Optimal insurance under the insurer’s risk constraint

In this paper, we impose the insurer’s risk constraint on Arrow’s optimal insurance model. The insured aims to maximize his/her expected utility of terminal wealth, under the constraint that the insurer wishes to control the expected loss of his/her terminal wealth below some prespecified level. We solve the problem, and it is shown that when the insurer’s risk constraint is binding, the solution to the problem is not linear, but piecewise linear deductible. Moreover, it can be shown that the insured’s optimal expected utility will increase if the insurer increases his/her risk tolerance.