Optimal Insurance for a Minimal Expected Retention : The Case of an Ambiguity-Seeking Insurer

In the classical Expected-Utility framework, a problem of optimal insurance design with a premium constraint is equivalent to a problem of optimal insurance design with a minimum expected retention constraint. When the insurer has ambiguous beliefs represented by a non-additive probability measure, as in Schmeidler (1989), this equivalence no longer holds. A problem of optimal insurance design with a premium constraint determined based on the insurer's ambiguous beliefs is not equivalent to a problem of optimal insurance design with a minimum expected retention determined based on the insurer's ambiguous beliefs. Recently, Amarante, Ghossoub, and Phelps (2014) examined the problem of optimal insurance design with a premium constraint when the insurer has ambiguous beliefs. In particular, they showed that when the insurer is ambiguity-seeking, with a concave distortion of the insured's probability measure, then the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer's distortion function. In this paper, we examine the problem of optimal insurance design with a minimum expected retention constraint, in the case where the insurer is ambiguity-seeking. We obtain the aforementioned result of Amarante, Ghossoub, and Phelps (2014) and the classical result of Arrow (1971) as special cases