On dependence consistency of CoVaR and some other systemic risk measures
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.
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Gauthier, Céline & Lehar, Alfred & Souissi, Moez, 2012. "Macroprudential capital requirements and systemic risk," Journal of Financial Intermediation, Elsevier, vol. 21(4), pages 594-618.
- Viral V. Acharya & Lasse H. Pedersen & Thomas Philippon & Matthew Richardson, 2010.
"Measuring systemic risk,"
1002, Federal Reserve Bank of Cleveland.
- Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
- Wei, Gang & Hu, Taizhong, 2002. "Supermodular dependence ordering on a class of multivariate copulas," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 375-385, May.
- Chen Zhou, 2009. "Are banks too big to fail?," DNB Working Papers 232, Netherlands Central Bank, Research Department.
When requesting a correction, please mention this item's handle: RePEc:arx:papers:1207.3464. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators)
If references are entirely missing, you can add them using this form.