  This paper is dedicated to the consistency of systemic risk measures with
respect to stochastic dependence. It compares two alternative 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 copulas, study their general dependence consistency and compare their
performance in several stochastic models. Our central finding is that
conditioning on X>=VaR_\alpha(X) gives a much better response to dependence
between X and Y than conditioning on X=VaR_\alpha(X). We prove general results
that relate the dependence consistency of CoVaR using conditioning on
X>=VaR_\alpha(X) to well established results on concordance 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=VaR_\alpha(X) is not dependence consistent. In
particular, if (X,Y) is bivariate normal, then CoVaR based on X=VaR_\alpha(X)
is not an increasing function of the correlation parameter. Similar issues
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>=VaR_\alpha(X) is an increasing
function of the dependence parameter.
