A state space representation of the chain ladder linear model

Abstract

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).

Continuous Chain Ladder: Reformulating and generalizing a classical insurance problem

“…The limited amount of data to estimate these underwriting period effects can lead to very volatile estimates, in which case the introdutione an appropriate level of smoothing can more stable and reliable results. Also having a similar aim but considering also smoothing in the development period are Zehnwirth (1989) and Verrall (1989), which use the Kalman filter.…”

State Space Models and the Kalman-Filter in Stochastic Claims Reserving: Forecasting, Filtering and Smoothing

“…The models in the papers of the first category are mostly distribution-free in contrast to the papers of the second category (Verrall (1989(Verrall ( ,1994, Ntzoufras and Dellaportas (2002)), in which incremental payments are assumed to follow a log-normal distribution. The logarithmic incremental payments are specified by the log-normal model:…”

“…The accident and development year parameters are further assumed to evolve as follows for the same reasons as in De Jong and Zehnwirth (1983): (5) was also named as the "linear CL model" by Verrall (1989Verrall ( ,1994, since it is very similar to an additive representation of the CL method (see also Kremer (1982)). In addition to the basic model (5), which is used by Verrall (1989) and Li (2006), Verrall (1994) and Ntzoufras and Dellaportas (2002) extend the basic model by integrating varying run-off evolutions.…”

“…Two of these models (Verrall (1989), Li (2006)) are based on the calendar year approach, and the other two models (Atherino et al (2010), Alpuim and Ribeiro (2003)) consider detached approaches. As for accident year and development year approaches, they can only be found in Taylor et al (2003) and De Jong and Zehnwirth (1983).…”

A generalized linear model with smoothing effects for claims reserving

“…Verrall (1989) also used the Kalman filter in order to smooth the estimates of α i and β j in (2.7). Barnett & Zehnwirth (2000) introduced a model which is referred to as the probabilistic trend family (PTF), and this model could be expressed using…”