## PREDICTIVE MODELING OF INSURANCE CLAIMS USING REVERSIBLE JUMP MARKOV CHAIN MONTE CARLO METHODS

ABSTRACT
There has been considerable amount of attention rendered to claims reserving methods over the last few decades in actuarial science. The commonly used method of estimating claims reserves is the chain ladder technique. The under-lying principle of the chain-ladder technique is that no underlying pattern to the run-o , and that each development year should be allocated a separate param-eter. Applicable to a wide range of data, the chain ladder could alternatively be condemned for having too many parameters and also assumptions have to be used to estimate reserves beyond the latest development year already observed. This research seeks to explain an approach to model the development of claims run-o , using reversible jump Markov Chain Monte Carlo (RJMCMC) method. The study uses claims data from a renowed Insurance Company in Ghana; Win-BUGS the tool used in simulating the reversible jump Markov Chain Monte Carlo (RJMCMC) method. The Bayesian methods are found to be better than the Over-dispersed Poisson model with lower predictive errors.

CHAPTER 1
Introduction
1.1       Background Study
The management of loss reserved that seals up future payments from claims is a basic and central department within the industry of insurance. This very department is technically known as loss reserves accounting as it represents an involvement of complicating calculations primarily because losses are continuous and comes in at any time ranging from currently to several years down the road. This concept of loss reserving is basically an estimation on an insurer’s liability from future claims; fundamentally, the allowance of an insurer to cover claims made against policies that it guarantees. Therefore when an insurer guarantees a new policy, there is immediately a recording of a premium payable articulating an asset and a claim obligation which represents a liability and it is this liability which is considered part of the unpaid losses account signifying the loss reserve. Principally, there is a fund set up for future compensation of policy holders. The amount is usually considered as provision for unsettled claims or simply claims reserves.

Modeling of claims reserves in non-life insurance is a subtle theme because the insurer requires an estimation precisely the amount of reserves to have in their stock; if the reserves are held in relatively higher amounts than actual, there is the possibility of lower pro tability which has the capacity to lower the competitive-ness of the company (insurer) on the business market. Meanwhile holding lower reserves has the potential of leaving the company in nancial turmoil primarily due to bankruptcy. The forwarding theme in the estimation of these reserves is to compute the future amounts payable from an insurer or reinsurer’s claims popularly known in actuarial science literature as Incurred But Not Reported (IBNR) claims according to Bornheutter and Ferguson (1972).

It is important in prediction of future claims to run the analysis separately from the insurer’s already dissimilar existent cases or portfolios which are dependent on the type of insurance. Classically, a structure that has been established in claim reserving is the representation of the historical payments from a single line of business in a triangular form as this allows practitioners to track the time development of payments. The estimation of the future claims are based on this triangular structure known as the run-o triangle. Mathematically, this is a ma-trix that contains claims or loss data where each row corresponds to a year of accident and each column corresponds to the delay; the delay being the time duration between the year of accident and the year that the claim was made. A lot of literature has been devoted to claims reserving using run-o triangles by England and Verrall (2000) and Gelman et al. (2000).

Two very fundamental yet simple statistical methods highly staged for forecast-ing claims reserves are the Chain Ladder Method (CLM) and the Bornhuetter-Ferguson method. The CLM makes use of data in two dimension array represent-ing happenings and development of claims with the upper left side of the modeling matrix containing known values (previous) and these are used to forecast the re-maining gures in the modeling matrix (future). The Bornhuetter-Ferguson is a Bayesian Method that incorporates an independently deprived prior estimate of ultimate expected losses as well as estimates generated by the same kind of modeling matrix. The credibility factor is used to weight the estimates with pref-erence to more reliable projections.

These fundamental and traditional methods such as the CLM and the Bornhuetter-Ferguson methods are deterministic in nature with no elements of probability at all as stated by England and Verrall (2002a). Over the past two and a half decades however, there has been a lot of increasing interest in stochastic reserv-ing methods although these have only been applied by a handful of experts in the eld of actuarial science England and Verrall (1999). Two things are nec-essary when modeling claims reserves; i.e. the provision of a ’best estimate’ of the considered reserve and secondly a provision of how precise this estimate is actually. Deterministic models are able to satisfy the rst necessity but fail to provide the second. These stochastic models in addition to the rst provision also present a precision of the estimate given by bringing on board the assessment of the variability of the claims reserves.

With reference to standards, the general approach is rstly specifying a model, i.e. nding an estimate of the outstanding claims under that model using the Maximum Likelihood and nally using the built model to nd the precision of the estimated structure. On the other hand stochastic claims reserving rst con-structs the model and a method that produces the actuary’s best estimate and then uses this model to assess the uncertainty of the estimate. There have been several research into nding the best estimate using the stochastic form of the Chain Ladder Method by Verrall (2000), Mack and Venter (2000) and England and Verrall (2000).

