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1 Managerial Finance Emerald Article: The firm-specific nature of debt tax shields and optimal corporate investment decisions Assaf Eisdorfer, Thomas J. O'Brien Article information: To cite this document: Assaf Eisdorfer, Thomas J. O'Brien, (2012),"The firm-specific nature of debt tax shields and optimal corporate investment decisions", Managerial Finance, Vol. 38 Iss: 6 pp Permanent link to this document: Downloaded on: References: This document contains references to 15 other documents To copy this document: permissions@emeraldinsight.com Access to this document was granted through an Emerald subscription provided by Emerald Author Access For Authors: If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service. Information about how to choose which publication to write for and submission guidelines are available for all. Additional help for authors is available for Emerald subscribers. Please visit for more information. About Emerald With over forty years' experience, Emerald Group Publishing is a leading independent publisher of global research with impact in business, society, public policy and education. In total, Emerald publishes over 275 journals and more than 130 book series, as well as an extensive range of online products and services. Emerald is both COUNTER 3 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download.

2 The current issue and full text archive of this journal is available at MF 38,6 560 Received September 2011 Revised December 2011 Accepted January 2012 The firm-specific nature of debt tax shields and optimal corporate investment decisions Assaf Eisdorfer and Thomas J. O Brien Department of Finance, University of Connecticut, Storrs, Connecticut, USA Abstract Purpose While an operation s unlevered value is objective, the value of the debt tax shield is subjective since it depends on the capital structure policy of the firm that owns the operation. The purpose of this paper is to explore the implications of this subjective nature of debt tax shield value for corporate investment decisions. Design/methodology/approach The study develops a simple theoretical model. Findings The paper shows that even a low probability of selling a project in the future to a firm with a different tax shield value can significantly affect a project s weighted average cost of capital (WACC) and total value. Practical implications Managers should be aware of this issue when making corporate investment decisions. Originality/value This is the first study to address the implication of the subjective nature of debt tax shield value. Keywords Tax shield, Cost of capital, Capital structure, Investments Paper type Conceptual paper Managerial Finance Vol. 38 No. 6, 2012 pp q Emerald Group Publishing Limited DOI / Introduction The weighted average cost of capital (WACC) is widely used to find the value of a corporate investment. As is very well known, discounting a project s expected after-tax cash flow stream using the WACC yields the correct total of the two components of the project s value: (1) the value of the unlevered project; and (2) the value of the tax shields of the corporate debt supported by the project. The good news of the WACC approach is that we conveniently by-pass having to find the two value components separately. And valuing the tax shield component separately can be very complex (Taggart, 1991; Arzac and Glosten, 2005; Cooper and Nyborg, 2006, 2007, 2008; Qi, 2011)[1]. But the bad news is that the WACC method, by combining the two components of a project s value into one number, obscures the significant quality difference between the two components. Given the project s cash flow stream, the unlevered project value component is objective and the same for any firm, but the value of the debt tax shields is firm-specific and thus may be different for different firms[2]. JEL classification G31, G32 The authors would like to thank Ian Cooper, John Knopf, Kjell Nyborg, Howard Qi, Lee Sanning, Piet Sercu, and anonymous referees for helpful comments and discussions.

