Javier Santibáñez*, Leire Alcañiz* y Fernando Gómez-Bezares*

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1 6 Javier Santibáñez*, Leire Alcañiz* y Fernando Gómez-Bezares* Cost of funds on the basis of Modigliani and Miller and CAPM propositions: a revision Coste de los fondos bajo las hipótesis de Modiagliani y Miller ABSTRACT When facing with any process involving financial valuation it is of a crucial importance to find and use the accurate discount rate. CAPM provides us with an accurate and very commonly used solution to solve the problem, but it requires the estimation of the Beta, which is quite difficult, especially when talking about non-quoted companies. For many years, text books have suggested to look for a quoted company with a strong similarity with the one to be priced in order to use its Beta. Nevertheless, this may bring up an important problem, since both companies could show different leverage, which would also affect to Taxes and Betas. Both academics and practitioners usually use the wildly renowned formulation by Hamada, but we think that this proposal may introduce important biases, insofar as it only takes Corporation Tax into account, and not Personal Income Tax. In this paper we introduce the effect of Personal Income Tax in a simple and original way that can be useful for Financial practice. Keywords: Cost of funds, Levered Beta, Hamada equation, Tax effects. JEL Classification: G31, G32. RESUMEN En el proceso de valoración de empresas y, en general, en cualquier tipo de valoración financiera, es muy importante acertar con el tipo de descuento. El CAPM aporta una solución elegante y muy utilizada para resolver el problema, pero se encuentra con la dificultad del cálculo de la Beta, sobre todo en empresas no cotizadas. Desde hace años, los libros de texto recomiendan buscar empresas cotizadas similares y utilizar sus Betas. Sin embargo, existe una dificultad: las empresas tienen diferente grado de apalancamiento, lo que también afecta a los impuestos y a la Beta. Académicos y practitioners utilizan para resolver este problema la conocida fórmula de Hamada, pero creemos que su planteamiento tiene el problema de considerar el impuesto de sociedades, pero no el de la renta de las personas físicas. Nosotros introducimos el impuesto sobre la renta de una manera que creemos que es sencilla y original, lo que puede ser de gran utilidad en la práctica financiera. Palabras claves: Coste de los fondos, Beta apalancada, Fórmula de Hamada, Efecto de los impuestos. Clasificación JEL: G31, G32. Recibido: 31 de enero de 2014 Aceptado: 20 de febrero de 2014 * Departamento de Finanzas de la Universidad de Deusto. Autor de contacto: Javier Santibáñez Grúber

2 COST OF FUNDS ON THE BASIS OF MODIGLIANI AND MILLER AND CAPM PROPOSITIONS: INTRODUCTION When trying to value a company on a cash-flow discount basis, it is of a crucial importance to estimate, among other variables, the appropriate discount rate. Assuming the conclusions of the Capital Asset Pricing Model (CAPM; Sharpe, 1970), the required return for any investment should depend on its systematic risk, which requires the estimation of the beta of the company, usually on the basis of a regression process between historical profitability of the studied company and the one provided by the market as a whole. However, an additional problem that we usually have to cope with is that not many companies are quoted in the market, so the process becomes a tough one when trying to calculate the beta of a non-quoted company. In those cases, professionals usually try to estimate the beta of the company to be priced on the basis of a similar one (this is known as levered Beta calculation); and it is very common to use the Hamada (1969) formulation, which implicitly assumes that we have to take the effect of Corporation Tax into account. Nevertheless, Miller (1977) made it clear that it is not only the Corporation Tax the one to be taken into account, but the Personal Income Tax as well, insofar as if the Tax System as a whole was properly designed (which means that the advantage for leverage in Corporation Tax should be eliminated in the Personal Income Tax, by treating better income coming from dividends than those coming from interest), no advantage would be reached by levering a company (which would take us back to Modigliani and Miller, 1958). While it is a very complex problem, as it is difficult to estimate accurately the Tax burden for dividends or interest in the Personal Income Tax 1, and even the real Corporation Tax rate that the companies are bearing (because of the effect of partial exemptions, etc.), we have to be aware of the distortion that we are introducing in the process when taking only Corporation Tax (Modigliani and Miller, 1963) into account. In this paper we will propose an alternative formulation to Hamada (1969) for the estimation of levered betas, in which Personal Income Tax is also taken into account (based on both Modigliani and Miller and CAPM propositions), which can be seen as a generalization of the Hamada one (that would remain as a particular case of the one that we pose) GENERAL FORMULATION Weighted Average Cost of Capital (WACC) is commonly calculated as follows: where: i k i k e WACC t D E Required rate of return for Debt Net cost of Debt Cost of Equity Weighted Average Cost of Capital Tax rate (Corporation Tax) Debt (Market value) Equity (Market value) (1) Let us assume stable conditions, no growth and that the company invests only the necessary amount to maintain the current value of assets (replacement investment, equal to depreciation, so as to keep constant its cash-flow generation capability); if we do so, the Market value of the firm (understood as Market value of assets) should be calculated as follows (present value of a perpetuity): (2) whereas in addition to the previously defined nomenclature, EBIAT is Earnings Before Interest and After Taxes, that is to say, generated profits by assets to pay the whole providers of funds; under the described conditions, it should be equal to total payments for financing. In addition: (3)

