Moral-hazard-free insurance contract design under the rank-dependent utility theory

This paper investigates a Pareto optimal insurance contract design problem within a behavioral finance framework. In this context, the insured evaluates contracts using the rank-dependent utility (RDU, for short) theory, while the insurer applies the expected value premium principle. The analysis incorporates the incentive compatibility constraint, ensuring that the contracts, called moral-hazard-free, are free from the moral hazard issues identified in Bernard et al. [4]. Initially, the problem is formulated as a non-concave maximization problem involving Choquet expectation. It is then transformed into a quantile optimization problem and addressed using the calculus of variations method. The optimal contracts are characterized by a double-obstacle ordinary differential equation for a semi-linear second-order elliptic operator with nonlocal boundary conditions, which seems new in the financial economics literature. We present a straightforward numerical scheme and a numerical example to compute the optimal contracts. Let θ and m₀ represent the relative safety loading and the mass of the potential loss at 0, respectively. We discover that every moral-hazard-free contract is optimal for infinitely many RDU-insured individuals if 0<θ<m₀/1-m₀. Conversely, certain contracts, such as the full coverage contract, are never optimal for any RDU-insured individual if θ>m₀/1-m₀. Additionally, we derive all the Pareto optimal contracts when either the compensation or the retention violates the monotonicity constraint.