Optimal proportional insurance under claim habit

In this paper, we study a two-period optimal insurance problem for a policyholder with mean-variance preferences who purchases proportional insurance at the beginning of each period. The insurance premium is calculated by a variance premium principle with a risk loading that depends on the policyholder’s claim history. We derive the time-consistent optimal insurance strategy in closed form and the optimal constant precommitment strategy in semiclosed form. For the optimal general precommitment strategy, we obtain the solution for the second period semi-explicitly and, then, the solution for the first period numerically via an efficient algorithm. Furthermore, we compare the three types of optimal strategies, highlighting their differences, and we examine the impact of the key model parameters on the optimal strategies and value functions.