Griffleiste_SL5.indd 246 16.03.2021 16: 22: 21 16.03.2021 16: 22: 21 Chapter 3 In the fixed-price procedure, which was common up to the mid-1990s, the seller agreed upon a fixed-price for the shares with the issuing bank. However, it must be stated that both the bank and the venture capital firm only have little knowledge about the actual demand for shares of the investors so that estimates are required. It can be assumed that the bank is interested in an issuing price that factors in a discount, which in turn seems adequate from their perspective. Potential reasons for this so-called underpricing, which the bank could possibly strive for, might be: 1. The bank - especially in case of a stipulated takeover of the shares - wants to minimize its placement risk and consequently wants to ascertain the sale of all shares by the discount. 2. There might be the potential for price advances (prices moving up) after the issuance because a stagnation or, even worse, a price drop (declining stocks) could lead to a reputational damage of the bank. A price increase potential should also be in the interest of the business in question, since stagnant or decreasing prices might also damage the image of the business on the capital market. For instance, the chances for future capital increases could be reduced. However, the interest of the venture capital company could be different - in particular at an agreed takeover - they might suffer a loss of sales by the underpricing. On the contrary, the book-building form presents an opportunity to reduce the uncertainty over demand for shares. After the conducted due diligence, the bank contacts potential institutional investors to inform them about the block of shares that is up for sale. By doing so, it will likely obtain an indication of which prices those investors expect. Based on this information, the bank and the seller ultimately determine a range for the arbitration value. While thar range must not violate the decision value of the seller as the lower price limit, it should also inspire speculative hope among the first (and following) buyers. Consequently, it should have a sufficient (price) gap to the decision values of the well-funded investors since they are (most likely) responsible for the success of the placement. The bank will try to set the complete price range below the estimated decision values of the investors as the upper price limits to reduce the placement risk and to virtually assure occurring price increases after the issuance (H ERING / O LBRICH 2002, p. 154). Since there are several (institutional and private) investors, this necessa- Methoden der Emissionspreisfindung Festpreisverfahren Preisspannenverfahren („Bookbuilding") Auktionsverfahren Figure 3.20: Methods of issuing price determination Methods of issuing price determination Fixed-price procedure Bookbuilding Auction process 3.2 Value Determination in Non-Dominated Conflict Situations 247 45520_Matschke_Griffleiste_SL5.indd 247 45520_Matschke_Griffleiste_SL5.indd 247 16.03.2021 16: 22: 21 16.03.2021 16: 22: 21 rily implies a range of different decision values. There are certainly interested investors whose decision values will be lower than the decision value of the seller so that they will be irrelevant for the placement. Conversely, there will likely be those whose decision value is above that of the seller so that they could become presumptive buyers of the shares and are consequently the addressees of the book-building process. After announcing the price range on the capital market, the bank and the seller introduce the business in question intensively and in detail to institutional investors at information events such as roadshows. In contrast, the interest of private investors is usually aroused by advertisements in the media or by customer pitches in the banks. The incoming subscriptions of the shares are collected and analyzed in an order book regarding the type of demand, the demanded quantities, and the expected prices. The subsequent determination of the prices by the bank does not have to be constrained to the highest possible price, but the aim could be also an appropriate combination of investors, according to the possibly existing objectives of the bank(s) and issuers. Such a mixture of investors might be sought concerning the national and international management or the probable holding period 1. within the group of the institutional investors and/ or 2. with respect to the relationship of the private and the institutional investors, where the allotment can be determined by a random selection (lottery). In comparison to the fixed-price variant, the bookbuilding process usually allows for an issuing price that is mostly in line with market requirements because its final determination is carried out only after including the subscription requests. M Pmin Pmax,oben Pmax,unten x 0 A B = Mindest-Emissionserlös aus Verkäufersicht x3 x1 Emissionspreis Emissionsvolumen C D E F G H J x2 K P L x4 Figure 3.21: Formal definition of the bookbuilding process Minimum issue proceeds from the seller’s perspective Issuing price Issue volume P max, above P max, below P P min 0 x 1 x x 4 x 2 x 3 Kapitel 1: Einführung 248 248 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 248 45520_Matschke_Griffleiste_SL5.indd 248 16.03.2021 16: 22: 23 16.03.2021 16: 22: 23 Chapter 3 The basic considerations of the book-building process are illustrated in the following formal example (cf. Figure 3.21). The seller intends to issue shares in the amount of x. The decision value per share is P min = 0A. The complete minimum issue proceeds from the seller’s perspective is represented graphically as the area 0ABx. The limits (interpreted as decision values of potential buyers) are estimated by the seller and the banking consortium to lie between P max,above and P max,below . The desired issue volume is decreased in the amount of (x - x 1 ) at the upper limit, whereas it is exceeded by (x 3 - x) at the lower limit. In both cases, the realizable complete issue proceeds would be lower than the required minimum issue proceeds of the seller: At P max,above it is represented by the area 0CDx 1 and at P max,below by the area 0EKx as long as the seller leaves the issue volume at an amount of x, as is subsequently assumed. The eligible price range in the example lies between the prices 0G and 0A, whereby the latter corresponds to the decision value of the seller. The price 0G would not inspire speculative hopes, whereas the price 0A (generally: below 0G) such a hope would exist due to a required rationing of the excess demand. If the price P = 0L per share is determined as arbitration value, there is a larger demand, namely x 4 that cannot be satisfied with the issue volume x. Thus, the excess demand (x 4 - x) has to be reduced by an allocation method, for instance in such a way that each investor willing to pay the price P = OL per share receives only one fraction x/ x 4 of the number of shares he wants. The realizable issue proceeds exceed the minimum issue proceeds from the perspective of the seller by the amount of ALMB. The auction method is based on the auction of the shares to be sold and is seldom used in Germany. In connection with the processing of main refinancing operations as liquidity-providing transactions of the national central banks in the Euro area, auction processes are used as tender processes. The Euro system can choose between two different approaches. The European Central Bank either specifies an interest rate and the volume is the object of the auction of the participants (quantity tender) or the desired amounts are obtained with notification of the minimum interest rate (interest rate tender, variable rate tender). The quantity tender corresponds to the already discussed fixed-price procedure. The interest rate tender can be subdivided into the Dutch process on the one hand and the American process on the other. The American process is characterized by an allocation at main refinancing rates of the participating banks until the total amount is achieved. Alternatively, the Dutch process focuses on the allocation at a market-clearing uniform rate. Figure 3.22 provides an overview of different auction processes. 3.2 Value Determination in Non-Dominated Conflict Situations 249 45520_Matschke_Griffleiste_SL5.indd 249 45520_Matschke_Griffleiste_SL5.indd 249 16.03.2021 16: 22: 23 16.03.2021 16: 22: 23 By analogy to the variable rate tender, consortium banks and sellers would determine a minimum price per share as price tender at an auction. Interested institutional and private investors could quote the desired number of shares and the associated maximum price per share in the subscription period. The allocation of the shares to the demanders takes place subsequently to the subscription period. The concrete arrangement of this allocation can be based on the Dutch or the American process. At the Dutch approach would involve determining a market-clearing uniform prise as the arbitration value. The feature of the American process is full price differentiation with regard to the demanders because the allotment follows their bid (offer). Therefore, this process would always start with the demander with the highest bid. This will continue with the following lower bids until the issue volume has been fully allocated. The variants of the price tender processes are represented in Figures 3.23 and 3.24, with a modified structure of Figure 3.21. The line DF in Figure 3.23 - as in Figure 3.21 - again represents the offers of the demanders and x denotes the issue volume of the seller. The market-clearing uniform price at the Dutch process results from the lowest offer that is still considered during the allotment of the issue volume. Under the conditions of this example, the result is P = 0G. The complete issue proceeds amount to 0GHx and exceed the minimum issue proceeds of the seller by the amount of AGHB. For the demanders, it remains a consumer surplus that amounts to GCDH. Using the American process, the consumer surplus should be skimmed off completely. The allocation is effected to the respective offers of the demanders until the projected issue volume x is achieved. Thus, the arbitration values differ individually and lie in the range GC or - on the demand curve - in the range DH, according to the Figure 3.24. The full realizable issue proceeds of the seller for the issue volume x amounts to 0CDHx. A consumer surplus does not remain for the demanders. To enforce the American approach, a very strong market position is required of the seller that will only exist in exceptional circumstances concerning the issuance of shares. Auktionsverfahren i. w. S. Mengentender Festpreisverfahren Preistender (Auktionsverfahren i. e. S.) Holländisches Verfahren Amerikanisches Verfahren Figure 3.22: Variants of auction processes Auction processes in the broader sense Quantity tender Price tender (auction processes in the narrower sense) American process Dutch process Fixed-price procedure Kapitel 1: Einführung 250 250 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 250 45520_Matschke_Griffleiste_SL5.indd 250 16.03.2021 16: 22: 24 16.03.2021 16: 22: 24 Chapter 3 Pmin Pmax,oben Pmax,unten P x 0 A B x3 x1 Emissionspreis Emissionsvolumen C D E F G H K = Mindest-Emissionserlös aus Verkäufersicht Minimum issue proceeds from the seller’s perspective Issuing price Issue volume P max, above P max, below P P min 0 x 1 x x 3 Figure 3.23: Formal definition of the Dutch price tender process Pmin Pmax,oben Pmax,unten x 0 A B x3 x1 Emissionspreis Emissionsvolumen C D E F G H K = Mindest-Emissionserlös aus Verkäufersicht Minimum issue proceeds from the seller’s perspective Issuing price Issue volume P max, above P max,below P min 0 x 1 x x 3 Figure 3.24: Formal definition of the American price tender process 3.2 Value Determination in Non-Dominated Conflict Situations 251 45520_Matschke_Griffleiste_SL5.indd 251 45520_Matschke_Griffleiste_SL5.indd 251 16.03.2021 16: 22: 26 16.03.2021 16: 22: 26 It could be problematic for both the bank and the company going public that the allocation by the auction principle does not take account of the distribution of the shares to different investors or investor groups or the presumed holding period. The allocation is instead only motivated by the price. Furthermore, there is a risk that the auction negatively affects the potential of an increase in prices. However, in comparison to the fixedprice procedure and the bookbuilding process, the venture capital firm might probably realize higher issue proceeds in many cases utilizing an auction method (Hering/ Olbrich 2002, p. 155). Within the process of the IPO (going public), the mediation function aims to fix of the issuing price in the sense of an arbitration value. Its amount results from the marginal prices of the parties, the collaboration and cooperation of the issuing bank during the price discovery mechanisms (e.g., fixed-price procedure, book-building, or auction processes) and the negotiation process. The latter is particularly influenced by the ability to participate in the negotiation and provide arguments in the price discussion. To achieve the most favorable determination of the arbitration value from their individual perspective, the conflicting parties purposefully introduce specific value propositions into the negotiation. Their generation is the main topic of the argumentation function that will be thoroughly examined in the fourth chapter. 3.2.3.2 Arbitration Value Determination at Auctions of Mergers & Acquisitions Another mediation function banks and the other consultants must be aware of is the so-called M&A auctions that become increasingly popular where the banks act as auctioneers. However, the mediation is primarily carried out with regard to the performance of the process and and not with a focus on the determination of an adequate price, which will be discussed below. At these auctions, businesses or business shares are sold in several (usually two to four) bidding rounds. Under competitive conditions, a group of presumptive buyers submits written offers (bids) in the respective bidding rounds that are generally not publicly available. Those investors make it to the next bidding round who have made attractive offers in the current round from the perspective of the seller. Accordingly, the pool of potential investors diminishes in each round - in contrast to the increasing highest offers in practice. At the transition to the next round, the qualified tenderers receive additional information about the business in question, for instance, they gain access to so-called (digital) data rooms and have the option to talk to the management. Additionally, further information can be provided upon request, such as, how their offer compares to those of rival bidders. The mention of concrete prices is most commonly avoided and the number of rival bidders is usually not revealed to increase competitive pressure (W EIHE 2004, p. 41). Figure 3.25 (W EIHE 2004, p. 42) outlines the exemplary process of an M&A auction. Kapitel 1: Einführung 252 252 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 252 45520_Matschke_Griffleiste_SL5.indd 252 16.03.2021 16: 22: 26 16.03.2021 16: 22: 26 Chapter 3 The open auction, the limited auction, and the dual track process are different forms of M&A auctions (cf. Figure 3.26). The open and the limited auction vary according to the group of addressed investors. If a relatively small number of potential investors (usually maximum five) is addressed, a limited auction is given. Conversely, an open auction addresses a large group of investors. The latter can again be subdivided into a public and a controlled auction. While at the controlled auction a large group of potential buyers is addressed personally, the selling intention at a public auction is primarily advertised in an official notice, in business journals, and on the internet. These forms of auctions are becoming increasingly popular, especially regarding the privatization by federal, state, and local authorities in Germany. Finally, the dual track process represents a form that prepares an IPO (going public) simultaneously with an open or limited auction (cf. Figure 3.26). The dual track process can be useful to those owners who want to divest their share capital either by the selling the business within an open or a limited auction) or by going public (including secondary public offering if necessary). Moreover, such owners want to make the final decision on one of the options only after the almost complete parallel preparation of both options. The process involves a great deal of expense and is thus primarily suited for businesses that expect high sales proceeds (W EIHE 2004, p. 45). Beispiel eines Auktionsprozesses Erstellung einer ausführlichen Investorenliste („long list“) Erstellung einer reduzierten Investorenliste („short list“) Versand der Vertraulichkeitsvereinbarung und des Kurzprofils Versand des Memorandums Gespräche mit Unternehmensleitung und Datenraum Unternehmensbesichtigung und Datenraum Endverhandlungen und Vertragsabschluß Mögliche Bietrunden Indikative Angebote Formalisierte Angebote Verbindliche Angebote Figure 3.25: Process of an M&A auction Example of an auction process Preparation of a detailed investor list (long list) Preparation of a reduced investor list (short list) Distribtion of a memorandum Пример течения аукционного процесса Пример течения аукционного процесса Final negotiations and conclusion of the contract Distribution of a confidentiality agreement and a short profile Management talks and data rooms Facility visit and data room Possible bidding rounds Indicative offers 2 Formalized offers Binding offers 3.2 Value Determination in Non-Dominated Conflict Situations 253 45520_Matschke_Griffleiste_SL5.indd 253 45520_Matschke_Griffleiste_SL5.indd 253 16.03.2021 16: 22: 26 16.03.2021 16: 22: 26 In summary, it is noticeable that the focus of an auction is on the price as a conflictresolution-relevant fact. In contrast to the IPO, the role of a mediating bank or a neutral arbitrator relates less to the active collaboration at the determination of the arbitration value, but rather to the adequate execution of the auction process. However, the character of the arbitration value is given by the determined minimum offer and finally by the highest offer named by the auctioneer at the end of the auction that wins the bid. Furthermore, it should noted that the participants in the auctions must always consider their individual marginal prices, even in the heat of the moment. These confidential marginal prices might change as available information about the valuation object increases, but they do not necessarily have to rise. The submitted bids generally represent argumentation values that must not exceed the bidders’ own concession limit if those are considered or eventual become binding. Offene Auktion Öffentliche Auktion Kontrollierte Auktion Begrenzte Auktion Dual-Track-Verfahren Börsengang Formen von M&A-Auktionen Figure 3.26: Forms of M&A auctions Forms of M&A auctions open auction Limited auction flotation Public auction controlled auction Dual track process Open auction Controlled auction Initial public offering Kapitel 1: Einführung 254 254 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 254 45520_Matschke_Griffleiste_SL5.indd 254 16.03.2021 16: 22: 27 16.03.2021 16: 22: 27 Chapter 3 3.3 Value Determination in Dominated Conflict Situations If an arbitrator is called upon to determine the arbitration value of the business as an independent third party in a dominated conflict situation , the following steps need to be taken for the determination in the broader sense within the matrix of the functional business valuation (cf. Figure 1.18 in Section 1.4.1, p. 47): Step 1 (field D of the matrix): Determination of the number of acceptable conflict resolutions from the perspective of each conflict party and the number of possible agreements (arbitration area) or - as far as no agreements are made - a derivative, that is, modified, agreement quantity determined by the arbitrator, Step 2 (field E of the matrix): Determination of the arbitration value (valuation in the narrower sense) by the arbitrator based on the (modified) agreement set, as adequate and acceptable conflict resolution, at least from the perspective of the party worth being protected, as well as Step 3 (field F of the matrix): Application of the arbitration value by the conflicting parties. The following passage makes particular mention of the differences that result within the steps made by a dominated conflict situation in comparison to a non-dominated conflict situation, as previously discussed in Section 3.2.1. These differences are recognizable if there are no acceptable conflict resolutions from the perspective of all conflicting parties with regard to the feature of rationality of action. Nevertheless, the determination of the arbitration value is still required. In case of dominated conflict situations, in which freedom of choice is prevented at least for one party, the first step of the matrix of business valuation might include a notification that there is no original agreement and hence no acceptable and rational resolution from the perspective of all conflicting parties. The agreement set E is empty. As far as the change of ownership of the business to be valuated is also possible without the consensus of all parties regarding the conditions under which such a change could occur, there is a need for mediation with a negative range of transactions (S IEBEN 1983, p. 541). In such a situation, an arbitration value proposed by the mediator presumes a non-rational behavior for at least one of the conflicting parties based on the available information. However, to avoid these expectations from that party whose interests should be protected, the impartial arbitrator has to define those conflict resolutions (with with no loss of generality of the statement, the parties worth being protected can be determined as and the other parties not worth being protected as that are consistent with the feature of rationality of action from the perspective of all parties worth being protected. In other words, the arbitration value must be compatible with rational action from the perspective of the party worth being protected (M ATSCHKE 1979, p. 49). It cannot generally be stated which interests should be protected in such a case, but this results inter alia from legislation. To determine the arbitration value, an agreement j = 1, ..., m' 1 ≤ m' < m; j = 1, ..., m' m'+1, ..., m) 3.3 Value Determination in Dominated Conflict Situations 255 45520_Matschke_Griffleiste_SL5.indd 255 45520_Matschke_Griffleiste_SL5.indd 255 16.03.2021 16: 22: 27 16.03.2021 16: 22: 27 set E ' modified by the feature of the interests worth being protected is taken as the new basis (M ATSCHKE 1979, p. 59): In the second step of the matrix of functional business valuation, an arbitration value is determined under the consideration of the feature of party-related adequacy by an impartial arbitrator within the agreement set E or alternatively - if E is a void set - within the modified agreement set E '. Finally in the third step of the matrix of functional business valuation, the arbitration value is actually used, where three alternative application possibilities can be found in dominated conflict situations: 1. The parties have to submit to the arbitration value, for instance, due to legislation. 2. With the determined arbitration value, the agreement conditionsthat have been selected in advance are reviewed. 3. The arbitration value can also serve as a recommendation, for example, in the sense of a basis for a court judgment. E ' = (s 1 , ..., s n ) | f j (s 1 , ..., s n ) ≥ N j (a opt ) ∀ j = 1, ..., m ' ∧ (s 1 , ..., s n ) ∈ S { } . Kapitel 1: Einführung 256 256 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 256 45520_Matschke_Griffleiste_SL5.indd 256 16.03.2021 16: 22: 27 16.03.2021 16: 22: 27 Chapter 3 3.4 Selected Control Questions Exercise 1 (20 Points) - Arbitration Value and Arbitration Value Determination a) Explain which tasks are pursued according to the mediation function. (2 points) b) Define the term arbitration value. Name and explain the characteristic features of the arbitration value in case of a non-dominated conflict situation. (8 points) c) Discuss the process of the arbitration value determination in case of a non-dominated conflict situation, according to the matrix of functional business valuation. (10 points) Exercise 2 (10 Points) - Arbitration Value Determination in Dominated Conflict Situations a) What is a dominated conflict situation? (3 points) b) How does the process of the arbitration value determination in dominated conflict situations differ from the arbitration value determination in non-dominated conflict situations concerning the matrix of functional business valuation? Discuss only the particularities that characterize dominated conflict situations. (7 points) Exercise 3 (10 Points) - Features of Party-Related Adequacy a) What is meant by the feature of party-related adequacy? (3 points) b) Determine the arbitration value in the sense of an arbitration price in case of a onedimensional, non-dominated, disjoint conflict situation of the acquisition/ sale type, according to the rule of absolutely equal division and the rule of relatively equal division if the decision value of the buyer is estimated at 100 GE and the one of the sellers at 80 GE. Briefly explain your approach. (4 points) c) How can the arbitration value ultimately be used by the conflicting parties in this conflict situation? (3 points) Exercise 4 (15 Points) - Combined Valuation Methods a) Identify three traditional combined valuation methods. Discuss their representation with J ACOB ’s normal form and their arbitration-theoretic interpretation. (9 points) b) Under which conditions are the combined valuation methods limited in their application for the arbitration value determination? Briefly illustrate three of the four cases. (6 points) 3.4 Selected Control Questions 257 45520_Matschke_Griffleiste_SL5.indd 257 45520_Matschke_Griffleiste_SL5.indd 257 16.03.2021 16: 22: 27 16.03.2021 16: 22: 27 Exercise 5 (15 Points) - MVA and EVA Methods Discuss market value added (MVA) method and the method of economic value added (EVA) method. Critically examine these methods and their implications as proposed by their advocates regarding value-oriented management. Exercise 6 (30 Points) - Arbitration Value at an Intial Public Offering (IPO) and at Auction Discuss the following topic in a short structured essay: “Arbitration values within the context of the IPO of young businesses and concerning auctions of Merger & Acquisition (M&A auctions).” Start with a table of contents. Exercise 7 (15 Points) - Auction Processes during the Initial Public Offering a) Systematize the types of auction processes in the broader sense and also plot them. Briefly discuss the main features of each process. (7 points) b) Describe one of the auction processes in the narrower sense in detail. Support your explanations by plotting the process. (8 points) Kapitel 1: Einführung 258 258 3 Mediation Function and Arbitration Value 45520_Matschke_Griffleiste_SL5.indd 258 45520_Matschke_Griffleiste_SL5.indd 258 16.03.2021 16: 22: 27 16.03.2021 16: 22: 27 Chapter 4: Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 259 45520_Matschke_Griffleiste_SL5.indd 259 16.03.2021 16: 22: 28 16.03.2021 16: 22: 28 Overview The overall objective of chapter 4 is to study the third main function of the functional business valuation, the argumentation function, which serves to determin the argumentation value of the business. This function represents the least theoretically permeated function. However, it is probably the most commonly used function in real life. This perception is further reinforced by observing the lagging managerial accompanying research for consulting products that make these valuation methods even more acceptable. These valuation methods represent suitable argumentation instruments in the negotiation process if the negotiation partner is not aware of the theoretical foundation of those methods. Section 4.1 will focus on the basics of the argumentation function and the features of the argumentation value. Then the determination of argumentation values is discussed (Section 4.2). After the presentation of the argumentation value determination within the context of the matrix of functional business valuation, several suitable valuation methods are analyzed in greater detail. Finally, selected questions conclude this chapter (Section 4.3). Learning objectives After studying this chapter, you should be able to 1. describe both the argumentation function and the argumentation value and enumerate the features of the argumentation value; 2. analyze possible representations of the determination of the argumentation value within the matrix of functional business valuation; 3. explore the argumentation factors used in the context of negotiations; 4. discuss the business valuation methods, which serve as an argumentation basis, and evaluate the mechanisms of these methods with regard to the strengthening or at least the preservation of a party’s negotation position if the other parties to the negotation also deploy argumentation values. 4.1 Grundlagen 260 260 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 260 45520_Matschke_Griffleiste_SL5.indd 260 16.03.2021 16: 22: 28 16.03.2021 16: 22: 28 Chapter 4 4.1 Basics The argumentation value is the result of a business valuation in the sense of the argumentation function (M ATSCHKE 1976, 1977a, and 1977b, M ATSCHKE / M UCHHEYER 1977, W AGENHOFER 1988a and 1988b, H AFNER 1993, G ORNY 2002, H ERING / O LBRICH 2002, B ARTHEL 2004, B RÖSEL 2004, B RÖSEL / B URCHERT 2004, H ERING / B RÖSEL 2004, B ARTHEL 2005, M ATSCHKE / B RÖSEL 2013a, p. 607). The argumentation value does not denote a single value size, but rather the totality of justifications (arguments) which one negotiating party reveals or makes available with the aim of improving their own bargaining position or even weakening the position of the negotiating partner and, in turn, ultimately obtaining a more favorable negotiation result (B ARTHEL 2005, p. 36, F REY / R APP 2011). The argumentation values represent partisan values, whose importance within the negotiation manifests itself in the crucial support of one party’s views to influence the opposing party (S EMANN 1970, B ARTHEL 2005, p. 33). This particular aspect, seeking a change in behavior or at least in the perception of the negotiation partner to gain advantages or avoid disadvantages, is one of the important reasons why the argumentation function is still the least theoretically permeated among the main function of business valuation. This theoretic neglect is all the more astonishing since the task at hand, to aid argumentation during negotiations, is likely to be practiced widely. We can assume that all reports presented at the negotiating table by one party were simply made available because the party hopes those reports will support its own bargaining position (B ARTHEL 2005, p. 32). All disclosed reports are understood here to constitute argumentation values. Therefore, the issue of influencing the opposing party with partisan biased information should be studied in business research. Business valuations are used as argumentation aids, something that must be considered during negotiations in which business valuations play a key role. Usually, a series of arguments is deliberately introduced in the negotiation process, either in form of supposed decision values or allegedly impartial arbitration values. As long as the negotiation seeks agreement, the change of ownership of the company should be realized under conditions that are as far as possible from one party’s own concession limit and as close as possible to the suspected concession limit of the opposite party. However, the derivation of appropriate argumentation values requires a party to not only have the knowledge of their own decision value and be able to make an educated guess about the opposing decision value but also an idea about the desired negotiation result. If negotiators know their own decision values and recognize them, argumentation values do not represent an instrument of overreaching (M ATSCHKE 1976, p. 520). It can be assumed that the conflicting parties adopt individually rational behavior and consequently would not approve a price that violates their decision values and hence the limits of their agreement area (G ORNY 2002, p. 13). 4.1 Basics 261 45520_Matschke_Griffleiste_SL5.indd 261 45520_Matschke_Griffleiste_SL5.indd 261 16.03.2021 16: 22: 29 16.03.2021 16: 22: 29 An argumentation value is always partisan, but that does not necessarily mean it is one-sided. Those parties using argumentation values aim to gain advantages or try to avoid possible disadvantages. The negotiating position of a party can be strengthened in various ways: 1. Arguments may be provided that allow the negotiation partner to make a concession or to agree to a specific negotiation outcome. 