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Tail Risk of Multivariate Regular Variation

Author

Listed:
  • Harry Joe

    (University of British Columbia)

  • Haijun Li

    (Washington State University)

Abstract

Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds.

Suggested Citation

  • Harry Joe & Haijun Li, 2011. "Tail Risk of Multivariate Regular Variation," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 671-693, December.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:4:d:10.1007_s11009-010-9183-x
    DOI: 10.1007/s11009-010-9183-x
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    References listed on IDEAS

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    1. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    2. Stan Alink & Matthias Löwe & Mario V. Wüthrich, 2007. "Diversification for general copula dependence," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 61(4), pages 446-465, November.
    3. Joe, Harry & Li, Haijun & Nikoloulopoulos, Aristidis K., 2010. "Tail dependence functions and vine copulas," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 252-270, January.
    4. Kousky, Carolyn & Cooke, Roger, 2009. "Climate Change and Risk Management: Challenges for Insurance, Adaptation, and Loss Estimation," RFF Working Paper Series dp-09-03-rev, Resources for the Future.
    5. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    6. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    7. Alink, Stan & Löwe, Matthias & Wüthrich, Mario V., 2005. "Analysis of the Expected Shortfall of Aggregate Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 25-43, May.
    8. Cheridito, Patrick & Delbaen, Freddy & Kupper, Michael, 2004. "Coherent and convex monetary risk measures for bounded càdlàg processes," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 1-22, July.
    9. Alink, Stan & Lowe, Matthias & V. Wuthrich, Mario, 2004. "Diversification of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 77-95, August.
    10. Embrechts, Paul & Neslehová, Johanna & Wüthrich, Mario V., 2009. "Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 164-169, April.
    11. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    12. repec:dau:papers:123456789/353 is not listed on IDEAS
    13. Joe, Harry & Hu, Taizhong, 1996. "Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions," Journal of Multivariate Analysis, Elsevier, vol. 57(2), pages 240-265, May.
    14. Li, Haijun, 2009. "Orthant tail dependence of multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 243-256, January.
    15. Dominik Kortschak & Hansjörg Albrecher, 2009. "Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 279-306, September.
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    Cited by:

    1. Cai, J., 2012. "Estimation concerning risk under extreme value conditions," Other publications TiSEM a92b089f-bc4c-41c2-b297-c, Tilburg University, School of Economics and Management.
    2. Li, Haijun & Hua, Lei, 2015. "Higher order tail densities of copulas and hidden regular variation," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 143-155.
    3. Li, Jinzhu, 2022. "Asymptotic results on marginal expected shortfalls for dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 146-168.
    4. Asimit, Alexandru V. & Furman, Edward & Tang, Qihe & Vernic, Raluca, 2011. "Asymptotics for risk capital allocations based on Conditional Tail Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 310-324.
    5. Juan-Juan Cai & John H. J. Einmahl & Laurens Haan & Chen Zhou, 2015. "Estimation of the marginal expected shortfall: the mean when a related variable is extreme," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 417-442, March.
    6. Bikramjit Das & Vicky Fasen-Hartmann, 2023. "On heavy-tailed risks under Gaussian copula: the effects of marginal transformation," Papers 2304.05004, arXiv.org.
    7. Li, Jinzhu, 2016. "Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 195-204.
    8. Das, Bikramjit & Fasen-Hartmann, Vicky, 2018. "Risk contagion under regular variation and asymptotic tail independence," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 194-215.
    9. Bernard, Carole & Czado, Claudia, 2015. "Conditional quantiles and tail dependence," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 104-126.
    10. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    11. Das Bikramjit & Fasen-Hartmann Vicky, 2019. "Conditional excess risk measures and multivariate regular variation," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 1-23, December.
    12. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Discussion Paper 2023-001, Tilburg University, Center for Economic Research.
    13. Haijun Li, 2018. "Operator Tail Dependence of Copulas," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1013-1027, September.
    14. Hansjörg Albrecher & Martin Bladt & Mogens Bladt, 2021. "Multivariate matrix Mittag–Leffler distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 369-394, April.
    15. Hua, Lei & Joe, Harry, 2011. "Second order regular variation and conditional tail expectation of multiple risks," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 537-546.
    16. Joe, Harry & Li, Haijun, 2019. "Tail densities of skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 421-435.
    17. Jaunė, Eglė & Šiaulys, Jonas, 2022. "Asymptotic risk decomposition for regularly varying distributions with tail dependence," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    18. Konstantinides, Dimitrios G. & Li, Jinzhu, 2016. "Asymptotic ruin probabilities for a multidimensional renewal risk model with multivariate regularly varying claims," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 38-44.
    19. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Other publications TiSEM 261583f5-c571-48c6-8cea-9, Tilburg University, School of Economics and Management.
    20. Haijun Li & Susan Xu & Way Kuo, 2014. "Asymptotic analysis of simultaneous damages in spatial Boolean models," Annals of Operations Research, Springer, vol. 212(1), pages 139-154, January.
    21. Asimit, Alexandru V. & Li, Jinzhu, 2016. "Extremes for coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 332-341.
    22. Xu, Maochao & Mao, Tiantian, 2013. "Optimal capital allocation based on the Tail Mean–Variance model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 533-543.
    23. Bikramjit Das & Vicky Fasen, 2016. "Risk contagion under regular variation and asymptotic tail independence," Papers 1603.09406, arXiv.org, revised Apr 2017.

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