# Valuation of callable bonds under progressive personal taxes and interest rate uncertainty.

* Several studies have offered tax-based rationales for the
widespread practice of attaching call provisions to corporate bonds.
Boyce and Kalotay [4] demonstrate that if there is a positive difference
between the tax rates of the corporation (borrower) and the typical
investor (lender), then there will be a net tax benefit from the call.
Their argument is that since the interest schedule on a callable bond and its future refunding bond can never increase, but may decrease, the
present value of the stream is higher for investors facing lower tax
rates (higher after-tax discount rates) and lower for corporations
(higher present value of interest tax shields). In a different vein,
Yawitz and Anderson [19] and Marshall and Yawitz [12] considered the net
tax benefit associated with the call premium paid when the bond is
called.(1) Since the call premium is taxed as a capital gain to
investor, but is deductible as an ordinary income expense to the firm,
they concluded that the expected tax benefit for a callable bond will be
higher than for an otherwise identical but noncallable bond. Recently,
however, Brick and Wallingford [5] have argued that the call premium tax
asymmetry may produce even greater tax benefits for noncallable debt
repurchased at market prices because there will be an expanded set of
profitable call opportunities. Thus, they argue that the call premium
tax asymmetry probably cannot explain the widespread practice of
attaching call provisions to bonds.

Although much has been learned from existing tax based analyses of callable bonds, the extant models suffer from both theoretical and practical deficiencies. First, most analyses do not consider the broader capital structure implications of their tax assumptions. For example, the assumption that the personal tax rate is below the corporate tax rate, which drives Boyce and Kalotay's conclusions, implies a corner leverage decision not-withstanding the call feature studied. To avoid such empirically unreasonable implications, the tax implications of particular bond characteristics should be studied under the benchmark conditions of Miller's [15] equilibrium analysis which leads to capital structure irrelevance.(2) Of course, the advantage of this approach is that it allows for the analysis of the tax effects of bond features (e.g., call provision) net of the tax gains from leverage. Second, the universally employed assumption that personal tax rates are fixed and equal across investors obscures the basic reality that future marginal personal tax rates are uncertain. This feature of the tax environment, which arises from the interaction between progressive personal taxation and uncertain future income (both wage and portfolio), induces a temporal association between future tax rates and interest rates, which may significantly affect the relative corporate aop personal valuation of the call.

The current analysis examines the value of the call under the realistic assumption that future marginal personal tax rates and interest rates are uncertain. Although the analysis does not depend on any particular covariance function between tax and interest rates, logic suggests that it is reasonable to assume that both variable are positively affected by the level of business conditions and therefore have a positive partial correlation. An upturn of the economy may be characterized by high marginal personal tax rates and interest rates, while a downturn may be associated with low tax interest rates. Indeed, inspection of the empirical record of nominal interest rates over the last three decades provides considerable evidence in support of a positive association between interest rates and economic activity. Short-term rates have generally been high during expansionary periods and low during recessions, while long-term rates have generally followed the same pattern (see, for example, Van Horne [18]).

While less direct evidence is available concerning the relationship between marginal personal tax rates and the state of the economy, a positive association is predicted by the recent capital structure models of Dammon [7] and Ross [17], and the subsequent multiperiod extension of those models, by Lewis [11]. For example, in Dammon's model, wherein investors face the same progressive tax function defined over uncertain taxable income, marginal personal tax rates are predicted to be highly correlated with aggregate per capita consumption. As such, marginal personal tax rate are also likely to be highly correlated with economic activity.(3) Consistent with expectations, empirical evidence on the correlation between marginal personal tax rates and interest rates indicates a positive association between the two variables.

Given a positive relationship between marginal personal tax rates and interest rates, the value of a call provision on a bond must include a "tax premium" which, ceteris paribus, makes the decision to issue a callable bond a negative sum game from the corporation's perspective. The intuition behind this result is straightforward. When tax and interest rates are positively correlated, investors will value bond income more highly in those states in which the firm will most likely call the bond. As a result, investors will require a tax premium for the inclusion of a call provision on a bond which cannot be recovered by firms. The implication is that, at least with respect to this realistic feature of the tax environment, tax arguments alone cannot rationalize the widespread practice of attaching call provisions to bonds.

I. The Model and thee Valuation of Noncallable Bonds

The framework for analysis is a simple multiperiod state-preference model. There are three dates:t = 0, t = 1,andt = 2. At time 1, there are two possible states of nature. Depending on which state is realized, the state of the economy is assumed fixed over the next period (from time 1 to time 2). Thus, all uncertainty is resolved at the end of the first period. Denote the value at time 0 of $1.00 (before-tax) in state 1 as [p.sub.01], and the value at time 0 of $1.00 (before-tax) is state as 2 as [p.sub.2]. Similarly, the time 1 value of $1.00 (before-tax) received at time 2, conditional on state 1 having occured at time 1, is denoted as [p.sub.11]. The corresponding price conditional on state 2 is denoted as [p.sub.12]. The only "macroeconomic" distinction between the two states at time 1 is the level of interest rates in the economy. Assume that the before-tax single-period riskless rate of interest in state 1, [r.sub.1], is greater than the before-tax single-period riskless interest rate in state 2, [r.sub.2]. Those interest rates both define and characterize the time 1 conditional state prices as

1 1 [p.sub.11] = ------------- < [p.sub.12] = -------------. (1)

1 + [r.sub.1] 1 + [r.sub.2]

On that basis, the restrictions on the time 0 before-tax state prices are as follows:

1 [p.sub.01] + [p.sub.02] = ---------------, and 0 < [p.sub.02] < [p.sub.01] < 1. (2)

1 + [r.sub.0]

The first condition in Equation (2) says that the sum of the two state prices equals the current single-period before-tax riskless discount factor. The second condition follows from the more favorable (re)investment opportunities investors face in state 1 at time 1 versus state 2 at time 1, i.e., [r.sub.1] > [r.sub.2]. Therefore, at time 0, investors will value $1.00 in state 1 more highly than $1.00 in state 2.

Investor tax rates are assumed to be heterogeneous, as in Miller [15], but also state dependent. Accordingly, at time 1, an investor's tax rate will be either [[Tau].sub.1] in state 1 or [[Tau].sub.2] in state 2. The tax rates are progressive in the sense that the state 1 tax rate, [[Tau].sub.1], is greater than the state 2 tax rate, [[Tau].sub.2]. Furthermore, given the heterogeneity of investor tax rates, [[Tau].sub.1] and [[Tau].sub.2] need not be the same across different investors. (Note that the simple progressive structure of the tax rates ensures a positive correlation with the time 1 interest rates which define the state prices). As in Miller, assume that equity income is not taxed, and that corporations are taxed at the marginal rate [[Tau].sub.C], which is spanned by investors tax rates. Finally, also as in Miller, assumed that investors are prohibited from engaging in tax arbitrage and therefore cannot, by short sales or other means, drive their tax rates to zero.

The Miller capital structure equilibrium is determined where the marginal investor's tax rate is equal to the corporate tax rate. The identification of that investor at time 0 is considerably more complicated when investors have stated-dependent time 1 tax rates. Assuming that corporate debt is riskless, and therefore pays the same relative amount of taxable cash flow in either state 1 or state 2, a Miller-type equilibrium will emerge at time 0 in which the marginal investor is identified as having a weighted average time 1 tax rate equal to the corporate tax rate. Thus, as in the Miller analysis, capital structure irrelevance results and tax clienteles form in which investors will hold either equity or debt. This correspondence with the Miller framework will breakdown if, for example, bonds are risky and therefore possibly pay different relative quantities of taxable cash flow across the two states at time 1. With state-dependent investor tax rates, bonds with different relative state-dependent taxable cash flow will no longer be perfect substitutes. The resulting tax clienteles will generally lead to a breakdown of the Miller debt invariance proposition.[4] Therefore, in order to sharpen the focus of the analysis on the value of the call feature on corporate bonds and to conduct the analysis to the fullest extent possible in the original Miller framework, corporate debt is assumed to be risk-free.

Since the level of investor tax rates is changing at time 1, the marginal investor at time 0 will not be the same marginal investor in either state 1 or state 2 at time 1. As a consequence, the capital structure equilibrium at time 1 will differ from that at time 0. To allow for such capital structure rearrangements, assume that at time 1 the corporation buys back its time 0 original issue bonds and finances the repurchase by issuing single-period debt which matures at time 2. Correspondingly, investors sell the bonds back to the firm, realize any capital gain/loss on the transaction, and rebalance their portfolios. The implication of this process is that the market price of debt in both states 1 and 2 at time 1 will reflect the corporate tax rate, since the marginal investor in either state will, as per the Miller equilibrium, be in the same tax bracket as the firm. Exhibit 1 summarize the tax and interest rate environment and displays the timing of trading in the bond market.