2. The negotiating party can receive information by which a) the arguments of the opponent’s party are invalidated, b) offers of negotiations are rejected for obvious reasons, or c) offers of negotiations may be modified to favor one party’s own benefit. The usage of argumentation values is not limited to the actual negotiation process with the opposing party, such as that covering the acquisition and the sale of a business. They can also be deployed in negotiations to signal tactical intention between internal conflict parties (M ATSCHKE 1977a, p. 91). After the negotiation, it is conceivable that argumentation values might be required, for instance, to justify a business acquisition or the termination of a negotiation process to supervisory bodies or shareholders. Any use in the run-up to the actual negotiation assumes that decisions are not made by only one person. Argumentation values can then serve to secure the acceptance of higer-level decision-makers (such as group management) or within a decision-making body (such the board of directors). Moreover, argumentation values might be useful to induce those high-level decision-makers to make decisions that accord with the desired outcome of the proposer, such as commencing negotiations or acquiring a business. In addition to this decision-influencing, business valuations can also facilitate the delimitation of responsibilities in internal negotiation processes. The intention might be to distribute the responsibility for a risky decision at an early stage to as many parties as possible, so that one party’s future bargaining position is affected as little as possible if decision subsequently proves unwise (sharing of responsibility); alternatively, those, who originally drove the acquisition of a property and subsequently assumed the line responsibility for that acquisition took precautions for that eventuality when advocating business valuation to ensure subsequent activity was not burdened from the outset with “a mortgage difficult to redeem” (avoiding responsibility). Figure 4.1 summarizes some options for the application of argumentation values (B ARTHEL 2005, p. 37). 262 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 262 45520_Matschke_Griffleiste_SL5.indd 262 16.03.2021 16: 22: 29 16.03.2021 16: 22: 29 Chapter 4 The aspect of the delimitation of responsibilities is considered next. This aspect relates to only one party to the conflict and precedes any conflict with other parties (in terms of time). The point of departure for an argumentative business valuation aiming to avoid responsibility might be the valuation-relevant performance (profit or economic benefit) in the eyes of the valuation subject. Performance is regarded here in sense of (accounting) profit and for reasons of simplification, it is assumed that the profit is fully and immediately available to the buyer company. Figure 4.2 showcases three different representations for the same issue, whereby each time the entire valuation-relevant profit contribution of the object of purchase is addressed from the perspective of the valuation subject and realistically estimated. After the transaction, the object of purchase will Figure 4.1: Use of argumentation values in business valuation Gebrauch von Argumentationswerten der Unternehmensbewertung Verschaffung von Vorteilen und/ oder Vermeidung/ Abschwächung von Nachteilen auf die gegnerische Konfliktpartei bezogen primär auf den Verhandlungsprozeß bezogen Entkräften von Argumenten Zurückweisen von Angeboten Modifikation der Verhandlungsführung Beeinflussung durch Information Beeinflussung der Vorteilsverteilung Beeinflussung des Einigungsbereichs manipulativ kooperativ primär auf das Verhandlungsergebnis bezogen Zustimmung oder Zugeständnisse im Hinblick auf das Verhandlungsresultat (Einigung oder Abbruch) auf die eigene Konfliktpartei bezogen dem externen Konflikt vorgelagert Entscheidungsbeeinflussung Abgrenzung von Verantwortlichkeiten Teilung der Verantwortung Abwälzung der Verantwortung dem externen Konflikt nachgelagert Rechtfertigung Rechenschaft Use of argumentation values in business valuation Providing advantages and/ or avoiding/ mitigating disadvantages With regard to the opposing party With regard to one’s own party before the actual negotiation after the actual negotiation I Rejection of offers I Invalidating/ weakening of arguments and the main focus on the negotiation result and the main focus on the negotiation process Approval or concession with regard to the negotiation result (agreement or termination) I Decision-influencing II Justification Modification of the negotiations Influencing by information Influencing of the benefit distribution Influencing of the agreement area Manipulatively Cooperatively Delimitation of responsibilities I Accountability Sharing of responsibility Buck-passing of responsibility 4.1 Basics 263 45520_Matschke_Griffleiste_SL5.indd 263 45520_Matschke_Griffleiste_SL5.indd 263 16.03.2021 16: 22: 30 16.03.2021 16: 22: 30 continue to be legally independent but economically managed by the purchaser (as a 100 % subsidiary of the purchaser). In representation A, the valuation-relevant profit contribution is reported as a total amount. In representation B, it is composed of two values. One amount is expected from the company in the case of an independent continuation of the purchase object by the buyer company. In addition, there is an anticipation of increased profits as a result of economies of scope (synergy effects) after the acquisition and integration of the object of purchase. Representation B corresponds to a usual allocation and procedure for the profit estimation for business valuations. The representations A and B are completely sufficient if it were only a question of determining the appropriate decision value from the point of view of the buyer company with an assumed profit objective. However, if the business valuation is also be used as an internal argumentation aid to delimitate later responsibilities, both representation A and representation B are not very expedient, because, for instance, representation B does not clearly show whether the supposed profit increase is to be expected in the object of purchase itself or in other areas of the buyer company. But representation C is a suitable as an argumentation aid to support the delimiting of responsibilities (in the sense of avoiding responsibility) for a future profit realization. There, the entire valuation-relevant profit contribution is allocated to different corporate divisions in which it is created and reported. If those responsible for the valuation must also bear responsibility for that purchase object after the actual acquisition, their immediate responsibility is limited to the double-framered area. This profit could - within an overall realistic estimate - then be valued rather pessimistically by those people, whereas the expected increase in profits in other divisions of the company is estimated optimistically. Therefore, the internal arguers could try to achieve both their desired acquisition of the object of purchase and avoid the responsibility for later profit realizations. However, the argumentative use of business valuations is not focused on influencing internal decisions or on internal buck-passing, but rather on the negotiations with external parties. Such negotiations include specifying the conditions under which the Figure 4.2: Distribution of the valuation-relevant profit contribution considering an internal delimitation of responsibilities A B C Gesamter bewertungsrelevanter Gewinnbeitrag nach Erwerb Gewinn bei selbständiger Fortführung des Kaufobjekts Positive Synergieeffekte Gewinn des Kaufobjekts nach Integration Gewinnsteigerung in anderen Bereichen des Käuferunternehmens nach Integration Valuation-relevant profit contribution after acquisition Profit increase in other corporate divisions of the buying company after integration Positive synergy effects (as a residual amount) Profit on object of purchase after integration Profits from independent continuation of the acquired object 264 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 264 45520_Matschke_Griffleiste_SL5.indd 264 16.03.2021 16: 22: 30 16.03.2021 16: 22: 30 Chapter 4 change of ownership should occur. In case of a purchase/ sale, the price to be paid that plays a particularly key role, whereas in case of a (de)merger, the distribution of property rights (shares) will probably play the most prominent role. In such negotiations, a business valuation as an argumentation aid would be dispensable if the parties made unsubstantiated proposals with their respective tender prices and agreement would ultimately emerge on a sequence of price concessions on both sides. In such a case, it would be sufficient for a rational negotiation of the parties that they know and also consider their respective decision values. The buyer must know their maximum price and the seller must know their minimum price. Figure 4.3 outlines the negotiaiton situation described above in which a business valuation as an argumentation aid is dispensable because the agreement results from a sequence of unsubstantiated price offers. The continuous presentation was chosen for simplificity (M ATSCHKE 1969, p. 60). The negotiation period is plotted on the abscissa, whereas the negotiation result is displayed on the ordinate. The price claims (asking prices) of the seller decline as the negotiation period grows longer, while the price offers (bid prices) of the buyer will likely increase. After a certain negotiation period, the demand corresponds to the offer. Both parties have agreed on a certain price. This price is acceptable from the persepective of both parties because it is below the price ceiling (upper price limit) of the buyer (P max ) as well as above the bottom price (lower price limit) of the seller (P min ). However, during the negotiations the parties are only aware of their own decision limit. The negotiation situation depicted in Figure 4.3 in the manner of an oriental bazaar, seems to apply for negotiations over the price of a company in only exceptional Figure 4.3: Negotiation as a mere sequence of price offers Zeit Preisangebote des Käufers Preisforderungen des Verkäufers Verhandlungsdauer Pmax Pmin P Preis Price Time Price claims of the seller (Asking price) Price offer of the buyer Duration of negotiation 4.1 Basics 265 45520_Matschke_Griffleiste_SL5.indd 265 45520_Matschke_Griffleiste_SL5.indd 265 16.03.2021 16: 22: 30 16.03.2021 16: 22: 30 cases, such as M&A auctions. It is more characteristic for conflict resolution processes occurring during an acquisition/ sale of a company (M ATSCHKE 1977a, p. 95) that • the parties substantiate and justify their offers based on business valuation results, • the agreement is not achieved by direct price concessions, but by a cooperative search of the correct parameters of a specific business valuation method that is both accepted and applied by the parties, and • the price concessions of the parties particularly refer to those valuation parameters. In such negotiation situations, the decision value of the respective party does not constitute a basis for a negotiation because a party should not argue on the basis of their own decision value if they do not want to weaken their negotiation position considerably. Therefore, the parties have to justify their offers as convincingly as possible while not making themselves vulnerable, while simultaneously trying to impede or even prevent inference with their own decision value based on the presented justification. To solve this problem, the parties resort to argumentation values of the business, which might therefore also be regarded as communications media (K USSMAUL 1996, p. 267) applied to bridge the conflicts of interest over the price between buyer and seller and to reach an agreement. At this point, it is even hypothesized that argumentation values are accepted by the conflicting parties as part of the laws of the game of a purchase negotiation and that they - despite their apparent intention to influence - do not constitute any instruments of overreaching as long as the parties know and honor their own decision value in the negotiation process. Concerning the intention to influence, the following differentiation is appropriate: 1. An argumentation is performed with regard to the benefit distribution within the assumed agreement area, and/ or 2. An argumentation is performed with respect to influencing that assumed agreement area. In the first case, the argumentation values serve to separate as much as possible for a party from the assumed realizable benefit in the amount of the difference between the upper and lower price limit. For this purpose, for example, argumentation values that lie in proximity of the presumed decision value of the other negotiation party may be introduced in the negotiation to underpin corresponding price claims. A fundamental factor of the preparations for negotiations is the thorough analysis of the decision situation of the other party. Their interests are recognized and their concession limits are estimated. Such argumentation values have, for reasons of credibility and owing to their usefulness as an instrument of influence, to be construed realistically and must not show up as exorbitant prices. Those who seek to impress the negotiation partner based on unilateral arguments that are not transformed into a realistic value may be victims of a fundamental fallacy. But the intention to influence can also - as the second case expresses - be aimed at changing the area of agreement itself in a way that is beneficial for the chosen negotiation strategy, through; 1. a rather manipulative intent with regard to a shift of the area of agreement toward the suspected decision value of the opposing party, or 2. a cooperative intention to create an agreement area or to expand an existing one and thereby enlarging the potential benefits for both parties in case of an agreement (B ARTHEL 2005, p. 34). The cooperative use of argumentation values is particularly significant. That use might include information in a communication with the negotiation partner about an argumentation value, which may be used to expand the agreement area. The information 266 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 266 45520_Matschke_Griffleiste_SL5.indd 266 16.03.2021 16: 22: 31 16.03.2021 16: 22: 31 Chapter 4 could also create the conditions for an agreement under which the negotiation partner is induced to commit to a revision of their own decision value that proves favorable in terms of agreement-seeking endeavors. For instance, the seller could point the buyer to integration options that the buyer was not aware of and that would also allow an increase to the maximum payable price so that a previously rather unacceptable price claim of the seller would become more acceptable. In the context of considering the argumentation function of business valuation, it is therefore not recommended to start from the assumption of the less expedient picture of the negotiation as an overreaching event, but rather of the positively connoted image of the negotiation as a cooperative benefit enhancement event. With regard to the latter point of view, the joint creation, and the aligned perception of additional benefits are occupy center stage of negotiation efforts. In this way, the given conflict of interest between the negotiating parties regarding the distribution of benefits can be mitigated, less stressed and eventually probably bridged. To negotiate creatively means to discover differences and to coordinate those in such a way that so-called cooperation profits (benefits/ gains) arise (S IEBE 1996, p. 206). In this sense, argumentation values should be introduced into the negotiation process. Below, the numerous features of the argumentation values are systematically summarized. They may be classified into three main features, the feature of camouflage (disguise), the feature of party-relatedness, and the feature of conflict resolution orientation. Figure 4.4 outlines a possible classification of the features of the argumentation value based on B RÖSEL (2004, p. 518). Because argumentation values are not introduced in a negotiation in their actual form, but as alleged or supposed decision or arbitration values, the feature of camouflage applies to them. It is part of the mimicry of the argumentation value that it denies its genuine character (M ATSCHKE 1977a, p. 102). It has to be considered that depending on Figure 4.4: Main features of argumentation values Merkmale des Argumentationswertes Merkmal der Tarnung Merkmal der Parteienbezogenheit Merkmal der Bezogenheit auf Entscheidungswerte Merkmal der Beeinflussung Merkmal der Konfliktlösungsorientierung Merkmal der Information Merkmal der Flexibilität Merkmal der Glaubwürdigkeit Feature of camouflage Feature of party-relatedness Feature of a conflict resolution orientation Feature of credibility Feature of flexibility Feature of information Feature of influencing Feature of the relatedness to decision values Main features of the argumentation value 4.1 Basics 267 45520_Matschke_Griffleiste_SL5.indd 267 45520_Matschke_Griffleiste_SL5.indd 267 16.03.2021 16: 22: 31 16.03.2021 16: 22: 31 the justification of bids and offers, drawing conclusions on the decision value of the arguing party might be complicated and perhaps even entirely impossible. According to the feature of party orientation, argumentation values are values of one negotiation side, designed to a certain negotiation situation and also to a specific negotiation partner to strengthen the negotiating position in the negotiation process (M ATSCHKE 1977a, p. 91). The consequences of this characteristic are the feature of the relatedness to decision values and the feature of influencing. The latter states that a behavioral change of the negotiation partner is sought with the aid of the argumentation function of business valuation. The opposing party should be induced by argumentation values to even consciously concede certain conflict resolution issues or desired negotiation results. The feature of relatedness to decision values aims at two directions. First, a party’s own decision value is the last line of retreat for the argumentation value (C OENENBERG / S IEBEN 1976. p. 4076, C OENENBERG 1992, p. 92); hence presenting argumentation values in the negotiation should not directly reveal a party’s own decision value nor should the opposite side be able to draw valuable conclusions about it. Nevertheless, argumentation values should be directed toward the alleged decision value of the opposing party (M ATSCHKE 1976, p. 521, G ORNY 2002, p. 156). The feature of conflict resolution orientation comes into play because argumentation values are usually introduced to the negotiation with the purpose of reaching an agreement or non-agreement concerning a change of ownership of the valuation object (i.e., the company). If a conflicting party is explicitly not seeking agreement in a negotiation, the presentation of astronomical price expectations and the corresponding price justifications might be suitable argumentation values to exclude agreement and thus accelerate conflict resolution, in the sense of a termination. However, it is assumed here that an agreement is envisaged in the conflict situation, although this is by no means mandatory in practice. Since argumentation values are usually not equivalent to price quotations that are simply floated arbitrarily but represent justified price expectations instead, they serve to bridge the existing conflicts of interest between the negotiation parties, especially concerning the price level and also other parameters, and finally to achieve a conflict resolution. This can be done by a cooperative search for conflict-resolution-relevant issues supported by argumentation values and a subsequent agreement on the characteristics of these parameters (M ATSCHKE 1977a, p. 96). The conflict resolution orientation is reflected in the subordinate features of information, flexibility, and credibility. The feature of information distinguishes the argumentation values because the negotiating parties try to justify their offers with these values (M ATSCHKE 1977a, p. 96). As a result, the negotiating partner gains information about the asking price of the other party and might be able to infer the negotiating tactics chosen by that party. Furthermore, the negotiating partners might be able to acquire previously unknown information about the valuation object from the use of both external and internal argumentation values in particular, if they are introduced into the process as arbitration values by independent appraisers/ valuators (M ATSCHKE 1976, p. 521). As already stated, information might be deliberately provided to the negotiation partner to expand the assumed agreement area, that is, the difference between one party’s own decision value and the estimated decision value of the opposing party. However, it will most likely be provided to induce a change of the decision value of the opposing party (M ATSCHKE 1977a, p. 98). In this way, the feature of information correlates with the feature of influence. The feature of flexibility describes the ability of argumentation values to consider newly obtained information and intermediate results of the negotiation (S IEBEN 1993, p. 268 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 268 45520_Matschke_Griffleiste_SL5.indd 268 16.03.2021 16: 22: 31 16.03.2021 16: 22: 31 Chapter 4 4319). Within the context of the argumentation function, the applied valuation methods should be both easy to manage and customizable because these methods allow several starting points for an argumentation, especially to appear trustworthy to the negotiating party. This emphasizes the close link to the feature of credibility discussed below (C OENENBERG / S IEBEN 1976, p. 4076, C OENENBERG 1992, p. 92). Finally, the argumentation value only proves its usefulness if the feature of credibility applies (C OENENBERG / S IEBEN 1976, p. 4076, C OENENBERG 1992, p. 92, S IEBEN 1993, p. 4319). Hence, argumentation values should represent convincing, less vulnerable, and realistic values. Their determination is tolerated and they are ultimately accepted by the opposing party as substantiated offers. With regard to the renown of the profession of certified public accountants (external auditors), their prepared valuation reports are quite suited as long as the opposing party does not realize their limited usability for decision purposes. Accordingly, the valuation reports of auditors are used as argumentation values if it is beneficial for the conflicting parties, , because essentially it can be assumed that the facts and figures of the report have not been compiled to deceive or defraud (M ATSCHKE 1976, p. 521). In Germany, the involvement of auditors in the business valuation process is common practice. However, on an international level, other groups of experts who (would claim) they possess broad expertise in the field of business valuation (e.g., business valuation specialists or certified valuation analysts) might also be involved in the business valuation process. Accordingly, references herein to auditors should be read as including such experts where appropriate. 4.1 Basics 269 45520_Matschke_Griffleiste_SL5.indd 269 45520_Matschke_Griffleiste_SL5.indd 269 16.03.2021 16: 22: 32 16.03.2021 16: 22: 32 4.2 Value Determination 4.2.1 Steps of Determination within the Matrix of Functional Business Valuation 4.2.1.1 Overview According to the argumentation function, the argumentation value results from the subsequent valuation steps that can be represented in the “matrix of functional business valuation” (cf. Figure 1.18 in Section 1.5.1): Step 1 (field G of the matrix): Determination of a party’s decision value and determination of the assumed opposing decision value (included in both field A and field B of the matrix of functional business valuation), Step 2 (field H of the matrix): Selection of argumentation factors and the correspoding argumentation value determination (valuation in the narrower sense), and Step 3 (field I of the matrix): Application of the argumentation values in the negotiation. 4.2.1.2 The Steps in Detail 4.2.1.2.1 First Step Argumentation values - as analyzed in Section 4.1 - are characterized by an orientation towards a decision value that results from the feature of party relation. A party’s own decision value, which should not be disclosed, represents the last line of retreat for the argumentation values used. In order to finally achieve a negotiation result that is as close as possible to the opposing concession limit, that is, the decision value of the opposing party (or also to obtain an extension of the assumed argumentation area), an educated guess about the other party’s decision value has to be made. Before the actual argumentation values can be determined, a party’s decision value has to be calculated or approximated and the opposing decision value has to be estimated as precisely as possible (step 1 and field G of the matrix). Regarding the determination of the decision values, it is necessary to procede analogously to the fields A and B of the matrix. This has been extensively discussed in the second chapter. Refering to the estimation of the opposing party’s decision value, the perspective of the negotiation partner should generally be adopted. This entails that assumptions about the target system and their decision field have to be made and that height, range, dispersion (standard deviation), and interdependencies of the potentially relevant future performances (and interest rates) for the opposing party must be considered. These assumptions represent the foundation for the investment-theoretic transformation into the alleged decision value of the negotiation partner. 270 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 270 45520_Matschke_Griffleiste_SL5.indd 270 16.03.2021 16: 22: 32 16.03.2021 16: 22: 32 Chapter 4 4.2.1.2.2 Second Step In the second step - under consideration of the determined decision values - the valuation in the narrower sense is performed within the argumentation function (step 2 and field H of the matrix) by the selection of the argumentation factors and the corresponding determination of the different argumentation values. With regard to the facts used to influence the negotiation partner as shown in Figure 4.5 (B RÖSEL / B URCHERT 2004, p. 356), it is possible distinguished between hard and soft factors (B RÖSEL / B URCHERT 2004, p. 354). The argumentation value is represented by the entirety of justifications (arguments) applied by a negotiation party to strengthen its own negotiation position or to weaken the position of the negotiation partner and hence to achieve a more favorable negotiation result. Finally, all sorts of different argumentation values result, depending on whether the argumentation factors are used individually or in combination. It is assumed that all facts that cause a direct or indirect change of the decision field or of the target system of a valuation subject are classified as hard argumentation factors of the original and derivate facts relevant to conflict resolution in the negotiation. The original facts relevant to conflict resolution are classified to the original argumentation factors. As already mentioned, investment and employment obligations/ commitments raised in the negotiation are included in this category, in addition to price offers and proposals of payment modifications. Derivative facts relevant to conflict resolution are classified under the derivative argumentation factors, especially the used valuation parameters and the applied valuation methods (H ERING / O LBRICH 2002, p. 155, M ATSCHKE / M UCHEYER 1977). They indirectly change the decision field and represent the (material) negotiation basis. Additionally, they lead to a direct support of the negotiation process. If the parties try to discuss the valuation methods with their negotiation partners, economically based valuation methods must be skillfully presented to the opposing par- Argumentationsfaktoren der Unternehmensbewertung «Harte» Argumentationsfaktoren Ausprägungen originärer konfliktlösungsrelevanter Sachverhalte Ausprägungen derivativer konfliktlösungsrelevanter Sachverhalte Bewertungsgrößen Bewertungsverfahren «Weiche» Argumentationsfaktoren Kommunikationsunterstützende Inhalte Formen der Kommunikationsunterstützung Figure 4.5: Argumentation factors of business valuation Argumentation factors of business valuation Hard argumentation factors Soft argumentation factors Characteristics of original conflict-resolution-relevant facts Characteristics of derivative conflict-resolution-relevant facts Communication-supporting content Forms of communication support Valuation parameters Valuation methods 4.2 Value Determination 271 45520_Matschke_Griffleiste_SL5.indd 271 45520_Matschke_Griffleiste_SL5.indd 271 16.03.2021 16: 22: 32 16.03.2021 16: 22: 32 ty. For this purpose, parties tend particularly to apply those methods with a strong reputation, regardless of whether they are effective for determining a decision value (due to decision-oriented aspects). Examples of these methods are the neoclassical valuation models related to financial theory (e.g., variants of the DCF methods or the option price model), originating from idealized assumptions like information efficiency, perfection and completeness of the markets and trying to determine a rather mystical objective (impartial) exchange value of the business as a hypothetical market value. Finally, these methods enjoy great popularity in practice despite their unsuitability for the determination of the decision value long being established in business literature (S CHNEIDER 1998, B ÖCKING / N OWAK 1999, K RAG / K ASPERZAK 2000, p. 112, H ERING / O LBRICH 2002, p. 156, M ATSCHKE / B RÖSEL 2003, p. 162, H ERING / B RÖSEL 2004, p. 936, W AMELING 2004, p. 82, H ERING 2005, and 2014, p. 297). If the opposing party accepts the widespread methods related to finance theory, they represent a profitable reservoir for all sorts of argumentation values (H ERING 2014, p. 222). On the contrary, if the opposing party is influenced by valuation parameters, the parties try to present the operands that were used within the valuation models related to investment theory (or other methods), such as, the future performances or the interest rates considered, in their favor. It is therefore important that the conflicting parties know the so-called impact mechanisms (mode of action/ operation, effect), that is, the impact on the business value to be determined of changes to each parameter. In the simplest case, increases in future performance and decreases of the calculation interest rates lead to higher values and corresponding decreases of the future performances, while increased interest rates lead to lower values. Below, the argumentation with valuation parameters is discussed for an one-dimensional conflict situation of the type acquisition/ sale from the perspective of a presumptive buyer - described here as valuation subject. In a simplified negotiation situation, the objective of the presumptive buyer is to keep the sole original conflict-resolution-relevant fact - the price payable - as low as possible with the aid of the valuation parameters cash flows, interest rates (either internal or required rate of return), and valuation period. These are derivative conflict-resolution-relevant facts. The negotiation partner - the presumptive seller - is often supported by an auditor in the negotiation. It is assumed that the business - the valuation object - will not generate cash flows (g) indefinitely but only for a limited number of years (T). Positive or negative synergies (economies of scope) for the valuation object are neither realizable for the presumptive buyer nor for the presumptive seller. Having scrutinized the past figures presented by the presumptive seller, the presumptive buyer is aware that the future cash flows are unlikely to be as high as in the past. It is assumed that the valuation subject has already determined a decision value before the negotiation by themselves or by a third party. The final result from the following negotiating now depends on the negotiating skills of the parties and their knowledge of the formal connections between the valuation parameters within the valuation methods. It is in any case recommended not to be naive. Just as poker players literally conceal their cards, this also applies figuratively to the negotiation about the business in question. The argumentation of the presumptive buyer should certainly not indicate their own decision value; this, of course, also holds for the negotiation partner. The alleged decision values, arbitration values, or objective values introduced in the negotiation by the opposing party or by the commissioned auditor should be examined critically by the presumptive buyer because essentially they have characteristics of argumentation values. 