A host of stochastic claims reserving models has been documented by Scollnik (2001) with majority of the constructed stochastic models being based on existing deterministic claims reserving models. The primary stage in the e ort towards the widespread application of stochastic reserving methods was to demonstrate how the most often applied practical approaches can be formulated in statistical models which was encouraged by England and Verrall (2000) who included the use of the Chain Ladder technique.

1.2       Problem Statement
Within the application of the Chain Ladder method, there is no underlying pat-tern in the run-o and each development year is given a separate parameter. The merit is that the Chain Ladder technique can be applicable to a wide range of data with the demerit being that the methodology has relatively voluminous parameters. The implication here is that some other assumption has to be ap-plied in the modeling of any potential claims beyond the latest development year already observed. Technically, this is referred to as the modeling of the tail by actuaries, or otherwise application of tail factors.

This problem of modeling the run-o tail has been investigated by many re-searchers using a whole range of techniques. The latest is the use of Markov Chain Monte Carlo (MCMC) technique through the application of Bayesian methods which has totally changed the trends involved in modeling run-o tails England and Verrall (2001a), Gelman et al. (2000), Lunn et al. (1998) and Lunn and Aarons (1995). In spite of this innovation there is still no room for the con-sideration of models which are trans-dimensional in nature. This means that it is impossible to consider models where the actual number of variables are not known Lunn et al. (2009a)

1.3       Objectives of the study
Based on the stated problem, this research will aimed at investigating the mod-eling the tail of a run-o triangle using a reversible jump Markov Chain Monte Carlo (RJMCMC) method. The speci c objectives will be:

To determine the developments factors of a chain ladder technique method.

To estimate the reserves of a local claims run-o data using chain ladder method.

To model the tail of a local claims run-o data using a reversible jump Markov Chain Monte Carlo (RJMCMC) method.

To nd the prediction error of the reserve estimates given by the both the chain ladder method and reversible jump Markov Chain Monte Carlo method.

1.4       Research Questions
The following research questions are posed in relation to addressing the problems stated in the modeling of claims run-o triangle and the objectives outlined for this research.

What are the developments factors of the chain ladder technique.?

How is the reserves of a local claims run-o data estimated using chain ladder method?

How is the tail of local claims run-o data modeled using reversible jump Markov Chain Monte Carlo(RJMCMC) method?

What is the prediction error of the reserve estimates given by both the chain ladder and the reversible jump Markov Chain Monte Carlo method?

1.5       Research Methodology
The research will investigate the modeling of tail of run-o using a reversible jump Markov Chain Monte Carlo (RJMCMC) method. The study will therefore narrow on by reviewing stochastic modeling of a claims run-o triangle that are existent in literature and consider speci cally the stochastic model used in this study. Secondly an overview of Markov Chain Monte Carlo (RJMCMC) methods will be outlined. The foregoing methodologies will be employed and a numerical example will be outlined. The research will feature the use of WinBUGS as the tool for running the required simulations.

1.6       Significance of the Research
As recent as the concept of stochastic claims reserve modeling is by serving as a signi cant task needed within general insurance actuary, its practicality has been downplayed with majority of the practitioners sticking to very traditional meth-ods as a means of claims reserve modeling. As discussed within the introduction however, these have the disadvantage of not providing the exact precision of the estimate provided. To some extent this tuned may be played with experience but why does try luck with experience when a few steps within modeling of the same phenomenon can help bridge the gap. This research o ers an extension of that part of literature by providing an overview of the entire stochastic claims modeling process and secondly providing a justi cation for its application to be now widespread within the experts’ environment.

1.7       Limitation of Study
The insurance sector, just as others consists of the life and non-life insurance; as the properties of life insurance, most importantly are distinct from non-life. This thesis exclusively dealt with the non-life insurance. Run o triangles usually arise particularly in non-life insurance where it may take some time after a loss until the full extent of the claims which have to be paid are known. Another limitation is that the reserves are unadjusted for in ation and undiscounted. The term reserve is used as a synonym for the sum of the Reported But Not Settled (RBNS) and the Incurred But Not Reported (IBNR).

1.8       Organization of the Study
The study contains ve chapters altogether. The rst chapter is the introduction which provides a brief background study, the problem statement, objectives based on which the study was organized, the research questions, the research methodol-ogy, signi cance and how the entire research has been organized. The literature review of mathematical models in both deterministic and stochastic claims reserve modeling is featured in chapter two. The third chapter deals with the general research design and precisely how the methods used in the research are explored and method of data analysis together with the organizational presentation of re-sults. The actual presentation and analysis of the models are documented in the fourth chapter along with the necessary discussion. The nal chapter is chapter ve which provides the summary and ndings, conclusion and recommendations for the future study.

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Item Type: Ghanaian Topic  |  Size: 58 pages  |  Chapters: 1-5
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