Fast Simulation of Multifactor Portfolio Credit Risk

Abstract

This paper develops rare-event simulation methods for the estimation of portfolio credit risk-the risk of losses to a portfolio resulting from defaults of assets in the portfolio. Portfolio credit risk is measured through probabilities of large losses, which are typically due to defaults of many obligors (sources of credit risk) to which a portfolio is exposed. An essential element of a portfolio view of credit risk is a model of dependence between these sources of credit risk: large losses occur rarely and are most likely to result from systematic risk factors that affect multiple obligors. As a consequence, estimating portfolio credit risk poses a challenge both because of the rare-event property of large losses and the dependence between defaults. To address this problem, we develop an importance sampling technique within the widely used Gaussian copula model of dependence. We focus on difficulties arising in multifactor models-that is, models in which multiple factors may be common to multiple obligors, resulting in complex dependence between defaults. Our importance sampling procedure shifts the mean of the common factor to increase the frequency of large losses. In multifactor models, different combinations of factor outcomes and defaults can produce large losses, so our method combines multiple importance sampling distributions, each associated with a shift in the mean of common factors. We characterize "optimal" mean shifts. Finding these points is both a combinatorial problem and a convex optimization problem, so we address computational aspects of this step as well. We establish asymptotic optimality results for our method, showing that-unlike standard simulation-it remains efficient as the event of interest becomes rarer.

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

“…In particular, considerable effort has been devoted towards developing efficient MC methods for estimating VaR in realistic credit risk and market models, see e.g. Glasserman (2004), Glasserman et al (2008), and the references therein. Unfortunately, VaR suffers from two drawbacks.…”

“…Accordingly, variance reduction techniques become very important in practical applications. For example, Glasserman (2004) and Glasserman et al (2008) discuss efficient estimators for loss probabilities by applying the importance sampling method exponential twisting. An analogous approach can be used to obtain numerically efficient MC estimators for SR, if variance reduction techniques are combined with stochastic approximation techniques.…”

“…Otherwise, additional importance sampling of the factor variables Z may turn out to be helpful, cf. Glasserman et al (2008). Importance sampling has led to a considerable variance reduction, cf.…”

“…The recent literature on VaR focusses mostly on the efficient estimation of the probability of large losses, see e.g. Glasserman (2004), Kang & Shahabuddin (2005), and Glasserman, Kang & Shahabuddin (2008).…”

Monte Carlo Computation in Finance

“…This makes it worthwhile to pursue variance reduction: see [46,Ch. 9] for general techniques and [8,25,48,49,51] for techniques specific to credit risk.…”