  This paper considers the problem of measuring the credit risk in portfolios
of loans, bonds, and other instruments subject to possible default under
multi-factor models. Due to the amount of the portfolio, the heterogeneous
effect of obligors, and the phenomena that default events are rare and mutually
dependent, it is difficult to calculate portfolio credit risk either by means
of direct analysis or crude Monte Carlo under such models. To capture the
extreme dependence among obligors, we provide an efficient simulation method
for multi-factor models with a normal mixture copula that allows the
multivariate defaults to have an asymmetric distribution, while most of the
literature focuses on simulating one-dimensional cases. To this end, we first
propose a general account of an importance sampling algorithm based on an
unconventional exponential embedding, which is related to the classical
sufficient statistic. Note that this innovative tilting device is more suitable
for the multivariate normal mixture model than traditional one-parameter
tilting methods and is of independent interest. Next, by utilizing a fast
computational method for how the rare event occurs and the proposed importance
sampling method, we provide an efficient simulation algorithm to estimate the
probability that the portfolio incurs large losses under the normal mixture
copula. Here the proposed simulation device is based on importance sampling for
a joint probability other than the conditional probability used in previous
studies. Theoretical investigations and simulation studies, which include an
empirical example, are given to illustrate the method.
