Conditional mean risk sharing with settlement delays in peer-to-peer disability insurance

This paper develops a dynamic conditional mean risk-sharing mechanism for peer-to-peer disability insurance when claim amounts are not observed at occurrence time. The model is motivated by cooperative arrangements such as Dutch bread funds, where participants mutually protect each other against the economic consequences of disability. We consider a two-state model in which participants move between active and disabled states, and where the ultimate cost of a disability claim is determined by the duration of the disability spell. Since this duration is only known at recovery, the conditional mean risk-sharing allocation cannot be implemented immediately when the claim is reported. The proposed mechanism requires active participants to contribute an initial reserve when a disability spell starts. This reserve is then dynamically updated as information about the claim development becomes available, and it converges at settlement to the terminal conditional mean risk-sharing allocation. We show that the dynamic reserve is a conditional expectation of the corresponding thinned individual loss under the information used by the pool. This representation yields martingale, conditional full-allocation and actuarial fairness properties. Most importantly, the cumulative contribution paid by each participant is smaller in convex order than the corresponding stand-alone incurred-and-reserved disability cost. Hence, every risk-averse participant is willing to join the pool. Numerical illustrations show how reserves and final adjustments depend on disability duration and heterogeneous recovery rates.