Simulation-analytical approach for calculating VaR contributions in credit portfolios

Risk capital allocation involves the decomposition of the overall portfolio risk in a credit portfolio into marginal risk contributions associated with individual obligors. The Value-at-Risk contribution (VaRC) measures how much each obligor contributes to the overall portfolio VaR. We propose two forms of simulation-analytical approach to calculate VaRC in Gaussian copula credit portfolios with correlation between probability of default (PD) and loss given default (LGD). Our method resolves the challenge of computing the expectation of the default loss of an individual obligor conditional on an event with zero probability mass. By employing ingenious analytical and simulation procedures, we recast the VaRC calculation to involve simulating the distribution function of the random portfolio loss, thereby avoiding numerical instabilities in the common kernel estimation method of computing expectations conditional on a zero-probability event. The inclusion of an analytical component in our algorithms helps reduce simulation effort when compared with the full simulation algorithms. Numerical experiments demonstrate that our proposed algorithms perform favorably well in terms of accuracy and computational efficiency, outperforming typical direct Monte Carlo simulation methods such as the iterative cross entropy method.