The value of tax shields is NOT equal to the present value of tax shields

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Journal of Financial Economics 73 (2004) 145–165

The value of tax shields is NOT equal to the present value of tax shields$ Pablo Fernandez* IESE Business School, University of Navarra, Camino del Cerro del Aguila 3, Madrid 28023, Spain Received 11 December 2001; accepted 23 October 2002

Abstract The value of tax shields 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 constant growth companies, the value of tax shields in a world with no leverage cost is the present value of the debt, times the tax rate, times the unlevered cost of equity, discounted at the unlevered cost of equity. This result arises as the difference of two present values and does not mean that the appropriate discount for tax shields is the unlevered cost of equity. r 2003 Elsevier B.V. All rights reserved. JEL classification: G12; G31; G32 Keywords: Value of tax shields; 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 $

I thank my colleague Jos!e Manuel Campa and Charles Porter for their wonderful help revising previous manuscripts of this paper, and an anonymous referee for very helpful comments and suggestions. I also thank Rafael Termes and my colleagues at IESE for their sharp questions that encouraged me to explore valuation problems. *Fax: 34-91-3572913. E-mail address: [email protected] (P. Fernandez). 0304-405X/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jfineco.2002.10.001

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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 for perpetuities is equal to the tax rate times the value of debt. For constant growth companies, 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 ðKu Þ: This does not mean that the appropriate discount for tax shields is the unlevered cost of equity, since the amount being discounted is higher than the tax shield (it is multiplied by the unlevered cost of equity and not the cost of debt). Rather, this result arises as the difference between two present values. 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), Myers (1974), and Ruback (2002), are incorrect for growing perpetuities. One could argue that working out the present value of the tax shields themselves would not necessarily be wrong, provided the appropriate discount rate were used (reflecting the riskiness of the tax shields). The problem here is that the riskiness of the tax shields is hard to evaluate because it represents the difference in the present value of two flows with different risk: the taxes paid by the unlevered company and those paid by the levered company. In this context, evaluating the riskiness of tax shields for a firm is conceptually as hard as evaluating the riskiness of the difference between Microsoft’s expected equity cash flows and GE’s expected equity cash flows. We can evaluate the riskiness of the expected equity cash flows of each company, but it is difficult (and hardly makes any sense) to try to evaluate the riskiness of the difference between two expected equity cash flows. Perhaps the most important issue to take into account when reading this paper is that the term ‘‘discounted value of tax shields’’ in itself is senseless. The value of tax shields is the difference between the present values of two separate cash flows each with its own risk. The paper is organized as follows. Section 2 follows a new method to prove that the value of tax shields for perpetuities in a world without leverage costs is equal to the tax rate times the value of debt (DT). Section 3 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 4 proves that 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. Section 5 revises and analyzes the existing financial literature on the value of tax shields. Most of the existing approaches, including Harris and Pringle (1985), Miles and Ezzell (1980), Myers (1974), and

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Ruback (2002) result in either inconsistent results regarding the cost of equity to the levered and unlevered firm or estimated tax shields that are too low. Section 6 contains a numerical example for a hypothetical firm to clarify the previous sections. Finally, Section 7 concludes.

2. The value of tax shields for perpetuities in a world without leverage costs The value of tax shields for a perpetuity is DT; where D indicates the value of the debt today and T is the tax rate. The value of debt ðDÞ plus that of the equity ðEÞ of the levered company is equal to the value of the unlevered company ðVu Þ plus VTS, the value of tax shields due to interest payments E þ D ¼ Vu þ VTS:

ð1Þ

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: Vu þ Gu ¼ E þ D þ GL ;

ð2Þ

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: Vu þ Gu ¼ E þ D þ GL þ Leverage Cost > E þ D þ GL :

ð3Þ

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 VTS ¼ Gu  GL :

ð4Þ

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 ðGu Þ and the PV of the taxes paid by the levered company ðGL Þ: It is quite easy to prove that, in perpetuity, the profit after tax of the levered company ðPATL Þ is equal to the equity cash flow (ECF): PATL ¼ ECF:

ð5Þ

This is so because the allowed depreciation deduction is exactly equal to the cash actually spent to replace capital equipment that wears out.

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In a perpetuity, then, the free cash flow (FCF) is equal to the profit before tax of the unlevered company ðPBTu Þ multiplied by one minus the effective tax rate: FCF ¼ PBTu ð1  TÞ:

ð6Þ

We will call FCF0 the company’s free cash flow if there were no taxes. FCF0 is equal to the profit before taxes of the unlevered company ðPBTu Þ: FCF0 ¼ PBTu :

ð7Þ

From (6) and (7) it is clear that the relation between after-tax free cash flow and the company’s free cash flow if there were no taxes is FCF ¼ FCF0 ð1  TÞ:

ð8Þ

The taxes paid every year by the unlevered company ðTaxesU Þ are TaxesU ¼ T PBTu ¼ T FCF0 ¼ T FCF=ð1  TÞ:

ð9Þ

Designate KTU as the required return to tax in the unlevered company, and KTL as the required return to tax in the levered company. The taxes paid by the unlevered company are proportional to FCF0 and FCF. Consequently, the taxes of the unlevered company have the same risk as FCF0 (and FCF), and hence must be discounted at the rate Ku (the required return to unlevered equity). In the unlevered company, the required return to tax ðKTU Þ is equal to the required return to equity ðKu Þ: This is only true for perpetuities. KTU ¼ Ku :

ð10Þ

The present value of the taxes paid by the unlevered company ðGu Þ is the present value of the taxes paid every year ðTaxesU Þ discounted at the appropriate discount rate ðKu Þ: As Vu ¼ FCF=Ku ; Gu ¼ TaxesU =KTU ¼ T FCF=½ð1  TÞKu  ¼ T Vu =ð1  TÞ:

