ABSTRACT
We show that the value of tax shields is the difference between the present values of two different cash flows with their own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company.
For perpetuities without costs of leverage, the value of tax shields is equal to the tax rate times the value of debt.
For any company, we claim that the value of the tax shield in a world with no leverage cost is the present value of the debt (D) times the tax rate (T) times the required return to the unlevered equity (Ku), discounted at the unlevered cost of equity (Ku): VTS = PV[Ku; D T Ku]. Please note that it does not mean that the appropriate discount for the tax shields is the unlevered cost of equity. We discount D T Ku, which is not the tax shield. This expression arises as the difference of two present values each with different risk.
JEL Classification: G12, G31, M21
1. Introduction
Following a new method, we prove that the value of tax shields for perpetuities in a world without cost of leverage is equal to the tax rate times the value of debt (DT). We reach this result without doing any assumption about the riskiness of the tax shields. This result is the same as in ModiglianiMiller (1963), but the reasoning behind it and the implications involved are quite different indeed. The increase in the company’s value due to the use of debt is not the present value of the tax shield 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 are the present values of 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 also addressed. We prove that, in perpetuities, the required return to taxes in the unlevered company is equal to the required return to equity in the unlevered company. It is also proven that the required return to taxes in the levered company is equal to the required return to equity.
We first show that the value of tax shields is the difference between the present values of two different cash flows with their own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company. This implies as a first guideline that, for the particular case of a perpetuity and a world without costs of leverage, the value of tax shields is equal to the tax rate times the value of debt.
In a world without leverage cost, the total value of the levered company is equal to the total value of the unlevered company. Total value is the enterprise value (often called value of the firm) plus the present value of taxes. When leverage cost do exist, the total value of the levered company is lower than the total value of the unlevered company.
The value of tax shields for perpetuities should be lower than the tax rate times the value of debt when costs of leverage do exist. A second guideline for the appropriateness of the valuation method should be that the value of tax shields without taxes and in the presence of leverage cost is negative.
Secondly, we analyze eight valuation theories to estimate the present value of tax shields and show their performance relative to the proposed guidelines. We are able to see there is but one theory that provides consistent results in a world without leverage cost. For constant growth companies, we claim that the value of the tax shield in a world with no leverage cost is the present value of the debt (D) times the tax rate (T) times the required return to the unlevered equity (Ku), discounted at the unlevered cost of equity (Ku):
VTS = PV[Ku; D T Ku] = D T Ku / (Kug)
Please note that it does not mean that the appropriate discount for the tax shields is the unlevered cost of equity. We discount D T Ku, which is higher than the shield. This expression arises as the difference of two present values each with different risk.
One referee said, “in practice I do not see why the approach of working out the present value of the tax shields themselves would necessarily be wrong, provided the appropriate discount rate was used (reflecting the riskiness of the tax shields)”. But it is hard to reflect the riskiness of the tax shield (D Kd T, being Kd the cost of debt) because it is the difference of two cash 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, and calculate the discount rate K* that accomplish with:
VTS = PV[Ku; D T Ku] = PV[K*; D T Kd]
We could find that for growing perpetuities K* = Kd + g – Kd g / Ku. Obviously this expression has not any intuition behind. And it is impossible to find a closed form solution for K* in a general case.
Although Copeland et al. (2000) claim that “the finance literature does not provide a clear answer about which discount rate for the tax benefit of interest is theoretically correct,” we show that we can provide some clear answers on that topic.
The rest of the paper is organized as follows.
In Section 2 we follow a new method to prove that the value of tax shields for perpetuities in a world without cost of leverage is equal to the tax rate times the value of debt (DT).
In Section 3 we derive the relationship between the required return to assets (Ku) and the required return to equity (Ke) for perpetuities in a world without leverage costs. The corresponding relationship between the beta of the levered equity ( L), the beta of the unlevered equity ( u) and the beta of debt ( d) is as well derived.
In Section 4 we revise the financial literature about the value of tax shields. We find eight different theories that value the tax shields in different ways.
Conclusions are in Section 5.
Appendix 1 contains the dictionary of the initials used in the paper.
2. The value of tax shields for perpetuities in a world without leverage cost is DT.
It is assumed that the debt’s market value (D) is equal to its book value[1]. First, we show that in a world without leverage cost, the value of tax shields for a perpetuity is DT. The formula for the adjusted present value [1] indicates that the value of the debt today (D) plus that of the equity (E) of the levered company is equal to the value of the equity of the unlevered company (Vu) plus the value of tax shields due to interest payments (VTS)[2].
