M&A and the Simulation-Based Valuation of Companies with an Uncertain Exit Price and Special Rights

Gleißner, Werner; Wolfrum, Marco; Dorfleitner, Gregor Article M&A and the Simulation-Based Valuation of Companies with an Uncertain Exit Price and Special Rights Credit and Capital Markets – Kredit und Kapital Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Gleißner, Werner; Wolfrum, Marco; Dorfleitner, Gregor (2024) : M&A and the Simulation-Based Valuation of Companies with an Uncertain Exit Price and Special Rights, Credit and Capital Markets – Kredit und Kapital, ISSN 2199-1235, Duncker & Humblot, Berlin, Vol. 57, Iss. 1/4, pp. 185-221, https://doi.org/10.3790/ccm.2025.1455401 This Version is available at: https://hdl.handle.net/10419/324957 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Credit and Capital Markets, 57 (2024) 1 – 4: 185 – 221 https://doi.org/10.3790/ccm.2025.1455401 Scientific Papers Open Access– Licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0). Duncker & Humblot · Berlin M&A and the Simulation-Based Valuation of Companies with an Uncertain Exit Price and Special Rights Werner Gleißner*, Marco Wolfrum**, Gregor Dorfleitner*** Abstract This article presents a new methodological approach to value private equity investments based on simulation. The valuation relies on ‘imperfect replication’. This method does not presuppose the perfection of the capital market and is essentially built on measuring the risk. The approach turns out to be easy to implement. Firm specific characteristics as well as and existing special rights can be depicted and modelled. The proposed methodology is of immediate practical usefulness as it can help to find decision support for concrete investment situations. Also, during the investment period it can be used for monitoring. The originality of the research lies in the combination of Monte Carlo simulation, multiple methods, relevant risk measures and risk-value models. Keywords: Company valuation, share valuation, exit price, risk analysis, Monte Carlo simulation, incomplete replications JEL Classification: G17, G24, G32, G33, G34 Acknowledgement: We would like to express our sincere gratitude to an anonymous reviewer for their invaluable feedback and constructive suggestions. Their insights significantly contributed to improving the quality and clarity of this paper. I. Introduction and overview A key challenge for private-equity and venture-capital companies is estimating a realistic range of possible future exit prices for an investment, particularly * Prof. Dr. Werner Gleißner is a member of the board of the FutureValue Group AG and honorary professor for business administration, esp. risk management, at the TU Dresden. He is also a board member of the European Association of Certified Valuators and Analysts (EACVA). E-Mail: [email protected]. ** Marco Wolfrum is a partner of FutureValue Group AG and deputy chairman of the board of the RMA Risk Management & Rating Association e. V. as well as managing director of the RMA Rating & Risk Academy GmbH. He is a lecturer in risk management at various universities. E-Mail: [email protected]. *** Prof. Dr. Gregor Dorfleitner is Professor of Finance and Director of the Center of Finance at the University of Regensburg. 186 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 in smalland medium-sized enterprises (SMEs1). The current value or the maximum acceptable purchase price (the subjective decision value2) depends on this range of possible future sale prices, and specifically on the expected value of the sale price and the ‘sale price risks’, which are investment risk, expressed, for instance, in the standard deviation of this price. Uncertain future sale prices are relevant to the decision, e. g. regarding the purchase of a company. Therefore, traditional ‘multiple methods’, which deal with the current form of the valuation level (multiple) and profitability (EBIT: earnings before interest and taxes) of the company, are insufficient on their own due to their lack of future reference.3 In light of the empirically proven imperfections4 of the capital market, the prices that can be achieved in an upcoming transaction (and also the stock-market prices) often deviate from the company value as calculated based on the model.5 As many empirical studies show, transaction prices for non-listed companies, private equity (PE) and, especially, venture capital (VC) are determined using comparative methods, in particular multiple methods. The often-realised ‘low’ transaction multiples (related to EBIT or EBITDA) or the high ‘return requirements’ show that the valuation6 takes into account: • theabove-averageprobabilityofinsolvency,whichisusuallyparticularlyhigh for SMEs and venture-capital investments, and/or • company-specific(idiosyncratic)risks Above-average earnings and insolvency risks in smaller companies, especially SMEs, are one explanation for the ‘size effect’ regularly found in empirical studies.7 Everything else being equal, smaller companies with higher earnings and insolvency risk have higher costs of capital and lower enterprise value.8 The Business Judgment Rule9 makes it important to consider the risks of a target company, including that of insolvency. When making a business decision 1 On the peculiarities of valuating SMEs, see, for instance, Coulon (2022) and Damodaran (2011), who specifically deal with the importance of risk analysis and risk simulation. 2 See Matschke (1972). 3 See Bhojraj/Lee (2002) on criticisms of multiple methods. 4 See, for instance, Haugen (2002); Gromb/Vayanos (2010); Shleifer/Vishny (1997); Jegadeesh/ Titman (2011); Joyce/Mayer (2012) and Zhang (2020). 5 See Calhoun (2020). 6 See, for instance, Kerins etal. (2004) and Müller (2004). 7 See Grabowski (2018). 8 See also Blitz (2020) for an overview of the current empirical research, as well as Fama/French (2015, 2018) and Elgammal etal. (2020). 9 See e. g. Leach (2014). When buying or selling a shareholding, a ‘business decision’ within the meaning of the Business Judgment Rule (Section 93 of the German Stock Corporation Act) must generally be assumed. The application of this provision is subject to M&A and the Simulation-Based Valuation of Companies 187 Credit and Capital Markets, 57 (2024) 1 – 4 to purchase a target company, the buyer’s board of directors should be aware of how their own company’s risk exposure changes if they acquire the target company. This requires much more than just due diligence or the consideration of stock-return fluctuations. It necessitates the analysis and aggregation of the risks of the target company, and the consideration of additional transaction-specific risks (e. g., those arising due to uncertain synergies and integration costs) and changes in the risk position as a result of the financing of the acquisition.10 Methods for quantitative risk analysis, risk aggregation and simulation-based valuation can capture the implications of a decision to purchase an investment and the effects on the risk-return profile of the buyer’s company.11 The simulation-based method we propose in this article, is suitable for adequately considering the ‘special rights’ of individual investors that are typical of venture-capital investments when valuing a share.12 Based on the uncertain exit price, it is possible to determine individual investors’ share of the exit price, taking into account existing special contractual regulations such as liquidity-preference regulations. The share of the exit price depends on the level of the exit price. This is the basis for a risk-adequate assessment of individual ‘equity tranches,’ including special rights. We show how realistic ranges of possible future sale prices can be determined based on risk analysis and simulation methods, and, in turn, how risk-adjusted discount rates for the valuation of a (potential) investment can be determined.13 With the methods we present, it is possible, based on the findings of quantitative risk analysis, to derive risk-adjusted return requirements (discount interest rates) or cost-of-capital and fundamental values directly from the ‘earning risks’, without resorting to unavailable historical capital-market data as with the capital-asset pricing model (CAPM).14 The effects of insolvency risks (probability of insolvency, rating) are also taken into account in the valuation calculation.15 When estimating prices, we utilize the multiple method to take into account the influence of market imperfections on the prices. Our valuation rationally the condition that the VC company is a corporation, but this should not be a larger restriction in practice. 10 On the methods, see Gleißner/Ernst (2019); Dorfleitner/Gleißner (2018); Dorfleitner (2022) and Ernst (2022a) on the assumptions and fundamentals of the methods used. 11 See Gleißner (2019) on the risks of M&A transactions. 12 See Jenkinson etal. (2019) and Cederburg/Stoughton (2018). 13 The structure of the company in the case study is based on Gleißner/Wolfrum (2008), but the valuation methods used in this article, which were not known at the time, were not used. 14 See, fundamentally, Gleißner (2011) for the method. 15 See Morris (2009); Gleißner (2010); Knabe (2012); Saha/Malkiel (2012); Lahmann etal. (2019) and Franken etal. (2020). 188 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 values previously little-considered information about the risks of the valuation object (and does not assume the perfection of the capital market, unlike, for example, a valuation based on the CAPM). We also take up the concepts of semi-investment-theoretical valuation paradigm, which is based on the method of imperfect replication, as well as simulation-based valuation methods. Summarizing, the article is the first to 1. demonstrate the application of both concepts to purchase decisions in the context of mergers and acquisitions, 2. combine the simulation-based valuation method with standard market concepts for estimating possible transaction prices (multiple method), 3. compare the valuation based on the standard deviation risk measure with that based on the semi-standard deviation, and 4. integrate a consistent valuation of the special rights of individual shareholders. Section II, discusses the theoretical underpinnings of our approach. SectionIII presents our model, including the concept of risk-based valuation using ‘imperfect replication’ and applies this to an uncertain exit price with standard deviation and semi-standard deviation as risk measures and a realistic model for the EBIT development over time. The new valuation approach is applied in a case study in Section IV. Based on certain valuation assumptions and, we undertake a risk-adequate assessment of the company as a whole and a consistent share valuation (taking special rights into account). A comparison with conventional valuation methods illustrates the superiority of the new approach. A general discussion in the last section concludes the paper. II. Valuation with uncertain exit prices: Theory 1. Existing approaches The standard theory of business valuation is represented by discounted cash flow methods, which involve taking the expected cash flows for each period and discounting