Tail-sensitive insurance pricing: An economic extension of the Esscher principle

With the increasing importance of managing extreme losses in insurance and financial markets, tail risk has become a central concern for premium setting. This study examines how tail risk and insurers’ risk aversion affect insurance contract pricing by incorporating tail information into the classical Esscher premium framework and introducing the tail Esscher premium principle (TEPP). Theoretical analysis shows that the TEPP preserves key properties, including monotonicity, translation invariance, and scaling behavior. At the same time, the tail-sensitivity parameter provides an economically interpretable measure of risk aversion toward extreme losses. Importantly, we emphasize that observed insurance premiums may deviate from underlying risk due to regulatory constraints and market frictions; in this context, the TEPP provides a transparent, risk-based benchmark for pricing extreme losses and assessing potential pricing distortions. Numerical illustrations based on gamma and standardized Student-t distributions demonstrate how tail conditioning and risk aversion jointly shape premium levels across different loss environments. Consistent with these predictions, an empirical analysis using five-year motor and homeowners insurance claims reveals pronounced tail behavior and strong sensitivity of premiums to tail thresholds, particularly in the presence of extreme events often associated with climate risks. Our results provide insurers and risk managers with a transparent, operational framework for tail-sensitive pricing in both market-based and regulatory-constrained environments.