Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-07T20:45:21.874Z Has data issue: false hasContentIssue false

CHAIN LADDER AND ERROR PROPAGATION

Published online by Cambridge University Press:  19 April 2016

Ancus Röhr*
Affiliation:
Helvetia Insurance Switzerland, St. Alban-Anlage 26, 4002 Basel, Switzerland E-mail: ancus.roehr@helvetia.ch

Abstract

We show how estimators for the chain ladder prediction error in Mack's (1993) distribution-free stochastic model can be derived using the error propagation formula. Our method allows for the treatment of the general case of the prediction error of the loss development result between two arbitrary future horizons. In the well-known special cases considered previously by Mack (1993) and Merz and Wüthrich (2008), our estimators coincide with theirs. However, the algebraic form in which we cast them is new, considerably more compact and more intuitive to understand. For example, in the classical case treated by Mack (1993), we show that the mean squared prediction error divided by the squared estimated ultimate loss can be written as jû2j, where ûj measures the (relative) uncertainty around the jth development factor and the proportion of the estimated ultimate loss that it affects. The error propagation method also provides a natural split into process error and parameter error. Our proofs identify and exploit symmetries of “chain ladder processes” in a novel way. For the sake of wider practical applicability of the formulae derived, we allow for incomplete historical data and the exclusion of outliers in the triangles.

Type
Research Article
Copyright
Copyright © Astin Bulletin 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Buchwalder, M., Bühlmann, H., Merz, M. and Wüthrich, M. (2006) The mean square error of prediction in the chain ladder reserving method (Mack and Murphy Revisited), ASTIN Bulletin, 36 (2), 521542.Google Scholar
Bühlmann, H., De Felice, M., Gisler, A., Moriconi, F. and Wüthrich, M.V. (2009) Recursive credibility formula for chain ladder factors and the claims development result. ASTIN Bulletin, 39 (1), 275306.Google Scholar
Dahms, R. (2008) A loss reserving method for incomplete data. Mitteilungen-Bulletin SAV, 1–2, 127148.Google Scholar
Gisler, A. (2013) “Die rasante Entwicklung der Mathematik in der Versicherung: eine Zeitreise über die letzten 30 Jahre”. Presentation given at the annual conference of the Swiss Actuarial Association in Winterthur, 7 September 2013.Google Scholar
Ku, H.H. (1966) Notes on the use of propagation of error formulas. Journal of Research of the National Bureau of Standards - C. Engineering and Instrumentation, 70C (4), 263273.Google Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23 (2), 213225.CrossRefGoogle Scholar
Mack, T. (2008) The prediction error of Bornhuetter/Ferguson. ASTIN Bulletin, 38 (1), 87103.Google Scholar
Matitschka, H. (2010) Prognosefehler im Overdispersed Poisson Modell für Abwicklungsdreiecke. Blätter DGVFM, 31, 291306.Google Scholar
Merz, M. and Wüthrich, M.V. (2008). Modelling the claims development result for solvency purposes. CAS E-Forum Fall 2008, 542–568.Google Scholar
Merz, M. and Wüthrich, M.V. (2014) Claims run-off uncertainty: The full picture, http://ssrn.com/abstract=2524352Google Scholar
Salzmann, R. and Wüthrich, M.V. (2010) Cost-of-capital margin for a general insurance liability runoff, ASTIN Bulletin, 40/2, 415451.Google Scholar
Taylor, G.C. and Ashe, F.R. (1983) Second moments of estimates of outstanding claims. Journal of Econometrics, 23, 3761.Google Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Non-Life Insurance. New York: Wiley.Google Scholar