Extensions to the Gaussian copula: random recovery and random factor loadings

Abstract

This paper presents two new models of portfolio default loss that extend the standard Gaussian copula model yet preserve tractability and computational efficiency. In one extension, we randomize recovery rates, explicitly allowing for the empirically well-established effect of inverse correlation between recovery rates and default frequencies. In another extension, we build into the model random systematic factor loadings, effectively allowing default correlations to be higher in bear markets than in bull markets. In both extensions, special cases of the models are shown to be as tractable as the Gaussian copula model and to allow efficient calibration to market credit spreads. We demonstrate that the models – even in their simplest versions – can generate highly significant pricing effects such as fat tails and a correlation “skew ” in synthetic CDO tranche prices. When properly calibrated, the skew effect of random recovery is quite minor, but the extension with random factor loadings can produce correlation skews similar to the steep skews observed in the market. We briefly discuss two alternative skew models, one based on the Marshall-Olkin copula, the other on a spread-dependent correlation specification for the Gaussian copula.