In this paper, we provide an axiomatic approach to general premium
principles 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
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. 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.
