Bargaining Over Equity's Share in the Bankruptcy Reorganization of Large, Publicly Held Companies

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model. THE TRADITIOKAL BLACK-SCHOLES (1973) and Merton (1974) contingentclaims-based approach to valuing corporate debt has become an integral part of the theory of corporate finance. In this approach, interest rates are assumed to be constant, and the default risk of a bond is modeled using option pricing theory. This framework for valuing risky debt has been applied in a number of articles including Geske (1977), Ingersoll (1977a, 1977131, Merton (1977), Smith and Warner (1979), and many others. One of the drawbacks of this approach is that default is assumed to occur only when the firm exhausts its assets. This is clearly unrealistic, since firms usually default long before the firm's assets are exhausted. In addition, Jones, Mason, and Rosenfeld (1984) and Franks and Torous (1989) show that this aspect of the model implies credit spreads much smaller than actual credit spreads. In an important article, Black and Cox (1976) relax this assumption and allow default to occur when the value of the firm's assets reaches a lower threshold. This feature makes the model consistent with *Both authors are from the Anderson Graduate School of Management, University of California a t Los Angeles. This is a substantially revised version of an earlier working paper entitled "Valuing Risky Debt: A New Approach. " We are grateful for the comments and suggestions of Brian Betker, Brad Cornell, Darrell DuEe, Julian Franks, Mark Grinblatt, Robert Heinkel, David Hirshleifer, Hayne Leland, Andrew Lo, Victor hlakarov, Richard Rendleman, Ehud Ronn, Mark Rubinstein, Chester Spatt, Walter Torous, Bruce Tuckman, Justin Wood, Josef Zechner, and seminar participants at the Amex Options and Derivatives Colloquium, the University of British Columbia, the University of California at Berkeley, Duke University, the European Institute for Financial Analysis and Portfolio Management, the University of Rochester, Southern Methodist University, Stanford University, the University of Strathclyde, Texas Christian University, UCLA, and the European Finance Association and Western Finance Association meetings. We are particularly grateful for generous financial support by the Milken Institute for Job and Capital Formation. All errors are our responsibility. The Journal of Finance either net-worth or cash-flow-based insolvency. By incorporating this more realistic default condition, the Black and Cox model is able to generate credit spreads more consistent with those observed in corporate debt markets. Despite this advantage, the Black and Cox model shares some of the other limitations of the traditional Black-Scholes-Merton framework for valuing risky debt. Specifically, this framework assumes that interest rates are constant. This assumption is difficult to justify in a valuation model for risky fixed-income securities. In addition, this framework assumes that assets are allocated among corporate claimants according to strict absolute priority rules if the firm defaults. However, recent evidence by Franks and Torous (1989, 1994) Eberhart, Moore, and Roenfeldt (1990), 1,oPucki and Whitford (1990), Weiss (1990), Betker (1991, 1992), and others shows that strict absolute priority is rarely upheld in distressed reorganizations. This article develops a simple new approach to valuing risky debt by extending the Black and Cox (1976) model in two ways. First, this model incorporates both default risk and interest rate risk. Second, this approach explicitly allows for deviations from strict absolute priority. In developing the model architecture, our objective is to be able to provide tractable valuation models for risky debt securities. Accordingly, we present the simplest possible specification for the model rather than the most general. This has the important advantage of allowing us to derive simple closed-form expressions for both risky fixed-rate and floating-rate debt. These closed-form expressions provide a number of new insights about the properties of corporate debt prices. We first apply our framework to value risky discount and coupon bonds. We show that the credit spreads implied by the model are consistent with many of the properties of actual credit spreads. For example, the model implies credit spreads comparable in magnitude to actual spreads and allows the term structure of credit spreads to be either monotone increasing or hump shaped. An important implication of our results is that credit spreads for firms with similar default risk can vary significantly if the assets of the firms have different correlations with changes in interest rates. This property of the model has the potential to explain why bonds with similar credit ratings but in different industries or sectors have widely differing credit spreads. Finally, we show that the properties of high-yield bonds can be very different from those of less risky debt. For example, the duration or interest rate sensitivity of a high-yield bond may actually increase as it gets closer to its maturity date. We then derive closed-form expressions for the value of risky floating-rate debt. We find that the price of a floating-rate bond can be an increasing function of the maturity of the bond in some situations. Similarly, the value of floating-rate debt can be an increasing function of the level of interest rates. These results illustrate that the properties of floating-rate debt are fundamentally different from those of fixed-rate debt. In general, the price of a floating-rate bond need not equal its par value even on coupon payment dates because of the risk of default. Approach to Valuing Risky Fixed and Floating Rate Debt 791 Using Moody's corporate bond yield averages, we examine whether the implications of our model are consistent with the actual properties of credit spreads. As implied by the model, we find that credit spreads are strongly negatively related to the level of interest rates. Furthermore, changes in interest rates account for the majority of the variation in credit spreads for most of the bonds in the sample. This drives home the importance of allowing for interest rate risk in addition to default risk in valuing risky debt securities. We also find that the differences in the duration of bonds across industries and sectors is consistent with the differences in correlations with changes in the interest rate. These results provide supporting evidence for the empirical implications of our valuation model. There are a number of other articles focusing on the valuation of corporate securities that allow for both default risk and interest-rate risk. These include Ramaswamy and Sundaresan (19861, Hull and White (1992), Maloney (1992), Jarrow and Turnbull (1992a, 1992b, 1992c, 1992d), Kim, Ramaswamy, and Sundaresan (1993), Ginzburg, Maloney, and Willner (1993), Shimko, Tejima, and Deventer (1993), Genotte and Marsh (1993), and Nielsen, Saa-Requejo, and Santa-Clara (1993). This article distinguishes itself from each of these other contributions in that it is the only one that provides closed-form valuation expressions for risky coupon bonds as well as risky floating-rate debt. In addition, it is the only one that jointly allows for (a) default before the firm exhausts all its assets, (b) complex capital structures including multiple issues of debt, (c) deviations from strict absolute priority, and (d) empirical evidence supporting the implications of the model. The remainder of this article is organized as follows. Section I presents the basic valuation framework. Section I1 derives a valuation model for risky fixed-rate debt and examines its implications for the risk structure of interest rates. Section I11 presents the valuation model for floating-rate debt. Section IV presents the results of the empirical analysis. Section V summarizes the article and makes concluding remarks. I. The Valuation Framework In this section, we extend the Black and Cox (1976) model to develop a simple continuous-time valuation framework for risky debt that allows for both default risk and interest rate risk. This framework is then used in later sections to derive closed-form valuation expressions for a variety of risky corporate debt securities. The basic assumptions of this framework parallel those of Black and Scholes (1973), Merton (1974), and Black and Cox (1976), and are discussed individually below. ASSUMPTION 1: Let V designate the total value of the assets of the firm. The dynamics of V are given by where a is a constant and Z, is a standard Wiener process. 792 The Journal of Finance ASSUMPTION 2: Let r denote the short-term riskless interest rate. The dynamics of r are given by d r = ( 5 pr)d t + qdZ,, (2) where C, p, and q are constants and Z, is also a standard Wiener process. The instantaneous correlation between dZ1 and dZ, is pdt. This assumption about the dynamics of r is drawn from the term structure model of Vasicek (1977). Although consistent with many of the observed properties of interest rates, these dynamics can allow negative interest rates. There are several reasons, however, why this assumption may be justifiable in the context of this model. First, the probability of negative interest rates occurring is small for realistic parameter values.