Business Valuation for a Company Purchase: Application of Valuation Formulas

International Review of Business Research Papers Vol. 11. No. 1. March 2015 Issue. Pp. 1 – 10 Business Valuation for a Company Purchase: Application ...
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International Review of Business Research Papers Vol. 11. No. 1. March 2015 Issue. Pp. 1 – 10

Business Valuation for a Company Purchase: Application of Valuation Formulas Thomas Hering, Christian Toll and Polina K. Kirilova In this paper a potential buyer (valuation subject) is interested in purchasing a company (valuation object). Thereby, he strives for maximum income and acts in a real imperfect market. In order to sustain his economic interest, he must conduct a business valuation. To help him determine what maximum price he may afford to pay without the transaction proving disadvantageous, we implement an investment theory-based method. The purpose of our paper is to show how formulas derived from the state marginal price model are used to fulfil the described valuation task under realistic imperfect market conditions. As a main conclusion, the correct business value can usually not be calculated using the partial-analytic future earnings method.

JEL Codes: D46, G31 and G34

1. Introduction Before engaging in a negotiation, a presumptive buyer needs to know the value of his company. With this knowledge, he can estimate whether the transaction is advantageous or not at a certain price. The purchase promotes the interest of the potential buyer (valuation subject) as long as the price paid for the acquired company (valuation object) does not exceed the subjective value associated with it. The valuation process depends on the target function (usually wealth or income maximisation) as well as on the decision field, which is constituted of all opportunities for action available to the valuation subject. The buyer‟s calculations are based on the expected uncertain future cash streams. A business valuation helps judge a given price‟s economic adequacy. The presumptive buyer has to compute his subjective decision value as a maximum price (marginal or critical price). He may just afford to pay this price without suffering an economic disadvantage (Laux & Franke 1969; Matschke 1975; Hering 2014; Toll 2011; Hering, Toll & Kirilova 2013). Under realistic imperfect market conditions, the decision value can be calculated applying general or partial investment theory-based models. In a fictitious perfect capital market, the Fisher-Separation (Fisher 1930) would hold and the marginal price would be relatively easy obtained applying a partial model (Ballwieser & Hachmeister 2013, pp. 13 ff.; Hering 2014, pp. 37 ff.). The future earnings value would then be calculated without minding the entire complex decision field of the valuation subject, as the interest rate would be exogenous. In a real imperfect market, it is inevitable to consider the interdependent 

Prof. Dr. Thomas Hering, Chair of Business Administration, esp. Investment Theory and Business Valuation, Fern-Universität in Hagen, Germany, Email: [email protected]  Dr. Christian Toll, Chair of Business Administration, esp. Investment Theory and Business Valuation, Fern-Universität in Hagen, Germany, Email: [email protected]  Dipl.-Kffr. Polina K. Kirilova, Chair of Business Administration, esp. Investment Theory and Business Valuation, Fern-Universität in Hagen, Germany, Email: [email protected]

Hering, Toll & Kirilova investment, financing and consumption decisions simultaneously. The valuation subject‟s consumption preference, expressed in the predefined structure of withdrawals, is no longer separable from the money‟s time value. It affects the temporal distribution and the level of the individual withdrawals as well as the investment and financing decisions. Thus, the period-specific shadow prices (the endogenous marginal interest rates), which are required for the partial model, can only be determined as a by-product of the general model solution (Hirshleifer 1958; Dean 1969). The purpose of our paper is to introduce a decision value calculation, which can handle the realistic conditions (market imperfections), rather than relying on perfect market assumptions. We show how valuation formulas derived from the so-called “state marginal price model” can be employed to compute the maximum affordable price for a corporate purchase. For the sake of simplicity, all modelling is done under the premise of certainty. The presumptive buyer pursues income maximisation. Our paper is organised as follows: In chapter two the difference from previous studies and the relevance of our paper will be explained. On this basis, we introduce an innovative way to compute the critical price for a company purchase under realistic conditions using both the state marginal price model and valuation formulas in chapter three. To show the proposed business valuation method‟s practical applicability, we will present an example in chapter four. Finally, chapter five summarises our findings and discusses the limitations of the applied model.