3 8 where S is net sales, C is disbursement operating costs, AM is depreciation and amortization (of tangible and intangible fixed assets), and EBIT is Earnings Before Interest and Taxes. 3. MODIGLIANI AND MILLER PROPOSITIONS (1958 and 1963) In their wildly renowned paper of 1958, Modigliani and Miller proposed that the Financial Structure does not have any impact on the Weighted Average Cost of Capital (and therefore, neither on the value of the firm). The line of reasoning is quite simple: the capital funds providers of the company as a whole would ask for a return depending on the risk of the assets; so if we assume the assets remain the same, no change should be expected on the required rate of return for financing them, even if we use different proportions of Debt and Equity. In other words, in competitive and efficient Markets two identical assets could not be priced differently (that is what would happen if different Financial Structures for the same assets had different costs). Modigliani and Miller proposal can be summed up in figure 1. Nevertheless, Corporation Tax changes it all, since it brings about a distortion due to the different treatment of interests and dividends. In fact, interests are deductible (provoking tax savings) from the Tax base, while dividends are not; therefore, all the rest remaining the same, if it is true that under perfect Market conditions the Financial Structure would be irrelevant, the imperfection mentioned above would cause an interest for leverage. The effect of the Corporation Tax advantage can be easily shown. If we assume a non leveraged company (fully Equity financed), the free cash-flow (that under the formerly proposed conditions could be used for paying dividends) can be expressed as follows: (4) whose present value should be calculated using a discount rate considering the Assets risk (R). This way, the Enterprise value (that in a non-leveraged company would be both Assets value and Equity value) should be estimated as follows:

4 COST OF FUNDS ON THE BASIS OF MODIGLIANI AND MILLER AND CAPM PROPOSITIONS:... 9 (5) So if we now equalize expressions (8) and (9) we reach (10): If we now consider the possibility for the company to borrow money, the free cash-flow to be distributed among the funds providers after Corporation Tax would be: (6) where I refers to the Interests to be paid on the Debt (so it is calculated by multiplying the interest rate by the total amount of Debt), and as can be seen, we face a new situation insofar as there is an additional item in the formula which is related to the savings that the interest payment causes in the Corporation Tax; if we assume both that this Tax shield can always be accomplished and that lenders do not assume any risk, this part of the income should be discounted at the Risk-free rate, and so: (7) It can be seen that the Enterprise value can be obtained by adding to the non-leveraged one a term that grows with Debt (the formerly mentioned Tax shield), so there is an interest for the company to increase leverage to its maximum. An important consequence of the former is that now the Weighted Average Cost of Capital is no longer independent from the leverage, but decreases as Debt rises. In fact, on the basis of expression (2) we can reach equation (8): (8) And if we now take equation (7) into account it is easy to see that: (9) (10) where the cost reduction effect related to leverage appears very clearly. In addition, a different formula for estimating the cost of Equity can be raised on the basis of the former; so, if we make equal expressions (1) and (10) we have equation (11): 4. CAPM (11) CAPM (Capital Asset Pricing Model) is built on the basis of Markowitz Model hypothesis, and sets out a fundamental relationship between a given asset and the required rate of return to finance it. The model highlights the fact that the total risk of any asset can be split up into Systematic risk and Diversifiable (Non-systematic) risk. Systematic risk is due to the Market, that is to say, it is given by the fact that any asset is in some way related to the general economic course, and because of that, it has a certain amount of non Diversifiable risk; Non-systematic is the specific risk of the investment (the portion of risk that the Market is unable to explain), so it can be avoided with a proper diversification. If this is the case and we assume risk-averse individuals, nobody would take avoidable risks, so we all would eliminate Diversifiable risks and only Systematic risk would remain. The classical measure for Systematic risk is Beta, which can be calculated as follows:

