In their daily reserving tasks, general insurance actuaries often use a mix of chain ladder, Bornhuetter-Ferguson, and Cape Cod methods to estimate the ultimate reserve amounts. These methods are usually applied on cumulative triangles and a payment or incurred pattern is derived from the application of the chain ladder method on these cumulative triangles. This pattern is then used in the Bornhuetter-Ferguson or Cape Cod method. Mack (2008) demonstrated that the stochastic model underlying this method should be based on incremental triangles. In addition, Saluz (2015) used a stochastic model for the Cape Cod method also based also on incremental triangles. Following on the works of Mack and Saluz, this paper will redevelop the chain ladder model on incremental triangles and unify the stochastic models of chain ladder, Bornhuetter-Ferguson, and Cape Cod into a single model. Based on this unified model, we will see that the first three moments (mean, variance and skewness) of the reserve risk distribution are defined by the relative position of the percentage of incremental claim to ultimate versus the claim pattern defining the best estimate. Such study of the position of incremental claims versus best estimate pattern should be looked at on the individual claims level in order to have all the information of the claims development. The second part of this paper will therefore focus on the ways in which such pattern study on individual claims can be done. As a conclusion, the moments of the reserve risk distribution will be derived using individual claims information. A numerical example of such a study on individual claims will illustrate how the moments of the reserving risk distribution can be estimated.
Eric Dal Moro (Mon,) studied this question.