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From risk reduction to risk elimination by conditional mean risk sharing of independent losses

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  • Denuit, Michel
  • Robert, Christian Y.

Abstract

This paper studies diversification effects resulting from pooling insurance losses according to the risk allocation rule proposed by Denuit and Dhaene (2012). General comparison results are established for conditional expectations given sums of independent random variables. It is shown that these expectations decrease in the number of terms comprised in the conditioning sums. Additional inequalities are obtained under regression dependence in the sum. These general results are used to derive the monotonicity of the respective contributions of the participants with respect to the convex order, showing that increasing the number of participants is always beneficial under conditional mean risk sharing. New convergence results are obtained, showing that the variance of individual contributions tends to zero in many interesting cases. This provides actuaries with conditions ensuring that the risk can be fully eliminated within the pool, at the limit.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:108:y:2023:i:c:p:46-59
    DOI: 10.1016/j.insmatheco.2022.11.003
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    References listed on IDEAS

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    1. Denuit, Michel & Robert, Christian Y., 2021. "Efron’s asymptotic monotonicity property in the Gaussian stable domain of attraction," LIDAM Reprints ISBA 2021029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Denuit, Michel & Dhaene, Jan, 2012. "Convex order and comonotonic conditional mean risk sharing," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 265-270.
    3. Mohd Ma’Sum Billah, 2019. "Islamic Insurance Products," Springer Books, Springer, number 978-3-030-17681-5, June.
    4. Denuit, Michel & Robert, Christian Y., 2022. "Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses," LIDAM Reprints ISBA 2022025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Denuit, Michel, 2020. "Investing in your own and peers’ risks: the simple analytics of P2P insurance," LIDAM Reprints ISBA 2020026, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Michel Denuit & Christian Y. Robert, 2022. "Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 693-711, June.
    7. Adnan Malik & Karim Ullah, 2019. "Introduction to Takaful," Springer Books, Springer, number 978-981-32-9016-7, June.
    8. Denuit, Michel & Robert, Christian Y., 2021. "Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models," LIDAM Reprints ISBA 2021020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. 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.
    10. Abdikerimova, Samal & Feng, Runhuan, 2022. "Peer-to-peer multi-risk insurance and mutual aid," European Journal of Operational Research, Elsevier, vol. 299(2), pages 735-749.
    11. Denuit, Michel & Robert, Christian Y., 2020. "Large-Loss Behavior of Conditional Mean Risk Sharing," LIDAM Reprints ISBA 2020021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Denuit, Michel, 2019. "Size-Biased Transform And Conditional Mean Risk Sharing, With Application To P2p Insurance And Tontines," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 591-617, September.
    13. Denuit, Michel & Robert, Christian Y., 2022. "Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses," LIDAM Reprints ISBA 2022021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. 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).
    15. Cornelis Van Bochove, 2011. "Expectation of a uniform random variable with uniform observation errors after selection of the highest observations," Statistical Papers, Springer, vol. 52(4), pages 971-977, November.
    16. Denuit, Michel, 2010. "Positive dependence of signals," LIDAM Discussion Papers ISBA 2010025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    17. Michel Denuit & Christian Y. Robert, 2022. "Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1953-1985, September.
    18. 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.
    19. Xiaofeng Shao & Jingsi Zhang, 2014. "Martingale Difference Correlation and Its Use in High-Dimensional Variable Screening," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1302-1318, September.
    20. Denuit, Michel & Robert, Christian Y., 2021. "Risk sharing under the dominant peer-to-peer property and casualty insurance business models," LIDAM Discussion Papers ISBA 2021001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    21. Saumard, Adrien & Wellner, Jon A., 2018. "Efron’s monotonicity property for measures on R2," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 212-224.
    22. Denuit, Michel & Robert, Christian Y., 2021. "From risk sharing to pure premium for a large number of heterogeneous losses," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 116-126.
    23. Denuit, Michel, 2019. "Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines," LIDAM Discussion Papers ISBA 2019010, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    24. Denuit, Michel, 2019. "Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines," LIDAM Reprints ISBA 2019038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    25. 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).
    26. Denuit, Michel & Robert, Christian Y., 2020. "Large-Loss Behavior Of Conditional Mean Risk Sharing," ASTIN Bulletin, Cambridge University Press, vol. 50(3), pages 1093-1122, September.
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    Cited by:

    1. Denuit, Michel & Robert, Christian Y., 2023. "Conditional mean risk sharing of independent discrete losses in large pools," LIDAM Discussion Papers ISBA 2023010, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Michel Denuit & Christian Y. Robert, 2024. "Conditional Mean Risk Sharing of Independent Discrete Losses in Large Pools," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-22, December.
    3. Denuit, Michel & Ortega-Jimenez, Patricia & Robert, Christian Y., 2024. "No-sabotage under conditional mean risk sharing of dependent-by-mixture insurance losses," LIDAM Discussion Papers ISBA 2024019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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