An Approximation for Credit Portfolio Losses

Mixture models play an important role in the modeling of portfolio losses. In these models the risk of default of individual obligors (indexed by i ∈ {1, . . . ,m}) depends on an underlying set of common economic factors, denoted Ψ. Given these factors, the losses due to default li of individual obligors are assumed to be stochastically independent. Dependence between different obligors stems only from dependence of the individual default probabilities on the set of factors. These models are used for both risk management of credit portfolios and valuation of multi-name credit derivatives. The current article investigates both issues. The numerical evaluation of the portfolio loss distribution is usually based on the two-stage structure of mixture models. For instance, in order to sample from the loss distribution by standard Monte Carlo, one generates first a realization of the systematic factor variable Ψ. In a second step one generates a sequence of independent variates l̂i, 1 ≤ i ≤ m, according to the conditional distribution of (li)1≤i≤m given Ψ. Standard Monte Carlo can be quite slow, and so various numerical techniques for estimating the distribution of the total loss of a portfolio in mixture models have been developed. In this paper we focus on the the second stage, i.e. the conditional loss distribution given the underlying factors, and propose an alternative way for evaluating the conditional distribution of the total loss L = ∑m i=1 li. Our approximation is based on a central limit theorem; error bounds can be derived from the Berry-Esseen-inequality. We compare the numerical performance of our method relative to the standard Vasicek-approximation and the true loss distribution obtained by standard Monte Carlo methods. It turns out that our suggested approximation technique often provides more accurate results than the Vasicek-approximation while being computationally less expensive