Chain Ladder Method Research Articles

Chain ladder, marginal sum and maximum likelihood estimation

The chain ladder method is one of the most famous methods used in reserving. It exploits all data from the run-off triangle and provides simple estimates of the expected ultimate aggregate claims. Simplicity of an estimator is important for its application in practice, but its performance usually depends upon the stochastic mechanism generating the data. In the present paper we consider two stochastic models which reflect certain elementary ideas on the occurrence and the delay in reporting of claims. We show that in these models the chain ladder estimators of the expected ultimate aggregate claims result from classical statistical estimation principles.

Non-optimal prediction by the chain ladder method

The chain ladder method is one of the most common methods for reserving, and various attempts have been made to justify it in a stochastic model. A particularly interesting model was proposed by Mack (1993, 1994a,b): Under the assumptions of his model, Mack proved that the chain ladder predictors of non-observable aggregate claims are unbiased, and Schmidt and Schnaus (1996) proved that the chain ladder predictor for the first non-observable calendar year is also optimal in the sense that it minimizes expected squared error loss over a wide class of unbiased predictors. In the present paper, it is shown that optimality fails for the chain ladder predictor for the second non-observable calendar year.

092038 (M40, B50) Non-optimal prediction by the chain ladder method : Schmidt K.D.,Dresdner Schriften zur Versicherungsmathematik, 3, 1996

092038 (M40, B50) Non-optimal prediction by the chain ladder method : Schmidt K.D.,Dresdner Schriften zur Versicherungsmathematik, 3, 1996

An Extension of Mack's Model for the Chain Ladder Method

AbstractThe chain ladder method is a simple and suggestive tool in claims reserving, and various attempts have been made aiming at its justification in a stochastic model. Remarkable progress has been achieved by Schnieper and Mack who considered models involving assumptions on conditional distributions. The present paper extends the model of Mack and proposes a basic model in a decision theoretic setting. The model allows to characterize optimality of the chain ladder factors as predictors of non-observable development factors and hence optimality of the chain ladder predictors of aggregate claims at the end of the first non-observable calendar year. We also present a model in which the chain ladder predictor of ultimate aggregate claims turns out to be unbiased.

081038 (M42) An extension of Mack's model for the chain ladder method: Schmidt K.D., Schnaus A., Institut für Mathematische Stochastik, Technische Universität Dresden, Dresden, Institut für Mathematische Stochastik, Dresdner Schriften zur Versicherungsmathematik, 4, 1995

081038 (M42) An extension of Mack's model for the chain ladder method: Schmidt K.D., Schnaus A., Institut für Mathematische Stochastik, Technische Universität Dresden, Dresden, Institut für Mathematische Stochastik, Dresdner Schriften zur Versicherungsmathematik, 4, 1995

Which stochastic model is underlying the chain ladder method?

The usual chain ladder method is a deterministic claims reserving method. In the last years, a stochastic loglinear approximation to the chain ladder method has been used by several authors especially in order to quantify the variability of the estimated claims reserves. Although the reserves estimated by both methods are clearly different, the loglinear approximation has been called ‘chain ladder”, too, by these authors. In this note, we show that a different distribution-free stochastic model is underlying the chain ladder method, i.e. yields exactly the same claims reserves as the usual chain ladder method. Moreover, a comparison of this stochastic model with the above mentioned loglinear approximation reveals that the two models rely on different philosophies of the claims process. Because of these fundamental differences the loglinear approximation deviates from the usual chain ladder method in a decisive way and should therefore not be called ‘chain ladder’ any more. Finally, in the appendix it is shown that the loglinear approximation is much more volatile than the usual chain ladder method.

A simple parametric model for rating automobile insurance or estimating IBNR claims reserves

It is shown that there is a connection between rating in automobile insurance and the estimation of IBNR claims amounts because automobile insurance tariffs are mostly cross-classified by at least two variables (e.g. territory and driver class) and IBNR claims run-off triangles are always cross-classified by the two variables accident year and development year. Therefore, by translating the most well-known automobile rating methods into the claims reserving situation, some known and some unknown claims reserving methods are obtained. For instance, the automobile rating method of Bailey and Simon produces a new claims reserving method, whereas the model leading to the rating method called “marginal totals” produces the well-known IBNR claims estimation method called “chain ladder”. A drawback of this model is the fact that it is designed for the number of claims and not for the total claims amount for which it is usually applied.As an alternative for both, rating and claims reserving, we describe a simple but realistic parametric model for the total claims amount which is based on the Gamma distribution and has the advantage of providing the possibility of assessing the goodness-of-fit and calculating the estimation error. This method is not very well known in automobile insurance—although a satisfactory application is reported—and seems to be completely unknown in the field of claims reserving, although its execution is nearly as simple as that of the chain ladder method.

A Simple Parametric Model for Rating Automobile Insurance or Estimating IBNR Claims Reserves

AbstractIt is shown that there is a connection between rating in automobile insurance and the estimation of IBNR claims amounts because automobile insurance tariffs are mostly cross-classified by at least two variables (e.g. territory and driver class) and IBNR claims run-off triangles are always cross-classified by the two variables accident year and development year. Therefore, by translating the most well-known automobile rating methods into the claims reserving situation, some known and some unknown claims reserving methods are obtained. For instance, the automobile rating method of Bailey and Simon produces a new claims reserving method, whereas the model leading to the rating method called “marginal totals” produces the well-known IBNR claims estimation method called “chain ladder”. A drawback of this model is the fact that it is designed for the number of claims and not for the total claims amount for which it is usually applied.As an alternative for both, rating and claims reserving, we describe a simple but realistic parametric model for the total claims amount which is based on the Gamma distribution and has the advantage of providing the possibility of assessing the goodness-of-fit and calculating the estimation error. This method is not very well known in automobile insurance—although a satisfactory application is reported—and seems to be completely unknown in the field of claims reserving, although its execution is nearly as simple as that of the chain ladder method.

A state space representation of the chain ladder linear model

In a recent paper, Kremer (1982) has shown how the classical chain ladder method for estimating outstanding claims on general insurance business is strongly related to a two-way analysis of variance. It can be argued that the estimation methods in a standard chain ladder analysis are inefficient from a statistical viewpoint and that an analysis of variance is more appropriate. Once the chain ladder method is identified with a standard statistical method, the well-known statistical theory can be used to the advantage of the claims reserver. For a further discussion of the use of main stream statistical theory applied to the least squares estimation of the linear model which is close to the chain ladder method, the reader is referred to Renshaw (1989).

The run-off-triangle: Least squares — against chainladder estimations

To estimate the lower, unknown, part of a run-off-triangle is one of the many problems an actuary has to face. Often the chain-ladder method is used. The predictions can be improved by using a least-squares model. Maximum likelihood-estimators can be obtained. An inequality coefficient is introduced to discriminate between models.

IBNR-claims and the two-way model of ANOVA

Abstract A method for estimating the expected value of IBNR-c1aims is derived by transforming the classical multiplicative model into a two-way model of Analysis of Variance, in which easy tractable results of Mathematical Statistics can be applied. The resulting method is shown to be strongly connected to the classical chain ladder method.