Efficient estimation of large portfolio loss probabilities in t-copula models

We consider the problem of accurately measuring the credit risk of a portfolio consisting of loans, bonds and other financial assets. One particular performance measure of interest is the probability of large portfolio losses over a fixed time horizon. We revisit the so-called t-copula that generalizes the popular normal copula to allow for extremal dependence among defaults. By utilizing the asymptotic description of how the rare event occurs, we derive two simple simulation algorithms based on conditional Monte Carlo to estimate the probability that the portfolio incurs large losses under the t-copula. We further show that the less efficient estimator exhibits bounded relative error. An extensive simulation study demonstrates that both estimators outperform existing algorithms. We then discuss a generalization of the t-copula model that allows the multivariate defaults to have an asymmetric distribution. Lastly, we show how the estimators proposed for the t-copula can be modified to estimate the portfolio risk under the skew t-copula model.