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Asymptotic results on tail moment for light-tailed risks

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  • Wang, Bingjie
  • Li, Jinzhu

Abstract

In this paper, we focus on the asymptotic behavior of a recently popular risk measure called the tail moment (TM), which has been extensively applied in the field of risk theory. We conduct the study under the framework in which the individual risks of a financial or insurance system follow convolution equivalent or Gamma-like distributions. Precise asymptotic results are obtained for the TM when the individual risks are mutually independent or have a dependence structure of the Farlie-Gumbel-Morgenstern type. Moreover, based on some specific scenarios, we give an asymptotic analysis on the relative errors between our asymptotic results and the corresponding exact values. Since the model settings in this paper are not covered by traditional ones, our work fills in some gaps of the asymptotic study of the TM for light-tailed risks.

Suggested Citation

  • Wang, Bingjie & Li, Jinzhu, 2024. "Asymptotic results on tail moment for light-tailed risks," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 43-55.
    • Handle: RePEc:eee:insuma:v:114:y:2024:i:c:p:43-55
    DOI: 10.1016/j.insmatheco.2023.11.001
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    References listed on IDEAS

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    1. Jiang, Jun & Tang, Qihe, 2008. "Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 431-436, December.
    2. Hashorva, Enkelejd & Li, Jinzhu, 2013. "ECOMOR and LCR reinsurance with gamma-like claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 206-215.
    3. 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.
    4. Li, Jinzhu, 2022. "Asymptotic results on marginal expected shortfalls for dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 146-168.
    5. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    6. 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.
    7. Yang, Yang & Jiang, Tao & Wang, Kaiyong & Yuen, Kam C., 2020. "Interplay of financial and insurance risks in dependent discrete-time risk models," Statistics & Probability Letters, Elsevier, vol. 162(C).
    8. Joseph Kim, 2010. "Conditional Tail Moments of the Exponential Family and Its Related Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(2), pages 198-216.
    9. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    10. 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.
    11. Claudia Klüppelberg & Miriam Isabel Seifert, 2019. "Financial risk measures for a network of individual agents holding portfolios of light-tailed objects," Finance and Stochastics, Springer, vol. 23(4), pages 795-826, October.
    12. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    13. 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.
    14. Bawa, Vijay S. & Lindenberg, Eric B., 1977. "Capital market equilibrium in a mean-lower partial moment framework," Journal of Financial Economics, Elsevier, vol. 5(2), pages 189-200, November.
    15. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    16. Oliver Kley & Claudia Klüppelberg & Gesine Reinert, 2018. "Conditional risk measures in a bipartite market structure," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(4), pages 328-355, April.
    17. Chen, Yiqing & Liu, Jiajun, 2022. "An asymptotic study of systemic expected shortfall and marginal expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 105(C), pages 238-251.
    18. Kim, Joseph H.T. & Kim, So-Yeun, 2019. "Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 145-157.
    19. Marri, Fouad & Moutanabbir, Khouzeima, 2022. "Risk aggregation and capital allocation using a new generalized Archimedean copula," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 75-90.
    20. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    21. Ignatieva, Katja & Landsman, Zinoviy, 2021. "A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 437-465.
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    More about this item

    Keywords

    Asymptotics; Tail moment; Risk measure; Convolution equivalence; Gamma-like distribution;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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