On Belief Singularity and the Optimality of Deductible Indemnity Schedules

Abstract

In Ghossoub (2019), we studied a problem of optimal insurance design under belief heterogeneity. We examined the effect of the divergence in beliefs, and the singularity between the beliefs, on optimal indemnities; and we provided a closed-form characterization of optimal indemnity schedules. In this paper, we build upon the work of Ghossoub (2019) and introduce a belief singularity metric to examine the effect of the level of singularity between the beliefs on the shape of optimal indemnity schedules. Our main result shows that for any level x of belief singularity with respect to the insurer's belief Q, there exists a probability measure P exhibiting a level x of belief singularity with respect to Q, such that an optimal indemnity for a decision maker with belief P is a linear deductible with Q-probability 1, and full insurance with Q-probability 0. In this sense, we show that from the perspective of the insurer, linear deductible indemnity schedules are robust to belief singularity, which gives a possible explanation of the popularity of these contracts in practice.