3 In this paper, we comment on the firm-specific nature of debt tax shields and the implications for corporate investment decisions. We assert that the standard WACC method implicitly assumes either:. the tax benefits of the debt financing by the investing firm will last forever (or until the investing firm is bankrupt); or. the potential future buyers of the project will get the same tax shield value as the original investor and will thus value the project the same as the original investor. The nature of debt tax shields 561 We contend that the investing firm may wish to, or may need to, sell the project later to a buyer for whom the component value of the debt tax shield is different. We show how this prospect affects a project s value. We also show an example of how a firm may include this effect by an adjustment to the WACC, based on the Miles and Ezzell (1980) approach to tax shield value, in order to take advantage of the WACC s desirable feature of valuing a project without having to value the components separately[3]. Our analysis is motivated by the observation that firms often divest a division or an operation within a few years after the acquisition. There are several reasons why a project s potential future buyers might have a different tax shield value than the original investor. The potential buyer could be a firm with higher financial distress costs, and thus the project will support a lower optimal debt level for the buyer than for the original owner. Or, the managers of potential buyers may have other managerial preferences for maintaining a different proportion of debt to value. Note that in reality, many industrial projects are of interest to only a small number of buyers, often in competition with each other. In an efficient market environment it may be reasonable to assume that the investing firm won the project originally because it could have raised more debt than its competitors, resulting in relatively higher debt tax shield. Thus, selling the project in the future would be on average to a competing firm with lower tax shield benefits. Another scenario where the potential buyers will place a lower value on the tax shields than the initial investor may be found in the world of private equity. Consider a private equity firm with a relatively high degree of financial leverage who uses the standard WACC to estimate a value for one of its corporate holdings. If the private equity firm decides to spin-off the holding to the public, the spin-off value will be lower if the publicly-traded company uses less debt and thus has a lower tax shield value. Consistent with this scenario, recent studies provide evidence that private equity firms have significantly higher financial leverage ratios than their public counterparts (Brav, 2009; Axelson et al., 2010). More empirical evidence consistent with our argument is provided by the work of Dittmar (2004) on the capital structures of spin-offs. She finds that a divested subsidiary typically has a substantially lower leverage ratio than the parent firm, but similar to a comparable non-spin-off firm. While the sale of a project to a firm that would have a lower value of the debt tax shields seems like a common scenario, our analysis also applies in general, i.e. to a firm who might sell the project in the future to a firm that would have a higher value of the debt tax shields. Overall, we contend that the possibility of selling a project in the future means that the market will implicitly place a lower value on the project than that estimated by a firm with high debt tax shields, given the firm s WACC;

4 MF 38,6 562 similarly the market will implicitly place a higher value on the project than that estimated by a firm with low debt tax shields, given that firm s WACC. 2. The scenario Our analysis is based on the following scenario: consider a homogeneous industry where all firms have the same unlevered cost of capital, k u. Firm A is analyzing an investment project that would grow its business. The project might be the purchase of a smaller firm in the industry or the organic investment into a new plant and operation. Firm B is a representative other firm in the industry. Firm B is analyzing the same project. Only one firm can make the investment. We assume that for any firm, the project would require the same immediate investment outlay and add the same incremental expected after-tax operating cash flow. Thus, the project would add the same unlevered value to any firm. We also assume that the firms face the same corporate tax rate. Each firm would finance the project with a combination of debt and equity, but Firm A would use a higher ratio of debt to value than Firm B. Letting denote the ratio of debt to value, A. B. The reason why Firm A uses a higher proportion of debt is not really important to the analysis, but perhaps Firm A s size may allow easier access to debt markets and the use of a higher proportion of debt before the costs of financial distress start to outweigh the benefits