5 10 (12) where COV (R j, R m ) is the Covariance between the studied asset and Market returns; and VAR (R m ) is the Variance of the Market returns. The model poses a positive linear relationship between the required return on any investment and its relevant risk (measured with Beta), that can be summarized in the following expression (Security Market Line, SML; the model requires adjustment to the purposed line to any asset correctly valued): (13) where E (R j ) is the expected return on the asset, R f is the Risk-free rate and the Premium is the difference between the expected return for the Market and the Risk-free rate. Under CAPM conditions, all Assets, Equity and Debt should perfectly adjust to equation (13), so: for estimating a company Beta on the basis of observed Betas in the Market. Again, the idea is quite simple: it would be very difficult to estimate Equity Beta in a nonquoted company, since we have no historical data about its returns, so Covariance with the Market would remain unknown 3 ; but in some cases, it could be derived from the Beta of a quoted company. This proposal poses one important problem: even if the activity of both companies were the same, leverage could be totally different, so we have to face an adjusting process in which first of all, Beta of the observed company assets must be inferred from its Equity one (taking both leverage and Beta of Debt into account); and in a second step, Equity Beta of the studied company would be estimated (again, taking its leverage and the Beta of Debt into account). This process is commonly known as levered Beta calculation. Formulation to be used is quite simple to derive. Let us remember that when equaling expressions (1) and (10) (that is to say we accept Modigliani and Miller 1963 propositions) we have: (14) (15) (16) where β a, β i and β e are the Systematic risk measures for the Assets (assuming non-leveraged company), for Debt and for Equity. 5. INTRODUCING CAPM INTO THE MODIGLIANI AND MILLER PROPOSALS (1958 and 1963) Clearing R we reach expression (17): (17) Combining CAPM and Modigliani and Miller propositions takes us to the classical formulas commonly used If we now take into account CAPM (expressions 15 and 16), equation (17) becomes the following:

6 COST OF FUNDS ON THE BASIS OF MODIGLIANI AND MILLER AND CAPM PROPOSITIONS: (18) Let us compare expression (18) with equation (14), that indicates the required rate of return on assets in a nonleveraged company under CAPM conditions; it is easy to derive equation (19) in order to estimate Beta of assets (assumed that both Beta of Equity and Debt and the leverage of the observed company are all known): (19) We then only have to clear Beta of Equity in expression (19), as can be seen in equation (20): (20) A particular case of the former happens if we assume fully guaranteed Debt (no risk taken by Debt providers; which means to assume Beta of Debt equal to zero): it is the formula of Hamada (1969) for estimating Beta of Equity: (21) NOTE: the whole formulation is applicable for the original proposal by Modigliani and Miller (1958; we only have to assume Tax Corporate rate equal to zero). 6. THE EFFECT OF PERSONAL INCOME TAX (MILLER, 1977) In the work of 1963, Modigliani and Miller assume that there is only Corporation Tax, which introduces a distortion, insofar as Debt and Equity are not treated the same. However in 1977, Miller points out that if the Tax System as a whole was properly designed, it should not interfere with the financial decisions of the companies; to put it another way, the function of the Tax System is not to provide companies with the proper Financial Structure, but to drain money from the Economic System in order to redistribute wealth, accomplish public investments, etc. If so, and if we assume no other Market imperfections, we would go back to the original proposal: Financial Structure is not relevant in order to put value on the company (the Financial Objective of the firm), since the advantage of Debt in Corporation Tax disappears when considering the Tax System as a whole. Let us revise the previous formulation, taking now into account that the Personal Income Tax treats differently Debt income and Equity income (interests and dividends) 4. Be m the Tax rate for dividends in Personal Income Tax, and n the one for interests (assuming that, obviously, m < n). Let us now first consider a non-leveraged company. The generated income to be distributed among funds providers can be seen in equation (22):

