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Panjer class revisited: one formula for the distributions of the Panjer (a,b,n) class

Published online by Cambridge University Press:  26 September 2022

Michael Fackler Open the ORCID record for Michael Fackler [Opens in a new window]
Michael Fackler*
Affiliation:
Actuary (DAV), Munich, Germany
*
E-mail: michael_fackler@web.de
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Abstract

The loss count distributions whose probabilities ultimately satisfy Panjer’s recursion were classified between 1981 and 2002; they split into six types, which look quite diverse. Yet, the distributions are closely related – we show that their probabilities emerge out of one formula: the binomial series. We propose a parameter change that leads to a unified, practical and intuitive, representation of the Panjer distributions and their parameter space. We determine the subsets of the parameter space where the probabilities are continuous functions of the parameters. Finally, we give an inventory of parameterisations used for Panjer distributions.

Keywords

Panjer classUnified representationDiscrete loss distributionBinomial series
Type
Original Research Paper
Information
Annals of Actuarial Science , Volume 17 , Issue 1 , March 2023 , pp. 145 - 169
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Institute and Faculty of Actuaries

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References

Albrecher, H., Beirlant, J. & Teugels, J.L. (2017). Reinsurance: Actuarial and Statistical Aspects. Wiley.CrossRefGoogle Scholar
Fackler, M. (2009). Panjer class united – one formula for the Poisson, Binomial, and Negative Binomial distribution. In ASTIN Colloquium 2009.Google Scholar
Fackler, M. (2011). Panjer Class United: one formula for the probabilities of the Poisson, Binomial, and Negative Binomial distribution. Anales del Instituto de Actuarios Españoles, 17, 1–12.Google Scholar
Fackler, M. (2017). Experience rating of (re)insurance premiums under uncertainty about past inflation. PhD thesis, Universität Oldenburg, urn:nbn:de:gbv:715-oops-33087.Google Scholar
Gerhold, S., Schmock, U. & Warnung, R. (2010). A generalization of Panjer’s recursion and numerically stable risk aggregation. Finance and Stochastics, 14(1), 81–128.CrossRefGoogle Scholar
Hess, K.T., Liewald, A. & Schmidt, K.D. (2002). An extension of Panjer’s recursion. ASTIN Bulletin, 32(2), 283–297.CrossRefGoogle Scholar
Johnson, N.L. & Kotz, S. (1969). Distributions in Statistics: Discrete Distributions. Houghton Mifflin.Google Scholar
Johnson, N.L., Kotz, S. & Kemp, A.W. (2005). Univariate Discrete Distributions. Wiley.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. & Willmot, G.E. (2008). Loss Models: From Data to Decisions. Wiley.CrossRefGoogle Scholar
Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 12(1), 22–26.CrossRefGoogle Scholar
Panjer, H.H. & Willmot, G.E. (1992). Insurance Risk Models. Society of Actuaries.Google Scholar
Schröter, K.J. (1990). On a family of counting distributions and recursions for related compound distributions. Scandinavian Actuarial Journal, 1990(2–3), 161–175.CrossRefGoogle Scholar
Sundt, B. (2002). Recursive evaluation of aggregate claims distributions. Insurance: Mathematics and Economics, 30(3), 297–322.Google Scholar
Sundt, B. & Jewell, W.S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12(1), 27–39.CrossRefGoogle Scholar
Willmot, G.E. (1988). Sundt and Jewell’s family of discrete distributions. ASTIN Bulletin, 18(1), 17–29.CrossRefGoogle Scholar
Willmot, G.E. & Panjer, H.H. (1987). Difference equation approaches in evaluation of compound distributions. Insurance: Mathematics and Economics, 6(1), 43–56.Google Scholar