of the debt tax shields. To keep the analysis somewhat simple, we further assume that the project s expected after-tax operating cash flow stream is a level perpetuity of CF per year[4]. Thus, the unlevered value of the project, to any investing firm, is V u ¼ CF/k u. 3. Review of the standard approach Let us now briefly review the well-known standard WACC approach to capital budgeting. The overall value of the project, V, is the present value of the project s expected after-tax operating cash flows, using the WACC (denoted k) as the discount rate. Thus, the overall value of the project, V ¼ CF/k, is firm-specific, because a firm s WACC depends on the firm s. Since the corporate tax rate is the same for both Firms A and B, and since A. B, it follows directly that the WACC for Firm A is less than the WACC for Firm B. That is, k A, k B. Note that this condition holds even if Firm A s cost of debt is higher than Firm B s. Since k A, k B, the overall value of the project is higher for Firm A than for Firm B. That is, V A. V B. The project s value is the sum of the unlevered project value and the present value of the future tax shields provided by the interest on the debt used to finance the project. Even though the WACC approach implies that the specific amounts of the two individual components are not calculated, it is useful to think about the components conceptually in our analysis. We denote the present value of the tax shields as PVTS. Since A. B, it follows that Firm A will have a higher PVTS than Firm B, because Firm A uses a higher proportion of debt financing for the project than Firm B. That is, PVTS A. PVTS B. Regardless of the correct value for PVTS, discounting CF using the WACC always gives the correct V. Letting I denote the project s immediate investment outlay, the conventional NPV of the project for Firm A is NPV A ¼ V A 2 I. Likewise, the conventional NPV of the project for Firm B is NPV B ¼ V B 2 I. Since NPV A. NPV B in our scenario, Firm A will accept the project in some cases where Firm B will not. If the two firms are competing

5 for the project, Firm A will win the competition by making a bid of I or slightly more than V B, whichever is higher. 4. Firm-specific tax shield value and investment decisions Now we come to our extension of the standard analysis for the possibility that the accepting firm may sell the project in the future. Assume that Firm A may decide at some time in the future that it is not able to, or does not wish to, maintain the project any more, and thus will want to sell it to another firm. We argue that in this case the original decision by Firm A on accepting the project should take into consideration the possibility of selling the project later to a firm for which the debt tax shields are lower. For this analysis, assume that Firm B is the representative potential buyer. We argue that Firm A should make the original accept/reject decision incorporating the maximum that Firm B would be willing to pay for the project, V B, and the probability, p, that Firm A sells the project in the future. Thus, we say that Firm A should make the original accept/reject decision based on the project s adjusted value, which we denote as V * A. To see the basic idea, assume for now a simplistic case where the only possible time to sell the project is immediately after the purchase. That is, if the project is not sold at this time, there is no chance of selling again at any time. Given the probability of selling p, V * A would be pv B þ (1 2 p)v A. In this case, it is fairly easy to see that if V B is less than I, there is some chance that the adjusted value of the project for Firm A would also be less than I. For a numerical example of this case, assume I ¼ 99, CF ¼ 9.5, and k u ¼ Assume PVTS A ¼ 5 and PVTS B ¼ 0, so V A ¼ 100 and V B ¼ 95; and k A ¼ and k B ¼ k u ¼ If p ¼ 0.40, the adjusted value of the project for Firm A is V * A ¼ 0.40(95) þ 0.60(100) ¼ 98. When considering the possibility of having to sell the project immediately after adopting it, Firm A should reject the project because the NPV is negative: ¼ 21. The nature of debt tax shields The general case: annual probability of project sale More realistically, assume that Firm A can sell the project at any time in the future. Specifically, at the end of each year Firm A will sell the project to Firm B with probability p, and retain the project with probability (1 2 p). We assume that this probability is independent of the level of Firm A s debt tax shields or the value that the project may be sold for[5]. If Firm A retains the project, it receives another year of after-tax cash flow and tax shield. If Firm A does not sell the project, it would receive the cash flow and tax shield for sure for every time period. Even if a spin-off occurs, the first cash flow and tax shield will be received with 100 per cent probability. What is the adjusted value of the project to Firm A today? The future payoffs to Firm A can be described in the following way. At the end of the first year, Firm A receives the project s annual cash flow and tax shield with probability 1, plus the project s selling price with probability p. At the end of the second year, Firm A receives the annual cash flow and tax shield with probability (1 2 p), plus the project s selling price with probability p (1 2 p). At the end of the third year, Firm A receives the cash flow and tax shield with probability (1 2 p) 2, plus the project s selling price with probability p (1 2 p) 2, and so forth. Shaffer (2006) was our guide in this approach. At time 0, the amounts of cash flow, tax shield, and selling price that Firm A will receive are unknown. The cash flow is expected to be CF and the tax