2. Literature Review The scientific debate between the advocates of the finance- and the investment-based valuation theories has been going on for more than sixty years (for a comprehensive overview, see Brösel, Toll & Zimmermann 2012). However, the apologists of the AngloSaxon finance-based valuation theory (e.g., Fisher 1930; Markowitz 1952; Modigliani & Miller 1958; Black & Scholes 1973; Cox, Ross & Rubinstein 1979; Rodrigues 2013; Trigeorgis & Ioulianou 2013; Sweeney 2014) seem not to acknowledge the existence of a feasible theory for imperfect market conditions. Accordingly, they assume a fictitious perfect market. Subsequently, their methods cannot take into account the individual expectations of the specific valuation subject. Instead, they pursue the futile quest for the one “true” value that has to be generally valid (Hering, Toll & Kirilova 2013). Therefore, they are not able to compute the critical price under realistic market conditions. This valuation task can only be fulfilled using investment theory-based business valuation methods. That is why we introduce an alternative way to conduct a business valuation for a company purchase that considers both existing market imperfections and individual expectations of the presumptive buyer. In order to calculate the decision value for a company acquisition, the state marginal price model will be introduced below (Hering 2014, pp. 45 ff.). This model combines the advantages of the mixed integer model of Laux and Franke (1969) with the two-step procedure of Jaensch (1966, p. 138) and Matschke (1975, pp. 253 ff. and 387 ff.). Laux and Franke (1969, pp. 207-210) calculated the marginal price of a certain cash stream within an imperfect capital market by applying the multi-period simultaneous planning approaches of Hax (1964) and Weingartner (1963). In this manner, they introduced an obviously advantageous price into their linear optimisation model. Subsequently, Laux and Franke varied this price parametrically until changing the valuation object‟s ownership becomes disadvantageous. This means that the variable 2

Hering, Toll & Kirilova representing the valuation object is no longer part of the optimal investment and financing programme (Laux & Franke 1969, pp. 208 f.). As a result, the Laux and Franke model requires a numerically extensive mixed-integer parametric optimisation. The models of Jaensch (1966) and Matschke (1975) handled this problem by determining the decision value in a two-step procedure. The first step is to calculate – as a so-called base programme – the investment and financing programme, which maximises the target function value (income size EN or asset value GW) under unchanged property conditions regarding the valuation object. Subsequently, in a second step, the valuation object has to be integrated into the presumptive buyer‟s investment programme. Then, the maximum affordable price as an immediate payment is to be determined. Hence, the decision field is changed by adding the valuation object at a price p and additionally supplemented by the condition that at least the target function contribution of the base programme must be achieved again. The result of this second step is the so-called valuation programme with its optimal value p* that indicates the requested upper price limit (i.e. decision value or marginal price). As opposed to Laux and Franke‟s (1969), the models of Jaensch (1966) and Matschke (1975) suffer from the blemish that the imperfect capital market is not considered over time. Instead, a single accumulated success number is assigned to each multi-period investment and financing object (Matschke 1975, pp 253 ff.). The state marginal price model combines the advantages of these models in a way that allows determining the marginal price under imperfect capital market conditions by setting up a base and a valuation approach without being dependent on the mixed-integer parametric optimisation as the model of Laux and Franke. The English-speaking scientific literature up to now provides only a few papers discussing the marginal price calculus (Hering, Olbrich & Steinrücke 2006; Olbrich, Brösel & Hasslinger 2009; Matschke, Brösel & Matschke 2010; Brösel, Matschke & Olbrich 2012; Hering, Toll & Kirilova 2014). It is surprising that only Matschke, Brösel & Matschke (2010) mention valuation formulas for the calculation of the maximum affordable price under realistic imperfect market conditions. To create knowledge in the international scientific society our paper introduces an elaborate example how an acquirer computes the maximum affordable price using valuation formulas.

