# 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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## Abstract

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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## Notes

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.

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## Author information

Correspondence to Christian Toll.

## Appendices

### 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}
(32)

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}
(33)

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}
(34)

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}
(35)

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}
(36)

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}
(37)

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}
(38)

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}
(39)

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}
(40)

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}
(41)

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}
(42)

### 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