ABSTRACT
This note builds on the paper of Farber, Gillet and Szafarz (2006). The WACC is a discount rate widely used in corporate finance. However, the correct calculation of the WACC rests on a correct valuation of the tax shields. The value of tax shields depends on the debt policy of the company. Many authors, (e.g. Inselbag and Kaufold (1997), Booth (2002), Cooper and Nyborg (2006), Farber, Gillet and Szafarz (2006)) consider that debt policy may only be framed in terms of maintaining a fixed market value debt ratio (Miles-Ezzell assumption) or a fixed dollar amount of debt (Modigliani-Miller assumption).
JEL classification: G12; G31; G32
Keywords: WACC; Required return to equity; Value of tax shields; Company valuation; APV; Cost of equity
I. INTRODUCTION
The value of tax shields (VTS) defines the increase in the company's value as a result of the tax saving obtained by the payment of interest. However, there is no consensus in the existing literature regarding the correct way to compute the VTS. Modigliani and Miller (1963), Myers (1974), Brealey and Myers (2000) and Damodaran (2006) propose to discount the tax savings due to interest payments on debt at the cost of debt ([r.sub.D]), whereas Harris and Pringle (1985) and Ruback (2002) propose discounting these tax savings at the cost of capital for the unlevered firm ([r.sub.A]). Miles and Ezzell (1985) propose discounting these tax savings the first year at the cost of debt and the following years at [r.sub.A].
Equation (18) of Farber, Gillet and Szafarz (2006) is a general formulation of the WACC:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
where E is the value of the equity, D is the value of the debt, Vu is the value of the unlevered equity, VTS is the value of tax shields, and [r.sub.E], [r.sub.D], [r.sub.A] and [r.sub.TS] are the required returns to the expected cash flows of equity, debt, assets (free cash flow) and tax shields.
This equation is correct if the required returns are always constant over time. Farber, Gillet and Szafarz (2006) consider that [r.sub.TS] can be [r.sub.D] (as Modigliani-Miller) or [r.sub.A] (as Harris-Pringle). These two scenarios correspond to two different financing strategies: the first one is valid for a company that has a preset amount of debt and the second one should be valid for a company that has constant debt ratio in market value terms. However, as Miles and Ezzell (1985) and Arzac and Glosten (2005) prove, the required return for the tax shield ([r.sub.TS]) of a company with a constant debt ratio in market value terms is [r.sub.D] for the tax shields of the first period and [r.sub.A] thereafter. It is not possible to derive a debt policy such that the appropriate discount rate for the tax shields is [r.sub.A] in all periods. [D.sub.t] = L x ([D.sub.t] + [E.sub.t]) implies that [D.sub.t] is also proportional to [FCF.sub.t]. The Miles and Ezzell (1985) correct formula for the VTS of a perpetuity growing at a rate g is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
Formula (2) is identical to formulae (21) of Miles and Ezzell (1985), (13) of Arzac and Glosten (2005) and (7) of Lewellen and Emery (1986). Formula (2) arises from considering that [r.sub.TS] = [r.sub.D] in the first period (t = 1) and [r.sub.TS] = [r.sub.A] for the following periods (t > 1). However, Farber, Gillet and Szafarz (2006) assume, as Harris and Pringle (1985), that [r.sub.TS] = [r.sub.A] in all periods. The formula for the VTS offered by Harris and Pringle (1985) and implied by Farber, Gillet and Szafarz (2006) in their equations (28) and (29) is:
[VTS.sup.HP] = [Dr.sub.D] x T/([r.sub.A] - g) (3)
If debt is adjusted continuously, not only at the end of the period, then the ME formula (2) changes to
VTS = D x [rho] x T/([kappa] - [gamma]) (4)
where [rho] = ln(1+ [r.sub.D]), [gamma] = ln(1+g), and [kappa] = ln(1+ [r.sub.sub.A]). Equation (4) is quite similar to equation (3) (but then [r.sub.D], g and [r.sub.A] should also be expressed in continuous time). However, (3) is incorrect for discrete time: (2) is the correct formula.