ð11Þ

For the levered company, taking into consideration Eq. (5), the taxes paid each year ðTaxesL Þ are proportional to the equity cash flow (ECF): TaxesL ¼ T PBTL ¼ T PATL =ð1  TÞ ¼ T ECF=ð1  TÞ:

ð12Þ

PBTL and PATL are the profit before and after tax of the levered company. Because taxes paid by the levered company are proportional to ECF, they have the same risk as the ECF and thus must be discounted at the rate Ke : So, in the case of perpetuities, the tax risk is identical to the equity cash flow risk, and the required return to tax in the levered company ðKTL Þ is equal to the required return to equity ðKe Þ: Again, this is only true for perpetuities. KTL ¼ Ke :

ð13Þ

The relation between profit after tax (PAT) and profit before tax (PBT) is PAT ¼ PBTð1  TÞ: Then the present value of the taxes paid by the levered company is equal to: GL ¼ TaxesL =KTL ¼ T ECF=½ð1  TÞKe  ¼ T E=ð1  TÞ:

ð14Þ

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The increase in the company’s value due to the use of debt is not the present value of tax shields due to interest payments, but rather the difference between Gu and GL ; which are the present values of two cash flows with different risks VTS ¼ Gu  GL ¼ ½T=ð1  TÞðVu  EÞ:

ð15Þ

Because Vu  E ¼ D  VTS; we have VTS ¼ Value of tax shields ¼ DT:

ð16Þ

This result is far from being a new idea. Brealey and Myers (2000), Modigliani and Miller (1963), Taggart (1991), and Copeland et al. (2000), 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, a; and then discounting that flow at the same rate a: At first glance, a could be anything, related or unrelated to the company that we are valuing. Modigliani and Miller (1963) argue that a is the risk-free rate ðRF Þ: Myers (1974) assumes that a is the cost of debt ðKd Þ and says that the value of tax shields is the present value of the tax savings ðDKd TÞ discounted at the cost of debt ðKd Þ: I represent the value of tax shields as the difference between Gu and GL ; which are the present values of two cash flows with different risks: the taxes paid by the unlevered company and the taxes paid by the levered company. In Section 4, I prove that the correct a is the required return to unlevered equity ðKu Þ: Eq. (16) is the difference between the present value of the taxes of the unlevered company, PV½Ku ; TaxesU ; and the present value of the taxes of the levered company, PV½Ke ; TaxesL : VTS ¼ PV½Ku ; TaxesU   PV½Ke ; TaxesL :

ð17Þ

The annual difference between the cash flows to investors in a leveraged firm and the cash flows to investors in an all-equity firm is precisely equal to the interest tax shield TaxesU  TaxesL ¼ DKd T:

ð18Þ

For perpetuities, and only for perpetuities, it can be argued that the risk of this difference is Kd : Therefore, the present value of this difference should be DKd T=Kd ¼ DT : But for growing companies, the risk of the interest tax shield is not Kd or Ku ; as will be seen in Section 4.

3. Relation between the required return to assets ðKu Þ and the required return to equity ðKe Þ Knowing the value of tax shields in a perpetuity ðDTÞ; and considering that Vu ¼ FCF=Ku ; we can rewrite Eq. (1) as follows: E þ Dð1  TÞ ¼ FCF=Ku :

ð19Þ

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Taking into consideration that the relation between ECF and FCF for perpetuities is FCF ¼ ECF þ DKd ð1  TÞ

ð20Þ

and that E ¼ ECF=Ke ; we find that the relation between the required return to assets ðKu Þ and the required return to equity ðKe Þ in a world without leverage costs is Ke ¼ Ku þ ðKu  Kd ÞDð1  TÞ=E:

ð21Þ

This relation can also be expressed in terms of the systematic risk of each cash flow, i.e., the betas. The formulas relating the betas with the required returns are Ke ¼ RF þ bL PM ;

Ku ¼ RF þ bu PM ;

Kd ¼ RF þ bd PM :

RF is the risk-free rate and PM is the market risk premium. The relation between the beta of the levered equity ðbL Þ; the beta of the unlevered equity ðbu Þ; and the beta of debt ðbd Þ is bL ¼ bu þ ðbu  bd ÞDð1  TÞ=E:

ð22Þ

4. The value of tax shields for growing perpetuities For growing perpetuities, we can calculate the value of taxes for the unlevered and levered firm. In the unlevered company with constant growth, the relation between taxes and free cash flow is different from Eq. (9), obtained for perpetuities: TaxesU ¼ TPBTu ¼ T½FCF þ gðWCR þ NFAÞ=ð1  TÞ ¼ T½FCF þ gðEbv þ DÞ=ð1  TÞ;

ð23Þ

where WCR represents net working capital requirements, NFA is net fixed assets, Ebv is the equity book value, and g is the constant growth rate. For any company, WCR þ NFA ¼ Ebv þ D: From Eq. (23) we cannot establish any clear relation between the required return to taxes and the required return to assets ðKu Þ as we had done for perpetuities in Eq. (9) since taxes are no longer proportional to free cash flow. Similarly, in the levered company with constant growth, the relation between taxes and equity cash flow is different from Eq. (12) for perpetuities: TaxesL ¼ TðECF þ gEbv Þ=ð1  TÞ:

ð24Þ

Again, a clear relation cannot be established between the required return to taxes and the required return to equity as for perpetuities in Eq. (12). A derivation analogous to the one done in Section 2 for a perpetuity could be done for a growing firm in the extreme situation in which the firm is all-debt financed. In this situation, ECF ¼ E ¼ 0; and all the risk of the assets is borne by the debt ðKd ¼ Ku Þ: In a growing perpetuity, the debt cash flow is DKd  gD: As ECF ¼ 0; and Kd ¼ Ku ; free cash flow in Eq. (24) is FCF ¼ DKd ð1  TÞ  gD: The debt cash flow is DKu  gD ¼ FCF þ DTKu :

