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Predictive Stop-Loss Premiums

Published online by Cambridge University Press:  29 August 2014

Werner Hürlimann
Werner Hürlimann*
Affiliation:
Switzerland
*
Allgemeine Mathematik, Winterthur-Leben, Römerstrasse 17, CH-8401 Winterthur.
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Abstract

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Based on a representation of the aggregate claims random variable as linear combination of counting random variables, a linear multivariate Bayesian model of risk theory is defined. In case of the classical risk theoretical assumptions, that is conditional Poisson likelihood counting variates and Gamma structural density, the model is shown to identify with a Bayesian version of the collective model of risk theory. An interesting multivariate credibility formula for the predictive mean is derived. A new type of recursive algorithm, called three-stage nested recursive scheme, allows to evaluate the predictive density and associated predictive stop-loss premiums in an effective way.

Keywords

Aggregate claimsstop-lossBayesian modellingcollective risk theoryrecursive algorithm
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 23 , Issue 1 , May 1993 , pp. 55 - 76
Copyright
Copyright © International Actuarial Association 1993

References

REFERENCES

Alting von Geusau, B. (1990) The shovelboard approach revisited. XXII ASTIN Colloquium, Montreux.Google Scholar
Ammeter, H. (1948) A generalization of the collective theory of risk in regard to fluctuating basic probabilities. Scandinavian Actuarial Journal.CrossRefGoogle Scholar
Ammeter, H. (1949) Die Elemente der kollektiven Risikotheorie von festen und zufallsartig schwankenden Grundwahrscheinlichkeiten. Mitteilungen der Vereinigung Schweiz. Vers. math., 35–95.Google Scholar
Beard, R.E., Pentikäinen, T. and Pesonen, E. (1984) Risk Theory, the stochastic basis of insurance, third edition. Chapman and Hall.Google Scholar
Bertram, J. and Feilmeier, M. (1987). Anwendung numerischer Methoden in der Risikotheorie. Schriftenreihe Angewandte Versicherungsmathematik, Heft 16. Verlag Versicherungswirtschaft, Karlsruhe.Google Scholar
Gerber, H.U. (1979) An introduction to mathematical Risk Theory. Huebner Foundation for Insurance Education. University of Pennsylvania.Google Scholar
Hürlimann, W. (1990a) On linear combinations of random variables and Risk Theory. In: Methods of Operations Research 63, XIV Symposium on Operations Research, Ulm, 1989, 11–20Google Scholar
Hürlimann, W. (1990b) Pseudo compound Poisson distributions in Risk Theory. ASTIN Bulletin 20, 57–79.CrossRefGoogle Scholar
Hürlimann, W. (1991) Negative claim amounts, Bessel functions, Linear Programming and Miller's algorithm. Insurance: Mathematics and Economics 10, 9–20.Google Scholar
Jewell, W.S. (1974) The credible distribution. ASTIN Bulletin 7, 237–269.CrossRefGoogle Scholar
Jewell, W.S. (1986) Introduction to Bayesian Statistics and Credibility Theory. Operations Research Center, University of California, Berkeley, presented at the Summer School of the Association of Swiss Actuaries, Gwatt, Switzerland.Google Scholar
Jewell, W.S. (1974) Up the misty staircase with Credibility Theory. Mitteilungen der Vereinigung Schweiz. Vers. math., 281–312.Google Scholar
De Jong, P. (1983). The mean square error of a randomly discounted sequence of uncertain payments. In: De Vylder, F., Goovaerts, M., Haezendonck, J. Premium Calculation in Insurance, NATO ASI Series, Series C: Mathematical and Physical Sciences vol. 121, 449–459.Google Scholar
Van Klinken, J. (1960) Actuariële Statistiek, Syllabus at the Institute of Actuarial Sciences and Econometrics of the University of Amsterdamm, pp. 13–15 (In Dutch).Google Scholar
Klugman, S.A. (1989) Bayesian modelling of mortality catastrophes. Insurance: Mathematics and Economics 8, 159–164.Google Scholar
London, D. (1988) Survival Models and their Estimation, secondedition. ACTEX Publications, Winsted and New Britain, Connecticut.Google Scholar
Norberg, R. (1987) A note on experience rating of large group life contracts. Mitteilungen der Vereinigung Schweiz. Vers. math., 17–34.Google Scholar
Panjer, H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
De Pril, N. (1986) On the exact computation of the aggregate claims distribution in the individual life model. ASTIN Bulletin 16, 109–112.CrossRefGoogle Scholar
De Pril, N. (1989) The aggregate claims distribution in the individual model with arbitrary positive claims. ASTIN Bulletin 19, 9–24.CrossRefGoogle Scholar
Wolthuis, H. and Van Hoek, I. (1986) Stochastic models for Life Contingencies. Insurance: Mathematics and Economics 5, 217–254.Google Scholar