When insurance claims are governed by fat-tailed distributions considerable uncertainty about the value of the tail-index is often inescapable. In this paper, using the theory of risk aversion, a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) is established and its properties investigated. Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims, for which the ratio of the actual claims is an unbiased as well as a consistent estimator. Whereas thin-tailed claim premiums are determined largely by the first two moments of the claims distribution, fat-tailed claim premiums are determined by the first two extremes. The context of risk-aversion leads to a natural model for incorporating tail-index uncertainty into premiums, which nevertheless leaves the basic ratio structure unaltered. To illustrate the theory, possible ‘premiums’ for US hurricane data are examined, which utilize the consistent pattern of observed extremes.