Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt

This paper derives a tax-adjusted discount rate formula with a constant proportion leverage policy, investor taxes, and risky debt. The result depends on an assumption about the treatment of taxlosses in default. We identify the assumption that justiftes the textbook approach of discounting interest tax shields at the cost of debt. We contrast this with an alternative assumption that leads to the Sick (1990) result that these should be discounted at the riskless rate. These two approachesrepresent polar cases. Each generates its results by using a different simplifying assumption, and we explain what determines the correct treatment in practice. We also discuss implementation of the valuation procedure using the capital asset pricing model. DOI: https://doi.org/10.1111/j.1755-053X.2008.00016.x Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-61726 Published Version Originally published at: Cooper, Ian; Nyborg, Kjell G (2008). Tax-adjusted discount rates with investor taxes and risky debt. Financial Management, 37(2):365-379. DOI: https://doi.org/10.1111/j.1755-053X.2008.00016.x Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt Ian A. Cooper and Kjell G. Nyborg* This paper derives a tax-adjusted discount rate formula with a constant proportion leverage policy, investor taxes, and risky debt. The result depends on an assumption about the treatment of tax losses in default. We identify the assumption that justiftes the textbook approach of discounting interest tax shields at the cost of debt. We contrast this with an alternative assumption that leads to the Sick (1990) result that these should be discounted at the riskless rate. These two approaches represent polar cases. Each generates its results by using a different simplifying assumption, and we explain what determines the correct treatment in practice. We also discuss implementation of the valuation procedure using the capital asset pricing model. The treatment of the tax saving from debt in company valuation has recently been of renewed interest (Fernandez, 2004; Cooper and Nyborg, 2006). However, there is still no standard approach to calculating tax-adjusted discount rates when debt is risky. There are two factors that complicate the analysis: 1) the yield spread on corporate debt and 2) investor taxes. The yield spread can be a significant part of the cost of capital for some firms, especially with the high leverage and low equity risk premia that are often used. Investor taxes can also have a significant impact. Evidence from the United States prior to the tax reforms of 2003 is inconclusive about the degree to which investor taxes offset the corporate tax advantage to borrowing. Fama and French (1998) present empirical evidence consistent with a large impact of investor taxes, whereas Kemsley and Nissim (2002) find little efFect. ' However, the recent changes to the US Tax Code have increased the impact of investor taxes on the value of leveraged firms. Also, countries with imputation taxes have an efFect of investor taxes built directly into their tax systems (Rajan and Zingales, 1995). There are several alternative approaches to these issues. Assuming a constant leverage ratio. Miles and Ezzell (1980) derive a well-known formula for tax-adjusted discount rates. The MilesEzzell ("ME") formula allows for risky debt, but not investor taxes. Taggart (1991) derives a variation of it that allows for investor taxes but assumes that corporate debt is riskless. Sick (1990) develops a valuation formula that takes into account both investor taxes and the yield spread on debt. He contrasts this with a version of the ME formula given in Brealey and Myers (2003), which includes an adjustment for investor taxes. According to Sick, the formula given by Brealey and Myers (2003) differs from his " . . . by the incorrect treatment of risky debt, as well as the failure to recognize that tax shields should be discounted at a cost of equity..." (Sick, 1990, We are grateful to an anonymous referee. Huang Wei Bin. and Bill Christie (the editor) for helpful suggestions. All remaining errors are ours. 'Ian A. Cooper is Professor at the London Business School in London, UK. Kjell G. Nyborg is DnBNOR Professor of Finance at the Norwegian School of Economics in Bergen, Norway and a research fellow ofCEPR. 'Graham (2000) finds that the net tax advantage to debt is less than the full corporate tax rate because of nondebt tax shields. The valuation consequences of this are more complicated as it implies that the tax rate that should be used in valuation is state and time dependent. Financial Management • Summer 2008 • pages 365 379 366 Financial Management « Summer 2008 p. 1441). The Sick approach is also different from the standard ME formula without investor taxes. Sick's formula differs from theirs because he discounts tax shields at a different rate. These different approaches can lead to economically significantly different discount rates and values. Understanding when the various formulas apply is, therefore, important. In this paper, we first clarify the source of the difference between the approaches of Sick (1990) and ME. We describe the underlying assumptions that can be used to justify their different methods. Which approach is appropriate depends upon the tax position of insolvent firms and the tax treatment of debt write-downs. We then use the assumption that is consistent with the ME approach to derive a formula for tax-adjusted discount rates that includes investor taxes and risky debt. The formula is the same as Taggart's (1991) when debt is risk free. It is similar, but not identical, to Brealey and