We consider the optimal proportional reinsurance from an insurer’s point of view to maximize the expected utility and minimize the value at risk. Under the general premium principle, we prove the existence and uniqueness of the optimal strategies and Pareto optimal solution, and give the relationship between the optimal strategies. Furthermore, we study the optimization problem with the variance premium principle. When the total claim sizes are normally distributed, explicit expressions for the optimal strategies and Pareto optimal solution are obtained. Finally, some numerical examples are presented to show the impact of the major model parameters on the optimal results.

Recently, Wang's premium principle (Wang, 1995, 1996) has been discussed by many authors. Considerable attention has been given to the conditions under which Wang's premium principle can be reduced to the standard deviation premium principle. In this paper, we have got two results on this problem. One is that the natural set is a location-scale family if Wang's premium principle can be reduced to the SD premium principle for all surjective distortions. The other is that the natural set is a location-scale family for all power distortions.