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Recursive Evaluation of a Family of Compound Distributions*

  • Harry H. Panjer (a1)
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Compound distributions such as the compound Poisson and the compound negative binomial are used extensively in the theory of risk to model the distribution of the total claims incurred in a fixed period of time. The usual method of evaluating the distribution function requires the computation of many convolutions of the conditional distribution of the amount of a claim given that a claim has occurred. When the expected number of claims is large, the computation can become unwieldy even with modern large scale electronic computers.

In this paper, a recursive definition of the distribution of total claims is developed for a family of claim number distributions and arbitrary claim amount distributions. When the claim amount is discrete, the recursive definition 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 sufficiently large portfolios.

Results for some specific distributions have been previously obtained using generating functions and Laplace transforms (see Panjer (1980) including discussion). The simple algebraic proof of this paper yields all the previous results as special cases.

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The author is grateful to the referee for pointing out an error in the original draft. This researcli was supported by the Natural Sciences and Engineering Research Council of Canada.

Footnotes
References
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Adelson, R. M. (1966). Compound Poisson Distributions, Operations Research Quarterly, 17, 7375.
Panjer, H. H. (1980). The Aggregate Claims Distribution and Stop-Loss Reinsurance, Trans, of the Society of Actuaries, XXXII.
Sundt, B. and Jewell, W. S. (1981). Further Results on Recursive Evaluation of Compound Distributions. Astin Bulletin 12.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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