them using a risk-adjusted discount rate, using the CAPM. This approach is well-documented in nearly every corporate finance textbook,16 so we refrain from providing a traditional literature review on the conventional solutions to this problem. However, to categorise our approach, it is important to distinguish between the various valuation functions, with particular emphasis on the argumentation 16 See e. g. Berk/DeMarzo (2023) for a contemporary presentation and the references herein for further academic literature. M&A and the Simulation-Based Valuation of Companies 189 Credit and Capital Markets, 57 (2024) 1 – 4 value and the decision value, as they are especially relevant to the field of application under consideration.17 While the argumentation value is calculated for negotiation situations to justify one’s own price expectations, the decision value represents the maximum acceptable purchase or sale price for the subject of valuation, for example, a marginal price. Valuation methods rooted in financing theory are founded on the neoclassical hypothesis of perfect capital markets. These methods calculate discount rates based on historical fluctuations in share returns using the CAPM. The assumptions of the CAPM lead to an equivalence between price and value.18 In contrast, factor modelling methods consider multiple factors– not just the beta factor of the CAPM– to explain the expected share returns.19 The theoretical foundation for these methods often draws upon Ross’s arbitrage pricing theory (1976). Many methods involve price estimation.20 Valuation multiples are derived from the prices of comparable companies, and these multiples are subsequently used to estimate the value of the company in question (see II.2.).21 So far, none of the methods mentioned allow for the calculation of decision values; this is instead facilitated by investment-theoretical valuation methods. These methods consider the individual information, actionable options, and constraints of the asset being valued.22 Notably, they do not rely on the assumption of perfect capital markets. Instead, valuation is based on ‘valuation principles derived from subjective value theory’23, where value is understood as a ‘subject-object-object relationship’.24, 25 Summarizing Olbrich etal. (2015, p. 34), subjective business valuation is grounded in the concept of valuation as an entity and its future orientation. These aspects provide a robust decision-making basis for economic subjects. The solid theoretical foundation of these methods is offset by the significant calculation effort required for the ‘total model, ’26 which allows for the consideration of numerous alternative investment options for the valuation subject. Con17 See Matschke (1975) on the decision value and Matschke (1979) and Olbrich etal. (2015, pp. 29 – 32) on the ‘functional business valuation theory’. 18 For criticism of the approach, see also Olbrich etal. (2015). 19 See Fama/French (1993, 2015, and 2018); Swade etal. (2023). 20 See Schüler (2020); Krolle etal. (2005). 21 For ‘theory-based’ multiplier methods, see e. g. Richter (2005); Kelleners (2004) and Herrmann (2002). 22 See Hering (2000). 23 See Olbrich etal. (2015, p. 17). 24 See Olbrich etal. (2015, p. 18), with reference to Matschke etal. (2010). 25 See also Olbrich etal. (2015, pp. 25 – 27) on the inadequate capture of uncertainty by the ß factor of the CAPM. 26 See Hering/Toll (2013). 190 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 versely, simple heuristic variants, known as partial models27, are easier to apply. However, these partial models rely on predetermined discount rates and do not provide a methodology that would, for example, incorporate insights from the company’s risk analysis. Klingelhöfer etal. (2009) provide an example of using these methods to determine a ‘maximum affordable settlement’ 28, thereby establishing a decision value.29 The calculation requires linear optimisation, which takes into account existing investment and financing options. For the first time, Hering etal. (2012) combined this method with a Monte Carlo simulation to value venture capital.30 Building on this work, Hering etal. (2013a) presented a model for simulative company valuation that addresses capital market imperfections and ambiguous cash flows associated with the asset, ultimately determining a range of subjective marginal prices. This valuation also required linear optimisation. In response to critiques of traditional valuation methods based on DCF and CAPM, Olbrich etal. (2015) developed a similar method specifically designed for M&A valuations. The semi-investment-theoretical valuation paradigm31 draws inspiration from investment-theoretical valuation concepts and integrates them with risk-value models.32 However, it also accepts certain simplifications that are common in financial valuation theory.33 In contrast to the strict focus on a specific business subject34, semi-investment-theoretical valuation paradigm embraces simplifications regarding the options considered. For instance, it typically evaluates two alternative investment options for the asset: a risk-free investment and a broad stock market index. TheMonte Carlo simulation aggregates the company’s risks using quantitative analysis based on corporate planning (risk aggregation)35, thereby accounting for financing restrictions and insolvency risks.36 Based on the simplified as27 See Hering/Toll (2013). 28 See Klingelhöfer etal. (2009, p. 302). 29 This holds especially for the valuation of changes in voting rights. 30 They utilize a simulation-based version of the state marginal quota model. 31 See Gleißner/Follert (2022) and Gleißner/Ernst (2024). 32 See Dorfleitner/Gleißner (2018), discusses below. 33 See Gleißner/Follert (2022). 34 See Olbrich etal. (2015) and note II.1. 35 For risk aggregation using Monte Carlo simulation in risk management, see e. g. Hunziker (2021); Gleißner (2019) and Vanini/Rieg (2021). 36 For the basic idea, see Coenenberg (1970) and Gleißner (2019); Gleißner/Ernst (2023) and Ernst (2022a, 2022b). Supplementary Hering etal. (2013a) on simulation in the context of investment theory and Matschke/Brösel (2013, pp. 277 – 253) on dealing with uncertainty. M&A and the Simulation-Based Valuation of Companies 191 Credit and Capital Markets, 57 (2024) 1 – 4 sumption of reliable environmental parameters, this approach yields a single reliable company value, rather than a range of values. The integration of a risk-value model37, which employs a risk measure to capture the risk content of cash flows, facilitates a comprehensive transformation of uncertainty concerning those cash flows. Alongside the limitation of alternative investment options, this uncertainty transformation through the risk-value model is the key feature that distinguishes semi-investment-theoretical valuation paradigm from its predecessors. Even independently of published case studies, surveys indicate that the use of simulation-based valuation methods has increased significantly in recent years, particularly for complex valuation problems. According to Gleißner etal. (2024), a survey of German valuation professionals revealed that 80 % of respondents already utilise such methods. 2. Theoretical background to our approach Our approach for the risk-adequate valuation of an investment company to be sold at an uncertain exit price integrates valuation and price-estimation methods.38 To ensure a strong connection between theoretical foundations and practical relevance, our approach focuses on three theoretical concepts: multiple valuation, risk-value model valuation, and Monte-Carlo simulation. Multiple valuation is a straightforward approach that is widely used in practice. It relies on deriving the valuation from a similar or comparable company based on the principle of proportionality.39 We chose to use this concept not for its theoretical elegance, but because provides an accurate representation of reality when selling private equity shares in a company. Schüler (2020) provides an overview of the conceptual requirements and implementation of company valuations using multiples, highlighting which multiples are suitable for accurate valuations and how the multiples used in practice are interrelated (e. g., enterprise value/sales or enterprise value/EBITDA). Additionally, empirical studies have been conducted by Cheng/McNamara (2000) and Chullen et al. (2015). Leveraged buyouts (LBOs) demonstrate that the total company value (enterprise value), based on median figures, is approximately seven times the EBITDA. This finding comes from an analysis of valuation multiples for European LBOs dur37 See Gleißner/Dorfleitner (2018) on the derivation of the valuation equations using incomplete replication. 38 See Hering etal. (2012) on the valuation of venture capital. 39 See Berk/DeMarzo (2023, ch. 9). 192 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 ing the period from 2000 to 2003,40 revealing significant fluctuations in valuation over time.41 Most older studies on multiple valuations primarily focus on industry multiples, often incorporating additional factors such as historical growth.42 Previous empirical studies have demonstrated that the price-earnings ratio can be effectively explained by industry affiliation, as it provides a suitable representation of risk and earnings growth.43 The estimation accuracy of industry-related valuation multiples can be further enhanced by using return on equity as an additional selection criterion alongside industry affiliation.44 These investigations, which use controlled multiples based on fundamental factors, yield significantly improved results and indicate that industry affiliation does not contribute additional explanatory power compared to the fundamental explanatory factors.45 The second, more theoretically oriented, conceptual framework used in our analysis is the risk-value model of valuation, as outlined by Dorfleitner/Gleißner (2018). This approach involves comparing the investment to be valued with a reference investment that has equal risk and expected end value. Notably, this methodology avoids unrealistic assumptions and does not require extensive information. Its grounding in rational decision theory makes it particularly well-suited for addressing the valuation problem at hand. Fundamental ideas for risk-adequate valuation, which do not rely on the assumption of a perfect capital market, stem from investment theory (see II.1.).46 These methods have emerged in competition with valuation theories based on financing theory, which assume the perfection of the capital market (for example, the CAPM). Due to the high complexity associated with classical investment theory methods,47 alternative valuation approaches, known as semi-in40 Accordingly, the ratio of the total enterprise value to the difference between EBITDA and investments in property, plant and equipment is around 10. See Richter (2005, p. 181). 41 The financial investors finance the total company value with an average of 63 % debt, corresponding to 4.6 times the EBITDA, see also Roosenboom (2012). 