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A QUANTITATIVE STUDY OF CHAIN LADDER BASED PRICING APPROACHES FOR LONG-TAIL QUOTA SHARES

Published online by Cambridge University Press:  24 March 2015

Ulrich Riegel*
Affiliation:
Swiss Re Europe S.A., Niederlassung Deutschland, 81911 München, Germany E-mail: Ulrich_Riegel@swissre.com

Abstract

Pricing approaches for long-tail quota shares are often based on the chain ladder method. Apart from IBNR calculation, common pricing methods require volume measures for accident years in the observation period, and for the quotation period. In practice, in most cases restated premiums are used as the volume measures. The prediction error of the chain ladder method is an important part of the prediction uncertainty of these pricing approaches. There are, however, two sources of uncertainty that are not addressed by the chain ladder model: the stochastic volatility of the claims in the first development year; and the restatement uncertainty, the risk that the restated premium is not a good volume measure. We extend Mack's chain ladder model to cover these two sources of uncertainty, and calculate the mean-squared error of chain ladder pricing approaches with arbitrary weights for the accident years in the observation period. Then we focus on the problem of finding optimal weights for the accident years. First, we assume that the parameters for restatement uncertainty are given, and provide recursion formulas to calculate approximately-optimal weights. Second, we describe a maximum likelihood approach that can be used to estimate the restatement uncertainty.

Type
Research Article
Copyright
Copyright © Astin Bulletin 2015 

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References

Braun, Ch. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. Astin Bulletin, 34 (2), 399423.Google Scholar
Buchwalder, M., Bühlmann, H., Merz, M. and Wüthrich, M. V. (2006) The mean square error of prediction in the chain ladder reserving method (Mack and Murphy revisited). Astin Bulletin, 36 (2), 521542.Google Scholar
Frishman, F. (1971) On the Arithmetic Means and Variances of Products and Ratios of Random Variables. Springfield NA: National Technical Information Service.CrossRefGoogle Scholar
Haas, A. (2013) Rentabilität von Kraftfahrt-Quoten im Niedrigzins-Umfeld. Versicherungswirtschaft, 2003 (22), 3235.Google Scholar
Jones, B. D. (2002) An introduction to premium trend. CAS Study Note.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory. Boston: Kluwer Academic Publishers.Google Scholar
Klugman, S. A., Panjer, H. A. and Willmot, G. E. (2004) Loss Models: From Data to Decisions. 2nd Edition. New York: Wiley.Google Scholar
Mack, Th. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. Astin Bulletin, 23 (2), 213225.CrossRefGoogle Scholar
Mack, Th. (2002) Schadenversicherungsmathematik. Karlsruhe: VVW.Google Scholar
Riegel, U. (2014) A bifurcation approach for attritional and large losses in chain ladder calculations. Astin Bulletin, 44 (1), 127172.Google Scholar
Wüthrich, M. V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: John Wiley & Sons.Google Scholar