Monte Carlo Methods for Portfolio Credit Risk

The financial crisis of 2007 – 2009 began with a major failure in credit markets. The causes of this failure stretch far beyond inadequate mathematical modeling (see Donnelly and Embrechts [2010] and Brigo et al. [2009] for detailed discussions from a mathematical finance perspective). Nevertheless, it is clear that some of the more popular models of credit risk were shown to be flawed. Many of these models were and are popular because they are mathematically tractable, allowing easy computation of various risk measures. More realistic (and complex) models come at a significant computational cost, often requiring Monte Carlo methods to estimate quantities of interest. The purpose of this chapter is to survey the Monte Carlo techniques that are used in portfolio credit risk modeling. We discuss various approaches for modeling the dependencies between individual components of a portfolio and focus on two principal risk measures: Value at Risk (VaR) and Expected Shortfall (ES). The efficient estimation of the credit risk measures is often computationally expensive, as it involves the estimation of small quantiles. Rare-event simulation techniques such as importance sampling can significantly reduce the computational burden, but the choice of a good importance sampling distribution can be a difficult mathematical problem. Recent simulation techniques such as the cross-entropy method [Rubinstein and Kroese, 2004] have greatly enhanced the applicability of importance sampling techniques by adaptively choosing the importance sampling distribution, based on samples from the original simulation model. The remainder of this chapter is organized as follows. In Section 2 we describe the general model framework for credit portfolio loss. Section 3 discusses the crude and importance sampling approaches to estimating risk measures via the Monte Carlo method. Various applications to specific models (including Bernoulli mixture models, factor models, copula models and intensity models) are given in Section 4. Many of these models capture empirical features of credit risk, such as default clustering, that are not captured by the standard Gaussian models. Finally, the Appendix contains the essentials on rare-event simulation and adaptive importance sampling.