Adaptive Premiums for Evolutionary Claims in Non-life Insurance

Rapid growth in heavy-tailed claim severity in commercial liability insurance requires insurer response by way of flexible mechanisms to update premiums. To this end in this paper a new premium principle is established for heavy-tailed claims, and its properties investigated. Risk-neutral premiums for heavy-tailed claims are consistently and unbiasedly estimated by the ratio of the first two extremes of the claims distribution. That is, the heavy-tailed risk-neutral premium has a Pareto distribution with the same tail-index as the claims distribution. Insurers must predicate premiums on larger tail-index values, if solvency is to be maintained. Additionally, the structure of heavy-tailed premiums is shown to lead to a natural model for tail-index imprecision (demonstrably inescapable in the sample sizes with which we deal). Premiums which compensate for tail-index uncertainty preserve the ratio structure of risk-neutral premiums, but make a 'prudent' adjustment which reflects the insurer's risk-profile. An example using Swiss Re's (1999) major disaster data is used to illustrate application of the methodology to the largest claims in any insurance class.