England and Verrall - Predictive Distributions of Outstanding Liabilities in General Insurance

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  Submitted to “Annals of Actuarial Science” - Confidential until published  1 PREDICTIVE DISTRIBUTIONS OF OUTSTANDINGLIABILITIES IN GENERAL INSURANCE B Y P.D. E  NGLAND AND R.J. V ERRALLABSTRACTThis 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 procedurefor 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 areconstructed as Bayesian generalised linear models using the approach described byDellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well knownmodel introduced by Mack (1993). Mack's model is useful, since it can be used with data setsthat exhibit negative incremental amounts. The techniques are illustrated with examples, andthe resulting predictive distributions from both the bootstrap and Bayesian methods arecompared.KEYWORDSBayesian, 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   Submitted to “Annals of Actuarial Science” - Confidential until published  2 1. I  NTRODUCTION   The “holy grail” of stochastic reserving techniques is to obtain a predictive distributionof outstanding liabilities, incorporating estimation error from uncertainty in the underlyingmodel parameters and process error due to the underlying claims generating process. Withmany of the stochastic reserving models that have been proposed to date, it is not possible toobtain that distribution analytically, since the distribution of the sum of random variables isrequired, 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 Carlotechniques. We are unaware of any papers in the academic literature comparing the twoapproaches until now, and as such, this paper aims to fill that gap, and highlight thesimilarities and differences between the approaches. Bootstrapping has been considered byAshe (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), deAlba (2002), England & Verrall (2002), Ntzoufras & Dellaportas (2002), Verrall (2004) andVerrall & 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 andshow that, with non-informative prior distributions, predictive distributions can be obtainedthat are very similar to those obtained using bootstrapping methods. Thus, Bayesian methodscan be seen as an alternative to bootstrapping. We limit ourselves to using non-informative prior distributions to highlight the similarities to bootstrapping, in the hope that a goodunderstanding of the principles and application of Bayesian methods in the context of claimsreserving will help the methods to be more widely applied, and make it easier to move on toapplications where the real advantages of Bayesian modelling become apparent.We believe that Bayesian methods offer considerable advantages in practical terms, anddeserve greater attention than they have received so far in practice. Hence, a further aim of this paper is to show that the Bayesian approach is only a short step away from the popular  bootstrapping methods. Once that step has been made, the Bayesian framework can be usedto explore alternative modelling strategies (such as modelling claim numbers and amountstogether), 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 theBayesian papers cited above, and we believe that there is scope for actuaries to progress fromthe basic stochastic reserving methods, which have now become better-understood, to moresophisticated approaches.Bootstrapping has proved to be a popular method for a number of reasons, including: −   The ease with which it can be applied −   The fact that bootstrap estimates can often be obtained in a spreadsheet −   The possibility of obtaining predictive distributions when combined with simulation for the process error.However, it is not without its difficulties, for example: −   A small number of sets of “pseudo” data may be incompatible with the underlyingmodel, and may require modification.  Submitted to “Annals of Actuarial Science” - Confidential until published  3 −   Models that require statistical software to fit them, and do not have an equivalenttraditional method, are more difficult to implement. −   There is a limited number of combinations of residuals that can be used when generating pseudo data, which is a potential issue with smaller data sets. −   The method is open to manipulation, and may not always be implemented appropriately.The final item in the list above could also be seen as a benefit, and partly explains the popularity of the method, since actuaries can extend the methodology, while broadly obeyingits spirit, but losing any clear link between the bootstrapping procedure and a well specifiedstatistical model.When using bootstrapping to help obtain a predictive distribution of outstanding claims,it is a common misunderstanding that the approach is “distribution-free”. Furthermore, sincethe publication of England & Verrall (1999), some readers have incorrectly associated“bootstrapping”, in this