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If the firm is all-debt financed, Ebv ¼ 0: In this situation, Eq. (23) is transformed into TaxesU ¼ T½FCF þ gðEbv þ DÞ=ð1  TÞ ¼ T½DðKu  gÞ  DTKu þ gDÞ=ð1  TÞ ¼ DTKu

ð25Þ

and Eq. (24) is transformed into TaxesL ¼ TðECF þ gEbv Þ=ð1  TÞ ¼ 0:

ð26Þ

The appropriate discount rate for the taxes of the unlevered company is Ku because the unlevered taxes are proportional to the interest rate payments (in this case DKu ). Then Eq. (17) is transformed into VTS ¼ Gu  GL ¼ ½TaxesU =ðKTU  gÞ  ½TaxesL =ðKTL  gÞ ¼ DTKu =ðKu  gÞ:

ð27Þ

Once again, the value of tax shields is not the present value of tax shields due to the payment of interest but rather the difference between Gu and GL ; which represent the present values of two cash flows with different risk. The appropriate way to do an adjusted present value analysis with a growing perpetuity is to calculate the VTS as the present value of DTKu (not the interest tax shield) discounted at the unlevered cost of equity ðKu Þ: VTS ¼ PV½Ku ; DTKu :

ð28Þ

This result can be derived for the general case by noticing that the relation between the equity cash flow of the unlevered firm and the free cash flow in a company with an expected growth rate of g is Vu ¼ FCF=ðKu  gÞ:

ð29Þ

By substituting (29) into (1), we get E þ D ¼ FCF=ðKu  gÞ þ VTS:

ð30Þ

The relation between the equity cash flow and the free cash flow is FCF ¼ ECF þ DKd ð1  TÞ  gD:

ð31Þ

By substituting (31) into (30), we get E þ D ¼ ½ECF þ DKd ð1  TÞ  gD=ðKu  gÞ þ VTS:

ð32Þ

As ECF ¼ EðKe  gÞ; (32) can be rewritten as E þ D ¼ ½EðKe  gÞ þ DKd ð1  TÞ  gD=ðKu  gÞ þ VTS:

ð33Þ

Multiplying both sides of Eq. (33) by ðKu  gÞ; we get ðE þ DÞðKu  gÞ ¼ ½EðKe  gÞ þ DKd ð1  TÞ  gD þ VTSðKu  gÞ:

ð34Þ

Eliminating gðE þ DÞ on both sides of Eq. (34), we have ðE þ DÞKu ¼ ½EKe þ DKd ð1  TÞ þ VTSðKu  gÞ:

ð35Þ

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Eq. (35) can be rewritten as D½Ku  Kd ð1  TÞ  EðKe  Ku Þ ¼ VTSðKu  gÞ:

ð36Þ

Dividing both sides of Eq. (36) by D (debt value), we get, ½Ku  Kd ð1  TÞ  ðE=DÞðKe  Ku Þ ¼ ðVTS=DÞðKu  gÞ:

ð37Þ

If ðE=DÞ is constant, the left-hand side of Eq. (36) does not depend on growth ðgÞ because for any growth rate ðE=DÞ; Ku ; Kd ; and Ke are constant. We know from Eq. (16) that for g ¼ 0; VTS ¼ DT: Then Eq. (37) applied to perpetuities ðg ¼ 0Þ is ½Ku  Kd ð1  TÞ  ðE=DÞðKe  Ku Þ ¼ TKu :

ð38Þ

Subtracting (38) from (37), we get 0 ¼ ðVTS=DÞðKu  gÞ  TKu : Solving for the present value of tax shields, we get VTS ¼ DTKu =ðKu  gÞ:

ð280 Þ

By substituting (28) into (35), we get Eq. (21). This implies that Eq. (21), which relates the cost of capital for the levered and unlevered firm, also applies to growing perpetuities. In fact, Eq. (21) also applies to the general case of any type of growth of the firm. An alternative way of deducing Eq. (28) is by noticing that Eq. (36) has to hold true when the company is all-debt financed, that is, when E ¼ 0 and ECF ¼ 0 (the equity cash flow is zero). In this extreme situation, all the risk of the assets is borne by the debt ðKd ¼ Ku Þ; and Eq. (36) is D½Ku  Ku ð1  TÞ ¼ VTSðKu  gÞ:

ð39Þ

Note that (39) is exactly the same as Eq. (28). Eq. (28) is valid if the cost of debt ðrÞ is equal to the required return to debt ðKd Þ: In this situation, the value of the debt ðDÞ is equal to the nominal or book value of the debt ðNÞ: When the value of the debt ðDÞ is not equal to the nominal or book value of the debt ðNÞ; Eq. (28) changes into VTS ¼ ½DTKu þ TðNr  DKd Þ=ðKu  gÞ:

ð40Þ

5. Comparison to alternative measures 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 how these proposed methods result in inconsistent valuations of the tax

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shield or an inconsistent relation between the cost of capital of the unlevered and levered firm. 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 þ D ¼ Vu ; if T ¼ 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 ðRF Þ; i.e., VTS ¼ PV½RF ; DTRF  ¼ DT: 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 tax savings ðDTKd Þ at the cost of debt ðKd Þ: 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 ¼ PV½Kd ; DTKd : This approach has also been recommended in later papers in the literature, such as Luehrman (1997). 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 interest tax savings ðKd TDÞ at the required return to unlevered equity ðKu Þ; i.e., VTS ¼ PV½Ku ; DKd T: 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 ðKu Þ: 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 Ve! lezPareja (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 ðDKd TÞ: According to the capital cash flow (CCF) method, the value of the debt today ðDÞ plus that of the shareholders’ equity ðEÞ is equal to the capital cash flow (CCF) discounted at the weighted average cost of capital before tax ðWACCBT Þ: E þ D ¼ PV½WACCBT ; CCF:

