On dependence consistency of CoVaRand some other systemic risk measures

Abstract

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.

On Conditional Value at Risk (CoVaR) for tail-dependent copulas

“…Furthermore, as was observed in [33] Th.3.4 for the L/P setting, modi ed CoVaR is compatible with the concordance ordering of copulas. The same is valid for the P/L setting.…”

“…To switch to the alternative Loss/Pro t (L/P) approach (applied for example in [5,18,33]) when random variables are modelling losses from the nancial investments, actuarial risks or high water levels in hydrology, for example, it is is enough to change the sign of the variables L = −X, and remember that, by convention, the subscript is changed. The signi cance level α is replaced by the condence level c = − α, that is VaRc(L) = VaRα(X).…”

“…Besides the issues described above, there are also some more practical drawbacks of the standard CoVaR pointed out for example by Mainik and Schaanning in [33], which motivate a modi cation of the original de nition. The main objection is due to the fact that the standard CoVaR is not compatible with concordance ordering.…”

“…This leads to the following de nition of CoVaR (see [4] and [33] Th. 3.1.a for L/P approach, compare [3] for conditional quantile setting)…”

“…Modi ed CoVaR was introduced by Girardi and A.T. Ergün in 2013 ( [16]) and Mainik and Schaanning in 2014 ( [33]), both in the L/P setting.…”