6 MF 38,6 564 shield is expected to be TS A. The expected ex-cash flow selling price for the project at any future time is V B. Thus, we can view Firm A s stream of future payments in terms of three components. The first component is based on the project s expected unlevered cash flow stream. At time 0, Firm A expects this component to be CF at the end of year 1, CF (1 2 p) at the end of year 2, CF (1 2 p) 2 at the end of year 3, and so forth. The second component is based on Firm A s expected stream of annual future tax shields. At time 0, given Firm A s initial level of debt financing of the project, Firm A expects a tax shield stream of TS A at the end of year 1, TS A (1 2 p) at the end of year 2, TS A (1 2 p) 2 at the end of year 3, and so forth. The third component is the proceeds from the sale of the project to Firm B, which at time 0 Firm A expects to be pv B at the end of year 1, p (1 2 p)v B, at the end of year 2, p (1 2 p) 2 V B at the end of year 3, and so forth. It is tempting to think we can value the first and second components simultaneously by capitalizing the unlevered cash flow component using Firm A s WACC, as we do in a standard analysis. But, as we will explain in more detail later, this approach is not correct here. So in order to value the cash flow and tax shield component streams correctly in our model, we need to apply a specific model of tax shield valuation. As Cooper and Nyborg (2008) show, different tax shield valuation models follow from different assumptions. A relatively simple and reasonable model that we use here is the Harris and Pringle (1985) extension of the Miles and Ezzell (1980) model, in which a project s future tax shields have the same risk class as the unlevered project. Of course, the unlevered cost of capital, k u, is the correct discount rate for unlevered cash flows, regardless of the model of tax shield value. In our valuation problem where Firm A sells the project in any future year with probability p, and the cash flow and tax shield are either fully realized or 0, the correct discount rate to apply to the unlevered cash flow component steam is k u, and the present value of the unlevered cash flow component stream at time 0 is CF/(k u þ p)[6]. Since the tax shields are assumed to have the same risk class as the unlevered cash flows, Firm A s expected tax shield stream is also discounted at k u. Thus, the present value of Firm A s expected tax shield component stream at time 0 is TS A /(k u þ p). As V B is a potential payment to Firm A when selling the project, for the same considerations discussed above the discount rate for this component should be k u as well. The present value of this component at time 0 is thus pv B /(k u þ p)[7]. Thus, the adjusted value of the project to Firm A is: V * A ¼ CF=ðk u þ pþþts A =ðk u þ pþþpv B =ðk u þ pþ ð1þ The same analysis also applies in general, i.e. to a firm who might sell the project in the future to a firm that would have a higher value of the debt tax shields. In fact, Firm B could be the original investor in the project and should take into consideration the possible future sale to the higher tax shield firm, Firm A. Without considering this possibility, a firm with low tax shield value might reject a project that should be accepted[8]. 4.2 Alternative formulation of V * A In the previous section we derive the project s adjusted value to Firm A, V * A, using the present value of all potential payments that the firm will receive in the future (i.e. cash flow from the project, debt tax shield, and proceeds from selling the project).