ð41Þ

The definition of WACCBT is WACCBT ¼ ðEKe þ DKd Þ=ðE þ DÞ

ð42Þ

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because (42) is the discount rate that ensures that the value of the company obtained using the following two expressions is the same: PV½WACCBT ; CCF ¼ PV½WACC; FCF:

ð43Þ

But Ruback (1995, 2002) assumes that WACCBT ¼ Ku : With this assumption, Ruback gets the same valuation as Harris and Pringle (1985) because E þ D ¼ PV½Ku ; CCF ¼ PV½Ku ; FCF þ PV½Ku ; DKd T ¼ Vu þ PV½Ku ; DKd T:

ð44Þ

Note that PV½Ku ; DKd T is the VTS according to Harris and Pringle (1985). Ruback (2002, p. 91) also shows that the relation between the beta of the levered equity ðbL Þ; the beta of the unlevered equity ðbu Þ; and the beta of debt ðbd Þ consistent with (44) is bL ¼ bu þ ðbu  bd ÞD=E:

ð45Þ

Therefore, all of Ruback’s results (the relation between bL and bu using no taxes, and Ku being the appropriate discount rate for capital cash flows) come from his method of estimating VTS, which is the same as that of Harris and Pringle. The enterprise value ðE þ DÞ according to my proposed method is E þ D ¼ Vu þ PV½Ku ; DKu T ¼ PV½Ku ; FCF þ DKu T:

ð46Þ 1

The difference between this method and Ruback’s is PV½Ku ; DðKu  Kd ÞT: 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 ½D=ðD þ EÞ 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 ¼ PV½Ku ; DTKd ð1 þ Ku Þ=ð1 þ Kd Þ: 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. Damodaran (1994, p. 31) argues that if all the business risk is borne by the equity, then the formula relating the levered beta ðbL Þ with the asset beta ðbu Þ is 1 If we start with the leverage beta of a company and we unlever that beta, Eqs. (44) and (46) provide the same answer only if the debt/equity ratio of the company from which we take the levered beta equals the debt/equity ratio of the company we are valuing.

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bL ¼ bu þ ðD=EÞbu ð1  TÞ: This formula is exactly the formula in Eq. (22) assuming that bd ¼ 0: One interpretation of this assumption is (Damodaran, 1994, p. 31) that ‘‘all of the firm’s risk is borne by the stockholders (i.e., the beta of the debt is zero).’’ In some cases, it can be reasonable to assume that the debt has a zero beta. But then, as assumed by Modigliani and Miller (1963) the required return to debt should be the risk-free rate. This relation for the levered beta appears in many finance books and is widely used by many consultants and investment bankers as an attempt to introduce some leverage cost in the valuation: for a given risk of the assets ðbu Þ; this formula results in a bL that is higher (and consequently Ke is higher and results in a lower equity value) than with Eq. (22). In general, it is hard to accept that the debt has no risk and that the return on the debt is uncorrelated with the return on assets of the firm. From Damodaran’s expression for bL it is easy to deduce the relation between the required return to equity and the required return to assets, i.e., Ke ¼ Ku þ ðD=EÞð1  TÞðKu  RF ). Although Damodaran does not mention what the value of tax shields should be, his formula relating the levered beta with the asset beta implies that the value of tax shields is VTS ¼ PV½Ku ; DTKu  DðKd  RF Þð1  TÞ: Finally, a common way of calculating the levered beta with respect to the asset beta (often used by consultants and investment banks) is the following (see, e.g., Ruback, 1995, p. 5): bL ¼ bu ð1 þ D=EÞ:

ð47Þ

It is obvious that given the same value for bu ; a higher bL (and a higher Ke and a lower equity value) is obtained than according to Eq. (22) or according to Damodaran (1994).2 Formula (47) relating the levered beta with the asset beta implies that the value of tax shields is VTS ¼ PV½Ku ; DTKd  DðKd  RF )]. Given its widespread use in the industry, we will call this method the Practitioners’ method. Given the large number of alternative methods existing in the literature to calculate the value of tax shields, Copeland et al. (2000, p. 482) assert that ‘‘the finance literature does not provide a clear answer about which discount rate for the tax benefit of interest is theoretically correct.’’ They further conclude, ‘‘We leave it to the reader’s judgment to decide which approach best fits his or her situation.’’ We propose two ways to compare and differentiate among the different approaches. One way is to calculate the value of tax shields for perpetuities according to the different approaches. A second way is to check the implied relation between the unlevered and levered cost of equity in each of the different approaches. The levered cost of equity should always be higher than the cost of assets ðKu Þ; since equity cash flows are riskier than the free cash flows. Table 1 summarizes the implications of these approaches for the value of tax shields in perpetuities. Table 2 summarizes the implications for the relation between 2

We interpret Eq. (47) as an attempt to introduce still higher leverage cost in the valuation. For a given risk of the assets ðbu Þ; by using Eq. (47) we obtain a higher bL (and consequently a higher Ke and a lower equity value) than with the approach of Damodaran (1994) or Eq. (22). Damodaran and practitioners impose a cost of leverage, but they do so in an ad hoc way.

