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1 Value of tax shields (VTS): 3 theories with some sense Pablo Fernandez, Professor of Finance IESE Business School, University of Navarra fernandezpa@iese.edu Camino del Cerro del Aguila Madrid, Spain xlnmapgsjv October 7, 207 The value of tax shields (VTS) defines the increase in the company s value as a result of the tax saving obtained by the payment of interest. However, there is no consensus in the existing literature regarding the correct way to compute the VTS. Most authors think of calculating the VTS in terms of the appropriate present value of the tax savings due to interest payments on debt, but Modigliani and Miller propose discounting the tax savings at the risk-free rate (R F ), and Miles and Ezzell (and many others) propose discounting these tax savings the first year at the cost of debt and the following years at Ku. We develop valuation formulae for companies that maintain a fixed book-value leverage ratio and argue that, for most companies, and especially when calculating residual values, this assumption is more realistic than those of MM and ME. We obtain an intermediate value between those of MM and ME. If a company targets its leverage in market-value terms, it has less value than if it targets the leverage in book-value terms. How could a manager target leverage in market-value terms? However, many authors identify constant leverage with a constant market leverage ratio. 0. Definition of VTS. General expression of the value of tax shields 2. Valuation of a firm whose debt policy is determined by a book-value ratio 3. Valuation of firms under alternative financing strategies 4. Required return to equity and WACC 5. A numerical example 6. The correlation between the tax shields and the free cash flow 7. Conclusions References Appendix. VTS equations according to the main theories JEL classification: G2; G3; G32 Keywords: value of tax shields, required return to equity, WACC, company valuation, APV, cost of equity I thank Enrique Arzac, José Manuel Campa, Ian Cooper, Javier Estrada, Mark Flannery, John Graham, Yilmaz Guney, Stephen Penman, Scott Richardson, Michael Roberts and Zhaoxia Xu for very helpful comments to earlier manuscripts of this note, and Rafael Termes for his sharp questions that encouraged me to explore valuation problems. Alberto Ortiz did a wonderful work as research assistant. CH39-

2 There is a considerable body of literature on the VTS (Value of Tax Shields). We will 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). 0. Definition of VTS Unlevered company means company without Debt. The term Enterprise value usually refers to the sum of the Value of the shares (E) and the value of the Debt (D). What are the differences between the levered company and the unlevered company? If we assume that the default risk does not change, the only difference is that the levered company pays lower taxes (because of the interest tax shield). Therefore, it must be true that: E + D = Vu + VTS. The last identity is usually called Adjusted present value (APV) : the value of the debt (D) plus that of the shareholders equity (E) is equal to the value of the unlevered company s shareholders equity (Vu) plus the value of the tax shield (VTS). The VTS is the present value of the tax savings due to the payment of interest, but it is also the difference between the present value of taxes paid by the unlevered firm (Gu) and the present value of taxes paid by the levered firm (GL). We will see that there are several theories for calculating the VTS. Ku is the required return to equity in the debt-free company. Vu is given by: Vu 0 = PV 0 [Ku t ; FCF t ] Unlevered company Gu Levered company GL Levered company Debt D VTS VTS = D + E Vu = Gu - GL Vu Shares E Vu We can calculate the present value of tax saving arising from the use of debt (tax shields). It is usually named Value of Tax Shields (VTS). If all interests paid are tax-deductible, the tax shield of a year is the product of the interest paid (N r) times the tax rate (T). In section, we derive the general formula for the value of tax shields. In section 2, we apply this formula to a company that maintains a constant book-value leverage ratio. In section 3, we compare the valuation with that of other financing strategies. In section 4, we calculate the required return to equity and the WACC. Section 5 is a numerical example. In section 6, the correlation between the tax shields and the free cash flow is discussed. Section 