Abstract

The value of tax shields depends upon the nature of the stochastic process of the net increase of debt, and does not depend upon the nature of the stochastic process of the free cash flow.

The value of tax shields in a world with no leverage cost is the tax rate times the debt plus the tax rate times the present value of the net increases of debt. This expression is the difference between the present values of two different cash flows, each with their own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company.

For perpetual debt, the value of tax shields is the debt times the tax rate. When the company forecast to repay the actual debt without issuing new debt, the value of tax shields is the present value of the interest times the tax rate, discounted at the required re turn to debt.

*JE**L classification: *G12; G31; G32

*Keywords**: *Value of tax shields, present value of the net increases of debt, required return to equity, leverage cost, unlevered beta, levered beta,

1. Introduction

There is no consensus in the existing literature regarding the correct way to compute the value of tax shields. Most authors think of calculating the value of the tax shield in terms of the appropriate present value of the tax savings due to interest payments on debt, but Myers (1974) proposes to discount the tax savings at the cost of debt, whereas Harris and Pringle (1985) propose discounting these tax savings at the cost of capital for the unlevered firm. Reflecting this lack of consensus, Copeland et al. (2000, p. 482) claim that “the finance literature does not provide a clear answer about which discount rate for the tax benefit of interest is theoretically correct.” In this paper, I show that a consistent way to estimate the value of the tax savings is not by thinking of them as the present value of a set of cash flows, but as the difference between the present values of two different sets of cash flows: flows to the unlevered firm and flows to the levered firm.

I show that the value of tax shields in a world with no leverage cost is the tax rate times the debt plus the tax rate times the present value of the net increases of debt. This expression is the difference between the present values of two different cash flows, each with their own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company.

For perpetual debt, the value of tax shields is equal to the tax rate times the value of debt. When the company forecast to repay the actual debt without issuing new debt, Myers (1974) applies, and the value of tax shields is the present value of the interest times the tax rate, discounted at the required return to debt. For constant growth companies, and under certain assumptions, the value of tax shields in a world with no leverage costs is the present value of the debt times the tax rate times the required return to the unlevered equity, discounted at the unlevered cost of equity (Ku). The paper also shows that some commonly used methodologies for calculating the value of tax shields, including Harris and Pringle (1985), Miles and Ezzell (1980), and Ruback (2002), are incorrect for growing perpetuities.

The paper is organized as follows. Section 2 follows a new method to prove that the value of tax shields in a world without leverage costs is equal to the tax rate times the value of debt (DT) plus the tax rate times the present value of the net increases of debt. Section 2 also applies this general result to specific situations, and derives the relation between the required return on assets and the required return on equity for perpetuities in a world without leverage costs. The corresponding relation between the beta of the levered equity, the beta of the unlevered equity, and the beta of debt is also derived. Section 3 revises and analyzes the existing financial literature on the value of tax shields. Most of the existing approaches, including Harris and Pringle; Miles and Ezzell; and Ruback, result in inconsistent results regarding the present value of the net increases of debt. Finally, Section 4 concludes.

2. Value of tax shields and the stochastic process of net debt increases

The present value of debt (D) plus that of the equity (E) of the levered company is equal to the value of the unlevered company (Vu) plus VTS, the value of tax shields due to interest payments:

In the literature, the value of tax shields defines the increase in the company’s value as a result of the tax saving obtained by the payment of interest. In this definition, the total value of the levered company is equal to the total value of the unlevered company. If leverage costs do not exist, then Eq. (1) could be stated as follows:

(2) where Gu is the present value of the taxes paid by the unlevered company and GL is the present value of the taxes paid by the levered company. Eq. (2) means that the total value of the unlevered company (left- hand side of the equation) is equal to the total value of the levered company (right-hand side of the equation). Total value is the enterprise value (often called the value of the firm) plus the present value of taxes. Please note that Eq. (2) assumes that expected free cash flows are independent of leverage. When leverage costs do exist, the total value of the levered company is lower than the total value of the unlevered company. A world with leverage cost is characterized by the following relation:

Leverage cost is the reduction in the company’s value due to the use of debt. From (1) and (2), it is clear that VTS is

Note that the value of tax shields is not the present value (PV) of tax shields. It is the difference between the PVs of two flows with different risk: the PV of the taxes paid by the unlevered company (Gu) and the PV of the taxes paid by the levered company (GL).

It is quite easy to prove that the relation between the profit after tax of the levered company (PATL) and the equity cash flow (ECF) is:

Being:

The relation between the free cash flow (FCF) and the profit aft er tax of the unlevered company (PATu) is:

Equation (9) means that the present value of the taxes paid by the unlevered company (Gu) is the present value of the taxes paid every year (TaxesU).

This result is far from being a new idea. Brealey and Myers (2000), Modigliani and Miller (1963), Taggart (1991), Copeland et al. (2000), Fernandez (2004) and many others report it. However, the way of deriving it is new. Most of these papers reach this result by arguing that the appropriate way of computing the value of the tax shield is to consider a certain flow DT multiplied by some measure of cost of funds, and then discounting that flow at the same rate. At first glance,could be anything, related or unrelated to the company that we are valuing. Modigliani and Miller (1963) argue that is the risk-free rate (RF ). Myers (1974) assumes that is the cost of debt (Kd) and says that the value of tax shields is the present value of the tax savings (D·Kd·T) discounted at the cost of debt (Kd). Fernandez (2004) argue that is the required return to unlevered equity (Ku).

