An Extension of Panjer's Recursion

Abstract

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {q n } n∈N 0 satisfies the recursionq n for all n ∈ N 0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n ∈ N 0 with n ≥ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n ∈ N 0 with n ≥ k for arbitrary k ∈ N 0 . Our result are of interest in catastrophe excess-of-loss reinsurance.

Panjer recursion versus FFT for compound distributions

“…The class given by (7) includes, inter alia, the l-truncations of the negative binomial and Poisson distributions, which are interesting in a reinsurance context. An overview of higher order Panjer recursions is given by Hess et al (2002); furthermore, Sundt (1992) discusses recursions of the kind…”

“…Panjer and Wang (1993) examined the stability of the Panjer recursion; Willmot (1988), Sundt (1999) and Hess et al (2002) have given generalizations of the Panjer recursion; Hermesmeier (1999, 2000) have investigated the propagation of discretization errors through compounding and established an improved FFT based procedure using an exponential change of measure. The latter contribution is quite substantial since it essentially eliminates the so called aliasing error, which is the fundamental deficit that arises through the use of the discrete Fourier transform.…”

Managing inventory systems of slow-moving items

“…The parameters a and b are presumed to satisfy a < 0 and b = −q · a for some q ∈ N + for the binomial distribution, a = 0 and b > 0 for the Poisson distribution, and 0 < a < 1 and b > −a for the NB distribution (Hess et al, 2002).…”

On excess-of-loss reinsurance

“…Other, more involved holonomic functions Q(z) will be discussed in the framework of mixed Poisson processes below. Note that the Q(z) satisfying (4) generalize (6) by allowing recursions for {p n } of higher order and by replacing the factor a+b/n by general rational functions (see also [19,23,30] and more recently [8…”