Collaborative Insurance with Stop-Loss Protection and Team Partitioning

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-xCrossref, 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.2012.04.005Crossref, 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:232135Crossref , 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.2020.11.006PubMed, 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.2020.23Crossref, 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.