Initially, suppose that the firm issues a two-period default-free noncallable bond. Choosing a coupon payment C, such that the bond sells for par, B, gives B = [p.sub.01][C (1 - [[Tau].sub.C] + [B.sub.1] - [[Tau].sub.C]([B.sub.1] - B)] + [p.sub.02] [C (1 - [[Tau].sub.C]) + [B.sub.2] - [[Tau].sub.C] ([B.sub.2] - B)], (3) where [B.sub.1] and [B.sub.2] are the time 1 market values of debt in states 1 and 2, and are equal to [B.sub.1] = [p.sub.11] [C (1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.1])], (4) and [B.sub.2] = [p.sub.12] [C(1 - [[Tau].sub.C] + B - [[Tau].sub.C](B - [B.sub.2])], (5) respectively. Both [B.sub.1] and [B.sub.2] reflect the corporate tax rate, in accordance with bond market equilibrium in states 1 and 222 at time 1.(5) In Equation (3), note that since the bond sells for par, and assuming that [r.sub.1] > [r.sub.0] > [r.sub.2], it must hold that [B.sub.1] < B and [B.sub.2] > B. As such, if state 1 occurs at time, the firm buys back discount debt and the gain is considered for tax purposes as ordinary income. Alternatively, if state 2 occurs, the firm buys the debt back at a premium and realizes a tax-deductible loss.(6) Note that while repurchasing debt and incurring a capital gains tax liability may not be optimal, repurchasing premium debt and realizing the loss for tax purposes is optimal (see Mauer and Lewellen [13]). Importantly, however, the reason that the firm recapitalizes at time 1 is to its capital structure in response to changes in the net tax advantage of debt occasioned by state-dependent investor tax rates.

The marginal investor at time 0 is identified as that investor who is indifferent between tax-exempt equity and taxable corporate bonds at the price B, Let [[Tau].sub.m1] and [[Tau].sub.m2] denote that investor's time 1 state-dependent tax rates. To satisfy the indifference requirement, those tax rates must equate the after-tax stream of payments received from the bond to its par value. Accordingly, B = [p.sub.01][C (1 - [[Tau].sub.m1]) + [B.sub.1] - [[Tau].sub.m1]([B.sub.1] - B)] + [p.sub.02][C(1 - [[Tau].sub.m2]) + [B.sub.2] - [[Tau].sub.m2]([B.sub.2] - B)]. (6) As in the Miller equilibrium, heterogeneity of investor tax rates implies that tax clienteles will form. Those investors who, based on their own tax rates, value corporate bonds that at lower values than B prefer tax-exempt equity. Those who value the bonds at higher values than B will prefer bonds. Note that since investors sell the bonds at equilibrium prices at time, the bond value at time 0 does not reflect investors tax liabilities at time 2.(7)

The time 0 capital structure equilibrium is established where the corporate tax rate, [[Tau].sub.C] falls between the marginal investors's time 1 personal tax rates [[Tau].sub.m1] and [[Tau].sub.m2]. When that condition is met, the corporation will be different between noncallable debt and equity.(8) Thus, setting Equation (3) equal to Equation (6), and solving for the corporate tax rate, gives [[Tau].sub.C] = [Alpha][[Tau].sub.m1] + (1 - [Alpha]) [[Tau].sub.m2] (7) where

Since [[Tau].sub.m2] < [[Tau].sub.m1], it is clear from Equation (7) that [[Tau].sub.m2] < [[Tau].sub.m1]. Given the identification of the time 0 marginal investor in noncallable bonds, the analysis now turns to the valuation of callable bonds. The critical issue to investigate is whether under state-dependent investor tax rates that same marginal investor places a higher value on the call provision than does the corporation.

II. Callable Bond Valuation

A. The Call Tax Premium

Suppose that instead of issuing a noncallable bond, the firm issues a callable bond. Since the focus is not the determination of the optimal time to call the bond (see Mauer [14]), assume that the bond will be called if the interest rate (or rather, the state price) falls (rises) to [r.sub.2] ([p.sub.12]) at time. Additionally, initially assume that the call price is the par value of the bond B. Thus, there are no tax consequences associated with any call premium paid when the bond is called. Denoting [C.sup.*] as the coupon payment which allows the callable bond to sell for par at time 0, gives B = [p.sub.01][[C.sup.*](1 - [[Tau].sub.m1] + [B.sub.C1] - [[Tau].sub.m1]([B.sub.C1] - B)] +[p.sub.02][[C.sup.*](1 - [[Tau].sub.m2]) + B], (9) where [B.sub.C1] = [p.sub.11] [[C.sup.*](1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.C1], (10) and is equal to the market value of the callable bond in state 1 at time 1. Note that this formulation assumes that the bond is called an instant after the coupon is received in state 2 at time 1. Futhermore, [C.sup.*] is determined in Equation (9) as the coupon payment which would make the marginal investor of the previous section indifferent between purchasing a callable or a noncallable bond at time 0.

Since a callable bond can be viewed as a noncallable bond on which the purchaser (the bondholder) has written a call option held by the issuer (the borrowing company), the callable bond value in Equation (9) can be decomposed into a noncallable bond with a coupon of [C.sup.*], minus the value of the call option. Thus, to compute the investor perceived call value, we simply take the difference between a noncallable bond evaluated at a coupon of [C.sup.*] (from Equation (6)) and the callable bond value in Equation (9). Performing the indicated calculation gives [V.sub.M] = [p.sub.02] [([B.sub.C2] - B)(1 - [[Tau].sub.m2)], (11) where [B.sub.C2] = [p.sub.12] [[C.sup.*](1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.C2])]. (12) Realize that [V.sub.M] is the call value which would make the marginal investor indifferent between purchasing a noncallable bond with a coupon of C, and a callable bond with a coupon of [C.sup.*]. That value reflects the difference between that the investor would have received in state 2 if the bond was not called, [B.sub.C2] - [[Tau].sub.m2] ([B.sub.C2] - B), and the call price, B. Thus, [V.sub.M] measures the cost of the call.

The value of the call provision from the standpoint of the corporate issuer is derived in a similar fashion. The objective is to compare the payments on a callable bond to those on an otherwise identical, but noncallable bond.(9) If state 1 occurs at time, the firm will not exercise the call. Instead, the firm will repurchase the discount debt at market prices, just as an the case of noncallable debt analyzed above. However, if state 2 occurs, the firm will call the bond for B. The net benefit of the call to the firm is the difference between what it would have paid if the bond was not called, [B.sub.C2] - [[Tau].sub.C] ([B.sub.C2] - B), and the call price, B. The value of that difference, using the current state 2 price, is equal to [V.sub.F] = [p.sub.02][{B.sub.C2] - [[Tau].sub.C]([B.sub.C2] - B)} - B] = [p.sub.02][([B.sub.C2] - B)(1 - [[Tau].sub.C])], (13) which quantity measures the ex ante benefit of the call to shareholders.

The main result of the analysis is obtained by comparing [V.sub.F] and [V.sub.M]. Since [V.sub.M] is the call value which would make the marginal bondholder indifferent between callable and noncallable debt, the only way in which shareholders of the firm would be similarly indifferent is if [V.sub.F] were equal to [V.sub.M]. If, on the other hand, [V.sub.F] is less than [V.sub.M], the issuing callable debt is a negative sum game from the corporation's perspective. Taking the difference between Equation (13) and Equation (11) gives [Delta] = [V.sub.F] - [V.sub.M] = [p.sub.02][([B.sub.C2] - B)([[Tau].sub.m2] - [[Tau].sub.C])], (14) which is negative because of the simultaneous determination of low interest rates (which imply [B.sub.C2] > B) and low tax rates (which imply [[Tau].sub.m2] < [[Tau].sub.C]) in state 2 at time 1. Thus, when investors tax rates are positively correlated with interest rates, the after-tax value of bond income will be greater than the after-tax value of equity income in states wherein the firm is likely to call the bond. Accordingly, investors will require a "tax premium" for the inclusion of a call provision on a bond.

