Robustness of Conditional Moments: An Application to Premium Calculation of Reinsurance Treaties

In this study, the tail probability of a class of distributions commonly used in assessing the severity of insurance losses was examined. Without specifying any particular distribution, the use of an algebraic functional form Cx−α to approximate the tail behavior of the distributions in the class was demonstrated. Norwegian fire insurance data were examined, and the algebraic functional form was applied to derive the expected loss of a reinsurance treaty that covers all losses exceeding a retention limit. It was shown that (1) the expected loss is insensitive to the parameter α for a high retention limit (e.g., a catastrophe treaty), and (2) with a low retention limit (e.g., a largest claim treaty), a reliable estimate of the parameter α and a sound judgment on the maximum potential loss of the treaty could provide useful and defensible summary statistics for pricing the treaty. Thus, when dealing with the losses of certain reinsurance treaties, it was concluded that knowledge of a specific probability distribution is not critical, and the summary statistics derived from the model are robust with respect to a large class of loss distributions.