Insurance: Mathematics and Economics 13 (1993) 15-22 North-Holland 15 Annuity distributions A new class of compound Poisson distributions Colin M. Ramsay zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA University of Nebraska-Lincoln, Lincoln NE, USA Received June 1992 Revised April 1993 Abstract: A discrete random variable N is said to have an annuity distribution if its probabilities satisfy the recursion ~,,=p~_~(a+b/c,,), n=1,2,3,... where a and b are real constants and c, = (1 - e -““)/(es - l), -m < S < 0~. Condi- tions for the existence of these three-parameter distributions are given. It is proved that the probability generating function of N satisfies a functional equation, This functional equation is used to prove that when N is an unbounded random variable then it is a compound Poisson random variable. It is also shown that the cumulants of N can be expressed as differences of Lambert series. Keywords: Probability generating function; Functional equa- tion; Cumulants; Lambert series. 1. Introduction In the field of actuarial risk theory, a well- known family of discrete distributions is the one which has probabilities satisfying the recursion zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Pk=Pk-l a+; , ( i k=l,2,3 ,..., (1) where p,, > 0 and Cz=, pk = 1, and a and b are real constants. This two-parameter family consists of the Poisson (a = 01, the binomial (a < 0 and b = -a(m + 1) for some positive integer m) and the negative binomial (0 < a < 1) distributions. These distributions have been extensively used in the context of modeling the claim number pro- cess and the aggregate claims process. In the latter context, Panjer (1981) and Sundt and Jew- Correspondence to: Cohn M. Ramsay, Actuarial Science De- partment, University of Nebraska-Lincoln, 310 Burnett Hall, Lincoln, NE 685880307, USA. ell (1981) have shown that this family yields a simple recursion for the aggregate claims distri- bution when the claim size distribution is arith- metic. Recently, Ramsay (1989) introduced a new real valued three-parameter (a, b, 6) family of dis- crete distributions, called annuity distributions. This family, which contains the family defined by equation (1) as a special case, the probabilities satisfying the recursion p,=p,-, a+b i I , n=1,2 9.1.) c, (21 where c,, n = 1,2,, . . . , is a continuous function of the parameter 6, ( 1 -e-“s 1 - - c, = es - 1 if S#O bz if S=O Note that c, has the same form as the present value of an n-year annuity certain at force of interest S [see Kellison (1991, chapter 311, hence the name of the distribution. However, I must emphasize that the parameter S should not, in general, be viewed as a force of interest; it is simply the extra parameter which is used to ex- tend the family of distributions in equation (1). Unfortunately, as will be seen, annuity distribu- tions are difficult to work with because of the complicated nature of their probabilities and cu- mulants. The primary objective of this paper is to offer annuity distributions for consideration in cases where the Poisson, binomial and negative bino- mial may seem inappropriate. To this end, the properties of annuity distributions are explored. In particular the conditions needed to ensure the existence of these distributions are given. It is proved that, under certain conditions, the proba- bility generating function (pgf) of N satisfies a functional equation. Using this functional equa- tion, it is proved that if N is unbounded, i.e., 0167-6687/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved