# Consistent valuation and cost of capital expressions with corporate and personal taxes.

* The adjusted present value, adjusted discount rate and flows to
equity valuation methods represent three different approaches to valuing
firms and other assets.(1) If they are to be used interchangeably, all
three methods should result in identical values for a given asset;
achieving such consistency in practice can prove elusive for several
reasons.

First, each method incorporates the asset's business risk and the tax effects of its financing mix in a different way, and each relies on a different cost of capital measure. It is thus necessary to understand how these measures are related to one another. Second, the interrelationships among different cost of capital measures are not unique. There are several distinct sets of valuation and cost of capital expressions, each derived under differing assumptions about the asset's cash flow and financing pattern and the applicable tax regime. Consistent valuation, of course, requires cost of capital expressions that are all based on the same assumptions. Finally, in many practical situations, it is cumbersome or even impossible to use all of the valuation methods. In such cases, they are not interchangeable, and the analyst should know which one is superior.

The potential for confusion resulting from this array of techniques and assumptions is heightened by the fragmented approach to the topic in textbooks and the literature. The literature has dealt most extensively with the case in which financing affects value only through corporate taxes.(2) Three distinct sets of cost of capital expressions have been derived for this case, each resting on a different assumption about the riskiness of future debt tax shields. For the case that includes both corporate and personal taxes, however, a complete set of analogous expressions has yet to be derived and differences in assumptions about the risk of debt tax shields have not received substantial emphasis.(3) The purpose of this paper is to fill that gap.

Section I describes the three valuation methods, and Section II surveys and summarizes existing results on valuation with corporate but not personal taxes. This includes two cases in which all future debt tax shields are known with certainty and two in which they are uncertain. Section III introduces personal taxes and derives cost of capital expressions for different assumptions about the risk of future debt tax shields. Section IV offers recommendations for choosing among the different valuation approaches, and Section V summarizes the principal findings.

I. Three Valuation Methods

All three basic valuation methods seek to discount after-corporate-tax cash flows at pre-investor-tax discount rates, but they make different adjustments for the effects of financing. Myers' [22] adjusted present value (APV) method calls for first computing a base-case value under the assumption of 100% equity financing, and then separately adding the present values of any costs and benefits from the actual financing package. The base-case value is calculated by discounting the asset's expected after-corporate-tax operating cash flows, [C.sub.n] for each period n at an all-equity, or unlevered, discount rate, r.(4)

The adjusted discount rate (ADR) method discounts expected operating cash flows, [C.sub.n], at a rate that reflects the asset's financing combination. Both the APV and ADR methods thus discount the same cash flows. However, the APV method adjusts for financing in one or more separate discounted cash flow terms, while the ADR method does so entirely in the discount rate.

Finally, the flows to equity (FTE) method calculates equity value directly by discounting cash flows to the equityholders at a cost of equity capital. The cash flows are in turn calculated by subtracting after-corporate-tax financing charges from the all-equity cash flows, [C.sub.n], and thus they represent actual cash flows to shareholders. By contrast, the cash flows used under the other two methods are hypothetical, all-equity cash flows.

A limitation of the analysis, which should be noted at the outset, is that all debt is assumed free of default risk. This assumption is made in part to maintain comparability with the bulk of the literature and in part to avoid unnecessary complexity. It does imply, however, that tax factors are the only effect of capital structure to be incorporated, while factors such as bankruptcy, agency and information costs are ignored.(5)

II. Valuation With Corporate But No

Personal Taxes

The notation used throughout the paper is summarized in Exhibit 1. The valuation and cost of capital expressions that appear most frequently in textbooks and other finance literature include only the corporate tax effects of debt, and these appear in Exhibit 2. There are six basic relationships: the APV relationship between total value and unlevered value; the overall cost of capital expressed as a weighted average of the cost of equity and the cost of debt; the overall cost of capital as a function of the unlevered cost of capital; the levered cost of equity as a function of the unlevered cost; the cost of equity as a function of beta; and the levered beta as a function of the unlevered beta. These relationships can take on three different forms, as shown in the three panels of Exhibit 2, depending on what is assumed about the time pattern and risk of the firm's interest tax shields. Exhibit 1. Summary of Notation E = market value of equity D = market value of debt V = E + D = total market value of firm [r.sub.m] = equilibrium expected return on market

portfolio of equity securities [r.sub.e] = cost of equity for an individual firm [r.sub.fe] = cost of risk-free equity [r.sub.fd] = cost of risk-free debt r = unlevered, or all-equity, cost of capital [r.sub.*] = adjusted, or overall, cost of capital [T.sub.p] = personal tax rate on income from bonds [T.sub.pe] = effective personal tax rate on income from

equity [T.sub.c] = corporate tax rate [Mathematical Expression Omitted]

advantage of corporate debt [C.sub.n] = expected value of period n after-corporate-tax

operating cash flow, [C.sub.n] [Beta.sub.U] = beta, or systematic risk, of an unlevered firm [Beta.sub.L] = beta of otherwise equivalent levered firm

A. The Case of Constant, Perpetual Debt

The cost of capital expressions in Panel A of Exhibit 2 are probably the most widely known (e.g.[12], [21], [27],[30]), and they have been shown to give consistent results under all three valuation methods ([1],[6],[10], [16],[22]). Since they show explicitly the relationship between the cost of capital and leverage, they also afford the flexibility to determine the valuation effects of alternative financing plans. Unfortunately, as Myers [22] has pointed out, all of the relationships in Panel A hold simultaneously only under restrictive assumptions: the stream of expected operating cash flow must be a level perpetuity, and the firm's outstanding debt (and hence its annual interest tax shield) must be known and constant forever.[6] Thus, most of the expressions in Panel A, though frequently cited in textbooks and elsewhere, are really of rather limited usefulness in practical valuation situations.

B. Finite Assets Lives with Known Debt

Schedules

Myers [22] has shown that the APV approach to valuation can be generalized to allow for finite and uneven operating cash flow streams. In particular, if the schedule of future debt levels is currently known with certainty, the value at time t of an asset whose useful life ends at time N is given by

[Mathematical Expression Omitted]

However, while this provides a natural generalization of Equation (2A.1) in Exhibit 2, there are no finite-life analogues for Equations (2A.2), (2A.3), (2A.4) or (2A.5) that are generally valid when the schedule of outstanding debt is certain. Thus, its not feasible to use either the adjusted discount rate or flows to equity methods in this case.(7) [Mathematical Expression Omitted]

C. The Miles-Ezzell Analysis

Alternatively, one can argue that it may not make the best sense to assume that future debt levels are known. For example, Fama's [9] analysis implies that one can justify the use of risk-adjusted discount rates to value a level perpetuity by assuming that each period's expected operating cash flows follow a geometric random walk. In that event, however, firm value also follows a random walk, and it seems inconsistent to assume that the level of debt remains constant with certainty even in the face of a changing firm value.

In an attempt to address this inconsistency and at the same time derive adjusted discount rates that are valid for finite asset lives and uneven cash flow streams, Miles and Ezzell [18] started with the premise that the firm maintains a constant debt-to-value ratio. The current debt level, which is based on current firm value, is known, so in the absence of default risk the interest tax shield at the end of the first period is also certain. Thus, it is justifiable to discount the first period's interest tax shield at [r.sub.fd], the risk-free debt rate. However, future firm values, and hence future debt levels, are currently uncertain. If the firm maintains a constant debt-to-value ratio, future firm value will be perfectly correlated with the value of the operating cash flow stream, and therefore all interest tax shields beyond the first period should be discounted at r, the unlevered cost of capital.

Based on this reasoning, Miles and Ezzell ([18], [19]) derived the set of valuation and cost of capital expressions in Panel B of Exhibit 2. Unlike those in Panel A, the Miles-Ezzell expressions give consistent results under any of the three valuation methods for both perpetual and finite-lived assets. However, these results will generally differ from those derived from Panel A, even in the perpetuity case, because of their different assumption about the risk of debt tax shields.

Another point that emerges from contrasting Panels A and B is the role of the weighted average cost of capital, which can be used in either case. This is because the market's valuation of debt tax shields is captured in the cost of equity, [r.sub.e]. If an estimated cost of equity correctly reflects investors' assumptions about the risk of the debt tax shields, whatever those assumptions may be, the weighted average cost of capital will yield a valid adjusted discount rate.

D. The Harris-Pringle Analysis

Harris and Pringle [14] have proposed that all debt tax shields, including the first year's, be treated as risky and discounted at r, which leads to the set of equations in Panel C of Exhibit 2. These can be thought of as the analogues of the Miles-Ezell expressions when the firm adjusts its debt level continuously to the target ratio. For example, the Miles-Ezzell cost of capital expression (2B.3) can be rearranged to produce:

[Mathematical Expression Omitted]

Then, dividing each period into arbitrarily small sub-periods, gives

[Mathematical Expressions Omitted]

Hence, expression (2C.3) is the continuous-time version of (2B.3).

