Optimal Insurance Indemnity for the Insured with Low Risk Tolerance

This study focuses on the optimal insurance for the insured with low risk tolerance from the perspective of rank-dependent utility. Dissimilar to the widely employed assumption that the initial wealth of the insured can sufficiently cover any incurred premium and maximal possible loss, we assume that the policyholder may not possess sufficient initial wealth, thus exhibiting a low risk tolerance. Hence, we establish a corresponding insurance model in which the insured’s initial wealth is designed in such a manner that it only needs to cover the premium. Further, the model considers an introduced risk level that is less than the maximal possible loss. Owing to the lack of the condition of the sufficiency of initial wealth, the original optimization issue could not be converted into a canonical quantile-optimization issue, as usual, and the quantile formulation could not be obtained directly. To overcome this challenge, we apply the calculus of variations method employing a variable upper limit that is related to the maximal risk level. Further, the explicit optimal solution bearing Yaari’s dual criterion is given, demonstrating that the optimal insurance policy with sufficient initial wealth represents a special case. Finally, numerical examples are presented to compare the indemnity with different risk levels.