[1] E + D = Vu + VTS
VTS is the term used to define the increase in the company’s value as a result of the tax savings obtained by the payment of interest (value of tax shields). As we deal for perpetuities, Vu = FCF/Ku, where FCF is the free cash flow of the company.
In a world without leverage cost, the total value of the levered company is equal to the total value of the unlevered company. A world without leverage cost is characterized by the following relationship:
[2] Vu + Gu = E + D + GL
Please note that equation [2] assumes that free cash flows are independent of leverage.
Vu is the value of the unlevered company. Gu is the present value of the taxes paid by the unlevered company. E is the equity value and D is the debt value. GL is the present value of the taxes paid by the levered company. Equation [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 value of the firm) plus the present value of taxes.[3]
From [1] and [2], it is clear that the VTS (value of tax shields) is:
[4] VTS = Gu – GL
We should note that the value of tax shields (VTS) is not (and it will be seen that this is the main error of many papers on this topic) the PV of the tax shield. It is the difference between two PVs of two flows with different risk: the PV of the taxes paid in the unlevered company (Gu) and the PV of the taxes paid in the levered company (GL).[4]
It is easy to prove that, in a perpetuity, the profit after tax of the levered company (PATL) is equal to the equity cash flow (ECF):
[5] PATL = ECF
This happens in perpetuities, as the allowed depreciation deduction is exactly equal to cash actually spent to replace capital equipment that wears out.
In a perpetuity, the free cash flow (FCF) is equal to the profit before tax of the unlevered company (PBTu) multiplied by (1T), with T being the tax rate.
[6] FCF = PBTu (1 T)

We will call FCF the company’s free cash flow if there were no taxes. The FCF is equal to the

profit before taxes of the unlevered company (PBTu).

[7] FCF = PBTu.
From [6] and [7] it is clear that the relationship between the free cash flow and the company’s free cash flow if there were no taxes is:

[8] FCF = FCF (1 T)
The taxes paid every year by the unlevered company (TaxesU) are:
[9] TaxesU = T PBTu = T FCF0 = T FCF / (1T)
We designate KTU as the required return to taxes in the unlevered company. KTL is the required return to taxes in the levered company.

The taxes paid by the unlevered company are proportional to FCFand FCF. Consequently, the

taxes of the unlevered company have the same risk as FCF (and FCF), and hence must be discounted at the rate Ku. In the unlevered company, the required return to taxes (KTU) is equal to the required return to equity (Ku). This is only true for perpetuities.
[10] KTU = Ku
The present value of the taxes paid by the unlevered company (GU) is the present value of the taxes paid every year (TaxesU) discounted at the appropriate discount rate (Ku):
[11] GU = TaxesU / KTU = T FCF / [(1T) Ku] = T Vu / (1T)
For the levered company, taking into consideration equation [5], the taxes paid every year (TaxesL) are proportional to the equity cash flow (ECF):
[12] TaxesL = T PBTL = T PATL / (1T)= T ECF / (1T)
PBTL and PATL are the profit before and after tax of the levered company.
The taxes paid by the levered company are proportional to ECF. Consequently, the taxes of the levered company 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 consequently the required return to taxes in the levered company (KTL) is equal to the required return to equity (Ke). This is only true for perpetuities.
[13] KTL = Ke
Then, the present value of the taxes of the levered company, that is, the value of the taxes paid to the
Government is equal to[5]:
[14] GL = TaxesL / KTL = T ECF / [(1T) Ke] = T E / (1T)
The increase in the company’s value due to the use of debt is not the present value of the tax shield due to interest payments, but the difference between GU and GL, which are the present values of two cash flows with different risks:
[15] VTS = GU – GL = [T / (1T)] (Vu – E)
As Vu – E = D – VTS, this gives:
[16] VTS = Value of tax shields = DT
Note that equation [16] is the difference between the present value of the taxes of the unlevered company (PV[Ku; TaxesU]) and the present value of the taxes of the levered company (PV[Ke; TaxesL]).
[17] VTS = PV[Ku; TaxesU] – PV[Ke; TaxesL]
D Kd T are the tax savings in every year. It is obvious that:
[18] TaxesU – TaxesL = D Kd T
One problem of equation [16] is that (for perpetuities) it can be understood as DT = D T/ . At first glance, can be anything, related or unrelated to the company that we are valuing. In Section 3 it will be seen that Modigliani and Miller (1963) assume that is riskfree 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 T Kd) discounted at the cost of debt (Kd). But it has been shown that the value of tax shields is 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.