Myers' (2003) formula. It turns out that Brealey and Myers' (2003) intuitive formula is not exactly correct as it does not include all the effects of investor taxes. Finally, we provide consistent expressions for the asset beta and discuss implementation using the capital asset pricing model I. The Tax-Adjusted Discount Rate We operate under the Miles and Ezzell (1980) leverage assumption that leverage is maintained at a constant proportion of the market value of the firm. This is the most realistic simple leverage policy, and also the one that is consistent with the use of the weighted average cost of capital {WACC). The ME formula applies to any profile of cash flows as long as the company maintains constant market value leverage. It provides a relationship between the leveraged discount rate, Ri, and the unleveraged rate, Rij. We analyze a firm with expected pre-tax cash flows C,, at dates t= 1 , . . . , r . Between these dates, leverage remains fixed. After each cash flow, leverage is reset to a constant proportion, Z,, of the leveraged value of the firm. The two rates, Ru and RL, are defined implicitly as the discount rates that give the correct unleveraged and leveraged values when the after-tax operating cash flows are discounted:^ t=\,...,T (1) i=r+l r Vu = Y. <̂ '(1 ^c)/(l + RLJ Í = 1,..., r, (2) where Ft/, is the unleveraged value of the firm and Vu its leveraged value. We assume that the representative investor has tax rates of Tpo on interest income and TPE on the total of equity income and capital gains. The use of a single rate for equity returns is a simplification, but our focus here is not the details of the investor tax system. Analysis of the implications of differential investor income and capital gains rates can be found in Lewellen and Lewellen (2005). With our assumptions, the increase in the after-tax cash flow to the representative investor resulting from an incremental dollar of corporate interest is: (1 Tc){\ TPE). (3) ^More complex capital structure issues in the presence of investor taxes are discussed in Emery and Gehr (1988). 'We assume that tax is levied on the operating cash flows. A more complex treatment does not alter the results. Cooper & Nyborg • Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt 367 This result is standard. We define the related variable, T*, by: T* = Ts/i\ Tpo). (4) Thus: (i-rc)(.-7-„) 1 Following Taggart (1991), we define the required return on riskless equity as The first equality results from setting the after-investor-tax returns on riskless debt and riskless equity equal to each other. The second equality follows from Equation (5). Note that if Tpo and TpE are equal, then RFE = RFThe equation that relates Ä y and/?i is one of the most important in valuation. It is used whenever discount rates are adjusted to reflect a different amount of leverage. Even when adjusted present value methods are used to take into account the tax saving from debt, this relationship may first be used to derive the unlevered required return. The original ME formula that relates Ru and RL was derived with an informal treatment of risky debt and no investor taxes. It is dependent upon the "cost of debt" where this could be interpreted either as the yield or the expected return. For clarity, we define the yield on the debt as YD, and the expected return on the debt as RD.^ The difference is the effect of expected default.^ Miles and Ezzell (1980) present the relationship between Ry and Ri as „ „ LYDTC{\+RU) ^ = ^ ( 7 ) where their "cost of debt" has been interpreted here as its yield. Brealey and Myers (2003) generalize this to the case of investor taxes by using T* rather than TQ. Sick (1990) derives a different relationship:* RL R U r . (9) I -IKFE The key difference, as noted by Sick, is that his result in Equation (9) involves the use of the riskless equity rate rather than the cost of debt that is in Equation (8). If the investor tax rate is set to zero, we see that Sick's formula also differs from ME's in Equation (7). ••in particular, Y,) is the coupon on debt issued at par. 'See Cooper and Davydenko (2007) for a more extensive discussion of this point. 'The notation in Sick (1990) is slightly different, but the result can be derived by simple substitution. Equation (9) is also the formula derived by Taggart ( 1991 ) when debt is riskless. 368 Fitiaticial Management » Summer 2008 II. The Assumptions Underlying the Different Approaches In this section, we show that the difference between the Sick (1990) and ME formulas concerns an assumption about the tax treatment of insolvent firms. Sick assumes that an insolvent firm will make a tax payment equal to the gain from writing off its debt multiplied by the tax rate. If the write-off is total, the entire principal ofthe debt will be taxed when the firm defaults. As we show below, the assumption implicit in the ME approach is that no such payment will be made. The issue in choosing between the two approaches is, therefore, whether insolvent firms can be expected to make such tax payments. At the end of this section, we discuss the factors that affect this. We show the impact ofthe different assumptions using a one-period two-state model. To focus on the fundamental difference between the two approaches, in this illustration there are no investor taxes. Later we derive the results including investor taxes. Table I demonstrates the valuation of the debt tax shield under two alternative assumptions about the tax savings from interest. The valuation is by no-arbitrage, using a riskless bond and the risky bond to span the tax saving. Panel A shows the prices and payoffs ofthe riskless and risky bonds. The risky bond either