Calculating Premiums for Extreme-Tail Risks with Liability Claims and Nuclear Verdicts

Purpose: When insurance claims, particularly liability claims and nuclear verdicts, are governed by fat-tailed distributions, considerable uncertainty exists about the value of the tail index. Material and Methods: Using the theory of risk aversion, this paper establishes a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) and investigates its properties. Findings: Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims. This structure provides a natural model for incorporating tail-index uncertainty into premiums. The theory is illustrated through possible ‘premiums’ for liability claims and nuclear verdicts, utilizing the consistent pattern of observed extremes. Implications to Theory, Practice and Policy: By integrating statistical methods for tail index estimation and addressing the inherent uncertainty, the power principle offers a robust framework for determining premiums in high-risk environments. The paper concludes with practical implications for insurers, highlighting the need for advanced risk management strategies and regulatory considerations in dealing with extreme liability claims.