7 We now show that V * A can be derived also by calculating the potential tax shield loss of Firm A due to the possible future sale to Firm B. Assuming that if not selling the project, the constant annual tax shield of Firm A will be paid forever, thus the perpetual annual tax shield must be equal to PVTS A k u. Similarly, the perpetual annual tax shield of Firm B equals PVTS B k u. Hence, if Firm A will sell the project to Firm B at year t, it will lose a perpetuity that starts at year t þ 1 and equals the difference between the annual tax shields: (PVTS A 2 PVTS B ), discounted by k u. As the probability of selling the project at time t is p(1 2 p) t2 1, the present value of this tax shield loss at time 0 is p(pvts A 2 PVTS B )/(k u þ p). The project s adjusted value to Firm A is given therefore by subtracting the present value of the potential loss from the value of the project to Firm A under the standard APV analysis (i.e. the value of the unlevered firm plus the present value of the tax shields): The nature of debt tax shields 565 V * A ¼ V u þ PVTS A 2 pðpvts A 2 PVTS B Þ=ðk u þ pþ ð2þ We show in Appendix 1 that the expression of V * A in equation (2) is equal to the one in equation (1). 4.3 Numerical example Assume again I ¼ 99, CF ¼ 9.5, k u ¼ 0.10, PVTS A ¼ 5, PVTS B ¼ 0, so TS A ¼ 0.50, V A ¼ 100, V B ¼ 95, k A ¼ and k B ¼ k u ¼ Assume p ¼ Using equation (1), the adjusted value of the project for Firm A is V * A ¼ 9.5/(0.10 þ 0.05) þ 0.50/(0.10 þ 0.05) þ 0.05(95)/(0.10 þ 0.05) ¼ þ 3.33 þ ¼ When considering the possibility of selling the project in any future year with probability of 5 per cent, Firm A should reject the project because the NPV is negative: ¼ For the same parameter values, Figure 1 shows the adjusted value, V * A, as a function of the probability of sale, p. Most of the potential reduction in value due to possible selling occurs fairly quickly for relatively low probabilities. This result emphasizes the relevance of our argument; even a very low probability of not maintaining the project in the future could have a significant effect on the real value of the project, and thus should be considered in the investment decision. Note that in our example, discounting Firm A s expected unlevered cash flow component stream at Firm A s WACC would result in the incorrect value of 9.5/(0.095 þ 0.05) þ 0.05(95)/(0.10 þ 0.05) ¼ þ ¼ As we said above, this tempting approach does not give the correct combined value of cash flow and tax shield component streams as it does in the standard analysis. Now we explain this point. The reason the WACC does not work is that it is based on a constant debt-to-value ratio of A, based on the value of Firm A under the standard analysis, V A. In our analysis with the possibility of future sale of the project, Firm A s assumed debt level is based on A and V A, but A is not the ratio of debt to value of the combined cash flow and tax shield component stream, nor is A constant. Thus, the conditions that enable the WACC to work correctly in a standard analysis do not hold in the valuation here. 5. Adjusted WACC In a standard textbook NPV analysis, the existing firm s WACC is used to discount cash flows for projects in the same risk class of the firm s existing assets,

8 MF 38,6 100 V A I 97 Figure 1. The adjusted value of the project to Firm A 96 V A * Annual probability of selling the project Note: The figure shows the adjusted value of the project to Firm A,V * A = CF/ (k u + p) + TS A /(k u + p) + pv B /(k u + p) (equation (1)), as a function of p, the annual probability of selling the project to Firm B, using the parameter values: I = 99, CF = 9.5, k u = 0.10, PVTS A = 5, PVTS B = 0, so TS A = 0.50, V A = 100, and V B = 95 presuming the project s debt-to-value ratio will be equal to the existing firm s debt-to-value ratio. But in our setting where the possibility of selling affects the project s value to Firm A, the correct cost of capital for capitalizing the project s CF is not Firm A s WACC for existing assets. In this section, we present a formula that managers may use to find a project s adjusted WACC, k *. The formula we derive for k * is based on inputs that Firm A s managers are presumed to know:. Firm A s WACC for existing assets, k A ;. Firm B s WACC for existing assets, k B ;. the probability of project sale, p; and. the project s unlevered cost of capital, k u. The formula is based on Harris and Pringle (1985), who apply the Miles and Ezzell (1980) approach to tax shield value. In that model, managers may find k u using a simple formula based on k A, the tax rate, the cost of debt, and A. Also see Cooper and Nyborg (2007))[9]. Our formula for Firm A s adjusted WACC for the project is: k * A ¼ k Aðk u þ pþ ½k u þ pðk A =k B ÞŠ ð3þ Equation (3) is derived from equation (1) as follows. Rearranging equation (1), we have that k u þ p ¼ (CF þ TS A þ pv B )/V * A. Using the definition of the WACC, we may use the relation V * A /V A ¼ k A /K * A to get that k u þ p ¼ (K * A /k A)(CF þ TS A þ pv B )/V A.