context, exclusively with the model presented in that paper (the chainladder model represented as the over-dispersed Poisson model described in Renshaw &Verrall (1998)). One of the aims of this paper is to correct those misconceptions, and describe bootstrapping as a general procedure, which, if applied consistently, can be used to obtain theestimation error (standard error) of well specified models. In addition, England (2002)showed that when forecasting into the future, bootstrapping can be supplemented by asimulation approach to incorporate process error, giving a full predictive distribution. The procedure for using bootstrap methods to obtain a predictive distribution for outstandingclaims is summarised in Figure 1.The procedure for obtaining predictive distributions using Bayesian techniques has manysimilarities to bootstrapping, and is summarised in Figure 2. The starting point is also a well-specified statistical model. However, instead of using bootstrapping to incorporate estimationerror, Markov chain Monte Carlo (MCMC) techniques can be used to provide distributions of the underlying parameters instead. The final forecasting stage is identical in both paradigms.Comparison with Figure 1 shows that the principal difference between the twoapproaches is at the second stage, and that as long as the underlying statistical model can beadequately defined, either methodology could be used. In this paper, we stress the importanceof starting with a well-defined statistical model, and show that where the procedures inFigure 1 and Figure 2 are followed, it is possible to apply bootstrapping and Bayesiantechniques to models that hitherto have not been tried, such as Mack’s model (Mack, 1993).Several stochastic models used for claims reserving can be embedded within theframework of generalised linear models (GLMs). This includes models for the chain-ladder technique, that is, the over-dispersed Poisson and negative binomial models, and the methodsuggested by Mack (1993). It also applies to some models including parametric curves, suchas the Hoerl curve, and models based on the lognormal distribution (see Section 8). In allcases, a similar procedure can be followed in order to apply bootstrap and Bayesian methodsto obtain the estimation error of the reserve estimates. If the process error is included in a waythat is consistent with the underlying model, the results will be analogous to results obtainedanalytically from the same underlying model. A further aim of this paper is to illustrate this by example, comparing results obtained analytically with results obtained using bootstrap andBayesian approaches.This paper is set out as follows. Section 2 contains some basic definitions. Section 3 briefly outlines the stochastic reserving methods that are considered in this paper, and Section4 summarises how predictions and prediction errors can be calculated analytically. Section 5considers a general procedure for bootstrapping generalised linear models, and describes howthe procedure can be implemented for the models introduced in Section 3. Section 6considers Bayesian modelling and Gibbs sampling generally, before introducing the  Submitted to “Annals of Actuarial Science” - Confidential until published  4application to Bayesian generalised linear models. Section 6 also describes how the Bayesian procedure can be implemented for the models introduced in Section 3. Examples are providedin Section 7, where the results of the bootstrap and Bayesian approaches are compared. Adiscussion appears in Section 8, and concluding remarks in Section 9.For readers only interested in bootstrapping, Section 6 can be ignored, and for readersonly interested in Bayesian methods, Section 5 can be ignored. 2. T HE C HAIN L ADDER  T ECHNIQUE   For ease of exposition, we assume that the data consist of a triangle of observations. Thestochastic methods described in this paper can also be applied to other shapes of data, and theassumption of a triangle does not imply any loss of generality. Thus, we assume that the dataconsist of a triangle of incremental claims: 1,11,21,2,12,1,1 , ,,,, nnn CCC CC C  − ……  This can be also written as { } :1,,;1,,1 ij Cinjni = = − + … … , where n is the number of srcin years. ij C  is used to denote incremental claims, and ij  D is used to denote thecumulative claims, defined by:  DC  ijik k  j = = ∑ 1 .The aim of the exercise is to populate the missing lower portion of the triangle, andextrapolate beyond the maximum development period where necessary. One traditionalactuarial technique that has been developed to do this is the chain-ladder technique, whichforecasts the cumulative claims recursively using ,2,12 ˆˆ iniinini  DD λ  − + − + − + = , and ,,1 ˆˆˆ ijijj  DD λ  − = , 3,4,,.  jninin = − + − + …  where the fitted development factors, denoted by { } λ   j  jn :,, = 2 … , are given by  , λ   jijinjijinj  D D = =− +−=− + ∑∑ 11111 .
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