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PREDICTIVE DISTRIBUTIONS OF OUTSTANDING LIABILITIES IN GENERAL INSURANCE BY P.D. ENGLAND AND R.J. VERRALL ABSTRACT This paper extends the methods introduced in England & Verrall (2002), and shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or Bayesian techniques for clearly defined statistical models. A general procedure for bootstrapping is described, by extending the methods introduced in England & Verrall (1999), England (2002) and Pinheiro et al (2003). The analogous Bayesian estimation procedure is implemented using Markov-chain Monte Carlo methods, where the models are constructed as Bayesian generalised linear models using the approach described by Dellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well known model introduced by Mack (1993). Mack's model is useful, since it can be used with data sets that exhibit negative incremental amounts. The techniques are illustrated with examples, and the resulting predictive distributions from both the bootstrap and Bayesian methods are compared. KEYWORDS Bayesian, Bootstrap, Chain-ladder, Dynamic Financial Analysis, Generalised Linear Model, Markov chain Monte Carlo, Reserving risk, Stochastic reserving. CONTACT ADDRESS Dr PD England, EMB Consultancy, Saddlers Court, 64-74 East Street, Epsom, KT17 1HB. E-mail: peter.england@emb.co.uk 1 1. INTRODUCTION The “holy grail” of stochastic reserving techniques is to obtain a predictive distribution of outstanding liabilities, incorporating estimation error from uncertainty in the underlying model parameters and process error due to the underlying claims generating process. With many of the stochastic reserving models that have been proposed to date, it is not possible to obtain that distribution analytically, since the distribution of the sum of random variables is required, taking account of estimation error. Where an analytic solution is not possible, progress can still be made by adopting simulation methods. Two methods have been proposed that produce a simulated predictive distribution: bootstrapping, and Bayesian methods implemented using Markov chain Monte Carlo techniques. We are unaware of any papers in the academic literature comparing the two approaches until now, and as such, this paper aims to fill that gap, and highlight the similarities and differences between the approaches. Bootstrapping has been considered by Ashe (1986), Taylor (1988), Brickman et al (1993), Lowe (1994), England & Verrall (1999), England (2002), England & Verrall (2002), and Pinheiro et al (2003), amongst others. Bayesian methods for claims reserving have been considered by Haastrup & Arjas (1996), de Alba (2002), England & Verrall (2002), Ntzoufras & Dellaportas (2002), Verrall (2004) and Verrall & England (2005). England & Verrall (2002) laid out some of the basic modelling issues, and in this paper, we explore further the methods that provide predictive distributions. A general framework for bootstrapping is set out, and illustrated by applying the procedure to recursive models, including Mack’s model (Mack, 1993). With Bayesian methods, we set out the theory and show that, with non-informative prior distributions, predictive distributions can be obtained that are very similar to those obtained using bootstrapping methods. Thus, Bayesian methods can be seen as an alternative to bootstrapping in practical applications. We limit ourselves to using non-informative prior distributions to highlight the similarities to bootstrapping, in the hope that a good understanding of the principles and application of Bayesian methods in the context of claims reserving will help the methods to be more widely applied, and make it easier to move on to applications where the real advantages of Bayesian modelling become apparent. By focusing on non-informative prior distributions, we acknowledge that we are presenting a very limited view of the possibilities and power of Bayesian inference. We believe that Bayesian methods offer considerable advantages in practical terms, and deserve greater attention than they have received so far in practice. Hence, a further aim of this paper is to show that the Bayesian approach with no prior information is only a short step away from the popular bootstrapping methods. Once that step has been made, the Bayesian framework can be used to explore alternative modelling strategies (such as modelling claim numbers and amounts together), and incorporating prior opinion (for example, in the form of manual intervention, or a stochastic Bornhuetter-Ferguson method). Some of these ideas have been explored in the Bayesian papers cited above, and we believe that there is scope for actuaries to

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