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Table 1 Comparison of value of tax shields (VTS) in perpetuities Only three out of the seven approaches compute correctly the value of the tax shield in perpetuities as DT. The other four theories imply a lower value of the tax shield than DT. Theories Correct method Damodaran (1994) Practitioners Harris and Pringle (1985), Ruback (1995) Myers (1974) Miles and Ezzell (1980) Modigliani and Miller (1963)

VTS

VTS in perpetuities

(28) PV½Ku ; DTKu  PV½Ku ; DTKu  DðKd  RF Þð1  TÞ PV½Ku ; DTKd  DðKd  RF Þ PV½Ku ; DTKd  PV½Kd ; DTKd  PV½Ku ; DTKd ð1 þ Ku Þ=ð1 þ Kd Þ PV½RF ; DTRF 

DT oDT oDT oDT DT oDT DT

Note: Ku ¼ unlevered cost of equity; Kd ¼ required return to debt; T ¼ corporate tax rate; D ¼ debt value; RF ¼ risk-free rate; PV½Ku ; DTKu  ¼ present value of DTKu discounted at the rate Ku :

Table 2 Comparison of the relation between Ke (levered cost of equity) and Ku (unlevered cost of equity) for growing perpetuities The approaches of Modigliani and Miller (1963) and Myers (1974) do not always result in a higher cost of equity ðKe Þ than cost of assets ðKu Þ: Myers (1974) obtains Ke lower than Ku if the value of tax shields is higher than the value of debt. This happens when the growth rate ðgÞ is higher than the after-tax cost of debt, i.e., g > Kd ð1  TÞ: Modigliani and Miller (1963) also provide the inconsistent result of Ke being lower than Ku if the value of tax shields is higher than D½Ku  Kd ð1  TÞ=ðKu  gÞ: This happens when leverage, the tax rate, the cost of debt, or the market risk premium are high. Theories Correct method Damodaran (1994) Practitioners Harris and Pringle (1985), Ruback (1995) Myers (1974) Miles and Ezzell (1980) Modigliani and Miller (1963)

Ke (levered cost of equity)

Ke oKu

(21) Ke ¼ Ku þ ðD=EÞð1  TÞðKu  Kd Þ Ke ¼ Ku þ ðD=EÞð1  TÞðKu  RF Þ Ke ¼ Ku þ ðD=EÞðKu  RF Þ Ke ¼ Ku þ ðD=EÞðKu  Kd Þ

No No No No

Ke ¼ Ku þ ðD  VTSÞðKu  Kd Þ=E

Yes, if g > Kd ð1  TÞ No

  D TKd Ke ¼ Ku þ ðKu  Kd Þ 1  E 1 þ Kd   D VTS a Ku  Kd ð1  TÞ  ðKu  gÞ Ke ¼ Ku þ E D

Yes, if VTS > D½Ku  Kd ð1  TÞ=ðKu  gÞ

Note: D ¼ debt value; E ¼ equity value; g ¼ growth rate; Kd ¼ required return to debt; RF ¼ risk-free rate; T ¼ corporate tax rate; VTS ¼ value of tax shields. a Valid only for growing perpetuities.

the cost of assets and the cost of equity in growing perpetuities. Table 1 shows that only three out of the seven approaches compute the value of tax shield in perpetuities as DT. The other four approaches imply a lower value of tax shields than DT.

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Table 2 shows that not all of the approaches also satisfy the relation between the cost of equity and the cost of assets. The Modigliani and Miller (1963) and Myers (1974) approaches do not always give a higher cost of equity than a cost of assets. Myers obtains Ke lower than Ku if the value of the tax shields is higher than the value of debt. This happens when DTKd =ðKd  gÞ > D; that is, when the growth rate is higher than the after-tax cost of debt: g > Kd ð1  TÞ: Please note also that in this situation, as the value of tax shields is higher than the value of debt, the equity ðEÞ is worth more than the unlevered equity ðVu Þ: This hardly makes any economic sense. Modigliani and Miller also provides the inconsistent result of Ke being lower than Ku if the value of the tax shields is higher than D½Ku  Kd ð1  TÞ=ðKu  gÞ: This happens when either the leverage, the tax rate, the cost of debt, or the market risk premium are high. The only approach that provides consistent results for both of these cases is the one proposed above. Appendix A is a compendium of the main valuation formulae according to the different theories.

6. An example of a perpetuity In this section, I provide a numerical example of how to calculate the value of tax shield for a simple firm under the different approaches. Let us start by assuming that the cash flows generated by the company are perpetual and constant (there is no growth). The company must invest to maintain its assets at a level that enables it to ensure constant cash flows: this implies that the book depreciation is equal to the replacement investment. The firm’s balance sheet, income statement, cash flows, and other valuation parameters are shown in Table 3. The breakdown of the value of the unlevered and levered companies between all the stakeholders in the firm is summarized in Fig. 1. The company has three main stakeholders: shareholders, bondholders, and the government. The value of the tax shield is exactly the difference between the present value of the government claim of the levered firm and the present value of the government claim of the unlevered firm. Table 4 reports the estimated value of the equity for the levered firm, the present value of tax shields, the cost of equity, the WACC for the levered and unlevered firm, and the implied value of the levered beta for each of the seven approaches. My proposed method, Modigliani and Miller (1963), and Myers (1974) all compute the value of tax shield to be $200 ðDT ¼ 500 0:4 ¼ 200Þ: All the other methods result in a value of the tax shield that is too low. Now let us assume that this hypothetical firm grows at a constant rate of 5%, i.e., g ¼ 5%: The firm’s balance sheet, income statement, cash flows and other valuation parameters are in Table 5. Note that the balance sheet and the income statement are identical to those in Table 3 (without growth). However, the cash flows vary. Table 6 reports the estimated value of the equity for the levered firm, the present value of tax shields, the cost of equity, the WACC, the WACC before taxes, the implied value of the levered beta, and the debt-to-equity ratio for each of the seven approaches. My proposed method computes the value of the tax shield at $400. The

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Table 3 Valuation of a hypothetical firm without growth: balance sheet, income statement, cash flows and valuations This table presents the valuation of a hypothetical firm without growth using APV (adjusted present value). The table also shows that four alternative valuations (equity cash flows discounted at the required return to equity, free cash flows discounted at the WACC, capital cash flows discounted at the WACC before taxes, and value of tax shields as the difference of two net present values) provide the same valuation. As shown in Fernandez (2002, Chapter 21), the APV provides the same value as the other most commonly used methods for valuing companies by cash flow discounting. Balance sheet Working capital requirements (WCR) Net fixed assets TOTAL ASSETS Debt ðDÞ Capital (book value)

