Determination of Market Values and Risk Premia of Multi-national Enterprises and Its Application to Transfer-pricing

Valuing a multi-national enterprise (MNE) using the discounted cash flow method (DCF) requires the joint determination of the market value of its equity (MVE) together with the equity risk premium (ERP) the firm should earn, since the latter is part of the discount rate used in the calculation of the MVE. This paper presents a theoretical derivation of how MVE and ERP can be calculated simultaneously under fairly general conditions and an application example. Besides firm data on free cash flow to equity the only external data needed are the risk-free rate of interest and a parameter indicating the required market risk premium per return volatility. The method presented allows for consistent valuation in particular of those firms that are not publicly listed and where ownership shares are not publicly traded. It also allows comparing the cash flows themselves to market returns on equally risky assets. This latter possibility is useful in transfer pricing, where the profit levels of dependent subsidiaries of MNEs are frequently under investigation.

to equity (FCFE), i.e. after deduction of any costs of debt financing, then the discount rate represents the cost of equity financing or the required (minimum) expected return on equity financing. (Note 5) Note that this interpretation implies a second role for the discount rate as cost of equity financing. Namely, the investor expects that future cash flows as a percentage of the market value of the equity invested (the MVE) will be at least as high. Hence when profits of individual firms are viewed as returns on equity invested, CAPM can also be used to compare individual firms' profits against a market benchmark. (Note 6) One of the main conclusions of the CAPM theory is that an adequate remuneration for the risks assumed by an equity investment is given by the market risk premium multiplied by the covariance of the returns on the equity invested with the market return. (Note 7) Since that covariance contains a measure of the volatility of the returns on the equity invested, an adequate equity risk premium (ERP) is also a function of the volatility of the returns on the equity invested. In fact, empirical analyses using historical financial markets data show that the ERP paid by the capital market for the assumption of risk corresponds to a multiple of the standard deviation of the Returns on Equity (RoE). (Note 8) While these empirical results are derived from data on investments in financial markets, the same principles should also apply when an investor finances an enterprise directly. As a consequence, the pricing of an enterprise's products should be set such that the resulting profits can be expected to adequately remunerate the firm's equity investors for the risks they have taken in financing the enterprise. Recent research shows that this is in fact the case and that firm's average RoEs tend to increase with the volatility of those RoEs. (Note 9)

Modeling: Simultaneous Determination of Market Value and Risk Premium
This section presents a simple theoretical model that can be solved simultaneously for the MVE and the ERP and a numerical example of its application.
According to the standard convention in the CAPM, the required return for any asset i, r i , can be expressed as: where r f denotes the risk-free rate of interest, r m denotes the market return, σ im and ρ im denote the covariance and the correlation coefficient, respectively, between firm i's return on equity and the market return, σ i denotes the standard deviation of asset i's return, σ m denotes the standard deviation of the market return, and σ 2 m denotes the variance of the market return.
Suppose asset i is a particular firm financed with a debt to equity ratio of δ (Note 10) and taxed at rate τ, then equation (2) becomes (Note 11) define α i as: (1 (1 ) ) ( ) For the firm i, let C i be its contemporary FCFE, r i its required return on equity (the applicable discount rate), and g i the expected growth rate of C i . Firm i's market value of equity will then be given by V i : Furthermore, let σ Ci be the standard deviation of FCFE i then the required return on equity can be expressed as (Note 12) Electronic copy available at: https://ssrn.com/abstract=1942845 Simultaneous solution of equations (4) and (5) then yields Note that equations (4) and (5) form a unique well-defined solution as long as the following parameter condition is satisfied: Condition (9) implies that for a well-defined solution to exist, a high-growth cash flow must also exhibit a relatively high volatility (and a low-growth cash flow a low volatility). A proof for the uniqueness of the derived solution is given in appendix 2.
A numerical example of this solution method is presented and illustrated graphically below. To solve for the market value of equity V i and the cost of equity r i simultaneously, we solve equations (4) and (5) simultaneously. This corresponds to finding graphically the unique intersection between the two equations. In order to show them together in a single figure, one of them has to be inversed. Hence, define the inverse of equation (5) as Then we can show equations (4) and (10) as well as the equilibrium solution graphically (Note 14). For a yearly cash flow of EUR 10m growing at a rate of 2% pa with a cash-flow volatility of EUR 5m pa, a risk-free rate of 5% pa and a risk parameter of 1, the market value of equity will be EUR 166.667m and the cost of equity will be 8% pa, and the risk premium will be 1*5/166.667 = 3 percentage points. This solution is shown in figure 1 below. The risk parameter of 1 is assumed to be estimated externally (Note 15); however, it can also be derived from underlying market parameters according to equation (2'). In our example, the parameters are: a debt/equity ratio of 1, a tax rate of 30%, a volatility of the market return of 5% pa, a market risk premium (the difference between market return and risk-free return) of 5% pa, and a correlation between the firm's equity return and the market return of 0.588.   As cash flow volatility increases, the equilibrium market value falls whereas the equilibrium return on equity rises.