3. The Methodology and Model The state marginal price model determines the decision value in a two-step procedure. The first step is to compute the base investment and financing programme, which maximises the target function value (income size EN or asset value GW) under unchanged property conditions regarding the valuation object. Afterwards, in a second step, the acquired company has to be integrated into the investment programme of the presumptive buyer. Then, the maximum affordable price is to be calculated. Therefore, the decision field is changed by adding the acquired company at a price p and additionally complemented by the condition that at least the target function contribution of the base programme must be reached again. The second step results in the valuation programme with its optimal value p* representing the requested upper price limit. In the following sections it is assumed that the valuation subject pursues the target income maximisation, striving for the greatest possible size EN of a structured withdrawal stream 3

Hering, Toll & Kirilova (Hering 2014, pp. 48 ff.; Toll 2011, pp. 49 ff.). The actual amount of the desired withdrawal at time t then results from the intended temporal structure predetermined by the consumption preference. Thus, the size EN gets converted into a stream of withdrawals with the help of the weightings w t . The autonomous cash flow bt results from the predetermined payments (for example, from current business operations and existing loan obligations) and is independent of the available objects j. To ensure the company‟s existence beyond the planning horizon n, the last withdrawal w n  EN must contain additionally to the normal income also the present value of a perpetual annuity, allowing the continuation of the desired dividend level beyond the planning horizon. Furthermore, the following assumptions are made: The planning period extends n years, whereas t = 0 defines the decision point in time. In the baseline situation, investment and financing objects j are available for the valuation subject (j = 1, ..., m). This also includes the opportunities of borrowing money, investing money in financial assets as well as an unlimited cash management. The cash stream of the object j is determined as follows: gj := (gj0, gj1, … , gjt, … , gjn). Each investment or financing object j could be realised xjmax times. The liquidity constraints have to ensure that at any time t, the sum of all realised investment and financing object cash flows, as well as from autonomous payments, suffice to enable the desired withdrawal. The variables EN and xj are limited to nonnegative quantities. The base programme (without the company purchase in question) results from the linear optimisation approach “max Entn” presented in Figure 1 (Hax 1964, pp. 435 ff.; Franke & Laux, 1968, p. 755; Matschke, Brösel & Matschke 2010, pp. 13 f.; Lerm, Rollberg & Kurz 2012, p. 265; Hering, Toll & Kirilova 2013, p. 42). The simplex algorithm calculates the optimal solution EN*. Acquiring the company at a price p is then only economically viable if the valuation programme yields at least the optimal target function value EN* of the base programme (Hering 2014, p 51; Toll 2011, p. 51). If the presumptive buyer comes in possession of company K, he receives its cash stream gK := (0, gK1, gK2, ... , gKt, ... , gKn). In exchange, he has to pay the price p at time t = 0. The decision value is then to be determined. The presumptive buyer has to know what price he can just afford, without the acquisition putting him into a worse position than if he had implemented the available base programme. The answer can be found with the valuation approach “max U” in Figure 1 (Hering 2014, p. 52; Toll 2011, p. 52). The simplex algorithm provides not only the marginal price p* but also the buyer‟s optimal investment and financing programme (valuation programme). Figure 1: Base and valuation approach max. Entn; Entn := EN m

 g j0  x j

max. U; U := p

 b0

j1

m

 g j0  x j + p

 b0

j1

m

 g jt  x j  w t  EN  bt j1

m

 g jt  x j  w t  EN  bt + gKt

 t  {1, 2, ... , n}

j1

–EN  –EN* xj 

max xj

xj, EN  0

max

xj  x j xj, EN, p  0

 j  {1, 2, ... , m}  j  {1, 2, ... , m}

4

Hering, Toll & Kirilova Now we will discuss the valuation formulas (Laux & Franke 1969, pp. 210-218; Hering 2014, pp. 53 ff.; Toll 2011, pp. 52 ff.) derived from the state marginal price model. The socalled „complex valuation formula“ can be derived by means of the duality theory of linear optimisation. This formula allows partial-analytic calculation of the maximum affordable price p*. The following equation can only be solved if the endogenous marginal interest rates it are known: n

p* =

 gKt  t



t 1

n

 bt  t 

t 0

future earnings value of the valution object t



Cj> 0

xmax  Cj j

with  t   1  i  1

and C j 

 w t  EN*  t

.