3. Relationship between the required return to assets (Ku) and the required return to equity
(Ke) in a world without leverage costs
Knowing the value of the tax shield (DT), and considering that Vu = FCF/Ku, we may rewrite equation [1] as [19]
[19] E + D (1T) = FCF/Ku
[20]Taking into consideration that the relationship between ECF and FCF for perpetuities is equation
[20] FCF = ECF + D Kd (1T) and that E = ECF/Ke, we may find the relationship between the required return to assets (Ku) and the required return to equity (Ke) in a world without leverage costs as:
[21] Ke = Ku + (KuKd) D (1 – T) /E
And the relationship between the beta of the levered equity ( L), the beta of the unlevered equity ( u) and the beta of debt ( d) is:
[22] L = u + ( u – d) D (1 – T) /E
4. Literature review: 8 theories
There is a considerable body of literature on the discounted cash flow valuation of firms. We will now discuss the most salient papers, concentrating particularly on those that proposed different expressions for the present value of the tax savings due to the payment of interest or value of tax shields (VTS). The main problem of most papers is that they consider the value of tax shields (VTS) as the present value of the tax savings due to the payment of interest. We have already argued that the value of tax shields (VTS) is the difference of two present values: the present value of taxes paid by the unlevered firm minus the present value of taxes paid by the levered firm.
Modigliani and Miller (1958) studied the effect of leverage on the firm’s value. Their proposition 1 (1958, formula 3) states that, in the absence of taxes, the firm’s value is independent of its debt, i.e.,
[23] E + D = Vu, if T = 0.
E is the equity value, D is the debt value, Vu is the value of the unlevered company and T is the tax rate. In the presence of taxes and for the case of a perpetuity, they calculate the value of tax shields (VTS) by discounting the present value of the tax savings due to interest payments of a riskfree debt (T D RF) at the riskfree rate (RF). Their first proposition, with taxes, is transformed into Modigliani and Miller (1963, page 436, formula 3):
[24] E + D = Vu + PV[RF; DT RF] = Vu + D T
DT is the value of tax shields (VTS) for a perpetuity. This result is the same as our equation [16]. But this result is only correct for perpetuities. Discounting the tax savings due to interest payments of a hypothetical riskfree debt at the riskfree rate provides inconsistent results for growing companies.
Myers (1974) introduced the APV (adjusted present value). According to Myers, the value of the levered firm is equal to the value of the firm with no debt (Vu) plus the present value of the tax savings due to the payment of interest (VTS). Myers proposes calculating the VTS by discounting the tax savings (D T Kd) 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. Then, according to Myers (1974):
[25] VTS = PV [Kd; D T Kd]
Luehrman (1997) also recommends to value companies using the Adjusted Present Value and calculates the VTS as Myers. This theory accomplish formula [16], which implies VTS = DT for perpetuities. But it provides inconsistent results for companies other than perpetuities. For example, for growing perpetuities, when g > Kd (1T), being g the growth rate, the required return to equity (Ke) is smaller than the required return to assets (Ku). Obviously, an acceptable theory should provide us a required return to equity (Ke) higher than the required return to assets (Ku).
Miller (1977) assumes no advantages of debt financing: “I argue that even in a world in which interest payments are fully deductible in computing corporate income taxes, the value of the firm, in equilibrium will still be independent of its capital structure.” According to Miller (1977) the value of the firm is independent of its capital structure, that is,
[26] VTS = 0.
According to Miles and Ezzell (1980), a firm that wishes to keep a constant D/E ratio must be valued in a different manner from the firm that has a preset level of debt. For a firm with a fixed debt target [D/(D+E)] they claim that the correct rate for discounting the tax savings due to debt (Kd T Dt1) is Kd for the tax savings during the first year, and Ku for the tax savings during the following years.
The expression of Ke is their formula 22:
[27] Ke = Ku + D (Ku – Kd) [1 + Kd (1T)] / [(1+Kd) E]
Although Miles and Ezzell do not mention what the value of tax shields should be, formula [27] relating the required return to equity with the required return for the unlevered company implies that
[28] VTS = PV[Ku; T D Kd] (1+Ku)/(1+Kd).
Lewellen and Emery (1986) also claim that the most logically consistent method is Miles and Ezzell. The problem with this theory is that it does not accomplish formula [16], which implies VTS = DT for perpetuities. It provides a VTS lower than DT. The difference could be attributed to the leverage cost. But if this is the case, leverage cost also exist when there are no taxes. In this situation this theory should provide a negative VTS and it is not the case.
