Working Paper. WP No 613 October, 2005 THE VALUE OF TAX SHIELDS DEPENDS ONLY ON THE NET INCREASES OF DEBT

Transcription

1 CII Working Paper WP No 63 October, 5 THE VALUE O TAX SHIELDS DEPENDS ONLY ON THE NET INCREASES O DEBT The value of tax shields, the risk of the increases of debt and the risk of the increases of assets Pablo ernández* * Professor of inancial Management, PricewaterhouseCoopers Chair of inance, IESE IESE Business School University of Navarra Avda. Pearson, 834 Barcelona, Spain. Tel.: (34) ax: (34) Camino del Cerro del Aguila, 3 (Ctra. de Castilla, km. 5,8) 83 Madrid, Spain. Tel.: (34) ax: (34) Copyright 5 IESE Business School.

2 The CII, International Center for inancial Research, is an interdisciplinary center with an international outlook and a focus on teaching and research in finance. It was created at the beginning of 99 to channel the financial research interests of a multidisciplinary group of professors at IESE Business School and has established itself as a nucleus of study within the School s activities. Ten years on, our chief objectives remain the same: ind answers to the questions that confront the owners and managers of finance companies and the financial directors of all kinds of companies in the performance of their duties Develop new tools for financial management Study in depth the changes that occur in the market and their effects on the financial dimension of business activity All of these activities are programmed and carried out with the support of our sponsoring companies. Apart from providing vital financial assistance, our sponsors also help to define the Center s research projects, ensuring their practical relevance. The companies in question, to which we reiterate our thanks, are: Aena, A.T. Kearney, Caja Madrid, undación Ramón Areces, Grupo Endesa, Telefónica and Unión enosa.

3 THE VALUE O TAX SHIELDS DEPENDS ONLY ON THE NET INCREASES O DEBT The value of tax shields, the risk of the increases of debt and the risk of the increases of assets Abstract The value of tax shields depends only on the nature of the stochastic process of the net increases of debt. The value of tax shields in a world with no leverage cost is the tax rate times the current debt plus the present value of the net increases of debt. By applying this formula to specific situations, we show that Modigliani-Miller (963) should be used when the company has a preset amount of debt; ernández (4), when the company maintains a fixed book-value leverage ratio; and Miles-Ezzell (98), when the company maintains a fixed market-value leverage ratio. JEL classification: G; G3; G3 Keywords: Value of tax shields, present value of the net increases of debt, required return to equity

4 THE VALUE O TAX SHIELDS WITH A IXED BOOK-VALUE LEVERAGE RATIO THE VALUE O TAX SHIELDS, THE RISK O THE INCREASES O DEBT AND THE RISK O THE INCREASES O ASSETS 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 Modigliani-Miller (963) propose to discount the tax savings at the risk-free rate, whereas Harris and Pringle (985) propose discounting these tax savings at the cost of capital for the unlevered firm. Miles and Ezzel (985) propose discounting these tax savings the first year at the cost of debt and the following years at the cost of capital for the unlevered firm. Reflecting this lack of consensus, Copeland et al. (, p. 48) 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 the value of tax shields depends only upon the nature of the stochastic process of the net increase of debt. More specifically, we prove that the value of tax shields in a world with no leverage cost is the tax rate times the current debt, plus the tax rate times the present value of the net increases of debt. By applying this formula to specific situations, we show that the Modigliani-Miller (963) formula should be used when the company has a preset amount of debt; ernández (4), when the company expects the increases of debt to be as risky as the free cash flows (for example, if the company wants to maintain a fixed book-value leverage ratio); and Miles-Ezzell (98), only if debt will be always a multiple of the equity market value D t L E t. We will argue that although D t L E t provides a computationally elegant solution, it is not a realistic one. What is more, we have not seen any company that follows this financing policy. It makes much more sense to characterize the debt policy of a company with expected constant leverage ratio as a fixed book-value leverage ratio rather than as a fixed marketvalue leverage ratio because. the debt does not depend on the movements of the stock market,. it is easier to follow for unlisted companies, and 3. managers should prefer it because the value of tax shields is higher. Myers (974) propose to discount it at the cost of debt (Kd).

5 Although Cooper and Nyborg (6) disagree, this paper shows that ernández s (4) formula (8) (VTS PV[Ku; D T Ku]) is valid, but only under the assumption that the increases of debt are as risky as the free cash flows. The increases of debt are as risky as the free cash flows if the company maintains a fixed book-value leverage ratio.. General expression of the value of tax shields 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 the value of tax shields due to interest payments (VTS): E D Vu VTS. () 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. If leverage costs do not exist, then Eq. () could be stated as follows: Vu Gu E D GL () 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. () 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. Eq. () assumes that expected free cash flows are independent of leverage. rom () and (), it is clear that VTS is VTS Gu GL (4) Note that the value of tax shields 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 relationship between the profit after tax of the levered company (PAT L ) and the equity cash flow (EC) is: EC t PAT Lt - A t D t (5) Notation being, A t Increase of net assets in period t (Increase of Working Capital Requirements plus Increase of Net ixed Assets); D t D t D t- Increase of Debt in period t. 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 Gu E D GL Leverage Cost > E D GL (3) Leverage cost is the reduction in the company s value due to the use of debt.

