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On dependence consistency of CoVaR and some other systemic risk measures

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  • Georg Mainik
  • Eric Schaanning
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    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 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.

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    File URL: http://arxiv.org/pdf/1207.3464
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1207.3464.

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    Date of creation: Jul 2012
    Date of revision: Aug 2012
    Handle: RePEc:arx:papers:1207.3464

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    Web page: http://arxiv.org/

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    References

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    1. Wei, Gang & Hu, Taizhong, 2002. "Supermodular dependence ordering on a class of multivariate copulas," Statistics & Probability Letters, Elsevier, Elsevier, vol. 57(4), pages 375-385, May.
    2. Viral V. Acharya & Lasse H. Pedersen & Thomas Philippon & Matthew Richardson, 2010. "Measuring systemic risk," Working Paper 1002, Federal Reserve Bank of Cleveland.
    3. Chen Zhou, 2009. "Are banks too big to fail?," DNB Working Papers, Netherlands Central Bank, Research Department 232, Netherlands Central Bank, Research Department.
    4. Gauthier, Céline & Lehar, Alfred & Souissi, Moez, 2012. "Macroprudential capital requirements and systemic risk," Journal of Financial Intermediation, Elsevier, Elsevier, vol. 21(4), pages 594-618.
    5. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
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    Cited by:
    1. Girardi, Giulio & Tolga Ergün, A., 2013. "Systemic risk measurement: Multivariate GARCH estimation of CoVaR," Journal of Banking & Finance, Elsevier, Elsevier, vol. 37(8), pages 3169-3180.
    2. Danielsson, Jon & James, Kevin & Valenzuela, Marcela & Zer, Ilknur, 2014. "Model Risk of Risk Models," Finance and Economics Discussion Series, Board of Governors of the Federal Reserve System (U.S.) 2014-34, Board of Governors of the Federal Reserve System (U.S.).
    3. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.

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