7 concludes.. General expression of the value of tax shields The value of the debt today (D 0 ) is the present value of the future stream of interest minus the present value of the future stream of the increases of debt (D t ): D PV Interest PV () 0 0 t 0 Dt The value of tax shields (VTS) defines the increase in the company s value as a result of the tax saving obtained by the payment of interest: it is the present value of the interest times the tax rate, 0 0 t 0 0 Dt VTS TPV Interest T D TPV (2) The author knows that this is very difficult to assume because it is not true. The levered company has a higher probability of not surviving and, consequently, the debt not only produces increases of value (tax savings) but also a decrease of value (higher probability of disappearing). But we start assuming that the probability of disappearing is equal in the levered and in the unlevered company. CH39-2

3 Equation (2), valid for perpetuities and for companies with any pattern of growth, shows that, if the tax shields are always deductible and T applies to all states, the value of tax shields depends only upon the nature of the stochastic process of the net increase of debt, the existing debt (D 0 ) and the tax rate (T) 2. We will see that the nature of the stochastic process of the net increase of debt is very different if the debt is proportional to the book-value of equity than if the debt is proportional to the market-value of equity. The riskiness of the increases of debt is smaller in the first case than in the second case and, consequently, the value today of the future stream of the increases of debt is higher. 2. Valuation of a firm whose debt policy is determined by a book-value ratio Suppose a firm s debt policy is determined by a book-value ratio. In this situation, D t = K Ebv t, being D the debt, Ebv the book-value of equity, and A is the book-value of assets: A t = Ebv t + D t. The Free Cash Flow (FCF) is the difference between the Profit After Tax of the unlevered company (PATu), and the increase of the book-value of the assets (A): FCF t = PATu t - A t. For a firm in a constant risk class we assume that Ku is the common appropriate discount rate for the expected values of FCF t, PATu t and A t. Then, in a perpetuity growing at a constant rate g, the value of the unlevered company (Vu) may be written as: FCF ( g) PATu ( g) - ga Vu (3) Ku - g Ku - g The present value of the expected increases of assets is: A ga / Ku - g PV0 t 0 (4) As A t = Ebv t + D t, and D t = K Ebv t, then D t = A t K / (+ K), and: D gd / Ku - g PV0 t 0 (5) Substituting (5) in (2) we get the VTS of a company that maintains a fixed book-value leverage ratio: DKEbv D Ku T VTS 0 0 (6) Ku - g Although equation (6) may be read as the present value of D 0 KuT growing at g, it does not represent it. Rather, equation (6) is the present value of the tax shields (D 0 R F T) growing at a rate g and discounted at a growing rate, being that rate R F for t = Valuation of firms under alternative financing strategies Appendix deals with the main theories about the VTS and their equations. If the company has a preset amount of debt, D t is known with certainty today and MM (Modigliani- Miller) applies: the appropriate discount rate for the expected value of D t is R F, the risk-free rate, and PV D gd / (R g) (7) 0 t 0 F Substituting (7) in (2), we get: MM D0R F T VTS0 (8) R F g Equation (7) provides a higher value than equation (5) and, consequently, the VTS according to MM is higher than the VTS according to equation (6). If the firm s debt policy is determined by a market-value ratio (instead of a book-value one), then the amount of debt is proportional to the market-value of equity 4 (E) and the present value of the expected increase of debt in period t (as D t- is known in period t-) is: 2 Booth (2007) discusses situations in which the tax shields are not always deductible and uses a parameter (a decreasing function of debt) instead of T. 3 Equation (6) is equal to equation (28) in Fernandez (2004), to equation (4) in Booth (2007), and to equation () in Massari, Roncaglio and Zanetti (2007). CH39-3

