Optimal insurance for repetitive natural disasters under moral hazard

This study provides novel insights into the design of insurance contracts for repetitive losses from natural disasters, such as typhoons, hurricanes, and wildfires, which have become more frequent in recent years due to climate change. This study considers ex-ante and ex-post moral hazards. When a loss occurs twice, an individual's share of each loss, which is the difference between the loss and insurance coverage, is a non-decreasing function of the loss. The optimal insurance for each loss involves full insurance up to a certain limit and partial insurance above that limit. Optimal partial insurance is represented by non-linear coinsurance, which can be either convex or concave in each loss, depending on absolute risk aversion (ARA), prudence, and the shape of the loss distribution. Although the two losses are independent, the insurer adjusts its estimate of the insured's effort level by observing the second loss. Consequently, insurance coverage for the second loss is adjusted in addition to any additional reward or penalty for the first loss. Insurance coverage for the second loss also involves a fixed reimbursement (upper limit) or deductible depending on whether the partial insurance is concave or convex in loss, similar to that for the first loss. However, in the case of decreasing ARA (DARA), the optimal form of partial insurance for the first and second losses can be different, with a deductible versus a fixed reimbursement (or vice versa). We illustrate the results under a loss distribution belonging to a one-dimensional exponential family with a canonical form and specific utility functions.