2.2. Debt of one year maturity but perpetually rolled-over

3. Value of net debt increases implied by the alternative theories

There is a considerable body of literature on the discounted cash flow valuation of firms. This section addresses the most salient papers, concentrating particularly on those papers that propose alternative expressions for the value of tax shields (VTS). The main difference between all of these papers and the approach proposed above is that most previous papers calculate the value of tax shields as the present value of the tax savings due to the payment of interest. Instead, the correct measure of the value of tax shields is the difference between two present values: the present value of taxes paid by the unlevered firm and the present value of taxes paid by the levered firm. We will show ho w these proposed methods result in inconsistent valuations of the tax shields.

Modigliani and Miller (1958, 1963) study the effect of leverage on firm value. Their famous Proposition 1 states that, in the absence of taxes, the firm’s value is independent of its debt, i.e., E + D = Vu, if T = 0. In the presence of taxes and for the case of a perpetuity, but with zero risk of bankruptcy, they calculate the value of tax shields by discounting the present value of the tax savings due to interest payments on risk- free debt at the risk-free rate

As indicated above, this result is the same as our Eq. (16) for the case of perpetuities, but it is neither correct nor applicable for growing perpetuities. Modigliani and Miller explicitly ignore the issue of the riskiness of the cash flows by assuming that the probability of bankruptcy was always zero.

Myers (1974) introduces the APV (adjusted present value) method in which the value of the levered firm is equal to the value of the firm with no debt plus the present value of the tax savings due to the payment of interest. Myers proposes calculating the VTS by discounting the expected tax savings (D·Kd·T) at the cost of debt (Kd). The argument is that the risk of the tax savings arising from the use of debt is the same as the risk of the debt. The value of tax shields is VTS = PV[E{D·Kd·T}; Kd]. This approach has also been recommended in later papers in the literature, such as Luehrman (1997). On section 2.5 we have shown that this expression is correct only when the company forecast to repay the actual debt without issuing new debt.

Harris and Pringle (1985) propose that the present value of the tax savings due to the payment of interest should be calculated by discounting the expected interest tax savings (D·Kd·T) at the required return to unlevered equity (Ku), i.e., VTS = PV[E{D·Kd·T}; Ku]. Their argument is that the interest tax shields have the same systematic risk as the firm’s underlying cash flows and, therefore, should be discounted at the required return to assets (Ku). Furthermore, Harris and Pringle believe that “the MM position is considered too extreme by some because it implies that interest tax shields are no more risky than the interest payments themselves” (p.242). Ruback (1995, 2002), Kaplan and Ruback (1995), Brealey and Myers (2000, p.555), and Tham and Vélez-Pareja (2001), this last paper following an arbitrage argument, also claim that the appropriate discount rate for tax shields is Ku, the required return to unlevered equity.

Ruback (2002) presents the Capital Cash Flow (CCF) method and claims that the appropriate discount rate is Ku; Arditti and Levy (1977) suggested that the firm’s value could be calculated by discounting the capital cash flows instead of the free cash flow. The capital cash flows are the cash flows available for all holders of the company’s securities, whether debt or equity, and are equivalent to the equity cash flows plus the cash flows corresponding to the debt holders. The capital cash flow is also equal to the free cash flow plus the interest tax shield (D·Kd·T). According to the Capital Cash Flow (CCF) method, the value of the debt today (D) plus that of the shareholders’ equity (E) is equal to the expected capital cash flow (CCF) discounted at the weighted average cost of capital before tax (WACCBT ):

A large part of the literature argues that the value of tax shields should be calculated in a different manner depending on the debt strategy of the firm. A firm that wishes to keep a constant D/E ratio must be valued in a different manner from a firm that has a preset level of debt. Miles and Ezzell (1980) indicate that for a firm with a fixed debt target (i.e., a constant [D/(D+E)] ratio), the correct rate for discounting the tax savings due to debt is Kd for the first year and Ku for the tax savings in later years. Lewellen and Emery (1986) also claim that this one is the most logically consistent method. Although Miles and Ezzell do not mention what the value of tax shields should be, this can be inferred from their equation relating the required return to equity with the required return for the unlevered company in Eq. (22) in their paper. This relation implies that VTS = PV[E{D·T·Kd}; Ku] (1 + Ku)/(1 + Kd). Inselbag and Kaufold (1997), and Ruback (2002) argue that if the firm targets the dollar values of debt outstanding, the VTS is given by the Myers (1974) formula. However, if the firm targets a constant debt/value ratio, the value of the tax shields should be calculated according to Miles and Ezzell (1980). Finally, Taggart (1991) proposes to use Miles and Ezzell (1980) if the company adjusts to its target debt ratio once a year and Harris and Pringle (1985) when the company continuously adjusts to its target debt ratio.

The paper also shows that to discount the expected tax shields at the required return to unlevered equity, as suggested by Harris and Pringle (1985), Miles and Ezzell (1980), and Ruback (2002), is not consistent

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I thank my colleague José Manuel Campa and Charles Porter for their wonderful help revising previous manuscripts of this paper, and an anonymous referee for very helpful comments. I also thank Rafael Termes and my colleagues at IESE for their sharp questions that encouraged me to explore valuation p roblems.

[*]Contact information:

IESE Business School, University of Navarra. Camino del Cerro del Aguila 3. 28023 Madrid, Spain. E-mail: fernandezpa@iese.edu

[1] If the nominal value of debt (N) is not equal to the value of debt (D), becaause the interest rate (r) is different from the required return to debt flows (Kd), equation (12) is: VTS0 = T· D0 + T· PV0[DNt]. The relationship between D and N is: D0 **= **PV0 [DNt] + PV0[Nt·rt].

[3 ]We use Kd for not complicating the notation. It should be Kdt, a different rate following the yield curve. Using Kd we may also think on a flat yield curve.