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, that is, what is the maximum and minimum value-at-risk (VaR) of a portfolio of risky loans that can be justified given a certain amount of available information. Puccetti and Rüschendorf [2012a. “Computation of Sharp Bounds on the Distribution of a Function of Dependent Risks”. Journal of Computational and Applied Maths 236, 1833–1840] and Embrechts, Puccetti, and Rüschendorf [2013. “Model Uncertainty and VaR Aggregation”. Journal of Banking and Finance 37, 2750–2764] propose the rearrangement algorithm (RA) as a general method to approximate VaR bounds when the loss distributions of the different loans are known but not their interdependence (unconstrained bounds). 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 using dependence information. We propose a modification of the RA that makes it possible to approximate sharp VaR bounds 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. A numerical study shows that the use of moment information makes it possible to significantly improve the (unconstrained) VaR bounds. However, VaR assessments of credit portfolios that are performed at high confidence levels (as it is the case in Solvency II and Basel III) remain subject to significant model uncertainty and are not robust.