Academic journal article Journal of Risk and Insurance

Bootstrap Methodology in Claim Reserving

Academic journal article Journal of Risk and Insurance

Bootstrap Methodology in Claim Reserving

Article excerpt

ABSTRACT

In this article, we use the bootstrap technique to obtain prediction errors for different claim-reserving methods, namely, the chain ladder technique and methods based on generalized linear models. We discuss several forms of performing the bootstrap and illustrate the different solutions using the data set from Taylor and Ashe (1983), which has already been used by several authors.

INTRODUCTION

The prediction of an adequate amount to face the responsibilities assumed by an insurance company is a major subject in actuarial science. Despite its well-known limitations, the chain ladder technique (see for instance Taylor (2000) for a presentation of this technique) is the most widely applied claim-reserving method. Moreover, in recent years, considerable attention has been given to the discussion of possible relationships between the chain ladder and various stochastic models (Mack, 1993, 1994; Mack and Venter, 2000; Verrall, 1991, 2000; Renshaw and Verrall, 1994; England and Verrall, 1999; etc.).

The bootstrap technique has proved to be a very useful tool in many fields and can be particularly interesting to assess the variability of the claim-reserving predictions and to construct upper limits at an adequate confidence level. Some applications of the bootstrap technique to claim reserving can be found in Lowe (1994), England and Verrall (1999), and Taylor (2000).

The application of the bootstrap technique to claim reserving is not straightforward and, in our opinion, the applications found in the actuarial literature were not the most adequate.

The main issue to be treated in this article concerns the definition of the residuals to be considered in the bootstrap procedure. We also discuss two different methodologies of performing the bootstrap.

The problem of claim reserving can be summarized in the following way: given the available information about the past, how can we obtain an estimate of the future payments (or the number of claims to be reported) due to claims occurred in those years? Furthermore, we need to determine a prudential margin, which is to say, we want to estimate an upper limit for the reserve with an adequate level of confidence.

Let [C.sub.ij] represent either the incremental claim amounts or the number of claims arising from accident year i and development year j and let us assume that we are in year n and that we know all the past information, i.e., [C.sub.ij] (i = 1, 2, ..., n and j = 1, 2, ..., n + 1 - i). The available data present a characteristic pattern, which can be seen in Figure 1. From now on, and without loss of generality, we consider that the [C.sub.ij] are the incremental claim amounts.

More than to predict the individual values, [C.sub.ij] (i = 2, 3, ..., n and j = n + 2 - i, n + 3 - i, ..., n), we are interested in the prediction of the rows total, [C.sub.i*] (i = 2, 3, ..., n), i.e., the amounts needed to face the claims occurred in year i and especially in the aggregate prediction, C, which represents the expected total liability. Keep in mind that we want to obtain upper limits to the forecasts and to associate a confidence level to those limits.

In "Generalized Linear Models and Claim-Reserving Methods" we present a brief review of generalized linear models (GLM) and their application to claim reserving, whereas in "The Bootstrap Technique" we discuss the application of the bootstrap technique. In "An Application" we illustrate the two different methods presented in "The Bootstrap Technique" section to the data set provided in Taylor and Ashe (1983), which has already been used by several authors.

GENERALIZED LINEAR MODELS AND CLAIM-RESERVING METHODS

Following Renshaw and Verrall (1994) we can formulate most of the stochastic models for claim reserving by means of a particular family of generalized linear models (see McCullagh and Nelder, 1989, for an introduction to GLM). …

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