Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-7j4dq Total loading time: 0.422 Render date: 2022-09-29T09:33:41.544Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS

Published online by Cambridge University Press:  12 August 2016

Greg Taylor*
Affiliation:
UNSW Business School, Level 6, West Lobby, UNSW Business School Building E12, UNSW Sydney 2052Australia E-Mail: gregory.taylor@unsw.edu.au

Abstract

The cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

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

England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8 (iii), 443518.CrossRefGoogle Scholar
England, P.D., Verrall, R.J. and Wüthrich, M.V. (2012) Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method. Annals of Actuarial Science, 6 (2), 258281.CrossRefGoogle Scholar
Gisler, A. and Müller, P. (2007) Credibility for additive and multiplicative models. Paper presented to the 37th ASTIN Colloquium, Orlando FL, USA. See http://www.actuaries.org/ASTIN/Colloquia/Orlando/Presentations/Gisler2.pdf.Google Scholar
Gisler, A. and Wüthrich, M.V. (2008) Credibility for the chain ladder reserving method. Astin Bulletin, 38 (2), 565597.CrossRefGoogle Scholar
Hachemeister, C. A. and Stanard, J.N. (1975) IBNR claims count estimation with static lag functions. Spring meeting of the Casualty Actuarial Society.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J.P. (2008) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika, 95, 979986.CrossRefGoogle Scholar
Kuang, D., Nielsen, B. and Nielsen, J.P. (2009) Chain-ladder as maximum likelihood revisited. Annals of Actuarial Science, 4 (1), 105121.CrossRefGoogle Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. Astin Bulletin, 23 (2), 213225.CrossRefGoogle Scholar
McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models, 2nd ed. London, UK: Chapman & Hall.CrossRefGoogle Scholar
Merz, M., Wüthrich, M.V. and Hashorva, E. (2013) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7 (1), 325.CrossRefGoogle Scholar
Renshaw, A.E. and Verrall, R.J. (1998) A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4 (iv), 903923.CrossRefGoogle Scholar
Shi, P., Basu, S. and Meyers, P.P. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16 (1), 2951.CrossRefGoogle Scholar
Taylor, G. (2000) Loss Reserving: An Actuarial Perspective. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
Taylor, G. (2009) The chain ladder and Tweedie distributed claims data. Variance, 3, 96104.Google Scholar
Taylor, G. (2011) Maximum likelihood and estimation efficiency of the chain ladder. Astin Bulletin, 41 (1), 131155.Google Scholar
Taylor, G. (2015) Bayesian chain ladder models. Astin Bulletin, 45 (1), 7599.CrossRefGoogle Scholar
Tweedie, M. C. K. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions, Proceedings of the Indian Statistical Golden Jubilee International Conference (ed. Ghosh, J.K. and Roy, J.), pp. 579604. Indian Statistical Institute.Google Scholar
Verrall, R.J. (2000) An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: Mathematics and Economics, 26 (1), 9199.Google Scholar
Verrall, R.J. (2004) A Bayesian generalised linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8 (3), 6789.CrossRefGoogle Scholar
Wüthrich, M.V. (2003) Wüthrich M V (2007). Claims reserving using Tweedie's compound Poisson model. Astin Bulletin, 33 (2), 331346.CrossRefGoogle Scholar
Wüthrich, M.V. (2007) Using a Bayesian approach for claims reserving. Variance, 1 (2), 292301.Google Scholar
Wüthrich, M.V. (2012) Discussion of “A Bayesian log-normal model for multivariate loss reserving” by Shi-Basu-Meyers. North American Actuarial Journal, 16 (3), 398401.Google Scholar
Wüthrich, M.V. and Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance, Chichester, UK: John Wiley & Sons Ltd.Google Scholar
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *