A Theoretical Inspection of the Market Price for Default Risk

Abstract

While there are now a number of empirical studies on the subject, very little is known on the market price for default risk from a theoretical perspective. This paper is a first step in the direction of an equilibrium model for the pricing of defaultable securities in an incomplete market setup. We first provide an explicit characterization of the set of equivalent martingale measures consistent with no arbitrage in the presence of default risk, as well as a necessary and sufficient condition for a convenient separation between adjustments for market risk and default risk. That result allows us to spell out an unambiguous definition of the market price for default risk as the logarithm of the ratio of the risk-adjusted probability of default to the original probability of default. It also suggests the following question: how should the original probability of default be adjusted to account for agents' risk-aversion? We address this question in a dynamic continuous-time equilibrium setup, and obtain a defaultable version of a standard consumption-based capital asset pricing model. In particular, we confirm the intuition that the correlation between default risk and market risk is a key ingredient of the equilibrium price for default risk, and obtain a quantitative estimate of the magnitude of the effect. Our model is consistent with empirical findings in that it predicts that the term structure of credit spreads can be upward sloping with a non-zero intercept. The theory is illustrated by an application to the valuation of employee compensation packages, which may be regarded as peculiar, yet natural, examples of defaultable securities.