makes its promised payment ofl + YD if it does not default, or defaults completely. Panel B indicates the valuation by no-arbitrage of the tax savings from interest assumed by Sick (1990). The tax impact of borrowing is TcYp per dollar ofthe face value ofthe bond if the company is solvent, but the company pays Tc as a consequence ofthe debt if it defaults. The latter payment is caused by the gain from writing oif the bond. To replicate the tax consequences ofthe debt, the replicating portfolio consists of an amount Tc ofthe risky bond and riskless borrowing of Tc/(1 + RF). This gives the same payoff as the incremental tax effect ofthe debt. The value ofthe tax saving, PVTSs, is simply the value of this replicating portfolio: PVTSs = TcRp/il + RF). (10) Panel C shows the no-arbitrage valuation ofthe ME tax savings from interest. This differs from the Sick (1990) tax savings in that there is no tax payment when the company is in default, as the company is assumed to pay no tax in this state. The replicating portfolio consists of TCYD/{\ -\YD) invested in the risky bond. The resulting value ofthe tax shield from debt is simply the value of this investment: PVTSME ^ TcYo/il + YD). (11) Although Table I uses a single-period model, this valuation procedure is also correct in the multi-period setting we use in the remainder of this paper. As seen above, both polar approaches generate simple results at the expense of simplifying assumptions. The correct treatment in practice will depend on which is closer to the actual tax impact of debt for insolvent companies. An important factor here is how the tax code treats debt write-downs. As discussed by Miller (1991), in the United States, a cancellation of indebtedness (COD) gives rise to a tax liability (IRC Section 61(a)(12)). This is also what is assumed by Sick (1990), but not by the ME approach. However, the tax code also grants an exception from tax on COD if the firm is insolvent in the sense that liabilities exceed assets (see Miller, 1991; United States Tax Court, 2006).' Thus, tax on COD is an issue only for firms seeking debt forgiveness 'Firms for which tax on COD is an issue can avoid the tax by reorganizing in Chapter 11 (Miller, 1991). The evidence suggests though that tax on COD is not an issue for many firms that reorganize in Chapter 11. For example, Betker ( 1995) finds that in none ofthe 41 pre-packaged Chapter 1 l's in his sample would the firm actually incur tax on COD, and Gilson (1997) provides some evidence that potential tax on COD does not affect the amount of debt reductions incurred by reorganizing distressed firms. Cooper & Nyborg » Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt 369 Table I. Illustration of the Difference in Tax Shield Valuation with Alternative Approaches to the Tax Treatment of Insolvent Firms Time 0 Value Time 1 Cash Flow If Solvent If Insolvent Panel A. Primitive Assets Riskless asset 1 \ -\RF \ + RF Bond 1 \-\-YD • 0 Panel B. No-Arbitrage Valuation of Tax Saving with Tax Payment in Insolvency (Sick, 1990) Tax saving PVTSs Te YD -Te Replicating portfolio Te Tel{\ -h RF) Te (1 -IYD) Te -Te Panel C. No-Arbitrage Valuation of Tax Saving with No Tax Payment in Insolvency (Miles and Ezzell, 1980) Tax saving PVTSME/BM TeYp 0 Replicating portfolio 7'C>'D/(1 + >'D) TeYD 0 without being legally insolvent. Both the Sick and ME approaches implicitly assume that debt write-downs occur only in the case of insolvency. Consequently, Sick's approach is not consistent with US tax law, whereas the ME approach is. In the remainder of this paper, we derive valuation formulas under the ME assumption. Therefore, we assume that the tax savings from risky debt should be valued using Equation (11). As illustrated in Table I, this represents one extreme assumption about the treatment of the COD. This assumption is closer to the US Tax Code than Sick's (1990) assumption, but the latter may be more appropriate in other jurisdictions. Sick's (1990) analysis may be relevant in jurisdictions where tax law does not provide an exception from tax on COD to insolvent firms. In such cases, Sick's approach is correct if insolvent firms are in tax-paying positions. However, based on US evidence, insolvent firms typically do not pay taxes (Gilson, 1997). Therefore, the principal mechanism for the incremental tax payment from COD in the insolvent state that Sick assumes must be a reduction in the value of carried-forward tax losses. Whether this is realistic depends on a variety of factors, including the efficiency of the market for tax loss transfers. Imperfections in this market would imply that Sick's approach needs modification even when COD is taxed when the firm is insolvent. The ME approach is most relevant in jurisdictions such as in the United States where tax law provides an exception from tax on COD for insolvent firms. However, if default implies only a partial loss on the bond, the ME approach is also incomplete in that setting. In general, it is likely that the correct treatment lies somewhere between the two polar cases. ̂ 'There is an interesting theoretical problem with the combination of a constant leverage policy and default. If a firm continuously maintains a constant leverage ratio and there are no jumps in its value, it will never default. Of course, default is possible without jumps if leverage is adjusted only periodically, as in the original setup of Miles and Ezzell (1980). 370 Financial Management • Summer 2008 III. The Relationship between Leveraged and Unleveraged Rates Appendix A demonstrates that under the standard ME assumptions with the inclusion of risky debt and investor taxes, the relationship between RL and Ry is: