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On dependence consistency of CoVaR and some other systemic risk measures

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  • Georg Mainik
  • Eric Schaanning

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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  • Georg Mainik & Eric Schaanning, 2012. "On dependence consistency of CoVaR and some other systemic risk measures," Papers 1207.3464, arXiv.org, revised Aug 2012.
  • Handle: RePEc:arx:papers:1207.3464
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    2. 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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    7. Reboredo, Juan C. & Ugolini, Andrea, 2015. "Systemic risk in European sovereign debt markets: A CoVaR-copula approach," Journal of International Money and Finance, Elsevier, vol. 51(C), pages 214-244.
    8. Tobias Adrian & Markus K. Brunnermeier, 2016. "CoVaR," American Economic Review, American Economic Association, vol. 106(7), pages 1705-1741, July.
      • Tobias Adrian & Markus K. Brunnermeier, 2008. "CoVaR," Staff Reports 348, Federal Reserve Bank of New York.
      • Tobias Adrian & Markus K. Brunnermeier, 2011. "CoVaR," NBER Working Papers 17454, National Bureau of Economic Research, Inc.
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    10. Yesol Huh, 2014. "Machines vs. Machines: High Frequency Trading and Hard Information," Finance and Economics Discussion Series 2014-33, Board of Governors of the Federal Reserve System (U.S.).
    11. Zulu, Thulani & Manguzvane, Mathias Mandla & Bonga-Bonga, Lumengo, 2023. "Assessing the contribution of South African Insurance Firms to Systemic Risk," MPRA Paper 116815, University Library of Munich, Germany.
    12. Areski Cousin & Elena Di Bernardino, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.
    13. Markus Brunnermeier & Simon Rother & Isabel Schnabel & Itay Goldstein, 2020. "Asset Price Bubbles and Systemic Risk," The Review of Financial Studies, Society for Financial Studies, vol. 33(9), pages 4272-4317.
    14. Girardi, Giulio & Tolga Ergün, A., 2013. "Systemic risk measurement: Multivariate GARCH estimation of CoVaR," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 3169-3180.
    15. Hélène Cossette & Mélina Mailhot & Étienne Marceau & Mhamed Mesfioui, 2016. "Vector-Valued Tail Value-at-Risk and Capital Allocation," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 653-674, September.
    16. Bokusheva, Raushan, 2014. "Improving the Effectiveness of Weather-based Insurance: An Application of Copula Approach," MPRA Paper 62339, University Library of Munich, Germany.
    17. Reboredo, Juan C. & Ugolini, Andrea, 2015. "A vine-copula conditional value-at-risk approach to systemic sovereign debt risk for the financial sector," The North American Journal of Economics and Finance, Elsevier, vol. 32(C), pages 98-123.
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