Moral-hazard-free insurance: mean-variance premium principle and rank-dependent utility theory

TL;DR

This paper develops a framework for designing moral-hazard-free insurance contracts under rank-dependent utility and mean-variance premium principles, solving a complex optimization problem with novel mathematical techniques and numerical methods.

Contribution

It introduces a new approach to moral-hazard-free insurance design using rank-dependent utility and mean-variance principles, with a novel solution method involving calculus of variations.

Findings

Derived the optimal contract as a second-order integro-differential equation.

Proposed an effective numerical method for computing optimal contracts.

Provided an analytical example illustrating the theoretical results.

Abstract

This paper investigates a Pareto optimal insurance problem, where the insured maximizes her rank-dependent utility preference and the insurer is risk neutral and employs the mean-variance premium principle. To eliminate potential moral hazard issues, we only consider the so-called moral-hazard-free insurance contracts that obey the incentive compatibility constraint. The insurance problem is first formulated as a non-concave maximization problem involving Choquet expectation, then turned into a concave quantile optimization problem and finally solved by the calculus of variations method. The optimal contract is expressed by a second-order ordinary integro-differential equation with nonlocal operator. An effective numerical method is proposed to compute the optimal contract assuming the probability weighting function has a density. Also, we provide an example which is analytically solved.