The Mack chain ladder and data granularity for preserved development periods

This paper is concerned with the choice of data granularity for the application of the Mack chain ladder model to forecast a loss reserve. It is a sequel to a related paper by Taylor, which considers the same question for the EDF chain ladder model. As in the earlier paper, it considers the question as to whether a decrease in the time unit leads to an increase or decrease in the variance of the loss reserve estimate. The question of whether a Mack chain ladder that is valid for one time unit (here called mesh size) remains so for another is investigated. The conditions under which the model does remain valid are established. There are various ways in which the mesh size of a data triangle may be varied, two of them of particular interest. The paper examines one of these, namely that in which development periods are preserved. Two versions of this are investigated: 1. the aggregation of development periods without change to accident periods; 2. the aggregation of accident periods without change to development periods. Taylor found that, in the case of the Poisson chain ladder, an increase in mesh size always increases the variance of the loss reserve estimate (subject to mild technical conditions). The case of the Mack chain ladder is more nuanced in that an increase in variance is not always guaranteed. Whether or not an increase or decrease occurs depends on the numerical values of certain of the age-to-age factors actually observed. The threshold values of the age-to-age factors at which an increase transitions to a decrease in variance are calculated. In the case of a change in the mesh of development periods, but with no change to accident periods, these values are computed for one particular data set, where it is found that variance always increases. It is conjectured that data sets in which this does not happen would be relatively rare. The situation is somewhat different when changes in mesh size over accident periods are considered. Here, the question of an increase or decrease in variance is more complex, and, in general terms, the occurrence of an increase in variance with increased mesh size is less likely.