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Estimating the Price of Default Risk

Gregory R. Duffee, Board of Governors of the Federal Reserve System

1996
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Finance and Economics Discussion Series
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A rm's instantaneous probability of default is modeled as a square-root diusion process. The parameters of these processes are estimated for 188 rms, using both the time series and cross-sectional (term structure) properties of the individual rms' bond prices. Although the estimated models are moderately successful at bond pricing, there is strong evidence of misspecication. The results indicate that single factor models of instantaneous default risk face a signicant c hallenge in matching
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... in key features of actual corporate bond yield spreads. In particular, such models have diculty generating both relatively at yield spreads when rms have low credit risk and steeper yield spreads when rms have higher credit risk. JEL Classications: G12, G13 for helpful conversations, and seminar participants at an NBER Derivatives Project meeting for helpful comments. The analysis and conclusions of this paper are those of the author and do not indicate concurrence by other members of the research sta, by the Board of Governors, or by the Federal Reserve Banks. 1 One illustration of this lack of success is that very few papers have e v en attempted to price particular instruments. The standard reference is Jones, Mason, and Rosenfeld (1984) , who nd that, even for rms with very simple capital structures, a Merton-type model is unable to price investment-grade corporate bonds better than a naive model that assumes no risk of default. 2 Researchers also continue to extend Merton-style models to address these problems. Recent examples are Longsta and Schwartz (1995) and Zhou (1996) . 1 obligated bond payments follows a square-root diusion process. To simplify the estimation procedure, I assume that each rm's instantaneous default probability process is independent of default-free interest rates. The goal of this exercise is to determine which features of the data are well-described by this model, and which features are inconsistent with the model. In this vein, there are a n umber of issues to consider. How w ell does this single-factor process capture the crosssectional and time-series patterns in corporate bond yields? Is it appropriate to model rms' default probabilities as independent of default-free rates? I also consider the ability of the model to price instruments other than corporate bonds; in particular, credit derivatives with payos tied to multiple defaults on corporate bonds. On average, the model ts corporate bond prices reasonably well. For the typical rm, the root mean squared error in yields is in the neighborhood of 18 basis points. However, the model fails in a numb e r o f w a ys. Some of these failures can be tied to the square-root specication of volatility, but others appear to be more generally applicable to single-factor descriptions of the evolution of default probabilities over time. The most basic failure is the inability of the model to simultaneously t both the level and the slope of the term structure of investment grade bond yield spreads (over Treasuries). For the typical rm and maturities examined in this paper, the observed term structure of yield spreads is positively sloped, and the slope is positively related to the level of spreads. For example, when the level of a typical investment-grade rm's term structure of spreads is low, it is also close to horizontal. When the level rises, so does the slope. (Earlier research has observed the same pattern in the term structures of aggregate yield spreads on investment-grade bonds.) This pattern implies that under the equivalent martingale measure, the drift in instantaneous default risk must be positively related to its level. The problem is that the size of the positive relation that is needed to explain the variation in the slope of the term structure is inconsistent with the very at term structures that are observed when the instantaneous default probability i s l o w. Roughly speaking, even if a rm currently has a very low instantaneous default risk, the possibility (under the equivalent martingale measure) of a future explosive increase in this risk is too large to be consistent with a very at term structure of yield spreads. Single factor models also are of limited use in pricing derivative instruments that have payos tied to the joint default behavior of rms with closely tied prospects (say, S&Ls in Texas during the 1980s). The diculty is that these models are unable to generate default correlations across rms that are much larger than zero. The next section discusses some of the previous empirical work in this area. It is followed by a brief description of the model and the estimation procedure. The fourth and fth sections report the results of estimating the default-free interest rate process and rms' default processes, respectively. The sixth section looks at the relation between credit ratings and default risk and the seventh section looks at credit derivative pricing. The eighth section concludes the paper. Prior empirical evidence Academic research i n to credit risk pricing, including the present paper, relies almost 2 exclusively on the contingent-claims framework. It is worth noting that the assumptions underlying this framework are probably not satised. Although credit risk has existed ever since nancial contracts were traded, in many respects credit risk is the new frontier of asset pricing on Wall Street. Because the stochastic behavior of credit risk is not well-understood by market participants, identifying potential arbitrage opportunities is dicult. In addition, markets for credit risk are typically illiquid and asymmetric information problems abound. In order to understand the eects that credit risks have on asset prices, it makes sense to focus on those markets where credit risk is important. By this metric, it is natural to look at corporate bond prices. The pricing of credit risk is much more important for corporate bonds than for interest rate swaps, because default risk should have a v ery small eect on swap prices (Sorensen and Bollier 1994, Due and Huang 1996) . Moreover, the market for seasoned corporate bonds is relatively more liquid than the (practically nonexistent) market for seasoned swaps. Madan and Unal (1994) examine another nancial instrument for which credit risk is important: Certicates of deposit issued by roughly 300 thrift institutions over January 1987 to December 1991. They used a model of instantaneous default risk to relate the time variation in average thrift CD rates to variations in average thrift stock prices, core deposit ratios, and returns to an index of low-grade bonds. Grinblatt (1995) explores the relation between mid-market interest-rate swap spreads and LIBOR, although he argues that this relation is driven by a liquidity yield to Treasury securities instead of default risk. He estimates two dierent single-factor forms of this spread (square-root and Gaussian). The use of yield indexes, instead of rm-specic yields, is justied in Grinblatt's work because of his maintained assumption that default risk is essentially irrelevant in the determination of swap spreads. Due and Singleton (1995b) also analyze mid-market swap quotes, modeling swap yields with a two-factor square-root process. Because they are unwilling to assume that swap quotes are unrelated to default risk, their use of mid-market yields on newly-issued swaps is explicitly justied by assuming that all swap counterparties maintain a credit rating o f A u n til the swap matures or default occurs. Although clearly at odds with reality, a n y errors induced by this assumption are small because swap yields are not very responsive t o v ariations in default risk. Nielsen and Ronn (1996) assume that instantaneous corporate yield spreads over defaultfree yields follow a geometric random walk, while default-free interest rates are described by a one-factor model with deterministic time-varying coecients. They t their model to both corporate bond yield indexes and swap spreads. Their estimation procedure uses only cross-sectional data, not time-series data, thus they cannot identify the true (i.e., physical) processes, and their estimated parameters change from day to day. The use of refreshed yield indexes is more dicult to justify when analyzing corporate bond yield spreads than when analyzing swap yields. Duee (1996) nds that empirically, refreshed corporate bond yield indexes underrespond to actual variations in credit quality. This result is not surprising, considering that corporate bond yields are quite sensitive t o v ariations in credit quality, but refreshed yield indexes hold constant one measure of credit quality (credit ratings). 3

doi:10.17016/feds.1996.29
fatcat:o5as22prffgv5k3aja7u6jszky