6 3 Similarly, the relationship between the profit after tax of the unlevered company (PATu) and the free cash flow (C) is: C t PATu t A t (6) The taxes paid every year by the unlevered company (Taxes U ) are Taxes Ut [T/( T)] PATu [T/( T)] (C t A t ) (7) or the levered company, taking into consideration Eq. (5), the taxes paid each year (Taxes L ) are: Taxes Lt [T/( T)] (EC t A t D t ) (8) are: PV [ ] is the present value operator. The present values at t of equations (7) and (8) Gu [T/( T)] (Vu PV [ A t ]) (9) G L [T/( T)] (E PV [ A t ] PV [ D t ]) () The increase in the company s value due to the use of debt is the difference between Gu (9) and G L (), which are the present values of two cash flows with different risks: VTS Gu G L [T/(-T)] (Vu E PV [ D t ]) () As, according to equation (), Vu E D VTS, then VTS [T/( T)] (D VTS PV [ D t ]). And the value of tax shields is: VTS T D T PV [ D t ] () Equation () is valid for perpetuities and for companies with any pattern of growth 3. More importantly, this equation shows that the value of tax shields depends only upon the nature of the stochastic process of the net increase of debt. The problem of equation () is how to calculate PV [ D t ], which requires to know the appropriate discount rate to apply to the expected increase of debt. 4 3 Equation () may also be deduced in a very straightforward way. The value of the debt today is the present value of the interest minus the present value of the increases of debt: D PV [int t ] PV [ D t ]. As the value of tax shields is the present value of the interest times the tax rate, VTS T PV [int t ] T D T PV [ D t ]. 4 If the nominal value of debt (N) is not equal to the value of debt (D), because the interest rate (r) is different from the required return to debt flows (Kd), equation () is: VTS T D T PV [ N t ]. The relationship between D and N is: D PV [N t r t ] PV [ N t ]. rom this equation we may also arrive at equation () because PV [N t r t ] D PV [ N t ]. Multiplying both sides by T, we get equation (): VTS PV [N t r t T] T D T PV [ N t ].

7 4 We may not know what are the correct values of Gu and G L, but we know the value of the difference, provided we can value PV [ D t ], the present value of the net debt increases 5.. VTS in specific situations To develop a better understanding of the result in (), we apply it in specific situations and show how this formula is consistent with previous formulae under restrictive scenarios... Debt of one-year maturity but perpetually rolled over As in the previous case, E [D t ] D, but the debt is expected to be rolled over every year. The appropriate discount rate for the cash flows due to the existing debt is Kd 6. Define K ND as the appropriate discount rate for the new debt (the whole amount) that must be obtained every year, then: Present value of obtaining the new debt every year 7 D / K ND Present value of the principal repayments at the end of every year 8 D ( K ND ) / [(Kd) K ND ] PV [ D t ] is the difference of the last two expressions. Then: PV [ D t ] D (K ND Kd) / [(Kd) K ND ] (3) In a constant perpetuity (E [C t ] C ), it may be reasonable that, if we do not expect credit rationing, K ND Kd, which means that the risk associated with the repayment of the current debt and interest (Kd) is equivalent to the risk associated with obtaining an equivalent amount of debt at the same time (K ND )... Debt is proportional to the Equity market value This is the assumption made by Miles and Ezzell (98) and Arzac and Glosten (5), who show that if D t L E t, then the value of tax shields for perpetuities growing at a constant rate g is: 5 ernández (4) neglected to include in Equations (5) to (4) terms with expected value equal to zero. And he wrongly considered as being zero the present value of a variable with expected value equal to zero. Due to these errors, Equations (5) to (7), Tables 3 and 4, and igure of ernández (4) are correct only if PV [ A t ] PV [ D t ]. 6 We use Kd so as not to complicate the notation. It should be Kd t, a different rate following the yield curve. Using Kd we may also think of a flat yield curve. 7 Present value of obtaining the new debt every year D /(K ND ) D /(K ND ) D /(K ND ) 3... because D E{D t }, where D t is the new debt obtained at the end of year t (beginning of t). 8 The present value of the principal repayment at the end of year is D /(Kd) The present value of the principal repayment at the end of year is D/[(Kd)( K ND )] The present value of the principal repayment at the end of year t is D/[(Kd)( K ND ) t- ] Because D E{D t }, where D t is the debt repayment at the end of year t.

