A nonlinear state marginal price vector model for the task of business valuation. A case study: The dimensioning of IT-service companies under nonlinear synergy effects

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In the present contribution we present a nonlinear extension of the innovative linear investment-oriented company valuation method and so-called state marginal price vector model of Toll which represents a two-step procedure separated into a base and a valuation approach. As novel aspect we address nonlinear synergy effects in M&A’s. For this purpose we introduce a nonlinear framework within a semi-discrete convex optimization approach. As capital market assumption we simulate an imperfect market. To demonstrate the usefulness of the method, we address a case study of a merger of two IT-service companies. The related valuation and dimensioning of capacities is done by solving a multi-period newsvendor model under stochastic demand. The demonstrated nonlinear framework is shown to be suitable for a wide range of business valuation tasks.

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  1. 1.

    To bridge the gap between linear and nonlinear programming, we can analyze Eq. (26) and observe that \(h(Q) = P(Q) - Q \, \partial _Q P(Q)\). Hence, considering an unbounded and linear and, thus, homogeneous function of degree \(k=1\) we can conclude that the related object is marginal or unrealized using the Euler relation \(Q \, \partial _Q P = k \, P(Q)\) for a homogeneous function \(P(\lambda \, Q) = \lambda ^k \, P(Q)\) of degree k since \(h(Q) = (1-k) \, P(Q) = 0\), which is indeed zero.


  1. Albach H (1962) Investition und Liquidität. Gabler, Wiesbaden

  2. Alfares HK, Elmorra HH (2005) The distribution-free newsboy problem: Extensions to the shortage penalty case. Int J Prod Econ 93–94(1):465–477

  3. Allais M (1953) Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine. Econometrica 21(4):503–546

  4. Angelus A, Porteus EL (2002) Simultaneous capacity and production management of short-life-cycle, produce-to-stock goods under stochastic demand. Manage Sci 48(3):399–413

  5. Arrow KJ (1964) The role of securities in the optimal allocation of risk-bearing. Rev Econ Stud 31(2):91–96

  6. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont

  7. Bronstein IN, Semendjajew KA (1991) Convex optimization. In: Handbook of mathematics (German edition, Ergänzende Kapitel), 6th edn. Harri Deutsch, Frankfurt, pp 135–137

  8. Brösel G, Matschke MJ, Olbrich M (2012a) Valuation of entrepreneurial businesses. Int J Entrep Ventur 4(3):239–256

  9. Brösel G, Toll M, Zimmermann M (2011) What the financial crisis reveals about business valuation. Manag Econom 5(10):27–39

  10. Brösel G, Toll M, Zimmermann M (2012b) Lessons learned from the financial crisis unveiling alternative approaches within valuation and accounting theory. Financ Rep 4(4):87–107

  11. Casimir RJ (2002) The value of information in the multi-item newsboy problem. Omega 30(1):45–50

  12. Coleman L (2014) Why finance theory fails to survive contact with the real world: A fund manager perspective. Crit Perspect Account 25(3):226–236

  13. Copeland JE, Koller T, Murrin J (1990) Valuation. Wiley, New York

  14. Damodaran A (2011) Applied corporate finance, 3rd edn. Wiley, Hoboken

  15. Debreu G (1959) Theory of value. Yale University Press, New Haven

  16. Dixit AK, Pindyck RS (1993) Investment under uncertainty. Princeton University Press, Princeton

  17. Gallego G, Moon I (1993) The distribution free newsboy problem: Review and extensions. J Oper Res Soc 44(8):825–834

  18. Garman MB, Ohlson JA (1981) Valuation of risky assets in arbitrage-free economies with transactions costs. J Financ Econ 9(3):271–280

  19. Griva I, Nash SG, Sofer A (2009) Linear and nonlinear optimization, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia

  20. Hax H (1964) Investitions- und Finanzplanung mit Hilfe der linearen Programmierung. Schmalenbachs Zeitschrift für betriebswirtschaftliche Forschung 16(6):430–446

  21. Hering T (2000) Das allgemeine Zustands-Grenzpreismodell zur Bewertung von Unternehmen und anderen unsicheren Zahlungsströmen. Die Betriebswirtschaft 60(3):362–378

  22. Hering T, Olbrich M, Steinrücke M (2006) Valuation of start-up internet companies. Int J Technol Manag 33(4):406–419

  23. Hering T, Toll C (2015) Application of alternative valuation formulas for a company sale. Glob Econ Finance J 8(2):14–30

  24. Hering T, Toll C, Kirilova PK (2014a) Acquiring a company: Assessing the maximum affordable price. World Rev Bus Res 4(3):35–44

  25. Hering T, Toll C, Kirilova PK (2014b) How to compute a decision-oriented business value for a company sale. J Account Finance Econ 4(1):43–52

  26. Hering T, Toll C, Kirilova PK (2015a) Business valuation for a company purchase: Application of valuation formulas. Int Rev Bus Res Pap 11(1):1–10

  27. Hering T, Toll C, Kirilova PK (2015b) Selling a company: Assessing the minimum demandable price. Glob Rev Account Finance 6(1):19–26

  28. Hering T, Toll C, Kirilova PK (2016) Assessing the maximum expendable quota for a milestone financing provided by a venture capitalist. Int J Entrep Ventur 8(1):102–117

  29. Hertz DB (1964) Risk analysis in capital investment. Harvard Bus Rev 42(1):95–106

  30. Höck M (2005) Dienstleistungsmanagement aus produktionswirtschaftlicher Sicht. Gabler, Wiesbaden

  31. Höck M (2008) Ein Planungsansatz zur Kapazitätsdimensionierung von IuK-Techniken. Zeitschrift für Planung und Unternehmenssteuerung 19(2):143–158

  32. Hurd CC (1954) Simulation by computation as an operations research tool. J Opt Soc Am 2(2):205–207

  33. Inwinkl P, Schneider G (2008) Unternehmensbewertung und Zustands-Grenzpreismodelle bei Agency-Problemen. Betriebswirtschaftliche Forschung und Praxis 60(3):276–292

  34. Inwinkl P, Kortebusch D, Schneider G (2009) Das allgemeine Zustands-Grenzpreismodell zur Bewertung von Unternehmen bei beidseitigen Agency-Konflikten. Betriebswirtschaftliche Forschung und Praxis 61(4):403–421

  35. Khouja M (1999) The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega 27(5):537–553

  36. Koller T, Goedhart MH, Wessels D (2010) Valuation, 5th edn. Wiley, Hoboken

  37. Laux H, Franke G (1969) Zum Problem der Bewertung von Unternehmungen und anderen Investitionsgütern. Unternehmensforschung 13(3):205–223

  38. Lerm M, Rollberg R, Kurz P (2012) Financial valuation of start-up businesses with and without venture capital. Int J Entrep Ventur 4(3):257–275

  39. Lintner J (1965) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev Econ Stat 47(1):13–37

  40. Luenberger DG (2016) Linear and nonlinear programming, 4th edn. Springer, New York

  41. Markowitz HM (1952) Portfolio selection. J Finance 7(1):77–91

  42. Matschke MJ (1975) Der Entscheidungswert der Unternehmung. Gabler, Wiesbaden

  43. Matschke MJ, Brösel G (2013) Unternehmensbewertung, Funktionen—Methoden—Grundsätze, 4th edn. Springer, Wiesbaden

  44. Matschke MJ, Brösel G, Matschke X (2010) Fundamentals of functional business valuation. J Bus Valuat Econ Loss Anal 5(1):1–39

  45. Mossin J (1966) Equilibrium in a capital asset market. Econometrica 34(4):768–783

  46. Mullins DW (1982) Does the capital asset pricing model work? Harvard Bus Rev 60(1):105–114

  47. Myers SC (1974) Interactions of corporate financing and investment decisions—implications for capital budgeting. J Finance 29(1):1–25

  48. Olbrich M, Brösel G, Hasslinger M (2009) The valuation of airport slots. J Air Law Commer 74(4):897–917

  49. Olbrich M, Quill T, Rapp DJ (2015) Business valuation inspired by the Austrian school. J Bus Valuat Econ Loss Anal 10(1):1–43

  50. Pfaff D, Pfeiffer T, Gathge D (2002) Unternehmensbewertung und Zustands-Grenzpreismodelle. Betriebs-wirtschaftliche Forschung und Praxis 54(2):198–210

  51. Porteus EL (1990) Stochastic inventory theory. In: Heyman DP, Sobel MJ (eds) Handbooks in operations research and management science, vol 2. North-Holland, Amsterdam, pp 605–652

  52. Rapp DJ (2015) Boom and bust: The role of business valuation in the recent financial crisis. J Prices Mark 4(1):86–93

  53. Rossi R, Prestwich S, Tarim SA, Hnich B (2014) Confidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand. Eur J Oper Res 239(3):674–684

  54. Salazar RC, Sen SK (1968) A simulation model of capital budgeting under uncertainty. Manag Sci 15(4):B161–B179

  55. Sankarasubramanian E, Kumaraswamy S (1983) Optimal ordering quantity for pre-determined level of profit. Manag Sci 29(4):512–514

  56. Sharpe WF (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. J Finance 19(3):425–442

  57. Silver EA, Pyke DF, Peterson R (1998) Inventory Management and Production Planning and Scheduling, 3rd edn. Wiley, New York

  58. Strang G (1986) Introduction to applied mathematics. Wellelsley-Cambridge Press, Wellesley

  59. Tobin J (1958) Liquidity preference as behavior towards risk. Rev Econ Stud 25(2):65–86

  60. Toll C (2010) Unternehmensbewertung bei Vorliegen verhandelbarer Zahlungsmodalitäten. Betriebs-wirtschaftliche Forschung und Praxis 62(4):384–411

  61. Weingartner HM (1963) Mathematical programming and the analysis of capital budgeting problems. Prentice Hall, Englewood Cliffs

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Correspondence to Christian Toll.


A Convex optimization in a nutshell

A convex objective function \(F(\mathbf{v})\) of \(n+1\) variables under m convex constraints \(\mathbf{f}(\mathbf{v})\le \mathbf{0}\) can be formulated as a Lagrange function:

$$\begin{aligned} \phi (\mathbf{v},\mathbf{u}) = F(\mathbf{v}) + \mathbf{u}\cdot \mathbf{f}(\mathbf{v}) \qquad \qquad \mathbf{v}\in {\mathbb R}_0^{+ \, n+1}, \mathbf{u}\in {\mathbb R}_0^{+ \, m} \end{aligned}$$

where \(\mathbf{v}\) and \(\mathbf{u}\) are the corresponding non-negative primal and dual variables, respectively (Bronstein and Semendjajew 1991). For the primal optimization problem this function is minimized with respect to \(\mathbf{v}\) and for the dual optimization problem maximized with respect to \(\mathbf{u}\). The resulting saddle point problem defines the interval for the optimal solution as follows (please observe the fine symmetry also with regard to the Kuhn-Tucker conditions):

$$\begin{aligned} \phi (\bar{\mathbf{v}},\mathbf{u}) = \min \limits _{\mathbf{v}} \phi (\mathbf{v},\mathbf{u}) \le \phi (\bar{\mathbf{v}},\bar{\mathbf{u}}) \le \max \limits _{\mathbf{u}} \phi (\mathbf{v},\mathbf{u}) = \phi (\mathbf{v},\bar{\mathbf{u}}) . \end{aligned}$$

In the optimum this interval is mapped onto the optimal point

$$\begin{aligned} \phi (\bar{\mathbf{v}},\bar{\mathbf{u}}) = \min \limits _{\mathbf{v}}(\max \limits _{\mathbf{u}} \phi (\mathbf{v},\mathbf{u})) = \max \limits _{\mathbf{u}} (\min \limits _{\mathbf{v}} \phi (\mathbf{v},\mathbf{u})) . \end{aligned}$$

The sufficient conditions for the existence of a global optimum are called Karush-Kuhn-Tucker or Kuhn-Tucker conditions, see Luenberger (2016) or Strang (1986), p. 724, which can be formulated for the primal problem as follows:

$$\begin{aligned} \text{ a) } \mathbf{f}(\bar{\mathbf{v}}) =\bar{\partial }_{\mathbf{u}} \phi (\bar{\mathbf{v}},\mathbf{u}) \le \mathbf{0},\qquad \text{ b) } \bar{\mathbf{u}} \cdot \bar{\partial }_{\mathbf{u}} \phi (\bar{\mathbf{v}},\mathbf{u}) = 0,\qquad \text{ c) } \mathbf{u}\ge \mathbf{0} \end{aligned}$$

with a function \(\mathbf{f}(\mathbf{v})\) convex in \(\mathbf{v}\). Here, \(\partial _{\mathbf{v}} \phi (\mathbf{v},\mathbf{u})\) denotes the gradient of \(\phi \) with respect to \(\mathbf{v}\) with the notation \((\frac{\partial \phi }{\partial v_1},\ldots ,\frac{\partial \phi }{\partial v_{n+1}})\). \(\bar{\partial }_{\mathbf{v}}(\cdot )\) means \(\partial _{\mathbf{v}} (\cdot ) \Big |_{\mathbf{v}= \bar{\mathbf{v}}}\).

The Kuhn-Tucker conditions for the dual optimization problem read as

$$\begin{aligned} \text{ a) } \bar{\partial }_{\mathbf{v}} \phi (\mathbf{v},\bar{\mathbf{u}}) \ge \mathbf{0}\qquad \text{ b) } \bar{\mathbf{v}} \cdot \bar{\partial }_{\mathbf{v}} \phi (\mathbf{v},\bar{\mathbf{u}}) = 0,\qquad \text{ c) } \mathbf{v}\ge \mathbf{0} . \end{aligned}$$

If we consider a linear equation system like in the linear programming case, the given primal optimization problem is simply rearranged to obtain its corresponding dual optimization problem and vice versa. Let \(\max _{\mathbf{v}} \mathbf{c}\cdot \mathbf{v}\) be the linear objective function under the linear constraints \({\mathbb H} \cdot \mathbf{v}\le \mathbf{b}\) such that the Lagrange function can be formulated as \(\phi (\mathbf{v},\mathbf{u}) = -\mathbf{c}\cdot \mathbf{v}+ \mathbf{u}\cdot ({\mathbb H} \cdot \mathbf{v}- \mathbf{b})\). Rearranging terms we obtain the expression \(\phi (\mathbf{v},\mathbf{u}) = -\mathbf{b}\cdot \mathbf{u}+ \mathbf{v}\cdot (-\mathbf{c}+ {\mathbb H}{}^T \cdot \mathbf{u})\). Hence, by considering the Kuhn-Tucker conditions of the dual optimization problem the Lagrange function \(\min _{\mathbf{u}} \mathbf{b}\cdot \mathbf{u}\) must hold under the linear constraints \({\mathbb H}{}^T \cdot \mathbf{u}\ge \mathbf{c}\).

B The newsvendor model revisited

We are able to transform the profit function \(P(Q) = E(Q)-L(Q)\) by using E(Q) from Eq. (18) and L(Q) from Eq. (19) as

$$\begin{aligned} P(Q)= & {} - c \, Q + \int ^Q_0 (p_{NV} \, x - c_H \, (Q-x)) \, d\psi (x) + \int ^\infty _Q (p_{NV} \, Q - c_S \, (x-Q)) \, d\psi (x)\nonumber \\= & {} - c \, Q + p_{NV} \, \mu - c_H \, (Q-\mu ) + (p_{NV} + c_S + c_H)\, \int ^\infty _Q (Q-x)\, d\psi (x) \end{aligned}$$

