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Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

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  • Cai, Jun
  • Wang, Ying
  • Mao, Tiantian

Abstract

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.

Suggested Citation

  • Cai, Jun & Wang, Ying & Mao, Tiantian, 2017. "Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 105-116.
    • Handle: RePEc:eee:insuma:v:75:y:2017:i:c:p:105-116
    DOI: 10.1016/j.insmatheco.2017.05.004
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    References listed on IDEAS

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    1. 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.
    2. Zhu, Li & Li, Haijun, 2012. "Tail distortion risk and its asymptotic analysis," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 115-121.
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    4. 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.
    5. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    6. Cossette, Hélène & Mailhot, Mélina & Marceau, Étienne, 2012. "TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 247-256.
    7. Belles-Sampera, Jaume & Guillen, Montserrat & Santolino, Miguel, 2016. "What attitudes to risk underlie distortion risk measure choices?," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 101-109.
    8. Landsman, Zinoviy & Valdez, Emiliano A., 2005. "Tail Conditional Expectations for Exponential Dispersion Models," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 189-209, May.
    9. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Multivariate tail conditional expectation for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 216-223.
    10. Dhaene, J. & Henrard, L. & Landsman, Z. & Vandendorpe, A. & Vanduffel, S., 2008. "Some results on the CTE-based capital allocation rule," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 855-863, April.
    11. 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.
    12. Belles-Sampera, Jaume & Guillén, Montserrat & Santolino, Miguel, 2014. "GlueVaR risk measures in capital allocation applications," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 132-137.
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    Cited by:

    1. 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.
    2. Shuo Gong & Yijun Hu & Linxiao Wei, 2022. "Risk measurement of joint risk of portfolios: a liquidity shortfall aspect," Papers 2212.04848, arXiv.org.
    3. Cai, Jun & Wang, Ying, 2021. "Optimal capital allocation principles considering capital shortfall and surplus risks in a hierarchical corporate structure," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 329-349.
    4. Baishuai Zuo & Chuancun Yin, 2022. "Doubly truncated moment risk measures for elliptical distributions," Papers 2203.01091, arXiv.org.
    5. Cornilly, D. & Rüschendorf, L. & Vanduffel, S., 2018. "Upper bounds for strictly concave distortion risk measures on moment spaces," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 141-151.
    6. Ling, Chengxiu, 2019. "Asymptotics of multivariate conditional risk measures for Gaussian risks," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 205-215.
    7. Wentao Hu & Cuixia Chen & Yufeng Shi & Ze Chen, 2022. "A Tail Measure With Variable Risk Tolerance: Application in Dynamic Portfolio Insurance Strategy," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 831-874, June.
    8. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    9. Sainan Zhang & Huifu Xu, 2022. "Insurance premium-based shortfall risk measure induced by cumulative prospect theory," Computational Management Science, Springer, vol. 19(4), pages 703-738, October.
    10. Shushi, Tomer & Yao, Jing, 2020. "Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 178-186.
    11. Tomer Shushi, 2018. "Towards a Topological Representation of Risks and Their Measures," Risks, MDPI, vol. 6(4), pages 1-11, November.

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