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Asymptotic Analysis of Multivariate Tail Conditional Expectations

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  • Li Zhu
  • Haijun Li

Abstract

Tail conditional expectations refer to the expected values of random variables conditioning on some tail events and are closely related to various coherent risk measures. In the univariate case, the tail conditional expectation is asymptotically proportional to Value-at-Risk, a popular risk mea-sure. The focus of this paper is on asymptotic relations between the multivariate tail conditional expectation and Value-at-Risk for heavy-tailed scale mixtures of multivariate distributions. Explicit tail estimates of multivariate tail conditional expectations are obtained using the method of regular variation. Examples involving multivariate Pareto and elliptical distributions, as well as application to risk allocation, are also discussed.

Suggested Citation

  • Li Zhu & Haijun Li, 2012. "Asymptotic Analysis of Multivariate Tail Conditional Expectations," North American Actuarial Journal, Taylor & Francis Journals, vol. 16(3), pages 350-363.
    • Handle: RePEc:taf:uaajxx:v:16:y:2012:i:3:p:350-363
    DOI: 10.1080/10920277.2012.10590646
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    Cited by:

    1. Andres Mauricio Molina Barreto & Naoyuki Ishimura, 2023. "Remarks on a copula‐based conditional value at risk for the portfolio problem," Intelligent Systems in Accounting, Finance and Management, John Wiley & Sons, Ltd., vol. 30(3), pages 150-170, July.
    2. Li, Jinzhu, 2022. "Asymptotic results on marginal expected shortfalls for dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 146-168.
    3. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    4. Yang Yang & Shuang Liu & Kam Chuen Yuen, 2022. "Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2600-2621, December.
    5. 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.
    6. 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.
    7. Haoyu Chen & Kun Fan, 2022. "Tail Value-at-Risk-Based Expectiles for Extreme Risks and Their Application in Distributionally Robust Portfolio Selections," Mathematics, MDPI, vol. 11(1), pages 1-16, December.
    8. Oliver Kley & Claudia Kluppelberg & Gesine Reinert, 2015. "Conditional risk measures in a bipartite market structure," Papers 1510.00616, arXiv.org.
    9. 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.
    10. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Discussion Paper 2023-001, Tilburg University, Center for Economic Research.
    11. Hua, Lei & Joe, Harry, 2014. "Strength of tail dependence based on conditional tail expectation," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 143-159.
    12. Jaunė, Eglė & Šiaulys, Jonas, 2022. "Asymptotic risk decomposition for regularly varying distributions with tail dependence," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    13. Leipus, Remigijus & Paukštys, Saulius & Šiaulys, Jonas, 2021. "Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 170(C).
    14. 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.
    15. Asimit, Alexandru V. & Li, Jinzhu, 2016. "Extremes for coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 332-341.
    16. 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.
    17. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    18. Ulrich K. Müller & Yulong Wang, 2017. "Fixed- Asymptotic Inference About Tail Properties," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1334-1343, July.
    19. 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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