[Paper Review] Optimal moral-hazard-free reinsurance under extended distortion premium principles

Zhuo Jin, Zuo Quan Xu+1 more|arXiv (Cornell University)|Apr 18, 2023

Insurance and Financial Risk ManagementEconomics, Econometrics and Finance3 citations

TL;DR

This paper develops an optimal moral-hazard-free reinsurance contract under an extended distortion premium principle, where the distortion function need not be concave. It establishes existence and characterizes the optimal reinsurance using a double-obstacle problem, deriving conditions that yield the optimal retention function in (semi)closed form for three examples, ensuring incentive compatibility and minimal ruin probability under a diffusion risk model.

ABSTRACT

We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study three examples and obtain the optimal contract in (semi)closed form.

Motivation & Objective

• To address moral hazard in reinsurance by enforcing incentive compatibility on indemnity functions.
• To model reinsurance premiums using an extended distortion principle that allows non-concave distortion functions.
• To minimize the probability of lifetime ruin for an insurer under a diffusion risk model.
• To derive necessary and sufficient conditions for optimal reinsurance under these constraints.
• To provide explicit (semi)closed-form solutions for specific examples, validating the general framework.

Proposed method

• Imposes an incentive compatibility (IC) constraint on indemnity functions to eliminate moral hazard, requiring 0 ≤ I(z) − I(z′) ≤ z − z′ for all z ≥ z′ ≥ 0.
• Uses an extended distortion premium principle with a non-concave distortion function g and a loading factor θ₀ ≥ 0, generalizing standard principles.
• Models the insurer’s risk process as a diffusion process and formulates the optimization problem as minimizing the probability of lifetime ruin.
• Characterizes the optimal retention function H*(z) = z − I*(z) via a double-obstacle problem, leading to an optimal implicit differential equation (OIDE).
• Applies Fubini’s theorem and variational analysis to derive necessary and sufficient optimality conditions involving a signed measure Φ.
• Solves the OIDE using a threshold-based structure: H*(z) = min{d, z}, with d determined by conditions on the survival function and distortion function.

Experimental results

Research questions

• RQ1 Under what conditions does an optimal moral-hazard-free reinsurance contract exist when the premium is calculated via an extended distortion principle?
• RQ2 How can the optimal reinsurance contract be characterized when the distortion function is non-concave and the IC constraint is enforced?
• RQ3 What is the structure of the optimal retention function H*(z) under the double-obstacle problem framework?
• RQ4 How do the parameters of the distortion function and the loading factor influence the optimal reinsurance design?
• RQ5 Can the optimal contract be derived in (semi)closed form for specific examples, and what are the resulting forms?

Key findings

• An optimal moral-hazard-free reinsurance contract always exists under the extended distortion premium principle and the IC constraint.
• The optimal retention function H*(z) is characterized by a double-obstacle problem, leading to an optimal implicit differential equation (OIDE).
• The optimal H*(z) takes a threshold form: H*(z) = min{d, z}, where d is determined by the condition that the derivative of the value function vanishes at the threshold.
• For the example with a power distortion function g(p) = p^{1/ρ}, the optimal contract is derived in semi-closed form, with d satisfying a specific equation involving the survival function and parameters.
• The optimal contract ensures that the insurer’s derivative of retained loss H*(z) is 1 when Φ(z) < 0, 0 when Φ(z) > 0, and in [0,1] when Φ(z) = 0, confirming the OIDE structure.
• The value function v(a*) is expressed as an integral involving the optimal retention function and the distortion function, providing a closed-form expression for the minimal ruin probability.