Risk sharing under the dominant peer-to-peer property and casualty insurance business models

This paper purposes to formalize the three business models dominating peer-to-peer (P2P) property and casualty insurance: the self-governing model, the broker model and the carrier model. The former one develops outside the insurance market whereas the latter ones may originate from the insurance industry, by partnering with an existing company or by issuing a new generation of participating insurance policies where part of the risk is shared within a community and higher losses, exceeding the community's risk-bearing capacity are covered by an insurance or reinsurance company. The present paper proposes an actuarial modeling based on conditional mean risk sharing, to support the development of this new P2P insurance offer under each of the three business models. In addition, several specific questions are also addressed in the self-governing model. Considering an economic agent who has to select the optimal pool for a risk to be shared with other participants, it is shown that uniform comparison of the Lorenz or concentration curves associated to the respective total losses of the pools under consideration allows the agent to decide which pool is preferable. The monotonicity of the respective contributions of the participants is established with respect to the convex order, showing that increasing the number of participants is always beneficial under conditional mean risk sharing.