The intuition behind this result can be made transparent by comparing the cash flow consequences of the call to the marginal investor and to the firm. The difference between the cash flows from not calling versus calling to the firm and to the marginal investor is [{[B.sub.C2] - [[Tau].sub.C]([B.sub.C2] - B)} - B] - [{[B.sub.C2] - [[Tau][.sub.m2] ([B.sub.C2] - B)} - B] (15) which quantity simplifies to [[Tau].sub.m2]([B.sub.C2] - B) - [[Tau].sub.C]([B.sub.C2] - B) = ([B.sub.C2] - B)([[Tau].sub.m2] - [[Tau].sub.C]). (16) An examination of the left side of Equation (16) reveals that the differential cash flow impact of the call is a function only of the taxes that would have been paid had the bond not been called. Since [B.sub.C2] > B and [[Tau].sub.m2] < [[Tau].sub.C] the capital incurred is smaller than the tax deduction that the firm could claim by repurchasing premium debt at market prices. As a consequence, the cost of the call to the marginal investor is higher than the benefit of the call to the firm. This, of course, leads to a negative call tax-premium.(10)

Thus, with state-dependent marginal investor tax rates and the absence of any other advantages associated with callable bonds, noncallable bonds will dominate callable bonds. the implication is that corporations would only be willing to supply noncallable bonds. Furthermore, under the conditions in Section 1 above, in equilibrium, there would be no optimal capital structure for any individual firm, but there would be a determinate aggregate amount of noncallable debt in the economy.

These conclusions contrast with those provided by Boyce and Kalotay [4], who argue that there is a tax advantage to callable debt. The difference in the results is directly attributable to the underlying tax assumptions. Boyce and Kalotay assume that all investors have the same constant tax rate which is lower than the corporate tax rate. In such a setting, there will always be a tax advantage to having a call provision on a bond. Indeed, everything else held constant, all firms would also be predicted to be 100% debt financed, since there would then be a strictly positive net tax advantage to debt.

There is however, a qualification to the results obtained above. An implicit assumption of the analysis is that the call features on corporate bonds are identical and that therefore the relative quantities of taxable cash flow supplied by callable bonds are equal in states 1 and 2 at time. If this were not the case, then it may be impossible to find a bondholder who would be marginal in all callable bonds. As such, definitive statements about the relative value of the call feature could not be made since bonds with different call features could be priced at different implicit personal tax rates, Here identical call features are conveniently assured by the assumption that the call price is the face value of the debt. However, differential taxable call premiums in call prices may lead to a violation of this condition. Fortunately, actual call premium in call prices are relatively homogeneous across corporate bonds and are therefore not likely to engender significant differences in relative quantities of taxable cash flows across bonds. Thus, the conclusions are likely to hold even when bond call provisions contain call premiums.

B. An Illustrative Example

To see the process by which a negative call tax-premium arises, consider the following numerical example. Assume: [p.sub.01] = 0.727, time 0 price of $1.00 (before-tax) in state 1 at time 1; [p.sub.02] = 0.812, time 0 price of $1.00 (before-tax) in state 2 at time 1; [r.sub.1 ] = 0.12, before-tax single-period riskless interest rate in state 1 at time 1; [r.sub.2] = 0.03, before-tax single-period riskless interest rate in state 2 at time 1; [[Tau].sub.C] = 0.35, marginal corporate tax rate; and [[Tau].sub.m1] = 0.45 marginal investor's tax rate in state 1 at time 1.

The time 0 state prices determine, from Equaltion (2), a current rate of interest ([r.sub.1]) equal to 10%. The higher price for state 1 reflects the fact that state 1 is more favorable to the investor than state 2, since [r.sub.1] is greater than [r.sub.2]. The noncallable bond coupon payment C, which satisfies Equation (3), at a par value of B = 100, is computed to be 15.4. The corresponding state 1 and 2 bond prices are [B.sub.1] = 97.41 and [B.sub.2] = 110.30. Equation (7) searches for the tax rates of the "marginal investor". To find a unique solution, the marginal investor's state 1 tax rate ([[Tau].sub.m1]) is assumed to be 45% which, along with a marginal corporate tax rate ([[Tau].sub.C]) of 35%, identifies the investor's state 2 tax rate ([[Tau].sub.m2] as 15%. (The weight, [Alpha]. which satisfies Equation (7) at those tax rates is 0.666).

With the same time 0 marginal investor identified, the callable bond can now be valued. Substituting the marginal investor's tax rates into Equation (9), and assuming a call price investor is indifferent between the noncallable and callable bond at a coupon payment ([C.sup.*]) on the latter of 17.2. The corresponding callable bond prices in states 1 and 2 are equal to [B.sub.C1] = 98.92 and [B.sub.C2] = 112.01, respectively. The investor perceived call value [V.sub.M] (from Equation (11)), comes out to be 1.85 and the value of the call to the firm [V.sub.F] (from Equation (13), equals 1.42. Accordingly, the call tax-premium ([Delta]), from Equation (14), equals -0.43, or 0.43% of the bond's par value. Note that the magnitude of the tax premium is arbitrarily detrmined by the parameters employed and by the use of a two-period framework. Increasing the number of periods will increase the tax premium.

C. State-Dependent Marginal Corporate Tax Rates

In addition to depending on the correlation between interest rates and marginal investor tax rates, the magnitude of the call tax-premium also depends on the spread between the marginal corporate tax rate and the marginal investor's tax rate in state 2 at time 1. Although if simplifies the analysis to assume that firms face constant marginal corporate tax rates, and that therefore the spread is determined by variation in the marginal investor's tax rate, it is not necessary to the result. In a setting in which the marginal corporate tax rate also varied over time, the magnitude of the call tax-premium would depend on the relative sensitives of both corporate and personal tax rates to the state of the economy.

In practice, the marginal corporate tax rate is probably less sensitive than are personal tax rates to economic conditions. This is a result of the tax-loss carryforward and carryback provisions for corporations. For example, even when taxable corporate income would be zero in an aconomic recession, corporations can use carryforward/back provisions so as not to lose tax shields. Current tax law allows corporations to carry losses back three years to offset against previous years' taxable income or to carry losses forward 15 years. Thus, the maximum loss in corporatetax shield value is the time value of money, and this occurs only when the previous years' taxable income is insufficient to offset the current year's losses. The implication is that the marginal corporate tax rate would be expected to be more stable over time than personal tax rates, and that therefore the call tax-premium would continue to be negative.

D. Asymmetric Taxation of the Call Premium

The analysis above assumes that the bonds are called at par in order to sharpen the focus on the call tax-premium. However, in practice most call provisions require that bonds be redeemed at par plus a call premium, which is typically equal t one year's interest and may decline over the life of the bond according to a prearranged schedule. The important feature of this call premium is that the corporation can immediately deduct it for tax purposes when the bonds are called. Accordingly, if the call price in the model includes a premium above the bond's par value, then there will be an additional differential tax effect which influences the value of the call. Since the call premium is tax deductible to the corporation and treated as a capital gain from the standpoint of the bondholder, exercise of the call will reduce the aggregate tax liability of both parties at the expense of the government.

The problem with this line of reasoning, however, is that one cannot therefore conclude that because of the potential offsetting tax benefit associated with the call premium paid, callable bonds would be preferred over noncallable bonds. The reason is that any premium paid above the bond's par value, whether the bond has a call provision or not, is tax deductible to the corporation as an ordinary income expense and taxable to bondholders as a capital gain. As a consequence, this tax benefit is not unique to callable bonds and therefore could only justify the existence of call provisions if superior tax benefits were achievable vis-a-vis the alternative of simply repurchasing noncallable bonds at market prices.

Along those lines, a recent series of papers (Emery and Lewellen [8], Brick and Wallingford [5], and Mauer and Lewellen [13]) have demonstrated that the tax benefits associated with repurchasing noncallable premium debt are likely to be superior. For example, using simulation analysis, Brick and Wallingford demonstrate that under almost any reasonable economic environment, the call premium tax asymmetry produces greater expected tax benefits for noncallable debt repurchased at market prices. The reason is that the set of profitable call opportunities for a noncallable bond repurchased at market price will, in general, be greater than those of a callable bond repurchased at fixed call premiums. This will be especially true if those fixed call premiums decrease over the life of the bond, as is typical of most call provisions. Consequently, although there may be tax benefits associated with bonds callable at fixed call premiums which are sufficient to overturn the negative tax premium examined here, there is no reason to believe that those benefits can rationalize the existence of call provisions.

III. Empirical Evidence on the Association Between Interest Rates and Personal Tax Rates

As is apparent in Equation (14), the economic significance of the call tax-premium is critically dependent on the correlation between marginal personal tax rates and market rates of interest. That the correlation is positive is intuitive and reasonable, and one possible line of economic reasoning for such as relation has been given in the introduction. Nonetheless, it is important to empirically assess the relationship.

One approach would be to infer marginal investor tax rates from taxable corporate and nontaxable municipal bond yields and then compute correlations between those (implied) tax rates and interest rates. Unfortunately, this approach is subject to criticism on the grounds that the marginal tax rates are themselves derived from interest rates, which may lead to spurious correlations. A better approach would be to correlate interest rates with a time-series of marginal tax rates derived from individuals' income tax statements. Fortunately, the required time-series of tax rates exists in the literature.

