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

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