8 5 VTS DKd T ( Ku) (Ku g) ( Kd) (4) Substituting (4) in (), we get: [ D ] PV t D (Kd - Ku) g( Kd) (Ku g)( Kd) (5) or the no growth case (g ), equation (5) is: PV [ D t ] D (Kd-Ku) / [Ku(Kd)] <. Comparing this expression with equation (3), it is clear that Miles and Ezzell imply that K ND Ku. The Miles-Ezzell setup works as if the company pays all the debt (D t- ) at the end of every period t and simultaneously raises all new debt D t. The risk of raising the new debt is similar to the risk of the free cash flow and, hence, the appropriate discount rate for the expected value of the new debt is Ku. However, to assume D t L E t is not a good description of the debt policy of any company because if a company has only two possible states of nature in the following period, it is clear that under the worst state (low share price) the leveraged company will have to raise new equity and repay debt, and this is not the moment companies prefer to raise equity. Under the good state, the company will have to take a lot of debt and pay big dividends. D t L E t provides a computationally elegant solution (as shown in Arzac-Glosten, 5), but unfortunately not a realistic one. urthermore, we have not seen any company that follows this financing policy. In Appendix we prove that if D t L E t, then the appropriate discount rate for the expected taxes is equal for the levered and the unlevered firm (Ku) for t>..3. Debt is proportional to the Equity book value It makes more sense to characterize the debt policy of a growing company with expected constant leverage ratio as a fixed book-value leverage ratio rather than as a fixed market-value leverage ratio because:. the debt does not depend on the movements of the stock market,. it is easier to follow for unlisted companies, and 3. managers should prefer it because the value of tax shields is higher. If D t K Ebv t, where Ebv is the book value of equity, then D t K Ebv t. The increase in the book value of equity is equal to the profit after tax (PAT) minus the equity cash flow. According to equation (5), Ebv t PAT Lt EC t A t D t D t / K (6)

9 6 In this situation, the increase of debt is proportional to the increases of net assets, and the risk of the increases of debt is equal to the risk of the increases of assets: D t A t / (/K) (7) If α is the appropriate discount rate for the expected increases of assets, then the present value of the increases of debt of a constant growing perpetuity is PV [ D ] t gd ( α g) (8) And, substituting (8) in (), the VTS is: VTS D α T ( α g) (9).4. Debt is proportional to the Equity book value. Debt increases are as risky as the free cash flows If we also assume that the risk of the increases of net assets is equal to the risk of the free cash flow, then the increases of the debt are as risky as the free cash flows (α Ku). In this situation, the correct discount rate for the expected increases of debt is Ku, the required return to the unlevered company. In the case of a constant growing perpetuity, PV [ D t ] g D / (Ku-g), and the VTS is Equation (8) in ernández (4): VTS D Ku T / (Ku-g) ().5. The company has a preset amount of debt In this case, the appropriate discount rate for the D t (known with certainty today) is R, the risk-free rate. In this situation, Modigliani-Miller (963) applies and the VTS for a growing perpetuity, according to equation (), is: VTS D T T g D / (R -g) T D R / (R -g) () Note that, in the case a growing perpetuity, Modigliani-Miller is just one case of section.3, in which α R. 3. Value of net debt increases implied by other authors Table I summarizes the implications of several approaches for calculating the value of tax shields. rom equation (), the present value of the increases of debt is: PV [ D t ] (VTS T D ) / T

10 7 Applying this equation to the theories mentioned, we may construct the predictions that each of these theories have for PV [ D t ]. As we have already argued, Modigliani-Miller (963) should be used when the company has a preset amount of debt; ernández (4), when we expect the increases of debt to be as risky as the free cash flow (for example, if the company wants to maintain a fixed book-value leverage ratio); and Miles-Ezzell (98), only if debt will be a multiple of the equity market value D t L E t. If the company maintains a fixed book-value leverage ratio and the risk of the increases of assets is different than the risk of the free cash flow, then the formulas of section.3 (and Appendix A) should be applied. ieten et al. (5) argue that the Modigliani-Miller formula may be applied to all situations. We have shown that it is valid only when the company has a preset amount of debt. Cooper and Nyborg (6) affirm that equation (8) violates value-additivity. It does not because Equation () holds. They use only the cost of debt (R ) or the cost of the unlevered equity (Ku) to discount the expected value of tax shields. We have seen that there are also other debt policies, such as when the firm wants to maintain a fixed book-value leverage ratio. 4. A numerical example and a closer look at the discount rates Appendixes, and 3 derive additional formulae for the three theories discussed in this paper: Miles-Ezzell, ernández and Modigliani-Miller, applied to growing perpetuities. Table II is a summary of the main formulae. Table III contains the main valuation results for a constant growing company. It is interesting to note that according to Miles-Ezzell, the present value of the increases of debt is negative. It is negative if g< (Ku- R )/(R ). irst, we derive the expression for the value of tax shields. Table IV contains the value of the tax shields (VTS) according to the different theories as a function of g and α. The results change dramatically when g increases. It may be seen that Modigliani-Miller is equivalent to a constant book-value leverage ratio (D t L Ebv t ), when α R 5%. ernández (4) is equivalent to D t L Ebv t when α Ku 9%. Second, we derive the appropriate discount rates for the increases of debt. It is interesting to note that while two theories assume a constant rate (Modigliani-Miller assume R and ernández assumes Ku), Miles-Ezzell assume one rate for t and Ku for t>. The appropriate discount rate for the increase of debt at t is, according to Miles-Ezzell, equation (A.): K D g ( Ku) ( R ) g ( R ) R Ku In our example, K D.5%.

