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Denuit ( 2019 ,  2020a ) demonstrated that conditional mean risk sharing introduced by Denuit and Dhaene ( 2012 ) is the appropriate theoretical tool to share losses in collaborative peer-to-peer insurance schemes. Denuit and Robert ( 2020a ,  2020b ,  2021 ) 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.

Collaborative Insurance with Stop-Loss Protection and Team Partitioning