Default Risk and Diversification: Theory and Applications*

Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity’s diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of “diversifiable default risk”. The equivalence between the empirical and martingale default intensity that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. The consistency of this diversifiable risk assumption is illustrated in the context of existing studies on corporate bonds, interest rate swaps, and mortgage-backed securities. Reduced-form models of defaultable securities, which view the default of corporate bond issuers as an unpredictable event, have become a popular tool in credit risk modeling. A key advantage of this approach is that it brings into play the machinery of classical term structure modeling techniques. This is convenient for the econometric specification of models for credit risky bonds as well as for the pricing of credit derivatives. The strong analogy with ordinary term structure modeling which will be briefly recalled in the next section allows for specifications of default intensities and short rates using for example the affine term structure machinery of which the models by Cox, Ingersoll and Ross (1985) and Vasicek (1977) are the classical examples.1 Pricing bonds and derivatives in this framework requires only the evolution of the state variables under an equivalent martingale measure. However, in order to understand the factor risk premia in bond markets (and to utilize time-series information in the empirical estimation), a joint specification of the evolution of the state variables under the “physical measure” and the equivalent martingale measure is required. The structure of these risk premia is well understood in for example the affine models of the term structure. A key concern in our understanding of the corporate bond market is the form (and size) of the risk premia for default risk. Since the reduced-form approach allows us to model default risk using standard term structure machinery, it is natural to use the same structure for the risk premia of the intensity processes as we would use for the short rate process in ordinary term structure models. This choice has led to an interpretation of the drift adjustment on the state variables underlying the martingale default intensity as a “default risk premium” or “price of default risk”.2 Recent examples of this approach are the empirical works by Duffie and Singleton (1997), Duffee (1999), and Liu, Longstaff and Mandell (2001). The last authors proceed a step further along the risk premium interpretation by computing the expected returns of defaultable bonds using these drift adjustments. We show in this paper, that this specification for the default risk premia implies a strong restriction on the set of possible risk premia. The fact that the intensity process is not just an affine function of diffusion state variables but is also the compensator of a jump process allows for a much richer class of risk premia. The critical distinction is really whether agents only price variations in the default intensity, which then must be pervasive, or they also price the default event itself. This insight can be derived from existing works such as Back (1991) and Jarrow and Madan (1995). Through the well-known connection between the state-price density and the marginal utility of (for example) a representative investor or a single optimizing agent, it is easy to see that the For more general works on affine models, see for example Duffie and Kan (1996) and Dai and Singleton (2000). To be precise, the martingale intensity is usually assumed to depend on short rate factors. This is to capture the systematic dependence of credit spreads on the default-free term structure. The drift adjustments on these short rate factors are appropriately interpreted as interest rate risk premia. However, usually one more state variable is included for the intensity and its risk adjustment is given the interpretation of a default risk premium.