The relationships in Panel C have a simpler form than those in Panel B, and as a practical matter, they do not yield valuations that are very different. Like the Miles-Ezzell equations, they are applicable to finite and uneven cash flow streams as well as level perpetuities, and they will give identical results under any of the three valuations methods. As in the other two panels of Exhibit 2, the weighted average cost of capital is a valid overall cost of capital expression, but here it reflects the relationship between [r.sub.e] and r that is expressed in (2C.4).

III. Valuation With Corporate and

Personal Taxes

In this section, we introduce personal taxes and derive valuation and cost of capital expressions analogous to those discussed in Section II. However, care must be taken in specifying investors' required rates of return. While these are often analyzed on a pretax basis, it is after-tax returns that ultimately drive the valuation process, and thus the equilibrium relationships among after-tax returns must be understood. In the remainder of this section, we assume that all investors pay taxes on income from debt securities at the rate [T.sub.p] and on income from equity securities at the rate [T.sub.pe].(8) For simplicity, we also assume that all investors are subject to the same tax rates.

A. After-Tax Rate of Return Relationships

In equilibrium, debt and equity securities of comparable risk must offer identical after-tax returns, or else investors will be motivated to rearrange their portfolios. In particular, suppose we have a risk-free debt security offering a pre-tax return per period of [r.sub.fd] and a risk-free equity security offering a pre-tax return per period of [r.sub.fe].(9) These returns will be set in the market so that:

[r.sub.fd](1 - [T.sub.p]) = [r.sub.fe](1 - [T.sub.pe]).

More generally, a tax-adjusted capital asset pricing model (CAPM) can be derived (e.g., [5], [11]), characterizing the after-tax returns on securities of both different risk and different tax treatment. Suppose, for example, that all risky assets are in the form of equity and that the expected pre-tax return to investors from this market equity portfolio is [r.sub.m]. If [r.sub.ej] is the pre-tax return on firm j's equity, then in equilibrium,(10)

[r.sub.ej](1 - [T.sub.pe])-[r.sub.fd](1 - [T.sub.p)=[beta.sub.j][r.sub.m] (1 - [T.sub.pe]-[r.sub.fd](1 - [T.sub.p])], (5)

where [beta.sub.j] is the systematic risk measure. Using Equation (4), this can be simplified to:

[r.sub.ej] = [r.sub.fe] + [beta.sub.j]([r.sub.m - r.sub.fe]).

The important point is that, in general, rates of return should be compared on an after-personal-tax basis. Pre-tax comparisons are valid only if all rates apply to instruments that receive identical tax treatment. Thus [r.sub.m], which is taxed at the rate [T.sub.pe], is directly comparable with [r.sub.fe], as in Equation (6), but not with [r.sub.fd], which is taxed at the rate [T.sub.p].

B. The Case of Constant, Perpetual Debt

For the level perpetuity case with constant debt, Miller [20] has shown the Equation (3A.1) in Exhibit 3 holds, where [G.sub.L] is the net tax benefit from corporate debt. This is the basis for the APV approach in this case. The remaining equations in Panel A of Exhibit 3 follow from (3A.1) and from basic valuation definitions.(11)

Two points are worth special note. First, one moves from the equations in Panel A of Exhibit 2 to those in Panel A Exhibit 3 by making two adjustments: (i) [T.sub.c], the corporate tax rate, is replaced by [G.sub.L], the net tax advantage to corporate debt; (ii) [r.sub.fd], the risk-free debt rate, is replaced by [r.sub.fe], the risk-free equity rate.(12) Indeed, it will be seen below that the same rule applies to every equation in Exhibit 3. In particular, if the CAPM is to be used to estimate a cost of equity, consistency requires that it be an after-personal-tax CAPM, as embodied in Equation (3A.5). Second, the assumptions that the operating cash flow stream is a level perpetuity and that debt is known and constant forever imply that the equations in this panel, like those in Panel A of Exhibit 2, are of limited practical use.

C. Finite Asset Life With a Known Debt

Schedule

As in Section II, one way to obtain more broadly applicable valuation expressions is to retain the assumption that future debt levels are known but allow for finite and uneven cash flow streams. Beginning with the APV approach, it might seem natural to generalize the analysis of Section II by simply substituting [G.sub.L] in place of [T.sub.c] in Equation (1) above. However, that would be incorrect. The reason is that the debt tax shields in the second term of Equation (1) are cash flows to the equityholders and should be discounted at an equityholders' opportunity cost. When there are only corporate taxes, investors would not distinguish between risk-free debt and risk-free equity, so it is perfectly appropriate to discount the second term in Equation (1) at the bondholders' opportunity cost, [r.sub.fd]. But when there are both corporate and personal taxes that is no longer the case. Instead, as shown in the Appendix, the correct generalization of Equation (1) is:

[Mathematical Expression Omitted]

Thus, not only must [G.sub.L] be substituted for [T.sub.c] in moving from Equation (1) to Equation (7), but also [r.sub.fe] must be substituted for [r.sub.fd]. The second term in Equation (7) might best be thought of as the present value of an annual financing subsidy. As in Brealey and Myers [4], the value of a subsidy can be calculated as the present value of the annual difference between unsubsidized and subsidized debt service charges (after corporate taxes), discounted at the unsubsidized market rate (also after-corporate-tax). Thus, if we let S represent the second term in Equation (7), (4) plus the definition of [G.sub.L] from Exhibit 1 can be used to write S as:

[Mathematical Expression Omitted]

That is, a firm wishing to issue a riskless claim on itself could issue either risk-free equity at the rate [r.sub.fe], or risk-free debt at the tax-subsidized rate [r.sub.fd](1 - [T.sub.c]). The net advantage to debt is simply the present value of this opportunity cost saving.

As when there are only corporate taxes, there are no adjusted discount rate expressions that will produce the same asset value given by Equation (7) in this case.

Thus the APV method is the only one of the three that gives theoretically correct results when the schedule of debt outstanding is known with certainty but the asset's life is finite.

D. Debt Adjusted Once Per Year to a Constant

Debt-to-Value Ratio

The second approach to obtaining more generally applicable results, as in Miles and Ezzell [18], is to assume that the firm's debt ratio is known and that the

(1) The adjusted discount rate method is also known as the weighted average cost of capital, or WACC method [15], while the flows to equity method is also known as the equity residual income method [10] or the equity residual value method [25]. [2] See [1], [10], [12], [14], [15], [18], [19], [21], [22], and [30] for discussions of this care. (3) Existing results for this case can be found in [4], [6], [8], [13], [16], [23] and [28]. (4) [C.sub.n] represents the expected after-tax cash flow the company would have gotten in period n if it had been entirely equity-financed, with no debt service charges deducted. (5) Some f the consequences of risky debt have been analyzed in [7], [23], [28] and [31]. In the absence of bankruptcy costs, Sick [28] has obtained similar results to this paper's but it should be emphasized that the cost of debt must then be interpreted as an expected, rather than a promised rate of return. It is incorrect to substitute the contractual debt rate for the risk-free rate. Similarly, under the APV approach, certainty-equivalent interest payments should be used in place of contractual interest payments. (6) This framework can also accommodate constant perpetual growth, as in Lewellen and Emery [16], but only level perpetuities are considered here to avoid excessive complication. (7) The CAPM can be used to compute a cost of equity for the current period, given an accurate estimate of [beta], but it is still impossible to use the FTE method. Future debt levels are known, but future firm value and hence future leverage are random. Thus future levels of [beta] are also random, so the appropriate cost of equity for valuing future cash flows cannot be determined today. (8) As in Miller [20], [T.sub.pe] should be interpreted as an effective tax rate, or as the uniform annual tax rate that would produce tax payments having the same present value as the pattern of actual tax payments on equity income. (9) In a CAPM context, [r.sub.fe] can be thought of as the pre-tax return on a zero-beta equity portfolio. Alternatively, [r.sub.fe] could be interpreted as the cost of equity for an entirely equity-financed firm with riskless assets. For the analysis that follows, it is not necessary that a specific, riskless equity security exist as long as one can be created synthetically. (10) A relatively simple proof can be constructed using the same steps found in Rubinstein [27], but with end-of-period investor wealth calculated after all taxes. (11) Specifically, Equation (3A.2) follows from the definitions E = [C - [r.sub.fd] D (1-[T.sub.c])] / [r.sub.e] and V = C/r*. Equation (2A.3) is obtained from equating (3A.2) and (3A.3) and solving for [r.sub.e.] Equation (3A.6) follows from writing (3A.5) for a levered and an unlevered firm and equating (3A.4) and (3A.5) for the levered firm. (12) These two rules are also valid for the weighted average cost of capital (3A.2), although not written in that form in Exhibit 3. That is because, from Equation (4) and the definition of [G.sub.L], [r.sub.fe](1-[r.sub.fd] (1-[T.sub.c)]. actual debt level is adjusted once per year this target ratio. Extending the Miles-Ezzell analysis to include both corporate and personal taxes thus leads to Equation (3B.1) in Exhibit 3 as the basic valuation relationship. Iterating backward from the end of the firm or asset's life, as shown in the Appendix, leads to an adjusted discount rate expression, (3B.3(), which is also consistent with the weighted average cost of capital, (3B.2).Expressions (3B.4) and (3B.6) follow in turn from equating (3B.2) with (3B.3).