42 The best estimation results for market prices can probably be achieved via a combination of DCF company-valuation methods and market-oriented multiples (Herrmann, 2002, p. 31; Kaplan/Ruback, 1996, p. 45; DeAngelo, 1989, p. 93; Bruner etal., 1998, p. 13). 43 See e. g., Alford (1992). 44 See Alford (1992). 45 See Herrmann (2002); Richter (2005) and Kelleners (2004). 46 See Matschke etal. (2010); Hering etal. (2013a); Hering etal. (2014); Matschke/Brösel (2021). 47 See Matschke etal. (2010) and Matschke etal. (2020). M&A and the Simulation-Based Valuation of Companies 199 Credit and Capital Markets, 57 (2024) 1 – 4 therefore, actual bankruptcy).73 Because the exit price is paid on the equity, the debt capital T D  must be repaid in accordance with the company’s value. This results in the following equation: (6)  ( ) max 0; , EXIT TT T P m EBIT D= ×-   where m  represents the stochastic multiple. This is a ‘price estimate’ in an imperfect market, and this exit price may deviate from a ‘reasonable’ fundamental value (although, as is well known, there is no such thing as a ‘true’ value).74 The exit price estimate is obtained from capital-market data (e. g., stock-market prices) or realised transaction prices of ‘comparable’ companies using a comparison method. The comparison method determines a potential market price (‘stock-exchange price’) likely to be achievable on the market. Such methods are therefore also referred to as market-oriented valuation methods. In the multiplier method, the potential market price P  is determined by multiplying a specific (size-related) parameter X of the company to be valued by a factor, m, that depends on the selected reference value and is usually industry-specific.75 =×- () ,PX m X D where  P is the estimated market price, X is the parameter,76 m is the industry-specific factor77 and D is debt capital. EBITDA, EBIT, cash flow or sales are commonly used to operationalize X. In terms of suitability for calculating multiples, Liu etal. (2002) rank the possibilities as follows: (1) earnings forecasts, (2)historical earnings, (3) cash flow and book value of equity, and (4) sales.78 73 See Gleißner (2010, 2019) and Franken etal. (2020) on the significance of the probability of insolvency, which is only touched upon here. 74 Although this should not exist in perfect markets, see Shleifer/Vishny (1997) for such incorrect valuations; Haugen (2002) and Campbell/Shiller (1998). 75 The debt capital FK is deducted if an enterprise value is calculated with ‘m · X’. 76 Examples are EBITDA or EBIT. 77 One example is the quasi-reciprocal discount factor. See Cochrane (2011). 78 Baker/Ruback’s (1999) empirical study of industry multiples for the S&P 500 index in 1995 shows that multiples based on EBITDA lead to better estimates of market prices than those based on EBIT or sales. In addition, in this empirical study, the formation of a harmonic mean turns out to be a suitable method for calculating the valuation multiple; see also Kaboth et al. (2022); Herrmann/Richter (2003); Yoo (2006) and Schreiner (2007). 200 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 3. An alternative risk measure: the semi-standard deviation While the standard deviation measures positive and negative fluctuations around the expected value, the possibility of negative deviations can be regarded as more relevant to the decision to buy or sell a company, as these determine the company’s equity requirements and, thus, the utilisation of the (scarce) risk-coverage potential.79 Therefore, the semi-standard deviation (7) ( ) ( )( ) ( ) 1 22 max ;0 .Z E EZ Z σ - =-   Could be more appropriate as a measure of risk.80 The importance of possible negative deviations from the plan and, especially, of possible losses in the valuation is well documented in research,81 which justifies the use of the semi-standard deviation risk measure.82 This is particularly true in the context of private-equity investments, where downward and upward deviations are not symmetrically distributed. The marginal price (decision value) at which a purchase becomes worthwhile from the point of view of a potential investor can again be determined using Equation (5) with () () .. R σ - =. In the case of a log-normal distribution, T σ λ - can be determined by simulation (see the case study in Section IV). For simplicity, it is assumed here that the investor does not have to spend any supplemental equity for the company in addition to the purchase price in order to provide it with equity that is in line with the risk. If this were the case, the required equity capital would reduce the marginal price. This equity requirement can also be estimated directly from the simulation. Furthermore, the probability of insolvency is implicitly considered in this marginal price given that the total loss of the investment is also taken into account in the simulation in the event of unfavourable business development. 4. Special rights and share valuation With the procedure outlined here, a risk-adequate assessment is possible not only for the entire company but also for individual ‘equity tranches’ with their specific rights. Knowing the complete frequency distribution of the uncertain 79 See Gleißner (2022, pp. 428 – 488) on the so-called risk-coverage approach. 80 For the importance of downside risks and the ‘skewness’ of a distribution of results, see Kraus/Litzenberger (1976). 81 See, for instance, Kahnemann/Tversky (1979) on the basics of psychology. 