272 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 272 45520_Matschke_Griffleiste_SL5.indd 272 16.03.2021 16: 22: 33 16.03.2021 16: 22: 33 Chapter 4 Regarding the cooperative price search, it is assumed that the negotiation partners have agreed upon a method that relies on the present value approach. The formula for the determination of the business value using the present value BW (German: Barwert) generally reads as follows: The cash flows g t are to be discounted for each year t, g 1 for the first year, g 2 for the second year and so forth until g T for the last included year. It is allowed in the formula that all amounts g t vary from each other. The symbol i represents the interest rate (rate of return) and is assumed to remain constant throughout time. The planning period of the present value calculation extends over T years. The payment series is in arrears, that is, the cash flows g t arise at the end of each period. The factor 1/ (1 + i) t symbolizes the discount factor, which becomes continually smaller as t increases so that the present value of an equal amount will decrease all the more, the later it is expected. The present value BW hinges on three parameters: 1. the cash flow g to be discounted (which is to be set as low as possible from the perspective of the presumptive buyer), 2. the period T, for which the present value is determined (which is also to be minimized from the perspective of the buyer), and 3. the interest rate (rate of return) i (that should be as high as possible from the perspective of the presumptive buyer). According to the cash flows g included in the valuation, the presumptive buyer should at first insist on not choosing historical data as the basis, as auditors often do, but to select planning and forecasting calculations as the basis. These planning and forecasting calculations are not only lower according to the assumptions made in the given example but could also be easier to influence due to the prevailing uncertainty. From the perspective of the presumptive buyer, it does no harm to be pessimistic about the payments (and also about the gains/ profits) (the lower, the better) or about the payouts (and the expenses) (the higher, the better). If things are different later, it is likely to be due to the effective work of the new owner. It is essential for the argumentation that the presumptive buyer is able to appropriately assess the calculation mechanism, that is, that they know how the present value changes if a parameter changes. Again, the aim must be to calculate a present value that is as low as possible. It will usually not be possible to plan everything in detail until the end of the entire planning (valuation) period (of T years, according to the previous formula). Therefore, the whole period is generally divided into two; one in which periodic planning is in place, and a subsequent period in which the valuators must be satisfied with rough plans. This is considered in the following formula. The phase of detailed planning with different cash flows to be discounted comprises τ periods; the subsequent phase with rough plans and an assumed constant cash flow g includes n periods. Therefore, the entire planning period T is T = τ + n. The present value can be calculated as follows: BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 + g 3 (1 + i) 3 +…+ g t (1 + i) t +…+ g T − 1 (1 + i) T − 1 + g T (1 + i) T . BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 + g 3 (1 + i) 3 +…+ g τ (1 + i) τ Phase of detailed , differentiated planning ! " ####### $ ####### +…+ g τ+ 1 (1 + i) τ+ 1 + g τ+ 2 (1 + i) τ+ 2 +…+ g T =τ+ n (1 + i) T =τ+ n Phase of global, stable planning ! " ###### $ ###### . 4.2 Value Determination 273 45520_Matschke_Griffleiste_SL5.indd 273 45520_Matschke_Griffleiste_SL5.indd 273 16.03.2021 16: 22: 34 16.03.2021 16: 22: 34 This formula can be simplified, because with regard to the second phase with g = g τ +1 = g τ +2 = … = g T= τ +n , the term g/ (1 + i) τ can be factorized so that a specific financial term, the so-called present value annuity factor (M ATSCHKE 1993b, p. 175) presents itself: The present value annuity factor is an important amount influencing the result BW. It becomes larger, the longer the second phase lasts (= n), and it becomes smaller, the higher the applied interest rate i is set. The interest of the presumptive buyer must be to minimize the present value annuity factor as a multiplier of the term g/ (1 + i) τ . In other words, the second period n should be as low as possible and the interest rate (rate of return) as high as possible. The calculation mechanism regarding the interaction of n and i is represented in the following Figure 4.6. Both control parameters of the present value annuity factor vary in the range 1 ≤ n ≤10 and 5,0 % ≤ i ≤ 10,0 %: The entire planning period in this example would lie between τ + 1 ≤ T = τ + n ≤ τ + 10 years, where τ is the lenght of the observation period with detailed planning. The presumptive buyer should insist that the amount at the end of the phase of detailed planning is unchanged and carried over to the second phase. According to the desired decrease of the purchase price, the presumptive buyer should particularly strive for a decrease in profits (cash flows) within the argumentative determination of the business value in the second phase. This would then lead to two arithmetical possibilities: a) A general reduction takes place. b) A negative growth rate is considered. At the assumption of a general reduction of the amount g = (1 - α ) • g τ , the situation is formulated as follows, whereby α serves as the reduction coefficient: BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 +…+ g τ (1 + i) τ + g (1 + i) τ ⋅ 1 (1 + i) 1 + 1 (1 + i) 2 +…+ 1 (1 + i) n ⎡ ⎣⎢ ⎤ ⎦⎥ BW = g t (1 + i) t t = 1 τ ∑ + g (1 + i) τ ⋅ (1 + i) n − 1 i ⋅ (1 + i) n . 123456 5,0 % 6,0 % 0,9523810 1,8594104 0,9433962 1,8333927 2,7232480 3,5459505 2,6730119 3,4651056 4,3294767 5,0756921 4,2123638 4,9173243 7,0 % 8,0 % 0,9345794 1,8080182 0,9259259 1,7832647 9,0 % 10,0 % 0,9174312 1,7591112 0,9090909 1,7355372 2,6243160 3,3872113 2,5770970 3,3121268 4,1001974 4,7665397 3,9927100 4,6228797 2,5312947 3,2397199 2,4868520 3,1698654 3,8896513 4,4859186 3,7907868 4,3552607 789 10 5,7863734 6,4632128 5,5823814 6,2097938 7,1078217 7,7217349 6,8016923 7,3600871 Figure 4.6: Present value annuity factor, depending on the interest rate i and the observation period n 5,3892894 5,9712985 5,2063701 5,7466389 6,5152322 7,0235815 6,2468879 6,7100814 5,0329528 5,5348191 4,8684188 5,3349262 5,9952469 6,4176577 5,7590238 6,1445671 n ↓ i → 274 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 274 45520_Matschke_Griffleiste_SL5.indd 274 16.03.2021 16: 22: 36 16.03.2021 16: 22: 36 Chapter 4 At α = 0,2, this would lead to the calculation with 80 % of the amount that is valid at the end of the first planning phase for the entire second phase. In practice, this procedure should find “acceptance” and a reduction coefficient between 20 % (= 1/ 5) and 33,33 % (= 1/ 3) applied is not likely to encounter very strong resistance. If a “negative growth rate” is assumed for the second phase, the calculation of the present value BW is formally more challenging and less transparent for laypeople, which might trigger an intellectual resistance to the more complicated calculation. Formally, the situation is represented as follows: Fortunately, the expression in the square brackets can be transformed into the present value annuity factor of a finite, growing (here: negatively) annuity immediately. Then, the formula reads: By using this formula, it is apparent that both phases are multiplicatively interconnected by the expression in the large brackets when a negative growth rate is applied. Conversely, they are only additively related in case of a general reduction. From the perspective of the presumptive buyer the argumentative goal of the application of the negative growth rate must be to minimize the value in the square brackets. This is achieved if the parameters i (interest rate, rate of return) and w (negative growth rate) are set as high as possible, whereas the duration n (planning periods of the second phase) is minimized. The effects of the parameters are illustrated by the examples in Figure 4.7 with an interest rate amounting to i = 5 % p. a. and in Figure 4.8 with a required rate of return i = 10 % p. a. BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 +…+ g τ (1 + i) τ + (1 − α ) ⋅ g τ (1 + i) τ ⋅ 1 (1 + i) 1 + 1 (1 + i) 2 +…+ 1 (1 + i) n ⎡ ⎣⎢ ⎤ ⎦⎥ BW = g t (1 + i) t t = 1 τ ∑ + (1 − α ) ⋅ g τ (1 + i) τ ⋅ (1 + i) n − 1 i ⋅ (1 + i) n . g τ BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 + … + g τ (1 + i) τ + g τ ⋅ (1 − w ) (1 + i) τ+ 1 + g τ ⋅ (1 − w ) 2 (1 + i) τ+ 2 + … + g τ ⋅ (1 − w ) n (1 + i) τ+ n BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 + … + g τ (1 + i) τ ⋅ 1 + 1 − w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ + 1 − w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ 2 + … + 1 − w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ . BW = g 1 (1 + i) 1 + g 2 (1 + i) 2 + … + g τ (1 + i) τ ⋅ 1 − 1 − w 1 + i ⎛ ⎝⎜ ⎞ ⎠⎟ n + 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⋅ 1 + i 1 + w ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ . 4.2 Value Determination 275 45520_Matschke_Griffleiste_SL5.indd 275 45520_Matschke_Griffleiste_SL5.indd 275 16.03.2021 16: 22: 37 16.03.2021 16: 22: 37 On the basis of tangible figures, it can be estimated, which of the both initial assumptions (general reduction versus negative growth rate) is more favorable and should be used to support the argumentation, according to the respective data constellation. In this context, the significance of future performances and interest rates for periods lying far in the far future must not be underestimated. H ERING (2014, p. 40) emphasizes the relevance of these future performances on a perfect capital market under uncertainty using mathematical finance as follows: In order to understand the impact of the prevailing uncertainty in real decision fields, it is assumed that the payment stream corresponds to a perpetuity g. For the future performance value, that is, the outcome will be determined by BW = g/ i. In the periods from t = 0 through t = τ the cash flow can be reliably determined ( ), but after that period, the determination become uncertain with . Hence, the present value (future performance value) BW is divided into a certain and an uncertain part: 123456 5,0 % 10,0 % 1,9047619 2,7233560 1,8571429 2,5918367 3,4639888 4,1340851 3,2215743 3,7613494 4,7403627 5,2888996 4,2240138 4,6205833 15,0 % 20,0 % 1,8095238 2,4648526 1,7619048 2,3424036 25,0 % 30,0 % 1,7142857 2,2244898 1,6666667 2,1111111 2,9953569 3,4248127 2,7846885 3,1216674 3,7724674 4,0539022 3,3784133 3,5740292 2,5889213 2,8492295 2,4074074 2,6049383 3,0351639 3,1679742 2,7366255 2,8244170 789 10 5,7851949 6,2342239 4,9604999 5,2518571 6,6404883 7,0080608 5,5015918 5,7156501 Figure 4.7: Present value annuity factor of a (negatively) growing annuity at i = 5 % 4,2817304 4,4661627 3,7230698 3,8366246 4,6154650 4,7363288 3,9231426 3,9890610 3,2628387 3,3305991 2,8829447 2,9219631 3,3789994 3,4135710 2,9479754 2,9653169 n ↓ w → 123456 5,0 % 10,0 % 1,8636364 2,6095041 1,8181818 2,4876033 3,2536627 3,8099814 3,0353118 3,4834369 4,2904385 4,7053787 3,8500848 4,1500693 15,0 % 20,0 % 1,7727273 2,3698347 1,7272727 2,2561983 25,0 % 30,0 % 1,6818182 2,1466942 1,6363636 2,0413223 2,8312359 3,1877732 2,6408715 2,9206338 3,4632793 3,6761704 3,1240973 3,2720708 2,4636551 2,6797649 2,2990233 2,4630148 2,8271124 2,9275766 2,5673731 2,6337829 789 10 5,0637361 5,3732267 4,3955113 4,5963274 5,6405139 5,8713529 4,7606315 4,8950622 Figure 4.8: Present value annuity factor of a (negatively) growing annuity at i = 10 % 3,8406771 3,9677959 3,3796878 3,4579548 4,0660241 4,1419277 3,5148762 3,5562736 2,9960750 3,0427784 2,6760436 2,7029369 3,0746216 3,0963329 2,7200507 2,7309414 n ↓ w → g % g, g = g = % g. BW = g i = g ⋅ 1 + i ( ) τ − 1 i ⋅ 1 + i ( ) τ + % g ⋅ 1 i ⋅ 1 + i ( ) τ . 276 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 276 45520_Matschke_Griffleiste_SL5.indd 276 16.03.2021 16: 22: 38 16.03.2021 16: 22: 38 Chapter 4 The relative weight of the certain future performance value part is ((1 + i) τ - 1)/ (1 + i) τ and of the uncertain future performance value part is 1/ (1 + i) τ . Generally, τ is not very large so that at realistic rates of return, a large part of the future performance value is based on wishful thinking that uncertain cash flows will be attained indefinitely. For instance, if a specific period comprises τ = 5 years, the uncertain part still contributes almost 50 % to the present value (future performance value) event if the rate of return is set at i = 15 % p. a. (H ERING 2014, p. 41). As noted above, the present value BW essentially depends on the amount of the interest rates i, ceteris paribus. This connection is mostly not understood or is difficult to understand at best for laypeople. With regard to the argumentation function, the interest of the presumptive buyer is aimed at a required rate of return that is as high as possible. The larger the i, the lower the present value, BW. If auditors are engaged, they might possibly try to determine the rate of return from a so-called base interest rate BZ (German: Basiszinssatz) under consideration of corrections K (German: Korrekturen) in form of premiums (i.e., agio or surcharges) and/ or discounts (i.e., disagio or deductions): i = BZ + K. The base interest rate is derived by auditors what they decide is “an average achievable interest rate for long-term, risk-free investments”. However, an auditor usually makes no efforts to derive and prove the selected interest rate, which is usually introduced without substantiation. The trend today is to derive the interest rate from the capital asset pricing model (CAPM). In case of doubt, auditors also generally do not understand the CAPM, so an effective derivation remains elusive. Nevertheless, laypeople might be impressed, because they most likely do not know anything about it at all, and there is also an element of thinking that if something is fashionable, it must be “good”. Other approaches to ascertain the base interest rate could include applying the interest rates known to the negotiation partners from other contexts such as courts or supervisory authorities. Therefore, it is wise to pay attention, search, and compare. The highest base interest rate is always the “better” interest rate from the perspective of the presumptive buyer, and, in addition, a layperson might be satisfied with a high-interest rate as it constitutes a “worthwhile goal”. Simply put: It is quite often very easy to push up the interest rate, on the basis of thinking that an extra percentage point would not cause a problem because the mode of action is not understood. Particularly, laypeople also often ignore the fact that premiums (surcharges) to the rate of return can be converted into deductions of the amount to be discounted. If it is not possible to reduce the expected cash flows any further in the numerator, it is usually much easier to argumentatively increase the required rate of return in the denominator. Corrections in the form of discounts should be strictly refused from the perspective of the presumptive buyer. Here, auditors often argue for what they might term an “inflation deduction”. It is immediately clear even to a layperson that the inflation must be anticipated in the future and that remuneration is included in the interest rate for the risk of inflation. Therefore, the inflation rate is excluded from the (nominal) interest rate in order to set a real long-term interest rate. According to this conclusion, the base interest rate must be shortened by an inflation deduction, which is often pure invention and completely unrealistic. This argumentation is substantially and formally extremely unlikely and should not be accepted under any circumstances by the presumptive buyer: Nobody knows if there is any inflation at all and how high it might eventually become (M ATSCHKE 1986). Nobody knows if it is included in the interest rate being processed on the market, and if so, how. A subtraction of the base interest rate is in any case arithmetically incorrect to calculate a real interest rate. However, the genuine problem is the 4.2 Value Determination 277 45520_Matschke_Griffleiste_SL5.indd 277 45520_Matschke_Griffleiste_SL5.indd 277 16.03.2021 16: 22: 38 16.03.2021 16: 22: 38 implication of a positive growth rate arising from such an inflation deduction. In other words, increasing cash flows would ultimately increase the present value. This, however, would not be in the interest of the presumptive buyer. Consequently, the presumptive buyer should not refuse to accept premiums on the base interest rate, according to the motto, the more the merrier. To achieve the corresponding increases in the rates of return (interest rates) and ultimately also reductions of the argumentation value, it is suitable to use the derivative conflict-resolution-relevant fact, “risk”. If the presumptive buyer aims to present low argumentation values in the negotiation, he might try to convince the negotiation partner of the particularly significant risks of the transaction to consider them as premiums on the interest rate or as discounts on the future performances, that is, the cash flows, or ideally - if the seller is to accept this argumentation - as discounts on both the cash flows and as premiums on the interest rate. For this purpose, an innovative presumptive buyer can resort to different types of risk, in the context of foreign acquisitions: political and economic country risks, foreign currency risks, legal risks, business and financing risks (P EEMÖLLER / K UNOWSKI / H ILLERS 1999, p. 626). If such surcharges are not (yet) provided, they should be required. The risk premium RZ (German: Risikozuschlag) is well-known and usually deployed by auditors. One half of the base interest rate is a common standard, where little to no discussion with the auditor is likely because it constitutes a regular approach. At a base interest rate BZ = 6 % p.a., the risk premium amounts to RZ = 3 % p. a. so that i = 9 % p. a.. However, different surcharges may be demanded like an (im-)mobility premium MZ (German: Mobilitätszuschlag), due to the extremely limited (or non-existent) marketability of shares (saleability). Investments in a limited liability company are generally more immobile than investments in shares, meaning another premium to the base interest rate of up to 50 % of the base interest rate might be justifiable. The rate of return would then increase by 3 % p. a. to i = 12 % p. a. (= BZ + RZ + MZ = 6 % + 3 % + 3 %). If the presumptive buyer enforces this and finally agrees on a price that is far from their decision value, they can even sit back and relax. The role of the facts relevant to derivative conflict resolution might eventually occupy center stage. Extensions of the original facts are determined by the parties agreeing on certain extensions of derivative facts. In these cases, the derivative facts not only support the original variables through arguments but substitute for them in the conflict resolution process. Therefore, multi-dimensional conflict situations arise from conflict situations with only one original fact. The advantage of a substitution of one original fact - like the level of the price payable - with several derivative facts, such as “future performance”, “required rate of return”, “business valuation method”, or “risk premium” - is that it can be seen as offering a concession. Usually, the parties do not have to worry about a (substantial) weakening of their position in the case of an agreement. Only the agreement about all derivative facts ultimately determines the level of the price and the changes in the decision field. Concessions over one derivative fact may be partly, fully, or even overcompensated for by concessions of the opposing party on other derivative facts. The latter is particularly the case if the other party complies with the negotiation rules by responding to concessions. However, certain indivisibilities of the derivative variables must also be taken into account. Nonetheless, derivative conflict-resolution-relevant facts are quite adequate instruments within the argumentation function.While at first glance it might seem that the substitution of original facts or variables by several derivative variables could complicate an agreement between the parties, the 278 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 278 45520_Matschke_Griffleiste_SL5.indd 278 16.03.2021 16: 22: 39 16.03.2021 16: 22: 39 Chapter 4 opposite effect is to be expected after a refined analysis of the change caused by the substitution in the conflict situation. This might also be regarded as an opportunity to reduce complexity. Finally, the common substitution of the original variables by derivative variables represents the first step towards a conflict resolution because the parties have agreed upon how to reach an agreement value that is acceptable for both parties. At the same time, the argumentation scope for the parties is defined by the substitution. Arguments that do not refer to the substituting derivative variables do not seem very expedient. The substitution might contribute to the canalization and objectification of the conflict resolution process. Moreover, concessions on derivative variables can be seen as expressing an interest in (better) adversarial arguments. This should additionally facilitate the transition from confrontation to a cooperative search for an acceptable solution.Besides the substitution of original conflict-resolution-relevant facts by derivative conflict-resolution-relevant facts, it is possible in a multi-dimensional conflict situation that a party may fulfill the desires of the other party, according to a conflict-resolutionrelevant fact, at least within certain ranges. This is because an agreement on certain aspects has no (case of indifference) or even positive (case of harmony) consequences for their decision field (H INTZE 1992b, p. 415). This has already been discussed in Section 2.2.2. Finally, the concerned party can use these cases to assist negotiation by providing approvals for questioned facts and by buying concessions at other important conflict-resolution-relevant facts. The preparation of an actual conflict situation/ negotiation requires a model of this conflict situation that is as realistic as possible. Such a model is always a simplified, complexity-reducing figure. Even if the need for a complexity reduction is undisputed, the previous extensive one-dimensional perspective of the decision value as a critical price (marginal price) that separates the advantageous prices (conflict resolutions) for a party from non-beneficial ones nevertheless remains problematic. A conflict situation of the acquisition type and a sale that solely focuses on the (cash) price is not as frequent as is assumed in theory and in the practice of business valuation. This has become evident from the privatization of formerly “nationally owned” enterprises. Additionally, for conflict resolution processes of the acquisition/ sale type, it seems to be characteristic that the parties verify the level of the price payable with the aid of derivative conflict-resolution-relevant facts. In such conflict situations, in which the price to be paid is justified or even derived by the parties with the aid of derivative variables, mere knowledge of the marginal price is not sufficient as a basis for negotiation. Moreover, the parties must know, upon which extensions of the derivative conflict-resolution-relevant facts they might agree without undermining their marginal prices after an agreement on certain extensions of the derivative variables and on their formalized interaction. Hence, the knowledge of business valuation methods applied and especially the knowledge of their mode of action (effects) constitutes a crucial basis for a rational negotiation (M ATSCHKE / M UCHEYER 1977, p. 180). In contrast, the soft argumentation factors (B URCHERT 1998, B RÖSEL / B URCHERT 2004, p. 342) of business valuation aim at to support communication between the negotiation partners. Those factors also serve to facilitate the prospect of a successful business transaction. Starting points are the chosen contents and the forms of communication support. Examples of forms of communication support are the selection of media, the adherence of (customary) communication forms available for consideration, and also the behaviors generally that are followed by the conflicting parties during the negotiations. The behavioral patterns also show attitudes adopted consciously or uncon- 4.2 Value Determination 279 45520_Matschke_Griffleiste_SL5.indd 279 45520_Matschke_Griffleiste_SL5.indd 279 16.03.2021 16: 22: 39 16.03.2021 16: 22: 39 sciously by the parties during the negotiations that might have a dominant influence on the conflict resolution (B ARTHEL 2005, p. 37). These include the variations of the negotiation period, the rigid adherence on prior procedures, the (alleged) termination of negotiation, the consultation of independent appraisers, the determination of a certain procedure that does not allow for an exit position, the early release of supposedly achieved negotiation results, the transfer to other levels of decision-making, the transfer of the subject from the price to other facts like warranty, methods of payment and rights of withdrawal, and modifications of the valuation object (only a part of the valuation object will be bought, certain elements like pension obligations will not be adopted, or at the other extreme, further tangible assets might be added). This behavior also includes determining strategies and resolving conflicts at the expense of a third party such as the tax authorities as well as the conscious complication or the inadequate simplification of a negotiation, the use of delaying tactics, the determination of an agreement date, etc. The effectiveness and pervasive use (in terms of value) of this attitude is shown by the case of Vodafone/ Mannesmann, even if an unlawful influence cannot be hypothesized. Lastly, the possibility that the opposing party might adopt unethical tactics (e.g., an attrition policy, breach of confidentiality, commencing negotiations with licensors, suppliers, landlords, executives, etc.) must be anticipated (S EMANN 1979). The content supporting communication is based on the current need for action and the country in which a business is bought or where the negotiation partners are from. An example is the application of terms, phrases, and keywords as symbols in the communication. Those symbols are more than a necessary prerequisite to enable orientation around complex expectations. The interpretation is at least steered in a particular direction by the deliberate choice of cultural elements in the scope of communication with the negotiation partner. Nonetheless, the communication remains flexible since the significance of the symbols is derived from the interaction. However, it is important to note that humans act based on meanings behind the symbols that surround them. Business partners coming from far away, who speak the national language and are proficient in the customary forms of communication, are viewed more favorably than those who cannot speak the language or do not even try to. This also holds when, for instance, German negotiators can at least convey some knowledge of Russian culture or certain traditions in front of Russian negotiation partners. Expressing an interest in the national culture might already be interpreted as a supporting symbol. At the same time, the ability to handle drinking vodka from so-called cto gpamm (100 ml) glasses could be perceived as quite impressive by the Russian negotiation partner. This example also attests to the effect of a supporting symbol and probably has a positive impact on the business deal (B URCHERT 1998). 4.2.1.2.3 Third Step After the argumentation factors are selected and the corresponding argumentation values are determined, they are finally used in the negotiation (step 3 as well as field F of the matrix). As noted previously, the negotiations can be conducted with internal and primarily external partners. The phenomenon that is described by the term negotiation is procedural. The main feature of a price negotiation is seen in the sequential exchange of offers and counteroffers. By the ongoing exchange of different values, the parties at least tend to get gradually closer to their objectives (G ORNY 2002, p. 157). G ORNY (2002) proposes a model of negotiation based on the mathematical theory of discrete 280 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 280 45520_Matschke_Griffleiste_SL5.indd 280 16.03.2021 16: 22: 40 16.03.2021 16: 22: 40 Chapter 4 dynamic systems, according to which the offer strategies of the parties are determined interactively depending on the last offer of the negotiation partner. The use of argumentation values is determined by both decision values and the negotiation process and one party’s negotiation strategy. The main problem of the third step of the argumentation value determination in the broader sense can be seen in the selection of an offer strategy, optimizing the desired negotiation result (G ORNY 2002, p. 159) because the offer strategy particularly depends on the offer strategy of the opposing party. According to the feature of camouflage, the argumentation values are introduced in the negotiation process as alleged decision and arbitration values to influence the further negotiation process and ultimately also the negotiation result (K USSMAUL 1996, p. 267). Furthermore, the arguing party should consider that the justification of their offers does not enable the opposing party to draw any conclusions about their own decision value. 4.2 Value Determination 281 45520_Matschke_Griffleiste_SL5.indd 281 45520_Matschke_Griffleiste_SL5.indd 281 16.03.2021 16: 22: 40 16.03.2021 16: 22: 40 4.2.2 Selected Valuation Methods 4.2.2.1 Comparison Methods 4.2.2.1.1 Comparison Methods based on Single Valuation 4.2.2.1.1.1 Stock-and-Debt Method As mentioned above, business valuation methods play a significant role in negotiations. Generally, those methods are particularly well suited that are currently en vogue. They serve for argumentation as long as the negotiation partner is convinced of a positive correlation between popularity and usability and ultimately accepts the method(s). It was mentioned in the previous section that the knowledge of applied business valuation methods and especially the knowledge of their mode of action (effects) represents a fundamental basis for a rational negotiation. If a party uses a certain method, it must know what it is for and which argumentation patterns exist. For this purpose, selected valuation methods are presented that are generally unsuited to decision value determination but are quite widespread. With regard to single-valuation-oriented comparison methods, attempts are made to derive a business value (in the sense of a potential market price) from realized prices of individual business investments, particularly shares. In this context, other methods quite popular in the English-speaking world include (quotation-based) stock valuation or (stock) market value methods. Generally, three single-valuation-oriented methods are distinguished: The stock-and-debt method; the similar public company approach; and the initial public offering (IPO) approach (O LBRICH 2000, p. 454, M ANDL / R ABEL 2019, p. 82). For listed (quoted) acquisition objects the stock-and-debt method is usually prefered (M ORAL 1920, p. 139, M ÜNSTERMANN 1966, p. 136, B UCHNER / E NGLERT 1994, p. 1573, O LBRICH 2000, p. 455). The potential market price of the valuation object W BO (W; German: Wert) is determined via the intermediate stage of the so-called market capitalization (stock market value) of the valuation object. The number of issued shares of the valuation object (AA BO ; AA; German: Aktienanzahl; BO; German: Bewertungsobjekt) is multiplied by the stock market price of these shares (AK BO ; AK; Aktienkurs). Random price fluctuations can be addressed by using an average of historical share prices (several days/ weeks) rather than the stock market price on the valuation date (B UCHNER 1995, p. 404). The resulting value is often corrected by an individual control premium (PZ BO ) (PZ; German: Paketzuschlag): The basis of the “business value” calculation is the actual price for shares of the business to be valuated, which is a result from supply and demand on the stock market and only represents the subjective values of the buyers and sellers at the margin for a single share at a certain historical point in time. A control premium represents the premium on the stock market value that results from the possible impact on the business policy (P RATT / N ICULITA 2008, p. 383, O LBRICH 2000, p. 455) since stock valuation only refers to a single share that, of course, does not have any influence. Instead, a (common) stock gives its holder the right to vote and the right to collect a dividend. The right to assume an influence over the business acquired W BO = AA BO ⋅ AK BO + PZ BO . 282 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 282 45520_Matschke_Griffleiste_SL5.indd 282 16.03.2021 16: 22: 40 16.03.2021 16: 22: 40 Chapter 4 in addition to the dividend claim by buying the majority of the shares is honored by the presumptive buyer with a premium. Such a premium may result either from the buyer having to pay the seller a price premium for the bundle of shares or from the buyer having to buy up the shares successively on the public market, thereby triggering corresponding price increases which affect the purchase prices of the shares then to be acquired. (O LBRICH 2000, p. 455). If there is one investor (or several investors) who wishes to acquire the majority of the shares due to their interest in the business, a control premium of 20-50 % and perhaps more can be achieved, depending on the actual number and the intentions of those investors (B ORN 2003, p. 164). The idea of the market-oriented valuation is to explicitly exclude the discretion of the valuators by invoking the “objectivity” of the market. Hence, the market-oriented valuation methods consciously decide against the subjective component that is required in the sense of the principle of subjectivity in order to determine individual decision values (B ÖCKING / N OWAK 1999, p. 174). From a decision-oriented perspective, the value of a business, which is derived from the respective stock market price and possibly corrected by a (high) control premium, is not suitable for the determination of decision values. The corrected stock market values (determined using the stock-and-debt method) are independent of the subject without the consideration of the targets and the decision field of the valuation subject are based on historical data and might incorporate rationally not justifiable control premiums. However, these premiums are subjective after all so that the idea of a completely market-oriented valuation is undermined. According to the use of the argumentation function, the stock-and-debt method has several advantages. One advantage is the acceptance of these values in literature and in practice: The American valuation literature recognizes the stock market price as a concretization of the business value. This is justified by the significance of the capital market in the US economy (B UCHNER 1995, p. 403). Additionally, the stock market prices are characterized by their transparency and objectivity so that at least some negotiation partners might be impressed. Another advantage is the possibility to influence the argumentation value somewhat randomly by the “adjustment of the stock market price” and particularly by the determination of the control premium. Example 1: Stock-and-Debt Method The application is demonstrated using a simple example. It is assumed that a business has issued 54.500 shares (AA BO ). The price of a share (AK BO ) at the valuation date is 90 GE. The average price of the last three months is 85 GE, that one of the last month 105 GE. The valuation subject now has various options to determine argumentation values. Figure 4.9 illustrates the possible argumentation values, while only the control premiums (PZ BO ) of 20 % and 25 % respectively, are considered. The argumentation values lie between the minimum of 5.559.000 GE (as starting point for a buyer) and the maximum of 7.153.125 GE (as starting point for a seller). 