7 12 (22) R being the required rate of return for the non-levered company before taking Personal Income Tax into account; the Enterprise value would be calculated as follows: (23) In a leveraged company the income generated and distributed among funds providers after taxes is shown in expression (24): (24) If we again assume that tax shields can always be reached, both items in expression (24) should be discounted at different discount rates: first element is identical to the one corresponding to a non-levered company, so we have to use R (1 m); and the second one should be discounted using i (1 n) (since all taxes have been deducted in equation (24); so discount rates must be defined after taxes in order to avoid duplicities). Then: (25) On the basis of (25) it is easy to reach equation (26): (26) where tax advantage related to leverage disappears when (1 t) (1 m) = (1 n); and obviously, if m=n we go back to expression (7). Let us call T to the tax distortion in the whole system: equation (26) can be expressed therefore as follows: (27) (28) T is always lower or equal than t (if we assume that m < n): at worst, Personal Income Tax would not discriminate interests and dividends, which means T=t (Modigliani and Miller 1963). Let us remember equation (8): (8) If we clear EBIAT in equation (28) it is easy to reach expression (29): (29) If we now put it together with equation (8) we reach expression (30): (30) where now again can be seen clearly the cheapen effect related to leverage (assuming that the Tax System is not neutral T 0 ; however, the advantage is lower now, since T < t, so the Weighted Average Cost of Capital decreases less when taking Personal Income Tax into account; see expression (10) in comparison with (30)). On the other hand, if we make equal expressions (1) and (30) we can derive the formulation for estimating the required rate of return for a non-leveraged company:

8 COST OF FUNDS ON THE BASIS OF MODIGLIANI AND MILLER AND CAPM PROPOSITIONS: Let us deepen on what the former means. Remember that T was defined as the measure of the imperfection related to the Tax System as a whole (considering both Corporation and Personal Income Taxes): (31) Nevertheless, expression (31) is not a logical one, since the weights for both cost of Debt and Equity should add up to one (as happens under Modigliani and Miller propositions when taking only Corporation Tax into account; see expression (17)). This leads us to bring about a correction in expression (1) that seems to be reasonable. Let us remember equation (1) that has been used both in a world without taxes, and in a single Corporation Tax context: (1) where t is the Corporation Tax rate (that would be zero in a world without taxes). Under Miller (1977) conditions it must be said that if the Tax System as a whole was properly designed, dividend collectors should have an advantage over interest ones in Personal Income Tax (that is to say, m<n); and this different treatment would lead lenders to ask for a higher profitability than what they would expect if the aforementioned difference did not happen. To make it clearer, let us think about a non-leveraged company fully owned by a sole investor. Only by shifting Equity for Debt he or she could collect part of the profitability in the form of interests, which are deductible from the Corporation Tax Base; so the higher the leverage, the lower the Corporation Tax to be paid, and the higher the money collected by that sole owner. But if Debt incomes are treated worse in Personal Income Tax, the advantage reached in Corporation Tax would decrease, and it could disappear if the system as a whole was properly designed (that is to say, T=0). This way, the tax correction should be given by a tax shield considering the system as a whole (so we might use (1 T) instead of (1 t)). Expression (1) can be then rewritten as follows: (32) 5 so: (27) The former implies that requirement of Debt Providers changes: (33) In other words, the proposed formulation might be seen as a correction to the one we use when only Corporation Tax exists, insofar as the tax shield achieved in Corporation Tax could be minored by the Personal Income Tax effect (since the item (1 m)/(1 n) should logically be higher than 1 and the decrease of Debt cost is lower; and if we assume m=n we would go back to 1963 Modigliani and Miller case). Under the proposed conditions, we can equal expressions (32) and (30): (34) Equation (34) seems to be more reasonable; and we can also clear the cost of Equity in this proposal, coherent with Miller (1977): (35)

9 14 7. INTRODUCTION OF THE CAPM INTO THE MILLER (1977) PROPOSAL Introducing CAPM into the Miller proposal is again quite simple. As we said earlier, CAPM poses the following formulation in order to estimate the required rate of return on Assets, Debt and Equity: (14) (15) (16) Let us remember as well equation (34): (34) If we now substitute expressions (15) and (16) in equation (34), the required rate of return on the assets in a non-leveraged company would be as follows: (36) If we now compare expressions (36) and (14) we can derive equation (37), under the assumption that both Beta of Equity and Beta of Debt, as well as leverage (all data referred to the observed company) are known: (37) where the Beta of Equity for the studied company can be cleared: adapted to Miller (1977) propositions, which we call SGHAMM formula (2014). (39) (38) If we assume that Debt is fully guaranteed (Beta of Debt equal to zero), then we have the Hamada formulation NOTE: Observe that all formulation proposed is valid for 1958 Modigliani and Miller propositions (T=0) and also for 1963 proposal (by substituting T for t).

