Optimal risk sharing arrangements have been substantially studied in the literature, from the aspects of generalizing objective functions, incorporating more business constraints, and investigating different optimality criteria. This paper proposes an insurance model with multiple risk environments. We study the case where the two agents are endowed with the Value-at-Risk or the Tail Value-at-Risk, or when both agents are risk-neutral but have heterogeneous beliefs regarding the underlying probability distribution. We show that layer-type indemnities, within each risk environment, are Pareto optimal, which may be environment-specific. From Pareto optimality, we get that the premium can be chosen in a given interval, and we propose to allocate the gains from risk sharing equally between the buyer and seller.

最佳风险分担安排已在文献中进行了大量研究，从概括目标函数、纳入更多业务约束以及研究不同的优化标准等方面。本文提出了一种具有多种风险环境的保险模型。我们研究了两个代理被赋予风险价值或尾部风险价值的情况，或者当两个代理都是风险中性但对潜在概率分布具有异质信念时。我们表明，在每个风险环境中，层类型的赔偿都是帕累托最优的，这可能是特定于环境的。从帕累托最优，我们得到可以在给定的间隔内选择溢价，我们建议在买卖双方之间平均分配风险分担的收益。