Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T03:37:57.575Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itasca.Google Scholar
De Pril, N. (1989) The Aggregate Claims Distribution in the Individual Model with Arbitrary Positive Claims. ASTIN Bulletin 19, 9–24.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd ed. Wiley, New York.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation for Insurance Education, Philadelphia.Google Scholar
Gerber, H.U. (1984) Error Bounds for the Compound Poisson Approximation. Insurance: Mathematics and Economics 3, 191–194.Google Scholar
Hipp, C. (1985) Approximation of Aggregate Claims Distributions by Compound Poisson Distributions. Insurance: Mathematics and Economics 4, 227–232.Google Scholar
Hipp, C. (1986) Improved Approximation for Aggregate Claims Distributions in the Individual Model. ASTIN Bullletin 16, 89–100.CrossRefGoogle Scholar
Jewell, W.S. and Sundt, B. (1981) Improved Approximations for the Distribution of a Heterogeneous Risk Portfolio. Mitt. Ver. Schweiz. Vers. Math., 1981, 221–240.Google Scholar
Kornya, P.S. (1983) Distribution of Aggregate Claims in the Individual Risk Theory Model. Transactions of the Society of Actuaries 35, 823–836. Discussion 837–858.Google Scholar
Panjer, H.H. (1981) Recursive Evaluation of a Family of Compound Distributions. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
Sundt, B. (1985) On Approximations for the Distribution of a Heterogenous Risk Portfolio. Mitt. Ver. Schweiz. Vers. Math., 1985, 189–203.Google Scholar
von Chossy, R. and Rappl, G. (1983) Some Approximation Methods for the Distribution of Random Sums. Insurance: Mathematics and Economics 2, 251–270.Google Scholar