Barro and Sahasakul [12] calculte average marginal tax rates for each year from 1916-1980 from the federal individual income tax as reported in the Internal Revenue Service's Statistics of Income. Exhibit 2 reports correlation coefficients between their arithmetic series and three different annual interest rate series reported in the Federal Reserve Bulletin for the period 1945-1980. The interest rate series includes: high-grade (Aaa) corporate bonds, high-grade (Aaa) municipal bonds, and 4-6 month commercial paper.

Every correlation coefficient is positive and significant and they range from 0.69 for commercial paper and municipal bonds to 0.75 for high-grade corporate bonds. However, since both tax and interest-rate series display an upward trend over time, the correlations may be entirely driven by a common association with time. As such, it is important to recompute the correlations after correcting for trend in the time-series. The second row in Exhibit 2 displays adjusted correlation coefficients which are partial correlations between tax rates and interest rates after controlling for linear time trend in the series. Although the adjusted correlations are smaller than their unadjusted counterparts, they are still positive and generally significant. For example, the correlation between marginal personal tax rates and corporate bond yields goes from 0.75 to 0.39, with the latter coefficient still significant at the 1% level. This evidence is encouraging, since it provides empirical support for the assumption of a positive relationship between marginal investor tax rates and market rates of interest. The call tax-premium is therefore likely to be economically significant.

IV. Conclusions

The analysis examines the relative tax benefits of a call provision on a bond under the assumption that investors have progressive tax rates which covary with interest rates. When marginal bondholder tax rates and interest rates are positively correlated, bondholders place a higher value on the call than the firm. Thus, bondholders require an additional tax premium for the inclusion of a call provision on a bond which makes the issuance of a callable bond a negative sum game from the corporation's perspective. Empirical evidence on the extent to which tax rates and interest rates positively covary indicates that the call tax-premium is likely to be nontrivial. Finally, it is argued that callable bonds continue to carry a tax disadvantage even if marginal corporate tax rates are not constant and if there is a positive expected tax benefit associated with the asymmetric taxation of the call premium on the bond.

In conclusion, it should be noted that the role played by call provisions in resolving other market imperfections, such as informational asymmetry and/or agency problems, may be sufficient to rationalize their widespread use. For example, Bodie and Taggart [3] suggest that callable debt may mitigate incentives for firms to reject profitable investment opportunities if the gains from those investments would be captured by bondholders. Similarly, Barnea, Haugen, and Senbet [1] argue that call provisions may mitigate the agency problems of informational asymmetry and stockholder risk incentives. (1) The call premium is the difference between the call price and the par value of the bond. Typically, the call premium is equal to one year's interest and declines throughout the life of the bond according to a prearranged schedule. Call provisions are also deferred for a period of time. Standard deferred call periods are 5 years for public utility bonds and 10 years for industrial bonds. (2) A notable exception in this regard is the paper by Brick and Wallingford [5], which discusses the implications of the call premium tax asymmetry for the Miller equilibrium. (3) Under the Tax Reform Act of 1986, tax rates continue to be progessive although fewer brackets exist. (4) Park and Williams [16] and Zechner [20] provide detailed analyses of the conditions which lead to tax-based bondholder clienteles and the resulting implications for corporate capital structures. (5) Since the marginal investors in states 1 and 2 at time 1 have tax rates which are equal to the corporate tax rate, the time 1 market prices of debt, [B.sub.1] and [B.sub.2], depend only on the corporate tax rate. Additionally, since [B.sub.1] < B, the state 1 bond price reflects the capital gains tax liability that the marginal bondholder faces at time 2. On the other hand, since [B.sub.2] > B, the state 2 bond price reflects the realization at time 2 of the tax-deductible premium paid for the bond. (6) See Finnerty, Kalotay, and Farrell [9] for details on the tax rules appropriate to corporate repurchases of discount and premium debt. (7) Equation (6) includes the capital gains taxes paid upon sale of the bond at time 1. In particular, given that the marginal investor's tax basis for the computation of capital gains, is B, in state 1 in the investor realizes a tax-deductible loss, and in state 2 in the investor pays taxes on the capital gain. Note that the capital gains taxes are computed using the marginal investor's ordinary income tax rates [[Tau].sub.m1]] and [[Tau].sub.m2]. This, of course, is congruent with the Tax Reform Act of 1986 wherein the distinction between ordinary and capital gains tax rates was eliminated. (8) As in Miller [15], the assumptions are sufficient to generate an equilibrium in which capital structure is indeterminate at the level of the individual firm, but determinate at the level of the corporate sector as a whole. However, the time 0 capital structure equilibrium is unstable in the sense that a new equilibrium will be established at time 1. In particular, given state 1 at time 1, state-dependent investor tax rates shift up with the interest rate and the advantage of debt financing will be reduced. Accordingly, corporate supply-side adjustments will ensure that a new equilibrium will be established in which the aggregate supply of noncallable debt in the corporate sector will also be reduced. A similar, although opposite, adjustment will occur given the realization of state 2 at time 1. (9) See Kraus [10] and Marshall and Yawitz [12] for a similar approach to call valuation. (10) Note that the cost of the call to the marginal investor would actually be higher if the investor could have escaped taxation of the capital gain in state 2 at time 1 by following the tax-trading rules outlined in Constantinides and Ingersoll [6]. Thus, to the extent that the analysis assumes complete realization of all capital gains and losses - in Constantinides and Ingersoll's terminology, a "continuous realization policy" - the magnitude of the call tax-premium is understated.

References

[1.] A Barnea, R.A. Haugen, and L.W. Senbet, "A Rationale for Debt Maturity Structure and Call Provisions in the Agency Theoretic Framework," Journal of Finance (December 1980), pp. 1223-1234. [2.] R.J. Barro and C. Sahasakul, "Measuring the Average Marginal Tax Rate from the Individual Income Tax," Journal of Business (October 1983), pp. 419-452. [3.] Z. Bodie and R.A. Taggart, "Future Investment Opportunities and the Value of the Call Provision on a Bond," Journal of Finance (March 1976), pp. 1187-1200. [4.] W.M. Boyce and A.J. Kalotay, "Tax Differentials and Callable Bonds," Journal of Finance (September 1979), pp. 825-838. [5.] I.E. Brick and B.A. Wallingford, "The Relative Tax Benefits of Alternative Call Features in Corporate Debt," Journal of Financial and Quantitative Analysis (March 1985), pp. 95-105. [6.] G.M. Constantinides and J.E Ingersoll, "Optimal Bond Trading with Personal Taxes," Journal of Financial Economics (September 1984), pp. 299-335. [7.] R.M. Dammon, "A Security Market and Capital Structure Equilibrium Under Uncertainty With Progressive Personal Taxes," in A. Chen (ed.), Research in Finance, Vol. 7, Greenwich, CT, JAI Press, 1987. [8.] D.R. Emery and W.G. Lewellen, "Refunding Noncallable Debt," Journal of Financial and Quantitative Analysis (March 1984), pp. 73-82. [9.] J.D. Finnerty, A.J. Kalotay, and F. Farrell, The Financial Manager's Guide to Evaluating Bond Refunding Opportunities, Ballinger Publishing Company, 1988. [10.] A. Kraus, "An Analysis of Call Provisions and the Corporate Refunding Decision," Midland Corporate Finance Journal (Spring 1983), pp. 46-60. [11.] C.M. Lewis, "A Multiperiod Theory of Corporate Financial Policy Under Taxation," Journal of Financial and Quantitative Analysis (March 1990), pp. 25-43. [12.] W.J. Marshall and J.B. Yawitz, "Optimal Terms of the Call Provision on a Corporate Bond," Journal of Financial Research (Summer 1980), pp. 202-221. [13.] D.C. Mauer and W.G. Lewellen, "Debt Management Under Corporate and Personal Taxation," Journal of Finance (December 1987), pp. 1275-1291. [14.] D.C. Mauer, "Optimal Bond Call Policies Under Transaction Costs and Taxes," Working Paper, University of Wisconsin, 1991. [15.] M.H. Miller, "Debt and Taxes," Journal of Finance (May 1977), pp. 261-275. [16.] S.Y. Park and J.T. Williams, "Taxes, Capital Structure, and Bondholder Clienteles," Journal of Business (April 1985), pp. 203-224. [17.] S.A. Ross, "Debt and Taxes and Uncertainty," Journal of Finance (July 1985), pp. 637-657. [18.] J.C. Van Horne, Financial Market Rates and Flows, New Jersey, Prentice Hall, 1989. [19.] J.B. Yawitz and J.A. Anderson, "The Effect of Refunding on Shareholder Wealth," Journal of Finance (June 1977), pp. 1738-1746. [20.] J. Zechner, "Tax Clienteles and Optimal Capital Structure under Uncertainty," Journal of Business (October 1990), pp. 465-491.