10 COST OF FUNDS ON THE BASIS OF MODIGLIANI AND MILLER AND CAPM PROPOSITIONS: CONCLUSIONS Formulation most frequently used by practitioners to estimate the appropriate rate of return to be used in the assessment of assets or stocks is based on CAPM and Modigliani and Miller (1963) propositions, taking only the effect of Corporation Tax into account (Hamada, 1969). Insofar as it is also very usual that dividends and interests are not equally treated in the Personal Income Tax (in order to eliminate the advantage that Debt has in Corporation Tax; Miller, 1977), the aforementioned formulation introduces a distortion in the results to be reached. In this paper we have developed, on the basis of both CAPM (Sharpe, 1970) and Modigliani and Miller propositions (Modigliani and Miller, 1958 and 1963; and Miller, 1977), an alternative formulation which takes the effect of both Taxes (Corporation and Personal Income Tax) into account, and allows to overcome the distortion mentioned before. 9. BIBLIOGRAPHY BREALEY, R.A., S.C. MYERS and F. ALLEN (2007): Principles or corporate finance, McGraw Hill, New York, 9 th ed. GÓMEZ-BEZARES, F. (2014): Dirección financiera, Desclée de Brouwer, Bilbao, 5 th ed. GÓMEZ-BEZARES, F. (2012): Elementos de finanzas corporativas, Desclée de Brouwer, Bilbao. HAMADA, R.S. (1969): Portfolio analysis, market equilibrium and corporation finance, Journal of Finance, March, pp MILLER, M.H. (1977): Debt and Taxes, Journal of finance, 32, May, pp MODIGLIANI, F. and M.H. MILLER (1958): The cost of capital, corporation finance and the theory of investment, American economic review, 48, June, pp MODIGLIANI, F. and M.H. MILLER (1963): Corporate income taxes and the cost of capital: Acorrection, American economic review, 53, June, pp SHARPE, W.F. (1970): Portfolio theory and capital markets, McGraw Hill, New York. APPENDIX Let us deepen in an alternative way to reach the logic of expression (39). Be q the required return (before Personal Income Tax) for a stock with identical risk (after Taxes) than Debt. It should happen that: so: Let us remember that: so: If we now make equal (1) and (30) we have: Taking (A.4) into account: which takes us directly to the logic of (34). (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) Taking now the CAPM into account, and following the same process than before (from (14) to (39)): Substituting (A.10) and (A.11) in (A.8): (A.9) (A.10) (A.11)

11 16 (A.12) Taking now (A.9) into account: (A.13) (A.14) where the Beta of Equity for the studied company can be cleared: (A.15) (A.16) If we assume that Debt (or the proposed stock for which return q is required) is fully guaranteed (Beta of Debt equal to zero), we reach again (39): Notes (A.17) 1.- And even more in the Corporation Tax, when it is a company the owner of stock or debt (with different rates depending on the kind of company). Not to talk about the additional problems that arise when facing with international ownership. 2.- Deeper explanations and formulation of both Modigliani and Miller propositions and Capital Asset Pricing Model (CAPM) can be found in Gómez-Bezares (2014), chapters 5 and 7. Gómez-Bezares (2012) and Brealey, Myers and Allen (2007) may be consulted as well for some particular issues. Formulation related to the inclusion of CAPM in Miller s proposal (1977) can be seen as original. 3.- Accounting data could be used, but this approach shows important theoretical problems. 4.- This was the way the Tax System worked in Spain since 1995 to The changes introduced in Personal Income Tax in 2007 took us back to a more coherent situation with Modigliani and Miller (1963), insofar as interests and dividends are since then equally treated (Tax rate of 21% nowadays). Anyway, when trying to measure the impact of the distortion, we face a more complicated problem, if we also take into account that there is an exemption in Personal Income Tax for the first euros collected as dividends, and that taxation for dividends collected by companies or Mutual Funds has a different treatment. 5.- In this context, i should be understood as the required return for Debt assuming m=n=0; or, in general, if no discrimination was made in the Personal Income Tax (that is to say, (1 m) = (1 n)). This will be treated in more depth in the Appendix.

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