4 t t D0 ( g) D0( g) PV0 Dt (9) t t ( Ku) ( R F )( Ku) The sum of all the present values of the expected increases of debt is a geometric progression with growth rate = (+g)/(+ku). Therefore: D0 Ku R F PV 0 Dt g (0) t Ku g R F Equation (0) tells us that investors would require money to hold a security with payoff equal to D t if the expected growth rate (g) is smaller than (Ku R F ) / (+R F ). Obviously, this is not a sensible result for most companies. Substituting (0) in (2), we get the well known Miles-Ezzell (ME) formula 5 : ME D0R F T ( Ku) VTS0 () (Ku g) ( R F ) ME provide a computationally elegant solution (as shown in Arzac-Glosten, 2005), but it is not a realistic one. D t = L E t implies that if a company has only two possible states of nature in the following period, under the worst state (low share price) the company will have to raise new equity and repay debt. Under the good state, the company will have to issue debt and pay big dividends. This is not a good description of the debt policy of most companies 6. We claim that 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 instead of as a fixed market-value leverage ratio for the following reasons: the company is more valuable: () is smaller than (6); rating agencies focus on book-value leverage ratios; the amount of debt does not depend on the movements of the stock market; it is easier to follow for non quoted companies; and, the empirical evidence provides more support to the fixed book-value leverage ratio hypothesis. Table contains statistics about the financial leverage of 27 US firms: line measures the leverage in book-value terms and line 2 in market-value terms. The average of the standard deviation/average of the book leverage ratio [D / (D+Ebv)] for each company was 0.34, smaller than 0.49, the same coefficient of the market-value leverage ratio [D / (D+E)]. The coefficient of the book ratio was smaller than the coefficient of the market ratio for 258 of the 27 US companies analyzed 7. Table 2 compares correlation coefficients of the increases of debt with the increases of assets measured in book-value (line ) and in market-value terms (lines 2): the average and the median of the bookvalue correlation coefficients are higher (and the SD smaller) in book-value terms than in market-value terms. According to ME, the correlation between D and D+E) should be, but it is only 0.23 on average. Tables and 2 permit to conclude that debt is more correlated to the book-value of the assets than to their market-value. When managers have a target capital structure, it is usually in book-value terms (as opposed to market-value terms), in large part because this is what credit rating agencies pay attention to. Stonehill et al. (973) provide evidence that most managers do think in terms of book values. Myers (984) argued that firms set target book debt ratios because book asset values are proxies for the values of assets 4 This is the assumption made by Miles and Ezzell (985), and Arzac and Glosten (2005). 5 Equation () is identical to equations (2) in ME (985), (3) in Arzac and Glosten (2005) and (6) in Lewellen and Emery (986). However, Harris and Pringle (985), Ruback (2002) and Cooper and Nyborg (2006, equation 29) propose VTS = D Kd T / (Ku-g). This expression does not correspond to ME assumption: it is correct only in continuous time, but then Ku, Kd and g should be expressed also in continuous time. 6 Grinblatt and Titman (2002) argue that firms often pay down debt when things are going well, and do not alter debt when returns are low. 7 Antoniou, Guney and Paudyal (2006, table 2) also calculate the standard deviation and the mean of the leverage ratio measured in market-value terms and in book-value terms of companies in France, Germany and the UK. For all three countries, the average ratio of the standard deviation to the mean of the leverage ratio measured in market-value terms is higher than the same average in book-value terms. Xu (2006, Table I) and Flannery and Rangan (2006, Table ), also show the standard deviation and the mean of the leverage ratio measured in market-value terms and in book-value terms (although not detailed by company): again, the ratio of standard deviation to the mean of the leverage ratio measured in market-value terms is higher than in book-value terms. CH39-4