under the presumption that \(\psi (x)|_{-\infty }^0 \approx 0\). For the maximum at \(Q^*\)

$$\begin{aligned} P'(Q) \Big |_{Q = Q^*} = -c - c_H + (p_{NV} + c_S + c_H) \, \int ^\infty _{Q^*} d\psi (x) = 0 \end{aligned}$$

holds such that by using \(\Psi (Q) = \psi (x) \Big |^Q_{-\infty } = (1 - \psi (x)\Big |^\infty _Q)\) we obtain

$$\begin{aligned} \Psi (Q^*) = \psi (x \le Q^*) = \frac{p_{NV} + c_S - c}{p_{NV} + c_S + c_H} \end{aligned}$$

as a critical fractile of the distribution function \(\psi (Q)\), see Sankarasubramanian and Kumaraswamy (1983) or Silver et al. (1998), pp. 385–389. Due to

$$\begin{aligned} P''(Q) = (p_{NV} + c_S + c_H) (-\Psi '(Q)) \le 0 \end{aligned}$$

and \(\Psi '(Q) \ge 0\) we can conclude that P(Q) is concave with respect to Q. We see that the inequality \(c \ge -c_H\) must be satisfied since \(\Psi (Q) \le 1\) in Eq. (39). Moreover, \(k \le K\) holds by definition, see Eq. (20). If we examine the dual Kuhn-Tucker condition of the multi-period newsvendor model in Eq. (26) for \(\bar{Q}_t > 0\) in combination with the nonlinear cost term \(h_c \, Q_t^\beta \) we get for \(\bar{Q}_t < Q_t^*\)

$$\begin{aligned} d_{t+1} \, (c + c_H - (p_{NV} + c_S + c_H)\, (1-\Psi (\bar{Q_t})) + \beta \, h_c \, \bar{Q}_t^{\beta -1} - c_{ct+1}) + d_{t} \, c_{ct} = 0 . \end{aligned}$$

Recalling \(\frac{d_{t+1}}{d_t} = q^{-1}\) we can find a lower bound by assuming \((1 - \Psi (\bar{Q_t})) = 1\) as well as \(\beta = 1\) for \(\bar{Q}_t > \text{ exp }\left( {\frac{\text{ ln }(\beta )}{(1-\beta )}}\right) \) and can derive the following inequalities which have to be satisfied by the parameters:

$$\begin{aligned}&q \, k \le K : p_{NV} + c_S \ge c + h_c - K + q \, k \nonumber \\&q \, k > K : p_{NV} + c_S \ge c + h_c - k + q \, K . \end{aligned}$$

C Are utility functions derivable under practical circumstances?

The empirical derivation of consistent utility functions, which are stable over time, causes insurmountable difficulties for every-day business practice. The well-known paradox of Allais (1953) proves that even under a seemingly plausible preference order a utility function may not exist. Furthermore, because of the axiom of continuity of utility functions according to the well-established principles of Bernoulli it may be impossible to reject alternatives which may be life-threatening for a company, especially when associated only with a small probability. Consumption utility functions are for real decision models no practicable concept and are even more difficult to derive if we consider a multitude of owners. A target which either strives for maximizing end value or consumption spending, fixing one variable by maximizing the other, meets the requirement of operationality and flexibility and serves the interests of owners in view of their demand for additional wealth or income. It is furthermore consistent with empirically observable behavior of German corporate companies which preferably pay a fixed dividend and plow the rest back under the maxim of maximizing end value or stock price.

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Toll, C., Kintzel, O. A nonlinear state marginal price vector model for the task of business valuation. A case study: The dimensioning of IT-service companies under nonlinear synergy effects. Cent Eur J Oper Res 27, 1079–1105 (2019) doi:10.1007/s10100-018-0535-x

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  • Investment analysis
  • Company/business valuation
  • Nonlinear convex programming
  • Nonlinear synergy effects
  • Multi-period newsvendor/newsboy model
  • IT-service companies