In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.

在本文中，我们通过考虑保险净收益的局部损益效用，即窄框架，研究均值方差准则下的最优保险设计问题。我们将文献中现有的结果扩展到决策者具有均值方差偏好且对保险净收益的预期效用有约束的情况，其中保费由均值方差保费原则确定。我们首先证明了本文研究的主要问题最优解的存在性和唯一性。我们发现，如果对保费原则的“平均部分”施加的安全负荷严格为正，则最佳赔偿函数涉及免赔额。我们的主要结果表明，窄框架确实减少了保险需求。显式最优赔偿函数是在两个特殊的局部增益-损失效用函数——二次效用函数和分段线性效用函数——下导出的。作为衍生结果，Bowley 解也是在二次局部增益-损失效用函数下针对决策者和保险公司之间的 Stackelberg 博弈得出的。给出了几个数值例子来进一步分析窄框架对最优赔偿函数以及双方利益的影响。