9 Rearranging the last expression, using (CF þ TS A )/V A ¼ k u and V B /V A ¼ k A /k B,we get equation (3). In our numerical example, equation (3) tells us that the project WACC for Firm A is 0.095(0.10 þ 0.05)/[0.10 þ 0.05(0.095/0.10)] ¼ When the CF of 9.5 is capitalized using Firm A s adjusted WACC of , the result is the project s adjusted value for Firm A: 9.5/ ¼ Using the same set of parameter values, Figure 2 shows the value of K * A as a function of the probability of selling the project. Consistent with the pattern in Figure 1, the cost of capital is increasing very quickly to a level closer to k u, the unlevered cost of capital of the project. The difference between K * A and k A thus implies that ignoring possible future project sales can lead firms to accepting a project that should be rejected. The nature of debt tax shields Conclusions The traditional WACC method yields a value for a corporate investment that includes the present value of debt tax shields. Since this component of the value is not broken out in the WACC method, its firm-specific nature is not stressed in textbooks. The objective of our study is to highlight the firm-specific nature of debt tax shield values and demonstrate the potential implications for corporate investment decision-making. We point out that the traditional WACC analysis implicitly assumes a project s tax shield value for a firm that never sells the project. We show that this approach can produce poor investment decisions when we take into account the realistic scenario that the investing firm may later sell the project to another firm. We show the how this possibility can affect a project s value, given the possibility of selling the project to a firm that has a different tax shield value from the debt financing of the project. Both project value and project WACC may be significantly affected by the expected tax shield of the potential buyer, even if the probability of selling the project is very low k u k A * k A Annual probability of selling the project Note: The figure shows the project s adjusted cost of capital for Firm A, K * A = k A (k u + p)/[k u + p(k A /k B )] (equation 3), as a function of p, the annual probability of selling the project to Firm B, using the parameter values: I = 99, CF = 9.5, k u = 0.10, PVTS A = 5, PVTS B = 0, so TS A = 0.50, V A = 100, and V B = 95, and so k A = 0.095, and k B = 0.10 Figure 2. The adjusted WACC for the project to Firm A

10 MF 38,6 568 Our analysis suggests that an efficient market will take into account the future possible project sale when valuing a company. The market will implicitly place a lower value on a project than the value estimated by a firm with a high debt tax shield value, given the firm s WACC. Similarly the market will implicitly place a higher value on a project than the value estimated by a firm with a low debt tax shield value, given that firm s WACC. Perhaps this idea represents an opportunity for an empirical research. Notes 1. Graham (2000) estimates the average tax shield value to be 9.5 per cent of total firm value for US companies. See Cooper and Nyborg (2007) for an extensive review of the empirical literature on the estimation of tax shield values. 2. We might even sometimes overlook that tax shield value is part of the total value, especially since tax shield values are not shown on reported financial statements. For example, textbooks tell us that a firm s enterprise value is the value of underlying business assets unencumbered by debt and separate from any cash and marketable securities, and calculate enterprise value as the market value of equity plus debt minus cash (Berk et al., 2009). But this computation is actually for the total of (1) the unencumbered value of the underlying business, and (2) the value of the debt tax shields. 3. Note that the purpose of this study is not to improve tax shield valuation by understanding all of its risks (e.g. profitability shocks, changes in the tax code), but rather to highlight the firm-specific nature of debt tax shields and its implications for value. 4. In Appendix 2 we show that the main results extend to the case of constant growth of expected cash flows, and in fact strengthen the case we are making. 5. We can show that incorporating stochastic k u and p into the analysis does not change the implications of the results. 6. Adding p to k u in the denominator comes directly from the summation of an infinite geometric series. 7. We are grateful to Howard Qi for helping us to better understand the issues in this approach to valuation. 8. The same calculations apply for general salvage value V B : if the project with some hazard rate is simply liquidated piecemeal or in some fashion comes to a premature end with closing cash flow V B, equation (1) is appropriate. We are grateful to a referee for this point. 9. Ruback (2002) contends that a drawback to using the WACC is that value must be estimated simultaneously in order to determine the value weights. In principle, this argument would apply here, but our formula in equation (3) gets around this issue. References Arzac, E.R. and Glosten, L.R. (2005), A reconsideration of tax shield valuation, European Financial Management, Vol. 11, pp Axelson, U., Jenkinson, T., Strömberg, P. and Weisbach, M.S. (2010), Borrow cheap, buy high? The determinants of leverage and pricing in buyouts, working paper, London School of Economics, London. Berk, J., DeMarzo, P. and Harford, J. (2009), Fundamentals of Corporate Finance, Prentice-Hall, Englewood Cliffs, NJ. Brav, O. (2009), Access to capital, capital structure, and the funding of the firm, Journal of Finance, Vol. 64, pp