t 1

net present value of the base programme

net present value of the valuation programme ( excluding the valution object ) 1

n



n

 g jt t .

t 0

The valuation formula can be rearranged to: n

p* =

 gKt  t  t 1

n

 bt  t 

t 0



Cj> 0

xmax  Cj j

net present value of the valuation programme (including the valution object )

n

 w t  EN* t



.

t 1

net present value of the base programme

The maximum affordable price can be computed as the difference between the net present value of the valuation programme (including the valuation object) and the net present value of the base programme (which cannot be implemented anymore). Acquiring the company is only economically viable if the purchase price p does not exceed the net present value increase between the programmes. To emphasise the link to the future earnings value, the complex valuation formula can be adjusted as follows: n

p* =

 gKt  t t 1

future earnings value of the valution object



n

 bt  t  

t 0

Cj> 0

xmax  Cj  j

n

 w t  EN*  t

.

t 1

net present value difference due to restructuring from base to valuation programme

It is evident from the formulas above that the critical price p* does not always equal the future earnings value under imperfect capital market conditions. The net present value difference due to restructuring from base to valuation programme has to be additionally considered. This net present value difference vanishes if the period-specific marginal objects of the base programme correspond to those of valuation programme, because then the period-specific endogenous discount factors ρt do not change. In this case, the complex valuation formula can be reduced to the so-called „simplified valuation formula“:

5

Hering, Toll & Kirilova n

p* =

 gKt  t

= EK = future earnings value.

t 1

4. Exemplary Presentation and Findings Now, a fictitious example will be conceived to illustrate the procedure presented above. Firm A aspires to purchase company K. The management forecasts that company K generates in the planning period (n = 5) the cash stream (0, 30, 35, 40, 30, 20) and from the sixth year on a perpetual annuity in the amount of 10 monetary units (MU). Company A expects that the previous business activity leads to a perpetual deposit excess amounting to 100 MU. The perpetual annuities are taken into account in the example, using the generally estimated interest rate of 5% p.a. for t > n = 5, resulting in gK = (0, 30, 35, 40, 30, 220) and b = (0, 100, 100, 100, 100, 2 100). In order to reduce the complexity of the example, we assume that firm A has only a few investment and finance options. First, at t = 0 company A can invest in a tangible asset (e.g., a modernisation of the existing production lines), which is associated with the payment stream (–160, 15, 15, 15, 15, 315) and can be realised partially. Second, firm A is able to invest an unlimited amount of money in financial assets that promise a return of 5% p.a. The local bank provides at t = 0 a five-year zero coupon loan at an annual interest rate of 7% restricted to 100 MU. Furthermore, company A can debit a revolving line of credit at a short-term interest rate of 10% p.a. limited to 150 MU. Company A pursues income maximisation, striving for a uniform income stream, which has to be perpetuated at the planning horizon. In the baseline situation (without the acquisition of company K), a uniform income stream of the size EN* = 105.9257 MU can be obtained. Table 1 shows the base programme. At the end of the planning horizon, a deposit ensues in the amount of 2 118.5138 MU, which enables – at a rate of 5% p.a. – the intended perpetual annuity EN* from the sixth year on. Hence, withdrawals in the amount of 105.9257 MU p.a. can be executed for all time. Not only the zero coupon loan is fully utilised, but also every year additional short-term financing is required. Table 1: Base programme in the case of a credit limit Time bt Tangible asset Zero coupon loan Revolving line Repayment Withdrawal Account balance

t=0 0 –160 100 60

–60

t=1 100 15 0 56.9257 –66 –105.9257 –56.9257

t=2 100 15 0 53.5439 –62.6183 –105.9257 –53.5439

t=3 100 15 0 49.8240 –58.8983 –105.9257 –49.8240

t=4 100 15 0 45.7321 –54.8064 –105.9257 –45.7321

t=5 2 100 315 –140.2552 –50.3053 –105.9257 2 118.5138

In a second step, company K, accompanied by the cash stream gK, has to be integrated into the investment programme. Company A then has to find out what price it can just afford. According to the valuation approach, this marginal price p* is 190.9099 MU. The complete valuation programme (Table 2) can be described as follows: Company K is now included in the optimal investment and financing programme. As the price paid for company K ties up funds and debt financing is limited, a credit bottleneck takes place in the first year. This funding problem prohibits the tangible asset investment to be executed completely. In the valuation programme only 36.93% of the tangible asset investment can be engaged. Subsequently, both the zero coupon loan (100 MU) and the revolving line (150 MU) are exhausted at t = 0. In the following years, just like in the base programme, company A requires short-term debt financing (respectively 135.4, 114.3, 86.1 and 65.1 6