82 And it is precisely this risk measure that can – better than the standard deviation–capture the implication of the limitations of liability and the ‘clipping’ of the possible losses. M&A and the Simulation-Based Valuation of Companies 201 Credit and Capital Markets, 57 (2024) 1 – 4 exit price makes it easy to depict existing contractual agreements with individual investors (equity providers) and thereby independently determine the range of the ‘uncertain return flow’ for each equity provider. According to the given rules, the total exit price is split into the exit price shares of the n individual investors Exit i P (here, i = 1, …, n): (8) 1 . n Exit Exit i i PP = =å With the methods for a risk-based assessment outlined above, considering the expected value and risk of the returns to each investor I = 1, …, n, the investors’ subjective value can be calculated, taking all special rights into account. All that is necessary is a breakdown of the uncertain exit price in accordance with the contractual provisions. If, for example, an investor i, who financed the last equity increase of a company, has a ‘liquidity preference’ such that they are initially served by the uncertain exit price until his or her capital investment Ii is reached and then proportionately with a share αi of the exit price, the following payoff applies: (9) { }{ } max min , ; Exit Exit Exit ii i P PI P α =× . 5. A model for the EBIT development over time In practice, the collection of company-specific data for the valuation is based on a detailed due-diligence (including supported) detection of further risks that determine the planning security. The most important results are estimates of the expected value and risk of the stochastic sales growth rates 11 , , , T gg - ¼ and an estimate of the expected EBIT margin and the typical extent (standard deviation) of risk-related deviations from this forecast. Based on sales of the previous period 0 R of €15 million and sales growth g0 of 20 % in the previous period, TextitAI, Inc. expects a linear decrease in the sales growth rate of 2 % from the terminal value period (meaning that planning is done with 5€25 Plan R= million).83 The coefficient of variation of the sales growth rate (i. e., the ratio of the standard deviation to the expected value) should be constant at 25 % annually, with deviations from the expected sales growth rate assumed to be normally distributed. (10) ( ) ( ) ( ) 11 11 g tt t t tt R R g R Eg ε -- = += + +     , where ( ) 0; gg tt N εσ ~ . 83 For other models, see, for example, Schwartz/Moon (2000, 2001); Behm (2003) and Klobucnik/Sievers (2013). 202 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 At the beginning (i. e., at t = 0), the EBIT margin (EBITM) is –1 %; in the long term, a value of 7.5 % is expected ( 7.5%EBITM =). The traditional EBIT planned for t = T = 5 without simulation is therefore: · 25 · 7.5% Plan Plan Plan TT T EBIT R EBITM == = € 1.875 million. A mean-reverting process is assumed for the development of the EBITM. Such a risky process trends towards the EBITM . The further the value from the previous period  1t EBITM - deviates from this mean, the stronger the mean-reverting tendency. Formally, this can be described recursively as follows: (11)   ( ) 1. EBITM tt t EBITM EBITM a EBITM EBITM ε - =+ -+  Deviations from the expected trend value ( ) ε EBITM t E are assumed to be normally distributed with a standard deviation EBITM t σ . The standard deviation of the EBIT margin should decrease linearly in the planning period, from 3 % in the past to 1 % in the terminal value period.84 As an alternative to the simple stochastic process of the EBIT margin, more detailed planning can be carried out. The fluctuation range of the EBIT margin can be determined using simulation-based risk aggregation methods. Starting from identified and valuated individual risks, these methods enable the direct calculation of the typical risk-related range of EBIT development by calculating a representative number of risk-related future scenarios. The interest rate on borrowed capital i for interest on borrowed capital (D) should remain constant85 over the period under consideration and is assumed to be 5.1 %. The simulation is carried out recursively, that is, progressively from t–1 to t. The profit before tax in period t is determined according to the following equation: (12)   1 tt tt EBT R EBITM D i - =× - ×  . For the sake of simplicity, taxes are neglected here. Growth is to be financed by retaining the profit before tax, so no payouts are made. It is assumed that the capital employed at the beginning of the planning period, namely 00 o CE E D=+ , develops analogously to sales. In other words, the capital turnover remains constant, leading to the following equation: 84 Various data sources such as comparative industry values, the results of risk analyses or subjective estimates by experts can be used to estimate these parameters. 85 Assume that the company has a 5-year line of credit with a constant, deterministic interest rate. Strictly speaking, the interest cash flows (I) from the point of view of t=0 are a conditional distribution, the form of which depends on the risks realised up to t – 1. M&A and the Simulation-Based Valuation of Companies 203 Credit and Capital Markets, 57 (2024) 1 – 4 (13)   10 0 1 tt tt t RR CE CE CE R R- - ==  . If the EBT (earnings before taxes) is not sufficient for financing, additional outside capital is taken out. If the EBT exceeds the necessary investment amount, borrowed capital is repaid. This results in the following equation: (14)    1 1 t tt tt D D EBT CE CE - - = - +-  . The (net) borrowed capital can therefore also assume negative values. These can be regarded as liquid funds. Interest income is then generated with these, where for simplicity no distinction is made between the interest rate on borrowed capital and the interest on the credit balance (each of which = i). The debt capital D can thus be interpreted as a net bank liability. Given that distributions and capital increases are excluded, the equity at the end of period t is the sum of the equity at the beginning of the period and the EBT: (15)  1t tt E E EBT - =+  . To simulate EXIT P  according to formula (6), the EBIT at time t = T must be calculated. This is done according to the relationship already used in formula (13): (16)   tt t EBIT R EBITM=× . The triangular distributed multiple m is stochastically simulated independently of the other variables. 