Convexity Inequalities for the Swiss premium

Summary

The Swiss premium calculation principle, introduced by Bühlmann, Gagliardi, Gerber, StrÄub (1977), assigns to a given risk X (a random variable, mostly nonnegative in practice) a premium p, solution of the equation E f (X-z p) = f ((1-z) p), where f is a continuous strictly increasing function and z e [0,1]. The premium p = π (X, f, z) may depend on X, f, z.

Let g also be continuous strictly increasing. Then we prove that π (X, f, z) ≦ π (X, g, z) for all X iff g is convex in f. This result can be extended to the case where g is not necessarily assumed to be continuous strictly increasing, but then it must be stated a bit differently since π (X, g, z) may be meaningless.

The main application concerns sub-additivity. For fixed f, z the Swiss premium calculation principle is said to be sub-additive if

$$\pi \left( {X + Y, f, z} \right) \leqq \pi \left( {X, f, z} \right) + \pi \left( {Y, f, z} \right),$$

(1)

for all independent X, Y. When (A) is considered for risks with arbitrary signs, sub-additivity necessarily reduces to additivity. When (A) is considered only for nonnegative risks (positive sub-additivity), then that relation is closely connected with the concavity of log f‚. The practical meaning of positive sub-additivity is clear: the contractholder may not take advantage to split independent risks.