Value-at-Risk bounds with variance constraints

Recent literature deals with bounds on the Value-at-Risk (VaR) of risky portfolios when only the marginal distributions of the components are known. In this paper we study Value-at-Risk bounds when the variance of the portfolio sum is also known, a situation that is of considerable interest in risk management. We provide easy to calculate Value-at-Risk bounds with and without variance constraint and show that the improvement due to the variance constraint can be quite substantial. We discuss when the bounds are sharp (attainable) and point out the close connections between the study of VaR bounds and convex ordering of aggregate risk. This connection leads to the construction of a new practical algorithm, called Extended Rearrangement Algorithm (ERA), that allows to approximate sharp VaR bounds. We test the stability and the quality of the algorithm in several numerical examples. We apply the results to the case of credit risk portfolio models and verify that adding the variance constraint gives rise to significantly tighter bounds in all situations of interest. However, model risk remains a concern and we criticize regulatory frameworks that allow financial institutions to use internal models for computing the portfolio VaR at high confidence levels (e.g., 99.5%) as the basis for setting capital requirements. Key-words: Value-at-Risk, Convex order, Comonotonicity, Model risk, Rearrangement algorithm. ∗Carole Bernard, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada. (email: [email protected]). Carole Bernard acknowledges support from NSERC, the CAE research grant and the Humboldt Research Foundation. †Ludger Rüschendorf, University of Freiburg, Eckerstraße 1, 79104 Freiburg, Germany. (email: [email protected]). ‡Corresponding author: Steven Vanduffel, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Bruxelles, Belgium. (email: [email protected]). Steven Vanduffel acknowledges support from BNP Paribas.