Optimal insurance contract design with "No-claim Bonus and Coverage Upper Bound" under moral hazard

In this paper, an optimal insurance problem from the view of a risk-averse individual under moral hazard is considered. Based on the Principal-Agent theory, we introduce a combined incentive tool of "No-claim Bonus and Coverage Upper Bound" to encourage the insured to make a higher risk-reducing effort and obtain higher expected utility than in a basic contract. We confirm if marginal expected utility brought by the increase of riskreducing effort decreases at the critical point of transition between two contracts, the combined incentive tool can restrain the moral hazard of the insured. Moreover, Pareto efficiency improvement of the tool is visually displayed by two case examples.