Optimal Insurance Research With Background Risk And Belief Heterogeneity

Background risk refers to the risks faced by individuals that cannot be insured but can be quantified.Obviously,there is a relationship between background risk and insurable risk,which may further influence the optimal form and level of insurance contracts.In an insurance contract,the subjective probability of potential loss may be different between the insured and the insurer,so belief heterogeneity also affects the optimal strategy.Based on background risk and belief heterogeneity,this paper studies the optimal insurance strategy of the insured under the framework of expected utility,taking the expected utility function of maximizing the final wealth of the insured as the optimization criterion.This paper gives the necessary and sufficient conditions for the existence of the optimal solution under the belief heterogeneity of any form and the general dependency structure of insurable risk and background risk.By using this condition,the optimality of several relatively pure dependent structures is proved.That is,no insurance coverage is needed in the presence of extreme negative dependence because the insurable risk is fully hedged,and full coverage above a deductible(i.e.,stop-loss insurance)is chosen in the presence of positive dependence because the positively dependent background risk does not provide any hedge to the right-tail risk of the insurable loss.This makes perfect sense because a moderately negative dependent background risk provides partial,but not full,hedge to the insurable risk,leaving the unhedged part of the insurable risk to be covered by insurance.