Before revision:  We show that, for every premium principle, there exists a maximal risk measure capturing all risky components covered by the insurance prices. In a second step, we consider dual representations of convex risk measures consistent with the premium principle.
After revision:  We show that, for every premium principle, there exists a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex risk measures consistent with the premium principle.