5 in place. Graham and Harvey (200) show that 8% of the CFOs in their sample had a target range for their debt-equity ratio and that 39% of the companies did not have publicly traded common stock 8. Penman, Richardson and Tuna (2005) point out that measuring leverage with market values leads to perverse results. Table. Statistics of financial leverage variables for 27 US companies of the S&P Financial companies (90), zero debt companies (27), companies with no data in 99 (0) and companies with negative book-value of equity () were eliminated. Source of the data: Datastream is the average of the 27 Standard Deviations (one for each company) of Dt/(Dt +Ebvt) Ebv = book-value of equity. E = market-value of equity. D = book-value of debt average of the statistics for each company Average Standard line SD/Av. median (Av) Deviation (SD) D / (D+Ebv) D / (D+E) SD /Av. of D/(D +E) > SD /Av. of D / (D +Ebv) for 258 companies Table 2. Statistics of the correlations between the increases of debt and the increases of assets for 27 US companies of the S&P Average, median, standard deviation (SD), maximum and minimum of the 27 correlation coefficients (one for each company) Statistics of the 27 correlation coefficients Number of companies with line Correlations between: Average Median SD MAX min Correl < 0 Correl > 0 Dt and At Dt and (D+E)t A = book-value of assets = Ebv + D = book-value of equity + book-value of debt. We argue that when managers have a target capital structure, it is usually in book-value terms (as opposed to market-value terms), in large part because this is what credit rating agencies pay attention to. Credit rating is the second most important factor, after the maintenance of financial flexibility, in the decision to issue more debt (Graham and Harvey, 200). While rating agencies emphasize that both financial and non-financial factors matter, the academic literature has focused primarily on the ability of financial ratios to predict ratings. The existing literature (e.g., Blume, Lim and Mackinlay, 998) shows us that interest coverage and leverage (two ratios calculated with book values) have the most pronounced effect on credit ratings. These ratios have also been identified by the rating agencies themselves (e.g., Standard and Poor s, 2006) as key determinants of a company s credit rating. We also provide empirical evidence about the capital structure of firms and find that the constant book-value leverage ratio assumption fits the market data much better than the constant market-value leverage ratio assumption. Many authors consider that debt policy may only be framed in terms of maintaining a fixed market-value debt ratio or a fixed dollar amount of debt. We develop valuation formulae for companies that maintain a fixed book-value leverage ratio and argue that, for most companies, and especially when calculating residual values, this assumption is more realistic than the two previous ones. 4. Required return to equity and WACC For perpetuities with a constant growth rate (g), the relationship between expected values in t= of the free cash flow (FCF) and the equity cash flow (ECF) is: ECF 0 (+g) = FCF 0 (+g) D 0 R F (-T) + g D 0 (2) The value of the equity today (E 0 ) is equal to the present value of the expected equity cash flows. If Ke is the average appropriate discount rate for the expected equity cash flows, then E 0 = ECF 0 (+g) / (Ke-g), and equation (2) is equivalent to: E 0 (Ke-g) = Vu 0 (Ku-g) D 0 (R F g) + D 0 R F T (3) As, E 0 = Vu 0 D 0 + VTS 0, the general equation for the average Ke is: Ke Ku D0 / E0 Ku R F( T) VTS0 / E0 (Ku g) (4) Substituting (6), (8) and () in (4), we get Ke according to the different theories: DKEbv D0 MM D0 R ( T) g Ke Ku (Ku R F )( T) (5) Ke Ku (Ku R ) F F (6) E0 E R g 0 F 8 77% of the private companies had a target range for their debt-equity ratio. It may be calculated using the survey data provided by the authors in Graham and Harvey (2003). CH39-5

6 ME Ke D0 R F ( T) Ku (Ku R F ) E0 R F (7) (5) and (6) are equal for g=0. However, equation (6) provides a Ke smaller than Ku if g > R F (-T). The WACC is the appropriate discount rate for the expected free cash flows, such that D 0 +E 0 = FCF 0 (+g) / (WACC-g). The equation that relates the WACC and the VTS is: WACC Ku VTS0 (Ku - g) / D0 E0 (8) Substituting (6), (8) and () in (8), we get the WACC according to the different theories 9 : D D T WACC KEbv 0 Ku Ku (9) MM D0T R F (Ku - g) WACC Ku (20) (D0 E0 ) (D0 E0 ) (R F - g) ME D0T R F ( Ku) WACC Ku (2) (D0 E0 ) ( R F ) 5. A numerical example Table 3 contains the main valuation results for a constant company growing at a constant rate (3%). One of the results has no economic sense: according to MM, Ke (7.65%) is smaller than Ku (9%). Figure shows the Present Value of the expected increases of debt according to the three theories as a function of growth. According to ME the present value of the increases of debt is negative if the growth rate (g) is lower than 4.8%. Figure 2 shows the WACC as a function of growth: according to MM WACC is a decreasing