Although much has been learned from existing tax based analyses of callable bonds, the extant models suffer from both theoretical and practical deficiencies. First, most analyses do not consider the broader capital structure implications of their tax assumptions. For example, the assumption that the personal tax rate is below the corporate tax rate, which drives Boyce and Kalotay's conclusions, implies a corner leverage decision not-withstanding the call feature studied. To avoid such empirically unreasonable implications, the tax implications of particular bond characteristics should be studied under the benchmark conditions of Miller's [15] equilibrium analysis which leads to capital structure irrelevance.(2) Of course, the advantage of this approach is that it allows for the analysis of the tax effects of bond features (e.g., call provision) net of the tax gains from leverage. Second, the universally employed assumption that personal tax rates are fixed and equal across investors obscures the basic reality that future marginal personal tax rates are uncertain. This feature of the tax environment, which arises from the interaction between progressive personal taxation and uncertain future income (both wage and portfolio), induces a temporal association between future tax rates and interest rates, which may significantly affect the relative corporate aop personal valuation of the call.

The current analysis examines the value of the call under the realistic assumption that future marginal personal tax rates and interest rates are uncertain. Although the analysis does not depend on any particular covariance function between tax and interest rates, logic suggests that it is reasonable to assume that both variable are positively affected by the level of business conditions and therefore have a positive partial correlation. An upturn of the economy may be characterized by high marginal personal tax rates and interest rates, while a downturn may be associated with low tax interest rates. Indeed, inspection of the empirical record of nominal interest rates over the last three decades provides considerable evidence in support of a positive association between interest rates and economic activity. Short-term rates have generally been high during expansionary periods and low during recessions, while long-term rates have generally followed the same pattern (see, for example, Van Horne [18]).

While less direct evidence is available concerning the relationship between marginal personal tax rates and the state of the economy, a positive association is predicted by the recent capital structure models of Dammon [7] and Ross [17], and the subsequent multiperiod extension of those models, by Lewis [11]. For example, in Dammon's model, wherein investors face the same progressive tax function defined over uncertain taxable income, marginal personal tax rates are predicted to be highly correlated with aggregate per capita consumption. As such, marginal personal tax rate are also likely to be highly correlated with economic activity.(3) Consistent with expectations, empirical evidence on the correlation between marginal personal tax rates and interest rates indicates a positive association between the two variables.

Given a positive relationship between marginal personal tax rates and interest rates, the value of a call provision on a bond must include a "tax premium" which, ceteris paribus, makes the decision to issue a callable bond a negative sum game from the corporation's perspective. The intuition behind this result is straightforward. When tax and interest rates are positively correlated, investors will value bond income more highly in those states in which the firm will most likely call the bond. As a result, investors will require a tax premium for the inclusion of a call provision on a bond which cannot be recovered by firms. The implication is that, at least with respect to this realistic feature of the tax environment, tax arguments alone cannot rationalize the widespread practice of attaching call provisions to bonds.

I. The Model and thee Valuation of Noncallable Bonds

The framework for analysis is a simple multiperiod state-preference model. There are three dates:t = 0, t = 1,andt = 2. At time 1, there are two possible states of nature. Depending on which state is realized, the state of the economy is assumed fixed over the next period (from time 1 to time 2). Thus, all uncertainty is resolved at the end of the first period. Denote the value at time 0 of $1.00 (before-tax) in state 1 as [p.sub.01], and the value at time 0 of $1.00 (before-tax) is state as 2 as [p.sub.2]. Similarly, the time 1 value of $1.00 (before-tax) received at time 2, conditional on state 1 having occured at time 1, is denoted as [p.sub.11]. The corresponding price conditional on state 2 is denoted as [p.sub.12]. The only "macroeconomic" distinction between the two states at time 1 is the level of interest rates in the economy. Assume that the before-tax single-period riskless rate of interest in state 1, [r.sub.1], is greater than the before-tax single-period riskless interest rate in state 2, [r.sub.2]. Those interest rates both define and characterize the time 1 conditional state prices as

1 1 [p.sub.11] = ------------- < [p.sub.12] = -------------. (1)

1 + [r.sub.1] 1 + [r.sub.2]

On that basis, the restrictions on the time 0 before-tax state prices are as follows:

1 [p.sub.01] + [p.sub.02] = ---------------, and 0 < [p.sub.02] < [p.sub.01] < 1. (2)

1 + [r.sub.0]

The first condition in Equation (2) says that the sum of the two state prices equals the current single-period before-tax riskless discount factor. The second condition follows from the more favorable (re)investment opportunities investors face in state 1 at time 1 versus state 2 at time 1, i.e., [r.sub.1] > [r.sub.2]. Therefore, at time 0, investors will value $1.00 in state 1 more highly than $1.00 in state 2.

Investor tax rates are assumed to be heterogeneous, as in Miller [15], but also state dependent. Accordingly, at time 1, an investor's tax rate will be either [[Tau].sub.1] in state 1 or [[Tau].sub.2] in state 2. The tax rates are progressive in the sense that the state 1 tax rate, [[Tau].sub.1], is greater than the state 2 tax rate, [[Tau].sub.2]. Furthermore, given the heterogeneity of investor tax rates, [[Tau].sub.1] and [[Tau].sub.2] need not be the same across different investors. (Note that the simple progressive structure of the tax rates ensures a positive correlation with the time 1 interest rates which define the state prices). As in Miller, assume that equity income is not taxed, and that corporations are taxed at the marginal rate [[Tau].sub.C], which is spanned by investors tax rates. Finally, also as in Miller, assumed that investors are prohibited from engaging in tax arbitrage and therefore cannot, by short sales or other means, drive their tax rates to zero.

The Miller capital structure equilibrium is determined where the marginal investor's tax rate is equal to the corporate tax rate. The identification of that investor at time 0 is considerably more complicated when investors have stated-dependent time 1 tax rates. Assuming that corporate debt is riskless, and therefore pays the same relative amount of taxable cash flow in either state 1 or state 2, a Miller-type equilibrium will emerge at time 0 in which the marginal investor is identified as having a weighted average time 1 tax rate equal to the corporate tax rate. Thus, as in the Miller analysis, capital structure irrelevance results and tax clienteles form in which investors will hold either equity or debt. This correspondence with the Miller framework will breakdown if, for example, bonds are risky and therefore possibly pay different relative quantities of taxable cash flow across the two states at time 1. With state-dependent investor tax rates, bonds with different relative state-dependent taxable cash flow will no longer be perfect substitutes. The resulting tax clienteles will generally lead to a breakdown of the Miller debt invariance proposition.[4] Therefore, in order to sharpen the focus of the analysis on the value of the call feature on corporate bonds and to conduct the analysis to the fullest extent possible in the original Miller framework, corporate debt is assumed to be risk-free.

Since the level of investor tax rates is changing at time 1, the marginal investor at time 0 will not be the same marginal investor in either state 1 or state 2 at time 1. As a consequence, the capital structure equilibrium at time 1 will differ from that at time 0. To allow for such capital structure rearrangements, assume that at time 1 the corporation buys back its time 0 original issue bonds and finances the repurchase by issuing single-period debt which matures at time 2. Correspondingly, investors sell the bonds back to the firm, realize any capital gain/loss on the transaction, and rebalance their portfolios. The implication of this process is that the market price of debt in both states 1 and 2 at time 1 will reflect the corporate tax rate, since the marginal investor in either state will, as per the Miller equilibrium, be in the same tax bracket as the firm. Exhibit 1 summarize the tax and interest rate environment and displays the timing of trading in the bond market.

Initially, suppose that the firm issues a two-period default-free noncallable bond. Choosing a coupon payment C, such that the bond sells for par, B, gives B = [p.sub.01][C (1 - [[Tau].sub.C] + [B.sub.1] - [[Tau].sub.C]([B.sub.1] - B)] + [p.sub.02] [C (1 - [[Tau].sub.C]) + [B.sub.2] - [[Tau].sub.C] ([B.sub.2] - B)], (3) where [B.sub.1] and [B.sub.2] are the time 1 market values of debt in states 1 and 2, and are equal to [B.sub.1] = [p.sub.11] [C (1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.1])], (4) and [B.sub.2] = [p.sub.12] [C(1 - [[Tau].sub.C] + B - [[Tau].sub.C](B - [B.sub.2])], (5) respectively. Both [B.sub.1] and [B.sub.2] reflect the corporate tax rate, in accordance with bond market equilibrium in states 1 and 222 at time 1.(5) In Equation (3), note that since the bond sells for par, and assuming that [r.sub.1] > [r.sub.0] > [r.sub.2], it must hold that [B.sub.1] < B and [B.sub.2] > B. As such, if state 1 occurs at time, the firm buys back discount debt and the gain is considered for tax purposes as ordinary income. Alternatively, if state 2 occurs, the firm buys the debt back at a premium and realizes a tax-deductible loss.(6) Note that while repurchasing debt and incurring a capital gains tax liability may not be optimal, repurchasing premium debt and realizing the loss for tax purposes is optimal (see Mauer and Lewellen [13]). Importantly, however, the reason that the firm recapitalizes at time 1 is to its capital structure in response to changes in the net tax advantage of debt occasioned by state-dependent investor tax rates.

