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The present paper provides a unifying survey of some of the most important methods of loss reserving based on run-off triangles and proposes the use of a family of such methods instead of a single one.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.The notion of a development pattern turns out to be a unifying force in the comparison of various methods of loss reserving, including the chain-ladder method, the loss-development method, the Cape Cod method, and the additive method. For each of these methods, the predictors of the ultimate losses can be given the shape of Bornhuetter-Ferguson predictors.The process of arranging known methods of loss reserving under the umbrella of the extended Bornhuetter-Ferguson method requires the identification of prior estimators of the development pattern and the expected ultimate losses. This process can be reversed by combining components of different methods to obtain new versions of the extended Bornhuetter-Ferguson method.The Bornhuetter-Ferguson principle proposes the simultaneous use of various versions of the extended Bornhuetter-Ferguson method and a comparison of the resulting predictors in order to select best predictors and to determine prediction ranges.

Zusammenfassung: In diesem Artikel wird der Weg von einem univariaten gemischten Poisson-Prozess, der in vielen Bereichen zum Zählen von Ereignissen benutzt wird, zu einem bivariaten gemischten Poisson-Prozess aufgezeigt. Dazu werden einige Eigenschaften des bivariaten Prozesses angegeben. Im zweiten Teil der Arbeit wird gezeigt, wie mit Hilfe dieses Prozesses der Übergang von einem herkömmlichen Bonus-Malus-System in der Kraftfahrthaftpflichtversicherung zu einem Bonus-Malus-System mit Berücksichtigung der Schadenart beschritten werden kann. Dazu wird zuerst eine Modellprüfung der gegebenen Daten vorgenommen und sodann werden für verschiedene mischende Verteilungen die Verteilungsparameter geschätzt und Nettoprämien angegeben sowie die Prognosegenauigkeit getestet.Summary: In this paper we show that the model of the bivariate mixed Poisson process arises in a natural way from the univariate mixed Poisson process, which is used in several areas for counting certain events. Furthermore we state some properties of the bivariate process. In the second part of the paper we illustrate how by means of the bivariate mixed Poisson process a bonus-malus system handling different types of accidents can be derived from the classical bonus-malus system in third-party liability insurance. To this end we first check the model on the given data and then estimate distribution parameters and compute net premiums for different mixing distributions as well as test the prediction probabilities.

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