11 Cooper, I. and Nyborg, K. (2006), The value of tax shields IS equal to the present value of tax shields, Journal of Financial Economics, Vol. 81, pp Cooper, I. and Nyborg, K. (2007), Valuing the debt tax shield, Journal of Applied Corporate Finance, Vol. 19, Spring, pp Cooper, I. and Nyborg, K. (2008), Tax-adjusted discount rates with investor taxes and risky debt, Financial Management, Vol. 37, pp Dittmar, A. (2004), Capital structure in corporate spin-offs, Journal of Business, Vol. 77, pp Graham, J. (2000), How big are the tax benefits of debt?, Journal of Finance, Vol. 55, pp Harris, R.S. and Pringle, J.J. (1985), Risk-adjusted discount rates extensions form the average-risk case, Journal of Financial Research, Vol. 8, pp Miles, J.A. and Ezzell, J.R. (1980), The weighted average cost of capital, perfect capital markets and project life: a clarification, Journal of Financial and Quantitative Analysis, Vol. 15, pp Qi, H. (2011), Value and capacity of tax shields: an analysis of the slicing approach, Journal of Banking & Finance, Vol. 35, pp Ruback, R.S. (2002), Capital cash flows: a simple approach to valuing risky cash flows, Financial Management, Vol. 31, pp Shaffer, S. (2006), Corporate failure and equity valuation, Financial Analysts Journal, Vol. 62, January-February, pp Taggart, R.A. (1991), Consistent valuation and cost of capital expressions with corporate and personal taxes, Financial Management, Vol. 20, pp The nature of debt tax shields 569 Appendix 1 We show that V * A derived in equation (2), V* A ¼ V u þ PVTS A 2 p(pvts A -PVTS B )/(k u þ p), is equal to the V * A derived in equation (1), V* A ¼ CF/(k u þ p) þ TS A /(k u þ p) þ pv B /(k u þ p). As V u ¼ V B 2 PVTS B, the value of V * A in equation (2) can be expressed as follows: V * A ¼ V B 2 PVTS B þ PVTS A 2 pðpvts A 2 PVTS B Þ=ðk u þ pþ ¼ V B þ k u ðpvts A 2 PVTS B Þ=ðk u þ pþ As TS A ¼ PVTS A k u and TS B ¼ PVTS B k u : V * A ¼ V B þ TS A =ðk u þ pþ 2 TS B =ðk u þ pþ ¼ CF=ðk u þ pþþts A =ðk u þ pþþ½v B 2 CF=ðk u þ pþ 2 TS B =ðk u þ pþš And as V B k u ¼ CF þ TS B : V * A ¼ CF=ðk u þ pþþts A =ðk u þ pþþ½v B 2 V B k u =ðk u þ pþš ¼ CF=ðk u þ pþþts A =ðk u þ pþþpv B =ðk u þ pþ Thus, equations (1) and (2) are identical. Appendix 2 We derive the project s adjusted value and WACC for Firm A when the project s cash flow is expected to grow at a constant rate, g. The three components of Firm A s future payments are adjusted as follows. The project s expected unlevered cash flow stream will be CF, CF (1 2 p)(1 þ g), CF (1 2 p) 2 (1 þ g) 2, and so forth. Because the debt of Firm A should grow at the same rate as that of the cash flow in order to keep the same leverage ratio, the annual tax

12 MF 38,6 570 shield will grow at the same rate as well. That is, the expected stream of tax shields will be TS A, TS A (1 2 p)(1 þ g), TS A (1 2 p) 2 (1 þ g) 2, and so forth. The expected proceeds from the sale of the project to Firm B will also grow at a rate g; this is because the value of the project for Firm B, V B, increases proportionally with the annual project s cash flow and tax shield, as both grow at g. Also, since V B is the value of the project for Firm B at time 0, the expected proceeds from selling the project to Firm B at time 1 is V B (1 þ g). Hence, the stream of the expected proceeds from the sale of the project is pv B (1 þ g), p (1 2 p)v B (1 þ g) 2, p (1 2 p) 2 V B (1 þ g) 3, and so forth. Using the unlevered cost of capital, k u, to discount all components yields the project s adjusted value for Firm A: V * A ¼½CF þ TS A þ pv B ð1 þ gþš=ðk u þ p þ pg 2 gþ To derive Firm A s adjusted WACC for the project, rearrange equation (A1) as follows: k u þ p þ pg 2 g ¼ [CF þ TS A þ pv B (1 þ g)]/v * A. Using the definition of the WACC, the relation V * A /V A ¼ (k A 2 g)/(k * A 2 g) yields: k u þ p þ pg 2 g ¼ [(K * A 2 g)/(k A 2 g)][cf þ TS A þ pv B (1 þ g)]/v A. Rearranging the last expression, and using (CF þ TS A )/V A ¼ k u 2 g and V B /V A ¼ (k A 2 g)/(k B 2 g), we get the adjusted WACC of the project for Firm A: k * A ¼ðk A 2 gþðk u þ p þ pg 2 gþ=½k u 2 g þ pð1 þ gþððk A 2 gþ=ðk B 2 gþþš þ g Note that equations (1) and (3) are specific cases of equations (A1) and (A2) where g ¼ 0. In fact, the adjustment to the value of the project for Firm A is increasing with the growth rate. That is, measuring the adjustment to the project value by the ratio V * A /V A (the bigger the adjustment, the lower the ratio), it can be shown that the first derivative of this ratio with respect to g is negative. To illustrate the effect of the growth rate on the adjustment to the project value, consider the numerical example in Section 4.3. In the basic setup (where g ¼ 0), V A ¼ 100 and V * A ¼ 98.33, i.e. the adjustment ratio is For g ¼ 0.02, however, V A ¼ 125 and V * A ¼ 122.6; thus the adjustment ratio is And for g ¼ 0.05, V A ¼ 200 and V * A ¼ 194.9, resulting in an adjustment ratio of Corresponding author Thomas J. O Brien can be contacted at: Thomas.obrien@uconn.edu ða1þ ða2þ To purchase reprints of this article please reprints@emeraldinsight.com Or visit our web site for further details: