Robust loss reserving in a log-linear model

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Abstract

It is well known that the presence of outlier events can overestimate or underestimate the overall reserve when using the chain-ladder method. The lack of robustness of loss reserving estimators leads to the development of this paper. The appearance of outlier events (including large claims—catastrophic events) can offset the result of the ordinary chain ladder technique and perturb the reserving estimation. Our proposal is to apply robust statistical procedures to the loss reserving estimation, which are insensitive to the occurrence of outlier events in the data. This paper considers robust log-linear and ANOVA models to the analysis of loss reserving by using different type of robust estimators, such as LAD-estimators, M-estimators, LMS-estimators, LTS-estimators, MM-estimators (with initial S-estimate) and Adaptive-estimators. Comparisons of these estimators are also presented, with application of a well known data set.

Section snippets

Introduction on loss reserving

The estimation of claim reserves and outstanding claims is very important for the operation of insurance companies, as well as for the determination of the profit. In the actuarial literature there is a variety of papers on loss reserving techniques within the chain ladder framework, namely, Kremer (1982), Taylor and Ashe (1983), Mack, 1991, Mack, 1993, Mack, 1994, Renshaw (1989), Verrall, 1991, Verrall, 2000. The book by Taylor (2000) presents useful material on stochastic methods and

Robust inference

Very often, assumptions, made in statistics, i.e. normality, linearity, independence are at most approximations of reality. Robust regression models are useful for filtering linear relationships when the random variation in the data is not normal or when the data contain significant outliers (see Hampel et al., 1986).

In the following we present a summary of some of the most important robust estimators that appear in the statistical and actuarial literature. Some basic concepts of robust

The log-linear loss reserving models

In this section we revisit two similar models that is the basis for our robust loss reserving estimation: the model of Verrall (1991), who considered the estimation of claims reserves and outstanding claims when a log-linear model is applied and the model of Kremer (1982) who applied the estimation methods of analysis of variance (ANOVA) to the problem of estimating the expected value of IBNR-claims. Typical estimators of the log-normal mean and variance fail to be both efficient and robust. In

Robust log-linear loss reserving model estimation

In this section, we incorporate the robust estimators presented in Section  2 into loss reserving techniques. A robust algorithm for the robustification of the log-linear model of Verrall (1991) is derived as well as a robust estimation in the ANOVA setup of Kremer (1982).

Numerical illustrations

In what it follows we will illustrate the LS and our robust procedures using the data given in Taylor and Ashe (1983) (see Table B.1 in Appendix B). The advantage of using this data set is that is well known and it has been widely used in the actuarial literature (see for example Mack, 1993; England and Verrall, 1999, Verdonck et al., 2009). Our first step is to identify the existence of outlier events (if there are any) in the original data and apply a log-linear regression model, a two way

Overview of the results and concluding remarks

In this work we have shown how robust estimation techniques can be incorporated in a loss reserving framework, providing a fair value for the estimation of outstanding reserves. An overview of the main robust estimators has been given in the first part of the paper. Least squares estimators and robust estimators were applied to Ashe and Taylor data.

Table 6 presents an overview of the robust loss reserving estimation in a log-linear model with the original data (without artificial outliers) and

Acknowledgments

Georgios Pitselis acknowledges the financial support from BOF-SF Senior Fellowships of KU Leuven.

References (37)

  • R.D. Armstrong et al.

    Least-absolute values estimation for one-way and two-way tables

    Nav. Res. Logist. Q.

    (1979)
  • D. Bradu et al.

    Estimation in lognormal linear models

    J. Amer. Statist. Assoc.

    (1970)
  • M. Busse et al.

    Robust estimation of reserve risk

    ASTIN Bull.

    (2010)
  • S. Christofides

    Regression models based on log-incremental payments

  • England, P.D., Verrall, V.J., 2002. Stochastic claims reserving in general insurance. In: Presented to the Institute of...
  • D. Gervini et al.

    A class of robust and fully efficient regression estimators

    Ann. Statist.

    (2002)
  • F.R. Hampel et al.

    Robust Statistics; The Approach Based on Influence Functions

    (1986)
  • P.J. Huber

    Robust statistics: asymptotics, conjectures and Monte Carlo

    Ann. Statist.

    (1973)
  • View full text