Methods and Models of Loss Reserving Based on Run-Off Triangles: A Unifying Survey

The present paper provides a unifying survey of some of the most important methods and models of loss reserving which are based on run-off triangles. The starting point is the thesis that the use of run-off triangles in loss reserving can be justified only under the assumption that the development of the losses of every accident year follows a development pattern which is common to all accident years. This assumption can be viewed as a primitive stochastic model of loss reserving. The notion of a development pattern turns out to be a unifying force in the comparison of methods which to a large extent can be summarized under a general version of the Bornhuetter-Ferguson method. It is shown that the loss-development method and the chain-ladder method as well as the Cape-Cod method and the additive method can be viewed as special cases of the general Bornhuetter-Ferguson method. Some of these methods can be justified by general principles of statistical inference applied to suitable and more sophisticated stochastic models. It is shown that credibility prediction and Gauss-Markov prediction as well as maximum-likelihood estimation can contribute in a substantial way to the understanding of various methods of loss reserving.