Modeling Credit Spreads and Ratings Migration

This paper develops a theory of bond pricing in which a firm’s instantaneous probability of default is allowed to depend on its credit rating as well as on a latent systematic factor. We examine two versions of the model, one with fixed probabilities of ratings changes and another in which the probability of rating changes varies with the systematic factor. We estimate the model in a Bayesian context using monthly data from the Lehman Brothers Fixed Income Database over the period 1985 – 1998 and find strong evidence in favor of time-varying transition probabilities The pricing of defaultable bonds has always been an important topic in finance. Structural models such as Merton (1974) relate the price of a bond to variability of the value of the issuing firm. More recently, attention has been focused on how ratings affect pricing. This interest has been fueled by empirical work which has shown that the dynamics of the yield spread between lower rated bonds and safer bonds is a predictor of future stock price movements. Despite this interest in credit spreads there is still a lack of models that can fit both the dynamics and shape of credit spread curves. Jarrow, Lando, and Turnbull (1997) was the first paper to incorporate ratings changes into a bond pricing model. In that model ratings follow a Markov chain with default being an absorbing state of the chain. If default occurs before maturity then the bondholder receives some fraction δ of the face value. Any payment is assumed to be made at the maturity date, even in the event of a default. The usefulness of this assumption is that it allows the price to be written as P (t, T ) = Pr(t, T )E Q t (δ1{τD≤T} + 1{τD>T}) where Pr(t, T ) is the price of a default-free bond with the same maturity and face value and τ is the time of default. Credit spreads in this model are determined by δ, which is fixed, and the probability of default, which is easy to calculate in this Markov chain formulation. Unfortunately, since the probability of default is only a function of the current rating and the transition probabilities of the chain, this model implies that credit spreads do not vary with time. So although this model can fit any given initial credit spread curve it cannot capture the dynamics of credit spreads. Lando (1998) contains an extension of the Jarrow, Lando, and Turnbull (1997) model which allows the transition matrix to vary through time by modeling it as a matrix-valued function of a state variable. However recovery rates are assumed to be zero in this model. Rather than assuming that bondholders will receive a fraction of face value on the maturity date in the event of default, Duffie and Singleton (1999) assume that a fraction of market value is received at the time of default. This allows the price of a defaultable Keim and Stambaugh (1986) find that a yield spread has some forecasting power for stock and bond returns. Fama and French (1989) find that a quality spread (or default spread) variable can forecast long-horizon stock returns and low-grade corporate bond returns. Schwert (1989) finds that a quality spread variable is positively related to future stock market volatility. It is now common practice in empirical work to use this “quality spread” as a conditioning variable as in Ferson and Harvey (1991).