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Extreme value behavior of aggregate dependent risks

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  • Chen, Die
  • Mao, Tiantian
  • Pan, Xiaoqing
  • Hu, Taizhong

Abstract

Consider a portfolio of n identically distributed risks with dependence structure modeled by an Archimedean survival copula. Wüthrich (2003) and Alink et al. (2004) proved that the probability of a large aggregate loss scales like the probability of a large individual loss, times a proportionality factor. This factor depends on the dependence strength and the tail behavior of the individual risk, denoted by qnF, qnW and qnG according to whether the tail behavior belongs to the maximum domain of attraction of the Fréchet, the Weibull or the Gumbel distribution, respectively. We investigate properties of the factors qnW and qnG with respect to the dependence parameter and/or the tail behavior parameter, and revisit the asymptotic behavior of conditional tail expectations of aggregate risks for the Weibull and the Gumbel cases by using a different method. The main results strengthen and complement some results in Alink et al. (2004, 2005)Barbe et al. (2006), and Embrechts et al. (2009).

Suggested Citation

  • Chen, Die & Mao, Tiantian & Pan, Xiaoqing & Hu, Taizhong, 2012. "Extreme value behavior of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 99-108.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:99-108
    DOI: 10.1016/j.insmatheco.2011.10.008
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    References listed on IDEAS

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    1. Barbe, Philippe & Fougères, Anne-Laure & Genest, Christian, 2006. "On the Tail Behavior of Sums of Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 361-373, 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. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    4. 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.
    5. 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.
    6. 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.
    7. Shaked, Moshe & Shanthikumar, J. George, 1997. "Supermodular Stochastic Orders and Positive Dependence of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 86-101, April.
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    Citations

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    Cited by:

    1. Jaume Belles‐Sampera & Montserrat Guillén & Miguel Santolino, 2014. "Beyond Value‐at‐Risk: GlueVaR Distortion Risk Measures," Risk Analysis, John Wiley & Sons, vol. 34(1), pages 121-134, January.
    2. Chuancun Yin & Dan Zhu, 2015. "New class of distortion risk measures and their tail asymptotics with emphasis on VaR," Papers 1503.08586, arXiv.org, revised Mar 2016.
    3. Pan, Xiaoqing & Leng, Xuan & Hu, Taizhong, 2013. "The second-order version of Karamata’s theorem with applications," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1397-1403.
    4. Ibragimov, Rustam & Prokhorov, Artem, 2016. "Heavy tails and copulas: Limits of diversification revisited," Economics Letters, Elsevier, vol. 149(C), pages 102-107.
    5. Richards, Jordan & Tawn, Jonathan A., 2022. "On the tail behaviour of aggregated random variables," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    6. Bingzhen Geng & Yang Liu & Yimiao Zhao, 2024. "Value-at-Risk- and Expectile-based Systemic Risk Measures and Second-order Asymptotics: With Applications to Diversification," Papers 2404.18029, arXiv.org.
    7. Jonathan El Methni & Laurent Gardes & Stéphane Girard, 2014. "Non-parametric Estimation of Extreme Risk Measures from Conditional Heavy-tailed Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 988-1012, December.
    8. Jaume Belles-Sampera & Montserrat Guillén & Miguel Santolino, 2013. "“Beyond Value-at-Risk: GlueVaR Distortion Risk Measures”," IREA Working Papers 201302, University of Barcelona, Research Institute of Applied Economics, revised Feb 2013.

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    More about this item

    Keywords

    Archimedean copula; Conditional tail expectation; Extreme value distribution; Maximum domain of attraction; Regular variation; The supermodular order; Value-at-Risk;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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