The marginal investor at time 0 is identified as that investor who is indifferent between tax-exempt equity and taxable corporate bonds at the price B, Let [[Tau].sub.m1] and [[Tau].sub.m2] denote that investor's time 1 state-dependent tax rates. To satisfy the indifference requirement, those tax rates must equate the after-tax stream of payments received from the bond to its par value. Accordingly, B = [p.sub.01][C (1 - [[Tau].sub.m1]) + [B.sub.1] - [[Tau].sub.m1]([B.sub.1] - B)] + [p.sub.02][C(1 - [[Tau].sub.m2]) + [B.sub.2] - [[Tau].sub.m2]([B.sub.2] - B)]. (6) As in the Miller equilibrium, heterogeneity of investor tax rates implies that tax clienteles will form. Those investors who, based on their own tax rates, value corporate bonds that at lower values than B prefer tax-exempt equity. Those who value the bonds at higher values than B will prefer bonds. Note that since investors sell the bonds at equilibrium prices at time, the bond value at time 0 does not reflect investors tax liabilities at time 2.(7)

The time 0 capital structure equilibrium is established where the corporate tax rate, [[Tau].sub.C] falls between the marginal investors's time 1 personal tax rates [[Tau].sub.m1] and [[Tau].sub.m2]. When that condition is met, the corporation will be different between noncallable debt and equity.(8) Thus, setting Equation (3) equal to Equation (6), and solving for the corporate tax rate, gives [[Tau].sub.C] = [Alpha][[Tau].sub.m1] + (1 - [Alpha]) [[Tau].sub.m2] (7) where

(8) [Alpha] = [p.sub.01][C + ([B.sub.1] - B)] [p.sub.01][C + ([B.sub.1] - B]) + [p.sub.02][C + ([B.sub.2] - B )]

Since [[Tau].sub.m2] < [[Tau].sub.m1], it is clear from Equation (7) that [[Tau].sub.m2] < [[Tau].sub.m1]. Given the identification of the time 0 marginal investor in noncallable bonds, the analysis now turns to the valuation of callable bonds. The critical issue to investigate is whether under state-dependent investor tax rates that same marginal investor places a higher value on the call provision than does the corporation.

II. Callable Bond Valuation

A. The Call Tax Premium

Suppose that instead of issuing a noncallable bond, the firm issues a callable bond. Since the focus is not the determination of the optimal time to call the bond (see Mauer [14]), assume that the bond will be called if the interest rate (or rather, the state price) falls (rises) to [r.sub.2] ([p.sub.12]) at time. Additionally, initially assume that the call price is the par value of the bond B. Thus, there are no tax consequences associated with any call premium paid when the bond is called. Denoting [C.sup.*] as the coupon payment which allows the callable bond to sell for par at time 0, gives B = [p.sub.01][[C.sup.*](1 - [[Tau].sub.m1] + [B.sub.C1] - [[Tau].sub.m1]([B.sub.C1] - B)] +[p.sub.02][[C.sup.*](1 - [[Tau].sub.m2]) + B], (9) where [B.sub.C1] = [p.sub.11] [[C.sup.*](1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.C1], (10) and is equal to the market value of the callable bond in state 1 at time 1. Note that this formulation assumes that the bond is called an instant after the coupon is received in state 2 at time 1. Futhermore, [C.sup.*] is determined in Equation (9) as the coupon payment which would make the marginal investor of the previous section indifferent between purchasing a callable or a noncallable bond at time 0.

Since a callable bond can be viewed as a noncallable bond on which the purchaser (the bondholder) has written a call option held by the issuer (the borrowing company), the callable bond value in Equation (9) can be decomposed into a noncallable bond with a coupon of [C.sup.*], minus the value of the call option. Thus, to compute the investor perceived call value, we simply take the difference between a noncallable bond evaluated at a coupon of [C.sup.*] (from Equation (6)) and the callable bond value in Equation (9). Performing the indicated calculation gives [V.sub.M] = [p.sub.02] [([B.sub.C2] - B)(1 - [[Tau].sub.m2)], (11) where [B.sub.C2] = [p.sub.12] [[C.sup.*](1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.C2])]. (12) Realize that [V.sub.M] is the call value which would make the marginal investor indifferent between purchasing a noncallable bond with a coupon of C, and a callable bond with a coupon of [C.sup.*]. That value reflects the difference between that the investor would have received in state 2 if the bond was not called, [B.sub.C2] - [[Tau].sub.m2] ([B.sub.C2] - B), and the call price, B. Thus, [V.sub.M] measures the cost of the call.

The value of the call provision from the standpoint of the corporate issuer is derived in a similar fashion. The objective is to compare the payments on a callable bond to those on an otherwise identical, but noncallable bond.(9) If state 1 occurs at time, the firm will not exercise the call. Instead, the firm will repurchase the discount debt at market prices, just as an the case of noncallable debt analyzed above. However, if state 2 occurs, the firm will call the bond for B. The net benefit of the call to the firm is the difference between what it would have paid if the bond was not called, [B.sub.C2] - [[Tau].sub.C] ([B.sub.C2] - B), and the call price, B. The value of that difference, using the current state 2 price, is equal to [V.sub.F] = [p.sub.02][{B.sub.C2] - [[Tau].sub.C]([B.sub.C2] - B)} - B] = [p.sub.02][([B.sub.C2] - B)(1 - [[Tau].sub.C])], (13) which quantity measures the ex ante benefit of the call to shareholders.

The main result of the analysis is obtained by comparing [V.sub.F] and [V.sub.M]. Since [V.sub.M] is the call value which would make the marginal bondholder indifferent between callable and noncallable debt, the only way in which shareholders of the firm would be similarly indifferent is if [V.sub.F] were equal to [V.sub.M]. If, on the other hand, [V.sub.F] is less than [V.sub.M], the issuing callable debt is a negative sum game from the corporation's perspective. Taking the difference between Equation (13) and Equation (11) gives [Delta] = [V.sub.F] - [V.sub.M] = [p.sub.02][([B.sub.C2] - B)([[Tau].sub.m2] - [[Tau].sub.C])], (14) which is negative because of the simultaneous determination of low interest rates (which imply [B.sub.C2] > B) and low tax rates (which imply [[Tau].sub.m2] < [[Tau].sub.C]) in state 2 at time 1. Thus, when investors tax rates are positively correlated with interest rates, the after-tax value of bond income will be greater than the after-tax value of equity income in states wherein the firm is likely to call the bond. Accordingly, investors will require a "tax premium" for the inclusion of a call provision on a bond.

The intuition behind this result can be made transparent by comparing the cash flow consequences of the call to the marginal investor and to the firm. The difference between the cash flows from not calling versus calling to the firm and to the marginal investor is [{[B.sub.C2] - [[Tau].sub.C]([B.sub.C2] - B)} - B] - [{[B.sub.C2] - [[Tau][.sub.m2] ([B.sub.C2] - B)} - B] (15) which quantity simplifies to [[Tau].sub.m2]([B.sub.C2] - B) - [[Tau].sub.C]([B.sub.C2] - B) = ([B.sub.C2] - B)([[Tau].sub.m2] - [[Tau].sub.C]). (16) An examination of the left side of Equation (16) reveals that the differential cash flow impact of the call is a function only of the taxes that would have been paid had the bond not been called. Since [B.sub.C2] > B and [[Tau].sub.m2] < [[Tau].sub.C] the capital incurred is smaller than the tax deduction that the firm could claim by repurchasing premium debt at market prices. As a consequence, the cost of the call to the marginal investor is higher than the benefit of the call to the firm. This, of course, leads to a negative call tax-premium.(10)

Thus, with state-dependent marginal investor tax rates and the absence of any other advantages associated with callable bonds, noncallable bonds will dominate callable bonds. the implication is that corporations would only be willing to supply noncallable bonds. Furthermore, under the conditions in Section 1 above, in equilibrium, there would be no optimal capital structure for any individual firm, but there would be a determinate aggregate amount of noncallable debt in the economy.

These conclusions contrast with those provided by Boyce and Kalotay [4], who argue that there is a tax advantage to callable debt. The difference in the results is directly attributable to the underlying tax assumptions. Boyce and Kalotay assume that all investors have the same constant tax rate which is lower than the corporate tax rate. In such a setting, there will always be a tax advantage to having a call provision on a bond. Indeed, everything else held constant, all firms would also be predicted to be 100% debt financed, since there would then be a strictly positive net tax advantage to debt.

