A Copula Based Bayesian Approach for Paid–Incurred Claims Models for Non-Life Insurance Reserving

40 Pages Posted: 5 Jun 2017

See all articles by Gareth Peters

Gareth Peters

Department of Actuarial Mathematics and Statistics, Heriot-Watt University; University College London - Department of Statistical Science; University of Oxford - Oxford-Man Institute of Quantitative Finance; London School of Economics & Political Science (LSE) - Systemic Risk Centre; University of New South Wales (UNSW) - Faculty of Science

Alice Dong

The University of Sydney - School of Mathematics and Statistics

Robert Kohn

University of New South Wales - School of Economics and School of Banking and Finance

Date Written: December 9, 2012

Abstract

Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of hierarchical Bayesian paid-incurred-claims models, combining the claims reserving models of Hertig and Gogol. In the process we extend the independent log-normal model of Merz and Wuethrich by incorporating different dependence structures using a Data-Augmented mixture Copula paid-incurred claims model.

The usefulness of incorporating both payment and incurred losses into estimating of the full predictive distribution of the outstanding loss liabilities and the resulting reserves is demonstrated in the following cases:

(i) an independent payment data model;

(ii) the independent payment and incurred claims data model of Merz and Wuethrich;

(iii) a novel dependent lag-year telescoping block diagonal Gaussian copula payment and incurred claim data model incorporating conjugacy via transformation;

(iv) a novel data-augmented mixture Archimedean copula dependent payment and incurred claim data model.

Inference in such models is developed by adaptive Markov chain Monte Carlo sampling algorithms. These incorporate a data-augmentation framework utilised to efficiently evaluate the likelihood for the copula based payment and incurred claim model in the loss reserving triangles. The adaptation strategy of the Markov chain Monte Carlo is based on two components. The first component uses an adaptive strategy for learning the posterior structures for the parameters defined over a Euclidean space and the second component deals with an adaptive learning of the posterior for the covariance matrices restricted to the Riemann manifold corresponding to the space of positive definite matrices for the linear dependence structure specified for the payment and incurred claim model.

Keywords: Chain Ladder Models, Claims Reserving, Data Augmentation, Adaptive Markov Chain Monte Carlo

Suggested Citation

Peters, Gareth and Dong, Alice and Kohn, Robert, A Copula Based Bayesian Approach for Paid–Incurred Claims Models for Non-Life Insurance Reserving (December 9, 2012). Available at SSRN: https://ssrn.com/abstract=2980405 or http://dx.doi.org/10.2139/ssrn.2980405

Gareth Peters (Contact Author)

Department of Actuarial Mathematics and Statistics, Heriot-Watt University ( email )

Edinburgh Campus
Edinburgh, EH14 4AS
United Kingdom

HOME PAGE: http://garethpeters78.wixsite.com/garethwpeters

University College London - Department of Statistical Science ( email )

1-19 Torrington Place
London, WC1 7HB
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

University of Oxford Eagle House
Walton Well Road
Oxford, OX2 6ED
United Kingdom

London School of Economics & Political Science (LSE) - Systemic Risk Centre ( email )

Houghton St
London
United Kingdom

University of New South Wales (UNSW) - Faculty of Science ( email )

Australia

Alice Dong

The University of Sydney - School of Mathematics and Statistics ( email )

Sydney, New South Wales 2006
Australia

Robert Kohn

University of New South Wales - School of Economics and School of Banking and Finance ( email )

Australian School of Business
Sydney NSW 2052, ACT 2600
Australia
+61 2 9385 2150 (Phone)

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