Some Inequalities for Stop-Loss Premiums

Extract

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S⁻] < ∞.

Let then v(t) be a twice differentiate function with

and let z be a constant with o ≤ z ≤ 1.

We define the premium P as follows

or equivalently

Notation: v⁻(∞) = ∞.

The definitions (1) and (equivalently) (2) are meaningful because of the

Lemma: a) E[v(S − zQ)] exists for all Q∈(− ∞, + ∞).

b) The set{Q∣ − ∞ < Q < + ∞, E[v(S−zQ)]>v((1−z)Q)} is not empty.

Proof: a)

b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).

If v(− ∞) = c finite then E[v(S − zQ)] > c

(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.