T ¼ corporate tax rate ¼ 40% Kd ¼ cost of debt ¼ 7%

t¼0 0 2000 2000 500 1500

P&L

t¼1

Cash flows

EBITDA

520

Profit after tax (PAT)

171

Depreciation EBIT

200 320

þ Depreciation þ increase of Debt

200 0

35

 increase of WCR

285

 investments in fixed assets ECF (equity cash flow) FCF (free cash flow)

Interest paid ðIÞ ¼ DKd Profit before tax (PBT) Taxes (40%)

114

Profit after tax (PAT)

171

t¼1

0 200

CCF (capital cash flow)

171 192 206

Valuation parameters: g ¼ growth rate ¼ 0%; RF ¼ risk-free rate ¼ 6%; bu ¼ unlevered beta ¼ 1:00 PM ¼ market risk premium ¼ 4%; Ku ¼ unlevered cost of equity ¼ 6% þ 1:00 4% ¼ 10% Valuation. APV (Adjusted present value) Vu ¼ value of unlevered company ¼ FCF=Ku ¼ 192=0:1 ¼ 1920 Using Eq. (16), VTS ¼ DT ¼ 500 0:4 ¼ 200: Using Eq. (1), E þ D ¼ Vu þ VTS ¼ 1920 þ 200 ¼ 2120 Four alternative valuation methods Alternative 1. Equity cash flows discounted at the required return to equity ðKe Þ Using Eq. (21), Ke ¼ Ku þ ðKu  Kd ÞDð1  TÞ=E ¼ 0:1 þ ð0:1  0:07Þ500 0:6=E ¼ 0:1 þ 9=E E ¼ ECF=Ke ¼ 171=ð0:1 þ 9=EÞ: 0:1E þ 9 ¼ 171: E ¼ 1620: Ke ¼ 10:556%: E þ D ¼ 1620 þ 500 ¼ 2120 Alternative 2. Free cash flows discounted at the WACC WACC ¼ ½EKe þ DKd ð1  TÞ=ðE þ DÞ ¼ ½1620 0:10556 þ 500 0:07 ð1  0:4Þ=ð1620 þ 500Þ ¼ 9:057% E þ D ¼ FCF=WACC ¼ 192=0:09057 ¼ 2120 Alternative 3. Capital cash flows discounted at the WACCBT WACCBT ¼ ðEKe þ DKd Þ=ðE þ DÞ ¼ ð1620 0:10556 þ 500 0:07Þ=ð1620 þ 500Þ ¼ 9:717% CCF ðCapital cash flowÞ ¼ ECF þ debt cash flow ¼ 171 þ 35 ¼ 206 E þ D ¼ CCF=WACCBT ¼ 206=0:09717 ¼ 2120 Alternative 4. VTS ¼ Gu  GL ¼ present value of the taxes paid by the unlevered company  present value of the taxes paid by the levered company

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Table 3. (Continued) Using Eq. (9), TaxesU ¼ taxes paid by the unlevered firm ¼ TFCF=ð1  TÞ ¼ 0:4 192=0:6 ¼ 128 Using Eq. (10), KTU ¼ required return to tax in the unlevered company ¼ Ku ¼ 10% Using Eq. (11), Gu ¼ present value of the taxes paid by the unlevered company ¼ TaxesU =KTU ¼ 128=0:1 ¼ 1280 From Eq. (13), KTL ¼ required return to tax in the levered company ¼ Ke ¼ 10:556% Using Eq. (14), GL ¼ present value of the taxes paid by the levered company ¼ TaxesL =KTL ¼ 114=0:10556 ¼ 1080 Using Eq. (4), VTS ðvalue of tax shieldsÞ ¼ Gu  GL ¼ 1280  1080 ¼ 200 Using Eq. (1), E þ D ¼ Vu þ VTS ¼ 1920 þ 200 ¼ 2120

Unlevered company Without taxes With taxes

Levered company with taxes

GL 1,080

Gu 1,280 FCFo/Ku

3,200

VTS

Vu = FCF/Ku 1,920

E + D = Vu + VTS.

GL 1,080

D = 500

E = ECF / Ke 1,620

E+D= = FCF / WACC = CCF / WACC BT 2,120

1,620 + 500 = 1,920 + 200 = 2,120

Fig. 1. Distribution of the company’s total value between shareholders, bondholders, and the government for the example of Table 3. The value of tax shields (VTS) is the difference between two present values (PV) of two cash flows with different risk: the PV of the taxes paid in the unlevered company ðGu Þ and the PV of the taxes paid in the levered company ðGL Þ: VTS ¼ Gu  GL ¼ 1280  1080 ¼ 200: CCF ¼capital cash flow ¼ 206; D ¼value of the debt¼ 500; E ¼ value of equity¼ 1620; ECF ¼equity cash flow ¼ 171; FCF ¼ free cash flow¼ 192; FCF0 ¼free cash flow if there were no taxes ¼ 320; Kd ¼required return to debt¼ 7%; Ke ¼ levered cost of equity ¼ 10:556%; Ku ¼unlevered cost of equity¼ 10%; WACC ¼weighted average cost of capital¼ 9:057%; WACCBT ¼weighted average cost of capital before taxes ¼ 9:717%:

methods of Modigliani and Miller (1963) and Myers (1974) result in a value of the tax shields that is too high (even higher than the debt value) and, correspondingly, a higher value of equity than the correct one. Please recall that these two methods were the only methods that for the case of a constant perpetuity computed the same value of the tax shield. Furthermore, these two methods reach the implausible conclusion that the beta for the levered firm should be lower than the beta for the unlevered firm (see the fifth column of Table 6). All the other methods result in a value of the estimated tax shield in line with their results for the case of constant perpetuities, i.e., their estimate of the value of the tax shield is too low. As we assume that the value of debt is 500 in both the growth example and the no-growth example, the debt/equity

