IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v26y2024i4d10.1007_s11009-024-10106-w.html
   My bibliography  Save this article

Conditional Mean Risk Sharing of Independent Discrete Losses in Large Pools

Author

Listed:
  • Michel Denuit

    (Louvain Institute of Data Analysis and Modeling - LIDAM, UCLouvain)

  • Christian Y. Robert

    (CREST - Center for Research in Economics and Statistics, ENSAE)

Abstract

This paper considers a risk sharing scheme of independent discrete losses that combines risk retention at individual level, risk transfer for too expensive losses and risk pooling for the middle layer. This ensures that pooled losses can be considered as being uniformly bounded. We study the no-sabotage requirement and diversification effects when the conditional mean risk-sharing rule is applied to allocate pooled losses. The no-sabotage requirement is equivalent to Efron’s monotonicity property for conditional expectations, which is known to hold under log-concavity. Elementary proofs of this result for discrete losses are provided for finite population pools. The no-sabotage requirement and diversification effects are then examined within large pools. It is shown that Efron’s monotonicity property holds asymptotically and that risk can be eliminated under fairly general conditions which are fulfilled in applications.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10106-w
    DOI: 10.1007/s11009-024-10106-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-024-10106-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-024-10106-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, December.
    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. 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).
    4. 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.
    5. Franco Pellerey & Jorge Navarro, 2022. "Stochastic monotonicity of dependent variables given their sum," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 543-561, June.
    6. 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.
    7. Denuit, Michel & Hieber, Peter & Robert, Christian Y., 2022. "Mortality Credits Within Large Survivor Funds," ASTIN Bulletin, Cambridge University Press, vol. 52(3), pages 813-834, September.
    8. 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.
    9. Denuit, Michel & Hieber, Peter & Robert, Christian Y., 2022. "Mortality credits within large survivor funds," LIDAM Reprints ISBA 2022030, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Guillaume Carlier & Rose-Anne Dana, 2003. "Pareto efficient insurance contracts when the insurer's cost function is discontinuous," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(4), pages 871-893, June.
    11. Denuit, Michel & Dhaene, Jan & Ghossoub, Mario & Robert, Christian Y., 2023. "Comonotonicity and Pareto Optimality, with Application to Collaborative Insurance," LIDAM Discussion Papers ISBA 2023005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Dutang, Christophe & Goulet, Vincent & Pigeon, Mathieu, 2008. "actuar: An R Package for Actuarial Science," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 25(i07).
    13. 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).
    14. repec:dau:papers:123456789/5394 is not listed on IDEAS
    15. Mario Ghossoub & Giulio Principi & Ruodu Wang, 2024. "Allocation Mechanisms in Decentralized Exchange Markets with Frictions," Papers 2404.10900, arXiv.org.
    16. 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).
    17. 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).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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 & 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. 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.
    4. Zhanyi Jiao & Steven Kou & Yang Liu & Ruodu Wang, 2022. "An axiomatic theory for anonymized risk sharing," Papers 2208.07533, arXiv.org, revised May 2023.
    5. Denuit, Michel & Robert, Christian Y., 2023. "Endowment contingency funds for mutual aid and public financing," LIDAM Discussion Papers ISBA 2023009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Denuit, Michel & Dhaene, Jan & Ghossoub, Mario & Robert, Christian Y., 2023. "Comonotonicity and Pareto Optimality, with Application to Collaborative Insurance," LIDAM Discussion Papers ISBA 2023005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    8. 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).
    9. 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.
    10. 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.
    11. Fallou Niakh, 2023. "A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules," Papers 2303.05421, arXiv.org, revised Jul 2023.
    12. Denuit, Michel & Robert, Christian Y., 2021. "Polynomial series expansions and moment approximations for conditional mean risk sharing of insurance losses," LIDAM Discussion Papers ISBA 2021016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. 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).
    14. Denuit, Michel, 2019. "Investing in your own and peers' risks: The simple analytics of p2p insurance," LIDAM Discussion Papers ISBA 2019028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Denuit, M. & Robert, C.Y., 2020. "Ultimate behavior of conditional mean risk sharing for independent compound Panjer-Katz sums with gamma and Pareto severities," LIDAM Discussion Papers ISBA 2020014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Denuit, Michel & Robert, Christian Y., 2020. "Risk reduction by conditional mean risk sharing with application to collaborative insurance," LIDAM Discussion Papers ISBA 2020024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    17. 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.
    18. Denuit, Michel & Robert, Christian Y., 2020. "Conditional mean risk sharing for dependent risks using graphical models," LIDAM Discussion Papers ISBA 2020029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    19. Denuit, Michel & Robert, Christian Y., 2020. "Stop-loss protection for a large P2P insurance pool," LIDAM Discussion Papers ISBA 2020028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10106-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.