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A Functional Approach to Approximations for the Individual Risk Model

Published online by Cambridge University Press:  17 April 2015

Susan M. Pitts
Susan M. Pitts*
Affiliation:
University of Cambridge
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Abstract

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A functional approach is taken for the total claim amount distribution for the individual risk model. Various commonly used approximations for this distribution are considered, including the compound Poisson approximation, the compound binomial approximation, the compound negative binomial approximation and the normal approximation. These are shown to arise as zeroth order approximations in the functional set-up. By taking the derivative of the functional that maps the individual claim distributions onto the total claim amount distribution, new first order approximation formulae are obtained as refinements to the existing approximations. For particular choices of input, these new approximations are simple to calculate. Numerical examples, including the well-known Gerber portfolio, are considered. Corresponding approximations for stop-loss premiums are given.

Keywords

Total claim amount distributioncompound distributionsstop-loss premiumsFréchet derivatives
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 2 , November 2004 , pp. 379 - 397
Copyright
Copyright © ASTIN Bulletin 2004

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