ABSTRACT
Farber, Gillet and Szafarz (2006) propose a general formula for the WACC in which the expected return on the tax shield appears explicitly. The classical Modigliani-Miller and Harris-Pringle WACC formulas for specific debt policies are then derived from the general formula after having determined the corresponding tax shield expected returns. Replying to Fernandez' (2007) comment, this note explores, in addition, the validity of the general formula in the Miles-Ezzel setup with annual adjustment of the level of debt to maintain a constant market-value debt ratio.
JEL Classification: G30; G32; E22
Keywords: WACC; Value of tax shield
I. INTRODUCTION
In his comment, Fernandez (2007) claims that the general formulation of the WACC proposed in Farber, Gillet, Szafarz (FGS, 2006) is only correct under the assumption that the required returns are constant over time. Moreover, he argues that "it is not possible to derive a debt policy such that the appropriate discount rate for the tax shield is [r.sub.A] in all periods" (where rA is the cost of capital for the unlevered firm).
These statements are incorrect. First, the general formula of the WACC in FGS (2006) is derived from the equality between the average return on both side of the market-value balance sheet. It should be satisfied at any point in time and, more importantly, it does not imply that the WACC is a constant. Of course, if all variable entering the formula are constant, the resulting WACC is a constant. But, as shown in this paper, a constant WACC can also be obtained when some of the variables vary over time.
Second, as shown by several authors (see, for instance, Taggart, 1991 or Arzac and Glosten 2005), the expected return on the tax shield is equal to the cost of capital of the unlevered firm for a leverage policy with continuous rebalancing of the level of debt to maintain a fixed leverage ratio. This financial policy was initially analyzed by Harris and Pringle (1985). Fernandez' comment refers to the Miles and Ezzell (1980) setting where the firm rebalances its debt once a year instead of continuously, a case not considered in FGS (2006).
The rest of this paper proceeds as follow. Section 1 shows that the formulation of the WACC in FGS (2006) is indeed general and does not require assuming constant returns over time. Section 2 derives the classical WACC formulas of Modigliani Miller (1963) and Harris Pringle (1985), the latter corresponding to the Miles and Ezzel (1980) case with continuous rebalancing. Section 3 examines whether the general formula applies in a Miles-Ezzel setup with annual rebalancing and arbitrary cash-flows. Section 4 considers the use of the general formula under perpetual growth. General conclusions are presented in section 5.
II. THE GENERAL WACC FORMULA
The general formula for the WACC (Equation (18) in FGS, 2006) is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
where [r.sub.A], [r.sub.D] and [r.sub.TS] are the expected returns respectively on the unlevered firm, the debt and the tax shield, V is the value of the levered firm, VTS is the value of the tax shield, D is the value of the debt and TC is the corporate tax rate.
This formula is derived from the balance sheet identity and the definition of the weighted average cost of capital (for a similar presentation, see Berk and De Marzo, 2007). It should therefore be verified at any point in time, whatever returns are annually or continuously compounded.
To exhibit this, start with the accounting identity (time indices have been added for clarification):
[V.sub.t] = V[U.sub.t] + VT[S.sub.t] = [E.sub.t] + [D.sub.t] (2)
where V[U.sub.t] is the value of the unlevered firm.
Assuming, for simplicity, that the corporate tax rate is constant over time, the weighted average cost of capital is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
From (2), the expected returns on both side of the balance sheet should be equal:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
As:
V[U.sub.t] = [V.sub.t] - VT[S.sub.t] (5)
and, from (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
the general formula for the WACC is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
This equation is the general formula in FGS (2006).
III. THE MODIGLIANI-MILLER AND HARRIS-PRINGLE FORMULAS
Although Eq. (7) is fully general, its practical applicability requires additional conditions. Indeed, when the WACC is constant over time, the value of a levered firm can be computed by discounting with the WACC the unlevered free cash-flows. Therefore, we pay a special attention to the special cases that make the WACC constant. The resulting particular formulas can also be found in textbooks (Brealey et al., 2006; Ross et al., 2005). As in all papers devoted to this topic, we assume here that [r.sub.A,t] = [r.sub.A] and [r.sub.D,t] = [r.sub.D] are constant.
First, Modigliani and Miller (1963) assume that the level of debt D is constant. Then, as the expected after-tax cash-flow of the unlevered firm is fixed, VU is also constant. By assumption, [r.sub.TS] = [r.sub.D] and the value of the tax shield is [VTS.sub.t] = [T.sub.C]D. Therefore, the value of the firm V is a constant and the general WACC formula (7) simplifies to a constant WACC:
[WACC.sub.t] = [r.sub.A] - [r.sub.A][T.sub.C]L for all t (8)
with L = D/V.
Second, Harris and Pringle (1985) assume that the level of debt is rebalanced continuously so as to maintain a constant debt ratio [D.sub.t]/[V.sub.t] = L. Moreover, as the level of debt is always proportional to the value of the unlevered firm, the expected return on the tax shield is the same as the cost of capital of the unlevered firm ([r.sub.TS] = [r.sub.A]). As a consequence, this implies a constant WACC:
[WACC.sub.t] = [r.sub.A] - [r.sub.D] [T.sub.C]L for all t (9)
where [r.sub.A] and [r.sub.D] are continuously compounded returns (see: Taggart, 1991, and Arzac and Glosten, 2005). It is worth noting that formula (9) does not require the calculation of VTS. Moreover, Eq. (9) shows that, with constant [r.sub.A], [r.sub.D] and [T.sub.C], a constant WACC may only be result from a constant debt ratio L. Any other financial policy makes the WACC time-varying.