11 8 Table V contains the present value of the increases of debt in different periods and the sum of all of them. According to Miles-Ezzell, the present value of the increases of debt in every period is negative. We also prove that although the equity value of a growing perpetuity can be computed by discounting the expected value of the equity cash flow with a single rate Ke, the appropriate discount rates for the expected values of the equity cash flows are not constant. Table VI presents the appropriate discount rates for the expected values of the equity cash flows of our example. According to Miles-Ezzell, Ke t is 58.3% for t and 9% for the rest of the periods. We also derive the appropriate discount rates for the expected values of the taxes. If we assume that the appropriate discount rate for the increases of assets is Ku, then the appropriate discount rate for the expected value of the taxes of the unlevered company is also Ku. But the appropriate discount rate for the expected value of the taxes of the levered company (Ke TAXL ) is different according to the three theories. Table VII presents the appropriate discount rates for the expected values of the taxes in the initial periods for our example. According to Miles-Ezzell, Ke TAXLt is 9.79% for t and 9% for the rest of the periods. According to the other theories, Ke TAXLt grows with t. According to Modigliani-Miller and according to ernández, the taxes of the levered company are riskier than the taxes of the unlevered company. However, according to Miles- Ezzell, both taxes are equally risky for t > Is Ku independent of growth? Up to now we have assumed that Ku is constant, independent of growth. rom equation (6) we know that C t PATu t A t. If we consider that the risk of the unlevered profit after tax (PATu) is independent of growth, and that K PATu is the required return to the expected PATu, the present value of equation (6) is: ( g)c (Ku g) ( g)patu (K g) Vu PATu ( g)c Ku g ( g)patu ga (K PATu g) ( α g) ga ( α g) Table VIII contains the required return to the free cash flows (Ku) as a function of α (required return to the increase of assets) and g (expected growth). It may be seen that Ku is increasing in g if α < K PATu, and decreasing in g if α > K PATu 9 If the risk of the increase of assets is smaller than the risk of the free cash flows, then Miles-Ezzell provides a surprising result: the taxes of the levered company are less risky than the taxes of the unlevered company. This result contradicts Cooper and Nyborg (6), who maintain that Ku is decreasing in g.

12 9 7. Conclusions The value of tax shields depends only upon the nature of the stochastic process of the net increase of debt. More specifically, the value of tax shields in a world with no leverage cost is the tax rate times the current 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 its own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company. The critical parameter for calculating the value of tax shields is the present value of the net increases of debt. It may vary for different companies, but it may be calculated in specific circumstances. or perpetual debt, the value of tax shields is equal to the tax rate times the value of debt. When the debt level is fixed, Modigliani-Miller (963) applies, and the value of tax shields is the present value of the tax shields, discounted at the required return to debt. If the leverage ratio (D/E) is fixed at market value, then Miles-Ezzell (98) applies, with the caveats discussed. If the leverage ratio is fixed at book values and the increases of assets are as risky as the free cash flows (the increases of debt are as risky as the free cash flows), then ernández (4) applies. We have developed new formulas for the situation in which the leverage ratio is fixed at book values but the increases of assets have a different risk than the free cash flows.

13 Table I Present value of the increases of debt implicit in the most popular formulae for calculating the value of tax shields. VTS D T T PV [ D t ] Perpetuities growing at a constant rate g Authors VTS PV [ D t ] D R T ( Ku) g D D (Ku - R Miles-Ezzell (98) - Arzac-Glosten (5) (Ku g) ( R ) Ku g (Ku g)( R Modigliani-Miller (963) D R T (R g) g D R g ernández (4) D Ku T g D (Ku g) Ku g Constant book-value leverage Ku unlevered cost of equity T corporate tax rate D debt value today R risk-free rate α required return to the increases of assets D α T (α g) g D α g

14 Table II Main formulas in the appendixes for growing perpetuities Modigliani-Miller ernández (4) Miles-Ezzell D t fixed D t K Ebv t D t K E t D t K Cd t D t K C t D t K C t VTS D R T D Ku T DR T ( Ku) (R g) (Ku g) (Ku g) ( R ) K D R Ku g ( Ku) ( R ) g ( R ) R Ku K D R Ku Ku Ke D R T Ku (Ku R )( D D ) Ku (Ku R )( T) R E (R g) E T Ku (Ku R ) E R K TS R R R K TS R R ( Ku) gku( R ) ( Ku) g( R ) Ku Miles-Ezzell Modigliani-Miller ernández (4) D t L Ebv t (Ke g)( Ku) (Ku g) Ke (A.6) (Ke g)( R )( Ku) (Ku g)( R ) (Ke Ku)( g) (A3.3) (Ke g)( R )( Ku) (Ku g)( R ) (Ke Ku) (A.6) E (Ke g) (A.5) (Ku g) g R ( T) Vu D ( Ku) ( α) ( R )

