Pricing risk when distributions are fat tailed

Abstract

In this paper, insurance claims X on [0, ∞) with tail distributions which are O(x−δ) for some δ > 1 are considered. Markets are assumed arbitrageable, the insurer setting a premium P > E[X]. Setting a premium as a fixed quantile of the loss distribution presents difficulties; for Pareto distributions with F(x) = 1 – (x + l)–δ ‘ultimately' (as δ ↓ 1) E[X] is larger than any quantile. When δ is near 1, premiums determined by weighting outcomes and a rule analogous to the expected utility principle are highly sensitive to change in δ, which is generally unknown or known only approximately. Under these circumstances, to protect insurers' interests, strategies are needed which provide some ‘premium stability' across a range of δ-values. We introduce a class of pricing functions which are functionally dependent on the governing loss distribution, and which are themselves distribution functions. We demonstrate that they provide a coherent framework for pricing insurance premiums when the loss distribution is fat tailed, and enable some degree of premium stability to be established.