This paper studies estimation of the conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of run-off triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalisation of Akaike’s final prediction error. A direct application of this simple principle in the setting of a data-generating process given in terms of a sequence of general linear models yields an estimator of the conditional mean squared error of prediction that can be computed explicitly for a wide range of models within this model class. Mack’s distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount are shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data-generating processes and results in estimators that have been studied in the literature.

Although loss reserving has been deeply studied in the literature, there are still practical issues that have not been addressed a lot. One of them is the estimation of reserves during the year, which is necessary for forecasts or closings during the year. We will study the following question: What can be done for forecasts and closings during the year that goes along with the reserving at year end? In order to make it not too complicated, we will focus on the Chain-Ladder method introduced by Mack (1993). We will describe several methods that are used in practice. We will discuss advantages and disadvantages of these methods based on a simple deterministic example. Roughly spoken, we will see that you may shift development or accident periods, or may split development periods, but should not split accident periods.

We introduce a stochastic model for the development of attritional and large claims in long-tail lines of business and present a corresponding “chain ladder-like” IBNR method which allows the use of claims payment data for attritional and claims incurred data for large losses. We derive formulas for the mean squared error of prediction and apply the method to a German motor third party liability portfolio.

In this article we want to motivate and analyse a wide family of reserving models, called linear stochastic reserving methods (LSRMs). The main idea behind them is the assumption that the (conditionally) expected changes of claim properties during a development period are proportional to exposures which depend linearly on the past. This means the discussion about the choice of reserving methods can be based on heuristic reasons about exposures driving the claims development, which in our opinion is much better than a pure philosophic approach. Moreover, the assumptions of LSRMs do not include the independence of accident periods.

We will see that many common reserving methods, like the Chain-Ladder-Method, the Bornhuetter-Ferguson-Method and the Complementary-Loss-Ratio-Method, can be interpreted in this way. But using the LSRM framework you can do more. For instance you can couple different triangles via exposures. This leads to reserving methods which look at a whole bundle of triangles at once and use the information of all triangles in order to estimate the future development of each of them.

We will present unbiased estimators for the expected ultimate and estimators for the mean squared error of prediction, which may become an integral part of IFRS 4. Moreover, we will look at the one period solvency reserving risk, which already is an important part of Solvency II, and present a corresponding estimator.

Finally we will present two examples that illustrate some features of LSRMs.