6. Model discussion Our approach can be used to determine decision values or argumentation values (see II.1.86). To establish an argumentation value in a purchase negotiation, parameters are set cautiously, so that they result in a very low valuation that acts as the starting point for the price negotiations. When realistic parameters are applied from the perspective of the valuation object (the buyer), a decision value is obtained, indicating the maximum acceptable purchase price. In a negotiation context, our approach has the advantage that only a consensus on a probability distribution is necessary concerning uncertain assumptions. For instance, this may pertain to the future sales growth rate or the price multiple achievable at the exit time. 86 See also Follert etal. (2018). 204 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 When determining company values for private equity or venture capital investments, it is advantageous that the procedure is based on the familiar method of assessing adequate prices using EBIT or EBITDA multiples. Extending this existing approach to account for future uncertainties, such as the sales growth rate and the future valuation levels, represents a significant benefit. It is also advantageous that, unlike valuations based on financing theory using the CAPM, this method does not assume a perfect capital market; instead, it considers existing rating and financing restrictions. The combination of a Monte Carlo simulation for estimating an uncertain exit price, combined with the results of a quantitative analysis of the company’s risks, and the application of the semi-investment-theoretical valuation paradigm based on imperfect replication, further enhances the robustness of this approach. In the simple model presented in Chapter III, it may be seen as a disadvantage that the valuation does not fully incorporate integrated corporate planning. While adding this element would increase complexity, it is certainly feasible to implement. From the perspective of financial valuation theory, a notable disadvantage is the absence of the CAPM, which remains widely used in practice, particularly in legal company valuations. However, there is a growing demand for simulation-based methods in demanding valuation cases,87 and the CAPM is increasingly subject to critical scrutiny in legal contexts, such as when calculating compensation for minority shareholders in Germany.88 From the perspective of investment valuation theory, the model can be critiqued for its simplifications, particularly regarding alternative investment options, which do not account for potentially relevant information about the valuation subject’s opportunities and restrictions.89 For example, the model is not designed to evaluate a large number of simultaneously feasible investment opportunities within a contextual framework, making it unsuitable for determining an optimal portfolio that considers the interdependencies between individual VC or PE investments. For the sake of simplicity, the valuation assumes that all other valuation issues have been resolved and that only one investment opportunity needs to be evaluated in isolation. However, this limitation can be relaxed by determining the respective net present value rates for multiple investment alternatives, i. e., calculating the enterprise value (V) in relation to the initial investment (Io), which facilitates prioritisation. From this perspective, the calculated enterprise value is not a decision value does not serve as a decision value in the narrower sense, as 87 See the empirical survey in Gleißner etal. (2024). 88 For example, see Lauber (2014); Follert (2019); Quill (2020) and Gleißner/Follert (2022). 89 See again Hering/Toll (2013) and Hering etal. (2013b). M&A and the Simulation-Based Valuation of Companies 205 Credit and Capital Markets, 57 (2024) 1 – 4 it does not incorporate all information relevant to the valuation subject. Instead, the valuation result can be viewed as a ‘typified decision value’ rather than a ‘subjective decision value’.90 In contrast to the approaches of Hering etal. (2013a) and other frameworks in investment-theoretical valuation theory, our model, inspired by the ideas of Fama (1977), determines a specific value rather than a range of values. This value results from a complete transformation of uncertainty, achieved through the application of a risk-value model. The risks associated with cash flows are converted into a safe figure via the risk measure, while uncertainties related to environment parameters (e. g. the risk-free interest rate) are neglected in line with the principles outlined by Fama (1977). Therefore, semi-investment-theoretical valuation paradigm, similar to financial-theoretical valuation based on CAPM, aims to determine a definite value rather than a value range, provided the environment parameters are assumed to be stable. IV. The valuation of a PE investment: A case study with an uncertain exit