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On a family of risk measures based on largest claims

Author

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  • Castaño-Martínez, A.
  • Pigueiras, G.
  • Sordo, M.A.

Abstract

Given a set of n≥2 independent and identically distributed claims, the expected average of the n−i largest claims, with 0≤i≤n−1, is shown to be a distortion risk measure with concave distortion function that can be represented in terms of mixtures of tail value-at-risks with beta mixing distributions. This result allows to interpret the tail value-at-risk in terms of the largest claims of a portfolio of independent claims. As an application, we provide sufficient conditions for stochastic comparisons of premiums in the context of large claims reinsurance.

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  • Castaño-Martínez, A. & Pigueiras, G. & Sordo, M.A., 2019. "On a family of risk measures based on largest claims," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 92-97.
    • Handle: RePEc:eee:insuma:v:86:y:2019:i:c:p:92-97
    DOI: 10.1016/j.insmatheco.2019.02.003
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    1. Hu, Taizhong & Wei, Ying, 2001. "Stochastic comparisons of spacings from restricted families of distributions," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 91-99, May.
    2. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    3. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    4. Balakrishnan, Narayanaswamy & Belzunce, Félix & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2012. "Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 45-54.
    5. Sordo, M.A. & Bello, A.J. & Suárez-Llorens, A., 2018. "Stochastic orders and co-risk measures under positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 105-113.
    6. Schweizer, Nikolaus & Szech, Nora, 2017. "Revenues and welfare in auctions with information release," Journal of Economic Theory, Elsevier, vol. 170(C), pages 86-111.
    7. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    8. Alfred Müller & Marco Scarsini, 2001. "Stochastic Comparison of Random Vectors with a Common Copula," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 723-740, November.
    9. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2010. "Decision principles derived from risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 294-302, December.
    10. Berliner, Baruch, 1972. "Correlations between excess of loss reinsurance covers and reinsurance of the n largest claims baruch berliner," ASTIN Bulletin, Cambridge University Press, vol. 6(3), pages 260-275, May.
    11. Tong Li, 2005. "Econometrics of first-price auctions with entry and binding reservation prices," Journal of Econometrics, Elsevier, vol. 126(1), pages 173-200, May.
    12. Kremer, Erhard, 1983. "Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties," Insurance: Mathematics and Economics, Elsevier, vol. 2(3), pages 209-213, July.
    13. Kremer, Erhard, 1998. "Largest Claims Reinsurance Premiums under Possible Claims Dependence," ASTIN Bulletin, Cambridge University Press, vol. 28(2), pages 257-267, November.
    14. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    15. Navarro, Jorge & Spizzichino, Fabio, 2010. "On the relationships between copulas of order statistics and marginal distributions," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 473-479, March.
    16. Sordo, Miguel A. & Castaño-Martínez, Antonia & Pigueiras, Gema, 2016. "A family of premium principles based on mixtures of TVaRs," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 397-405.
    17. Kremer, Erhard, 1982. "Rating of Largest Claims and Ecomor Reinsurance Treaties for Large Portfolios," ASTIN Bulletin, Cambridge University Press, vol. 13(1), pages 47-56, June.
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    Cited by:

    1. Barmalzan, Ghobad & Akrami, Abbas & Balakrishnan, Narayanaswamy, 2020. "Stochastic comparisons of the smallest and largest claim amounts with location-scale claim severities," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 341-352.

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    More about this item

    Keywords

    Risk measure; Premium principle; Order statistics; Stop-loss order; Excess-wealth order; Reinsurance;
    All these keywords.

    JEL classification:

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

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