Yet another approximation for the total claims amount using the Weibull distribution

The accurate evaluation of the distribution of a compound sum is a crucial task in actuarial science and operational risk management. For non-life insurance companies, the total claims amount over a specific period can be represented as SN = X1 + ... + XN, where N denotes the number of occurring claims and Xi the i-th claim size (i = 1,...,N). The Xi’s are assumed to be iid positive random variables, typically continuous, and N is a counting random variable independent of the Xi’s. The evaluation of the distribution of SN is challenging: only in a few situations one can derive it analytically; in the other cases, one needs to resort to numerical methods, Monte Carlo simulations, or discrete/continuous approximations. Focusing on this latter technique, one common approach is to approximate the distribution of SN using normal, normal-power or translated Gamma distributions, whose parameter values are obtained by matching the same-order moments. An approximation of the total claims amount distribution by a three-parameter Weibull distribution is introduced, discussed, and assessed. This assessment considers different combinations of distributions for the claim frequency and size. The availability of relatively easy expressions for the first three non-central moments facilitates its use. However, care should be taken as the level of approximation might be unsatisfactory for some parts of the distribution under certain circumstances.