The chain-ladder method is one of the most popular claims reserving techniques. The aim of this study is to back-test the chain-ladder method. For this purpose, we use a stochastic scenario generator that allows us to simulate arbitrarily many upper claims reserving triangles of similar characteristics for which we also know the corresponding lower triangles. Based on these simulated triangles, we analyse the performance of the chain-ladder claims reserving method.

In the industry, generally, reserving actuaries use a mix of reserving methods to derive their best estimates. On the basis of the best estimate, Solvency 2 requires the use of a one-year volatility of the reserves. When internal models are used, such one-year volatility has to be provided by the reserving actuaries. Due to the lack of closed-form formulas for the one-year volatility of Bornhuetter-Ferguson, Cape-Cod and Benktander-Hovinen, reserving actuaries have limited possibilities to estimate such volatility apart from scaling from tractable models, which are based on other reserving methods. However, such scaling is technically difficult to justify cleanly and awkward to interact with. The challenge described in this editorial is therefore to come up with similar models like those of Mack or Merz-Wüthrich for the chain ladder, but applicable to Bornhuetter-Ferguson, mix Chain-Ladder and Bornhuetter-Ferguson, potentially Cape-Cod and Benktander-Hovinen — and their mixtures.

This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a ‘best estimate’; reporting the variability of claims reserves; and use in dynamic financial analysis models.

It has long been known that maximum likelihood estimation in a Poisson model reproduces the chain-ladder technique. We revisit this model. A new canonical parametrisation is proposed to circumvent the inherent identification problem in the parametrisation. The maximum likelihood estimators for the canonical parameter are simple, interpretable and easy to derive. The boundary problem where all observations in one particular development year or on particular underwriting year is zero is also analysed.

The separation method was introduced by Verbeek (1972) in order to forecast numbers of excess claims and it was developed further by Taylor (1977) to be applicable to the average claim cost. The separation method differs from the chain-ladder in that when the chain-ladder only assumes claim proportionality between the development years, the separation method also separates the claim delay distribution from influences affecting the calendar years, e.g. inflation. Since the inflation contributes to the uncertainty in the estimate of the claims reserve it is important to consider its impact in the context of risk management, too.

In this paper we present a method for assessing the prediction error distribution of the separation method. To this end we introduce a parametric framework within the separation model which enables joint resampling of claim counts and claim amounts. As a result, the variability of Taylor's predicted reserves can be assessed by extending the parametric bootstrap techniques of Björkwall et al. (2009). The performance of the bootstrapped separation method and chain-ladder is compared for a real data set.

The Benktander (1976) and Neuhaus (1992) credibility claims reserving methods are reconsidered in the framework of a credible loss ratio reserving method. As a main contribution we provide a simple and practical optimal credibility weight for combining the chain-ladder or individual loss ratio reserve (grossed up latest claims experience of an origin period) with the Bornhuetter-Ferguson or collective loss ratio reserve (experience based burning cost estimate of the total ultimate claims of an origin period). The obtained simple optimal credibility weights minimize simultaneously the mean squared error and the variance of the claims reserve. We note also that the standard Chain-Ladder, Cape Cod and Bornhuetter-Ferguson methods can be reinterpreted in the credible context and extended to optimal credible standard methods. The new approach is inspired from Mack (2000). Two advantages over the Mack method are worthwhile to be mentioned. First, a pragmatic estimation of the required parameters leads to a straightforward calculation of the optimal credibility weights and mean squared errors of the credible reserves. An advantage of the collective loss ratio claims reserve over the Bornhuetter-Ferguson reserve in Mack (2000) is that different actuaries come always to the same results provided they use the same actuarial premiums.

We develop a semi-parametric predictor of the IBNR reserve in a macro-model when the claim amount for a certain accident and development year can be expressed in a loglinear form composed of a deterministic part and a random error. We need to make assumptions only on the first two moments of the error, without any specified parametric assumption on its distribution. We give its properties, present its advantages and compare the estimates obtained with various predictors of the IBNR reserve, parametric and non-parametric, using a data set.