function of g. The constant leverage in book-value terms provides results that lie between those of MM and ME. Table 3. Example. Valuation of a constant growing company A 0 =,000; D 0 = 750; PATu 0 = 00; R F = 4%; Ku = 9%; T = 40%; g = 3%; Vu 0 =,26.67 Constant market-value leverage. Miles-Ezzell Constant book-value leverage. Fernandez Modigliani-Miller VTS ,200 PV0[Dt] ,250 Equity value (E0) , Ke 4.46%.45% 7.65% WACC 8.2% 7.38% 6.02% Figure. Present value of the expected increases of debt as a function of growth A 0 =,000; D 0 = 750; PATu 0 = 00; R F = 4%; Ku = 9%; T = 40% PV(incD)mm corresponds to equation (7); PV(incD)me corresponds to equation (0); PV(incD)cbvl corresponds to equation (5). PV (increases of Debt) PV(incD)mm PV(incD)me PV(incD)cbvl 0% % 2% 3% 4% 5% g 9 (9) is equal to formulae (20) of ME (980), (2) of Lewellen and Emery (986), and (8) of Stanton and Seasholes (2005). CH39-6

7 Figure 2. WACC as a function of growth A 0 =,000; D 0 = 750; PATu 0 = 00; R F = 4%; Ku = 9%; T = 40% WACCmm corresponds to equation (20); WACCme corresponds to equation (2); WACCcbvl corresponds to equation (9). 9% 8% WACC 7% 6% 5% WACCmm WACCme WACCcbvl 4% 0% % 2% 3% 4% 5% g With the constant growth model the market to book ratio is also constant, since all parameters are constant, and so the market and book debt ratios are constant through time. However, the riskiness of the future debt is completely different: under ME (debt is determined by a market-value ratio) debt and tax shields are riskier than if debt is determined by a book-value ratio. 6. The correlation between the tax shields and the free cash flow The relationship between the dividends (equity cash flows) and the free cash flow is: Div t = FCF t D t- R F (-T) + D t. (22) As managers do not like to change the dividends much, it is reasonable to expect that most companies will show a negative correlation between the FCF and the D. Table 4 confirms that prediction: 265 US companies (out of 27) have a negative correlation between D and FCF in the period , and the negative correlation is very strong. On the other hand, the correlation between D and A (shown in table 2) is strongly positive. If Debt is proportional to the book-value of equity, it is also proportional to the bookvalue of assets (A): D = LA; and D t = L A t. Substituting D t by L A t in (22), we get: D t (-L)/L = PATu t - Div t - D t- R F (-T) (23) Table 4. Statistics of the correlation between PATu, FCF and A and other variables for 27 US companies of the S&P Average, median, standard deviation (SD), maximum and minimum of the 27 correlation coefficients (one for each company) Statistics of the 27 correlation coefficients Number of companies with Correlations between: Average Median SD MAX min Correl < 0 Correl > 0 D t and FCF t -0,76-0,86 0,26 0,43 -, A/A and FCF/A -0,9-0,97 0,5 0,43 -, A/A and PATu/A 0,4 0,50 0,33 0,94-0, FCF/A and PATu/A -0, -0,3 0,38 0,89-0, interest t and FCF t - -0,2-0,25 0,30 0,74-0, D t /A t and PATu t /A t 0,8 0,23 0,34 0,89-0, Statistics of PATu, FCF and A is the average of the 27 SD of (PATut/At-) (one for each company) Average (Av) Number of companies with average average of the statistics for each company Standard Deviation (SD) SD/Av. median MAX min < 0 > 0 PATu/A FCF/A A/A Correlation between SD of FCF/A and SD of A/A = 0.98 In t-, D t- R F (-T) is known and Div t has already been announced. Consequently, D t and PATu t are positively correlated. If D t was a multiple of A t, then D t should be also positively correlated with PATu t. CH39-7

8 Table 4 tells us that 90 companies present a positive correlation between D t /A t and PATu t /A t, although these correlation coefficients are smaller than the ones between A t and PATu t. 7. Conclusions The value of tax shields (VTS) defines the increase in the company s value as a result of the tax saving obtained by the payment of interest. However, there is no consensus in the existing literature regarding the correct way to compute the VTS. Most authors think of calculating the VTS in terms of the appropriate present value of the tax savings due to interest payments on debt, but Modigliani and Miller (963, MM) propose discounting the tax savings at the risk-free rate (R F ) 0, and Miles and Ezzell (980, 985, ME) propose discounting these tax savings the first year at the cost of debt and the following years at Ku. 