4.2 Value Determination 283 45520_Matschke_Griffleiste_SL5.indd 283 45520_Matschke_Griffleiste_SL5.indd 283 16.03.2021 16: 22: 40 16.03.2021 16: 22: 40 4.2.2.1.1.2 Similar Public Company Approach If business values are derived from stock market prices of “comparable businesses”, the process is referred to as similar public company approach (M ANDL / R ABEL 1997, p. 259). This method is applied in particular if the businesses are not listed due to a lack of stock market quotations. Generally, within the scope of this method (with the exception of the previous analysis of the valuation object) the process of valuation is classified in the following sub-steps (B ÖCKING / N OWAK 1999, p. 171, L ORSON 2004, p. 228): 1. Selection of a “comparable business”: a) Search of a business in the same industry, b) Limitation of the businesses regarding qualitative and quantitative aspects, and c) Decision for a “comparable business”; 2. Determination of a temporary business value (in the broader sense): a) Determination of the market capitalization of the “comparable business”, b) Determination of performance indicators of the “comparable business”, and c) Calculation of a temporary business value (in the narrower sense); 3. Corrections for the determination of a persistent business value. If both last two sub-steps are summarized, the value of the valuation object (W BO ; W; German: Wert; BO; German: Bewertungsobjekt) results from a selected performance indicator of the valuation object (BG BO ; BG; German: Bezugsgröße) and the “comparable business” (BG VU ; VU; German: Vergleichsunternehmen) as well as the (possibly average) price of the share as a “marginal” market price of the “comparable business” (AK VU ) and the number of traded shares (AA VU ) are represented as follows. The interim result is corrected by a discount for lack of marketability (FA; German: Fungibilitätsabschlag) and a control premium (PZ; German: Paketzuschlag) (M ANDL / R ABEL 2002, p. 74): The significance and “determination” of the control premium (PZ) is comparable to the process of the stock-and-debt method, according to the acquired controlling majority of the business. With reference to the estimates of P RATT / N ICULITA (2008, p. 381), the average control premium in the US amounts to roughly 40 % of the calculated initial value. J UNG (1993, p. 309) states a margin between 32 % and 116 % (! ). Valuation with AK BO at the valuation date AK BO in GE AA BO in GE 90 90 54.500 54.500 Valuation with AK BO as the average price of the last month Valuation with AK BO as the average price of the last three months 105 105 54.500 54.500 85 85 54.500 54.500 Interim result in GE PZ BO 4.905.000 4.905.000 20% 25% W BO in GE 5.886.000 6.131.250 5.722.500 5.722.500 20% 25% 4.632.500 4.632.500 20% 25% 6.867.000 7.153.125 5.559.000 5.790.625 Figure 4.9: Argumentation values, according to the stock-and-debt method W BO = BG BO ⋅ AK VU ⋅ AA VU BG VU + PZ − FA. 284 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 284 45520_Matschke_Griffleiste_SL5.indd 284 16.03.2021 16: 22: 42 16.03.2021 16: 22: 42 Chapter 4 A discount for lack of marketability also referred to as fungibility discount (FA), is indicated for not-listed valuation objects due to an actual lack of marketability and limited (re-)saleability (G REENSIDE 1976). Such a discount might also be indicated due to other legal and contractual sales restrictions (B UCHNER 1995, p. 411, P RATT / N ICULITA 2008, p. 383). A discount for lack of marketability is determined based on the calculated initial value (i.e., without the consideration of the “control premium”), and 35 to 40 % are usual (L ORSON 2004, p. 230). The determination of the amount of the discount for lack of marketability is usually arbitrary and cannot be rationally justified. Both the selection of suitable comparable businesses and the formation of discounts for lack of marketability control premiums solely fall within the discretion of the valuator; they cannot be reproduced intersubjectively (O LBRICH 2000, p. 459). Depending on the negotiation situation, the premiums and the discounts are substantiated by the valuation subject, by, for instance, referring to various recommendations in the literature of practitioners. The quotient of market capitalization of the “comparable business” (MK VU ; MK; German: Marktkapitalisierung) and its performance indicator (BG VU ) can be regarded as (M VU ; M; German: Multiplikator): In literature and in practice there are virtually no limits regarding the selection of performance indicator BG; as B UCHNER (1995, p. 410) states: In addition to the performance indicator “profit” (EBIT, net income) other parameters are also taken into consideration, especially both cash flows and dividend payments. A certain degree of latitude arises for the valuator, according to the selection, scale/ scope, and weighting of the relevant key figures. Lastly, the experience and preferences of the valuer play a decisive role in the valuation process. The determination of the business value based on several multipliers offers a control of the achieved results, while a low range of values signals a certain reliability of the results. On the basis of the data of comparable businesses, it may be essential to adjust the determined key figures to the business to be valuated in a further step. This seems to be particularly appropriate if the selection of comparable businesses could not be solved to the full satisfaction of the valuator. According to the argumentation value to be determined, considerable scope for a valuator’s imagination exists. If the business values are determined on the basis of several performance indicators, they must be weighted appropriately: Depending on which argumentation value is desired, the valuation subject has to adjust the parameters. Adjustments result in dependence of the performance indicators (as represented by B UCHNER ) for the valuator with reference to 1. the choice of performance indicators, 2. the number of performance indicators, 3. the weighting of the performance indicators, and 4. the correction of the performance indicators. W BO = BG BO ⋅ AK VU ⋅ AA VU BG VU + PZ − FA W BO = BG BO ⋅ MK VU BG VU + PZ − FA W BO = BG BO ⋅ M VU + PZ − FA. W BO = AK VU ⋅ AA VU ⋅ α i ⋅ BG i BO BG i VU i = 1 I ∑ + PZ − FA with α i = 1 i = 1 I ∑ . 4.2 Value Determination 285 45520_Matschke_Griffleiste_SL5.indd 285 45520_Matschke_Griffleiste_SL5.indd 285 16.03.2021 16: 22: 42 16.03.2021 16: 22: 42 It must be stated that “potential market prices” are determined by stock valuation, which plays a particularly important role in the valuation literature of English-speaking jurisdictions. Since the capitalization of the stock market does not reflect the “right” market price of a business but results from a schematic multiplication of the stock market price of a share with the number of shares outstanding and since the (required) control premiums and discounts for lack of marketability cannot be objectified, the significance of the determined value (“potential market price”) as a decision value equals zero. In this situation it is important to note that the stock price movement cannot be predicted due to risky speculations, psychological influences (S HEFRIN 2000), panic reactions, and herd behavior (B IKHCHANDANI / H IRSHLEIFER / W ELCH 1992, L UX 1995, Z HU 2009). This method neglects both the subject and the future orientation of a valuation (B ÖCKING / N OWAK 1999, p. 174), but it is also assumed that the value of a business corresponds to its price. This approach and the initial public offering approach outlined in the following sections are often classified among the overall valuation methods. However, these methods suffer from the lack of differentiation between the valuation of a business as a whole, and the valuation of business shares. Hence, a connection between the price of a share in the sense of a price of the fractional ownership and the value of the business is established over the construct of stock market value (O LBRICH 2000, p. 459). Thus, this approach is classified among single-valuation-oriented methods. Further shortcomings of this approach emerge when examining the concept behind the valuation formula. The relation between the known “business value” of the comparable business W VU and the performance indicator of this business BG VU corresponds to the relation of the required value of the valuation object W BO and the corresponding known performance indicator of this business BG BO : These approaches not only misjudge the distinction between value and price, but there is a basic question concerning the comparability between businesses. The challenge of comparability is the mere fact that no two businesses are identical. On the contrary, the valuation itself shows that a major inequality must be considered in terms of comparison object(s) and the valuation object (L ORSON 2004, p. 233). Nevertheless, this method of argumentation is used in the negotiations as long as the negotiation partner accepts it. It results in scope for argumentation, in terms of calculating the discount for lack of marketability, the control premium, the “adjustment of the stock market price” as well as in terms of the performance indicator(s). However, the question arises as to how the negotiation partner might be convinced of the existence of a (listed) comparable business despite the unique character of the valuation object. Hence, it is to be pursued to the prevailing opinion of the authors expressing this opinion in the sense of an effective round of negotiation. According to B UCHNER (1995, p. 408), the of comparable businesses is of central importance in this approach and it is at the same time its most difficult and challenging part. Given that finding an identical business in the main key areas is improbable, B UCHNER proposes identifying comparable businesses that produce mainly the same products and offer them in corresponding markets and which demonstrate similar growth opportunities and risks. W BO BG BO = W VU BG VU ⇔ W BO W VU = BG BO BG VU ⇒ W BO = BG BO ⋅ W VU BG VU . 286 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 286 45520_Matschke_Griffleiste_SL5.indd 286 16.03.2021 16: 22: 43 16.03.2021 16: 22: 43 Chapter 4 Similar suggestions are made by L ÖHNERT / B ÖCKMANN (2019, p. 853): The decisive criterion is hence the identity or comparability of the industry or the respective business segment, as businesses of the same industry usually have similar growth potential, cycles, and operating risks. A similar business size might also be important because large enterprises usually have a different cost structure than medium-sized enterprises. Additionally, it should be noted that valuations may differ across national borders since the cost of debt and the cost of equity costs differ owing to differences in risk-free investments and risk premiums. A review of each comparable business should verify if the stock value offers only a limited amount of information due to limited trading volumes (P RATT / N ICULITA 2008, p. 269). S ANFLEBER -D ECHER (1992, p. 598) classifies the following four criteria: 1. equal or very similar business, 2. distribution channels, 3. profits in recent years, and 4. size. Additionally, the same study suggests databases listing possible “comparable businesses”. Depending on which aspects influence the choice of “comparable businesses”, a desired business may become a “comparable business”, in the sense of an argumentation objective. The comparison object required here must not be confused with the comparison object that is relevant in the context of decision value determination. At the decision value determination for the buyer, the comparison object corresponds (under the assumptions of the net present value calculus) to the combination of the investment objects that are equal in their utility and were displaced from the base program and to the additional financing opportunities that were not used in the base program. For the seller, the comparison object generally does not only consist of investment objects to be included but also the displaced financing opportunities (decision opportunities) of the base program. Example 2: Similar Public Company Approach (a Single Comparable Business) The scope for argumentation is illustrated by the following example. The valuation subject has identified a single comparable business in preparation to negotiations about the valuation object. In order to keep the sphere of action within reasonable limits, it is assumed for reasons of simplification that the stock market value was constant in the last months and the market capitalization (MK VU ) is 5.000.000 GE, that is, a constant price of 100 GE multiplied by 50.000 issued shares. According to the possible performance indicators, it is assumed that these last years were constant. In practice, the performance indicators of the last one, three, or five years are considered; but that calculation is omitted here (S ANFLEBER -D ECHER 1992, p. 600). The annual profit (net income, etc.) (JG; German: Jahresgewinn), the sales revenue (U; German: Umsatz), and the dividends (D; German: Dividende) are used as performance indicators. While the annual profit JG VU of the comparison object amounts to 500.000 GE, the annual profit of the valuation object JG BO is 430.000 GE. The sales revenue is 3.500.000 GE (U VU ) and 2.500.000 GE (U BO ), respectively; the dividends amount to 450.000 GE (D VU ) and to 400.000 GE (D BO ). The valuation subject considers a calculation involving the respective control premiums of 20 % and 25 % as well as the discounts for lack of marketability of 30 % and 40 %. According to the consideration of the performance-indicators, only the following options are to be considered: (a) only the annual profit, (b) only the dividend, (c) all three performance indicators in equal parts, and (d) the profit and the dividend with 40 % each and sales revenue with 20 %. 4.2 Value Determination 287 45520_Matschke_Griffleiste_SL5.indd 287 45520_Matschke_Griffleiste_SL5.indd 287 16.03.2021 16: 22: 44 16.03.2021 16: 22: 44 Based on the example in Figure 4.10, those determined argumentation values are represented that lie between the minimum of 3.284.233 GE and the maximum of 4.222.222 GE. Example 3: Similar Public Company Approach (several comparable businesses) An extension of the scope for argumentation is also usable for valuation subjects if several so-called comparable businesses are used, as is often applied in practice. This is illustrated with a further example below (B ÖCKING / N OWAK 1999, p. 173), in which only one performance-indicator is considered because the profit and valuation premiums and discounts are ignored. As a performance indicator, the profit for the valuation subject of 19.200.000 GE is taken. According to the similar public company approach five comparable businesses are used, of which the so-called price-earning-ratios (KGV; German: Kurs-Gewinn-Verhältnis) lie between 11,5 and 24,7. KGV specifies the multiple of the profit for the comparison object estimated at the stock market. Correspondingly, the following argumentation values result in the given range between 220.800.000 GE and 474.240.000 GE, which are represented in Figure 4.11. Performance indicator BG (a) Profit MK VU in GE Interim result in GE 5.000.000 5.000.000 4.300.000 4.300.000 (b) Dividend 5.000.000 5.000.000 4.300.000 4.300.000 5.000.000 5.000.000 4.444.444 4.444.444 PZ FA 20% 25% 40% 40% W BO in GE 3.440.000 3.655.000 20% 25% 30% 30% 20% 25% 40% 40% 3.870.000 4.085.000 3.555.556 3.777.778 (c) Profit, dividend, and sales in equal parts 5.000.000 5.000.000 4.444.444 4.444.444 5.000.000 5.000.000 4.105.291 4.105.291 (d) Profit and dividend at 40 % each and sales at 20 % 5.000.000 5.000.000 4.105.291 4.105.291 5.000.000 5.000.000 4.212.063 4.212.063 20% 25% 30% 30% 20% 25% 40% 40% 4.000.000 4.222.222 3.284.233 3.489.497 20% 25% 30% 30% 20% 25% 40% 40% 3.694.762 3.900.026 3.369.651 3.580.254 Figure 4.10: Argumentation values, according to the similar public company approach (a single comparable business) 5.000.000 5.000.000 4.212.063 4.212.063 20% 25% 30% 30% 3.790.857 4.001.460 288 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 288 45520_Matschke_Griffleiste_SL5.indd 288 16.03.2021 16: 22: 45 16.03.2021 16: 22: 45 Chapter 4 4.2.2.1.1.3 Initial Public Offering Approach According to the similar public company approach, the average prices of the shares of listed businesses are used as a reference for the valuation. The business values at the Initial Public Offering Approach, i.e., the IPO approach, are derived from issuing prices, that is, from share prices for the first public offering (B UCHNER 1995, p. 406 and 412, M ANDL / R ABEL 1997, p. 264). As with both of the previously presented approaches, this approach also originates from the American valuation practice. In American valuation practice, this approach is used in the context of public offerings of the valuation object. This approach differs from the method of a listed comparable business, according to the source of the comparison prices. At this point, according to the criticism and the scope for argumentation, reference is made to the statements of the previous sections. The value is derived from the price of the shares of the comparison object at the placement on the stock exchange under consideration of the corresponding performance indicators, control premiums, and possible discounts for lack of marketabilitys. 4.2.2.1.2 Comparison Methods based on Overall Valuation 4.2.2.1.2.1 Recent Acquisitions Approach The approaches that are classified as comparison methods based on overall valuation derive business values from realized transaction prices of “comparable businesses” (“the recent acquisitions approach”) and also determine business values with multiples, which mainly result from so-called experience rates of the respective business industry (“the market multiples approach”). The process at the recent acquisitions approach (O LBRICH 2000, p. 457, M ANDL / R ABEL 2019, p. 84) is similar to stock valuation. However, there is a major difference: The business value W BO is derived in the sense of a “potential market price” from the prices P VU , which do not refer to business shares. They do not represent prices of fractional ownership in form of share prices, but instead result from transactions, where a full change of ownership has occurred at the comparable business. For this purpose, the consideration of both a control premium and - as long as the comparison object is not a listed company - of a discount for lack of marketability is neglected. If the business value is determined based on several performance indicators, they have to be weighted: Comparable business 12 Profit of the valuation object in GE KGV of the comparable business 19.200.000 24,7 20,9 345 Figure 4.11: Argumentation values, according to the similar public company approach (several comparable businesses) 19,0 15,4 11,5 W BO in GE 474.240.000 401.280.000 364.800.000 295.680.000 220.800.000 4.2 Value Determination 289 45520_Matschke_Griffleiste_SL5.indd 289 45520_Matschke_Griffleiste_SL5.indd 289 16.03.2021 16: 22: 45 16.03.2021 16: 22: 45 In this context, it is also an overall valuation method that abstracts from the valuation subject and from the future. This comparison method is however based on the discredited assumption that the price of a business corresponds to its value (O LBRICH 2000, p. 457). M ANDL / R ABEL (2019, p. 85) point out that the comparison methods classified are rarely applicable in German-speaking jurisdictions because - in contrast to the USA - vast databases and other information sources cannot be used. They are of the opinion that these approaches are suitable for the estimation of achievable market prices (at least in the USA). In essence, the recent acquisitions approach presupposes those assumptions that should actually be proven first. Based on any method, the “speculative market prices” are estimated, as far as the market participants accept applying such methods to determine their individual prices to be negotiated. This process is not problematic if the parties involved are conscious of their own decision value and if required are no longer willing to play along (i.e., are comfortable with the termination of the negotiation). The problem with this method is that the necessity to form a subjective decision value is not expressly pointed out. If it is then suggested that on the basis of such methods an “acceptable” price can always be found “automatically”, the propagation of such methods might be a recipe for deception and overreaching, especially on the part of less experienced market participants. The prices available in the extensive databases relate to transactions in the past. These prices result from the subjective value conceptions of the transaction partners, which are in principle not identical with the valuation subject, as well as from the negotiating power and the negotiating skills of these contracting parties. Therefore, it is doubtful that from these data meaningful conclusions can be drawn on prices attainable in the present or in the future for a (different) valuation object (H ERING / O LBRICH 2009, p. 367). For the determination of decision values, these approaches are not suitable due to the absence of a concrete subject relation. With this method, the scope for argumentation mainly results from the choice of the listed comparable business and according to the choice, number, weighting, and potential “correction” of the performance indicators. If the transactions concerns occurred too far in the past, the arguer should consider mapping temporary extrapolated trends to enhance the scope for argumentation (B ÖCKING / N OWAK 1999, p. 174). 4.2.2.1.2.2 Market Multiples Approach According to the market multiples approach, business values are determined (also in the sense of potential market prices) by multiplying a specific performance indicator of the valuation object with an industry-specific factor, the so-called market multiples that are generally valid for the complete industry or the entire branch (M ANDL / R ABEL 1997, p. 265). In practice, small businesses like retailers, medical practices, or law offices are valuated with this method. Therefore, it is not the stock market prices or realized transaction prices, but so-called monetary profitability indicators are used, as for example, the profit and the sales (sales method), or so-called quantitative profitability indicators, such as the product quantity, the extension and sales area, and the number of customers, W BO = P VU ⋅ α i ⋅ BG i BO BG i VU i = 1 I ∑ with α i = 1 i = 1 I ∑ . 290 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 290 45520_Matschke_Griffleiste_SL5.indd 290 16.03.2021 16: 22: 46 16.03.2021 16: 22: 46 Chapter 4 and also the respective comparative sizes and industry-specific multiples that are derived from experience or as rules of thumb (M ANDL / R ABEL 2019, p. 85). This is indicative for the lack of a theoretic foundation of these approaches. Such multiples are published in various business magazines as well as by industry and trade organizations. It is conceivable that the diffusion of the multiples method and the publishing of the multiples values have a normative effect in the sense of an orientation on well-known multiples ranges, but nevertheless, the performance of generating and justifying prices can only be hypothesized (L ORSON 2004, p. 225). Additionally, this approach is based on numerous simplifications and historical data, neglecting both the subject and future relation (B ALLWIESER 1991). Nevertheless, Figure 4.12 (K NIEF 2005, p. 1360) outlines a selection of multiples. Hence, the scope for argumentation resulting from the application of this approach is to some extent recognizable. Industry Smallcap EBIT Multiple from to Consultancy services Software Telecommunications Media 5,3 5,5 7,8 11,0 6,0 5,5 9,0 7,5 Trade/ E-Commerce Transport & Logistics Engineering/ Electronic Vehicle construction 4,5 4,0 8,0 6,8 5,0 5,4 11,0 8,2 Machinery/ Plant engineering Chemistry Pharma business Textile and clothing 3,5 4,5 5,0 8,0 7,0 3,0 9,9 5,0 Sales Multiple from to Midand Largecap EBIT Multiple from to 0,60 0,65 1,30 1,80 0,68 0,55 1,38 1,23 6,0 6,0 10,3 9,5 6,0 5,5 8,6 8,5 0,35 0,37 0,80 0,92 0,60 0,48 1,37 0,73 5,0 5,8 7,8 9,4 6,3 6,0 10,2 8,8 0,60 0,55 1,00 0,95 0,63 0,30 1,25 0,80 4,5 5,7 8,8 9,7 7,7 4,3 13,0 7,2 Food Gas, electricity, water Environmental technology/ Recycling Construction and crafts 4,6 4,3 8,0 8,7 4,0 2,5 7,3 6,0 Figure 4.12: EBIT and sales multiples for the business value (as of September 2004) 0,63 0,50 0,95 1,33 0,43 0,65 1,00 0,87 6,2 6,3 9,4 10,7 5,1 3,2 8,0 9,3 Sales Multiple from to 0,75 0,90 1,93 1,75 0,70 0,40 1,25 1,00 0,53 0,48 0,83 1,13 0,57 0,48 1,23 0,70 0,43 0,58 0,75 1,15 0,83 0,33 2,15 0,73 0,54 0,90 0,93 2,03 0,73 0,41 1,23 1,40 4.2 Value Determination 291 45520_Matschke_Griffleiste_SL5.indd 291 45520_Matschke_Griffleiste_SL5.indd 291 16.03.2021 16: 22: 46 16.03.2021 16: 22: 46 4.2.2.2 Finance-theoretic Methods 4.2.2.2.1 Capital Market-theoretic Methods (DCF Methods) 4.2.2.2.1.1 Basics Those valuation methods based on insights from finance theory are classified among the capital market theoretic valuation methods and in the so-called methods of strategic valuation. While the real options theory is categorized as a strategic valuation method, the capital market theoretic valuation methods primarily comprise various discounted cash flow methods (DCF methods). The DCF methods (K RUSCHWITZ / L ÖFFLER 2006) - like the investment theoretic partial model “future performance value” - are based on the present value calculus, as the expected future cash flows are discounted to the valuation date, thus, the DCF methods are not based on investment theoretic models (e.g., the net present value method) as assumed in literature. They do not necessarily equate because the derivation of the used discount rates used based on capital market theoretic equilibrium models. In addition, a hypothetical market value of equity is determined that should serve as the business value (M ANDL / R ABEL 2019, p. 78). Various authors (e.g., C OENENBERG / S CHULTZE 2002, p. 601) classify the DCF methods and the income value method to the methods of future performance determination. This vague terminological definition might cause misunderstandings because the DCF methods are not per se a form of the future performance value method. However, the common pattern of such methods is their representation of variants of the present value calculus. It is not helpful if the content-related differences are neglected by the application of the same terminology. Doing so inevitably creates misunderstandings. It is important to bear in mind that formal similarities do not lead to content-related conformity. The adjusted present value (APV) approach and the weighted average cost of capital (WACC) approach are gross methods (or entity approaches). These methods, alongside the flow to equity (FTE) approach, which is a net method (or an equity approach) are discussed below in detail. Regarding the consideration of business taxes, there are two distinguished variants of the WACC approach: The first formula determines the financing advantages of taxation in the denominator, that is, in the weighted cost of capital [also called the free cash flow approach (or the FCF method)]. In the second variant, the tax effects are considered in the numerator, that is, they are subtracted from the cash flows. The latter approach is known as the total cash flow approach (or the TCF method). Those two are referred to as gross methods because the determined “market value” of the total capital GK must be reduced by the “market value” of debt capital FK (German: Fremdkapital) to derive the required “equity (market) value”, EK (German: Eigenkapital), which corresponds to the required business value UW (German: Unternehmenswert) (Equity approach). It is determined as follows: EK = GK - FK with GK = EK + FK (GK; German: Gesamtkapital). In contrast, within the net method, the required “equity (market) value”, EK, is determined directly. Figure 4.13 (B ALLWIESER / H ACHMEISTER 2016, p. 137) systematizes and outlines the following DCF methods. According to the single DCF methods, the statements below include consequences of taxation and assume a simple profit tax (or cash flow tax) system, which imposes a homogenous tax rate on profits (or cash flows). Further argumentation options result 292 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 292 45520_Matschke_Griffleiste_SL5.indd 292 16.03.2021 16: 22: 47 16.03.2021 16: 22: 47 Chapter 4 from the possible inclusion of personal taxes in the respective methods (M ANDL / R ABEL 1997, p. 166, B ALLWIESER / H ACHMEISTER 2016, p. 137). However, with regard to the market value-oriented DCF methods, it should be noted that the desired objectification and the implicit assumption of full profit distribution and also the (actual) taxation with various subjective elements and the related (and required) individual determination of an optimum dividend policy represent an unbridgeable contradiction (W AMELING 2004, p. 85). The DCF methods are all characterized by a synthesis of the capital market approach by M ODIGLIANI / M ILLER and the capital asset pricing model (CAPM), which is used for the valuation of the required rate of return. The main features of the capital market models are restated. In the next step, statements (K RAG / K ASPERZAK 2000, p. 85) are made on the methods that essentially refer to the presentation of these approaches and less to the detailed critical appraisal of the basic assumptions with regard to the decision-oriented business valuation. Based on unrealistic assumptions, M ODIGLIANI / M ILLER (1958) showed that the capital structure of a business has no influence on its total capital market value in a specific risk category (thesis of irrelevance): Hence, under these circumstances, the market value of the total capital of a purely self-financed business GK e corresponds to the market value of the total capital of a leveraged (debt-financing) company GK f , namely: GK e = GK f . The market value of the total capital GK is generally an output of the sum of the market value of debt, FK, and the market value of equity, EK, thus: GK = FK + EK. Businesses that do not differ regarding their investment risk and their expected cash flow X, but - with respect to a different finance structure - do in terms of their financial risk, have equal total capital market values in the assumed (market) equilibrium. In this equilibrium, the market value of total capital results by discounting the cash flow X that is available for all investors, that is, both equity and debt providers. For clarity, it is assumed that X is a perpetuity discounted at a risk-adjusted rate of return k: Figure 4.13: Variants of DCF methods DCF-Verfahren Bruttoverfahren («Entity»-Verfahren) «Weighted Average Cost of Capital»-Ansatz (WACC-Verfahren) Steuerberücksichtigung im Nenner: «Free Cash Flow»-Verfahren (FCF-Methode) Steuerberücksichtigung im Zähler: «Total Cash Flow»-Verfahren (TCF-Methode) «Adjusted Present Value»-Verfahren (APV-Verfahren) Nettoverfahren («Equity»-Verfahren) «Flow to Equity-Methode» (FTE-Methode) DCF methods Gross method Net method Flow to equity (FTE approach) Adjusted Present Value (APV approach) Weighted Average Cost of Capital (WACC approach) Consideration of taxes in the denominator: Free cash flow (FCF approach) Consideration of taxes in the numerator: Total cash flow (TCF approach) 4.2 Value Determination 293 45520_Matschke_Griffleiste_SL5.indd 293 45520_Matschke_Griffleiste_SL5.indd 293 16.03.2021 16: 22: 47 16.03.2021 16: 22: 47 The result - transformed - for k: Since it was supposed that equal cash flows (i.e., X e = X f ) are valid for businesses within the same risk category GK e = GK f , the risk-adjusted discount rate k also has to be independent from the capital structure: k e = k f . If a business is (to at least some extent) leveraged, it utilizes the weighted average cost of capital (WACC). Then, the discount rate k corresponds to the weighted average of the cost of debt i and the required rate of return on equity r f : The consideration of the discount rate of a self-financed business k e and a transformation of this equation for r f exemplify the pathway of the return on equity r f because it increases linearly with a rising debt-equity ratio (also known as leverage). It is assumed that i < k e . As the expected rate of return on equity r f increases linearly with the leverage, it corresponds to the required return r e (= k e ) of a fully self-financed business: and an additional risk premium that results depending on the debt-equity ratio: However, the thesis of irrelevance concerning the debt-equity ratio is no longer valid if the restrictions of taxation regarding the equal taxation of both equity and debt capital are relaxed. If profits (before interest and taxes) X are subjected to a homogeneous profit tax s in a simple tax system, debt financing will increase the market value of the total capital. This is due to the fact that interest payments for the debt capital reduce the tax base. Therefore, the convergence of the market value of the total capital of an entirely self-financed business GK e and of that of an leveraged business GK f is no longer given (M ODIGLIANI / M ILLER 1963, p. 436). Since the insolvency risk is assumingly excluded, the tax benefits in the amount of s • i • FK (tax shield) are valuated with the risk-free interest rate i (K RAG / K ASPERZAK 2000, p. 89): The weighted average cost of capital are also affected by the possible tax deductibility. The cost of debt i (here: equivalent to the risk-free rate) is reduced by the term (1 - s). This means that interest payments are fully tax-deductible: The increase of the cost of equity in contrast to a world without taxation must also be corrected. Hence, the term (1 - s) is introduced so that it increases more slowly if the GK = FK + EK = X k . k = X GK = X FK + EK . WACC = k f = i ⋅ FK GK f + r f ⋅ EK GK f = k e . k e = i ⋅ 0 GK e = EK + r e ⋅ EK GK e = EK = r e r f = r e + r e − i ( ) ⋅ FK EK . GK f = X ⋅ 1 − s ( ) r s e + s ⋅ i ⋅ FK i = GK e + s ⋅ FK. WACC = k s f = i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f . 