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From risk sharing to pure premium for a large number of heterogeneous losses

Author

Listed:
  • Denuit, Michel

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Robert, Christian Y.

    (ENSAE, Paris, France)

Abstract

This paper considers linear fair risk sharing rules and the conditional mean risk sharing rule for independent but heterogeneous losses that are gathered in an insurance pool. It studies the asymptotic behavior of individual contributions to total losses when the number of participants to the pool tends to infinity. It is shown that (i) insurance at pure premium is obtained for an infinitely large pool and (ii) the difference between the actual contribution and the pure premium becomes ultimately Normally distributed. The linear fair risk sharing rule approximating the conditional mean risk sharing rule is then identified, providing practitioners with a useful simplification applicable within large pools. Also, the approximate number of participants required to keep the volatility of individual contributions within an acceptable range is obtained from the established asymptotic Normality.

Suggested Citation

  • Denuit, Michel & Robert, Christian Y., 2021. "From risk sharing to pure premium for a large number of heterogeneous losses," LIDAM Reprints ISBA 2021001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvar:2021001
    DOI: https://doi.org/10.1016/j.insmatheco.2020.11.006
    Note: In: Insurance: Mathematics and Economics - Vol. 96, p. 116-126 (2021)
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    Citations

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    Cited by:

    1. Thomas Bernhardt & Ge Qu, 2021. "Wealth heterogeneity in a closed pooled annuity fund," Papers 2110.13467, arXiv.org, revised Aug 2022.
    2. Michel Denuit & Jan Dhaene & Christian Y. Robert, 2022. "Risk‐sharing rules and their properties, with applications to peer‐to‐peer insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(3), pages 615-667, September.
    3. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    4. Denuit, Michel & Robert, Christian Y., 2021. "Efron’s asymptotic monotonicity property in the Gaussian stable domain of attraction," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    5. Fallou Niakh, 2023. "A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules," Papers 2303.05421, arXiv.org, revised Jul 2023.
    6. Michel Denuit & Christian Y. Robert, 2021. "Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 24(2), pages 181-205, June.
    7. Denuit, Michel & Robert, Christian Y., 2021. "Stop-loss protection for a large P2P insurance pool," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 210-233.
    8. Denuit, Michel & Robert, Christian Y., 2023. "From risk reduction to risk elimination by conditional mean risk sharing of independent losses," Insurance: Mathematics and Economics, Elsevier, vol. 108(C), pages 46-59.

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