This paper reviews Panjer's recursion formula for evaluation of compound distributions and illustrates how it can be applied to life and general insurance problems.

In this paper we consider compound distributions where the counting distribution is a bivariate distribution with the probability function (Pn1,n2)n1,n2≥0 that satisfies a recursion in the form

We present an algorithm for recursive evaluation of the corresponding compound distributions and some examples of distributions in this class.

We consider three classes of bivariate counting distributions and the corresponding compound distributions. For each class we derive a recursive algorithm for calculating the bivariate compound distribution.

In this paper we study a transform introduced by De Pril (1989) for recursive evaluation of convolutions of counting distributions with a positive probability in zero. We discuss some cases where the evaluation of this transform is simplified and relate the transform to infinitely divisible distributions. Finally we discuss an algorithm presented by Dhaene & Vandebroek (1994) for recursive evaluation of convolutions.

Let an interval I ⊂ ℝ and subsets D0, D1 ⊂ I with D0 ∪ D1 = I and D0 ∩ D1 = Ø be given, as well as functions r0: D0 → I, r1: D1 → I. We investigate the system (S) of two functional equations for an unknown function f: I → [0, 1]: We derive conditions for the existence, continuity and monotonicity of a solution. It turns out that the binary expansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, but also for the elliptic integral of the first kind. Moreover, we use (S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasing functions whose derivative is 0 almost everywhere.

The recursive algorithm of Hesselager (1994) is extended to a more general class of counting distributions, which includes Sundt's (1992) class as well as all the mixed Poisson distributions discussed by Willmot (1993).

We consider compound distributions where the counting distribution has the property that the ratio between successive probabilities may be written as the ratio of two polynomials. We derive a recursive algorithm for the compound distribution, which is more efficient than the one suggested by Panjer & Willmot (1982) and Willmot & Panjer (1987). We also derive a recursive algorithm for the moments of the compound distribution. Finally, we present an application of the recursion to the problem of calculating the probability of ruin in a particular mixed Poisson process.

The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).

Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.

It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.