Optimal Incentive-Compatible Insurance With Background Risk

Abstract

In this paper, the optimal insurance design is studied from the perspective of an insured, who faces an insurable risk and a background risk. For the reduction of ex post moral hazard, alternative insurance contracts are asked to satisfy the principle of indemnity and the incentive-compatible condition. As in the literature, it is assumed that the insurer calculates the insurance premium solely on the basis of the expected indemnity. When the insured has a general mean-variance preference, an explicit form of optimal insurance is derived explicitly. It is found that the stochastic dependence between the background risk and the insurable risk plays a critical role in the insured’s risk transfer decision. In addition, the optimal insurance policy can often change significantly once the incentive-compatible constraint is removed.

Mean–variance Insurance Design With Counterparty Risk and Incentive Compatibility

“…We follow Chi and Tan (2021) and study the optimal insurance problem within a mean-variance framework. We focus on the following general problem with counterparty risk.…”

“…We remark that the optimal indemnity function within the mean-variance framework for additive background risk model has been derived by Chi and Tan (2021) through a constructive approach with stochastic ordering techniques. The marginal indemnity function formulation, which is obtained through the calculus of variations, not only helps us to solve the multiplicative background risk (or counterparty risk) model but also provides an alternative way to solve the additive background risk model.…”

“…The aforementioned studies are all within the expected utility framework. Recently, Chi and Tan (2021) extend the study into the mean-variance framework. Their methodology allows the derivation of the optimal indemnity function for a very general dependence structure between the insurable risk and background risk.…”

“…This is also named as the no-sabotage condition by Carlier and Dana (2003). Note that such condition is also considered by Chi and Wei (2020) and Chi and Tan (2021) where the additive background risk is the focus.…”

“…• Second, without assuming any specific dependence structure between the counterparty risk and the insurable loss, we derive the explicit structure of the optimal indemnity function based on the element-wise minimizer to our problem under some mild assumptions on the counterparty risk. We point out that the problem in Chi and Tan (2021) could also be solved using the marginal indemnity function approach, which provides an alternative approach to study the additive background risk model within the mean-variance framework.…”