Harris and Pringle (1985) propose that the present value of the tax savings due to the payment of interest (VTS) should be calculated by discounting the tax savings due to the debt (Kd T D) at the rate 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).
Then, according to Harris and Pringle (1985):
[29] VTS = PV [Ku; D Kd T]
Harris and Pringle (1985, page 242) say “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. The Miller position is too extreme for some because it implies that debt cannot benefit the firm at all. Thus, if the truth about the value of tax shields lies somewhere between the MM and Miller positions, a supporter of either Harris and Pringle or Miles and Ezzell can take comfort in the fact that both produce a result for unlevered returns between those of MM and Miller. A virtue of either Harris and Pringle compared to Miles and Ezzell is its simplicity and straightforward intuitive explanation.” Ruback (1995) reaches formulas that are identical to those of HarrisPringle (1985). Kaplan and Ruback (1995) also calculate the VTS “discounting interest tax shields at the discount rate for an allequity firm”. Tham and Vélez Pareja (2001), following an arbitrage argument, also claim that the appropriate discount rate for the tax shield is Ku, the required return to unlevered equity.
The problem with this theory is that it does not accomplish formula [16], which implies VTS = DT for perpetuities. It provides a VTS = D T Kd/Ku, lower than DT. The difference could be attributed to the leverage cost. But if this is the case, leverage cost also exist when there are no taxes. In this situation this theory should provide a negative VTS and it is not the case.
Damodaran (1994, page 31) argues that if all the business risk is borne by the equity, then the formula relating the levered beta[6] ( L) with the asset beta ( u) is:
[30] L = u + (D/E) u (1 – T).
It is important to note that formula [30] is exactly formula [22] assuming that d = 0. Although one interpretation of this assumption is that “all of the firm’s risk is borne by the stockholders (i.e., the beta of the debt is zero)”[7], we think it is difficult to justify that the debt has no risk and that the return on the debt is uncorrelated with the return on assets of the firm. We rather interpret formula [30] as an attempt to introduce some leverage cost in the valuation: for a given risk of the assets ( u), by using formula [30] we obtain a higher L (and consequently a higher Ke and a lower equity value) than with formula [22]. Equation [30] appears in many finance books and is often used by consultants and investment banks. Although Damodaran does not mention what the value of tax shields should be, his formula [30] relating the levered beta with the asset beta implies that the value of tax shields is:
[31] VTS = PV[Ku; D T Ku – D (Kd RF) (1T)]
This theory provides a VTS lower than DT for perpetuities. This theory also provides a negative VTS when there are no taxes. Comparing [31] to [24] for a perpetuity, it can be seen that [31] provides a VTS that is D(Kd RF)(1T)/Ku lower than [24]. We interpret this difference as leverage cost introduced in the valuation.
Another way of calculating the levered beta with respect to the asset beta is the following:
[32] L = u (1+ D/E).
We will call this method the Practitioners’ method, because consultants and investment banks often use it[8]. It is obvious that according to this formula, given the same value for u, a higher L (and a higher Ke and a lower equity value) is obtained than according to [22] and [30].
One should notice that formula [32] is equal to formula [30] eliminating the (1T) term. We interpret formula [32] as an attempt to introduce still higher leverage cost in the valuation: for a given risk of the assets ( u), by using formula [32] we obtain a higher L (and consequently a higher Ke and a lower equity value) than with formula [30].
Formula [30] relating the levered beta with the asset beta implies that the value of tax shields is:
[33] VTS = PV[Ku; T D Kd – D(Kd RF)]
This theory provides a VTS lower than DT for perpetuities. This theory also provides a negative VTS when there are no taxes. By comparing [33] to [31] for a perpetuity, it can be seen that [33] provides a VTS that is DT(Ku RF)/Ku lower than [31]. We interpret this difference as additional leverage cost introduced in the valuation.
Inselbag and Kaufold (1997) argue that if the firm targets the dollar values of debt outstanding, the VTS is given by the Myers (1974) formula. They also argue that if the firm targets a constant debt/value ratio, the Miles and Ezzell (1980) formula give the VTS.
Copeland, Koller y Murrin (2000) treat the Adjusted Present Value in their Appendix A. They only mention perpetuities and only propose two ways of calculating the VTS: Harris y Pringle (1985) and Myers (1974). They conclude “ we leave it to the reader’s judgment to decide which approach best fits his or her situation”. They also claim that “the finance literature does not provide a clear answer about which discount rate for the tax benefit of interest is theoretically correct.” It is quite interesting to note that Copeland et al. (2000, page 483) only suggest Inselbag and Kaufold (1997) as additional reading on Adjusted Present Value.