15 Table III Example. Valuation of a constant growing company C 5; A ; D 375; R 5%; Ku 9%; T 4%; g %. Modigliani-Miller ernández Miles-Ezzell D t fixed D t K Ebv t D t K E t D t K Cd t D t K C t D t K C t D Vu VTS E PV [ D t ] Gu PV [TAX Ut ] G L PV [TAX Lt ] Ke 9.83%.65%.7% Modigliani-Miller ernández Miles-Ezzell D t fixed D t K Ebv t D t K E t t t t t t t K Dt 5% 5% 9% 9% -.5% 9% K Dt 5% 5% 5.76% 5.47% 9% 9% Ke t 9.33% 9.344% 9.998%.5% 58.95% 9% K Et 9.867% 9.97%.7%.76% 9% 9% K Vut 9% 9% 9% 9% 9% 9% K VTSt 5% 5% 5.93% 5.956% 9% 9% K TSt 5% 5% 5% 5.76% 5% 9% K TAXUt 9% 9% 9% 9% 9% 9% K TAXLt 9.787% 9.84% 9.787% 9.87% 9.787% 9% Ke average 9.88%.647%.66%

16 3 Table IV Present value of the tax shields (VTS) according to the different theories as a function of g (expected growth) and α (required return to the increase of assets). D 375; R 5%; Ku 9%; T 4% g % % % 3% 4% 5% Miles-Ezzell Modigliani-Miller ernández (4) D t L Ebv t ; α5% D t L Ebv t ; α7% D t L Ebv t ; α9% D t L Ebv t ; α% D t L Ebv t ; α5% Table V Present value of the increases of debt in different periods and the sum of all of them. D 375;R 5%; Ku 9%; T 4%; g % PV ( D t) t t t3 t4 t5 t t t3 t4 t5 Sum Miles-Ezzell Modigliani-Miller ernández (4) D t L Ebv t ; α5% D t L Ebv t ; α7% D t L Ebv t ; α9% D t L Ebv t ; α% Table VI Appropriate discount rates for the expected values of the equity cash flows (Ke t ) C 5; D 375; R 5%; Ku 9%; T 4%; g %. Ke t t t t3 t5 t t t3 t5 Miles-Ezzell 58.3% 9.% 9.% 9.% 9.% 9.% 9.% 9.% Modigliani-Miller 9.33% 9.34% 9.36% 9.39% 9.48% 9.74%.7%.65% ernández (4).%.%.5%.%.8%.76%.55% 7.37% D t L Ebv t; α5% 9.33% 9.34% 9.36% 9.39% 9.48% 9.74%.7%.65% D t L Ebv t; α7% 9.67% 9.7% 9.7% 9.78% 9.95%.4%.% 7.4% D t L Ebv t; α9%.%.%.5%.%.8%.76%.55% 7.37% D t L Ebv t; α%.3%.33%.35%.4%.5%.88%.5% 5.54%

17 4 Table VII Appropriate discount rate for the expected value of the taxes of the levered company. α Ku 9%; C 5; D 375; R 5%; T 4%; g %. K TAXLt t t t3 t4 t5 t6 t7 t8 Miles-Ezzell 9.79% 9.% 9.% 9.% 9.% 9.% 9.% 9.% Modigliani-Miller 9.79% 9.8% 9.86% 9.9% 9.94% 9.99%.4%.9% ernández (4) 9.79% 9.8% 9.83% 9.85% 9.87% 9.89% 9.9% 9.94% Table VIII Ku as a function of g (growth) and α (required return to the increase of assets) if the required return to the profit after tax of the unlevered company (K PATu ) is fixed K PATu 9%; C 5; D 375; R 5%; T 4%. g % % % 3% 4% 5% 7% 9.% 9.57%.3%.47% 4.6% 89.% 8% 9.% 9.3% 9.49% 9.79%.9%.86% α 9% 9.% 9.% 9.% 9.% 9.% 9.% % 9.% 8.83% 8.67% 8.54% 8.43% 8.36% % 9.% 8.59% 8.6% 8.% 7.88% 7.84% 5% 9.% 8.37% 7.93% 7.65% 7.5% 7.55%

18 5 Appendix Derivation of formulas for Miles-Ezzell: D t K E t We are valuing a company with no leverage cost. The cost of debt is the risk-free rate (R ). The company is a growing perpetuity, which means that E [D t ] D (g) t, Value of tax shields The tax shield of the next period (t ) is known with certainty (D R T) and the appropriate discount rate is R. However, the appropriate discount rate for the expected tax shields if t is bigger than is Ku. VTS 3 DR T D( g)r T D( g) R T D ( g) R T R ( R )( Ku) 3 ( R )( Ku) ( R )( Ku) We have the sum of a geometric progression growing at a rate (g)/(ku), and VTS DR T ( Ku) (A.) (Ku g) ( R )... The appropriate discount rate for the expected increases of debt PV [ D ] g D ( K D ) D( g) D - ( Ku) ( R g ( Ku) ( R ) Some algebra permits to express K D (A.) 3 g ( R ) R Ku ) Equation (A.) is asymptotic in g (Ku- R )/( R ), in our example in g 3.846% g -3% -% % % % 3% 4% 5% 6% 7% K D M-E -5.% -6.5% -.% -38.8% -.5% -53.9% 89.% 357.8% 98.6% 39.% We define the appropriate discount rate of a random variable X as being the discount rate that, when applied to the expected value of the variable X, provides us with the present value of the variable: PV PV [ X ] [ X ] [ X ] E ( appropriate discount rate ) [ X ] E ( appropriate discount rate)( appropriate discount rate ) ollowing equation (): E t D t Vu t VTS t, as D t K E t, and because VTS t is a function of D t, it follows that D t, E t and VTS t have the same risk as Vu t, and the same appropriate discount rate as Vu t, that is Ku. 3 Note that if g, then K D -%. This does not make any economic sense because in this situation the expected value of the increase of debt is also.

