Optimal Insurance with Heterogeneous Beliefs and Disagreement About Zero-Probability Events

Abstract

Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub (2013) re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the monotone likelihood ratio condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub (2013) only gave a characterization of these optimal indemnity schedules in the special case of a monotone likelihood ratio. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result as well as one of the results of Ghossoub (2013) as corollaries. Finally, we formalize Marshall’s (1991) argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any” shape.