Hering, Toll & Kirilova MU). Although the tangible asset investment can only be partially executed (36.93%), company A is still able to provide the dividends of the base programme. Table 2: Valuation programme in the case of a credit limit Time bt + gKt Marginal price p* Tangible asset (36.93%) Zero coupon loan Revolving line Repayment Withdrawal Account balance

t=0 0 –190.9099

t=1 130

t=2 135

t=3 140

t=4 130

t=5 2 320

–59.0901

5.5397

5.5397

5.5397

5.5397

116.3337

100 150

0 135.3860 –165 –105.9257 –135.3860

0 114.3106 –148.9246 –105.9257 –114.3106

0 86.1276 –125.7416 –105.9257 –86.1276

0 65.1264 –94.7404 –105.9257 –65.1264

–140.2552

–150

–71.6390 –105.9257 2 118.5138

Comparing Table 1 and 2, it is apparent that the acquisition of K has caused structural changes between the base and valuation programmes. The period-specific marginal objects (commenced, but not entirely realised investment or financing activities) and the corresponding endogenous marginal interest rates of the base and valuation programmes differ. In the baseline situation, company A did not exhaust the credit line (10% p.a.) in any year, whereas in the valuation programme the credit line is exhausted in year one. That is why in the baseline situation the credit line is the marginal object in every year, whereas in the valuation programme only from the second year on. The marginal object of year one cannot be immediately recognised from Table 2. The partial-analytic determination of the critical price requires all period-specific endogenous marginal interest rates. So we need to compute i1. As no marginal object can be identified for year one only, i 1 must be explained as a mixed interest rate of other objects. Therefore, we have to take the partially realised tangible asset. Because the net present value of this marginal object vanishes, i1 can be calculated as follows: C = 160 

15 

 i1 =

15



15



15



15



315

1  i1  1  i1   1.1 1  i1   1.12 1  i1   1.13 1  i1   1.14

=0

15 15 15 315    1.1 1.12 1.13 1.14  1 = 1.67157511 = 167.157511%. 160

In consequence, the divergent interest rate structure leads to different net present values between the base and valuation programmes. As a result, the maximum affordable price can only be calculated using the complex valuation formula: n

p* =

n

 gKt  t   bt  t  t 1

t 0



Cj 0

xmax  Cj  j

n

 w t  EN* t

.

t 1

p* = 160.1355 + 1 066.6667 + 42.6911 + 51.2907 – 1 129.8740 = 190.9099 MU. The complex valuation formula confirms the result of the state marginal price model. To illustrate how changes in the decision field affect the maximum affordable price, the local bank additionally grants an unlimited overdraft facility at a short-term interest rate of 10% p.a. Since the changes in the decision field of company A do not influence the optimal decisions made in the baseline situation, the base programme shown in Table 1 remains valid. Due to the improved financing situation, the maximum affordable price for K 7

Hering, Toll & Kirilova in the valuation programme without credit limit is now 243.3440 MU (compared to 190.9099 MU with a credit limit). In the case of an unlimited overdraft facility, no financing bottleneck occurs, and the tangible asset investment can be engaged fully also in the valuation programme. This is financed by further short-term debt (the lucrative zero coupon loan is of course again exhausted) in each year (respectively 303.3, 294.6, 280.0, 258.9, 245.7 MU). Consequently, no investments in financial assets take place. Table 3 presents these results. Table 3: Valuation programme in the case of an unlimited overdraft facility Time bt + gKt Marginal price p* Tangible asset Zero coupon loan Revolving line Repayment Withdrawal Account balance

