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An Appropriate Way to Switch from the Individual Risk Model to the Collective One

Published online by Cambridge University Press:  29 August 2014

S. Kuon ,
M. Radtke  and
A. Reich
S. Kuon*
Affiliation:
The Cologne Re, Cologne
M. Radtke*
Affiliation:
The Cologne Re, Cologne
A. Reich*
Affiliation:
The Cologne Re, Cologne
*
Kölnische Rückversicherungs-Gesellschaft AG, Theodor-Heuss-Ring 11, Postfach 10 80 16, D-5000 Köln 1, Germany.
Kölnische Rückversicherungs-Gesellschaft AG, Theodor-Heuss-Ring 11, Postfach 10 80 16, D-5000 Köln 1, Germany.
Kölnische Rückversicherungs-Gesellschaft AG, Theodor-Heuss-Ring 11, Postfach 10 80 16, D-5000 Köln 1, Germany.
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Abstract

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For some time now, the convenient and fast calculability of collective risk models using the Panjer-algorithm has been a well-known fact, and indeed practitioners almost always make use of collective risk models in their daily numerical computations. In doing so, a standard link has been preferred for relating such calculations to the underlying heterogeneous risk portfolio and to the approximation of the aggregate claims distribution function in the individual risk model. In this procedure until now, the approximation quality of the collective risk model upon which such calculations are based is unknown.

It is proved that the approximation error which arises does not converge to zero if the risk portfolio in question continues to grow. Therefore, necessary and sufficient conditions are derived in order to obtain well-adjusted collective risk models which supply convergent approximations. Moreover, it is shown how in practical situations the previous natural link between the individual and the collective risk model can easily be modified to improve its calculation accuracy. A numerical example elucidates this.

Keywords

Individual risk modelcolletive risk modelmodified natural approximationaggregate claims distributionBerry-Esséen bound
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 23 , Issue 1 , May 1993 , pp. 23 - 54
Copyright
Copyright © International Actuarial Association 1993

References

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