Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses

This paper exploits the representation of the conditional mean risk sharing allocations in terms of size-biased transforms to derive effective approximations within insurance pools of limited size. Precisely, the probability density functions involved in this representation are expanded with respect to the Gamma density and its associated Laguerre orthonormal polynomials, or with respect to the Normal density and its associated Hermite polynomials when the size of the pool gets larger. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. The numerical method then consists in truncating the series expansions to a limited number of terms. This results in an approximation in terms of the first moments of the individual loss distributions. Compound Panjer-Katz sums are considered as an application. The proposed method is compared with the well-established Panjer recursive algorithm. It appears to provide the analyst with reliable approximations that can be used to tune system parameters, before performing exact calculations.