Simulating Correlated Default Processes Using Copulas: a Criterion-based Approach

Modeling correlated default risk is a new phenomenon currently sweeping through the credit markets. Little is known about the drivers of default risk at the portfolio level. This paper develops a methodology to assess alternative specifications of the joint distribution of default risk. Specifications are based on three criteria: level, asymmetry, and tail-dependence in the joint default distribution. The 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 1987-2000. Using copula functions, we separate the estimation of the marginal distributions from the joint distribution. We determine the appropriate choice of multivariate distribution based on a new metric for the assessment of joint distributions. This metric accounts for different aspects of default correlation, namely level, asymmetry and tail-dependence or extreme behavior. Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the fat-tailed 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 dependency 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. We believe this is one of the first papers to empirically compare joint default distributions using copulas for the U.S. market. SIMULATING CORRELATED DEFAULT PROCESSES 3