294 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 294 45520_Matschke_Griffleiste_SL5.indd 294 16.03.2021 16: 22: 49 16.03.2021 16: 22: 49 Chapter 4 leverage rises: According to the requirements of return for equity providers, the literature often referred to the CAPM. Again, it should be noted that this model is based on greatly idealized, but by no means equal assumptions (M ARKOWITZ 1952) as are the theses of irrelevance of M ODIGLIANI / M ILLER . Accordingly, the following contradiction is apparent: While the sizes of the DCF methods in the numerator are derived from (planned) annual financial statements and are thus at least somewhat subjective, the sizes in the denominator are derived in an objective way from the valuation object itself. Hence, the expected return r for the business U (German: Unternehmen) results from the risk-free interest rate i rf and a risk premium. The risk premium results in the equilibrium as the product of the so-called market risk premium, which is the difference between the expected market return r M and the risk-free interest rate i rf , and the object-specific beta factor β, which illustrates the systematic business-specific risk (i.e., the market-related risk of the valuation object) in comparison to the market portfolio: The beta factor β is the quotient of the covariance between the uncertain expected return of the valuation object for a single reporting period and the expected return of the market portfolio and the variance of return of the market portfolio: If the expected return of the valuation object corresponded to that of the market portfolio, hence at r = r M , the beta factor would amount to β = 1. The market portfolio results from a wide range of possible uncertain investments (shares, bonds, interests in non-listed companies like a limited liability company, real estates, exploitation rights, licenses, life assurances, gold, coins, stamps, art treasures, diamonds etc.) (H ERING 2017, p. 302). The non-systematic business-specific risk that arises due to competitive disadvantages and management errors could be eliminated by diversification by the investors in an efficient portfolio. Accordingly, it is not compensated by a risk premium. In summary, the CAPM requires the determination of β, i rf , and r M . According to the argumentation the requested data have to be derived or collected in an way acceptable to the negotiation partner. A pragmatic possibility for the determination of the cost of equity based on the CAPM is represented in Figure 4.14 (L ORSON 1999, p. 1330). Various sources are available for the valuation subject from which it can generate a range of different values of the single components. Further argumentation options offer considered or planned changes in the tax legislation, from which influences on the expected return of the investors may be assumed, according to the needs of the arguer. r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK . r business-specific ( ) return, expected by the equity providers & = i rf risk-free interest rate & + r M − i rf ( ) market risk premium ! " # $ # ⋅ β businessspecific risk & . β = σ r , r M σ r M 2 . 4.2 Value Determination 295 45520_Matschke_Griffleiste_SL5.indd 295 45520_Matschke_Griffleiste_SL5.indd 295 16.03.2021 16: 22: 51 16.03.2021 16: 22: 51 Alongside the dominating CAPM, “Arbitrage Pricing Theory” (APT) is used as an alternative approach to determine the risk-adequate cost of equity. Under APT, the systematic business risk is not aggregated and illustrated in the beta factor, but it is divided into a multitude of systematic risks. Additionally, the illustration of the market portfolio is waived. As the application of this method has not been established in business valuation and is thus limited in the scope of argumentation, it is not further explained here (K RAG / K ASPERZAK 2000, p. 96, H ERING 2014, p. 239). For further explanations see Section 4.2.2.2.1.3. As outlined below, valuators face a circularity problem if they apply DCF methods, that is, the determination of the required value or even the awareness, for instance at the determination of the weighted cost of capital is required. The issue leads proponents of these approaches to use a special device to reduce the complexity. In this context with the determination of the target capital structure and the determination of the debt capital, the DCF literature distinguishes between two ideal financing strategies. A common Figure 4.14: Pragmatic CAPM-based determination of the cost of equity r = irf + ( rM - irf ) · ß Risikoprämie des betrachteten Unternehmens Berechnungsformel Erläuterung Beispiel Gebräuchliche Datenquellen in Deutschland Rendite von Bundesanleihen (Laufzeit ≥ 10 Jahre) Aktienindex des Statistischen Bundesamtes DAX 30/ 80 empirische Studien Handelsblatt, Informationsdienste, Frankfurter Allgemeine Zeitung Operationalisierung Rendite festverzinslicher längerfristiger Wertpapiere von Emittenten unzweifelhafter Bonität (durchschnittliche) Normalrendite eines Marktindizes (geometrisches Mittel) (durchschnittliche) Renditedifferenz zur Entschädigung für das allgemeine unternehmerische Risiko Kovarianz der Aktienrendite des fokussierten Unternehmens mit der Rendite des Marktindizes (Renditeschwankungskoeffizient) systematisches Risiko des Unternehmens als Vielfaches des Marktrisikos Preis des Marktrisikos Rendite des Marktportefeuilles risikofreie Rendite risikoangepaßte Renditeforderung der Eigenkapitalgeber 11,6 % = 5 % + ( 11 % - 5 % ) · 1,1 Risk-adjusted rate of return on equity Risk-free rate Return of the market portfolio Market risk premium Systematic risk of the business as a multiple of the market risk (Average) difference in interest rates for compensation of general business risk Covariance of stock return of the focused business with the return of the market index Empirical studies Information services, newspapers Calculation Formula Explanation Operationalization Common data sources in Germany Return of long-term fixed-income securities of issuers of undoubted creditworthiness (Average) normal rate of return of a market index (geometric mean) Return of (≥ 10-year) German Treasury Bonds Stock index of the German Federal Statistical Office, DAX 30 Example 11,6 % = 5 % + (11 % - 5 %) • 1,1 r = i rf + (r M - i rf ) ú β Risk premium of the business in question 296 4 Argumentation Function and Argumentation Value Risk premium of the business in question 45520_Matschke_Griffleiste_SL5.indd 296 45520_Matschke_Griffleiste_SL5.indd 296 16.03.2021 16: 22: 55 16.03.2021 16: 22: 55 Chapter 4 feature of both is the absence of discussion on the reason these strategies should be pursued by the valuation subject or the respective management (I NSELBAG / K AUFOLD 1997, p. 114): 1. During the determination of a target capital structure, a fixed debt ratio FK/ GK f is given or assumed and consequently adhered to by the management. The debt ratio describes the relationship between the market value of debt capital FK and the market value of the total capital of the (leveraged) business GK f . As the strict observance of the target capital structure requires adjustment of the debt capital at alleged changes of the market value, this financing strategy is also called business valuedependent financing policy. The debt capital varies with the market value of the total capital. With regard to the consideration of business taxes in the models, uncertain tax benefits result due to uncertain interest payments. 2. In contrast, the determination of the debt capital assumes that the management of the business has predefined the debt capital due to contractually safeguarded agreements or owing to a commitment to financial sizes. This financing strategy is also referred to as autonomous financing policy because it does not depend on the business value. If the amount of debt capital is fixed, the interest payments are predetermined and the tax benefits are certain as long as the business can pay its interest and as long as the tax base is not already otherwise exhausted. The literature suggests a multitude of proposals to assist the determination of the cash flows that are relevant for business valuation within these approaches. Usually, the DCF methods use a cash flow before interest and taxes (CF) that is based on planned financial statements (both balance sheets and income statements [profit and loss statements]). Since payments are not explicitly regarded, this procedure is also known as the indirect method. Instead, gains (income, earnings) and expenses serve as the starting point for the cash flow derivation. With the help of various corrections (according to the specific proposals) that relate to non-cash expenses and income, a supposed cash flow amount is approximated. 4.2 Value Determination 297 45520_Matschke_Griffleiste_SL5.indd 297 45520_Matschke_Griffleiste_SL5.indd 297 16.03.2021 16: 22: 55 16.03.2021 16: 22: 55 In Figure 4.15 the proposal of K OLLER / G OEDHART / W ESSELS (2020) is presented. The free cash flow (FCF) represents the amount that is available to both equity and debt providers as an influx in a single period. According to K OLLER / G OEDHART / W ESSELS (2020) the FCF (9) is an outcome of the operating free cash flow (7) and the potential non-operating cash flow (8) of the business. The operating free cash flow (7) corresponds to the balance of operating gross cash flow (6) (operating result plus non-cash expenses) and net new investments (f). By using the elaborations of G ÜNTHER (1997, p. 112), who compares the 14 “most important discussed approaches in literature” of the free cash flow from his perspective, it is shown which scope of argumentation results regarding the determination of the free cash flow - with C OPELAND / K OLLER / M URRIN (now K OLLER / G OEDHART / W ESSELS ) as a quite popular example. The following Figure 4.16a and 4.16b, based on G ÜNTHER , present eight of these 14 approaches. Only those approaches that also referred to cash flows as free cash flows in the primary source have been considered. (1) Annual profit (2) (3) -/ + = Interest income/ interest expense Profit before interest after taxes (4) (5) (6) -/ + +/ - Increase/ decrease of tax liabilities Depreciation/ appreciation = Gross cash flow (a) (b) (c) (d) + Increase in working capital (amount of capital, which is invested in assets of net working capital by the business) Net new investments in fixed assets + +/ - Net new investments in goodwill Increase/ decrease of book values of other assets (e) (f) (6) -/ + = Increase/ decrease of other future interest-free liabilities Total net new investments Gross cash flow (f) (7) (8) (9) - = Total net new investments Operating free cash flow + = Non-operating cash flow Free Cash Flow (FCF) before financing Figure 4.15: Indirect determination of the free cash flow 298 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 298 45520_Matschke_Griffleiste_SL5.indd 298 16.03.2021 16: 22: 56 16.03.2021 16: 22: 56 Chapter 4 Author C OPELAND / K OLLER / M URRIN F ICKERT Year of publication Terminology 1990 free cash flow 1992 free cash flow Approach Definition Consideration of taxes Treatment of deferred taxes Anglo-Saxon operating result after taxes and before interest + depreciations - gross investments in fixed assets - increase in working capital following Anglo-Saxon terms profit before taxes b) - taxes - net investments in fixed assets - increase in net working capital (excluding liquidities) after taxes without deferred taxes after taxes no explicit reference Consideration of interest Basis of the result Extent of investments in fixed assets Extent of working capital before interest operating profit before interest operating profit replacement and expansion investments current assets a) less current liabilities replacement and expansion investments current assets (excluding cash and cash equivalents) less current liabilities and tax provisions a) Only the non-interest bearing part of the “working capital” is addressed explicitly. b) It can be inferred that taxes are not considered. c) This considers, for instance, imputed risks. d) Calculated from gross investments less imputed depreciation. H ACHMEISTER H ERTER 1995 free cash flow 1994 free cash flow continental European annual profit before taxes + interest expense + ∆special items with reserve elements + ∆pension provisions - ∆fixed assets - ∆working capital - cash taxes continental European operating result before taxes and after interest + ∆pension provisions + calculatory interest and additional costs c) - wealth and trade taxes - net investments in fixed assets d) - ∆net working capital after taxes without deferred taxes before income taxes without deferred taxes before interest annual profit before interest operating profit replacement and expansion investments current assets less non-interest-bearing debt capital replacement and expansion investments current assets less current non-interestbearing liabilities and provisions Figure 4.16a: Comparison of different free cash flow definitions 4.2 Value Determination 299 45520_Matschke_Griffleiste_SL5.indd 299 45520_Matschke_Griffleiste_SL5.indd 299 16.03.2021 16: 22: 56 16.03.2021 16: 22: 56 Even within the synoptically compared approaches in Figure 4.16a and 4.16b, which represent the free cash flow, here are significant differences regarding the definition of the free cash flow and in the terms used by the respective authors. According to the argumentation function, additional space for negotiation might be realizable from the following aspects: • the general treatment of taxation caused by different tax systems, • the consideration of interest (in particular at H ÖFNER / P OHL ), • the basis of the derivation of the free cash flow (in particular at H ACHMEISTER ), and • the extent of working capital. Author H ÖFNER / P OHL G OMEZ / W EBER Year of publication Terminology 1993 free cash flow 1989 - 1993 free cash flow (net cash flow) Approach Definition Consideration of taxes Treatment of deferred taxes continental European operating result before taxes and interest - interest expenses - taxes - investment e) continental European operating result before interest and after taxes + depreciations - net investments in fixed assets - investments in net working capital after taxes no explicit reference after taxes without deferred taxes Consideration of interest Basis of the result Extent of investments in fixed assets Extent of working capital after interest operating profit before interest operating profit not specified not specified replacement and expansion investments operational current assets less current liquidities (excluding cash and cash equivalents) e) Without further specification. f) The tax impact of the interest deduction is added back to the tax expenses. Figure 4.16b: Comparison of different free cash flow definitions S TEWART U NZEITIG / K ÖTHNER 1990 free cash flow 1995 free cash flow Anglo-Saxon operating result after taxes and before interest - increase in fixed assets - increase in working capital continental European operating result before taxes and interest + calculative costs + calculative depreciations + ∆long-term provisions - net investments - ∆stocks - ∆capital to be deducted after taxes f) without deferred taxes after taxes no explicit reference before interest operating profit before interest operating profit replacement and expansion investments not specified in more detail replacement and expansion investments stocks and inventories and current and non-interest-bearing debt capital e) 300 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 300 45520_Matschke_Griffleiste_SL5.indd 300 16.03.2021 16: 22: 57 16.03.2021 16: 22: 57 Chapter 4 In the negotiation, further potential for the arguer because practitioners often emphasize the notion that absolute academic rigor is not important at the determination because of the uncertainty of the future (G ÜNTHER 1997, p. 118). In contrast to the investment-theoretic valuation methods, it should be noted that in addition to the derivation of the cash flow from the planned financial statements the DCF methods consider cash flows that are available for distribution. Naturally, those cash flows do not have to correspond to the actual distribution (S CHULTZE 2003, p. 95). Figure 4.17 (B ALLWIESER / H ACHMEISTER 2016, p. 138) presents the relationships between the determined free cash flow and other cash flow variations (outlined in Figures 4.15 to 4.16). In Figure 4.17, the possibility of a direct cash flow determination is illustrated too (lines 1 to 3). The prognosis of the cash flows is generally organized in two phases. For the first phase, the so-called detailed phase, which encompasses a manageable period of three to five years, sufficiently detailed budget calculations are available. For this initial phase, several influencing variables are individually estimated to obtain the prognosis of the cash flows. The budget years of the distant period are usually based on more or less generalized forward projections of the detailed planning in the first phase. (1) Payments from the operating range (cash inflow) (2) (3) - = Payments from the operating range (cash outflow) Cash flow before interest after taxes (CF) (4) (5) (6) (7) - = Taxes from pure self-financing Operating cash flow (OCF) - -/ + Net amount of cash paid for investments and cash inflows from divestments Increase/ decrease in cash and cash equivalents (8) (9) (10) (11) = + Free cash flow (FCF) Business tax advantages due to deductibility of interest, so-called tax shield (TS) = - Total cash flow (TCF) Interest (12) (13) (14) Figure 4.17: Relationship between different cash flow terms + - Raising of credit (Borrowing) Loan repayment = Flow to equity (FTE) includes the tax burden under consideration of full self-financing; payable to equity and debt providers; is used in the first WACC approach (FCF method) and the APV approach. includes the tax burden with regard to a given financial structure; payable to equity and debt investors; is included in the second WACC approach (TCF method). determines the net cash flow (after interest and taxes) payable to the respective equity providers; is included in the equity approach. 4.2 Value Determination 301 45520_Matschke_Griffleiste_SL5.indd 301 45520_Matschke_Griffleiste_SL5.indd 301 16.03.2021 16: 22: 57 16.03.2021 16: 22: 57 4.2.2.2.1.2 Weighted Average Cost of Capital Approach The weighted average cost of capital (WACC) approach, a gross method, attempts to determine the “market value” of the total capital GK of the business by discounting the available free cash flows FCF with a weighted average cost of capital k (M ANDL / R ABEL 1997, p. 311, K RAG / K ASPERZAK 2000, p. 85, M ANDL / R ABEL 2019, p. 70). According to the type of collection of tax-based aspects; generally, two variants of the WACC approach can be distinguished: While the textbook formula of the WACC approach (defined as free cash flow) displays the tax-based financing benefits in the denominator, that is, in the discount rate k, the alternative approach of the total cash flow considers the tax effects in the numerator. In other words, they are subtracted from the relevant cash flows. Since the free cash flow approach considers tax-based financing advantages in the denominator (M ANDL / R ABEL 1997, p. 311), the free cash flow (FCF) has to be considered in the numerator. For the determination of the business value UW FCF (UW; German: Unternehmenswert) the expected free cash flow (FCF) that is available for all investors (and corresponds to that of a fictitious, entirely self-financed business, where usually the fictitious (income) taxes were considered which correspond to the ones of a self-financed business) is discounted with the adjusted weighted cost of capital rate and finally net of liabilities FK. This is because the relationship “market value of the total capital of the business = market value of equity + market value of debt” (GK = EK + FK) holds for this gross method: The weighted average cost of capital weight the tax-adjusted interest with the debt ratio FK/ GK f measured in “market values” and the required rate of return on equity with the equity ratio EK/ GK f , also expressed in “market values”. Hence, the sought-after value EK FCF (= EK) and the cost of capital are interdependent: Therefore, it can be written as: and if FCF is a perpetuity, the formula reads as follows: k s f UW FCF = EK FCF market value of equity of a leveraged firm ! "$ = FCF t 1 + k s f ( ) t t = 1 T ∑ market value of total capital GK = EK + FK ( ) ! " # $ # − FK market value of debt & . k s f (1 - s) ⋅ i r s f k s f k s f = i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK v . UW FCF = EK FCF = FCF t 1 + i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ t t = 1 T ∑ − FK UW FCF = EK FCF = FCF i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ − FK = FCF k s f − FK. 302 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 302 45520_Matschke_Griffleiste_SL5.indd 302 16.03.2021 16: 22: 59 16.03.2021 16: 22: 59 Chapter 4 If an estimation of the free cash flow is organized in two phases (phase 1 from t = 1 to τ ), UW FCF is determined as follows: The discounted cash flows concerning the end of the detailed phase (or the detailed planning horizon) are also defined as continuing value (CV) or terminal value (TV) (J UNG / M ANDL 2003, p. 45). The CV is calculated as follows: Since the resulting values UW (= EK) are already required for the determination of the weighted average cost of capital k, there is a circularity problem (H ERING / B RÖSEL 2004, p. 512). The cost of equity r included in the WACC must be known, but also the yet to be found debt-equity ratio also known as leverage (i.e., the relationship between the market value of debt and the market value of equity FK/ EK). If is inserted in , it results: In other words, through determination the debt ratio, the valuator has already “determined” (in some form) what is later independently “presented” to the negotiation partner as the main result. The valuator intends that the parter will astonished and impressed by this persuasiveness. This “magic trick” is most easily clarified by utilizing a perpetuity. Regardless of these interdependencies, advocates of the WACC approach recommend using the CAPM for the estimation of r and hence to pragmatically combine the multiperiodic WACC approach with a essentially one-period model. Example 4: Free Cash Flow Approach (Target Capital Structure FK/ GK f = 0,55) The determination of the argumentation value with the aid of the free cash flow approach is now presented with the aid of a transparent example: A simple profit tax system is assumed in which the annual profits at the end of each period are taxed with a UW FCF = EK FCF = FCF t 1 + k s f ( ) t t = 1 τ ∑ + FCF τ+ 1 k s f ⋅ 1 + k s f ( ) τ − FK = FCF t 1 + k s f ( ) t t = 1 τ ∑ + CV τ FCF 1 + k s f ( ) τ − FK. CV τ FCF = FCF τ+ 1 k s f . r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK k s f = i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f k s f = r s e ⋅ 1 − s ⋅ FK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ . UW FCF = EK FCF = FCF i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ − FK = FCF k s f − FK = GK f − FK. From FCF k s f ⋅ FK EK follows FCF k s f ⋅ 1 − FK GK f ⎡ ⎣⎢ ⎤ ⎦⎥ = GK f ⋅ 1 − FK GK f ⎡ ⎣⎢ ⎤ ⎦⎥ = GK f − FK. 4.2 Value Determination 303 45520_Matschke_Griffleiste_SL5.indd 303 45520_Matschke_Griffleiste_SL5.indd 303 16.03.2021 16: 23: 00 16.03.2021 16: 23: 00 constant tax rate of s = 50 %. Debt capital, which does not bear the risk of default, is borrowed at a (constant) certain interest rate of i = 8 % p. a. The constant rate of return on equity regarding an entirely equity financing is set at = 10 % p. a. The cash flows X t before interest and taxes amounts to 400 GE, 300 GE, and 500 GE in the first three periods (t = 1, 2 and 3). From the fourth period, a perpetuity of 400 GE is expected. The valuator assumes that the management aims for a target capital structure ZKS = FK/ GK f = 0,55 (ZKS; German: Zielkapitalstruktur). Conversely, if an autonomous financing policy was assumed at the WACC approach, the capital structure would change over time, which would theoretically lead to different period-specific required rates of return of the owners. This would inevitably increase the already existing circularity problem. At the target capital structure of 0,55, the cost of equity amounts to of 11,22 % p. a., which is the result of the calculation: Under consideration of the determined cost of equity of 11,22 % p. a. under the target capital structure, the WACC amounts to 7,25 % p. a. and can be calculated as follows: However, it can also be determined without explicitly considering interest rates: Before the business value can be calculated, the given cash flows before interest and taxes are transformed into the free cash flows: FCF t = X t · (1 - s). The sought UW FCF produces the given target capital structure as follows: Hence, the market value of equity amounts to 1.240,06 GE, whereas the total value of both equity and debt amounts to 1.240,06/ 0,44 = 2.755,68 GE. As mentioned, this “goal” can be attained more easily: r se r sf r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK = 0,1 + 0,1 − 0,08 ( ) ⋅ 1 − 0,5 ( ) ⋅ 0,55 0, 45 = 0,1122. r sf k s f = i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f = 0,08 ⋅ 1 − 0,5 ( ) ⋅ 0,55 + 0,1122 ⋅ 0, 45 = 0,0725. k s f = r s e ⋅ 1 − s ⋅ FK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ = 0,1 ⋅ 1 − 0,5 ⋅ 0,55 ( ) = 0,0725. UW FCF = 200 1,0725 + 150 1,0725 ( ) 2 + 250 1,0725 ( ) 3 + 200 0,0725 ⋅ 1,0725 ( ) 3 − 0,55 ⋅ GK f UW FCF = 186, 48 + 130, 40 + 202,65 + 2.236,15 − 0,55 ⋅ UW FCF 0, 45 2, 22 ⋅ UW FCF = 2.755,68 UW FCF = 1.240,06. FCF t (1 + k s f ) τ t = 1 τ ∑ + FCF τ+ 1 k s f ⋅ (1 + k s f ) τ ⎡ ⎣⎢⎢ ⎤ ⎦⎥⎥ ⋅ FK EK = FCF τ (1 + k s f ) τ t = 1 3 ∑ + FCF 3 + 1 k s f ⋅ (1 + k s f ) 3 ⎡ ⎣⎢⎢ ⎤ ⎦⎥⎥ ⋅ 1 − FK GK f ⎡ ⎣⎢ ⎤ ⎦⎥ 200 1,0725 + 150 1,0725 ( ) 2 + 250 1,0725 ( ) 3 + 200 0,0725 ⋅ 1,0725 ( ) 3 ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ ⋅ 1 − 0,55 ⎡⎣ ⎤⎦ = 1.240,06. 304 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 304 45520_Matschke_Griffleiste_SL5.indd 304 16.03.2021 16: 23: 02 16.03.2021 16: 23: 02 Chapter 4 Example 5: Free Cash Flow Approach (Calibration of the Beta Factor) The determination of in Example 4, the (constant) rate of return on equity of = 10 % p. a. under fully self-financing. This value plays a significant role in the following APV approach and causes serious difficulties during its determination because purely self-financed or unleveraged businesses do not usually exist. In this context, H ERING (2014, p. 266) points out that without considering the necessary relation formulated by M ODIGLIANI / M ILLER , the practice-oriented literature recommends estimating the cost of equity using the CAPM. However, there is no reason for the assumption that such a capital-market-oriented estimation produce to the same value as the M O- DIGLIANI -M ILLER formula (in the example: 11,22 % p. a.) since the M ODIGLIANI -M ILLER approach is based on other assumptions. By sticking to Example 4, the consultant advocating the WACC approach modifies the beta factor in such a way that = 12,5 % p. a. results for the valuation object. The presumptive owners subsequently decide to require this rate of return. To further accentuate the WACC approach the consultant continues to calculate: Finally, the calculation produces the business value of (only) 1.148,76 GE, which is derived from: H ERING (2014, p. 267) also expresses the following concern: Owing to the WACC approach not conceptionally providing the value = 10 % p. a. that is later included in the APV approach and determines in a different way, an accidental match with the results of other DCF variants seems rather unlikely. Conversely, how should or could APV user know to assume = 10,7931034 % p. a. in order to finally derive = 12,5 % p. a. and UW FCF = 1.148,76 GE. Hence, there is ample scope for argumentation. r sf r se r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK r sf r sf k s f = i ⋅ 1 − s ( ) ⋅ FK GK f + r s f ⋅ EK GK f = 0,08 ⋅ 1 − 0,5 ( ) ⋅ 0,55 + 0,125 ⋅ 0, 45 = 0,07825. UW FCF = 200 1,07825 + 150 1,07825 2 + 250 1,07825 3 + 200 0,07825 ⋅ 1,07825 3 − 0,55 ⋅ GK f UW FCF = 185, 49 + 129,03 + 199, 43 + 2.038,86 − 0,55 ⋅ UW FCF 0, 45 2, 22 ⋅ UW FCF = 2.552,79 UW FCF = 1.148,76. r se k sf r se r sf 4.2 Value Determination 305 45520_Matschke_Griffleiste_SL5.indd 305 45520_Matschke_Griffleiste_SL5.indd 305 16.03.2021 16: 23: 03 16.03.2021 16: 23: 03 Example 6: Free Cash Flow Approach (Target Capital Structure FK/ GK f = 0,6) In order to present the significance of choosing a particular target capital structure, Example 4 is slightly modified. It is now assumed - under ceteris-paribus-conditions - that the management aims for a target capital structure of ZKS = FK/ GK f = 0,6. This leads to the weighted average cost of capital of only 7 % p. a.: from which a business value of 1.141,71 GE results. The total market value of both equity and debt amounts to GK f = 1.141,71/ 0,4 = 2.854,29 GE. This is shown below: The modification of the (fictitious) target capital structure from 0,55 to 0,6 leads to a reduction of the business value from 1.240,06 GE to 1.141,71 GE. Example 7: Free Cash Flow Approach (Perpetuity) A third and last FCF example relates a business that generates a cash flow X of 400 GE before interest and taxes as a perpetuity. Moreover, a constant tax rate of s = 50 %, a (constant) certain interest rate of i = 8 % p. a., a required rate of return on equity at selffinancing of = 10 % p. a., and a target capital structure of ZKS = FK/ GK f = 0,55 are given. The direct determination of a weighted average cost of capital reads: Under consideration of FCF = X · (1 - s), the following business value is determined: After the comprehensive examples for the free cash flow approach, the second WACC variant - the total cash flow approach - is examined next (B ALLWIESER / H ACH- MEISTER 2016, p. 189). In this context, the tax-based (debt) financing advantages are regarded in the numerator. Thus, no adjustments need to be performed with respect to the k s f = r s e ⋅ 1 − s ⋅ FK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ = 0,1 ⋅ 1 − 0,5 ⋅ 0,6 ( ) = 0,07, UW FCF = 200 1,07 + 150 1,07 ( ) 2 + 250 1,07 ( ) 3 + 200 0,07 ⋅ 1,07 ( ) 3 − 0,6 ⋅ GK f UW FCF = 186,91 + 131,01 + 204,07 + 2.332, 28 − 0,6 ⋅ UW FCF 0, 4 2,5 ⋅ UW FCF = 2.854, 29 UW FCF = 1.141,71. r se k s f = r s e ⋅ 1 − s ⋅ FK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ = 0,1 ⋅ 1 − 0,5 ⋅ 0,55 ( ) = 0,0725. UW FCF = EK FCF = FCF k s f − FK = 200 0,0725 − 0,55 ⋅ UW FCF 0, 45 2, 22 ⋅ UW FCF = 2.758,62 UW FCF = 1.241,38. 306 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 306 45520_Matschke_Griffleiste_SL5.indd 306 16.03.2021 16: 23: 05 16.03.2021 16: 23: 05 Chapter 4 weighted average cost of capital. In other words, the term (1 - s) is omitted in the denominator: In the numerator, the total cash flow (TCF) is considered instead of free cash flow (FCF). The FCF is adjusted by the so-called tax shield (TS) that considers the tax deductibility of interest: For the determination of the business value UW TCF , the expected total cash flow TCF is discounted at the weighted average cost of capital k f . Once again, this rate does not consider the tax deductibility of interest. Finally, the TFC is corrected by the market value of the debt capital FK (Note: GK = EK + FK): In case that TCF is a perpetuity, it follows: If the total cash flow is organized in two phases (with phase 1 from t = 1 to τ ), the market value UW TCF can be calculated as follows: Example 8: Total Cash Flow Approach (Perpetuity) Since the TCF approach has no relevant significance in practice (B ALLWIESER / H ACH- MEISTER 2016, p. 192), only one example will illustrate that both variants of the WACC approach come to equal results if the valuators make equal assumptions. The cost of equity depends on the debt-equity ratio (leverage) and are calculated analogously to the FCF variant. Therefore, they amount to 11,22 % p. a.: Finally, UW TCF results after several calculation steps. Due to the consistent choice of parameters, it corresponds to UW FCF - in the amount of 1.241,38 GE - which was already calculated with the FCF variant of the WACC approach (cf. Example 7): k f = i ⋅ FK GK f + r s f ⋅ EK GK f . TCF = FCF + TS = FCF + s ⋅ i ⋅ FK. UW TCF = EK TCF market value of equity of a leveraged firm ! "$ = FCF t + s ⋅ i ⋅ FK 1 + i ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ t t = 1 T ∑ market value of total capital GK = EK + FK ( ) ! " ##### $ ##### − FK market value of debt & . UW TCF = EK TCF = FCF + s ⋅ i ⋅ FK i ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ − FK. UW TCF = EK TCF = TCF t + s ⋅ i ⋅ FK 1 + k f ( ) t t = 1 τ ∑ + TCF τ+ 1 + s ⋅ i ⋅ FK k f ⋅ 1 + k f ( ) τ − FK. r sf r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK = 0,1 + 0,1 − 0,08 ( ) ⋅ 1 − 0,5 ( ) ⋅ 0,55 0, 45 = 0,1122. UW TCF = EK TCF = FCF + s ⋅ i ⋅ FK i ⋅ FK GK f + r s f ⋅ EK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ − FK = 200 + 0,5 ⋅ 0,08 ⋅ FK 0,08 ⋅ 0,55 + 0,1122 ⋅ 0, 45 − FK 4.2 Value Determination 307 45520_Matschke_Griffleiste_SL5.indd 307 45520_Matschke_Griffleiste_SL5.indd 307 16.03.2021 16: 23: 07 16.03.2021 16: 23: 07 H ERING (2014, p. 268) states that if the assumption of a perpetuity is no longer valid, the variants of the WACC approach are exposed to the general problem of adjusting the model to period-specifically fluctuating cash flows. In order to continue the calculations with a constant WACC, the rather arbitrary assumption of a constant target capital structure is required at equal values of i, and s. Because UW and [and by the same token UW and are interdependent, the right way forward is an iterative solution of the valuation equations. Generally, the capital structure should not be determined by the idiosyncrasies of a model, but rather be the result of economic and rational decisions. Considering the intricateness, inconsistency, and the lack of a theoretical basis of the WACC approach, it is quite striking that many authors and consultants tend to prefer exactly this particular DCF variant. 