Application: Conclusions for the Valuation of Multi-national Firms
The method presented above allows the application of the DCF modeling with FCFEs leading to the derivation of an adequate ERP directly from the firm's own cash flow data; the only external data needed are the risk-free rate of interest and a parameter indicating the required market risk premium per return volatility. This allows for consistent valuation of firms including of those firms that are not publicly listed and where ownership shares are not publicly traded.
Besides valuation of a firm given its cash flows this method also allows comparing the cash flows themselves to market returns on equally risky assets. This latter possibility is potentially useful in transfer pricing, where the profit levels of dependent subsidiaries of MNEs are frequently under investigation. OECD transfer pricing guidelines, i.e. taxation guidelines with respect to income that derives from controlled transactions between subsidiaries and/or with owners within an MNE, stipulate that the pricing of these transactions and the resulting profits must be such that uncontrolled third parties would have agreed voluntarily to undertake such transactions; this is known as the arm's length standard (Note 16). In principle this implies that prices for goods and services are set at market prices and that profits should earn a market return that adequately remunerates individual risk. (Note 17) Examples for applications in transfer pricing include the pricing of adequate remuneration of contract manufacturers in the automobile industry as well as the determination of adequate profit shares between several risk-bearing co-entrepreneurs within a multi-national enterprise (Note 18).

Acknowledgement
The views expressed in this paper are those of the author and do not necessarily reflect those of the institutions he is affiliated with. Any information presented is of a general nature and does not address individual circumstances of Electronic copy available at: https://ssrn.com/abstract=1942845 www.ccsenet.org/ibr International Business Research Vol. 5, No. 12;2012 any particular person or entity. The author would like to thank Nils Holinski for helpful comments and suggestions as well as Nitish Maini and Keshav Goel for diligent research assistance; the usual disclaimer applies. Financial support by the International Centre for Economic Research (ICER), Torino, Italy, and by the Spanish Ministry of Education and Science (Grant No. ECO2008-06191) is gratefully acknowledged.

Appendix 1. DCF Volatility
Derivation of the variance of the market value of equity given constant growth in expected cash flows and their volatilities. Let the cash flow of firm i in period t j be where C i0 , the cash flow in period 0, is a random variable. Then the variance of the market value of equity is given as (Note 19) Hence we have: The volatilities (standard deviations) of the firm value and the cash flows are proportional to their respective discounted expected values. This corresponds to the heteroskedasticity often exhibited by empirical data.

Appendix 2. Uniqueness of the Solution for MVE and Roe
To show uniqueness of the solution given in equations (7) and (8), we compare the curvature of the cost-of-equity equation (6) with that of the inverse of the market-value-of-equity equation (5)  Note that given condition (9), equations (A.2.1) and (6) have an intersection given by equations (7) and (8). Furthermore, given equation (9)