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

Abstract

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 $0＜\theta ＜\frac{{m}_{0}}{1-{m}_{0}}$. Conversely, certain contracts, such as the full coverage contract, are never optimal for any RDU-insured individual if $\theta >\frac{{m}_{0}}{1-{m}_{0}}$. Additionally, we derive all the Pareto optimal contracts when either the compensation or the retention violates the monotonicity constraint.

Keywords

Pareto optimal/efficient insurance / Rank-dependent utility theory / Quantile optimization / Probability weighting/distortion function / Double-obstacle ordinary differential equation / Calculus of variations

Acknowledgements

The earlier version of this paper is available at [43] since May 2018. The author is grateful to the editor and anonymous referees for their constructive comments and suggestions which have helped to significantly improve the previous versions of this paper. The author also thanks Dr. Jing Peng for his help on calculating the numerical example in Section 4.3.

The author is grateful for comments from many conference/worksop/seminar participants, especially those at The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Taipei, 2018 Conference in Memory of Professor Xunjing Li in Changchun, SCFM 2018 in Qingdao, 2018 International Conference on Mathematical Finance & Symposium on the Role of Mathematical Finance on FinTech Business in Daejeon, 2018 Stochastic Analysis, Stochastic Control and New Developments in Weihai, The 8th Annual Conference of Financial Engineering & Financial Risk Management Branch of OR Society of China in Xi’an, 2018 Advanced Methods in Mathematical Finance in Angers, The Sixth Asian Quantitative Finance Conference in Guangzhou, The First Conference on Actuarial Science and Applications in Shanghai.

This author acknowledges financial support from the NSFC (Grant No. 11471276, 11971409), The Hong Kong RGC (GRF Grant No. 15202817, 15202421, 15204622 and 15203423), the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, the Research Centre for Quantitative Finance (1-CE03), and internal grants from The Hong Kong Polytechnic University.

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