Therefore, equations (14) and (28) of Farber et al (2006) for discrete time should be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
Equations (25) and (29) should be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
II. REQUIRED RETURN TO EQUITY AND WACC FOR PERPETUITIES WITH A CONSTANT GROWTH RATE
For perpetuities with a constant growth rate (g), the relationship between expected values in t=1 of the free cash flow (FCF) and the equity cash flow (ECF) is:
EC[F.sub.0](1+g) = FC[F.sub.0](1+g) - [D.sub.0] [r.sub.D] (1-T) + g [D.sub.0] (7)
The value of the equity today (E) is equal to the present value of the expected equity cash flows. If [r.sub.E] is the average appropriate discount rate for the expected equity cash flows, then E = EC[F.sub.0](1+g) /([r.sub.E] -g), and equation (7) is equivalent to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
And the general equation for the [r.sub.E] is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
Equation (9) is equivalent to equation (10) of Farber et al (2006) because VTS = D [r.sub.D]T/ ([r.sub.TS] - g).
The WACC is the appropriate discount rate for the expected free cash flows, such that [D.sub.0]+[E.sub.0]= FC[F.sub.0](1+g) / (WACC-g) . The equation that relates the WACC and the VTS is (10):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
Equation (10) is equivalent to equation (18) of Farber et al (2006) because VTS = D [r.sub.D] T / ([r.sub.TS] - g).
III. CONCLUSIONS
The WACC is a discount rate widely used in corporate finance. However, the correct calculation of the WACC rests on a correct valuation of the tax shields. The value of tax shields depends on the debt policy of the company. When the debt level is fixed, Modigliani-Miller applies, and the tax shields should be discounted at the required return to debt. If the leverage ratio is fixed at market value, then Miles-Ezzell applies. Other debt policies should be explored. For example, Fernandez (2006) develops valuation formulae for the situation in which the leverage ratio is fixed at book values and argues that it is more realistic to assume that a company maintains a fixed book-value leverage ratio than to assume, as Miles-Ezzell do, that the company maintains a fixed market-value leverage ratio because the company is more valuable, and because it is easier to follow for non quoted companies.
REFERENCES
Arzac, E.R., and L.R. Glosten, 2005, "A Reconsideration of Tax Shield Valuation," European Financial Management 11/4, pp. 453-461.
Booth, L., 2002, "Finding Value Where None Exists: Pitfalls in Using Adjusted Present Value," Journal of Applied Corporate Finance 15/1, pp. 8-17.
Brealey, R.A., and S.C. Myers, 2000, Principles of Corporate Finance, 6th edition, New York: McGraw-Hill.
Cooper, I.A., and K.G. Nyborg, 2006, "The Value of Tax Shields IS Equal to the Present Value of Tax Shields," Journal of Financial Economics 81, pp. 215-225.
Damodaran, A., 2006, Damodaran on Valuation, 2nd edition, New York: John Wiley and Sons.
Farber, A., R.L. Gillet, and A. Szafarz, 2006, "A General Formula for the WACC," International Journal of Business 11/2, pp. 211-218.
Fernandez, P., 2006, "A More Realistic Valuation: APV and WACC with Constant Book Leverage Ratio," Available at SSRN: http://ssrn.com/abstract=946090.
Harris, R.S., and J.J. Pringle, 1985, "Risk-adjusted discount rates extensions from the average-risk case," Journal of Financial Research 8, 237-244.
Inselbag, I., and H. Kaufold, 1997, "Two DCF Approaches for Valuing Companies under Alternative Financing Strategies and How to Choose between Them," Journal of Applied Corporate Finance 10, pp. 114-122.
Lewellen, W.G., and D.R. Emery, 1986, "Corporate Debt Management and the Value of the Firm," Journal of Financial and Quantitative Analysis 21/4, pp.415-426.
Miles, J.A., and J.R. Ezzell, 1985, "Reformulating Tax Shield Valuation: A Note," Journal of Finance 40/5, pp. 1485-1492.
Modigliani, F., and M. Miller, 1963, "Corporate Income Taxes and the Cost of Capital: a Correction," American Economic Review 53, pp.433-443.
Myers, S.C., 1974, "Interactions of Corporate Financing and Investment Decisions--Implications for Capital Budgeting," Journal of Finance 29, pp.1-25.
Ruback, R., 2002, "Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows," Financial Management 31, pp. 85-103.
Pablo Fernandez (a)
(a) IESE Business School, University of Navarra Camino del Cerro del Aguila 3, 28023 Madrid, Spain fernandezpa@iese.edu
