Denuit (2019 Denuit, M. 2019. Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines. ASTIN Bulletin 49:591–617.[Crossref], [Web of Science ®] , [Google Scholar], 2020a Denuit, M. 2020a. Investing in your own and peers’ risks: The simple analytics of P2P insurance. European Actuarial Journal 10:335–59. doi:https://doi.org/10.1007/s13385-020-00238-x[Crossref], [Web of Science ®] , [Google Scholar]) demonstrated that conditional mean risk sharing introduced by Denuit and Dhaene (2012 Denuit, M., and J. Dhaene. 2012. Convex order and comonotonic conditional mean risk sharing. Insurance: Mathematics and Economics 51:265–70. doi:https://doi.org/10.1016/j.insmatheco.2012.04.005[Crossref], [Web of Science ®] , [Google Scholar]) is the appropriate theoretical tool to share losses in collaborative peer-to-peer insurance schemes. Denuit and Robert (2020a Denuit, M., and C. Y. Robert. 2020a. Efron’s asymptotic monotonicity property in the Gaussian stable domain of attraction. Submitted. https://dial.uclouvain.be/pr/boreal/object/boreal:232135[Crossref] , [Google Scholar], 2020b Denuit, M., and C. Y. Robert. 2021. From risk sharing to pure premium for a large number of heterogeneous losses. Insurance: Mathematics and Economics 96: 116–26. doi:https://doi.org/10.1016/j.insmatheco.2020.11.006[PubMed], [Web of Science ®] , [Google Scholar], 2021 Denuit, M., and C. Y. Robert. 2020b. Large-loss behavior of conditional mean risk sharing. ASTIN Bulletin 50:1093–122. doi:https://doi.org/10.1017/asb.2020.23[Crossref], [Web of Science ®] , [Google Scholar]) studied this risk sharing mechanism and established several attractive properties including linear approximations when total losses or the number of participants get large. It is also shown there that the conditional expectation defining the conditional mean risk sharing is asymptotically increasing in the total loss (under mild technical assumptions). This ensures that the risk exchange is Pareto-optimal and that all participants have an interest to keep total losses as small as possible. In this article, we design a flexible system where entry prices can be made attractive compared to the premium of a regular, commercial insurance contract and participants are awarded cash-backs in case of favorable experience while being protected by a stop-loss treaty in the opposite case. Members can also be grouped according to some meaningful criteria, resulting in a hierarchical decomposition of the community. The particular case where realized losses are allocated in proportion to the pure premiums is studied.

德纽特 ( 2019 德纽特，M. 2019 年。规模偏向变换和条件平均风险分担，适用于 P2P 保险和联合养老保险。ASTIN 公告49：591 – 617。[交叉引用]，[Web of Science ®]  ，[谷歌学术]，2020a 德纽特，M. 2020a。投资自己和同行的风险：P2P 保险的简单分析。欧洲精算杂志10: 335 – 59。doi:https://doi.org/10.1007/s13385-020-00238-x [Crossref], [Web of Science ®]  , [Google Scholar] ) 证明了由 Denuit 和 Dhaene (2012 Denuit、M . 和J. Dhaene。2012 年。凸阶和单调条件均值风险分担。保险：数学和经济学51：265 - 70。doi:https://doi.org/10.1016/j.insmatheco.2012.04.005 [Crossref], [Web of Science ®]  , [Google Scholar] ) 是在协作对等网络中分担损失的适当理论工具保险计划。德纽特和罗伯特（2020a Denuit, M.和CY Robert。2020a。Efron 在高斯稳定吸引域中的渐近单调性。已提交。https://dial.uclouvain.be/pr/boreal/object/boreal:232135 [交叉引用]  ，[谷歌学术]，2020b Denuit, M.和CY Robert。 2021 年。从风险分担到大量异质损失的纯保费。保险：数学和经济学96：116-26。doi：https://doi.org/10.1016/j.insmatheco.2020.11.006 [PubMed]，[Web of Science®]  ，[Google Scholar]，2021 Denuit, M.和CY Robert。 2020b。条件均值风险分担的大损失行为。ASTIN 公告50：1093 – 122。doi：https://doi.org/10.1017/asb.2020.23 [Crossref]、[Web of Science®]  、[Google Scholar]) 研究了这种风险分担机制，并建立了几个有吸引力的特性，包括当总损失或参与者数量变大时的线性近似。那里还表明，定义条件平均风险分担的条件期望在总损失中逐渐增加（在温和的技术假设下）。这确保了风险交换是帕累托最优的，并且所有参与者都有兴趣使总损失尽可能小。在本文中，我们设计了一个灵活的系统，与常规商业保险合同的保费相比，进入价格可以更具吸引力，并且参与者在获得良好经验的情况下获得现金返还，同时受到止损条约的保护。相反的情况。成员也可以根据一些有意义的标准进行分组，导致社区的分层分解。研究了按纯保费比例分配已实现损失的特殊情况。