4.2.2.2.1.3 Adjusted Present Value Approach The adjusted present value approach (APV approach) is another gross method. Use of the APV leads to a tax-adjusted market value of the total capital GK of the business being determined in the capital market equilibrium and takes into account of revised assumptions by M ODIGLIANI / M ILLER (1963). At APV approach also results in the (business) tax-related aspects are represented separately (M ANDL / R ABEL 1997, p. 372, B ALL- WIESER / H ACHMEISTER 2016, p. 139). If 1. future free cash flows FCF (corresponding to the FCF variant of the WACC approach) before interest after taxes at fully self-financing are expected as a perpetuity, 2. the cost of equity when entirely self-financing, 3. a business-oriented tax rate s and the market value of debt capital FK are given, the result is the business value UW APV that corresponds the market value of equity EK APV from a) the market value of a (theoretically) unleveraged firm business , b) the tax shield that arises due to the deductibility of interest and is discounted at the interest rate i and c) the debt FK that has to be substracted from the total capital GK (GK = EK + FK): UW TCF = 200 + 0,5 ⋅ 0,08 ⋅ FK 0,0945 − FK UW TCF = 200 0,0945 + 0,5 ⋅ 0,08 ⋅ 0,55 ⋅ UW TCF 0, 45 0,0945 − 0,55 ⋅ UW TCF 0, 45 UW TCF = 2.116, 40 + 0,52 ⋅ UW TCF − 1, 22 ⋅ UW TCF 1,70 ⋅ UW TCF = 2.116, 40 UW TCF = 1.241,38. r s f , k sf k f ] r se FCF / r s e s ⋅ i ⋅ FK UW APV = EK APV market value of equity of a leveraged firm ! "$ = FCF r s e + s ⋅ i ⋅ FK i − FK = FCF r s e market value of a theoretically unleveraged firm & + s ⋅ FK tax shield from debt financing & market value of total capital GK = EK + FK ( ) ! " ### $ ### − FK market value of debt & 308 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 308 45520_Matschke_Griffleiste_SL5.indd 308 16.03.2021 16: 23: 09 16.03.2021 16: 23: 09 Chapter 4 If the free cash flows are organized in two phases and phase 1 lasts from t = 1 to τ , UW APV is determined as follows: Since the debt-capital-induced tax shield increases the present value of the (hypothetically) unleveraged firm complete debt-financing maximizes the market value under the given assumptions. However, M ODIGLIANI / M ILLER (1963, p. 442) pointed out that a transfer into reality is neither appropriate nor feasible. For the value determination in the APV approach, the assumed cost of equity is determined from an immediately unknowable required rate of return on equity of an unleveraged firm. Because it is not usually possible to observe unleveraged firms on the market, research proposes deriving the required rate of return for an unleveraged business from that of a leveraged business (W AMELING 2004, p. 81). For this purpose, the representatives of this approach refer to the CAPM. Again, generally, the CAPM assumes riskaversion, neglect of taxes, and a planning horizon of only one period. Nonetheless, the CAPM is still combined with the APV approach that usually valuates multi-period payment streams (cash flows), both without preference and under the explicit consideration of taxes. According to the M ODIGLIANI -M ILLER model expanded by taxes, the question concerning an optimal (market-value-maximized) capital structure without fully debt-financing remains unanswered in the APV approach. Whether these problems can be conclusively solved by the capital market theory (e.g., by a convincing multi-period CAPM), remains to be seen. However, the current state of this theory has at least the potential to produce manifold specific varieties for the argumentation function of business valuation (H ERING 2014, p. 265). Example 9: Adjusted Present Value Approach (Perpetuity) As this model may nevertheless be used for argumentation in practice, due to its prevalence, its application is illustrated using the input data of the Examples 7 (for the FCF variant of the WACC approach) and 8 (for the TCF variant of the WACC approach) from Section 4.2.2.2.1.2. In this context, the given rate of return on equity of = 10 % p. a. is used. The result then reads as follows: Due to corresponding input data, a business value of 1.241,38 GE results from the APV approach (as the result of Example 9). Hence, the same business value that has al- UW APV = FCF r s e − 1 − s ( ) ⋅ FK. UW APV = FCF t 1 + r s e ( ) t t = 1 τ ∑ + FCF τ+ 1 r s e ⋅ 1 + r s e ( ) τ + s ⋅ i ⋅ FK t − 1 1 + i ( ) t t = 1 τ ∑ + s ⋅ i ⋅ FK τ i ⋅ 1 + i ( ) τ − FK. (FCF / r s e ), r s e r s e UW APV = FCF r s e − (1 − s) ⋅ FK = 200 0,1 − (1 − 0,5) ⋅ 0,55 ⋅ UW APV 0, 45 UW APV = 2.000 − 0,61 ⋅ UW APV 1,61 ⋅ UW APV = 2.000 UW APV = 1.241,38. 4.2 Value Determination 309 45520_Matschke_Griffleiste_SL5.indd 309 45520_Matschke_Griffleiste_SL5.indd 309 16.03.2021 16: 23: 10 16.03.2021 16: 23: 10 ready been calculated in Section 4.2.2.2.1.2 using the FCF variant (Example 7) and the TCF variant (Example 8) of the WACC approach has been attained. K RAG / K ASPERZAK (2000, p. 108) argue that if the valuation is based on a businessvalue-oriented financing policy, predetermined debt capital is not seen as ideal prerequisites for the solution in the APV approach. However, this does not have to correspond with the economic reality, according to the target system of the business. Moreover, fluctuating debt capital has to be expected because otherwise the compliance with the (constant) target capital structure is not assured. The debt capital is only known at a time when the valuation process is accomplished. The so-called roll-back approach might offer a solution. It is a retrograde, iterative calculation process. The problem is rolled back successively from the later points in time to the valuation date. Since the calculation of the business value, particularly for argumentation purposes, can be provided directly with the WACC approach without taking a detour, it is strongly recommended to use the WACC at a business-value-oriented financing policy. Examples 10 and 11: Adjusted Present Value Approach (Predefined Debt Capital) The original WACC Examples 4 and 6 are now modified insofar as management substitutes a target capital structure with a predefined amount of debt for each period. Figure 4.18 shows the cash flows X before interest and taxes as well as the respective predefined debt capital for the respective Examples 10 and 11. If the formula for the calculation of UW APV is divided into the components market value of an unleveraged firm EK e , value of tax benefits from financing TS, and market value of debt capital FK at the valuation date, it follows: The market value of an unleveraged business EK e is calculated in these examples under consideration of required rate of return on equity of = 10 % p. a. Since FCF is defined as FCF = X · (1- s), using the tax rate s = 0,5 leads to: Cash flow X before interest and taxes in GE Debt capital in GE (Example 10) t 0 t 1 400 400 450 Debt capital in GE (Example 11) Figure 4.18: Input data of the APV examples (predefined debt capital) 400 400 t 2 t 3 300 500 500 600 t 4 to ∞ 400 600 450 550 550 UW APV = EK e + TS − FK. r s e EK e = FCF t 1 + r s e ( ) t t = 1 τ ∑ + FCF τ+ 1 r s e ⋅ 1 + r s e ( ) τ EK e = 200 1,1 + 150 1,1 2 + 250 1,1 3 + 200 0,1 ⋅ 1,1 3 = 1.996, 24. 310 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 310 45520_Matschke_Griffleiste_SL5.indd 310 16.03.2021 16: 23: 12 16.03.2021 16: 23: 12 Chapter 4 The market value of the tax shield TS in Example 10 is calculated as follows using the tax rate s = 0,5 and the interest rate i = 0,08 p. a.: Therefore, Example 10 shows the following result at the valuation date t = 0: Due to equal input data, the amount of 1.996,24 GE can also be used as the market value of the unleveraged business EK e in Example 11. The market value of the tax shield TS is calculated under consideration of the modified debt capital: Since debt capital is lower, lower interest payments have to be made. However, this reduces the market value of the tax shield. All in all, it results a lower business value for the owners in Example 11 than in Example 10, just because the business raises less debt capital in the future: 4.2.2.2.1.4 Net Method (Flow to Equity Approach) The equity approach (M ANDL / R ABEL 1997, p. 367), also known as the flow to equity approach (FTE) approach, represents a net method where the market value of equity EK FTE is calculated directly. In other words, there is no need to calculate the market value of the total capital GK as a first step. Therefore, the starting point is a net cash flow, namely the flow of equity (FTE) that is solely available to the respective business owners.With the already used cash flow X before interest and taxes, the FTE is calculated by subtracting interest payments from X. The formula reads as follows: Under consideration of the relationship the following relation between the free cash flow (FCF) and the flow of equity (FTE) can be established: This (net) cash flow, which is also based on both the M ODIGLIANI -M ILLER approach and the CAPM, is discounted at the required rate of return on equity reflects both the financing risk that depends on the respective capital structure and the operating investment risk. If the flow to equity (FTE) is a perpetuity, the market value of equity TS = s ⋅ i ⋅ FK t − 1 (1 + i) t t = 1 τ ∑ + s ⋅ i ⋅ FK τ i ⋅ (1 + i) τ TS = 0,5 ⋅ 0,08 ⋅ 400 1,08 + 0,5 ⋅ 0,08 ⋅ 450 1,08 2 + 0,5 ⋅ 0,08 ⋅ 500 1,08 3 + 0,5 ⋅ 0,08 ⋅ 600 0,08 ⋅ 1,08 3 TS = 284, 27. UW APV = EK e + TS − FK = 1.996, 24 + 284, 27 − 400 UW APV = 1.880,51. TS = 0,5 ⋅ 0,08 ⋅ 400 1,08 + 0,5 ⋅ 0,08 ⋅ 400 1,08 2 + 0,5 ⋅ 0,08 ⋅ 450 1,08 3 + 0,5 ⋅ 0,08 ⋅ 550 0,08 ⋅ 1,08 3 TS = 261,13. UW APV = EK e + TS − FK = 1.996, 24 + 261,13 − 400 UW APV = 1.857,37. FTE t = X t − i ⋅ FK t ( ) ⋅ 1 − s ( ) . FCF t = X t ⋅ 1 − s ( ) , FTE t = FCF t − i ⋅ FK t ⋅ 1 − s ( ) . r s f . r s f 4.2 Value Determination 311 45520_Matschke_Griffleiste_SL5.indd 311 45520_Matschke_Griffleiste_SL5.indd 311 16.03.2021 16: 23: 13 16.03.2021 16: 23: 13 EK FTE is represented as follows: If the estimates of the relevant cash flows are organized in two phases, where the first phase is defined from t = 1 to τ , UW FTE is determined by: According to the assumed financing strategies, the following should be considered: If an autonomous financing policy (determination of a target debt capital) was used, the capital structure would change over time and this would theoretically necessitate period-specific rates of return on equity and circularity problems would inevitably occur, as with the WACC approach. In the case of the value-oriented financing policy (determination of a target capital structure), it would be required to adjust the market value of debt periodically in order to ensure adherence to the desired target capital structure. This would cause difficulties with the determination of the FTE because the valuer has to be informed about the tax and interest payments originating from the cash flows X. K RAG / K ASPERZAK (2000, p. 110) conclude that in contrast to both methods of the entity approach (gross methods) it is not possible to solve the circularity problem analytically applying the equity approach. This problem exists independent of the assumed financing policy. The proposed solutions discussed in the literature either aim for making use of other valuation approaches or deploying the roll-back method. If an idealized financing strategy has to be regarded, then in terms of the argumentation function it is best to directly apply a gross method. H ERING (2014, p. 270) points out that the problems of the net method might cause doubts regarding the practicability of this approach. Example 12: Equity Method (Flow to Equity Approach) The equity method is demonstrated with a single example. This is primarily due to the low proliferation of this method in practice and the insignificance in the context of the argumentation function. The example is known from the previous sections as Example 7 (for the FCF variant of the WACC approach), as Example 8 (for the TCF variant of the WACC approach) as well as Example 9 (for the APV approach). The business in question generates a cash flow X of 400 GE before interest and taxes as a perpetuity. Once again, a constant tax rate of s = 50 %, a (constant) certain interest rate of i = 8 % p. a., a required rate of return on equity of = 10 % p. a. at self-financing and the aimed target capital structure ZKS = FK/ GK f = 0,55 are given. Then, the required rate of return on equity is calculated analogously to the WACC approach and amounts to 11,22 % p. a.: UW FTE = EK FTE Market value of equity capital of an indebted business ! "$ = FTE r s f . UW FTE = FTE t 1 + r s f ( ) t t = 1 τ ∑ + FTE τ+ 1 r s f ⋅ 1 + r s f ( ) τ = FTE t 1 + r s f ( ) t t = 1 τ ∑ + CV τ FTE 1 + r s f ( ) τ . r s e r s f r s f = r s e + r s e − i ( ) ⋅ 1 − s ( ) ⋅ FK EK = 0,1 + 0,1 − 0,08 ( ) ⋅ 1 − 0,5 ( ) ⋅ 0,55 0, 45 = 0,1122. 312 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 312 45520_Matschke_Griffleiste_SL5.indd 312 16.03.2021 16: 23: 15 16.03.2021 16: 23: 15 Chapter 4 Finally, after several calculation steps the following business value results: 4.2.2.2.1.5 Summary Overview Figure 4.19 presents a summary overview about the main components of the DCF methods. The DCF methods are summarized and critically analyzed with regard to their objectives and assumptions, their choice for a capital structure, the correspondence (consistency) of the results of the different DCF methods, and the respective parameters (operands). Objectives and assumptions The DCF methods - as already thoroughly discussed in Section 1.2.2 - are not suitable for the determination of a decision value because they pursue the idealized objective of a market value maximizing at an assumed capital market equilibrium (H ERING 2014, p. 207). In summary, all three presented DCF methods are based on the heuristic synthesis of the capital market-theoretic approaches of the CAPM and by M ODIGLIANI / M ILLER , which combine different, but generally unrealistic assumptions. Due to the consideration of these approaches focused on idealized and hypothetical assumptions, the DCF methods suffer major shortcomings regarding the solution of real economic valuation problems. Moreover, the pragmatic combination of the CAPM and the M O- DIGLIANI -M ILLER approach causes further difficulties since their conceptional assumptions not only differ significantly from each other but are ultimately incompatible (H E- RING 2014, p. 280). UW FTE = FTE r s f = X − i ⋅ FK ( ) ⋅ 1 − s ( ) r s f = 400 − 0,08 ⋅ FK ( ) ⋅ 1 − 0,5 ( ) 0,1122 UW FTE = 400 − 0,08 ⋅ 0,55 ⋅ UW FTE 0, 45 ⎛ ⎝⎜ ⎞ ⎠⎟ ⋅ 0,5 0,1122 = 200 − 0,0489 ⋅ UW FTE 0,1122 UW FTE = 200 0,1122 − 0,0489 ⋅ UW FTE 0,1122 = 1.782,17 − 0, 4356 ⋅ UW FTE 1, 4356 ⋅ UW FTE = 1.782,17 UW FTE = 1.241,38. 4.2 Value Determination 313 45520_Matschke_Griffleiste_SL5.indd 313 45520_Matschke_Griffleiste_SL5.indd 313 16.03.2021 16: 23: 16 16.03.2021 16: 23: 16 Calculation of the market value of equity Calculation of the market value of total capital Valuation-relevant cash flow Calculation of the valuation-relevant cash flow from the cash flow X before interest and taxes Gross methods WACC approach FCF approach TCF approach Net method APV approach Equity approach (FTE approach) Relevant cost of capital Figure 4.19: Overview of the DCF methods Indirect determination EK FCF = GK FCF − FK GK FCF = FCF k sf or GK FCF = FCF t (1 + k sf ) t t = 1 τ ∑ + FCF τ+ 1 k sf ⋅ (1 + k sf ) τ Indirect determination EK TCF = GK TCF − FK GK TCF = TCF k f or GK TCF = TCF t (1 + k f ) t t = 1 τ ∑ + TCF τ+ 1 k f ⋅ (1 + k f ) τ Free Cash Flow (FCF t ) FCF t = X t ⋅ (1 − s) Total Cash Flow (TCF t ) = FCF t + tax benefits of the business due to deductibility of debt (FK) interest TCF t = FCF t + s ⋅ i ⋅ FK t − 1 TCF t = X t ⋅ (1 − s) + s ⋅ i ⋅ FK t − 1 Indirect determination EK APV = GK APV − FK GK APV = FCF r se + s ⋅ FK oder GK APV = FCF t (1 + r se ) t t = 1 τ ∑ + FCF τ+ 1 r se ⋅ (1 + r se ) τ + s ⋅ i ⋅ FK t − 1 (1 + i) t t = 1 τ ∑ + s ⋅ i ⋅ FK τ i ⋅ (1 + i) τ Direct determination EK FTE = FTE r sf or EK FTE = FTE t (1 + r sf ) t t = 1 τ ∑ + FTE τ+ 1 r sf ⋅ (1 + r sf ) τ Free Cash Flow (FCF t ) FCF t = X t ⋅ (1 − s) Flow to Equity (FTE t ) = net cash flow payable to the respective owner FTE t = (X t − i ⋅ FK t ) ⋅ (1 − s) k sf = i ⋅ (1 − s) ⋅ FK GK f + r sf ⋅ EK GK f or k sf = r se ⋅ 1 − s ⋅ FK GK f ⎛ ⎝⎜ ⎞ ⎠⎟ k f = i ⋅ FK GK f + r sf ⋅ EK GK f or k f = r se − s ⋅ FK GK f ⋅ (r se − i) r se and i r sf = r se + (r se − i) ⋅ (1 − s) 314 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 314 45520_Matschke_Griffleiste_SL5.indd 314 16.03.2021 16: 23: 19 16.03.2021 16: 23: 19 Chapter 4 Choice of the Capital Structure Even if the market value of the total capital of a business is independent of the capital structure (financing structure) in the M ODIGLIANI -M ILLER world with respect to the thesis of irrelevance, a certain capital structure has to be predefined for the determination of the respective discount rate (cost of capital) so that the weighted average of the cost of equity and the cost of debt can be established. This means that the total capital valuation approach requires the predetermination of a constant debt ratio, that is, the constant target capital structure (or predefined market values of the debt capital) based on the market values of equity and debt. In doing so the allocation of the market value of total capital to the market value of equity and the market value of debt is closed. In other words, the result is already anticipated through this assumption. A theory-based derivation of this predefined target capital structure is not possible (H ERING 2014, p. 273, H ERING 2005). Hence, the mere assumption becomes of vital importance for the determination of the cost of capital and finally for the allocation of the market value of total capital to equity capital (market value of equity capital) and debt capital (market value of debt capital). This can be compared with a magician pulling a rabbit out of a hat. The WACC supporters hide the result at first, which will be presented later on, namely in the approach of the weighted average cost of capital, and thus, must conceal this fact skillfully. The same holds for arguers using this approach. The discount rate is derived from the cost of capital of the valuation object and under consideration of a pseudo-objectified capital market relation. Hence, the models are self-referential and for this reason - and due to further deficiencies as already mentioned - unsuitable for decision purposes. In contrast, if investment-theoretic methods are applied, the discount rate is derived from the best alternative of acquisition or sale of the business to be valuated (optimal alternative investment, comparison object). This is the principal difference compared to the market-value-oriented valuation. With reference to the market value of the total capital, which purpose it is to serve remains unanswered, that is, for whom, in which context, and for which problem this size might be important. The purpose orientation of this calculation is entirely neglected. A valuation abstracting from the capital structure is already known from the history of (German) business valuation (K OLBE 1954, K OLBE 1959). The issue at stake then was the question of whether the so-called total capital performance or the equity capital performance should be taken as the basis. The supporters of the approach of a total capital performance wanted to avoid that the amount of the business value is dependent on “the composition of total capital from equity and debt capital” (M ÜNSTERMANN 1966, p. 40). The financing was regarded as a problem, “which is not directly associated with the valuation. The consequences of adequate or inadequate financing must not influence the total value of a business” (M ÜNSTERMANN 1966, p. 40). It must however be mentioned that the financing structure then was not based on market values, but on nominal values (book values). The idea of an optimal debt-equity ratio (or of a debt-ratio) played a rather subtle role. 4.2 Value Determination 315 45520_Matschke_Griffleiste_SL5.indd 315 45520_Matschke_Griffleiste_SL5.indd 315 16.03.2021 16: 23: 20 16.03.2021 16: 23: 20 The Correspondence of the Valuation Results of the DCF Methods In case of assuming the entity approach, the market value of equity EK results from the difference of the market value of total capital GK and the market value of debt FK: There is a close relationship to the equity approach that determines the market value of equity directly without using the market value of the total capital. However, both approaches can only be transformed into each other if a perpetuity model is assumed. This was already shown in the Examples 7, 8, 9, and 12. Now, it will be presented more generally. For simplicity the consideration of taxes is neglected: If the market value of the total capital is determined on the basis of a certain debt ratio, the market value of equity can also be determined by the multiplication of the market value of total capital with the equity ratio: GK = EK + FK or EK = GK − FK. EK = GK − FK = EK + FK ( ) − FK = X k f − FK or EK = X r f ⋅ EK GK + i ⋅ FK GK − FK or EK = GK − FK = X − i ⋅ FK r f + i ⋅ FK i ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ − FK or due to X − i ⋅ FK = X EK EK = X EK r f + FK ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ − FK = X EK r f = EK. EK = X EK r f ⋅ EK GK + i ⋅ FK GK ⋅ EK GK or EK = X EK ⋅ GK r f ⋅ EK + i ⋅ FK ⋅ EK GK or EK = X EK r f ⋅ EK + i ⋅ FK ⋅ EK or 316 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 316 45520_Matschke_Griffleiste_SL5.indd 316 16.03.2021 16: 23: 23 16.03.2021 16: 23: 23 Chapter 4 Based on the so-called sum model, that is, the explicit consideration of a muli-period context, correspondence can no longer be generally proven. Consequently, the identity of cannot be established if no specific (arbitrary) target capital structure is predefined. The equality can always be technically induced by iteratively adjusting the debt ratio or the discount rate as the weighted average cost of capital to a given periodical capital structure until the market value of equity based on the entity approach and the market value of equity based on the equity approach are equal. However, such a technically (i.e., iteratively) defined debt ratio or discount rate is virtually deprived of its economic content. Regarding the fact that the capital market-theoretic valuation methods seek objectivity, it is striking that each DCF method usually leads to different valuation results in practice (H ERING / V INCENTI 2004, p. 351). Hence, the purpose of such an approximation is clear: The focus is not directed toward a theoretical foundation, but solely on the ability to defend potential weak points in the valuation that might arise from different valuation results. The induced equality of the results should cover conceptional shortcomings. In other words, the same contents are always reproduced in different ways. This harmonization serves one purpose, to protect the argumentation basis. For management consultancies, this is an extraordinarily important purpose. Additionally, the effort to show the equality of valuation results for the market value of equity is already known in the history of (German) business valuation. The fact that the results of different approaches can be transformed into each other, that is, different valuation approaches lead to EK = X EK r f ⋅ EK EK + i ⋅ FK EK or EK = X EK + X FK r f + i ⋅ X FK i EK = X EK + X FK r f + i ⋅ X FK EK ⋅ i or EK = X EK + X FK r f + X FK EK = X EK + X FK r f ⋅ EK + X FK ⋅ EK or EK = X EK + X FK X EK + X FK ⋅ EK = X EK r f = EK. EK = GK − FK = X t 1 + k ( ) t − t = 1 ∞ ∑ FK = X 1 + r f ⋅ EK GK + i ⋅ FK GK ⎛ ⎝⎜ ⎞ ⎠⎟ t t = 1 ∞ ∑ − FK and EK = X t EK 1 + r f ( ) t t = 1 ∞ ∑ 4.2 Value Determination 317 45520_Matschke_Griffleiste_SL5.indd 317 45520_Matschke_Griffleiste_SL5.indd 317 16.03.2021 16: 23: 24 16.03.2021 16: 23: 24 the same results, does not per se provide information about the reasonableness of comparing these methods: In other words, this is no quality seal. In the German literature of business valuation, this effort culminated in J ACOB ’ S normal form (J ACOB 1960, J ACOB 1970). J ACOB has shown that different valuation methods might be represented in a common formula, according to which the business value is formally calculated as the arithmetical mean (average) of the income value EW and the net asset value SW: with a as the method-specific weighting factor. Parameter (Operand) The term cash is king was coined by C OPELAND / K OLLER / M URRIN (2000, p. 73) with regard to the parameters used in the Anglo-Saxon approaches. In the investment theory of German-speaking jurisdictions, the payment stream orientation is also long established. As already discussed in Section 2.3.1.2.1, L ÜCKE recognized in 1955 that the valuation can also be based on performance sizes if the calculatory interest rates on the capital commitment are appropriately assessed. The consideration of payment sizes, that is, cash flows, is not a progress compared to investment-theoretic valuation methods because they use cash flows too. However, the process is different (W AMELING 2004, p. 84): The discounted cash flow methods at first regard the value level business environment and adjust the predicted (operating) cash flows in several ways in the further steps in order to finally receive the distributable cash flows. By discounting the cash flows, a full distribution is implicitly assumed without a critical examination of this blanket assumption. In contrast, the investment-theoretic approaches aim to discern the expected cash flows or, more generally, the economic benefit (utility) that the valuation subject is able to generate due to interests in the valuation object. This comprises not only the payments between business and business owner, but also those of or to third parties such as tax payments and tax returns and also payments from selling pre-emptive rights. While the subjectively estimated cash flows are relevant to the investment-theoretic approaches, the relevant payment streams are essentially determined using blanket assumptions under the finance-theoretic approaches. Almost every author makes a different proposal of which parameter should be chosen as the basis for valuation. After the definition of a specific concept, the valuers have only a little leeway to opt for sizes that they deem appropriate and to make use of their own information state. The fixation on the (isolated) cash flow of the valuation object as the starting point of the analysis bears the risk of overlooking important and valuation-relevant financial relations between the valuation object and the valuation subject. Those relations might be based on individual production or financing synergies. Each formula of the free cash flow that is not individually adjusted risks interpreting the valuation object as an isolated unit and ignoring potentially numerous interdependencies of the buying or the selling business. The very different diagrams for the calculation of the cash flow, which are recommended by many consultants, represent a step backward. In this light, they fall back on the state where the relevant net cash flow is individually based on the incorporation of the valuation object in a subjective decision field (H ERING 1999, p. 103). UW = (1 − a ) ⋅ SW + a ⋅ EW = SW + a ⋅ (EW − SW) 318 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 318 45520_Matschke_Griffleiste_SL5.indd 318 16.03.2021 16: 23: 25 16.03.2021 16: 23: 25 Chapter 4 4.2.2.2.2 Methods of Strategic Valuation For some time, different approaches of so-called strategic valuation have been discussed in the literature. They are based on the real options pricing theory (M YERS 1968, F ISCHER / H AHNENSTEIN / H EITZER 1999, K OCH 1999). In a valuation based on the option pricing theory, the business value (UW; German: Unternehmenswert) is determined as the sum of the so-called underlying (GW; German: Grundwert), a base value, and the option value (OW; German: Optionswert) in the sense of a strategic premium as follows: UW = GW + OW. This is why these approaches are also referred to as strategic business valuation methods. Advocates of this approach hold the opinion that traditional valuation methods produce values that are too low because the action alternatives that a buyer has in the context of the acquisition are not adequately considered (H ERTER 1992). The term real option was coined by M YERS (1977) in an effort to quantify the immaterial value components of businesses. M YERS determined the business value as the sum of real assets on the one hand and the connected real options on the other. During the development of the real options theory, four significant concepts can be recognized. These include the perspectives of real options as (M ÜLLER 2005, p. 33): 1. parts of the goodwill, 2. flexibility of a participant to arrange an irreversible action, 3. investment projects or objects with the character of options, and 4. strategic heuristics. Instead of undertaking a critical examination of the available instruments of this valuation theory and of (sometimes extremely high) stock prices, advocates of this approach sought explanations for the high market valuations (W AMELING 2004, p. 92). Especially in the context of the acquisition of young businesses in growing industries; such an acquisition increases the adaptability of the valuation subject to changing environmental conditions, for instance, by options of modification, delay, expansion, and/ or termination (K RAG / K ASPERZAK 2000, p. 117). Hence, the flexibility acquired should be considered during the value determination in the form of a strategic premium (M YERS 1984, p. 134). If the possibility of a free decision under uncertainty about the realization of irreversible investment action is available for the participant, analogies exist between the action situation of a potential investment and a call option on a share. This is why the action situations of a possible investment are also referred to as real options. The decision-maker has the right to acquire assets, but no obligation to do so (shares at the stock option or the present value of a real option) for a determined price (exercise price or strike price at the stock option or cash flows from the investment at the real option) at a specific moment in time (expiration date). An investment option is given. The valuation of this action oppportunity relies strongly on the possibility that the investment could be delayed and need not necessarily be realized, meaning the investment opportunity corresponds to the right, but not an obligation, to invest. If an analogy between the stock options and real action scenarios is assumed, the value of the action opportunity is determined by using valuation methods based on (financial) option pricing theory. The background to this assumption is that the valuation object reveals action opportunities to the valuation subject. The valuation subject can use these options depending on the occurrence of specific states, but there is no obligation to do so. This process is analogous to trading on the stock exchange: In addition to trading securities like shares, of the market also trades in derivatives like options. Option businesses comprise two 4.2 Value Determination 319 45520_Matschke_Griffleiste_SL5.indd 319 45520_Matschke_Griffleiste_SL5.indd 319 16.03.2021 16: 23: 25 16.03.2021 16: 23: 25 components, a so-called strike price that is valid for the transaction for a certain period (expiration date or maturity date) and a price for the option right (option price) itself, which is due at the time of the transaction. Analysts distinguish between a call option and a put option. The buyer of a call option generally wants to buy the shares and has the choice of acquiring the securities at the target date and paying the underlying strike price or allowing the option to lapse. By paying the option price, the buyer acquires the right from the seller of the option to demand the delivery of a predetermined number of shares at the strike price within the option period. Conversely, the buyer of a put option wants to sell the shares. By paying the put option price, the buyer acquires the right from the seller of the put option to demand the purchase of a certain amount of shares within the option period. The option business is organized in two steps at the stock exchange: 1. Acquisition (sale) of an option right and payment of the option price by the buyer and 2. Utilitization or non-utilitization of