Recursive Evaluation of a Family of Compound Distributions

Compound dlstributmns such as the compound Pmsson and the compound negative binomial are used extensively m the theory of risk to model the distributmn off the total claims incurred m a fixed period of time The usual method of evaluating the dlqtributmn functmn requires the computatmn of many convolutions of the conditional d~atnbutmn of the amount of a claim given that a clmm has occurred When the expected number of claims is large, the computatmn can become unwmldy even with modern large scale electronic computers In tlus paper, a recurs|xe definitmn of the distribution of total clmms is developed for a family of claml numbel distnbutmns and arbitrary claim amount distributions When the clam1 amount is discrete, the recursive dehnitmn can be used to compute the distribution of total claims without the use of convolutions. This can reduce the number of required computations by several orders of magnitude for sufhcmntlv large portfolios Results for some spemfic dlatnbutmna have been prevmusly obtained using generating functions and Laplace transforms (see PANJER (1980) including dlscussmn). The simple algebraic proof of this paper yields all the previous results as special case~