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. Many authors (e.g. Taggart (99), Inselbag and Kaufold (997), Booth (2002), Cooper and Nyborg (2006), Oded and Michel (2006), Farber et al (2006)) consider that debt policy may only be framed in terms of maintaining a fixed market-value debt ratio (ME) or a fixed dollar amount of debt (MM). MM and ME are two extreme cases that are not valid for most companies: MM should be used when the company has a preset amount of debt; ME should be used only when debt will be always a multiple of the equity market-value. We develop valuation formulae for companies that maintain a fixed book-value leverage ratio and argue that, for most companies, and especially when calculating residual values, this assumption is more realistic than those of MM and ME. We obtain an intermediate value between those of MM and ME. If a company targets its leverage in market-value terms, it has less value than if it targets the leverage in book-value terms. How could a manager target leverage in market-value terms? However, many authors identify constant leverage with a constant market leverage ratio. Flannery and Rangan (2006) point out that finance theory tends to downplay the importance of book ratios, and we argue that it makes more sense to characterize the debt policy of a company with expected constant leverage ratio as a fixed book-value leverage ratio (instead of as a fixed market-value leverage ratio). The empirical evidence also provides more support for the fixed book-value leverage ratio hypothesis. It means that when managers have a target capital structure, it is usually in book-value terms (as opposed to market-value terms), in large part because this is what credit rating agencies pay attention to. REFERENCES Antoniou, A., Y. Guney and K. Paudyal (2006), The Determinants of Debt Maturity Structure: Evidence from France, Germany and the UK, European Financial Management 2/ 2, pp Arzac, E.R and L.R. Glosten (2005), A Reconsideration of Tax Shield Valuation, European Financial Management /4, pp Blume, M., F. Lim, and A. MacKinlay (998), "The Declining Credit Quality of U.S. Corporate Debt: Myth or Reality", Journal of Finance 53, pp.,389-,43. Booth, L. (2002), Finding Value Where None Exists: Pitfalls in Using Adjusted Present Value, Journal of Applied Corporate Finance 5/, pp Booth, L. (2007), Capital Cash Flows, APV and Valuation, European Financial Management, 3/, forthcoming. Brealey, R.A. and S.C. Myers (2000), Principles of Corporate Finance, 6 th edition, New York: McGraw- Hill. Cooper, I. A. and K. G. Nyborg (2006), The Value of Tax Shields IS Equal to the Present Value of Tax Shields, Journal of Financial Economics 8, pp Myers (974), Luehrman (997), and Damodaran (2006, page 22) propose to discount it at the cost of debt (Kd). Brealey and Myers (2000, page 558) propose using the effective tax shield on interest (a figure lower than the corporate tax rate that takes account of the years in which the firm does not pay taxes), but they admit that we were unable to pin down an exact figure for the effective tax shield on interest. Damodaran (2006, page 25) proposes to deduct the present value of the bankruptcy cost. CH39-8

9 Copeland, T. E., T. Koller and J. Murrin (2000), Valuation: Measuring and Managing the Value of Companies, 3 rd edition, Wiley, New York. Damodaran, A. (2006), Damodaran on Valuation, 2 nd edition, New York: John Wiley and Sons. st edition: 994. Farber, A., R. L. Gillet and A. Szafarz, 2006, A General Formula for the WACC, International Journal of Business /2, pp Fernandez, P. (2004), The Value of Tax Shields Is NOT Equal to the Present Value of Tax Shields, Journal of Financial Economics 73/, pp Fernandez, P. (2007). A More Realistic Valuation: APV and WACC with constant book leverage ratio, Journal of Applied Finance, Fall/Winter, Vol.7 No 2, pp Flannery, M. J. and K. P. Rangan (2006), "Partial Adjustment Toward Target Capital Structures," Journal of Financial Economics 79, pp Graham, J.R. and C.R. Harvey (200), The Theory and Practice of Corporate Finance: Evidence from the Field, Journal of Financial Economics 60/2, pp Grinblatt, M. and S. Titman (2002), Financial Markets and Corporate Strategy, Irwin/Mc-Graw-Hill, 2 nd ed. Burr-Ride, Il. Harris, R.S. and J.J. Pringle (985), Risk adjusted discount rates extensions from the average-risk case, Journal of Financial Research 8, Inselbag, I. And H. Kaufold (997), Two DCF Approaches for Valuing Companies under Alternative Financing Strategies and How to Choose between Them, Journal of Applied Corporate Finance 0, pp Lewellen, W.G. and D.R. Emery (986), Corporate Debt