t=0 0 –243.3440 –160 100 303.3440

–303.3440

t=1 130

t=2 135

t=3 140

t=4 130

t=5 2 320

15 0 294.6041 –333.6784 –105.9257 –294.6041

15 0 279.9902 –324.0645 –105.9257 –279.9902

15 0 258.9149 –307.9893 –105.9257 –258.9149

15 0 245.7321 –284.8064 –105.9257 –245.7321

315 –140.2552 –270.3053 –105.9257 2 118.5138

As the purchase of company K in a world without a credit limit does not cause any structural changes between the base (Table 1) and the valuation (Table 3) programmes, the critical price can be calculated using the simplified valuation formula (future earnings method). The credit line is the marginal object for all years in both programmes. That‟s why the net present value difference vanishes. In this case, the critical price can be computed using the simplified valuation formula: n

p* =



gKt



t 1 1  i



t

=

30



35

1.1 1.12



40 1.13



30 1.14



220 1.15

= 243.3440 MU.

The simplified valuation formula confirms the result of the state marginal price model when the net present difference vanishes.

5. Summary and Conclusions The discussion above demonstrates that a business valuation cannot be executed completely detached from the individual expectations and plannings of the specific valuation subject (Matschke & Brösel 2013, p. 18). Appraisal always depends on the subjective aim and the decision field of the valuation subject. Even when the same company is being assessed from the perspectives of different valuation subjects the decision value may vary, especially if they pursue different types of prosperity maximisation. In this paper, we assumed that the valuation subject is interested in purchasing a company, pursuing income maximisation. It desires the greatest possible uniform income stream, which has to be perpetuated at the planning horizon. The example shows that even the same valuation subject may come to diverging limits of concession willingness regarding the same valuation object when the underlying decision field changes (Hering, Toll & Kirilova 2014, p. 41; Matschke & Brösel 2013, p. 368). The maximum affordable price for the very same cash stream depends on the available opportunities for action. We have shown that simply removing the short-term credit upper limit alters the critical price of the same company noticeably. Of course, the determination of the decision value using the state marginal price model has also engendered criticism (Koch 1982, pp. 25 ff.; Rollberg 2002, pp. 4 ff.; Ballwieser 1990, pp. 28 f.; Hering, Toll & Kirilova, 2013, p. 45). In a general model all investment and 8

Hering, Toll & Kirilova financing objects are directly included in a simultaneous optimisation approach. As this requires elaborate information gathering and processing, a centralised simultaneous planning with general models is often marked by complexity and clumsiness. Even if it were possible to develop a general model considering all data and interdependences, this model would suffer from a solution defect, since the optimal solution could not be found at economically viable expense. Moreover, a centralised simultaneous planning with general models is rather demotivating for subordinated operating units (divisions) because all decisions are made at management level (Koch 1982, p. 27; Rollberg 2002, p. 5; Hering, Toll & Kirilova, 2013, p. 45). While the operating units are only empowered to submit information upwards, the decision makers are hopelessly overburdened. Therefore, it is recommended to divide the general model into several simpler partial models. For this purpose, the upper management level has to delegate certain decision-making power visà-vis partial models. To ensure planning integrity, a link to the general model is still necessary, whereby the theoretical relations between general and partial models have to be taken into account. As shown, the future earnings method often fails to calculate the decision value. It is evident from the complex valuation formula that the critical price p* does not always equal the future earnings value under imperfect capital market conditions. Only when the net present value difference vanishes, the simplified valuation formula and the state marginal price model lead to the same result. But as a company purchase is typically accompanied by structural changes, the simplified valuation formula cannot be applied. Unfortunately, the presented valuation formulas suffer from the dilemma that they require information, which can be solely derived from the solution of the state marginal price model. However, when the state marginal price model has already been solved, the valuation task is already completed. A valuation formula is then of no further benefit, as it simply confirms the result of the state marginal price model. One way to evade this dilemma in large-scale enterprises is the approximate decomposition. In this case, the upper management level delegates certain decision-making power to the subordinate levels, which can base their decisions on the partial-analytic future earnings method.

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