Optimal insurance design with background risk and higher-order risk attitudes

What is it about?

We study an optimal insurance problem in the presence of background risk from the perspective of an insured with higher-order risk attitudes. We introduce several useful dependence notions to model positive dependence structures between the insurable risk and background risk. Under these dependence structures, we compare insurance contracts of different forms in higher-order risk attitudes and establish the optimality of stop-loss insurance form. We also explicitly derive the optimal retention level. Finally, we carry out a comparative analysis and investigate how the change in the insured's initial wealth or background risk affects the optimal retention level.

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Why is it important?

The optimal insurance design with background risk is a very challenging problem in insurance and actuarial science, especially when the contract is asked to satisfy the principle of indemnity and have the non-decreasing retained loss function. More specifically, the dependence between the insurance risk and background risk plays a critical role in the optimal insurance design. Under some assumption of the positive dependence, we show that the stop-loss insurance is an optimal choice of an insured with higher-order risk attitudes.

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