160

Modigliani and Miller Myers Correct formula Damodaran Miles and Ezzell Harris, Pringle and Ruback Practitioners

VTS value of tax shields

E equity value

Ke levered cost of equity (%)

Beta levered

D=E Debt to equity ratio (%)

WACC weighted average cost of capital (%)

WACCBT WACC before taxes (%)

200.00 200.00 200.00 170.00 143.93 140.00 90.00

1620.00 1620.00 1620.00 1590.00 1563.93 1560.00 1510.00

10.56 10.56 10.56 10.75 10.93 10.96 11.32

1.138889 1.138889 1.138889 1.188679 1.233507 1.240385 1.331126

30.86 30.86 30.86 31.45 31.97 32.05 33.11

9.057 9.057 9.057 9.187 9.303 9.320 9.552

9.717 9.717 9.717 9.856 9.981 10.000 10.249

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g ¼ 0% (no growth)

P. Fernandez / Journal of Financial Economics 73 (2004) 145–165

Table 4 Value of tax shields (VTS) for a hypothetical firm with no growth: application of the seven theories to the example in Table 3. Beta unlevered ¼ 1; Ku ¼ 10%; D ¼ 500 Modigliani and Miller (1963), Myers (1974), and I compute the value of the tax shield equal to 200 ðD=T ¼ 500 0:4 ¼ 200Þ; the correct value. All the other methods result in a value of the tax shield that is too low.

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Table 5 Valuation of a hypothetical firm with growth: Balance sheet, income statement, cash flows, and valuation This table presents the valuation of a hypothetical firm with 5% growth using APV (adjusted present value). Three alternative valuations (equity cash flows discounted at the required return to equity, free cash flows discounted at the WACC, and capital cash flows discounted at the WACC before taxes) provide the same valuation. Balance sheet

t¼0 P&L

t ¼ 1 Cash flows

Working capital requirements Net fixed assets TOTAL ASSETS

0 2000 2000

EBITDA Depreciation EBIT

520 200 320

Debt ðDÞ Capital (book value)

500 1500

Interest payments 35 Taxes (40%) 114 Net income 171

T ¼ corporate tax rate ¼ 40% Kd ¼ cost of debt ¼ 7%

t¼1

Net income þ Depreciation þ increase of Debt

171 200 25

 increase of WCR 0  investments in fixed assets 300 ECF (equity cash flow) 96 FCF (free cash flow) CFd (debt cash flow) CCF (capital cash flow)

92 10 106

Valuation parameters g ¼ growth rate ¼ 5%; RF ¼ risk-free rate ¼ 6%; bu ¼ unlevered beta ¼ 1:00; PM ¼ market risk premium ¼ 4%; Ku ¼ unlevered cost of equity ¼ 6% þ 1:00 4% ¼ 10% Valuation 1. APV (Adjusted present value) Using Eq. (29) Vu ¼ value of unlevered company ¼ FCF=ðKu  gÞ ¼ 92=ð0:1  0:05Þ ¼ 1840 Using Eq. (27) VTS ¼ value of tax shields ¼ DTKu =ðKu  gÞ ¼ 500 0:4 0:1=ð0:1  0:05Þ ¼ 400 Using Eq. (1) E þ D ¼ Vu þ VTS ¼ 1840 þ 400 ¼ 2240 Three alternative valuation methods Alternative 1. Equity cash flows discounted at the required return to equity ðKe Þ Using Eq. (21), Ke ¼ Ku þ ðKu  Kd ÞDð1  TÞ=E ¼ 0:1 þ ð0:1  0:07Þ500 0:6=E ¼ 0:1 þ 9=E E ¼ ECF=ðKe  gÞ ¼ 96=ð0:1 þ 9=E  0:05Þ: 0:05E þ 9 ¼ 96: E ¼ 1740: Ke ¼ 10:517% Alternative 2. Free cash flows discounted at the WACC WACC ¼ ½EKe þ DKd ð1  TÞ=ðE þ DÞ ¼ ½1740 0:10517 þ 500 0:07 ð1  0:4Þ=ð1740 þ 500Þ ¼ 9:107% E þ D ¼ FCF=ðWACC  gÞ ¼ 92=ð0:09107  0:05Þ ¼ 2240 Alternative 3. Capital cash flows discounted at the WACCBT WACCBT ¼ ðEKe þ DKd Þ=ðE þ DÞ ¼ ð1740 0:10517 þ 500 0:07Þ=ð1740 þ 500Þ ¼ 9:732% CCF ðCapital cash flowÞ ¼ ECF þ debt cash flow ¼ 96 þ 10 ¼ 106 E þ D ¼ CCF=ðWACCBT  gÞ ¼ 106=ð0:09732  0:05Þ ¼ 2240

ratio is lower with growth because equity is higher. For this reason, the WACC of the growing firm (Table 6) is higher than the WACC of the firm with no growth (Table 4). One could say, ‘‘In practice I do not see why the approach of working out the present value of tax shields themselves would necessarily be wrong, provided

162

VTS value of tax shields

Modigliani and Miller Myers Correct formula Damodaran Miles-Ezzell Harris, Pringle, and Ruback Practitioners

1200.00 700.00 400.00 340.00 287.85 280.00 180.00

E equity value 2540.00 2040.00 1740.00 1680.00 1627.85 1620.00 1520.00

Ke levered cost of equity (%) 8.78 9.71 10.52 10.71 10.90 10.93 11.32

Beta levered

D=E Debt to equity ratio (%)

WACC weighted average cost of capital (%)

WACCBT WACC before taxes (%)

0.694882 0.926471 1.129310 1.178571 1.224337 1.231481 1.328947

19.69 24.51 28.74 29.76 30.72 30.86 32.89

8.026 8.622 9.107 9.220 9.324 9.340 9.554

8.487 9.173 9.732 9.862 9.982 10.000 10.248

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g ¼ 5%

P. Fernandez / Journal of Financial Economics 73 (2004) 145–165

Table 6 Value of tax shields (VTS) for a hypothetical firm with constant growth: application of the seven theories to the example of Table 5. Beta unlevered ¼ 1; Ku ¼ 10%; g ¼ 5% My method computes the value of the tax shield equal to $400. The methods of Modigliani and Miller (1963) and Myers (1974) result in a value of the tax shield that is too high (even higher than the debt value). All the other methods result in a value of the estimated tax shield that is too low.

