In this paper, we provide an axiomatic approach to general premium priciples giving rise to a decomposition into risk, as a generalization of the expected value, and deviation, as a generalization of the variance. We show that, for every premium priciple, 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 priciple. In particular, we show that the convex conjugate of the aforementioned maximal risk measure coincides with the convex conjugate of the premium principle on the set of all finitely additive probability measures. In a last step, we consider insurance prices in the presence of a not neccesarily frictionless market, where insurance claims are traded. In this setup, we discuss premium principles that are consistent with hedging using securization products that are traded in the market.