19 6 Appendix (continued) Now, for t: PV D ( g) D ( g) D [ ] [( g) ( g) ] D - ( Ku) ( R )( Ku) g D ( g). ( K D )( K D ) ( K D)( K D ) D ( g) D ( g) g D ( g) As -, it is obvious that K D Ku ( Ku) ( R )( Ku) ( K D ) ( Ku) The present value of all the expected increases of debt, using equation (), is: PV [ D t ] VTS /T - D. Substituting VTS using equation (A.): g( R ) Ku R PV [ D ] D t (A.3) (Ku g)( R ) Note that, PV [ D t ]< if g< (Ku- R )/( R ), in our example if g < 3.846% The appropriate discount rate for the expected equity cash flows The relationship between expected values at t of the free cash flow, the equity cash flow and the debt cash flow (Cd) is: E [EC ] E [C ] E [Cd ] D R T. This relationship is equivalent to: E (Ke-g) Vu (Ku-g) - D (R -g) D R T As E Vu - D VTS, then E Ke Vu Ku - D R VTS g D R T And the general equation for Ke is: D VTS Ke Ku [(Ku R ( T) ] (Ku g) (A.4) E E Substituting (A.) in (A.4): D Ke Ku E DR T ( Ku) D R [ ] T (Ku R ( T) Ku (Ku R ) E ( R ) E R (A.5) But this expression is the average Ke. It is not the required return to equity (Ke t ) for all periods. The value of the equity today is the sum of the present value of the equity cash flow of next period plus the present value of the equity value of next period: ( g)ec ( g)e [ ] [ ] E PV EC PV E ( Ke) ( Ku) ( g)ec ( Ku) E(Ke g)( Ku) (Ke g)( Ku) ( Ke) (A.6) E (Ku g) E (Ku g) (Ku g)

20 7 Appendix (continued) Substituting (A.5), we get: [ R ( T) ] D (Ku R ) ( Ku) Ke Ku (A.7) E (Ku g)( R ) D Ke M-E 9.% 8.8% 9.78% 44.4% 58.3% 63.65% 9.6% 8.6% 9.% 33.7% 584.7% 368.% or t: PV [ EC ] ( g) EC ( g) C DR ( T)( g) gd ( g) ( Ke )( Ke ) ( Ku) ( R )( Ku) g Ku ( R ) R Ku ( )( Ku) g ( R ) R Ku Comparing this equation with the one for t, it is clear that Ke Ku. The appropriate discount rate for the expected taxes If we assume that the appropriate discount rate for the expected increases of assets is also α, then (see equation (7)), the present value of the expected taxes of the unlevered company is: PV [ Taxex ] [ Taxex ] E U T ( g)c U ( KTAXU) T ( Ku) ga ( α) As E[Taxes U ][T/(-T)] [C (g) ga ], we can calculate K TAXU. [ Taxex U] [ Taxex ] E ( g)c ga ( KTAXU) ( Ku)( α) (A.8) PV U ( g)c ( α) ga ( Ku) If α Ku, then K TAXUt Ku According to (9): T g A Gu Vu (A.9) T ( α g) To calculate the appropriate discount rate for the expected taxes of the levered company we use equation (8): E [ ] [ Taxes ] T ( g)c ga D R ( T) PV Taxes L L ( KTAXL) T ( Ku) As E[Taxes L ] [ T/(-T)] [C (g) ga D R (-T)] ( α) ( R )

21 8 Appendix (continued) ( g)c ga DR ( T) KTAXL (A.) ( g)c ga DR ( T) ( Ku) ( α) ( R ) or t >, (for example, for t), the present value is: PV [ Taxes ] [ Taxes ] E L ( g) L ( KTAXL)( KTAXL ) It is obvious that K TAXL Ku if αku rom equation () we van calculate the present value of the levered taxes: T g A DR T ( Ku) G L Gu VTS Vu T ( g) (A.) α (Ku g) ( R ) Although K TAXUt and K TAXLt are not constant, we can calculate K TAXU and K TAXL such that G U Taxes U (g) / (K TAXU - g) and G L Taxes L (g) / (K TAXL - g). Some algebra permits to find: KTAXU Vu ( α g)ku gαa Vu ( α g) ga KTAXL E (Ke g) g(a D ) g ga VTS ( T) Vu ( α g) T The appropriate discount rate for the expected value of the unlevered equity (Vu) Vu C ( g) ( g)vu ( Ku) ( K ) Vu Vu( g)( Ku) Vu( g)( Ku) Vu( g)( Ku) ( KVu ) Ku Vu ( Ku) C ( g) Vu ( Ku) Vu (Ku g) Vu ( g) The appropriate discount rate for the expected value of equity (E) EC ( g) E ( g) E ( Ke ) ( K ) E

22 9 Appendix (continued) E( g) ( K E). Using (A.7) and knowing that EC (-g) E (Ke-g): E EC ( g) /( Ke ) E( g) ( g)( Ku) KE Ku. E(Ke g)(ku g) ( Ku) (Ku g) E (Ke g)( Ku) D t L E t is absolutely equivalent to D t M Vu t. In this case, D t X (C t - C t- ) X C t, where X D / C. As, according to Miles Ezzel, Vu t E t (L-L R T [(Ku)/(R )]/(Ku-g)), the relationship between L and M is: M L /(L-L R T [(Ku)/(R )]/(Ku-g)). It is easy to find the relationship between M and X: D M Vu M C (g)/(ku-g) M (D /X)(g)/(Ku-g), and X M (g)/(ku-g).