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163

the appropriate discount rate were used (reflecting the riskiness of the tax shields).’’ The problem is that it is hard to evaluate the riskiness of tax shield because it is the difference between two present flows (the taxes paid by the unlevered company and those paid by the levered company) with different risk. On the other hand, if we were to follow this approach, we could calculate the discount rate K  that accomplishes VTS ¼ DTKu =ðKu  gÞ ¼ DTKd =ðK   gÞ:

ð48Þ

The solution is K  ¼ Kd þ g  Kd g=Ku : Unfortunately, this expression does not lend itself to intuition.

7. Conclusions This paper shows that the increase in a company’s value due to the use of debt is not the present value of tax shields due to interest payments. It is the difference between the present value of the taxes of the unlevered company and the present value of the taxes of the levered company, which represent two separate cash flows each with their own risk. The issue of the riskiness of the taxes for both the unlevered and the levered company is addressed. In perpetuities, the required return to tax in the unlevered company is equal to the required return to equity in the unlevered company, and the required return to tax in the levered company is equal to the required return to equity. The value of tax shields is the present value of the debt times the tax rate times the required return to the unlevered equity, all discounted at the unlevered cost of equity. For a constant perpetuity, this results in a value of the tax shield equal to the value of debt times the tax rate. For the case of growing perpetuities, this number is lower than that proposed by either Myers (1974) or Modigliani and Miller (1963). The paper also shows that some of the alternative approaches suggested in the literature lead to inconsistent results. In particular, discounting tax shields at the cost of debt (Myers, 1974; Brealey and Myers, 2000; and many others) or at the risk-free rate (Modigliani and Miller, 1963) can result in estimating a lower cost of equity to the levered firm than that of the unlevered firm for growing perpetuities. This seems clearly unreasonable. On the other hand, to discount tax shields at the required return to unlevered equity, as suggested by Harris and Pringle (1985), Miles and Ezzell (1980), and Ruback (2002), results in an estimated value of tax shields that is too low.

Appendix A Table A.1 shows the main valuation theories according to the different theories.

164

Theories Correct

VTS

Ke

PV½Ku ; DTKu  DðKd  RF Þð1  TÞ

Ke ¼ Ku þ ðD=EÞð1  TÞðKu  RF Þ

PV½Ku ; TDKd  DðKd  RF Þ

Ke ¼ Ku þ ðD=EÞðKu  RF Þ

D bL ¼ bu þ bu (47) E

RF  Kd ð1  TÞ Ku  D ðE þ DÞ

Harris and

PV½Ku ; TDKd 

Ke ¼ Ku þ ðD=EÞðKu  Kd Þ

D bL ¼ bu þ ðbu  bd Þ (45) E

Kd T Ku  D ðE þ DÞ

Pringle

bL ¼ bu þ

bL ¼ bu þ

Dð1  TÞ bu E

WACCBT

Ku þ

DðKd  RF Þ ðE þ DÞ Ku

(1985), Ruback (1995, 2002) D  DVTS bL ¼ bu þ ðbu  bd Þ E     Miles and PV½Ku ; TDKd ð1 þ Ku Þ=ð1 þ Kd Þ D TKd D TKd Ke ¼ Ku þ ðKu  Kd Þ 1  bL ¼ bu þ ðbu  bd Þ 1  E E 1 þ K 1 þ K d d Ezzell (1980)     Modigliani and PV½RF ; DTRF  D VTS a D TKd VTSðKu  gÞ a Ke ¼ Ku þ bL ¼ bu þ  Ku  Kd ð1  TÞ  ðKu  gÞ bu  bd þ E D E DP P M M Miller (1963) Myers (1974)

a

PV½Kd ; TDKd 

Valid only for growing perpetuities.

Ke ¼ Ku þ ðD  VTSÞðKu  Kd Þ=E

Ku 

DVTSðKu  Kd Þ þ DKd T ðE þ DÞ

Kd T 1 þ Ku Ku  D ðE þ DÞ 1 þ Kd DKu  ðKu  gÞVTSa ðE þ DÞ

Ku  Ku 

DVTSðKu  Kd Þ ðE þ DÞ

DTKd ðKu  Kd Þ ðE þ DÞ ð1 þ Kd Þ

DKu  ðKu  gÞVTS þ DTKd a ðE þ DÞ

ARTICLE IN PRESS

Ke ¼ Ku þ ðD=EÞð1  TÞðKu  Kd Þ (21)

(1994) Practitioners

Dð1  TÞ ðbu  bd Þ (22) E

WACC

  DTðKu  Kd Þ DT Ku  Ku 1  ðE þ DÞ EþD   TðKu  RF Þ  ðKd þ RF Þ DT ðKd  RF Þð1  TÞ Ku  D þD Ku 1  ðE þ DÞ EþD ðE þ DÞ

PV½Ku ; DTKu  (28)

method Damodaran

bL

P. Fernandez / Journal of Financial Economics 73 (2004) 145–165

Table A.1 Main valuation formulas. Market value of the debt ¼ Nominal value

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