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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 given a stress event for a random variable , 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 ≥ VaR ( ) gives a much better response to dependence between and than conditioning on = VaR ( ). We prove general results that relate the dependence consistency of CoVaR using conditioning on ≥ VaR ( ) 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 = VaR ( ) is not dependence consistent. In particular, if ( , ) is bivariate normal, then CoVaR based on = VaR ( ) is not an increasing function of the correlation parameter. Similar issues arise in the bivariate model and in the model with margins and a Gumbel copula. In all these cases, CoVaR based on ≥ VaR ( ) is an increasing function of the dependence parameter.

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We propose two indicators for quantifying the potential exposure of financial institutions to indirect contagion arising from deleveraging of assets in stress scenarios. The first indicator, the Endogenous Risk Index (ERI) captures spillovers across portfolios arising from deleveraging in stress scenarios. The second indicator, the Indirect Contagion Index (ICI) measures the systemic importance of a bank by quantifying the loss its distressed liquidation would inflict on other institutions.Both are computable from portfolio holdings of financial institutions and measures of market depth for the assets held in the portfolio. We discuss the micro-foundation of these indicators and apply them to the analysis of the vulnerability of the European banking system to indirect contagion.Using data on portfolio holdings of European banks, we show that our indicators correlate to the magnitude of fire-sales losses in simulated stress scenarios, thus providing a simple to compute proxy for the outcome of stress tests. We also show that the information provided by our indicators on the systemic importance of banks is different from indicators based on size, thereby providing a measure of interconnectedness complementary to those currently used by supervisors.