Abstract

This paper is dedicated to the consistency of systemic risk mea-sures with respect to stochastic dependence. It compares two alter-native notions of Conditional Value-at-Risk (CoVaR) available in the current literature. These notions are both based on the conditional distribution of a random variable Y given a stress event for a random variable X, but they use different types of stress events. We derive representations of these alternative CoVaR notions in terms of cop-ulas, study their general dependence consistency and compare their performance in several stochastic models. Our central finding is that conditioning on X ≥ VaRα(X) gives a much better response to de-pendence between X and Y than conditioning on X = VaRα(X). We prove general results that relate the dependence consistency of CoVaR using conditioning on X ≥ VaRα(X) to well established results on con-cordance ordering of multivariate distributions or their copulas. These results also apply to some other systemic risk measures, such as the Marginal Expected Shortfall (MES) and the Systemic Impact Index (SII). We provide counterexamples showing that CoVaR based on the stress event X = VaRα(X) is not dependence consistent. In particu-lar, if (X,Y) is bivariate normal, then CoVaR based on X = VaRα(X) is not an increasing function of the correlation parameter. Similar is-sues arise in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X ≥ VaRα(X) is an increasing function of the dependence parameter.