The equations in Panel B of Exhibit 3 are applicable to both finite-and infinite-lived assets using any of the three valuation methods. In fact, they are exact analogues to the Miles-Ezzell equations in Panel B of Exhibit 2 with the same two adjustments that are made in Panel A: [T.sub.c] is replaced by [G.sub.L], and [r.sub.fd] is replaced by [r.sub.fe.].

E. Debt Adjusted Continuously to a Constant

Debt-to-Value Ratio

If the firm adjusts its capital structure continuously to a target debt ratio, all future debt tax shields, including the first year's, are risky. This implies that the basic valuation relationship is given by Equation (3C.1) in Exhibit 3, which generalizes the Harris-Pringle [14] analysis to include both corporate and personal taxes. Equation (3C.1) can then be used to derive the remaining expressions kin Panel C. Exhibit 3. These in turn produce identical firm values, for both perpetual and finite-lived assets, regardless of which valuation method is used.

A special feature of the continuous -adjustment case is that Equation (2C.6) is identical to Equation (3C.6). That is, the relationship between the levered and unlevered betas is the same, whether or not personal taxes are included in the analysis. As discussed in more detail in Section IV, Myers and Ruback [23] have exploited this fact to show that, under a special assumption about the firm's capital structure, the value of an asset will be the same, regardless of the assumed tax regime.

IV. Implementation Issues

A. A. Numeric Example

The use of these equations is illustrated by the example in Exhibit 4. The first step is to choose the appropriate set of valuation and cost of capital expressions. Since personal tax rates are present, this calls for the expression in Exhibit 3, rather than Exhibit 2. The asset has only a 10-period life, so this rules out the perpetuity expressions in Panel A, and if we further suppose that the firm adjusts to its target debt ratio once per period rather than continuously, this establishes Panel B as giving the relevant set of expressions.

Using the equations in Panel B, we can derive the values shown in the bottom half of Exhibit 4. For example, the adjusted discount rate method calls for using either Equation (4B.2) or (4B.3) to find the overall cost of capital, r(*) = 0.1593. Then, discounting the ten year's worth of operating cash flows at this rate gives V = 484.6. The same value can also be obtained with either of the other two valuation methods, but it should be noted that both are extremely cumbersome to use under the assumed capital structure adjustment scenario.

Exhibit 4. Numerical Example (the equations in Panel

A. Assumed Parameter Values

N = 10 periods [C.sub.n] = 100 for all n [T.sub.p] = 0.28 [T.sub.pe] = 0.18 [T.sub.c] = 0.34 rfd = 0.07 [r.sub.m] = 0.15 [beta.sub.U] = 1.2 D/V = 0.05

B. Derived Values

[G.sub.L] = 0.2483, from definition in Exhibit 1 rfe = 0.0615, from Equation (4) r = 0.1677, from Equation (3B.5) [r.sup.*] = 0.1593, from Equation (3B.3) or from

(3B.2), if (3B.4) is used to estimate [r.sub.e] [r.sub.e] = 0.2724, from Equation (3B.4) or from

(3B.5)

V = 484.6, from discounting [C.sub.n] at [r.sup.*] [beta.sub.L] = 2.38, from Equation (3B.6) E = 242.3, from discounting equity cash

flows at [r.sub.e]

To implement the flows to equity method, for instance, it is necessary to know the entire future schedule of debt service charges. Such a schedule can be derived bu using the ADR method to obtain a period-by-period schedule of asset values, as shown in Exhibit 5, and these can in turn be multiplied by 0.5 to obtain the schedule of outstanding debt, From this,the after-tax debt service charge (including principal repayment) can be calculated for each period and subtracted from the after-tax operating cash flow, [C.sub.n] to obtain the equity cash flows shown in Column 5 Exhibit 5. These cash flows can then be discounted at [r.sub.e] = 0.2724 to obtain E = 242.3, which is consistent with V = 484.6.

[TABULAR DATA OMITTED]

To implement the adjusted present value method, the debt schedule in Exhibit 5 can be used to calculate the schedule of effective debt tax shields, [r.sub.fe] [G.sub.L] [D.sub.n-1], as shown in Column j6, However, these must be discounted iteratively, using Equation (3B.1), since only the next period's cash flow is certain at any given time. Following this procedure, one can duplicate the period-by-period schedule of asset value given in Column 1.

Under the assumed capital structure adjustment process, the FTE and APV methods are not only cumbersome but redundant in this case. To implement either, we effectively need the entire schedule of asset values, but this is what we are trying to determine in the first place. Thus even though all three methods can yield identical results in principle in this case, the ADR method is definitely preferred.

A second point that emerges from the numerical example concerns the schedule of debt. Equations such as (3B.2), (3B.3), and (3B.4) allow one to calculate a current cost of capital that is predicated on an entire schedule of future asset values, as illustrated in Exhibit 5. In fact, however, asset value is random over time, because of the random nature of the expected operating cash flows. As firm value changes over time, the schedule of expected future debt levels needed to maintain a constant debt-to-value ratio will also be updated.

The example can also give a sense of the magnitude of the error resulting from inappropriate use of different valuation methods. Since the expressions in Panel A of Exhibit 2 are the most familiar from textbooks and the literature, it might be common to use those, even though the asset in question has a finite life and personal taxes are relevant. For the parameter values in Exhibit 4, Equation (2A.5) would give r = 0.166 and Equation (2A.3)(the MM cost of capital formula) would give the overall cost of capital as r(*) = 0.1378. Discounting the ten years' worth of operating cash flows at r(*) then gives the asset value as V = 526.2, an error of 8.6% relative to the true value. This error will increase with asset life, with the difference between [G.sub.L] and [T.sub.c], and with the difference between r and [r.sub.fe] For example, if we keep other parameter values the same but move from a 10-year to a perpetual asset life, the estimated asset value is 718.4 using [r.sup.*] = 0.1392, an error of 14.4% relative to the correct value of 627.7 that is obtained using r(*) = 0.1593.

Another error to which the analyst might be susceptible is inappropriate use of the APV method. Suppose the analyst knows the initial debt level is 242.3 but assumes this level will remain constant, rather than changing over the asset's life. All other parameter values are as given in Exhibit 4. If the APV method is implemented using Equation (1) in the text, this results in V = 513.2, an error of 5.9% relative to the true value of 484.6. Again, this error will increase with asset life, with the difference between [G.sub.L] and [T.sub.c], and with the difference between r and [r.sub.fe].

Other types of errors, however, may be quite negligible. Suppose, for instance, that the analyst uses the expressions in Panel C of Exhibit 3, rather than those in Panel B, even though the asset's capital structure is not adjusted continuously. From Equation (3C.3), this results in r(*) = 0.1591 and V = 485.0, an error of less than 0.1% from the true value.

B. The Choice of a Valuation Method

Exhibit 6 provides a roadmap that can be used to navigate through the different cost of capital expressions and the circumstances under which one of the three valuation methods is preferred to the others. For simplicity, Exhibit 6 is keyed to a tax regime that includes both corporate and personal taxes and thus to the equations in Exhibit 3, but an exactly analogous roadmap could be keyed to Exhibit 2. Given the relevant tax regime, the next question is whether the operating cash flow stream is best viewed as a perpetual annuity of if a finite life and/or uneven cash flows are important characteristics of the asset. Most important, the analyst decide whether future debt tax shields can be treated as a known schedule or whether they vary with future firm values.

If the schedule of debt tax shields is known in dollar terms, the APV method is always appropriate , and for finite asset lives it is the only one of the three methods that is even feasible. One the other hand, if debt financing is specified as a fraction of asset value, the ADR methods is always appropriate, while the other two methods are cumbersome at best. In the latter case, moreover, the numerical example suggests that it makes little practical difference whether the analyst uses the Miles-Ezzell analogue expressions in Exhibit 3, Panel B or the Harris-Pringle analogue expressions in Exhibit 3, Panel C. This may in turn lead to a preference for Exhibit 3, Panel C, because of the simpler form of the equations.

C. When Can We Avoid the Choice of a Tax

Regime?

