Modelling the Dynamic Relationship between Systematic Default and Recovery Risk

Abstract

Default correlation modelling is becoming the most popular problem in the field of credit derivatives pricing. An increase in default risk would cause the recovery rate to change correspondingly. Correlation between default and recovery rates has a noticeable effect on risk measures and credit derivatives pricing. After an introduction, we review the most recent literature covering default correlation and the relationship between default and recovery rates. We adopt the copula methodology to focus on estimating the default correlations rather than focus on modelling probabilities of default, we then use stress testing to compare the distributions of the probability of default under different copula functions. We develop a Gamma-Beta model to link the recovery rate directly with the individual probability of default, this is instead of an extended one factor model to relate them by a systematic common factor. One factor models are re-examined to explore correlated recovery rates under three distributions: the Logit-normal, the Normal and the Log-normal. By analyzing the results respectively obtained from these two classes of modelling scheme, we argue that the direct dependence (Gamma-Beta) model behaves better, in estimating the recovery rate given individual probability of default and in suggesting a better indication of their relationship. Finally, we apply default correlation and the correlated recovery rate to portfolio risk modelling. We conclude that if the recovery rates are independent stochastic variables, the expected losses in a large portfolio might be underestimated because the uncorrelated recovery risks can be diversified, so the correlation between default rate and recovery risk can not be neglected in the applications. Here, we believe the first time, the recovery rate depends on individual default probability by means of a closed formula.