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Optimal reinsurance with positively dependent risks

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  • Cai, Jun
  • Wei, Wei

Abstract

In the individual risk model, one is often concerned about positively dependent risks. Several notions of positive dependence have been proposed to describe such dependent risks. In this paper, we assume that the risks in the individual risk model are positively dependent through the stochastic ordering (PDS). The PDS risks include independent, comonotonic, conditionally stochastically increasing (CI) risks, and other interesting dependent risks. By proving the convolution preservation of the convex order for PDS random vectors, we show that in individualized reinsurance treaties, to minimize certain risk measures of the retained loss of an insurer, the excess-of-loss treaty is the optimal reinsurance form for an insurer with PDS dependent risks among a general class of individualized reinsurance contracts. This extends the study in Denuit and Vermandele (1998) on individualized reinsurance treaties to dependent risks. We also derive the explicit expressions for the retentions in the optimal excess-of-loss treaty in a two-line insurance business model.

Suggested Citation

  • Cai, Jun & Wei, Wei, 2012. "Optimal reinsurance with positively dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 57-63.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:57-63
    DOI: 10.1016/j.insmatheco.2011.10.006
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    References listed on IDEAS

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    1. Colangelo, Antonio & Hu, Taizhong & Shaked, Moshe, 2008. "Conditional orderings and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 358-371, March.
    2. Van Heerwaarden, A. E. & Kaas, R. & Goovaerts, M. J., 1989. "Optimal reinsurance in relation to ordering of risks," Insurance: Mathematics and Economics, Elsevier, vol. 8(1), pages 11-17, March.
    3. Block, Henry W. & Savits, Thomas H. & Shaked, Moshe, 1985. "A concept of negative dependence using stochastic ordering," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 81-86, April.
    4. Colangelo, Antonio & Scarsini, Marco & Shaked, Moshe, 2005. "Some notions of multivariate positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 13-26, August.
    5. Ohlin, Jan, 1969. "On a class of measures of dispersion with application to optimal reinsurance," ASTIN Bulletin, Cambridge University Press, vol. 5(2), pages 249-266, May.
    6. Denuit, Michel & Vermandele, Catherine, 1998. "Optimal reinsurance and stop-loss order," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 229-233, July.
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    Cited by:

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    8. Nicole Bauerle & Alexander Glauner, 2017. "Optimal Risk Allocation in Reinsurance Networks," Papers 1711.10210, arXiv.org.
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    15. Guerra, M. & de Moura, A.B., 2021. "Reinsurance of multiple risks with generic dependence structures," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 547-571.
    16. Cheung, K.C. & Chong, W.F. & Yam, S.C.P., 2015. "The optimal insurance under disappointment theories," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 77-90.
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    18. Bernard, Carole & Liu, Fangda & Vanduffel, Steven, 2020. "Optimal insurance in the presence of multiple policyholders," Journal of Economic Behavior & Organization, Elsevier, vol. 180(C), pages 638-656.
    19. Ghossoub, Mario & Zhu, Michael B., 2024. "Stackelberg equilibria with multiple policyholders," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 189-201.
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    22. Zhu, Yunzhou & Chi, Yichun & Weng, Chengguo, 2014. "Multivariate reinsurance designs for minimizing an insurer’s capital requirement," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 144-155.

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    More about this item

    Keywords

    IM30; IM52; IB92; Dependent risk; Positive dependence; PDS; Convex order; Individualized reinsurance treaty; Optimal reinsurance;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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