A General Importance Sampling Algorithm for Estimating Portfolio Loss Probabilities in Linear Factor Models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. In a sense we modify the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models, including the t copula. The basic idea is to apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors. Motivated by the cross-entropy method we determine optimal values of the tilt parameters by minimizing the Kullback-Liebler divergence of the candidate family from the ideal (i.e. zero-variance) importance sampling measure, leading to an intuitive moment-matching characterization. Finally, by exploiting the large portfolio approximation to the portfolio loss, we develop efficient and accurate approximations to the requisite moments. In the context of the Gaussian copula the proposed algorithm offers comparable performance to the Glasserman and Li (2005) algorithm, which is known to be asymptotically optimal. In the context of the t copula the proposed algorithm offers comparable performance for substantially less computational time than the Chan and Kroese (2010) algorithm, which also has good asymptotic properties