Claims reserving: A correlated Bayesian model

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Abstract

Estimation of adequate reserves for outstanding claims is one of the main activities of actuaries in property/casualty insurance and a major topic in actuarial science. The need to estimate future claims has led to the development of many loss reserving techniques. There are two important problems that must be dealt with in the process of estimating reserves for outstanding claims: one is to determine an appropriate model for the claims process, and the other is to assess the degree of correlation among claim payments in different calendar and origin years. We approach both problems here. On the one hand we use a gamma distribution to model the claims process and, in addition, we allow the claims to be correlated. We follow a Bayesian approach for making inference with vague prior distributions. The methodology is illustrated with a real data set and compared with other standard methods.

Introduction

The problem of claim reserving can be summarised in the following way. Given available information about the past, we want to estimate future payments due to claims occurred in those years. Since one of the largest liabilities of an insurance company is future claims, estimation of adequate reserves for outstanding claims is one of the main activities of actuaries in property/casualty insurance and a major topic in actuarial science. The need to estimate future claims has led to the development of many loss reserving techniques.

The oldest and most widely used of these techniques is the “chain-ladder”. Despite its well-known limitations, the chain ladder technique is still the most commonly applied claim reserving method. It is frequently used as a benchmark, due to its generalized use and ease of implementation. Originally it is a non-stochastic model. However stochastic versions of this method, or stochastic models that reproduce its results, have been proposed. In recent years, considerable attention has also been given to discuss possible relationships between the chain-ladder and various stochastic models, as well. Mack (1993) and England and Verrall (2002) present a comprehensive review of the various methods mostly from the classical statistics point of view. Bayesian stochastic formulations of the problem can be found in de Alba, 2002, de Alba, 2006, Ntzoufras and Dellaportas (2002) and Verrall (2004).

There are two important problems that must be dealt with in the process of estimating reserves for outstanding claims: one is to determine an appropriate model for the claims process, and the other is to assess the degree of dependence among claim payments in different calendar years and years of origin. In fact, Pinheiro et al. (2003) pointed out that the main problem is the choice of a model, and showed that in some cases the gamma model presents a better fit to the observed values than other models that are used more frequently. These authors do not consider correlation among claims. In fact, correlation among claims has barely been dealt with in the claims reserving literature. Only recently has it been incorporated into the models, Kremer (2005) and de Jong (2006).

We address both these problems here. On the one hand we propose a gamma distribution with a novel parametrization to model the claims process. Additionally, we allow claims to be correlated along development years and for this we propose a correlated gamma process, where the dependence between claims is achieved through a Poisson latent process.

The Bornhuetter–Ferguson (BF) method is indicated as an appropriate solution to situations where the claims data are very unstable and very sensitive to changes in the last available year that is used in forecasting outstanding claims (Taylor, 2004). It is also useful when the actuary is not confident that the future will behave as the past and hence introduces additional information besides using the known (historic) claims data to predict future claims. The use of prior information makes it amenable to a Bayesian formulation if a probabilistic model is assumed for the data. This has been done by Mack (2000) in a credibility framework, and by Verrall (2004) as a GLM, assuming an overdispersed Poisson distribution. We show in this paper that the BF (Bayesian) method can also be seen as a particular case of our model.

The structure of the paper is as follows. In Section 2 we describe the problem in more detail and include some approaches that have been presented in the literature. In Section 3 we consider the independence case and Section 4 deals with the dependence case. Section 5 includes relations of our model with the Bornhuetter–Ferguson model and others. We illustrate our models in Section 6 and we also include some concluding remarks in Section 7.

Section snippets

The problem

To state the problem we denote by Xij either the incremental claim amounts or the number of claims arising from year of origin i and paid in development year j and let us assume that we are in year n and that we know all the past information, that is, Xij for i=1,2,,n and j=1,2,,n+1i have been observed. These available data can be represented in terms of a run-off triangle as the one given in Table 1. The assumption that the data consist precisely of a triangle is made to simplify the

Independence case

We first assume that incremental claim amounts {Xij} follow an independent gamma process, that is, XijGa(αij,βij) independently for i,j=1,,n, such that E(Xij)=αij/βij. In order to give the parameters a meaningful interpretation, we take αij=αi for all j and βij=βj for all i. In this case, αi becomes the row factor and can be thought of as the (ultimate) mean total amount to be paid for those claims originating in year i, and βj becomes the column factor, which can be interpreted as follows:

Dependence case

In order to model the correlation between the Xij’s along development years, we use a correlated gamma process introduced by Nieto-Barajas and Walker (2002). The correlated gamma process {Xij} is constructed via a latent process {Zij} the following way: for each iXi1Ga(αi1,βi1) and for j=2,3,Zij|Xi,j1Po(γijXi,j1), and also Xij|ZijGa(αij+Zij,βij+γij), where the parameters αij,βij,γij>0 for all i and j. Notice that this process {Xij} is, for each i, a Markov process with the following

Relation with other methods

In this section we present the possible relation of our model with other models for claims reserving that exist in the literature. In particular we concentrate on the chain-ladder and the Borhuetter–Ferguson (BF) method and generalizations of it.

In addition to the triangle of incremental claims as the one presented in Table 1, there is a corresponding triangle of cumulative claims Cij defined as: Cij=k=1jXik. For estimating the reserves, the chain-ladder method is based on computing the

Illustration

The data set we have chosen to illustrate the application of our model is the well known Taylor and Ashe (1983) data set. These data have been used by many authors to compare their reserve estimates (see, for example, Verrall (1991), de Alba (2002) and Pinheiro et al. (2003)). The data consists of incremental claims in a 10×10 triangle and is presented in Table 2.

We first implemented the independence model, and to avoid numerical problems, we worked with the observations in millions. To define

Concluding remarks

In summary, we have proposed an independent gamma process to model the incremental claim amounts in a run-off triangle. We assume a gamma distribution for the incremental claims, which has been proposed as an adequate model for this type of data, with a very appealing parameterization which can be easily interpreted in claims reserving terms. This model has the advantage that we work in the original scale of the data (positive values), and in so doing we avoid possible complications arising

Acknowledgement

This research was supported by Asociación Mexicana de Cultura, A.C.–Mexico.

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