There is however, a qualification to the results obtained above. An implicit assumption of the analysis is that the call features on corporate bonds are identical and that therefore the relative quantities of taxable cash flow supplied by callable bonds are equal in states 1 and 2 at time. If this were not the case, then it may be impossible to find a bondholder who would be marginal in all callable bonds. As such, definitive statements about the relative value of the call feature could not be made since bonds with different call features could be priced at different implicit personal tax rates, Here identical call features are conveniently assured by the assumption that the call price is the face value of the debt. However, differential taxable call premiums in call prices may lead to a violation of this condition. Fortunately, actual call premium in call prices are relatively homogeneous across corporate bonds and are therefore not likely to engender significant differences in relative quantities of taxable cash flows across bonds. Thus, the conclusions are likely to hold even when bond call provisions contain call premiums.

B. An Illustrative Example

To see the process by which a negative call tax-premium arises, consider the following numerical example. Assume: [p.sub.01] = 0.727, time 0 price of $1.00 (before-tax) in state 1 at time 1; [p.sub.02] = 0.812, time 0 price of $1.00 (before-tax) in state 2 at time 1; [r.sub.1 ] = 0.12, before-tax single-period riskless interest rate in state 1 at time 1; [r.sub.2] = 0.03, before-tax single-period riskless interest rate in state 2 at time 1; [[Tau].sub.C] = 0.35, marginal corporate tax rate; and [[Tau].sub.m1] = 0.45 marginal investor's tax rate in state 1 at time 1.

The time 0 state prices determine, from Equaltion (2), a current rate of interest ([r.sub.1]) equal to 10%. The higher price for state 1 reflects the fact that state 1 is more favorable to the investor than state 2, since [r.sub.1] is greater than [r.sub.2]. The noncallable bond coupon payment C, which satisfies Equation (3), at a par value of B = 100, is computed to be 15.4. The corresponding state 1 and 2 bond prices are [B.sub.1] = 97.41 and [B.sub.2] = 110.30. Equation (7) searches for the tax rates of the "marginal investor". To find a unique solution, the marginal investor's state 1 tax rate ([[Tau].sub.m1]) is assumed to be 45% which, along with a marginal corporate tax rate ([[Tau].sub.C]) of 35%, identifies the investor's state 2 tax rate ([[Tau].sub.m2] as 15%. (The weight, [Alpha]. which satisfies Equation (7) at those tax rates is 0.666).

With the same time 0 marginal investor identified, the callable bond can now be valued. Substituting the marginal investor's tax rates into Equation (9), and assuming a call price investor is indifferent between the noncallable and callable bond at a coupon payment ([C.sup.*]) on the latter of 17.2. The corresponding callable bond prices in states 1 and 2 are equal to [B.sub.C1] = 98.92 and [B.sub.C2] = 112.01, respectively. The investor perceived call value [V.sub.M] (from Equation (11)), comes out to be 1.85 and the value of the call to the firm [V.sub.F] (from Equation (13), equals 1.42. Accordingly, the call tax-premium ([Delta]), from Equation (14), equals -0.43, or 0.43% of the bond's par value. Note that the magnitude of the tax premium is arbitrarily detrmined by the parameters employed and by the use of a two-period framework. Increasing the number of periods will increase the tax premium.

C. State-Dependent Marginal Corporate Tax Rates

In addition to depending on the correlation between interest rates and marginal investor tax rates, the magnitude of the call tax-premium also depends on the spread between the marginal corporate tax rate and the marginal investor's tax rate in state 2 at time 1. Although if simplifies the analysis to assume that firms face constant marginal corporate tax rates, and that therefore the spread is determined by variation in the marginal investor's tax rate, it is not necessary to the result. In a setting in which the marginal corporate tax rate also varied over time, the magnitude of the call tax-premium would depend on the relative sensitives of both corporate and personal tax rates to the state of the economy.

In practice, the marginal corporate tax rate is probably less sensitive than are personal tax rates to economic conditions. This is a result of the tax-loss carryforward and carryback provisions for corporations. For example, even when taxable corporate income would be zero in an aconomic recession, corporations can use carryforward/back provisions so as not to lose tax shields. Current tax law allows corporations to carry losses back three years to offset against previous years' taxable income or to carry losses forward 15 years. Thus, the maximum loss in corporatetax shield value is the time value of money, and this occurs only when the previous years' taxable income is insufficient to offset the current year's losses. The implication is that the marginal corporate tax rate would be expected to be more stable over time than personal tax rates, and that therefore the call tax-premium would continue to be negative.

D. Asymmetric Taxation of the Call Premium

The analysis above assumes that the bonds are called at par in order to sharpen the focus on the call tax-premium. However, in practice most call provisions require that bonds be redeemed at par plus a call premium, which is typically equal t one year's interest and may decline over the life of the bond according to a prearranged schedule. The important feature of this call premium is that the corporation can immediately deduct it for tax purposes when the bonds are called. Accordingly, if the call price in the model includes a premium above the bond's par value, then there will be an additional differential tax effect which influences the value of the call. Since the call premium is tax deductible to the corporation and treated as a capital gain from the standpoint of the bondholder, exercise of the call will reduce the aggregate tax liability of both parties at the expense of the government.

The problem with this line of reasoning, however, is that one cannot therefore conclude that because of the potential offsetting tax benefit associated with the call premium paid, callable bonds would be preferred over noncallable bonds. The reason is that any premium paid above the bond's par value, whether the bond has a call provision or not, is tax deductible to the corporation as an ordinary income expense and taxable to bondholders as a capital gain. As a consequence, this tax benefit is not unique to callable bonds and therefore could only justify the existence of call provisions if superior tax benefits were achievable vis-a-vis the alternative of simply repurchasing noncallable bonds at market prices.

Along those lines, a recent series of papers (Emery and Lewellen [8], Brick and Wallingford [5], and Mauer and Lewellen [13]) have demonstrated that the tax benefits associated with repurchasing noncallable premium debt are likely to be superior. For example, using simulation analysis, Brick and Wallingford demonstrate that under almost any reasonable economic environment, the call premium tax asymmetry produces greater expected tax benefits for noncallable debt repurchased at market prices. The reason is that the set of profitable call opportunities for a noncallable bond repurchased at market price will, in general, be greater than those of a callable bond repurchased at fixed call premiums. This will be especially true if those fixed call premiums decrease over the life of the bond, as is typical of most call provisions. Consequently, although there may be tax benefits associated with bonds callable at fixed call premiums which are sufficient to overturn the negative tax premium examined here, there is no reason to believe that those benefits can rationalize the existence of call provisions.

III. Empirical Evidence on the Association Between Interest Rates and Personal Tax Rates

As is apparent in Equation (14), the economic significance of the call tax-premium is critically dependent on the correlation between marginal personal tax rates and market rates of interest. That the correlation is positive is intuitive and reasonable, and one possible line of economic reasoning for such as relation has been given in the introduction. Nonetheless, it is important to empirically assess the relationship.

One approach would be to infer marginal investor tax rates from taxable corporate and nontaxable municipal bond yields and then compute correlations between those (implied) tax rates and interest rates. Unfortunately, this approach is subject to criticism on the grounds that the marginal tax rates are themselves derived from interest rates, which may lead to spurious correlations. A better approach would be to correlate interest rates with a time-series of marginal tax rates derived from individuals' income tax statements. Fortunately, the required time-series of tax rates exists in the literature.

Barro and Sahasakul [12] calculte average marginal tax rates for each year from 1916-1980 from the federal individual income tax as reported in the Internal Revenue Service's Statistics of Income. Exhibit 2 reports correlation coefficients between their arithmetic series and three different annual interest rate series reported in the Federal Reserve Bulletin for the period 1945-1980. The interest rate series includes: high-grade (Aaa) corporate bonds, high-grade (Aaa) municipal bonds, and 4-6 month commercial paper.

Every correlation coefficient is positive and significant and they range from 0.69 for commercial paper and municipal bonds to 0.75 for high-grade corporate bonds. However, since both tax and interest-rate series display an upward trend over time, the correlations may be entirely driven by a common association with time. As such, it is important to recompute the correlations after correcting for trend in the time-series. The second row in Exhibit 2 displays adjusted correlation coefficients which are partial correlations between tax rates and interest rates after controlling for linear time trend in the series. Although the adjusted correlations are smaller than their unadjusted counterparts, they are still positive and generally significant. For example, the correlation between marginal personal tax rates and corporate bond yields goes from 0.75 to 0.39, with the latter coefficient still significant at the 1% level. This evidence is encouraging, since it provides empirical support for the assumption of a positive relationship between marginal investor tax rates and market rates of interest. The call tax-premium is therefore likely to be economically significant.

