How Robust is the Value-at-Risk of Credit Risk Portfolios?

In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, i.e., what is the maximum and minimum Value-at-Risk (VaR) that can be justified given a certain amount of available information. Puccetti and Rüschendorf (2012b) and Embrechts et al. (2013) propose the rearrangement algorithm (RA) as a general method to approximate VaR bounds when the default probabilities, exposures and recovery rates of the different loans are known but not their interdependence. Their numerical results show that the gap between worst-case and best-case VaR is typically very high, a feature that can only be explained by lack of dependence information. Hence, sharpening the VaR bounds by considering the presence of dependence information is of great practical relevance. In this paper, we propose an efficient algorithm to approximate sharp VaR bounds for credit risk portfolios when besides the marginal distributions also higher order moments of the aggregate portfolio such as variance and skewness are available as sources of dependence information. We also give explicit sharp bounds for homogeneous credit risk portfolios. A numerical study shows that in all practical situations of interest, VaR assessments of credit portfolios that are performed at high confidence levels (as in Solvency II and Basel III) are subject to significant model uncertainty and thus not robust even with the additional moment information.