ON THE PRICING OF CORPORATE DEBT: THE RISK STRUCTURE OF INTEREST RATES*

The value of a particular issue of corporate debt depends essentially on three items: (1) the required rate of return on riskless (in terms of default) debt (e.g., government bonds or very high grade corporate bonds); (2) the various provisions and restrictions contained in the indenture (e.g., maturity date, coupon rate, call terms, seniority in the event of default, sinking fund, etc.); (3) the probability that the firm will be unable to satisfy some or all of the indenture requirements (i.e., the probability of default). While a number of theories and empirical studies has been published on the term structure of interest rates (item 1), there has been no systematic development of a theory for pricing bonds when there is a significant probability of default. The purpose of this paper is to present such a theory which might be called a theory of the risk structure of interest rates. The use of the term “risk” is restricted to the possible gains or losses to bondholders as a result of (unanticipated) changes in the probability of default and does not include the gains or losses inherent to all bonds caused by (unanticipated) changes in interest rates in general. Throughout most of the analysis, a given term structure is assumed and hence, the price differentials among bonds will be solely caused by differences in the probability of default. In a seminal paper, Black and Scholes 1 present a complete general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of “observable” variables. Therefore, the model is subject to direct empirical tests which they 2 performed with some success. Merton 5 clarified and extended the Black-Scholes model. While options are highly specialized and relatively unimportant financial instruments, both Black and Scholes 1 and Merton 5, 6 recognized that the same basic approach could be applied in developing a pricing theory for corporate liabilities in general. In Section II of the paper, the basic equation for the pricing of financial instruments is developed along Black-Scholes lines. In Section III, the model is applied to the simplest form of corporate debt, the discount bond where no coupon payments are made, and a formula for computing the risk structure of interest rates is presented. In Section IV, comparative statics are used to develop graphs of the risk structure, and the question of whether the term premium is an adequate measure of the risk of a bond is answered. In Section V, the validity in the presence of bankruptcy of the famous Modigliani-Miller theorem 7 is proven, and the required return on debt as a function of the debt-to-equity ratio is deduced. In Section VI, the analysis is extended to include coupon and callable bonds. The dynamics for the value of the firm, V, through time can be described by a diffusion-type stochastic process with stochastic differential equation α is the instantaneous expected rate of return on the firm per unit time, C is the total dollar payouts by the firm per unit time to either its shareholders or liabilities-holders (e.g., dividends or interest payments) if positive, and it is the net dollars received by the firm from new financing if negative; σ 2 is the instantaneous variance of the return on the firm per unit time; dz is a standard Gauss-Wiener process. Many of these assumptions are not necessary for the model to obtain but are chosen for expositional convenience. In particular, the “perfect market” assumptions (A.1-A.4) can be substantially weakened. A.6 is actually proved as part of the analysis and A.7 is chosen so as to clearly distinguish risk structure from term structure effects on pricing. A.5 and A.8 are the critical assumptions. Basically, A.5 requires that the market for these securities is open for trading most of time. A.8 requires that price movements are continuous and that the (unanticipated) returns on the securities be serially independent which is consistent with the “efficient markets hypothesis” of Fama 3 and Samuelson 9.11 Of course, this assumption does not rule out serial dependence in the earnings of the firm. See Samuelson 10 for a discussion. In closing this section, it is important to note which variables and parameters appear in (7) (and hence, affect the value of the security) and which do not. In addition to the value of the firm and time, F depends on the interest rate, the volatility of the firm's value (or its business risk) as measured by the variance, the payout policy of the firm, and the promised payout policy to the holders of the security. However, F does not depend on the expected rate of return on the firm nor on the riskȁpreferences of investors nor on the characteristics of other assets available to investors beyond the three mentioned. Thus, two investors with quite different utility functions and different expectations for the company's future but who agree on the volatility of the firm's value will for a given interest rate and current firm value, agree on the value of the particular security, F. Also all the parameters and variables except the variance are directly observable and the variance can be reasonably estimated from time series data. As a specific application of the formulation of the previous section, we examine the simplest case of corporate debt pricing. Suppose the corporation has two classes of claims: (1) a single, homogenous class of debt and (2) the residual claim, equity. Suppose further that the indenture of the bond issue contains the following provisions and restrictions: (1) the firm promises to pay a total of B dollars to the bondholders on the specified calendar date T; (2) in the event this payment is not met, the bondholders immediately take over the company (and the shareholders receive nothing); (3) the firm cannot issue any new senior (or of equivalent rank) claims on the firm nor can it pay cash dividends or do share repurchase prior to the maturity of the debt. For a given maturity, the risk premium is a function of only two variables: (1) the variance (or volatility) of the firm's operations, σ 2 and (2) the ratio of the present value (at the riskless rate) of the promised payment to the current value of the firm, d. Because d is the debt-to-firm value ratio where debt is valued at the riskless rate, it is a biased upward estimate of the actual (market-value) debt-to-firm value ratio. Since Merton 5 has solved the option pricing problem when the term structure is not “flat” and is stochastic, (by again using the isomorphic correspondence between options and levered equity) we could deduce the risk structure with a stochastic term structure. The formulae (13) and (14) would be the same in this case except that we