Optimal Premium Principles and Capital Allocation Principles - The General Case

Some recent papers, as Zaks et al.(2006), Frostig et al.(2007) and Dhaene et al.(2009), are dealing with capital allocation and premium pricing as optimization problems. In these papers the authors are considering some constraints on the convex function and they find an explicit solution in each case. In this paper we prove the existence of a unique solution where each class has a different convex function. The constraint might be the Value at Risk. Moreover, we give an explicit solution in case the convex functions have a second derivative. we are dealing with the primal and the dual problems as defined in Zaks et al.(2006). The last part of this paper consider these optimization problems from a utility point of view, here the calculations are made in terms of utility instead of money