We will consider an additional theory to calculate the value of the tax shields. We label this theory the NoCostsOfLeverage formula because it is the only formula that provides consistent results.
According to this theory, the VTS is the present value of D T Ku (not the interest tax shield) discounted at the unlevered cost of equity (Ku).
[34] PV[Ku; D T Ku]
Equation [34] is the result of applying equations [21] and [22] to the general case. For simplicity, we apply it to the constant growth case. In the constant growth case, we designate the growth rate by g. The relationship between the equity cash flow and the free cash flow is
[35] FCF = ECF + D Kd (1T) –g D
The relationship between the value of the equity of the unlevered firm and the free cash flow is
[36] Vu = FCF / (Ku – g)
By substituting [35] and [36] in [1], we get:
[37] E + D = [ECF + D Kd (1T) –g D ]/ (Ku – g) + VTS
The relationship between the equity cash flow and the equity value is
[38] ECF = E (Ke – g)
By substituting [38] in [37], and assuming that the relationship between the levered and the unlevered required return to equity is [21], we get:
[39] E + D = [E Ku + (KuKd) D (1 – T) – Eg + D Kd (1T) –g D ]/ (Ku – g) + VTS
Some algebra in [39] leads us to [34] applied to constant growth companies: VTS = D T Ku / (Kug)
This theory accomplish formula [16], which implies VTS = DT
Table 1 is the synthesis of the 8 theories mentioned about the value of tax shields applied to level perpetuities.
Table 1. Perpetuities. Value of tax shields (VTS) according to the eight theories.
Theories  Formula  VTS 
Modigliani&Miller  [24]  DT 
Myers  [25]  DT 
Miller (1977)  [26]  0 
MilesEzzell  [28]  TDKd(1+Ku)/[(1+Kd)Ku] 
HarrisPringle  [29]  T D Kd/Ku 
Damodaran  [31]  DT[D(KdRF)(1T)]/Ku 
Practitioners  [33]  D[RFKd(1T)]/Ku 
NoCostsOfLeverage formula  [34]  DT 
5. Conclusions
We claim that the value of the tax shield (VTS) in a world with no leverage cost is the present value of the debt (D) times the tax rate (T) times the required return to the unlevered equity (Ku), discounted at the unlevered cost of equity (Ku):
VTS = PV[Ku; D T Ku]
We also prove that:
To discount the tax shield at the required return to the unlevered equity, as suggested by Harris and Pringle (1985), provides inconsistent results.
To discount the tax shield at the cost of debt, as suggested by Myers (1974), Brealey and Myers (2000), and many others, provides inconsistent results.
To discount the tax shield at the cost of debt the first year and at the required return to the unlevered equity the following years, as suggested by Miles and Ezzell (1980), provides inconsistent results.
The increase in the company’s value due to the use of debt is not the present value of the tax shield 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 are the present values of 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.
We also find two theories that could provide consistent results in a world with leverage cost.
APPENDIX 1
Dictionary
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[*] I would like to thank my colleagues José Manuel Campa, Javier Estrada and Josep Faus for very helpful comments, and Teresa Modroño, Charlie Porter, Gabriel Rabassa and Laura Reinoso for their wonderful help revising previous manuscripts of this paper.
[1] This means that the required return to debt (Kd) is the same as the interest rate paid by the debt.
[2] As shown in Fernandez (2001), The APV provides always the same value as the other most commonly used methods for valuing companies by cash flow discounting: free cash flow discounted at the WACC; equity cash flows discounted at the required return to equity…
[3] When leverage cost 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 relationship:
[3] Vu + Gu > E + D + GL = E + D + GL + Leverage Cost Leverage Cost is the reduction in the company´s value due to the use of debt. This reduction in value has two main components: the increase in risk of the levered company (the levered company has a higher probability of default than the unlevered company), and the reduction of expected flows generated by the levered company (the levered company will have more difficulties to undertake new projects than the unlevered company)
[4] In the presence of leveraged cost, the value of tax shields is: VTS = Gu – GL – Leverage Cost
[5] The relationship between profit after tax (PAT) and profit before tax (PBT), is: PAT=PBT(1T).
[6] There is a link between the VTS and the relationship between the levered beta with the asset beta.
[7] See page 31 of Damodaran (1994)
[8] One of the many places where it appears is Ruback (1995), p. 5.
Previously published by the IESE Business School