the option right by the buyer; for utilitization: Acquisition or sale of the shares and payment of the strike price (depending on whether it is a call option or a put option). The representatives of the strategic valuation approach transfer this concept to real investment project (real options). Growth options (which are interpreted as call options), change options (call options or put options), or divestment options (put options) are associated with business acquisitions; however, that approach ignores those options not having underlying shares that are, in turn, not traded on a regulated market. The underlying (GW) is calculated with a valuation method based on the present value calculus. The expected future cash flows are discounted to the valuation date. Regarding the argumentation function, both investmentand capital market-theoretic methods could be used in this case. However, it is assumed that the capital market approaches (DCF methods) are more plausible in practice, especially because the strategic valuation itself is market-value-oriented (K RAG / K ASPERZAK 2000, p. 122). In contrast, the option value (OW) is determined by resorting to models that are based on the principle of arbitrage-free valuation and hence on the assumption of the general equilibrium theory (G ILLES / L E R OY 1991, C AMPBELL / L O / M C K INLAY 1997, H ERING 2014, p. 239). Under restrictive and idealized conditions, equal prices are given for equal asset positions. Any option (in the sense of a state-dependent cash flow) can be replicated (duplicated) by a portfolio of traded securities (cash flows) on the market. The portfolio generates equal (state-dependent) cash flows as the option itself. The price for the replication is P*. No investor would pay more than the price P* for the option under the assumed conditions. P* corresponds to the full replication value of a cash flow. However, no buyer of the option would be satisfied with less than the price P*. If the restrictions of the arbitrage-free valuation are maintained, the option value (OW) equals the price for the portfolio that replicated the option. In contrast to the pricing theory of securities (CAPM), the arbitrage-free valuation is both preference-free and distribution-free. Subjective probabilities of occurrence and risk-utility functions are not required; only the assumption of financial non-saturation (the more the better) is valid. However, the infinitely large uncertain state space only permits the arbitrage-free valuation of risky payment streams in a heuristic way. Hence, the valuation based on option pricing models is indeed preference-free, but the state space is reduced to an arithmetically manageable level by strict assumptions regarding the distribution. The heuristic simplification of these option models is because the arbi- 320 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 320 45520_Matschke_Griffleiste_SL5.indd 320 16.03.2021 16: 23: 26 16.03.2021 16: 23: 26 Chapter 4 trage-free valuation is limited to a reasonably defined section of the entire state space (H ERING 2014, p. 257). With regard to the assumptions of distribution, the model of C OX / R OSS / R UBINSTEIN and the B LACK -S CHOLES model are most often used. The model of C OX / R OSS / R UBIN- STEIN (1979) assumes a binomial distribution and is hence referred to as the binomial model. The period under consideration is divided into equal intervals t. The model assumes a discrete stochastic process for the uncertain development of the underlying share (present value of the expected cash flows). The underlying can either decrease or increase within an interval t by a (possibly divergent) fixed percentage. Alternative states are not possible. This means that the underlying can only have two different values in each (state-dependent) interval and that it either moves upward or downward (cf. Figure 4.20). In this figure (M ÜLLER 2004, p. 124) S represents the strike price of the underlying, whereas u indicates the possible growth factor at a positive development and d the factor at a negative development (with u > 1 and d < 1). After modeling the development of the underlying, the option value is determined starting at the end of the binomial tree (recursively) (E RNST 2002, p. 19). In this way, the decision tree is solved backward to the starting point. The so-called maximum value standard has to be maintained so that the option value from the previous period is determined from the maximum value of the option values at the exercising and non-exercising (abandoning) of the following period. Both the so-called American options and European options can be valuated with this model. While American options can be exercised (at any time) during the period, European options can only be exercised at the end of the period. Figure 4.20: Price movements in the binomial tree over n periods t = 0 t = 1 t = 2 S · u 1 · u 2 S · u 1 S S · d 1 S · u 1 · d 2 S · d 1 · u 2 S · d 1 · d 2 … … … … … … … … t = n 4.2 Value Determination 321 45520_Matschke_Griffleiste_SL5.indd 321 45520_Matschke_Griffleiste_SL5.indd 321 16.03.2021 16: 23: 27 16.03.2021 16: 23: 27 The B LACK -S CHOLES model (1973) supposes - based on the so-called W IENER process - a continuous stochastic value development of the underlying value and marks (under certain conditions) a continuous marginal case of the (time-discrete) binomial model. The W IENER process is the basis for the geometric B ROWNIAN motion which is crucial in the B LACK -S CHOLES model. Particles (e.g., pollen) applied to a drop of water move randomly. This was first observed in 1827 by the botanist B ROWN and mathematically examined in 1923 by the mathematician W IENER , and consequently the terms W IENER process and B ROWNIAN motion are often used synonymously (M ÜLLER 2004, p.116). The major idea of both models is the construction of a portfolio with value development that is independent of the uncertain prices of the underlying. Therefore, it can be discounted at the risk-free interest rate (H ERING 2014, p. 257). However, the B LACK -S CHOLES model is only suitable for the valuation of European options. The calculation of the business value UW using strategic valuation is illustrated below with a simple example (M ANDL / R ABEL 1997, p. 56). The valuation of a business to be acquired should be performed at period t = 0, the potential acquisition date. By using the DCF method, the valuation subject assessed an underlying (GW) in the amount of 300 GE in advance. Furthermore, the valuation subject identifies a scope of action (option) and determines its value (option value OW) in order to calculate the business value based on this relation UW = GW + OW. The following data are provided: If the valuation subject acquired the company, they would have the opportunity (option) to enter an innovative business segment at time t = 1. For this purpose, a single one-time investment payout of 90 GE is required at t = 1, which represents the underlying of the option. Whether the option was exercised by the valuer at t = 1 depends on the developments (states) in period 1, the time between t = 0 and t = 1. Principally, two scenarios are assumed at the valuation date; an optimistic scenario A and a pessimistic scenario B. In scenario A, if positive developments come into effect, a discounted cash flow of 150 GE will be available to the valuation subject at t = 1. However, if negative developments occur in period 1 (scenario B), a discounted cash flow of only 60 GE results at t = 1. Under such unfavorable market conditions, the valuation subject will not realize the investment at t = 1, due to the initial investment payout of 90 GE. Now, the valuation subject has to generate a portfolio with a similar risk structure, which correlates perfectly with the business. The existence of a share is assumed, with a price of 3,5 GE per share at t = 0, expecting a price of 6 GE per share at t = 0 at favorable market conditions and a price of 2 GE per share at unfavorable market development. Additionally, it is possible for the valuation subject to raise a credit for a single period at an interest rate i = 8 % p. a. The valuation subject has to compose a portfolio with the given stock and the fictitious borrowing, which generates a cash flow of 60 GE at t = 1 at favorable market conditions and a cash flow of 0 GE at t = 0 at unfavorable market conditions. Hence, the following conditions must be fulfilled, where A represents the number of shares and K symbolizes the credit amount: • at favorable market conditions: • at unfavorable market conditions: Assuming the aforementioned interest rate of i = 8 % p. a., the solution of this equation is A = 15 and K = 27,78 GE. This means that the portfolio consists of 15 shares and a credit amount of 27,78 GE at t = 0 and generates the same result as the option that is bought along with the business: A ⋅ 6 − K ⋅ (1 + i) = 60, A ⋅ 2 − K ⋅ (1 + i) = 0. 322 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 322 45520_Matschke_Griffleiste_SL5.indd 322 16.03.2021 16: 23: 28 16.03.2021 16: 23: 28 Chapter 4 • at favorable market conditions: • at unfavorable market conditions: This portfolio exactly replicates the option that is connected to the acquisition. Thus, the option must have the same value as the portfolio on the capital market. The described portfolio currently costs 24,72 GE: Thus, the option also has a value of 24,72 GE under the specified assumptions. Hence, the value of the (real) option represents the strategic premium. This value is added to the underlying value at the strategic valuation. In the example, the business value UW results as follows: UW = GW + OW = 300 + 24,72 = 324,72. The advantage of this model is primarily the fact that no probabilities of occurrence for a favorable or unfavorable development of the market and also no risk-adjusted interest rates are required. However, a business valuation based on option pricing theory presents serious method-specific deficiencies, resulting in numerous grounds for criticism (D IRRIGL 1994, H ERING 2014, p. 293): • Firstly, according to the assumption of a complete and perfect market, assumptions are made that are highly unrealistic. However, in order to assess a fair market prices of real options, these markets are essentially required. While the trading of stock options occurs on markets that are characterized by explicit regulations, implicit usages, standardized products, and an exchange supervisory authority like the SEC, real options are traded on highly imperfect markets, if they are traded at all (T RIGEORGIS 1996, p. 128). • Secondly, the artificial division of the value in an underlying value and an option value represents a methodically unnecessary differentiation because upon an appropriate application of the investment-theoretic methods presented in the second chapter - possibly extended by a flexible planning - it can be assumed that all valuation objects are real options in the sense of state-dependent payment streams. Hence, the revealed action options of the buyer are included in the value determination according to their significance. • Thirdly, the rather rigid assumptions regarding the distribution proposed in literature are another shortcoming: It is not apparent who guarantees to the valuation subject that the distribution assumed in these models can be found in reality and that it remains constant over time. The type of postulated stochastics of returns ranges from the assumption of binomial and trinomial decision trees to the modeling of time-continuous processes with or without so-called jumps. The already mentioned geometric B ROWNIAN motion describes a log-normal distribution, which cannot always be found in reality. Moreover, the geometric B ROWNIAN motion only permits positive quotations that are also not always given in reality. • Fourthly, the transferability of stock option models to real decision situations should be questioned generally. In the contract of stock options, the main parameters are determined. Conversely, contractual fixing of real options occurs quite seldom and thus, the major parameters and the influence of changes in the period are not determined. In negotiations, the consideration of optional (additional) components of the contract is usually possible through specific stipulations and the corresponding instruments (e.g., debtor warrant, option rights, or rights of withdrawal). If there is no contracting party, limitations relating to the writer (i.e., the seller of options) are vitally important. Upon conclusion of the contract, stock options are only unilaterally binding, whereas real options are not. 15 ⋅ 6 − 27,78 ⋅ (1 + 0,08) = 60, 15 ⋅ 2 − 27,78 ⋅ (1 + 0,08) = 0. 15 ⋅ 3,5 − 27,78 = 24,72. 4.2 Value Determination 323 45520_Matschke_Griffleiste_SL5.indd 323 45520_Matschke_Griffleiste_SL5.indd 323 16.03.2021 16: 23: 28 16.03.2021 16: 23: 28 • Fifthly, it is doubtful whether the valuation subject is capable of constructing a portfolio (with a similiar risk structure) that correlates perfectly with the business to be valuated. • Last but not least, one has to consider that this method is rarely applied in practice, which is primarily due to the complexity and missing operationalizability of the quite abstract option models (E RNST 2002, p. 17). Considering these flaws, the methods of option pricing theory are not suited for decision value determination; but due to the large scope of manipulations, they can serve as argumentation values. However, a consideration of strategic valuation within the context of the arbitration function of business valuation would be questionable. Then, the valuator could also play dice, according to H ERING (2014, p. 296). For the reasons stated, there is great danger of misapplication and misinterpretation of models referring to the real options’ valuation. This difficulty in interpretation, however, facilitates the application of those models within the argumentation function, assuming the ignorance of the negotiation partner regarding the aforementioned problems. 324 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 324 45520_Matschke_Griffleiste_SL5.indd 324 16.03.2021 16: 23: 29 16.03.2021 16: 23: 29 Chapter 4 4.3 Selected Control Questions Exercise 1 (40 Points) - Argumentation Values a) Define the term argumentation value. (4 points) b) Systematize the features of the argumentation value and discuss at least four features in detail. (14 points) c) How and with which objectives can argumentation values be used in internal conflicts? (10 points) d) The application of argumentation values is associated with the intent to influence. Explain the characteristics of this purpose with regard to negotiations with external conflicting parties. How can it be ensured that argumentation values do not represent instruments of overreaching? (12 points) Exercise 2 (20 Points) - Determination of Argumentation Values within the Matrix of Functional Business Valuation a) Explain the steps of the argumentation value determination within the matrix of functional business valuation. (9 points) b) Systematize the argumentation factors with respect to business valuation. (6 points) c) Discuss, how complexity might be reduced by the substitution of original conflictresolution-relevant facts by several derivative conflict-resolution-relevant facts. (10 points) d) What is the significance of substitution in the argumentation function? (5 points) Exercise 3 (10 Points) - Comparison Object Compare critically the so-called comparison objects of the investment-theoretic valuation methods to the comparison objects of the market-price-oriented methods and of the market-value-oriented methods. Exercise 4 (20 Points) - Stock-and-Debt Method a) Discuss the stock-and-debt method (stock valuation). Which argumentation options result from using this method? (8 points) b) A business has 650.000 issued shares (AA BO ) in total. The price per share (AK BO ) at the valuation date is 85 GE. The average price of the last three months is 105 GE, the average price of the last month is 90 GE. Show the scope for argumentation of the valuation subject if control premiums (PZ BO ) of 30 % and 40 % respectively are considered. Determine the respective argumentation values. (12 points) 4.3 Selected Control Questions 325 45520_Matschke_Griffleiste_SL5.indd 325 45520_Matschke_Griffleiste_SL5.indd 325 16.03.2021 16: 23: 29 16.03.2021 16: 23: 29 Exercise 5 (40 Points) - Similar Public Company Approach a) Explain similar public company approach. Which argumentation options result from applying this approach? (8 points) b) Explain the significance of discounts for lack of marketability and control premiums. (4 points) c) Which parameters can the valuator adjust by using this method? List six parameters and discuss them briefly. (9 points) d) In what ways can a comparison business be selected? (4 points) e) By preparing the negotiations of the valuation object BO, you have identified an allegedly comparable business VU. Both its stock market price and the number of issued shares are constant for months. The market capitalization (MK VU ) amounts to 4.000.000 GE. The sales revenue (U) and dividends (D) are considered as possible performance indicators. While the sales of the comparison object amount to U VU 3.000.000 GE, the sales of the valuation object U BO amount to 4.300.000 GE. To date, dividends amounted to 450.000 GE (D VU ) and to 400.000 GE (D BO ) respectively. You consider calculating your argumentation values with a control premium of 35 % and with respective discounts for lack of marketability of 40 % and 45 %. With regard to the performance indicators, you consider the following options: (a) only sales, (b) only dividend, and (c) all performance indicators at equal shares. Determine the argumentation values based on these scenarios. (15 points) Exercise 6 (50 Points) - Finance-theoretic Valuation Methods a) Systematize the variants of the DCF method. Briefly discuss three of the four methods. (20 points) b) Which assumptions are the basis of the thesis of irrelevance concerning the leverage? (5 points) c) On which assumptions is the CAPM based on? Explain the determination of the cost of equity within the CAPM and analyze its usage within other approaches that are based on assumptions of the M ODIGLIANI -M ILLER world. (15 points) d) Elucidate the method of strategic valuation and critically comment on its decisionoriented application. (10 points) Exercise 7 (55 Points) - Capital Market-theoretic Valuation Methods a) Discuss the determination of free cash flows. Explain the relationships between the different types of cash flows and their application within the DCF methods. Compare these parameters with those used in the investment-theoretic methods. (15 points) b) The business to be valuated is subject of a simple profit tax system, in which a constant tax rate of s = 40 % is imposed on the annual profits at the end of the period. Debt capital does not bear the risk of default, it is risk-free. The (constant) interest rate amounts to i = 6 % p. a., the (constant) required rate of return on equity at selffinancing amounts to = 9 % p. a. The cash flows X t before interest and taxes are (t = 1, 2, and 3): 400 GE, 600 GE, and 400 GE in the first three periods. From the fourth period, a perpetuity of 600 GE is expected. As a valuer, you assume that the management aims for a target capital structure ZKS - the debt ratio - of ZKS r se 326 4 Argumentation Function and Argumentation Value 45520_Matschke_Griffleiste_SL5.indd 326 45520_Matschke_Griffleiste_SL5.indd 326 16.03.2021 16: 23: 30 16.03.2021 16: 23: 30 Chapter 4 = FK / GK = 0,7. Determine the market value of equity EK using the free cash flow approach (FCF approach). (10 points) c) Now, you should valuate a business with the total cash flow approach (TCF approach). It generates a perpetual cash flow X of 300 GE before interest and taxes. The (constant) tax rate is s = 40 %, debt capital can be borrowed at a (constant) certain interest rate of i = 7 % p. a., the required rate of return on equity at entirely self-financing amounts to = 10 % p. a., and the target capital structure is ZKS = FK/ GK = 0,65. (10 points) d) Calculate the market value of equity of a business using the APV approach. Additionally, you have the following information: the tax rate is constant at s = 40 %, the constant interest rate amounts to i = 8 % p. a., and the required rate of return on equity at entirely self-financing is = 10 % p. a. However, instead of a target capital structure, the amount of debt capital is predetermined for each period. Figure 4.21 illustrates the cash flows X before interest and taxes and the debt capital. (15 points) e) Determine the business value with the equity approach (net method) based on the information provided in c). (5 points) Exercise 8 (10 Points) - Methods of Strategic Business Valuation Briefly present the method of strategic business valuation. Which criticisms arise from the perspective of the decision-oriented business valuation? r se r se Cash flow X before interest and taxes Debt capital t 0 t 1 200 200 250 Figure 4.21: Exercise example for the APV approach t 2 t 3 150 250 250 300 t 4 to ∞ 200 1 4.3 Selected Control Questions 327 45520_Matschke_Griffleiste_SL5.indd 327 45520_Matschke_Griffleiste_SL5.indd 327 16.03.2021 16: 23: 30 16.03.2021 16: 23: 30 45520_Matschke_Griffleiste_SL5.indd 328 45520_Matschke_Griffleiste_SL5.indd 328 16.03.2021 16: 23: 30 16.03.2021 16: 23: 30 Chapter 5: Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 329 45520_Matschke_Griffleiste_SL5.indd 329 16.03.2021 16: 23: 30 16.03.2021 16: 23: 30 Overview The fifth and final chapter discusses the principles of business valuation. These principles are supposed to guide and control the business valuation process in the sense of a supporting a consistent normative system (a system of norms or a standard system). The characteristics of that normative system are outlined at the beginning of Section 5.1. In the next step, the purposes the principles facilitate are presented in detail. At the end of Section 5.1, an overview of possible sources of the principles is outlined. Finally, in Section 5.2 principles of functional business valuation are determined deductively, based on functional business valuation theory. At the end of the chapter (Section 5.3), selected questions are presented to reinforce thorough study of the topic. Learning objectives After studying this chapter, you should be able to 1. describe the characteristics of the principles of business valuation; 2. explain, why they should be better referred to as principles of functional business valuation; 3. discuss the purposes of the principles of functional business valuation in detail; 4. define and critically evaluate the sources of the principles of valuation. 330 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 330 45520_Matschke_Griffleiste_SL5.indd 330 16.03.2021 16: 23: 31 16.03.2021 16: 23: 31 Chapter 5 5.1 Principles of Business Valuation as a Norm System 5.1.1 Characteristics Principles governing quality management are essential owing to the complexity of the subject matter and the resulting risks created, particularly by business valuation events targeting a change of ownership of the business to be valuated (M ATSCHKE 2017d). The principles governing business valuation can be defined (M ATSCHKE 2003, p. 3) as a normative system to control the process of business valuation that is as consistent as possible and thus serves as an approach to support the derivation of the result. However, expectations of such principles of business valuation should not be raised too high, because they will never be more than a specific description of a task-oriented approach. Conventions do not necessarily lead to task-oriented actions (M ATSCHKE in G OETZKE / S IEBEN 1977, p. 264). Moreover, principles of business valuation must not be confused with schematic instructions that guarantee the “correct” valuation or safeguard that different valuators determine approximately equal business values (M OXTER 1980, p. 454). Principles are used to control the behavior of the addressees. Therefore, they can be defined as supra-individual behavioral norms, where the characteristics are not always spotted immediately but become apparent from the context. The intended control of human behavior under specific context conditions postulated by those principles determines that they should be operable and consistent, and also exhibit a systemic character (F ISCHER -W INKELMANN 2003, p. 84). The question arises of what should be understood by the term normative system (a system of norms). According to systems theory (L UHMANN 2018) every system - including the normative system - is characterized by its elements as well as the relationships between these elements (M ATSCHKE 2003, p. 4). Such relationships can be, for example, of a formal nature (such as hierarchical relations of superiority, subordination, or equality) or of a logical nature. The relationships between the elements might be part of a nexus of conditions or they might be formulated unconditionally. The norms as system elements can be professional or legal standards and they can be of a general or a specific nature. A normative system can combine norms of different kinds. The norms can be dissected into subsystems, emphasizing the importance of the form of relationship. The norms can complement or compete with each other, so, under certain circumstances, they can replace each other or be alternatives. Normative systems can be distinguished according to who is either the norm setter or the norm addressee. Additionally, the degree of the norms’ commitment (their binding nature) and their enforceability can vary considerably, which is indicated by the use of terms such as standard, guideline, statement, or recommendation. Principles mainly refer to a category of problem situations and should generate a comprehensible and predictable solution pattern, based on assumptions and abstractions (K ÜNNEMANN 1985, p. 170). Such a pattern should be properly realizable in a particular case but should also permit verifiable behavior control. Their generally quite abstract formulations mean that principles claim a validity that extends beyond the individual case (P OOTEN 1999, p. 8). Verifiable behavior control means that the principles must not be so-called normative logical molds, that is, norms lacking substance. A behavioral 5.1 Principles of Business Valuation as a Norm System 331 45520_Matschke_Griffleiste_SL5.indd 331 45520_Matschke_Griffleiste_SL5.indd 331 16.03.2021 16: 23: 31 16.03.2021 16: 23: 31 norm that accepts all kinds of behavior in a concrete situation or prohibits only what is not possible lacks normative substance and hence has no regulative function (F ISCHER - W INKELMANN 2003, p. 85). M OXTER (1976b, p. 989, 1980, p. 455) refers to the dynamics of such a system: Principles of business valuation are not a self-contained system as opposed to the approved principles of technique or of the medical profession. The highly intensive research in this field often leads to peripheral corrections. Unfortunately, as is prevalent in science, a problem solved might illuminate several new ones. It would be no exaggeration to state that even today the number of unsolved problems is greater than that of solved ones (M OXTER 1980, p. 455). The situation remains flexible. Because the valuation function, which determines the respective business valuation, is behind the formulated business valuation principles, this chapter will refer to principles of functional business valuation. Kapitel 1: Einführung 332 332 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 332 45520_Matschke_Griffleiste_SL5.indd 332 16.03.2021 16: 23: 32 16.03.2021 16: 23: 32 Chapter 5 5.1.2 Purposes Principles of functional business valuation are used to govern the behavior of the valuators and in particular for quality management. The resulting purposes (M ATSCHKE / B RÖSEL 2013a, p. 766), which have to fulfill certain principles, can be divided into the purpose of complexity reduction, the tripartite protective purpose, the information purpose, and the purpose of communication support. This is outlined in Figure 5.1. Finally, these purposes must cumulatively focus on the respective function of business valuation. Due to the openness of the decision fields and the prevailing uncertainty, in reality, business valuations generally are poorly structured problems (A DAM 1983, O LBRICH 1999, p. 81, R OLLBERG 2001, p. 3). The causes can be found in four possible defects: 1. Business valuation is a target defective problem: Different objectives such as the divergent goals of several business owners compete with each other while at the same time those involved cannot know with absolute certainty what are the objectives of the owners. They may well be pursuing goals other than the usually assumed satisfaction of consumer needs, and they might have a multi-dimensional target system. 2. Additionally, business valuation is an ineffective problem: The type and quantity of valuation-relevant variables and restrictions are not predictable while there might also be ambiguity regarding the interdependencies between the different manifestations of the features and the level of the respective variables. For instance, the influence of competing businesses on the business to be valuated cannot be precisely determined. 3. Furthermore, business valuation represents a defective valuation problem: In an open decision field, the valuation-relevant variables are imperfect or not quantifiable at all. Future factors such as expected interest rates, positive and negative synergies, or the length of the planning period cannot be fully anticipated. 4. Finally, business valuation can also be regarded as an unsolvable problem: Even if all data and interrelations were given, there is no fully efficient method. Due to the complexity of a required company-wide total model, the optimum solution would not be determinable at an economically acceptable cost The defects described above require extensive (heuristic) reductions of complexity when addressing the valuation problem (B ALLWIESER 1990, p. 1) because even theoretically no definite value of a business can be determined ex-ante. Accordingly, for Purposes of the principles of functional business valuation Purpose of reduction of complexity Tripartite protective purpose Information purpose Purpose of communication support Figure 5.1: Purposes for principles of functional business valuation 5.1 Principles of Business Valuation as a Norm System 333 45520_Matschke_Griffleiste_SL5.indd 333 45520_Matschke_Griffleiste_SL5.indd 333 16.03.2021 16: 23: 32 16.03.2021 16: 23: 32 example, according to the decision function, the determination of reasonable value in terms of a range of different possible values is the main focus. The principles of functional business valuation should include rules for complexity reduction (P OOTEN 1999, p. 41, M OXTER 1980, p. 458), from which plausible assumptions and decisions can be derived in real valuation situations to narrow down the decision field and hence, to better cope with the complexity. The problem is to carry out unavoidable complexity reductions in as as “value-neutral” a manner as possible. In other words, the parties must find an “optimum” that represents an acceptable compromise between the interests of potential buyers or sellers and the necessity of simplification (M OXTER 1980, p. 454). Moreover, the principles of functional business valuation have a tripartite protective purpose, with regard to the frequent separation of valuators (e.g., according to the decision function as an external advisor) and the valuation addressee (e.g., in the context of the decision function in the sense of acting as the principal for the actual valuation subject “business owner”) and the possible separation of owner and management. Hence, primarily three groups of persons can be protected by the principles of functional business valuation. The valuator, who is either an external advisor or employee of the valuating company, benefits from the principles because their application protects them from avoidable mistakes, which M OXTER (1976a, p. 17, 1976b) calls malpractice (first characteristic of the protective purpose). In this context, M OXTER (1980, p. 455) makes a comparison with medicine: Even if an observer cannot describe all interdependencies between the determinants of the business value in a differentiated way, the available insight is sufficient to recognize fictitious determinants of the business value. Such fictitious determinants are similar to dubious therapies: As it is assumed with sufficient certainty in medicine that some of the therapies are contraindicated (and others ineffective), certain valuation parameters (might) lead to serious or systematic falsifications of the business value and allow room for coincidence and arbitrariness despite the fact that greater insights have already been acquired. This means that the protection needs of the business owner as the valuation subject must also be fulfilled (second characteristic of the protective purpose) by avoiding possible damage and providing the valuation addressee with a value adequate for the task at hand. With regard to minority shareholders, there is an additional interpretation of the second characteristic: a profitability or marginal price analysis requires that sufficient information is available. The separation between ownership and control (authority to dispose) leads to the fact that owners cannot usually examine or influence a decision on acquisition concerning its financial advantages. This is primarily due to an asymmetrical distribution of information (S CHWETZLER ET AL . 