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An Accurate Asymptotic Approximation for Experience Rated Premiums

Published online by Cambridge University Press:  17 April 2015

Riccardo Gatto
Riccardo Gatto*
Affiliation:
Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern Switzerland. Internet: gatto@stat.unibe.ch; http://www.stat.unibe.ch/~gatto
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Abstract

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In the Bayesian approach, the experience rated premium is the value which minimizes an expected loss with respect to a posterior distribution. The posterior distribution is conditioned on the claim experience of the risk insured, represented by a n-tuple of observations. An exact analytical calculation for the experience rated premium is possible under restrictive circumstances only, regarding the prior distribution, the likelihood function, and the loss function. In this article we provide an analytical asymptotic approximation as n → ∞ for the experience rated premium. This approximation can be obtained under more general circumstances, it is simple to compute, and it inherits the good accuracy of the Laplace approximation on which it is based. In contrast with numerical methods, this approximation allows for analytical interpretations. When exact calculations are possible, some analytical comparisons confirm the good accuracy of this approximation, which can even lead to the exact experience rated premium.

Keywords

Class of conjugate distributionscredibility theoryLaplace approximationloss functionposterior mode
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 1 , May 2004 , pp. 113 - 124
Copyright
Copyright © ASTIN Bulletin 2004

Footnotes

1

The author thanks an anonymous referee for thoughtful comments which improved the quality of this article.

References

Bailey, A.L. (1945) “A generalized theory of credibility”, Proceedings of the Casualty Actuarial Society, 32, 13–20.Google Scholar
Bailey, A.L. (1950) “Credibility procedures, Laplace’s generalizations of Bayes rule and the combination of collateral knowledge with observed data”, Proceedings of the Casualty Actuarial Society, 37, 7–23.Google Scholar
Bichsel, F. (1964) “Erfahrungstarifierung in der Motorfahrzeughaftpflicht-Versicherung”, MSV, 64, 119–130.Google Scholar
Bleistein, N. and Handelsman, R.A. (1986) Asymptotic Expansions of Integrals, second edition, Dover.Google Scholar
Bühlmann, H. (1964) “Optimale Prämienstufensysteme”, MSV, 64, 193–214.Google Scholar
Bühlmann, H. (1967) “Experience rating and credibility”, Astin Bulletin, 4, 199–207.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory, Springer.Google Scholar
Heilmann, W.-R. (1989) “Decision theoretic foundations of credibility theory”, Insurance: Mathematics and Economics, 8, 77–95.Google Scholar
Klugman, S.A. (1992) Bayesian Statistics in Actuarial Sciences, Kluwer.CrossRefGoogle Scholar
Lundberg, O. (1964), On random processes and their application to sickness and accidents statistics, Ph.D thesis, University of Stockholm (1940), second edition, Almquist & Wiksells.Google Scholar
Makov, U.E. (2001) “Principal applications of Bayesian methods in actuarial science: a perspective”, with discussions, North American Actuarial Journal, 5, 53–73.CrossRefGoogle Scholar
Mayerson, A.L. (1964) “The uses of credibility in property insurance ratemaking”, Giornale dell’Instituto Italiano degli Attuari, 27, 197–218.Google Scholar
Rohatgi, V.K. (1976), An Introduction to Probability Theory and Mathematical Statistics, Wiley.Google Scholar
Tierney, L. and Kadane, J.B. (1986), “Accurate approximations for posterior moments and marginal densities”, Journal of the American Statistical Association, 81, 82–86.CrossRefGoogle Scholar