Before revision:  The counterexamples for CoVaR based on the stress event X=VaR_\alpha(X) include inconsistency with respect to correlation in the bivariate Gaussian model .
After revision:  The counterexamples for 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 Gaussian model .