would replace “ exp [ − r τ ]” by the price of a riskless discount bond which pays one dollar at time τ in the future and “ σ 2 τ ” by a generalized variance term defined in 5. In the derivation of the fundamental equation for pricing of corporate liabilities, (7), it was assumed that the Modigliani-Miller theorem held so that the value of the firm could be treated as exogeneous to the analysis. If, for example, due to bankruptcy costs or corporate taxes, the M-M theorem does not obtain and the value of the firm does depend on the debt-equity ratio, then the formal analysis of the paper is still valid. However, the linear property of (7) would be lost, and instead, a non-linear, simultaneous solution, F = F [ V ( F ) , τ ] , would be required. Fortunately, in the absence of these imperfections, the formal hedging analysis used in Section II to deduce (7), simultaneously, stands as a proof of the M-M theorem even in the presence of bankruptcy. To see this, imagine that there are two firms identical with respect to their investment decisions, but one firm issues debt and the other does not. The investor can “create” a security with a payoff structure identical to the risky bond by following a portfolio strategy of mixing the equity of the unlevered firm with holdings of riskless debt. The correct portfolio strategy is to hold ( F v V ) dollars of the equity and ( F – F v V ) dollars of riskless bonds where V is the value of the unlevered firm, and F and F v are determined by the solution of (7). Since the value of the “manufactured” risky debt is always F, the debt issued by the other firm can never sell for more than F. In a similar fashion, one could create levered equity by a portfolio strategy of holding ( f v V ) dollars of the unlevered equity and ( f – f v V ) dollars of borrowing on margin which would have a payoff structure identical to the equity issued by the levering firm. Hence, the value of the levered firm's equity can never sell for more than f. But, by construction, f + F = V, the value of the unlevered firm. Therefore, the value of the levered firm can be no larger than the unlevered firm, and it cannot be less. Note, unlike in the analysis by Stiglitz 11, we did not require a specialized theory of capital market equilibrium (e.g., the Arrow-Debreu model or the capital asset pricing model) to prove the theorem when bankruptcy is possible. Contrary to what many might believe, the relative riskiness of the debt can decline as either the business risk of the firm or the time until maturity increases. Inspection of (33) shows that this is the case if d > 1 (i.e., the present value of the promised payment is less than the current value of the firm). To see why this result is not unreasonable, consider the following: for small T (i.e., σ 2 or τ: small), the chances that the debt will become equity through default are large, and this will be reflected in the risk characteristics of the debt through a large g. By increasing T (through an increase in σ 2 or τ), the chances are better that the firm value will increase enough to meet the promised payment. It is also true that the chances that the firm value will be lower are increased. However, remember that g is a measure of how much the risky debt behaves like equity versus debt. Since for g large, the debt is already more aptly described by equity than riskless debt. (E.g., for d > 1 , g > 1 2 and the “replicating” portfolio will contain more than half equity.) Thus, the increased probability of meeting the promised payment dominates, and g declines. For d < 1 , g will be less than a half, and the argument goes just the opposite way. In the “watershed” case when d = 1 , g equals a half; the “replicating” portfolio is exactly half equity and half riskless debt, and the two effects cancel leaving g unchanged. In closing this section, we examine a classical problem in corporate finance: given a fixed investment decision, how does the required return on debt and equity change, as alternative debt-equity mixes are chosen? Because the investment decision is assumed fixed, and the Modigliani-Miller theorem obtains, V, σ 2 , and α (the required expected return on the firm) are fixed. For simplicity, suppose that the maturity of the debt, τ, is fixed, and the promised payment at maturity per bond is $1. Then, the debt-equity mix is determined by choosing the number of bonds to be issued. Since in our previous analysis, F is the value of the whole debt issue and B is the total promised payment for the whole issue, B will be the number of bonds (promising $1 at maturity) in the current analysis, and F /B will be the price of one bond. In the usual analysis of (default-free) bonds in term structure studies, the derivation of a pricing relationship for pure discount bonds for every maturity would be sufficient because the value of a default-free coupon bond can be written as the sum of discount bonds' values weighted by the size of the coupon payment at each maturity. Unfortunately, no such simple formula exists for risky coupon bonds. The reason for this is that if the firm defaults on a coupon payment, then all subsequent coupon payments (and payments of principal) are also defaulted on. Thus, the default on one of the “mini” bonds associated with a given maturity is not independent of the event of default on the “mini” bond associated with a later maturity. However, the apparatus developed in the previous sections is sufficient to solve the coupon problem. Moreover, even for those cases where closed-form solutions cannot be found, powerful numerical integration techniques have been developed for solving equations like (7) or (41). Hence, computation and empirical testing of these pricing theories is entirely feasible. Note that in deducing (40), it was assumed that coupon payments were made uniformly and continuously. In fact, coupon payments are usually only made semi-annually or annually in discrete lumps. However, it is a simple matter to take this into account by replacing “ C ¯ ” in (40) by “ Σ i C ¯ i δ ( τ − τ i ) ” where δ( ) is the dirac delta function and τ i is the length of time until maturity when the i th coupon payment of C ¯ i dollars is made. We have developed a method for pricing corporate liabilities which is grounded in solid economic analysis, requires inputs which are on the whole observable; can be used to price almost any type of financial instrument. The method was applied to risky discount bonds to deduce a risk structure of interest rates. The Modigliani-Miller theorem was shown to obtain in the presence of bankruptcy provided that there are no differential tax benefits to corporations or transactions costs. The analysis was extended to include callable, coupon bonds.