Optimum insurance contracts with background risk and higher-order risk attitudes

In this paper, 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.