Efficient Algorithms for Calculating Risk Measures and Risk Contributions in Copula Credit Models

The numerical calculations of marginal risk contributions associated with the two risk measures, Value-at-Risk and Expected Shortfall, pose challenges due to the rare event character of multiple defaults among obligors in credit portfolios at a high confidence level. We explore various enhancements in several importance sampling algorithms to achieve better computational efficiency and accuracy for computing marginal risk contributions of the popular Gaussian copula model and t-copula model. The Glasserman-Li exponential twisting method is seen to have a close link with the saddle-point approximation method. We develop the hybrid saddle-point approximation method that inherits the merit of the nice analytical tractability exhibited by the saddle-point approximation framework. We enhance the efficiency of calculating the saddle-point via an interpolation procedure. We also propose new formulas to enhance the cross entropy method for the estimation of the optimal parameters in the t-copula model. Extensive numerical tests on computing risk contributions were performed on various copula models with multiple risk factors. Our enhanced exponential twisting method, cross entropy method, and hybrid saddle-point approximation method are seen to exhibit a high level of efficiency, accuracy, and reliability when compared with existing importance sampling