Management and the Value of the Firm, Journal of Financial and Quantitative Analysis 2/4, pp Luehrman, T. (997), Using APV: a Better Tool for Valuing Operations, Harvard Business Review 75, pp Massari, M., F. Roncaglio and L. Zanetti (2007), On the Equivalence Between the APV and the WACC Approach in a Growing Leveraged Firm, European Financial Management, Forthcoming. Miles, J.A. and J.R. Ezzell (980), The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: A Clarification, Journal of Financial and Quantitative Analysis 5, pp Miles, J.A. and J.R. Ezzell (985), Reformulating Tax Shield Valuation: A Note, Journal of Finance 40/5, pp Modigliani, F. and M. Miller (963), Corporate Income Taxes and the Cost of Capital: a Correction, American Economic Review 53, pp Myers, S.C. (974), Interactions of Corporate Financing and Investment Decisions Implications for Capital Budgeting, Journal of Finance 29, pp.-25. Myers, S.C. (984), The Capital Structure Puzzle, Journal of Finance 39/3, pp Oded, J. and A. Michel (2006), Reconciling Valuation Methodologies: The Importance of a Firm s Debt Rebalancing Policy, Boston University, Unpublished paper. Penman, S. H., S.A. Richardson and A.I. Tuna (2005), "The Book-to-Price Effect in Stock Returns: Accounting for Leverage", Unpublished paper. Ruback, R. (995), A Note on Capital Cash Flow Valuation, Harvard Business School, Ruback, R. (2002), Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows, Financial Management 3, pp Standard and Poor s (2006), Corporate Ratings Criteria, downloadable in Stanton, R. and M.S. Seasholes (2005), The assumptions and Math Behind WACC and APV Calculations, Unpublished paper, U.C. Berkeley. Stonehill, A., T. Beekhuisen, R. Wright, L. Remmers, N. Toy, A. Pares, A. Shapiro, D. Egan and T. Bates (973), Determinants of Corporate Financial Structure: A Survey of Practice in Five Countries, Unpublished paper. Taggart, R.A. Jr. (99), Consistent Valuation and Cost of Capital. Expressions with Corporate and Personal Taxes, Financial Management 20, pp Xu, Z. (2007), Do Firms Adjust Toward a Target Leverage Level?, Bank of Canada Working Paper CH39-9

10 Appendix. VTS equations according to the main theories Modigliani and Miller (958) studied the effect of leverage on the firm s value. Their proposition (958, equation 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 on a risk-free debt (T D R F ) at the risk-free rate (R F ). Their first proposition, with taxes, is transformed into Modigliani and Miller (963, page 436, equation 3): [24] E + D = Vu + PV[R F ; DT R F ] = Vu + D T DT is the value of tax shields (VTS) for perpetuity. This result is only correct for perpetuities. As Fernandez (2004 and 2007) demonstrates, discounting the tax savings due to interest payments on a riskfree debt at the risk-free rate provides inconsistent results for growing companies. Myers (974) 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 saving 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 saving arising from the use of debt is the same as the risk of the debt. Therefore, according to Myers (974): [25] VTS = PV [Kd; D T Kd] Luehrman (997) recommends valuing companies using the Adjusted Present Value and calculates the VTS in the same way as Myers. Fernandez (2007) shows that this theory yields consistent results only if the expected debt levels are fixed. D: Market value of the debt. N: Nominal value of the debt When D = N When D N Modigliani-Miller PV[R F ; T D R F ] Myers (974) PV[Kd; T D Kd] PV[Kd; T N r] Miles-Ezzell (980) PV[Ku; T D Kd] (+Ku)/(+Kd) PV[Ku t ; N t- r t T] ( + Ku) / (+ Kd) Fernandez (2007) PV[Ku; DTKu] PV[Ku; DTKu + T(Nr-DKd)] Damodaran (994) PV[Ku; DTKu - D (Kd- R F ) (-T)] PV[Ku; T N r +DT(Ku- R F ) - D(Kd- R F )] Harris-Pringle (985) PV[Ku; T D Kd] PV[Ku; T N r] Miller 0 Practitioners PV[Ku; T D Kd - D(Kd- R F )] PV[Ku; T N r - D(Kd- R F )] With-cost-of-leverage PV[Ku; D (KuT+ R F - Kd)] Miller (977) 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 (977), the value of the firm is independent of its capital structure, that is, [26] VTS = 0. According to Miles and Ezzell (980), 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. For a firm with a fixed debt target [D/(D+E)], they claim that the correct rate for discounting the tax saving due to debt (Kd T D t- ) is Kd for the tax saving during the first year, and Ku for the tax saving during the following years. The expression of Ke is their equation 22: [27] Ke = Ku + D (Ku - Kd) [ + Kd (-T)] / [(+Kd) E] Arzac and Glosten (2005) and Cooper and Nyborg (2006) show that Miles and Ezzell (980) (and their equation [27]) imply that the value of tax shields is : [28] VTS = PV[Ku; T D Kd] (+Ku)/(+Kd). Harris and Pringle (985) calculate the VTS by discounting the tax saving 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). According to them: [29] VTS = PV [Ku; D Kd T] Harris and Pringle (985, 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 Lewellen and Emery (986) also claim that the most logically consistent method is Miles and Ezzell. CH39-0