price 1. Basics and valuation assumptions The following simple case study of a non-listed participation in an investment fund shows how a simulation-based valuation can be carried out.91 In the case study, the marginal price P* is to be determined for TextitAI, Inc., with an assumed exit in T = 5 years. The operating capital of € 7.5 million is financed with equity at the level of 20 % (= €1.5 million). The current borrowed capital D0 is thus € 6 million. Two partners own the company. The founder (G) of the company and a venture-capital (VC) fund that joined after the company’s founding each own 50 % of the shares. With its entry into the company and the acquisition of the 50 % shareholder share, the VC fund contributed one million euros in equity. It has agreed to a special right (liquidity preference) for the intended exit, the sale to a strategic investor, and will receive the first million euros from the uncertain future exit price ( ) EXIT p exclusively. The excess amount is divided in half. The owners have two alternative investment options to choose from: quasirisk-free government bonds with an interest rate of rf = 3 %, or a broadly diversified stock index with an uncertain market return  m r . Their annual distribution is i.i.d. (independent and identically distributed) assumed to be log-normally 90 A distinction that is also to be made in Germany with the revision of the IDW S1 valuation standard (IDW ES 1, 2024). 91 See also Gleißner/Ernst (2019) and Ernst (2022a). 206 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 distributed, with an expected value e m r = 8 % and standard deviation σ m r = 20 %. Equation (4) produces market price of the risk T σ λ results for the risk-measure standard deviation as shown in Table 1. Table 1 Multi-period risk price depending on the period length T using the standard deviation as a risk measure and log-normally distributed market returns T 1 2 3 4 5 T σ λ 0.25 0.34 0.41 0.45 0.49 These values (as well as the ones in Table 2) result from a Monte Carlo simulation with the parameters given above. With the same parameters, the T σ λ - results shown in Table 2 are obtained for the risk measure of the (lower) standard deviation (σ). Table 2 Multi-period risk price depending on the period length T using the semi-standard deviation as a risk measure and log-normally distributed market returns T 1 2 3 4 5 T σ λ -0.33 0.44 0.51 0.57 0.60 In the example case, it is fixed that • thepredictedexittakesplaceinyearT = 5, • theriskdiversificationfactorisassumedtobed = 0.5,92 • atthetimeofexit,avaluationmultiple(m) based on EBIT with a minimum of 6, the most likely value of 8 and a maximum of 11 is assumed for the rated company (modelled with a triangular distribution). The most likely value is considered as the traditional planned value for the EBIT multiple m: 8.m= In addition, a sales multiple typical for the industry is known, namely mR = 0.7. Finally, the additional funding obligations of equity investors are excluded, so the exit cannot take negative values. With these specifications, the uniform val92 This means that half of the overall risk is assumed to be relevant to the assessor after diversification effects. In view of the imperfect diversification of most valuation subjects, this use of an average risk-diversification degree of d = 0.5 seems acceptable. M&A and the Simulation-Based Valuation of Companies 207 Credit and Capital Markets, 57 (2024) 1 – 4 uation equation for the marginal price (fundamental value) of the company is derived according to equation (5), whereby only two variables have to be estimated through using company-specific simulations: the expected value of the exit price, using a (stochastic) multiplier method; and the semi-standard deviation of the exit price, as a measure of the overall (operational) risk exposure. That is: (17) ( ) ( )( ) ( ) σ - =-   1 22 max ; 0 Exit Exit Exit P E EP P =  ( )( )  ( )( ) ( ) 1 22 max max 0; max 0; ;0 . tT tT tT tT E E m EBIT D m EBIT D == == = ×--×-   2. Risk simulation and exit price range Figure 1 shows the course of the expected values for the capital items. This graphic was created using the values given above via simulation.93 Figure 1: Expected course of the capital positions Based on the (manageable) informational input from the analysis, we perform a simulation followed by a valuation. Figure 2 shows the range of EBIT development. The values result from a Monte Carlo simulation that we carried out in Microsoft Excel with a supplemental add-in. 93 We use Excel and Crystal Ball. 0 2000 4000 6000 8000 10000 12000 14000 0 1 2 3 4 5 T€ Period (in years) Equity Debt Capital Employed 208 Werner Gleißner, Marco Wolfrum, Gregor Dorfleitner Credit and Capital Markets, 57 (2024) 1 – 4 Figure 2: Range of possible EBIT developments The initial objective is to perform a risk-based assessment of the company as a whole and assess the shares of the two shareholders, considering the special rights of the venture-capital investor. The immediate result of the simulations is the distribution of Exit P  at T = 5, calculated according to (6), from which the expected value and (semi-)standard deviation can be calculated. Figure 3 shows the results of the simulation. To determine a marginal price P*, the range of the uncertain exit price must be estimated. The simulation of the cash flow in the planning period (and the implicit underlying risk assessment) is mainly used to determine the possible states of the company (and the environment) at the time of exit and to derive an exit price from them. 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