23 Appendix Derivation of formulas if debt is proportional to the book value of equity D t K Ebv t, where Ebv is the book value of equity. Then D t K Ebv t and the relationship between D t and A t (increase of assets) is 4 D t A t / (/K). The appropriate discount rate for the expected increases of debt As D t A t / (/K), the appropriate discount rate for the increases of debt is the appropriate discount rate for the increases of assets. If the appropriate discount rate for the expected increases of assets is α, then the present value of the increases of assets for a g D growing perpetuity is PV [ D ] t ( α g) Value of tax shields In this situation, we can use equation (), VTS T D T PV [ D t ], and the value of the tax shields is: D α T VTS (A.) ( α g) If α Ku, then VTS T D Ku / (Ku-g) The appropriate discount rate for the expected value of the tax shields The tax shield of the next period (t ) is known with certainty (D R T) and the appropriate discount rate is R. The appropriate discount rate for the expected tax shield of t (K TS ) is: PV [ D R T] D ( g)r T D gd R T ( R )( K ) TS ( R ) ( R )( α) R ( α) gα( R ) KTS (A.) ( α) g( R ) If g, K TS R The present value of the tax shield of period t is: t D ( g) R T PV [ TS ] t ( R )( KTS )...( KTSt ) t D gd gd( g) gd ( g) R T... t t t t ( R ) ( R ) ( α) ( R ) ( α) ( R )( α) 4 The increase of the book value of equity is equal to the profit after tax (PAT) minus the equity cash flow. According to equation (5): Ebv t PAT Lt - EC t A t - D t D t / K In this situation, the increase of debt is proportional to the increases of net assets, and the risk of the increases of debt is equal to the risk of the increases of assets: D t A t / (/K)

24 Appendix (continued) This expression is the sum of a geometric progression with a factor X (g)( t R TD g X PV TSt (A.3) t ( R ) ( R ) ( α) [ X ] R )/( α). The solution is: [ ] [ ] PV ( KTSt ) PV t [ TSt ] ( α)(x ) g( R )(X ) ( g) X( α) [ TS ] t ( α)(x ) g( R )(X ) t When t tends to infinity, K TS MIN[α, ( R )(g)-] The appropriate discount rate for the expected equity cash flows Substituting (A.) in (A.4): D α T(Ku g) Ke Ku (Ku R ( T) E ( g) (A.4) α D Ke Ku ( T)(Ku R E Note that if α Ku: ) But this expression is the average Ke. It is not the required return to equity (Ke t ) for all the periods. But Ke t is not constant for t. or t, the relationship among the cash flows is: EC C - D R (-T) D Calculating the expected value at t for a growing perpetuity and substituting the value of K D α: PV [ EC ] ( g)ec ( Ke) ( g)c ( Ku) DR ( T) gd ( R ) ( α) E (Ke g) Vu (Ku g) DR ( T) gd ( Ke) ( Ku) ( R ) ( α) ( E(Ke g) Ke ) (A.5) (Ku g) g R ( T) Vu D ( Ku) ( α) ( R ) if α Ku: (Ke g)( R )( Ku) ( Ke) (A.6) (Ku g)( R ) (Ke Ku) It is obvious that: [ ECt ] [ EC ] PV ( Ket ) ( g) PV t

25 Appendix (continued) or t> PV t ( g) C t D ( g) R ( T) [ ECt ] t t ( Ku) ( R )( KTS )...( KTSt ) Using (A.3), if we define X (g)( R )/(α): PV t ( g) C D R ( T) ( R ( R ) ) t gd ( g) ( α) t g(x ) ( Ku)(X ) t gd ( g) [ ECt ] t t t ( Ku) ( α) Note that PV t [EC t ] < means only that PV t [C t D t ] < PV t [D t- R (-T)] The appropriate discount rate for the expected taxes (A.9) and (A.) also apply to this situation rom equation () we can calculate the present value of the levered taxes: T g A Dα( T) G L Gu VTS Vu T ( g) (A.7) α The appropriate discount rate for the expected value of tax shields (VTS) DTR VTS ( g) VTS PV [ TS] PV [ VTS] ; ( R ) ( K VTS) α( R )( g) ( K VTS) ( α gr ) or t. VTS PV [ TS ] PV [ TS ] PV [ VTS ] PV [ TS ] PV [ TS ] VTS ( g) ( K VTS ) ( K VTS )(VTS PV [ TS ] PV [ TS ]) VTS( g) ( K VTS)( K VTS ) The appropriate discount rate for the expected value of equity (E). E EC ( g) E( g) ( Ke ) ( K ) E