IV. MILES-EZZEL WITH ANNUAL REBALANCING AND ARBITRARY CASH-FLOWS
We now apply the general formula (7) to the Miles and Ezzel (1980) setup where the debt is rebalanced once a year and cash-flows are arbitrary, a case not considered in FGS (2006).
Formula (7) can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
With arbitrary cash-flows, both [r.sub.TS,t] and [VTS.sub.t] /[V.sub.t] may vary over time. Under what condition does the WACC remain constant?
From Eq. (10), the condition is: the product ([r.sub.TS,t] - [r.sub.A])([VTS.sub.t]/[V.sub.t]) is constant. This condition is met by the Miles-Ezzell leverage policy with annual rebalancing. Indeed, these authors obtain the following value for the tax shield at time t:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
The next year tax shield is discounted at the cost of debt whereas the expected future value of the tax shield is discounted at the cost of capital of the unlevered firm.
By definition, the expected return on the tax shield is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)
Using (11):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)
Therefore:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
Formula (14) shows that [r.sub.TS,t] is inversely related to the fraction of the value accounted by the tax shield. For instance, with a finite horizon T and [VTS.sub.T], the value of the tax shield at time T-1 is obtained from (11) as :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)
and:
[r.sub.TS,t-1] = [r.sub.D] (16
Since the dynamics of the unlevered cash-flows is arbitrary, [V.sub.t] may well vary with time, so that both [r.sub.TS,t] and [VTS.sub.t] / [V.sub.t] change as well. However, the WACC remains constant because:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
Replacing in the general formula (7), we obtain the Miles-Ezzell formula for the WACC with annual rebalancing:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)
V. MILES EZZEL WITH ANNUAL REBALANCING AND CONSTANT GROWTH
Consider now a firm whose (unlevered) free cash-flows grow at a constant rate g. In order to derive the expected return on the tax shield for this firm, let us start from the formula for [VTS.sub.t] obtained by Arzac and Glosten (2005) (their Eq. (13)):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)
Therefore, the constant ratio of the value of the levered firm accounted to the value the tax shield is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)
Replacing in (14), the constant expected return on the tax shield is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)
With perpetual constant growth, the expected return on the tax shield is constant over time and positively related to the growth rate. In the special case where g = 0, it simplifies to:
[r.sub.TS] = [r.sub.A] 1 + [r.sub.D] / (1 + [r.sub.A]) (22)
where the ratio (1+[r.sub.D])/(1+[r.sub.A]) captures the impact of annual rebalancing (as opposed to continuous rebalancing) on the expected return of the tax shield.
Finally, appropriate replacements in the general formula lead again to Eq. (18), the Miles-Ezzell formula with annual rebalancing.
VI. CONCLUSIONS
The formula proposed in FGS (2006) is general as claimed originally. It is a time-independent identity which does not depend on the model used to value the tax shield. The WACC formulas of Modigliani-Miller (constant debt) and Harris-Pringle (constant debt ratio with continuous rebalancing) can be seen as special cases of this general formula. Moreover, in the Miles-Ezzell setup (constant debt ratio with annual rebalancing), a constant WACC is obtained because variations in the expected return on the tax shield compensate variations in the fraction of the firm accounted by the value of the tax shield.
Whether the Miles and Ezzel model is realistic or not regarding firms' debt structure remains an open question lying outside the scope of the current paper. Further work could also address the discrete vs. continuous time scale assumptions and their consequence regarding the WACC calculations.
Finally, we thank Pablo Fernandez for pointing out the importance and complexity of these issues and offering us the possibility to clarify our point and exhibiting additional applications of the WACC general formula.
REFERENCES
Arzac, E.R., and L.R. Glosten, 2005, "A Reconsideration of Tax Shield Valuation", European Financial Management, 11/4, pp. 453-461.
Berk, J., and P. DeMarzo, 2007, Corporate Finance, Boston: Pearson-Addison Wesley.
Brealey, R., S. Myers and F. Allen, 2006, Corporate Finance, 8th ed., New York: McGrawHill.
Farber, A., R. Gillet, and A. Szafarz, 2006, "A General Formula for the WACC", International Journal of Business, 11/2, pp. 211-218.
Fernandez, P., 2007, "A General Formula for the WACC: a Comment", International Journal of Business, this issue.
Harris, R., and J. Pringle, 1985, "Risk-Adjusted Discount Rates--Extension form the Average-Risk Case", The Journal of Financial Research, 8/3, pp. 237-244.
Miles, J.A., and J.R. Ezzel, 1980, "The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: A Clarification" Journal of Financial and Quantitative Analysis, 15/3, pp. 719-730.
Modigliani, F., and M. Miller, 1963, "Corporate Income Taxes and the Cost of Capital: a Correction" American Economic Review, 53, pp. 433-443.
Ross, S., R. Westerfiel, and D. Jaffee, 2005, Corporate Finance, 7th ed., New York: McGraw Hill.
Taggart, R.A., 1991, "Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes", Financial Management, 20, pp. 8-20.
Andre Farber (a), Roland Gillet (b), and Ariane Szafarz (c)
(a) Professor of Finance, Universite Libre de Bruxelles Centre Emile Bernheim (Solvay Business School), CP 145/1 50, av. Roosevelt, 1050 Bruxelles, Belgium afarber@ulb.ac.be
(b) Professor of Finance, Universite de Paris 1-Pantheon-Sorbonne Gestion Sorbonne, 17, rue de la Sorbonne 75005 Paris, France roland.gillet@univ-paris1.fr
(c) Professor of Finance, Universite Libre de Bruxelles Centre Emile Bernheim (Solvay Business School), CP 145/1 50, av. Roosevelt, 1050 Bruxelles, Belgium aszafarz@ulb.ac.be
