Correlated Default Processes: A Criterion-Based Copula Approach

Abstract

Modeling correlated default risk is a new phenomenon currently sweeping through the credit markets. In this paper, we develop a methodology to model, simulate and assess the joint default process of hundreds of issuers. Our study is based on a data set of default probabilities supplied by Moody's Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period of 1987-2000 using copula functions. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for different aspects of default correlation, namely (i) level, (ii) asymmetry and (iii) tail-dependence and extreme behavior. Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the empirical joint distribution of defaults. A comparison of a jump model and a regime-switching model shows that the latter provides a better representation of the properties of correlated default. We also find that the skewed double-exponential distribution is the best choice for the marginal distribution of each issuer's hazard rate process, and combines well with the normal, Gumbel, Clayton and student's t copulas in the joint dependence relationship amongst issuers. As a complement to the methodological innovation, we show that (a) appropriate choices of marginal distributions and copulas are essential in modeling correlated default, (b) accounting for regimes is an important aspect of joint specifications of default risk, and (c) misspecification of credit portfolio risk can occur easily if joint distributions are inappropriately chosen. The empirical evidence suggests that improvements are indeed possible over the standard Gaussian copula used in practice.

Keywords: copula, tail-dependence, default

JEL Classification: C14, C15, G13