26 3 Appendix (continued) ( K E or t. E PV ) E E( g) EC ( g) /( Ke [ EC ] PV [ EC ] PV [ E ] PV [ EC ] PV [ EC ] E( g) ( K E ) ( K E)(E PV ) [ EC ] PV [ EC ]) E ( g) ( K E)( K E ) The appropriate discount rate for the expected value of debt (D) ( K [ D ] E ( K D D ( g)d D ) ( K ) ( R D ( g)( R )( ) ) α α g( R ) gd ) ( α)

27 4 Appendix 3 Derivation of formulas if D t is known with certainty at t. This is the Modigliani-Miller assumption. This situation can be analyzed as a special case of Appendix : that of a company with α R. The value of the tax shield is D R T VTS (A3.) (R g) The appropriate discount rate for the expected value of the tax shields is R The appropriate discount rate for the expected value of the equity cash flow Substituting (A3.) in (A.4), or substituting α by R in (A.4): D (Ku - g) Ke Ku (Ku R ( T) R T (A3.) E (R - g) But this expression is the average Ke. It is not the required return to equity (Ke t ) for all the periods. Substituting α by R in (A.5): E(Ke g)( Ku)( R ) ( Ke) (A3.3) Vu(Ku g)( R ) D( Ku) [ g R ( T) ] It may be expressed also as: (Ke g)( Ku)( R ) ( Ke) (Ku g)( R ) (Ke Ku)( g) or t, it is obvious that: PV [ EC ] PV [ EC ] ( g) ( Ke ) The appropriate discount rate for the expected taxes (A.9) and (A.) also apply to this situation. or t, the present value is: PV [ Taxex ] [ Taxex ] E L ( g) L ( KTAXL)( KTAXL ) rom equation () we can calculate the present value of the levered taxes: T g A DR T G L Gu VTS Vu T ( g) (A3.4) α (R g)

28 5 Appendix 3 (continued) The appropriate discount rate for the expected value of tax shields (VTS) Another way of finding the same result: VTS PV [TS ] PV [VTS ]. DTR VTS ( g) VTS ( R ) ( K VTS ) DR T (R g) DTR DR T ( R ) (R g) ( g) ( K VTS ) ( g) ( R ) (R g). ( g). (R g)( K VTS ) (R g)( R ) (R g)( R ). It is obvious that K VTS R K VTSt The appropriate discount rate for the expected value of equity (E) Calculating the present value of equation () at t : PV [VTS ] PV [E ] PV [D ]- PV [Vu ]. ( g) ( g) ( g) ( g) VTS E D Vu ( R ) ( K E) ( R ) ( Ku) E VTS D ( K E) ( R ) Vu (VTS D )( Ku) Vu ( R ) E (VTS D )Ku VuR ( Ku) ( R )( Ku) ( R )( Ku) E( R )( Ku) ( K E ) (E Vu )( Ku) Vu( R ) for t E E Vu ( K E )( K E ) ( R ) Vu ( Ku) (E Vu )( Ku) Vu ( R ) ( R ) ( Ku) E ( R ) ( Ku) ( K E)( K E ) (E Vu )( Ku) Vu ( R ) t t (E Vu )( Ku) Vu ( R ) ( K Et ) ( R )( Ku) t t (E Vu )( Ku) Vu ( R )

29 6 The appropriate discount rate for the expected value of debt (D) E ( K [ D ] D ( g)d D ) ( K ) ( R D gd ) ( R ( g)d ) ( R ) Then, K Dt R

30 7 References Arzac, E.R, Glosten, L.R., 5, A Reconsideration of Tax Shield Valuation, European inancial Management /4, Cooper, I.A., and Nyborg, K.G., 6, The Value of Tax Shields IS Equal to the Present Value of Tax Shields, Journal of inancial Economics, forthcoming. Copeland, T.E., Koller T., Murrin J.,, Valuation: Measuring and Managing the Value of Companies (Third edition. Wiley, New York). Damodaran, A., 994, Damodaran on Valuation (John Wiley and Sons, New York). ernández, P., 4, The value of tax shields is NOT equal to the present value of tax shields, Journal of inancial Economics 73/, ieten, P., Kruschwitz, L., Laitenberger, J., Löffler, A., Tham, J., Vélez-Pareja, I., and N. Wonder, 5, Comment on The value of tax shields is NOT equal to the present value of tax shields, Quarterly Review of Economics and inance 45, #, Harris, R.S., Pringle, J.J., 985, Risk-adjusted discount rates extensions from the averagerisk case, Journal of inancial Research 8, Miles, J.A., Ezzell, J.R., 98, The weighted average cost of capital, perfect capital markets and project life: a clarification, Journal of inancial and Quantitative Analysis 5, Modigliani,., Miller, M., 963, Corporate income taxes and the cost of capital: a correction, American Economic Review 53, Myers, S.C., 974, Interactions of corporate financing and investment decisions implications for capital budgeting, Journal of inance 9, -5. Ruback, R., 995, A note on capital cash flow valuation, Harvard Business School Case No Ruback, R.,, Capital cash flows: a simple approach to valuing risky cash flows, inancial Management 3, 85-3.