The inappropriateness of the Poisson distribution for forecasting has long been recognised by many, and collective risk algorithms (Panjer (1908), Heckman & Meyers (1983)) have been developed that work just as well with other frequency distributions, in particular the Negative Binomial. However, to calibrate a Negative Binomial model requires two parameters, equivalent to specifying both mean and variance. The author believes that one reason for the prevalence of Poisson models is lack of knowledge about how to objectively quantify the variance as well as the mean. This paper aims to contribute in this area.
The main reasons why the variance should exceed the expected number of claims are identified as parameter estimation error, heterogeneity, contagion, and future exposure uncertainty. While all these factors have long been recognised by some practitioners, this paper provides a framework for their systematic analysis and quantification. A mathematical model is developed in which these concepts are precisely defined, and statistical methods are developed for the quantification of these factors from claim frequency data. The model also shows how these factors interact to produce the overall variance forecasts.
It is not claimed that the particular form of model presented will be appropriate in all circumstances, but where necessary, modifications will often be possible within the general framework presented here.
Keywords: Bayesian forecasting, claim frequency, collective risk model, contagion, heterogeneity, maximum likelihood estimation, negative binomial distribution, parameter uncertainty, Poisson distribution, risk loading.
