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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.

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