On convex principles of premium calculation

The classical definition of a principle of premium calculation is generalized: risks with identical distributions do not necessarily lead to the same premium. In the first part (Sections 1–3) the theoretical properties of convexity are discussed; in particular, the gradient of a principle is introduced. It is noted that the more common principles are all convex. In the second part these notions are applied to solve two problems under rather general assumptions: (1) Optimal purchase of reinsurance: If the first insurer knows how the reinsurer determines his premium, what form and degree of reinsurance should he choose? (2) Optimal cooperation: How should n companies split up a given risk to minimize the total premium? The case where the optimal decompostion consists of constant quotas is described in detail. In general, there is a close connection with Pareto optimality on the one hand, and no trade equilibria on the other.