11 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 Harris and Pringle compared to Miles and Ezzell is its simplicity and straightforward intuitive explanation. Ruback (995, 2002) reaches equations that are identical to those of Harris-Pringle (985). Kaplan and Ruback (995) also calculate the VTS discounting interest tax shields at the discount rate for an all-equity firm. Tham and Vélez-Pareja (200), following an arbitrage argument, also claim that the appropriate discount rate for the tax shield is Ku, the required return to unlevered equity. Damodaran (994, page 3) argues that if all the business risk is borne by the equity, then the equation relating the levered beta ( L ) to the asset beta (u) is: [30] L = u + (D/E) u ( - T). Note that equation [30] is exactly equation [22] assuming that d = 0. 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). However, I think that it is difficult to justify that the debt has no risk (unless the cost of debt is the risk-free rate) and that the return on the debt is uncorrelated with the return on assets of the firm. I rather interpret equation [30] as an attempt to introduce some leverage cost in the valuation: for a given risk of the assets (u), by using equation [30] we obtain a higher L (and consequently a higher Ke and a lower equity value) than with equation [22]. Equation [30] appears in many finance books and is used by some consultants and investment banks. Although Damodaran does not mention what the value of tax shields should be, his equation [30] relating the levered beta to the asset beta implies that the value of tax shields is: [3] VTS = PV[Ku; D T Ku - D (Kd- R F ) (-T)] Another way of calculating the levered beta with respect to the asset beta is the following: [32] L = u (+ D/E). We will call this method the Practitioners method, because consultants and investment banks often use it (one of the many places where it appears is Ruback (995, page 5)). It is obvious that according to this equation, 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 equation [32] is equal to equation [30] eliminating the (-T) term. We interpret equation [32] as an attempt to introduce still higher leverage cost in the valuation: for a given risk of the assets (u), by using equation [32] we obtain a higher L (and consequently a higher Ke and a lower equity value) than with equation [30]. Equation [32] implies that the value of tax shields is: [33] VTS = PV[Ku; D T Kd - D(Kd- R F )]. VTS according to [33] is PV[Ku; D T (Ku- R F )] lower than [3]. Inselbag and Kaufold (997) argue that if the firm targets the dollar values of debt outstanding, the VTS is given by the Myers (974) equation. However, if the firm targets a constant debt/value ratio, the VTS is given by the Miles and Ezzell (980) equation. Copeland, Koller and 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 and Pringle (985) and Myers (974). 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 (997) as additional reading on Adjusted Present Value. According to Fernandez (2007), only three of the theories may be correct: - When the debt level is fixed, Modigliani-Miller or Myers apply, and the tax shields should be discounted at the required return to debt. - If the leverage ratio is fixed at market value, then Miles-Ezzell applies. - If the leverage ratio is fixed at book value, and the appropriate discount rate for the expected increases of debt is Ku, then Fernandez (2007) applies and the VTS is the present value of DTKu (not the interest tax shield) discounted at the unlevered cost of equity (Ku). [34] PV[Ku; D T Ku] With-Costs-Of-Leverage. This theory provides another way of quantifying the VTS: [35] VTS = PV[Ku; D Ku T D (Kd - R F )] One way of interpreting equation [35] is that the leverage costs are proportional to the amount of debt and to the difference between the required return on debt and the risk-free rate. [35] provides a VTS that is PV[Ku; D (Kd - R F )] lower than [34]. CH39-