2005, p. 110). Finally, the principles of functional business valuation should protect the management against any claims of the owners that might result from improper valuations and wrong decisions of the management (third characteristic of the protective purpose). Hence, the business judgment rule in the USA limits the scope of the management with regard to entrepreneurial decision-making. This facilitates the filing of claims for damages by the shareholders and supervisory bodies in case of violation of due diligence by the management. In connection with the business judgment rule, the fairness opinion (Z IMMERMANN 2016) also plays an important role in the USA. A fairness opinion is the statement of an independent expert on the financial adequacy of the transaction. It is essential that the statement for an individual party is made during the transaction process. A fairness opinion is not an instrument of pricing, but a recognition of the appropriateness of the offer price. An adequate price is a price at which an informed and rational Kapitel 1: Einführung 334 334 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 334 45520_Matschke_Griffleiste_SL5.indd 334 16.03.2021 16: 23: 33 16.03.2021 16: 23: 33 Chapter 5 shareholder would carry out the transaction of their own accord (S CHWETZLER ET AL . 2005, p. 107). In other words, the fairness opinion is the result of a comparison of the decision value from the perspective of the valuation subject with the price offer of the opposing party that might show the characteristics of an argumentation or arbitration value. Generally, a fairness opinion can consist of an opinion letter, a valuation memorandum, and a factual memorandum, whereby the content and scope are individually customizable. The opinion letter is the actual fairness opinion in form of a written statement concerning the price offer. The marginal price is compared to the tendered transaction price and that forms the basis for the profitability assessment. That assessment is often merely described as adequate or inadequate. The valuation memorandum expands the opinion letter and provides information on the appropriateness of the transaction price. Emphasis is laid on the valuation results, especially on the detailed description of the valuation methods and assumptions, such as the derivation of future cash flows and the cost of capital. For the purpose of verification, the complete basis for the information is documented in a factual memorandum. This includes both the detailed and confidential financial information and economic, legal, or fiscal parameters (S CHWETZLER ET AL . 2005, p. 108). A fairness opinion is not only used for a comparison of the decision value and price offer (e.g., in the sense of an argumentation or arbitration value), but itself has the characteristics of an argumentation value (Z IMMERMANN 2016). The estimate of an impartial expert leads to a reduction of information asymmetries and to an enhancement of credibility of the management (S CHWETZLER ET AL . 2005, p. 111). Eventually, it is up to the management as the usual principal determining whether to publish or withhold a fairness opinion is that of a conflict with their own interests (S CHWETZLER ET AL . 2005, p. 114). However, a fairness opinion only fulfills the described protective function if it corresponds to specified quality requirements. S CHWETZLER ET AL . (2005, p. 114) argue that compliance with the principles of business valuation during the preparation of fairness opinions is suitable to restrict the interest-driven exploitation of latitude possessed by the parties to the valuation process (Z IMMERMANN 2016). The purpose of information (P OOTEN 1999, p. 43), which the principles of functional business valuation have to fulfill, aims to reduce the information deficits of the valuators and potential interested parties and to achieve a goal-oriented behavioral influence. The valuators should have clarity about the adequacy of methods and the necessity of valuation-relevant information. The latter must partly be made available by the interested party. The principles serve the interested party by setting out the origin, nature and quality of the information provided. The principles serve the interested party by setting out the origin, nature and quality of the information provided. Additionally, the purpose of decreasing expectations derives from the information function for the addressee of the information. For example, the complexity reduction operation should diminish unrealistic and exaggerated expectations of the validity and significance of the valuation results on the part of the interested party. Since several people are involved in the valuation process, the principles of functional business valuation must also meet the purpose of communication support (P OOTEN 1999, p. 44). The principles particularly support the communication within the valuation process through the reduction of semantic problems. If the principles themselves and especially the valuation results are to be interpreted equally by the parties to the valuation, the principles must be complete, consistent, and without overlap and should also be formulated unambiguously. 5.1 Principles of Business Valuation as a Norm System 335 45520_Matschke_Griffleiste_SL5.indd 335 45520_Matschke_Griffleiste_SL5.indd 335 16.03.2021 16: 23: 33 16.03.2021 16: 23: 33 5.1.3 Sources With regard to the sources of principles of functional business valuation, it has to be differentiated between both legislation and jurisdiction and committees for standardization of interested business circles as well as science and research (M ATSCHKE 2003, p. 5, M OXTER 2003, p. 9). While legislation creates binding standards in the form of statutes governing opposing interests and judgments, jurisdiction ensures their enforcement. However, even such standards are reversible because nothing is for eternity. Changes in jurisdiction regarding problems of business valuation prove this impressively. Admittedly, it has to be considered that only a few lawyers possess profound knowledge in the field of business valuation. By the application of capital market-oriented models and with reference to the alleged objective market data desired values are generated that are more easily accepted by economic laymen (e.g., lawyers and judges) because these methods seem to be objective. With respect to the DCF methods, it is regularly concealed that they might lead to different results (F ISCHER -W INKELMANN 2003, p. 153). In this context, there is a risk that lawyers as well as economists are deceived by skillfully negotiating valuators: The market is often regarded as the highest authority of economic sciences. That is why at first glance it appears obvious to determine the market price (H ERING 2014, p. 9). Fortunately, the courts only interpret law to deliver verdicts rather than routinely making law. However, it is important to bear in mind that jurisdiction is significantly more influential in a case law legal system than it is in the codified continental European legal system. Standardization committees such as the National Association of Certified Valuators and Analysts (NACVA) in the field of business valuation seek a uniform course of action from their professional members. Hence, the rules to be recognized ultimately become accepted rules of professional action. The NACVA is the market leader in the training of valuation professionals in the United States and has more than 7.000 members. It has certified valuation professionals since 1990 and supports their daily work (W AL- TER 2005, p. I). The risk inherent in such committees setting principles (M OXTER 2003, p. 10) is that they see their task as being to standardize commercial practices. This would be unproblematic if the corresponding principles were just technical standards from which all protective functions follow. It would be idealistic to expect legal certainty and legal clarity of standardization committees because the different interests of the represented groups in such committees force compromises and necessitate considerable discretion. Science and research, if independent, can develop a theory-based task-adequate normative system (M OXTER 1976a, M OXTER 1976b, G OETZKE / S IEBEN 1977, M OXTER 1980, M OXTER 1983, P OOTEN 1999, M ATSCHKE 2003). These can provide benchmarks for the valuation of other systems of norms. Hence, contradictions or implications can be revealed, and reasonable, adequate solutions can be worked out. Science can be useful for legislation and jurisdiction both de lege lata (in currently applicable law) and de lege ferenda (future applicable law). Science can offer a critical counter to standardization committees and the strength of science is its independence; but even scientists are only human and tend to swim with, rather than against, the tide. The prevailing doctrine is a criterion that should not be overestimated and even in science, the truth sometimes struggles to overcome inertia. An observer would always be wise to preserve a certain skepticism and to acknow- Kapitel 1: Einführung 336 336 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 336 45520_Matschke_Griffleiste_SL5.indd 336 16.03.2021 16: 23: 34 16.03.2021 16: 23: 34 Chapter 5 ledge divergent views (M OXTER 1976b, p. 991). Scientists are not immune to trends and views, which can constitute a backward step, but nonetheless, alongside logic, trial and error are essential sources of knowledge in science. The same holds for the tension between thesis, antithesis, and synthesis. The accompanying research, however, concerning consulting products in the field of business valuation remains uncritical, because it knows nothing else or does not perceive any divergent views. Finally, the German idiom “only people who can sing the desired modern melodies remain in the choir” may illustrate the preceding analysis. These three sources - (1) legislation and jurisdiction, (2) standardization committees, and (3) science and research - should not be seen in isolation, but to represent a metaphorical water course system with visible and invisible connections that enable waters to ultimately combine, albeit that combining process can sometimes take a while and might temporarily lead to murky waters. Section 5.2 below outlines a theory-based system of principles based on the current findings of the functional business valuation. 5.1 Principles of Business Valuation as a Norm System 337 45520_Matschke_Griffleiste_SL5.indd 337 45520_Matschke_Griffleiste_SL5.indd 337 16.03.2021 16: 23: 34 16.03.2021 16: 23: 34 5.2 Principles of Functional Business Valuation The deductive determination of theory-based principles of functional business valuation is based on the functional business valuation theory. Hence, the determination of a system of principles by means of deduction only follows business considerations. The principles of business valuation are determined on the basis of the valuation purpose. That is quite different in the case of induction where it would be extrapolated from the individual opinions of business people to the principles of business valuation. What conclusions can be drawn for the principles of functional business valuation from the theoretical considerations discussed in the previous chapters? Figure 5.2 offers an overview of the (basic) principles of functional business valuation that results from the functional business valuation theory. The principles are discussed below. M ATSCHKE (in G OETZKE / S IEBEN 1977, p. 292) defined the valid overriding principle (leading principle), namely the principle of the consideration of the respective function of business valuation. M OXTER initially called this principle the principle of “careful task analysis” (1976a, p. 26) and later described it succinctly as the “principle of purpose adequacy” (1983, p. 5). It follows that further principles must be formulated according to the task or function. Only if this principle is accepted as the overriding principle can principles of business valuation that are adequate for purpose be generated with regard to every single function. Concerning the mediation (arbitration) function and the arbitration value, M ATSCHKE postulated two basic principles in 1979 (1979, p. 43 and 92). First, the principle of party-related adequacy because the arbitration value represents a “fair” value. Second, the principle of the rationality of action of the conflicting parties which requires the consideration of decision values in order to ensure that in case of a non-dominated conflict situation, the arbitration value represents a permissible conflict resolution for all conflicting parties, whereas in the case of a dominated conflict situation the interests of the dominated party are preserved too. The principle of the rationality of action represents the parenthesis to the principles of functional business valuation in the decision function. However, the point of view is quite different. The mediation function emphasizes the determination of a potential agreement set, whereas the decision function focuses on the set of acceptable conflict resolutions for a party where the decision value as the concession limit for this party is particularly relevant. Therefore, the valuation in the decision function must be viewed as a subjective business valuation. With regard to the argumentation function, B RÖSEL systematized the characteristics of the argumentation value in 2004, which can be classified as the principles of the argumentation function (2004, p. 518). Accordingly, argumentation values must not be introduced in the negotiations as such to correspond to their purpose. Hence, the argumentation value should be disguised as a decision value or also as an arbitration value (the principle of camouflage). Kapitel 1: Einführung 338 338 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 338 45520_Matschke_Griffleiste_SL5.indd 338 16.03.2021 16: 23: 35 16.03.2021 16: 23: 35 Chapter 5 Figure 5.2: (Basic) principles of functional business valuation Zwecke der Grundsätze funktionsgemäßer Unternehmensbewertung Komplexitätsreduktionszweck Ternärer Schutzzweck Informationszweck Kommunikationsunterstützungszweck Figure 5.1: Purposes for principles of functional business valuation Zweckadäquanzprinzip: Aufgabenadäquater Wert Entscheidungsfunktion: Entscheidungswert Generelle Prinzipien Grundsatz der Beachtung der realen Konfliktsituation Grundsatz der Beachtung der realen Ausgangssituation Grundsatz der Beachtung der Ziele Grundsatz der Beachtung des Entscheidungsfeldes Grundsatz der Subjektivität Grundsatz der Zukunftsbezogenheit Spezielle Prinzipien: Zukunftserfolgswert Prinzipien der Erfolgsabgrenzung Prinzip der Gesamtbewertung Synergieberücksichtigungsprinzip Zufluß- oder Ausschüttungsprinzip Prinzipien der Kapitalisierung Prinzip der Grenzzinsfüße Stichtagsprinzip Prinzipien der Risikooffenlegung Prinzip der Eingrenzung des Entscheidungswertes Prinzip der Risikoprofilbildung Vermittlungsfunktion: Arbitriumwert Grundsatz der Rationalität des Handelns der Konfliktparteien Grundsatz der Rationalität des Handelns der zu schützenden Partei Grundsatz der parteienbezogenen Angemessenheit Grundsatz der absolut gleichen Vorteilsverteilung (Mittelungsprinzip) Grundsatz der relativ gleichen Vorteilsverteilung Sonstige Gerechtigkeitsgrundsätze Argumentationsfunktion: Argumentationswert Grundsatz der Parteienbezogenheit Prinzip der Bezogenheit auf Entscheidungswerte Prinzip der Beeinflussung Grundsatz der Tarnung Grundsatz der Konfliktlösungsorientierung Prinzip der Information Prinzip der Flexibilität Prinzip der Glaubwürdigkeit Principle of considering the real conflict situation Principle of considering the real initial situation Principle of subjectivity Principle of future orientation Principle of accrued income Principles of capitalization Principles of risk disclosure Principle of considering the objectives Principle of considering the decision field Principle of overall valuation Principle of synergy consideration Principle of inflow or distribution Principle of marginal interest rates Principle of the effective date Principle of limitation of decision value Principle of risk profile development Principle of action rationality of the party to be protected Principle of the absolute equal benefit distribution/ division Principle of the relatively equal benefit distribution/ division Other principles of justice Principle of relatedness to decision values Principle of influencing Principle of information Principle of credibility Principle of the rationality of action of the conflicting parties Principle of party-related adequacy Principle of party-relatedness Principle of camouflage Principle of conflict resolution orientation General principles Specified principles: Future performance value Decision function: Decision value Mediation function: Arbitration value Argumentation function: Argumentation value Principle of purpose adequacy: adequate value Principle of flexibility Kapitel 1: Einführung 339 5.2 Principles of Functional Business Valuation 339 45520_Matschke_Griffleiste_SL5.indd 339 45520_Matschke_Griffleiste_SL5.indd 339 16.03.2021 16: 23: 36 16.03.2021 16: 23: 36 The principle of conflict resolution orientation holds that argumentation values should be designed in a way that bridges present conflicts of interest with regard to the price level, shareholdings, and other conflict-resolution-relevant facts. This principle is subdivided into three subordinate principles: the principle of information, the principle of flexibility, and the principle of credibility. The latter states that argumentation values are only useful if accepted by the negotiation partner, for example, if a business valuation uses methods and parameters that are accepted by the opposing party. With respect to the principle of flexibility, the argumentation values should include information obtained and interim results stipulated during the negotiation. The principle of information means that argumentation values must not only represent the given price and shareholding offer but be values justifying and substantiating the offers. Therefore, those argumentation values should support the information exchange. The negotiating parties must try to acquire information relevant to conflict resolution, such as, negotiation tactics and price expectations from the argumentation values of the negotiating partner. Meanwhile, each party will be seeking to provide the opposing party with information intended to precipitate the desired negotiation result. Last but not least, the principle of party-relatedness must be considered. That principle can be subdivided into the principles of influencing and the principle of the decision value reference. The principle of influencing states that the argumentation value convinces the negotiation partner to make concessions with regard to the desired negotiation result. However, the most important principle is the principle of relatedness of decision values. Thus, a party’s decision values have to be considered as the concession limits if argumentation values are introduced into the negotiation. Additionally, the argumentation values have to aligned with the opposing decision value. Similar to the principle of the rationality of action in the mediation function, the principle of reference to decision values serves as a parenthesis to the principles of functional business valuation in the decision function. The argumentation function focuses on the determination of room for argumentation. Finally, the principles of functional business valuation in the decision function can be classified into general and specific principles, which are relevant for the future performance value method and consequently also for the future performance value. According to the general principles of the decision function, it is essential for the determination of a task-adequate value to choose the real conflict situation as a starting point (principle of considering the real conflict situation). The relevance of this principle will be discussed with the aid of an example focusing on a joint conflict situation: If such a joint situation is given, it has to be considered that the decision value of one busifness is determined in association with the negotiation results regarding other businesses, provided that these results have effects on the decision field of the valuation subject. If a real conflict situation is not considered, incorrect decision values become likely. Furthermore, it is necessary to identify the real initial situation of the valuation subject (principle of considering the real initial situation). According to this principle, the real targets (principle of considering the objectives of the valuation subject) and the real action possibilities and restrictions of the conflicting party (principle of considering the decision field of the valuation subject) must be considered. The fact that it comes to typifications, that is, simplifications and complexity reductions (B ALLWIESER 1990), is inevitable with regard to the purpose of complexity reduction of the principles of functional business valuation. This issue not only concerns the conflict situation, as explained above, but also the objectives, the action options and the restrictions constraining the conflicting party (M ATSCHKE 1979, p. 113), for which the decision value is determined 340 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 340 45520_Matschke_Griffleiste_SL5.indd 340 16.03.2021 16: 23: 37 16.03.2021 16: 23: 37 Chapter 5 and for which the characteristics of the specified principles are described below. Such a typification must factor in that the information about the targets and action options, for example, is incorporated in the value determination that comes closest to the supposedly required but not attainable information (M OXTER 1983, p. 26). Moreover, the principle of subjectivity and the principle of future orientation have to be considered For the convenient determination of the decision value as a marginal price, the future performance value remains the essential initial value (M ÜNSTERMANN 1952, p. 214, M ÜNSTERMANN 1956, p. 1062, M ÜNSTERMANN 1976, p. 170, M ÜNSTERMANN 1980). The following specific principles of decision value determination are discussed regarding the conflict situation of the acquisition/ sale type. If a conflict situation of the merger/ demerger type is the issue, the principles must be adjusted accordingly. The future performance value method can be derived consistently from the general model of decision value determination and its concretization, the state marginal price model. In contrast, the DCF methods cannot be derived from investment theory and therefore are not suitable methods for the determination of the marginal price. Admittedly, they are still widely used in practice as argumentation values due to their high renown, but are likely to be replaced at some point by even more modern consulting products. In this context, the risks of misunderstandings in the Anglo-Saxon literature and its accompanying research should not be overlooked: Due to the formal similarities of DCF methods and the future performance value method, especially in terms of the discounting of future payments such as cash flows and income, content-related proximity or even a correspondence (equivalence) are sometimes suspected. However, this is a misjudgment, because formal similarities do not induce content-related equivalence. The efforts to consider the DCF methods and the future performance value method as equivalent stem partly from a recognizably defensive attitude toward the supposedly progressive DCF methods. It would be inaccurate to state that the DCF methods and the shareholder value philosophy are products of consulting companies that are constantly developing and introducing "new" products to the market. The efforts to “prove” the correspondence of the valuation results of the different DCF methods (K RUSCHWITZ / L ÖFFLER 2003) and of the alleged identity of DCF and investment-theoretic methods (B ORN 1996, B ORN 2003, p. 176) are naïve. That view, however, overlooks the fact that the variants considering different methodological manifestations of the free cash flow or the discount rate (capitalization rate, cost of capital) are legion. The specific principles of decision value determination result from the application of the future performance value method. Matschke has distinguished between the principles of accrued (deferred) income, capitalization, and risk disclosure (M ATSCHKE 2003, p. 25). They are closely related to the general principles of the decision function. The principle of accrued income refers to the size of future performance, using the formula of the future performance value method. Hence, the most important principle is the overall valuation principle. The business as a whole is to be valuated so that the cash flow of the whole business is considered as the future performance. Generally, the subjective planning of the party for which the valuation is carried out determines future performance. The overall valuation principle and the general principles of subjectivity and future orientation represent a trinity in business valuation. A practical approach in forecasting is not in conflict with the principle of the overall valuation if the (entire) cash flow is deconstructed (divided) into its individual elements, thereby also reflecting the purpose of complexity reduction of those principles. This includes, for instance, the practice to distinguish between operating assets and nonoperating assets, provided that the distinction follows the predicted subjective planning. 5.2 Principles of Functional Business Valuation 341 45520_Matschke_Griffleiste_SL5.indd 341 45520_Matschke_Griffleiste_SL5.indd 341 16.03.2021 16: 23: 37 16.03.2021 16: 23: 37 A permanent cash flow is allocated to the operating assets, according to the subjective going concern principle, whereas a temporary cash flow is assigned to the non-operating assets under consideration of the planned type, time, and intensity of divestiture (liquidation). Another practice in connection with forecasting, namely to distinguish between phases of different levels of detail regarding the forecast of cash flows, does not contradict the overall valuation principle. The near future can be estimated more accurately, whereas coarser methods and assumptions must be used for the more distant future. The future performance value method is a partial model. If income effects occur outside the valuation object that relate to its purchase or sale, they are not reflected in its cash flow. Conversely, in a total model, such performance-based differences would directly affect the size of the decision value as a marginal price. Therefore, if such a partial model is used and integration effects are expected, it is necessary to extend the cash flow of the valuation object by such positive or negative synergies. This is the content of the principle of synergy consideration. Hence, the valuation-relevant benefit stream, in the form of cash flows, results from the difference of the performances of the investment and financing program of the valuation subject with and without the business to be valuated. Since respective synergy effects (economies of scope) are generated by a synthesis of the future investment and financing program and the business to be valuated, synergy considerations reflect the principle of subjectivity, the principle of future orientation, and the overall valuation principle. The decision value determination is based on the subjective value theory, which has a long tradition in the field of economy. The value is only defined by satisfying need, thus creating utility (benefits). This includes the sum of all advantages that the business owner achieves due to the disposal of the business. In the individual case, these benefits can be very heterogeneous (M OXTER 1983, p. 75). In the context of the application of the future performance value method, there is a narrowing down to financial advantages because money is the most general means of satisfying needs. Money as a nominal good represents the real goods that could be acquired to satisfy needs. The business only satisfies the needs of its owners if they profit financially; therefore, the subjective business valuation focuses on cash flows to the owners. In other words, the determination of the future performance value relies on the withdrawals of those cash flows (payments to the owners) or their distribution. This example would also include payment savings occasioned by services rendered by the company to its owners. Hence, the influence of the business to be valuated on the satisfaction of needs of the owners can be measured by the initiated payment consequences. This is what is meant by the inflow or distribution principle. Taking the L ÜCKE theorem into account, profits can theoretically be considered a potential distribution, provided that double counting is avoided (L ÜCKE 1955). In partnerships, the equivalent of the entrepreneurial performance would have to be subtracted from the withdrawals as an employer’s salary. According to the future performance value method, the marginal price (or marginal price range) is estimated by discounting the future performances (cash flows, profits) at the valuation date; thus, the principle of capitalization must be observed. This means that the valuation is based on the conditions and future prospects of this effective date (effective date principle). The discount rates are estimated as marginal interest rates relating to the last intended use or procurement of money in the forecast period. In this context, reference is often made to the credit and debit interest rate. However, this could lead to a misunderstanding, because the terminology in this theoretical context should not be understood in the same way as banking terms, but rather as a marginal use or marginal procurement of means of payment (principle of the marginal interest rates). Kapitel 1: Einführung 342 342 5 Principles of Business Valuation 45520_Matschke_Griffleiste_SL5.indd 342 45520_Matschke_Griffleiste_SL5.indd 342 16.03.2021 16: 23: 38 16.03.2021 16: 23: 38 Chapter 5 Provided that there are no reorganizations with modifications of the net present value during the transformation from the base to the valuation program, the capitalized future performance corresponds to the marginal price. The last group of the analyzed specific principles concerns the principles of risk disclosure. This includes two principles: the principle of the limitation of decision value and the principle of risk profiling. The principle of the limitation of decision value refers to the risk arising from the method itself, because the future performance value method can only determine one lower and upper limit for the decision value, if the marginal interest rates in the base and valuation program do not coincide. The principle of risk profiling development is not intended to disclose such a method-related risk, but rather the forecast-related risk, because fundamental data of any business valuation come with risk and uncertainty in practice. Hence, it is not advisable to pursue risk solidifying or risk condensing in the form of risk discounts from the future performance and risk premiums to the discount rate (capitalization rate) because neither can be derived from the general theory of the decision value. Both principles also indicate the departure from the idea that the future performance value as the decision value in the sense of a marginal price represents a point size. From both a method-related and a forecast-related perspective, only those areas in which the decision value will lie can be specified realistically. Finally, it should be noted that it is not possible to consistently formulate principles of proper business valuation that are wholly satisfactory. There will always remain white spots in the area of which “procedural freedom” (degrees of freedom) is granted within certain limits (M OXTER 1976b, p. 991). 5.2 Principles of Functional Business Valuation 343 45520_Matschke_Griffleiste_SL5.indd 343 45520_Matschke_Griffleiste_SL5.indd 343 16.03.2021 16: 23: 38 16.03.2021 16: 23: 38 344 5 Principles of Business Valuation 5.3 Selected Control Questions Exercise 1 (20 Points) - Principles of Business Valuation as a Norm System a) Explain the characteristics of the principles of business valuation. (5 points) b) Which purposes are pursued with the principles of business valuation? Describe two purposes in more detail. (9 points) c) List possible sources for principles of business valuation and critically examine these sources. (6 points) Exercise 2 (30 Points) - Principles of Functional Business Valuation Discuss the topic “Valuation approaches of functional business valuation and principles of their functional determination” in a short, structured essay. Start with a table of contents. Exercise 3 (30 Points) - Principles of Functional Decision Value Determination Discuss the topic “Theory-based principles of functional decision value determination - basis, derivation and presentation” in a short, structured essay. Start with a table of contents. 45520_Matschke_Griffleiste_SL5.indd 344 45520_Matschke_Griffleiste_SL5.indd 344 16.03.2021 16: 23: 39 16.03.2021 16: 23: 39 References References A A DAM , D.: Entscheidungsorientierte Kostenbewertung, Wiesbaden, 1970. A DAM , D.: Planung in schlechtstrukturierten Entscheidungssituationen mit Hilfe heuristischer Vorgehensweisen, in: Betriebswirtschaftliche Forschung und Praxis. 1983. Pp. 484-494 A DAM , D.: Planung und Entscheidung, 4th ed., Wiesbaden, 1996. 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References 363 45520_Matschke_Griffleiste_SL5.indd 363 45520_Matschke_Griffleiste_SL5.indd 363 16.03.2021 16: 23: 55 16.03.2021 16: 23: 55 45520_Matschke_Griffleiste_SL5.indd 364 45520_Matschke_Griffleiste_SL5.indd 364 16.03.2021 16: 23: 55 16.03.2021 16: 23: 55 About the Authors M ATSCHKE , M ANFRED J ÜRGEN Personal Data: Dr. rer. pol., Dipl.-Volksw., born 1943, Emeritus Professor of the Chair of Business Valuation at the E RNST -M ORITZ -A RNDT -University of Greifswald. M ANFRED J ÜRGEN M ATSCHKE studied in Cologne from 1963 to 1968. He received his doc