As Hamada and Scholes [13] argue, it may not be obvious which tax regime discussed above is driving asset values. Additionally , if personal taxes are included, unobservable magnitudes, such as the next tax gain to leverage [G.sub.L] and the risk-free equity rate, [r.sub.fe], must be estimated to move from one cost of capital expression to another.

If the analyst wishes to avoid these problems altogether, there are two special cases in which the choice of the tax regime is irrelevant. The first occurs when all debt tax shields are discounted at r (as in Panel C of Exhibit 2 or 3), and firm's capital structure is characterized by (E/V) = [beta.sub.U] and (D/V) = (1 -[beta.sub.U)]. In that case, Myers an Ruback [23] have shown that, for both tax regimes, the adjusted discount rate is

[r.sup.*] = [r.sub.fd](1-[beta.sub.U])(1 - [T.sub.c])+[beta.sub.U][r.sub.m]. (9)

Expression (9) thus provides a useful cost of capital expression when the analyst is uncertain about which tax regime better reflects the true security valuation process. The tradeoff, however, is that the analyst must accept the assumed capital structure.

The second special case occurs when all cash flows are riskless. Ruback [26] has shown under very general conditions that riskless corporate cash flows can be valued by discounting them at the after-corporate-tax riskless interest rate, regardless of the relevant tax regime. This follows by an arbitrage argument from the observation that such a stream can support 100% of its own value in debt financing. The current results are consistent with this analysis,as can be seen from Exhibits 2 and 3. When there are only flow stream would all-equity-financed riskless cash flow stream would have an unlevered cost of capital r = [f.sub.fd.] If D/V = 1, then, all of the overall cost of capital expressions in Exhibit 2 reduce to r(*) = [r.sub.fd] (1 - [T.sub.c]). With corporate and personal taxes, kr = [r.sub.fe] for a riskless, unlevered stream, and if D/V = 1, the third equation in each panel of Exhibit 3 reduces to r(*) = [r.sub.fe] (1 - [G.sub.L]. However, from Equation (4) and the definition of [G.sub.L], [r.sub.fe] (1 - [G.sub.L]) = [r.sub.fd](1 - [T.sub.c]. The weighted average cost of capital also reduces to r(*) = [r.sub.fd] (1 - [T.sub.c]) in this case.

V. Summary and Conclusions

This paper has analyzed the conditions under which the adjusted present value, adjusted discount rate and flows to equity methods all lead to identical asset valuations in the presence of corporate and personal taxes. Its principle conclusions are:

(i) Consistent valuation can be achieved if (a) the asset's operating cash flows and debt service charges are level perpetuities or (b) the firm maintain a constant leverage ratio. Depending on which of these assumptions is appropriate, the analyst must use the valuation and cost of capital expressions from only one panel of expressions i Exhibit 3.

(ii) The expressions in Exhibit 3, which incorporate both corporate and personal taxes, differ from the more familiar expressions in Exhibit 2, which include only corporate taxes, in two ways: [T.sub.c], the corporate tax rate, is replaced by [G.sub.L], the net tax advantage to corporate debt, and [r.sub.fd], the risk-free debt rate, is replaced by [r.sub.fe], the risk-free equity rate.

(iii) Although consistent valuation is possible using any of the three methods when the firm maintains a constant leverage ratio, the adjusted present value and flows to equity methods are cumbersome, and the adjusted discount rate method is preferred.

(iv) When the asset has a finite life but the schedule of outstanding debt is known with certainly, there are no adjusted discount rate expressions for either the overall cost of capital or the cost of equity, so the adjusted present value method must be used in this case.

(v) The weighted average cost of capital is robust to changes in both the tax regime and in the perceived risk of interest tax shields. This is because the relationship between the levered cost of equity, [r.sub.e], and unlevered cost, r, changes to reflect both tax factors and tax shield risk, leaving the weighted average formula intact. To correctly estimate the weighted average starting from scratch, however, the analyst must use the correct relationship between r and [r.sub.e]. A special case of the weighted average cost of capital, proposed by Myers and Ruback [23], does not require specific knowledge of the relationship between r and [r.sub.e], but does require a specific capital structure assumption.

References

[ 1.] D.J. Ashton and D.R. Atkins, "Interactions in Corporate

Financing and Investment Decisions: A Further Comment,"

Journal of Finance (December 1978), pp. 1447-1453. [ 2.] R.P. and J.M. Collins, "The Relationship Between Before-and

After-Tax Yields on Financial Assets," Financial Review

(August 1988), pp. 313-331. [ 3.] F. Black and M.S. Scholes, "The Effects of Divided Yield and

Divided Policy on Common Stock Princess and Returns," Journal

of Financial economic (May 1974, pp.1-21. [ 4.] R.A. Brealey and S.C. Myers, Principle o Corporate Finance,

New York, McGraw-Hill, 4th edition, 1991. [ 5.] M.J. Brennan, "Taxes, Market Valuation and Corporate Financial

Policy," National Tax Journal (December 1970),pp.

417-427. [ 6.] D.R. Chambers, R.S. Harris, and J.J. Pringle, "Treatment of

Financing Mix in Analyzing Investment Opportunities," financial

Management (Summer 1982), pp. 24-41. [ 7.] T.E. Conine, Jr., "Corporate Debt and Corporate Taxes: An

Extension," Journal of Finance (September 1980), pp.1033-1037. [ 8.] J.R. Ezzell and W.A. Kelly, Jr., "An APV Analysis of Capital

Budgeting Under Inflation," Financial Management (Autumn

1984),pp.49-54. [ 9.] E. Fama, "Risk-Adjusted Discount Rates and Capital Budgeting

Under Uncertainty," Journal Economic (August

1977), pp.3-24. [10.] J.R.Franks, J.E. Broyles,and W.T. Carleton, Corporate Finance:

Concepts and Applications, Boston, Kent Publishing Co., 1985. [11.] R.H. Gordon and B.G. Malkiel, "Corporation Finance" in How

Taxes Affect Economic Behavior, H.J. Aaron and J.A. Pechman

(eds.), Washington, D.C., Brooking Institution, 1989. [12.] R.S. Hamada, "Porfolio Analysis, Market Equilibrium and Corporation

Finance," Journal of Finance (March 1969), pp. 13-31. [13.] R.S. Hamada and Myron S. Scholes, "Taxes and Corporate

Financial Management," Recent Advances in Corporate

Finance, E.I. Altman and M.G. Subrahmanyam (eds),

Homewood, IL. Richard D. Irwin, 1985. [14.] R.S. Harris and J.J. Pringle, "Risk-Adjusted Discount Rates - Extensions

from the Average-Risk Case," Journal of Financial

Research (Fall 1985), pp. 237-244. [15.] I. Inselbag and H. Kaufold, "How to Value Recapitalizations and

Leveraged Buyouts," Continental Bank Journal of Applied Corporate

Finance (Summer 1989), pp. 87-96. [16.] W.G. Lewellen and D.R. Emery, "Corporate Debt Management

and the Value of the Firm," Journal of Financial and Quantitative

Analysis (December 1986), pp. 415-426. [17.] R.W. Masulis, "The Impact of Capital Structure Change of Firm

Value: Some Estimates," Journal of Finance (March 1983), pp.

107-126. [18.] J. Miles and J.R. Ezzell, "The Weighted Average Cost of Capital,"

Perfect Capital Markets and Project Life: A Clarification,

Journal of Financial and Quantitative Analysis (September 1980),

pp. 719-730. [19.] J. Miles and J.R. Ezzell,"Reformulating Tax Shield Valuation:

A Note," Journal of Finance (December 1985), pp. 1485-1492. [20.] M.H. Miller, "Debt and Taxes," Journal of Finance (May 1977),

pp. 261-276. [21.] F. Modigliani and M.H. Miller, "Corporate Income Taxes and

the Cost of Capital: A Correction," American Economic Review

(June 1963), pp.433-443. [22.] S.C. Myers, "Interactions of Corporate Financing and Investment

Decisions - Implications of Capital Budgeting," Journal

of Finance (March 1974), pp. 1-25. [23.] S.C. Myers and R.S. Ruback,"Discounting Rules for Risky

Assets," Harvard Business School Working Paper, November

1988. [24.] S.C. Myers and S.M. Turnbull, "Capital Budgeting and the Capital

Asset Pricing Model: Good News and Bad News," Journal of

Finance (May 1977), pp.321-332. [25.] G.E. Pinches and L.W. Courtney, Lotus 1-2-3 for financial

Management, New York, Harper and Row, 1989. [26.] R.S. Ruback, "Calculating the Market Value of Risk-Free Cash

Flows," Journal of Financial Economics (March 1986), pp. 323-339. [27.] M.E. Rubinstein, "A Mean-Variance Synthesis of Corporate

Financial Theory," Journal of Finance (March 1973), pp.167-182. [28.] G.A. Sick, "Tax-Adjusted Discount Rates," Management Science

(December 1990), pp. 1432-1450. [29.] G.A. Sick, "A Certainly-Equivalent Approach to Capital

Budgeting," Financial Management (Winter 1986), pp. 23-32. [30.] R.A. Taggart, J.R., "Capital Budgeting and the Financing

Decision: An Exposition," Financial Management (Summer

1977), pp. 59-64. [31.] J. Yagil, "On Valuation, Beta and the Cost of Equity Capital,"

Journal of Financial and Quantitative Analysis (September 1982),

pp.441-448.

Appendix

I. Derivation of Equation (7)

Starting at the end of the valuation horizon, suppose the firm will pay a liquidating divided to its shareholders at time N, consisting of the after-tax operating cash flow, [C.sub.N.] minus the after-tax interest and principal on its beginning-of-period debt, [D.sub. N-1]. Personal taxes on equity income are levied against dividends plus capital gains, so beginning equity value, [E.sub.N.1], serves as a personal tax shield.

Letting [CEQ.sub.N] denote the pre-personal-tax certainty-equivalent of [C.sub.N], the value of the firm's equity and debt at N - 1 can be expressed as:

[Mathematical Expression Omitted]

Since these two discount rates must be the same in equilibrium, we can express beginning-of-period firm value, [V.sub.N-1] = [E.sub.N-1] + [D.sub.N-1], as

[Mathematical Expression Omitted]

Then using the definition of [G.sub.L], text Equation (4),and solving for [V.sub.N-1]:

[Mathematical Expression Omitted]

Iterating backward in similar fashion, the value of the firm at any time t is given by

[Mathematical Expression Omitted]

When risk-adjusted discount rate are appropriate (see [9], [24], [28]), (A5) can also be written in the more familiar form of Equation (7) in the text.

II. Derivation of Equation (3B.3) from Exhibit 3

First, write (A4) in its risk-adjusted form as

[Mathematical Expression Omitted]

If the ratio of debt to form value is a constant, we can express (A6) as

[Mathematical Expression Omitted]

[V.sub.N-2] is in turn equal to the sum of the values as of time N - 2 of three terms: [C.sub.N-1] , the net tax advantage of interest paid at time N - 1, and [V.sub.N-1]. Discounting [V.sub.n-1] at the unlevered cost of capital r, backward iteration of (A7) yields, for any time t:

[Mathematical Expression Omitted]

Setting the term in square brackets equal to (1 + r(*)) and solving for r(*) then gives Equation (3B.3) in Exhibit 3.

First, each method incorporates the asset's business risk and the tax effects of its financing mix in a different way, and each relies on a different cost of capital measure. It is thus necessary to understand how these measures are related to one another. Second, the interrelationships among different cost of capital measures are not unique. There are several distinct sets of valuation and cost of capital expressions, each derived under differing assumptions about the asset's cash flow and financing pattern and the applicable tax regime. Consistent valuation, of course, requires cost of capital expressions that are all based on the same assumptions. Finally, in many practical situations, it is cumbersome or even impossible to use all of the valuation methods. In such cases, they are not interchangeable, and the analyst should know which one is superior.

The potential for confusion resulting from this array of techniques and assumptions is heightened by the fragmented approach to the topic in textbooks and the literature. The literature has dealt most extensively with the case in which financing affects value only through corporate taxes.(2) Three distinct sets of cost of capital expressions have been derived for this case, each resting on a different assumption about the riskiness of future debt tax shields. For the case that includes both corporate and personal taxes, however, a complete set of analogous expressions has yet to be derived and differences in assumptions about the risk of debt tax shields have not received substantial emphasis.(3) The purpose of this paper is to fill that gap.

Section I describes the three valuation methods, and Section II surveys and summarizes existing results on valuation with corporate but not personal taxes. This includes two cases in which all future debt tax shields are known with certainty and two in which they are uncertain. Section III introduces personal taxes and derives cost of capital expressions for different assumptions about the risk of future debt tax shields. Section IV offers recommendations for choosing among the different valuation approaches, and Section V summarizes the principal findings.

I. Three Valuation Methods

All three basic valuation methods seek to discount after-corporate-tax cash flows at pre-investor-tax discount rates, but they make different adjustments for the effects of financing. Myers' [22] adjusted present value (APV) method calls for first computing a base-case value under the assumption of 100% equity financing, and then separately adding the present values of any costs and benefits from the actual financing package. The base-case value is calculated by discounting the asset's expected after-corporate-tax operating cash flows, [C.sub.n] for each period n at an all-equity, or unlevered, discount rate, r.(4)

The adjusted discount rate (ADR) method discounts expected operating cash flows, [C.sub.n], at a rate that reflects the asset's financing combination. Both the APV and ADR methods thus discount the same cash flows. However, the APV method adjusts for financing in one or more separate discounted cash flow terms, while the ADR method does so entirely in the discount rate.

Finally, the flows to equity (FTE) method calculates equity value directly by discounting cash flows to the equityholders at a cost of equity capital. The cash flows are in turn calculated by subtracting after-corporate-tax financing charges from the all-equity cash flows, [C.sub.n], and thus they represent actual cash flows to shareholders. By contrast, the cash flows used under the other two methods are hypothetical, all-equity cash flows.

A limitation of the analysis, which should be noted at the outset, is that all debt is assumed free of default risk. This assumption is made in part to maintain comparability with the bulk of the literature and in part to avoid unnecessary complexity. It does imply, however, that tax factors are the only effect of capital structure to be incorporated, while factors such as bankruptcy, agency and information costs are ignored.(5)

II. Valuation With Corporate But No

Personal Taxes

The notation used throughout the paper is summarized in Exhibit 1. The valuation and cost of capital expressions that appear most frequently in textbooks and other finance literature include only the corporate tax effects of debt, and these appear in Exhibit 2. There are six basic relationships: the APV relationship between total value and unlevered value; the overall cost of capital expressed as a weighted average of the cost of equity and the cost of debt; the overall cost of capital as a function of the unlevered cost of capital; the levered cost of equity as a function of the unlevered cost; the cost of equity as a function of beta; and the levered beta as a function of the unlevered beta. These relationships can take on three different forms, as shown in the three panels of Exhibit 2, depending on what is assumed about the time pattern and risk of the firm's interest tax shields. Exhibit 1. Summary of Notation E = market value of equity D = market value of debt V = E + D = total market value of firm [r.sub.m] = equilibrium expected return on market

portfolio of equity securities [r.sub.e] = cost of equity for an individual firm [r.sub.fe] = cost of risk-free equity [r.sub.fd] = cost of risk-free debt r = unlevered, or all-equity, cost of capital [r.sub.*] = adjusted, or overall, cost of capital [T.sub.p] = personal tax rate on income from bonds [T.sub.pe] = effective personal tax rate on income from

equity [T.sub.c] = corporate tax rate [Mathematical Expression Omitted]

advantage of corporate debt [C.sub.n] = expected value of period n after-corporate-tax

operating cash flow, [C.sub.n] [Beta.sub.U] = beta, or systematic risk, of an unlevered firm [Beta.sub.L] = beta of otherwise equivalent levered firm

A. The Case of Constant, Perpetual Debt

The cost of capital expressions in Panel A of Exhibit 2 are probably the most widely known (e.g.[12], [21], [27],[30]), and they have been shown to give consistent results under all three valuation methods ([1],[6],[10], [16],[22]). Since they show explicitly the relationship between the cost of capital and leverage, they also afford the flexibility to determine the valuation effects of alternative financing plans. Unfortunately, as Myers [22] has pointed out, all of the relationships in Panel A hold simultaneously only under restrictive assumptions: the stream of expected operating cash flow must be a level perpetuity, and the firm's outstanding debt (and hence its annual interest tax shield) must be known and constant forever.[6] Thus, most of the expressions in Panel A, though frequently cited in textbooks and elsewhere, are really of rather limited usefulness in practical valuation situations.

B. Finite Assets Lives with Known Debt

Schedules

Myers [22] has shown that the APV approach to valuation can be generalized to allow for finite and uneven operating cash flow streams. In particular, if the schedule of future debt levels is currently known with certainty, the value at time t of an asset whose useful life ends at time N is given by

[Mathematical Expression Omitted]

However, while this provides a natural generalization of Equation (2A.1) in Exhibit 2, there are no finite-life analogues for Equations (2A.2), (2A.3), (2A.4) or (2A.5) that are generally valid when the schedule of outstanding debt is certain. Thus, its not feasible to use either the adjusted discount rate or flows to equity methods in this case.(7) [Mathematical Expression Omitted]

C. The Miles-Ezzell Analysis

Alternatively, one can argue that it may not make the best sense to assume that future debt levels are known. For example, Fama's [9] analysis implies that one can justify the use of risk-adjusted discount rates to value a level perpetuity by assuming that each period's expected operating cash flows follow a geometric random walk. In that event, however, firm value also follows a random walk, and it seems inconsistent to assume that the level of debt remains constant with certainty even in the face of a changing firm value.

In an attempt to address this inconsistency and at the same time derive adjusted discount rates that are valid for finite asset lives and uneven cash flow streams, Miles and Ezzell [18] started with the premise that the firm maintains a constant debt-to-value ratio. The current debt level, which is based on current firm value, is known, so in the absence of default risk the interest tax shield at the end of the first period is also certain. Thus, it is justifiable to discount the first period's interest tax shield at [r.sub.fd], the risk-free debt rate. However, future firm values, and hence future debt levels, are currently uncertain. If the firm maintains a constant debt-to-value ratio, future firm value will be perfectly correlated with the value of the operating cash flow stream, and therefore all interest tax shields beyond the first period should be discounted at r, the unlevered cost of capital.

Based on this reasoning, Miles and Ezzell ([18], [19]) derived the set of valuation and cost of capital expressions in Panel B of Exhibit 2. Unlike those in Panel A, the Miles-Ezzell expressions give consistent results under any of the three valuation methods for both perpetual and finite-lived assets. However, these results will generally differ from those derived from Panel A, even in the perpetuity case, because of their different assumption about the risk of debt tax shields.

Another point that emerges from contrasting Panels A and B is the role of the weighted average cost of capital, which can be used in either case. This is because the market's valuation of debt tax shields is captured in the cost of equity, [r.sub.e]. If an estimated cost of equity correctly reflects investors' assumptions about the risk of the debt tax shields, whatever those assumptions may be, the weighted average cost of capital will yield a valid adjusted discount rate.

D. The Harris-Pringle Analysis

Harris and Pringle [14] have proposed that all debt tax shields, including the first year's, be treated as risky and discounted at r, which leads to the set of equations in Panel C of Exhibit 2. These can be thought of as the analogues of the Miles-Ezell expressions when the firm adjusts its debt level continuously to the target ratio. For example, the Miles-Ezzell cost of capital expression (2B.3) can be rearranged to produce:

[Mathematical Expression Omitted]

Then, dividing each period into arbitrarily small sub-periods, gives

[Mathematical Expressions Omitted]

Hence, expression (2C.3) is the continuous-time version of (2B.3).

The relationships in Panel C have a simpler form than those in Panel B, and as a practical matter, they do not yield valuations that are very different. Like the Miles-Ezzell equations, they are applicable to finite and uneven cash flow streams as well as level perpetuities, and they will give identical results under any of the three valuations methods. As in the other two panels of Exhibit 2, the weighted average cost of capital is a valid overall cost of capital expression, but here it reflects the relationship between [r.sub.e] and r that is expressed in (2C.4).

III. Valuation With Corporate and

Personal Taxes

In this section, we introduce personal taxes and derive valuation and cost of capital expressions analogous to those discussed in Section II. However, care must be taken in specifying investors' required rates of return. While these are often analyzed on a pretax basis, it is after-tax returns that ultimately drive the valuation process, and thus the equilibrium relationships among after-tax returns must be understood. In the remainder of this section, we assume that all investors pay taxes on income from debt securities at the rate [T.sub.p] and on income from equity securities at the rate [T.sub.pe].(8) For simplicity, we also assume that all investors are subject to the same tax rates.

A. After-Tax Rate of Return Relationships

In equilibrium, debt and equity securities of comparable risk must offer identical after-tax returns, or else investors will be motivated to rearrange their portfolios. In particular, suppose we have a risk-free debt security offering a pre-tax return per period of [r.sub.fd] and a risk-free equity security offering a pre-tax return per period of [r.sub.fe].(9) These returns will be set in the market so that:

[r.sub.fd](1 - [T.sub.p]) = [r.sub.fe](1 - [T.sub.pe]).

More generally, a tax-adjusted capital asset pricing model (CAPM) can be derived (e.g., [5], [11]), characterizing the after-tax returns on securities of both different risk and different tax treatment. Suppose, for example, that all risky assets are in the form of equity and that the expected pre-tax return to investors from this market equity portfolio is [r.sub.m]. If [r.sub.ej] is the pre-tax return on firm j's equity, then in equilibrium,(10)

[r.sub.ej](1 - [T.sub.pe])-[r.sub.fd](1 - [T.sub.p)=[beta.sub.j][r.sub.m] (1 - [T.sub.pe]-[r.sub.fd](1 - [T.sub.p])], (5)

where [beta.sub.j] is the systematic risk measure. Using Equation (4), this can be simplified to:

[r.sub.ej] = [r.sub.fe] + [beta.sub.j]([r.sub.m - r.sub.fe]).

The important point is that, in general, rates of return should be compared on an after-personal-tax basis. Pre-tax comparisons are valid only if all rates apply to instruments that receive identical tax treatment. Thus [r.sub.m], which is taxed at the rate [T.sub.pe], is directly comparable with [r.sub.fe], as in Equation (6), but not with [r.sub.fd], which is taxed at the rate [T.sub.p].

B. The Case of Constant, Perpetual Debt

For the level perpetuity case with constant debt, Miller [20] has shown the Equation (3A.1) in Exhibit 3 holds, where [G.sub.L] is the net tax benefit from corporate debt. This is the basis for the APV approach in this case. The remaining equations in Panel A of Exhibit 3 follow from (3A.1) and from basic valuation definitions.(11)

Two points are worth special note. First, one moves from the equations in Panel A of Exhibit 2 to those in Panel A Exhibit 3 by making two adjustments: (i) [T.sub.c], the corporate tax rate, is replaced by [G.sub.L], the net tax advantage to corporate debt; (ii) [r.sub.fd], the risk-free debt rate, is replaced by [r.sub.fe], the risk-free equity rate.(12) Indeed, it will be seen below that the same rule applies to every equation in Exhibit 3. In particular, if the CAPM is to be used to estimate a cost of equity, consistency requires that it be an after-personal-tax CAPM, as embodied in Equation (3A.5). Second, the assumptions that the operating cash flow stream is a level perpetuity and that debt is known and constant forever imply that the equations in this panel, like those in Panel A of Exhibit 2, are of limited practical use.

C. Finite Asset Life With a Known Debt

Schedule

As in Section II, one way to obtain more broadly applicable valuation expressions is to retain the assumption that future debt levels are known but allow for finite and uneven cash flow streams. Beginning with the APV approach, it might seem natural to generalize the analysis of Section II by simply substituting [G.sub.L] in place of [T.sub.c] in Equation (1) above. However, that would be incorrect. The reason is that the debt tax shields in the second term of Equation (1) are cash flows to the equityholders and should be discounted at an equityholders' opportunity cost. When there are only corporate taxes, investors would not distinguish between risk-free debt and risk-free equity, so it is perfectly appropriate to discount the second term in Equation (1) at the bondholders' opportunity cost, [r.sub.fd]. But when there are both corporate and personal taxes that is no longer the case. Instead, as shown in the Appendix, the correct generalization of Equation (1) is:

[Mathematical Expression Omitted]

Thus, not only must [G.sub.L] be substituted for [T.sub.c] in moving from Equation (1) to Equation (7), but also [r.sub.fe] must be substituted for [r.sub.fd]. The second term in Equation (7) might best be thought of as the present value of an annual financing subsidy. As in Brealey and Myers [4], the value of a subsidy can be calculated as the present value of the annual difference between unsubsidized and subsidized debt service charges (after corporate taxes), discounted at the unsubsidized market rate (also after-corporate-tax). Thus, if we let S represent the second term in Equation (7), (4) plus the definition of [G.sub.L] from Exhibit 1 can be used to write S as:

[Mathematical Expression Omitted]

That is, a firm wishing to issue a riskless claim on itself could issue either risk-free equity at the rate [r.sub.fe], or risk-free debt at the tax-subsidized rate [r.sub.fd](1 - [T.sub.c]). The net advantage to debt is simply the present value of this opportunity cost saving.

As when there are only corporate taxes, there are no adjusted discount rate expressions that will produce the same asset value given by Equation (7) in this case.

Thus the APV method is the only one of the three that gives theoretically correct results when the schedule of debt outstanding is known with certainty but the asset's life is finite.

D. Debt Adjusted Once Per Year to a Constant

Debt-to-Value Ratio

The second approach to obtaining more generally applicable results, as in Miles and Ezzell [18], is to assume that the firm's debt ratio is known and that the

(1) The adjusted discount rate method is also known as the weighted average cost of capital, or WACC method [15], while the flows to equity method is also known as the equity residual income method [10] or the equity residual value method [25]. [2] See [1], [10], [12], [14], [15], [18], [19], [21], [22], and [30] for discussions of this care. (3) Existing results for this case can be found in [4], [6], [8], [13], [16], [23] and [28]. (4) [C.sub.n] represents the expected after-tax cash flow the company would have gotten in period n if it had been entirely equity-financed, with no debt service charges deducted. (5) Some f the consequences of risky debt have been analyzed in [7], [23], [28] and [31]. In the absence of bankruptcy costs, Sick [28] has obtained similar results to this paper's but it should be emphasized that the cost of debt must then be interpreted as an expected, rather than a promised rate of return. It is incorrect to substitute the contractual debt rate for the risk-free rate. Similarly, under the APV approach, certainty-equivalent interest payments should be used in place of contractual interest payments. (6) This framework can also accommodate constant perpetual growth, as in Lewellen and Emery [16], but only level perpetuities are considered here to avoid excessive complication. (7) The CAPM can be used to compute a cost of equity for the current period, given an accurate estimate of [beta], but it is still impossible to use the FTE method. Future debt levels are known, but future firm value and hence future leverage are random. Thus future levels of [beta] are also random, so the appropriate cost of equity for valuing future cash flows cannot be determined today. (8) As in Miller [20], [T.sub.pe] should be interpreted as an effective tax rate, or as the uniform annual tax rate that would produce tax payments having the same present value as the pattern of actual tax payments on equity income. (9) In a CAPM context, [r.sub.fe] can be thought of as the pre-tax return on a zero-beta equity portfolio. Alternatively, [r.sub.fe] could be interpreted as the cost of equity for an entirely equity-financed firm with riskless assets. For the analysis that follows, it is not necessary that a specific, riskless equity security exist as long as one can be created synthetically. (10) A relatively simple proof can be constructed using the same steps found in Rubinstein [27], but with end-of-period investor wealth calculated after all taxes. (11) Specifically, Equation (3A.2) follows from the definitions E = [C - [r.sub.fd] D (1-[T.sub.c])] / [r.sub.e] and V = C/r*. Equation (2A.3) is obtained from equating (3A.2) and (3A.3) and solving for [r.sub.e.] Equation (3A.6) follows from writing (3A.5) for a levered and an unlevered firm and equating (3A.4) and (3A.5) for the levered firm. (12) These two rules are also valid for the weighted average cost of capital (3A.2), although not written in that form in Exhibit 3. That is because, from Equation (4) and the definition of [G.sub.L], [r.sub.fe](1-[r.sub.fd] (1-[T.sub.c)]. actual debt level is adjusted once per year this target ratio. Extending the Miles-Ezzell analysis to include both corporate and personal taxes thus leads to Equation (3B.1) in Exhibit 3 as the basic valuation relationship. Iterating backward from the end of the firm or asset's life, as shown in the Appendix, leads to an adjusted discount rate expression, (3B.3(), which is also consistent with the weighted average cost of capital, (3B.2).Expressions (3B.4) and (3B.6) follow in turn from equating (3B.2) with (3B.3).

The equations in Panel B of Exhibit 3 are applicable to both finite-and infinite-lived assets using any of the three valuation methods. In fact, they are exact analogues to the Miles-Ezzell equations in Panel B of Exhibit 2 with the same two adjustments that are made in Panel A: [T.sub.c] is replaced by [G.sub.L], and [r.sub.fd] is replaced by [r.sub.fe.].

E. Debt Adjusted Continuously to a Constant

Debt-to-Value Ratio

If the firm adjusts its capital structure continuously to a target debt ratio, all future debt tax shields, including the first year's, are risky. This implies that the basic valuation relationship is given by Equation (3C.1) in Exhibit 3, which generalizes the Harris-Pringle [14] analysis to include both corporate and personal taxes. Equation (3C.1) can then be used to derive the remaining expressions kin Panel C. Exhibit 3. These in turn produce identical firm values, for both perpetual and finite-lived assets, regardless of which valuation method is used.

A special feature of the continuous -adjustment case is that Equation (2C.6) is identical to Equation (3C.6). That is, the relationship between the levered and unlevered betas is the same, whether or not personal taxes are included in the analysis. As discussed in more detail in Section IV, Myers and Ruback [23] have exploited this fact to show that, under a special assumption about the firm's capital structure, the value of an asset will be the same, regardless of the assumed tax regime.

IV. Implementation Issues

A. A. Numeric Example

The use of these equations is illustrated by the example in Exhibit 4. The first step is to choose the appropriate set of valuation and cost of capital expressions. Since personal tax rates are present, this calls for the expression in Exhibit 3, rather than Exhibit 2. The asset has only a 10-period life, so this rules out the perpetuity expressions in Panel A, and if we further suppose that the firm adjusts to its target debt ratio once per period rather than continuously, this establishes Panel B as giving the relevant set of expressions.

Using the equations in Panel B, we can derive the values shown in the bottom half of Exhibit 4. For example, the adjusted discount rate method calls for using either Equation (4B.2) or (4B.3) to find the overall cost of capital, r(*) = 0.1593. Then, discounting the ten year's worth of operating cash flows at this rate gives V = 484.6. The same value can also be obtained with either of the other two valuation methods, but it should be noted that both are extremely cumbersome to use under the assumed capital structure adjustment scenario.

Exhibit 4. Numerical Example (the equations in Panel

B of Exhibit 3 are used to value an asset characterized by the assumed parameter values)

A. Assumed Parameter Values

N = 10 periods [C.sub.n] = 100 for all n [T.sub.p] = 0.28 [T.sub.pe] = 0.18 [T.sub.c] = 0.34 rfd = 0.07 [r.sub.m] = 0.15 [beta.sub.U] = 1.2 D/V = 0.05

B. Derived Values

[G.sub.L] = 0.2483, from definition in Exhibit 1 rfe = 0.0615, from Equation (4) r = 0.1677, from Equation (3B.5) [r.sup.*] = 0.1593, from Equation (3B.3) or from

(3B.2), if (3B.4) is used to estimate [r.sub.e] [r.sub.e] = 0.2724, from Equation (3B.4) or from

(3B.5)

V = 484.6, from discounting [C.sub.n] at [r.sup.*] [beta.sub.L] = 2.38, from Equation (3B.6) E = 242.3, from discounting equity cash

flows at [r.sub.e]

To implement the flows to equity method, for instance, it is necessary to know the entire future schedule of debt service charges. Such a schedule can be derived bu using the ADR method to obtain a period-by-period schedule of asset values, as shown in Exhibit 5, and these can in turn be multiplied by 0.5 to obtain the schedule of outstanding debt, From this,the after-tax debt service charge (including principal repayment) can be calculated for each period and subtracted from the after-tax operating cash flow, [C.sub.n] to obtain the equity cash flows shown in Column 5 Exhibit 5. These cash flows can then be discounted at [r.sub.e] = 0.2724 to obtain E = 242.3, which is consistent with V = 484.6.

[TABULAR DATA OMITTED]

To implement the adjusted present value method, the debt schedule in Exhibit 5 can be used to calculate the schedule of effective debt tax shields, [r.sub.fe] [G.sub.L] [D.sub.n-1], as shown in Column j6, However, these must be discounted iteratively, using Equation (3B.1), since only the next period's cash flow is certain at any given time. Following this procedure, one can duplicate the period-by-period schedule of asset value given in Column 1.

Under the assumed capital structure adjustment process, the FTE and APV methods are not only cumbersome but redundant in this case. To implement either, we effectively need the entire schedule of asset values, but this is what we are trying to determine in the first place. Thus even though all three methods can yield identical results in principle in this case, the ADR method is definitely preferred.

A second point that emerges from the numerical example concerns the schedule of debt. Equations such as (3B.2), (3B.3), and (3B.4) allow one to calculate a current cost of capital that is predicated on an entire schedule of future asset values, as illustrated in Exhibit 5. In fact, however, asset value is random over time, because of the random nature of the expected operating cash flows. As firm value changes over time, the schedule of expected future debt levels needed to maintain a constant debt-to-value ratio will also be updated.

The example can also give a sense of the magnitude of the error resulting from inappropriate use of different valuation methods. Since the expressions in Panel A of Exhibit 2 are the most familiar from textbooks and the literature, it might be common to use those, even though the asset in question has a finite life and personal taxes are relevant. For the parameter values in Exhibit 4, Equation (2A.5) would give r = 0.166 and Equation (2A.3)(the MM cost of capital formula) would give the overall cost of capital as r(*) = 0.1378. Discounting the ten years' worth of operating cash flows at r(*) then gives the asset value as V = 526.2, an error of 8.6% relative to the true value. This error will increase with asset life, with the difference between [G.sub.L] and [T.sub.c], and with the difference between r and [r.sub.fe] For example, if we keep other parameter values the same but move from a 10-year to a perpetual asset life, the estimated asset value is 718.4 using [r.sup.*] = 0.1392, an error of 14.4% relative to the correct value of 627.7 that is obtained using r(*) = 0.1593.

Another error to which the analyst might be susceptible is inappropriate use of the APV method. Suppose the analyst knows the initial debt level is 242.3 but assumes this level will remain constant, rather than changing over the asset's life. All other parameter values are as given in Exhibit 4. If the APV method is implemented using Equation (1) in the text, this results in V = 513.2, an error of 5.9% relative to the true value of 484.6. Again, this error will increase with asset life, with the difference between [G.sub.L] and [T.sub.c], and with the difference between r and [r.sub.fe].

Other types of errors, however, may be quite negligible. Suppose, for instance, that the analyst uses the expressions in Panel C of Exhibit 3, rather than those in Panel B, even though the asset's capital structure is not adjusted continuously. From Equation (3C.3), this results in r(*) = 0.1591 and V = 485.0, an error of less than 0.1% from the true value.

B. The Choice of a Valuation Method

Exhibit 6 provides a roadmap that can be used to navigate through the different cost of capital expressions and the circumstances under which one of the three valuation methods is preferred to the others. For simplicity, Exhibit 6 is keyed to a tax regime that includes both corporate and personal taxes and thus to the equations in Exhibit 3, but an exactly analogous roadmap could be keyed to Exhibit 2. Given the relevant tax regime, the next question is whether the operating cash flow stream is best viewed as a perpetual annuity of if a finite life and/or uneven cash flows are important characteristics of the asset. Most important, the analyst decide whether future debt tax shields can be treated as a known schedule or whether they vary with future firm values.

If the schedule of debt tax shields is known in dollar terms, the APV method is always appropriate , and for finite asset lives it is the only one of the three methods that is even feasible. One the other hand, if debt financing is specified as a fraction of asset value, the ADR methods is always appropriate, while the other two methods are cumbersome at best. In the latter case, moreover, the numerical example suggests that it makes little practical difference whether the analyst uses the Miles-Ezzell analogue expressions in Exhibit 3, Panel B or the Harris-Pringle analogue expressions in Exhibit 3, Panel C. This may in turn lead to a preference for Exhibit 3, Panel C, because of the simpler form of the equations.

C. When Can We Avoid the Choice of a Tax

Regime?

As Hamada and Scholes [13] argue, it may not be obvious which tax regime discussed above is driving asset values. Additionally , if personal taxes are included, unobservable magnitudes, such as the next tax gain to leverage [G.sub.L] and the risk-free equity rate, [r.sub.fe], must be estimated to move from one cost of capital expression to another.

If the analyst wishes to avoid these problems altogether, there are two special cases in which the choice of the tax regime is irrelevant. The first occurs when all debt tax shields are discounted at r (as in Panel C of Exhibit 2 or 3), and firm's capital structure is characterized by (E/V) = [beta.sub.U] and (D/V) = (1 -[beta.sub.U)]. In that case, Myers an Ruback [23] have shown that, for both tax regimes, the adjusted discount rate is

[r.sup.*] = [r.sub.fd](1-[beta.sub.U])(1 - [T.sub.c])+[beta.sub.U][r.sub.m]. (9)

Expression (9) thus provides a useful cost of capital expression when the analyst is uncertain about which tax regime better reflects the true security valuation process. The tradeoff, however, is that the analyst must accept the assumed capital structure.

The second special case occurs when all cash flows are riskless. Ruback [26] has shown under very general conditions that riskless corporate cash flows can be valued by discounting them at the after-corporate-tax riskless interest rate, regardless of the relevant tax regime. This follows by an arbitrage argument from the observation that such a stream can support 100% of its own value in debt financing. The current results are consistent with this analysis,as can be seen from Exhibits 2 and 3. When there are only flow stream would all-equity-financed riskless cash flow stream would have an unlevered cost of capital r = [f.sub.fd.] If D/V = 1, then, all of the overall cost of capital expressions in Exhibit 2 reduce to r(*) = [r.sub.fd] (1 - [T.sub.c]). With corporate and personal taxes, kr = [r.sub.fe] for a riskless, unlevered stream, and if D/V = 1, the third equation in each panel of Exhibit 3 reduces to r(*) = [r.sub.fe] (1 - [G.sub.L]. However, from Equation (4) and the definition of [G.sub.L], [r.sub.fe] (1 - [G.sub.L]) = [r.sub.fd](1 - [T.sub.c]. The weighted average cost of capital also reduces to r(*) = [r.sub.fd] (1 - [T.sub.c]) in this case.

V. Summary and Conclusions

This paper has analyzed the conditions under which the adjusted present value, adjusted discount rate and flows to equity methods all lead to identical asset valuations in the presence of corporate and personal taxes. Its principle conclusions are:

(i) Consistent valuation can be achieved if (a) the asset's operating cash flows and debt service charges are level perpetuities or (b) the firm maintain a constant leverage ratio. Depending on which of these assumptions is appropriate, the analyst must use the valuation and cost of capital expressions from only one panel of expressions i Exhibit 3.

(ii) The expressions in Exhibit 3, which incorporate both corporate and personal taxes, differ from the more familiar expressions in Exhibit 2, which include only corporate taxes, in two ways: [T.sub.c], the corporate tax rate, is replaced by [G.sub.L], the net tax advantage to corporate debt, and [r.sub.fd], the risk-free debt rate, is replaced by [r.sub.fe], the risk-free equity rate.

(iii) Although consistent valuation is possible using any of the three methods when the firm maintains a constant leverage ratio, the adjusted present value and flows to equity methods are cumbersome, and the adjusted discount rate method is preferred.

(iv) When the asset has a finite life but the schedule of outstanding debt is known with certainly, there are no adjusted discount rate expressions for either the overall cost of capital or the cost of equity, so the adjusted present value method must be used in this case.

(v) The weighted average cost of capital is robust to changes in both the tax regime and in the perceived risk of interest tax shields. This is because the relationship between the levered cost of equity, [r.sub.e], and unlevered cost, r, changes to reflect both tax factors and tax shield risk, leaving the weighted average formula intact. To correctly estimate the weighted average starting from scratch, however, the analyst must use the correct relationship between r and [r.sub.e]. A special case of the weighted average cost of capital, proposed by Myers and Ruback [23], does not require specific knowledge of the relationship between r and [r.sub.e], but does require a specific capital structure assumption.

References

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Homewood, IL. Richard D. Irwin, 1985. [14.] R.S. Harris and J.J. Pringle, "Risk-Adjusted Discount Rates - Extensions

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Perfect Capital Markets and Project Life: A Clarification,

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Appendix

I. Derivation of Equation (7)

Starting at the end of the valuation horizon, suppose the firm will pay a liquidating divided to its shareholders at time N, consisting of the after-tax operating cash flow, [C.sub.N.] minus the after-tax interest and principal on its beginning-of-period debt, [D.sub. N-1]. Personal taxes on equity income are levied against dividends plus capital gains, so beginning equity value, [E.sub.N.1], serves as a personal tax shield.

Letting [CEQ.sub.N] denote the pre-personal-tax certainty-equivalent of [C.sub.N], the value of the firm's equity and debt at N - 1 can be expressed as:

[Mathematical Expression Omitted]

Since these two discount rates must be the same in equilibrium, we can express beginning-of-period firm value, [V.sub.N-1] = [E.sub.N-1] + [D.sub.N-1], as

[Mathematical Expression Omitted]

Then using the definition of [G.sub.L], text Equation (4),and solving for [V.sub.N-1]:

[Mathematical Expression Omitted]

Iterating backward in similar fashion, the value of the firm at any time t is given by

[Mathematical Expression Omitted]

When risk-adjusted discount rate are appropriate (see [9], [24], [28]), (A5) can also be written in the more familiar form of Equation (7) in the text.

II. Derivation of Equation (3B.3) from Exhibit 3

First, write (A4) in its risk-adjusted form as

[Mathematical Expression Omitted]

If the ratio of debt to form value is a constant, we can express (A6) as

[Mathematical Expression Omitted]

[V.sub.N-2] is in turn equal to the sum of the values as of time N - 2 of three terms: [C.sub.N-1] , the net tax advantage of interest paid at time N - 1, and [V.sub.N-1]. Discounting [V.sub.n-1] at the unlevered cost of capital r, backward iteration of (A7) yields, for any time t:

[Mathematical Expression Omitted]

Setting the term in square brackets equal to (1 + r(*)) and solving for r(*) then gives Equation (3B.3) in Exhibit 3.

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Title Annotation: | Topics in Cost of Capital |
---|---|

Author: | Taggart, Robert A., Jr. |

Publication: | Financial Management |

Date: | Sep 22, 1991 |

Words: | 7080 |

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