Exhibit 2. Correlation Coefficients Between Annual Average Marginal Personal Tax Rates and Interest Rates Over the Period 1945-1980(a) High-Grade High-Grade (Aaa) (Aaa) 4-6 Month Corporate Municipal Commercial Type(b) Bonds Bonds Paper Unadjusted 0.75(*) 0.69(*) 0.69(*) Adjusted 0.39(*) 0.28(**) 0.22(***) (a) The annual average marginal tax rates are from Barro and Sahasakul [2, Table 2, column 2, pp. 434, 435]. The yields on corporate and municipal bonds, and the interest rate on commercial paper, are from various issues of the Federal Reserve Bulletin. (b) The adjusted coefficients are partial correlations between annual average marginal tax rates and interest rates after controlling for linear time trend. (*) Significant at the 1% level. (**) Significant at the 5% level. (***) Significant at the 10% level.

IV. Conclusions

The analysis examines the relative tax benefits of a call provision on a bond under the assumption that investors have progressive tax rates which covary with interest rates. When marginal bondholder tax rates and interest rates are positively correlated, bondholders place a higher value on the call than the firm. Thus, bondholders require an additional tax premium for the inclusion of a call provision on a bond which makes the issuance of a callable bond a negative sum game from the corporation's perspective. Empirical evidence on the extent to which tax rates and interest rates positively covary indicates that the call tax-premium is likely to be nontrivial. Finally, it is argued that callable bonds continue to carry a tax disadvantage even if marginal corporate tax rates are not constant and if there is a positive expected tax benefit associated with the asymmetric taxation of the call premium on the bond.

In conclusion, it should be noted that the role played by call provisions in resolving other market imperfections, such as informational asymmetry and/or agency problems, may be sufficient to rationalize their widespread use. For example, Bodie and Taggart [3] suggest that callable debt may mitigate incentives for firms to reject profitable investment opportunities if the gains from those investments would be captured by bondholders. Similarly, Barnea, Haugen, and Senbet [1] argue that call provisions may mitigate the agency problems of informational asymmetry and stockholder risk incentives. (1) The call premium is the difference between the call price and the par value of the bond. Typically, the call premium is equal to one year's interest and declines throughout the life of the bond according to a prearranged schedule. Call provisions are also deferred for a period of time. Standard deferred call periods are 5 years for public utility bonds and 10 years for industrial bonds. (2) A notable exception in this regard is the paper by Brick and Wallingford [5], which discusses the implications of the call premium tax asymmetry for the Miller equilibrium. (3) Under the Tax Reform Act of 1986, tax rates continue to be progessive although fewer brackets exist. (4) Park and Williams [16] and Zechner [20] provide detailed analyses of the conditions which lead to tax-based bondholder clienteles and the resulting implications for corporate capital structures. (5) Since the marginal investors in states 1 and 2 at time 1 have tax rates which are equal to the corporate tax rate, the time 1 market prices of debt, [B.sub.1] and [B.sub.2], depend only on the corporate tax rate. Additionally, since [B.sub.1] < B, the state 1 bond price reflects the capital gains tax liability that the marginal bondholder faces at time 2. On the other hand, since [B.sub.2] > B, the state 2 bond price reflects the realization at time 2 of the tax-deductible premium paid for the bond. (6) See Finnerty, Kalotay, and Farrell [9] for details on the tax rules appropriate to corporate repurchases of discount and premium debt. (7) Equation (6) includes the capital gains taxes paid upon sale of the bond at time 1. In particular, given that the marginal investor's tax basis for the computation of capital gains, is B, in state 1 in the investor realizes a tax-deductible loss, and in state 2 in the investor pays taxes on the capital gain. Note that the capital gains taxes are computed using the marginal investor's ordinary income tax rates [[Tau].sub.m1]] and [[Tau].sub.m2]. This, of course, is congruent with the Tax Reform Act of 1986 wherein the distinction between ordinary and capital gains tax rates was eliminated. (8) As in Miller [15], the assumptions are sufficient to generate an equilibrium in which capital structure is indeterminate at the level of the individual firm, but determinate at the level of the corporate sector as a whole. However, the time 0 capital structure equilibrium is unstable in the sense that a new equilibrium will be established at time 1. In particular, given state 1 at time 1, state-dependent investor tax rates shift up with the interest rate and the advantage of debt financing will be reduced. Accordingly, corporate supply-side adjustments will ensure that a new equilibrium will be established in which the aggregate supply of noncallable debt in the corporate sector will also be reduced. A similar, although opposite, adjustment will occur given the realization of state 2 at time 1. (9) See Kraus [10] and Marshall and Yawitz [12] for a similar approach to call valuation. (10) Note that the cost of the call to the marginal investor would actually be higher if the investor could have escaped taxation of the capital gain in state 2 at time 1 by following the tax-trading rules outlined in Constantinides and Ingersoll [6]. Thus, to the extent that the analysis assumes complete realization of all capital gains and losses - in Constantinides and Ingersoll's terminology, a "continuous realization policy" - the magnitude of the call tax-premium is understated.

References

[1.] A Barnea, R.A. Haugen, and L.W. Senbet, "A Rationale for Debt Maturity Structure and Call Provisions in the Agency Theoretic Framework," Journal of Finance (December 1980), pp. 1223-1234. [2.] R.J. Barro and C. Sahasakul, "Measuring the Average Marginal Tax Rate from the Individual Income Tax," Journal of Business (October 1983), pp. 419-452. [3.] Z. Bodie and R.A. Taggart, "Future Investment Opportunities and the Value of the Call Provision on a Bond," Journal of Finance (March 1976), pp. 1187-1200. [4.] W.M. Boyce and A.J. Kalotay, "Tax Differentials and Callable Bonds," Journal of Finance (September 1979), pp. 825-838. [5.] I.E. Brick and B.A. Wallingford, "The Relative Tax Benefits of Alternative Call Features in Corporate Debt," Journal of Financial and Quantitative Analysis (March 1985), pp. 95-105. [6.] G.M. Constantinides and J.E Ingersoll, "Optimal Bond Trading with Personal Taxes," Journal of Financial Economics (September 1984), pp. 299-335. [7.] R.M. Dammon, "A Security Market and Capital Structure Equilibrium Under Uncertainty With Progressive Personal Taxes," in A. Chen (ed.), Research in Finance, Vol. 7, Greenwich, CT, JAI Press, 1987. [8.] D.R. Emery and W.G. Lewellen, "Refunding Noncallable Debt," Journal of Financial and Quantitative Analysis (March 1984), pp. 73-82. [9.] J.D. Finnerty, A.J. Kalotay, and F. Farrell, The Financial Manager's Guide to Evaluating Bond Refunding Opportunities, Ballinger Publishing Company, 1988. [10.] A. Kraus, "An Analysis of Call Provisions and the Corporate Refunding Decision," Midland Corporate Finance Journal (Spring 1983), pp. 46-60. [11.] C.M. Lewis, "A Multiperiod Theory of Corporate Financial Policy Under Taxation," Journal of Financial and Quantitative Analysis (March 1990), pp. 25-43. [12.] W.J. Marshall and J.B. Yawitz, "Optimal Terms of the Call Provision on a Corporate Bond," Journal of Financial Research (Summer 1980), pp. 202-221. [13.] D.C. Mauer and W.G. Lewellen, "Debt Management Under Corporate and Personal Taxation," Journal of Finance (December 1987), pp. 1275-1291. [14.] D.C. Mauer, "Optimal Bond Call Policies Under Transaction Costs and Taxes," Working Paper, University of Wisconsin, 1991. [15.] M.H. Miller, "Debt and Taxes," Journal of Finance (May 1977), pp. 261-275. [16.] S.Y. Park and J.T. Williams, "Taxes, Capital Structure, and Bondholder Clienteles," Journal of Business (April 1985), pp. 203-224. [17.] S.A. Ross, "Debt and Taxes and Uncertainty," Journal of Finance (July 1985), pp. 637-657. [18.] J.C. Van Horne, Financial Market Rates and Flows, New Jersey, Prentice Hall, 1989. [19.] J.B. Yawitz and J.A. Anderson, "The Effect of Refunding on Shareholder Wealth," Journal of Finance (June 1977), pp. 1738-1746. [20.] J. Zechner, "Tax Clienteles and Optimal Capital Structure under Uncertainty," Journal of Business (October 1990), pp. 465-491.

Printer friendly Cite/link Email Feedback | |

Author: | Mauer, David C.; Barnea, Amir; Kim, Chang-Soo |
---|---|

Publication: | Financial Management |

Date: | Jun 22, 1991 |

Words: | 6844 |

Previous Article: | An evaluation of investment banker acquisition advice: the shareholders' perspective. |

Next Article: | A full-information approach for estimating divisional betas. |

Topics: |