Assessing Individual Unexplained Variation in Non-Life Insurance

We consider variation of observed claim frequencies in non-life insurance, modeled by Poisson regression with overdispersion. In order to quantify how much variation between insurance policies that is captured by the rating factors, one may use the coefficient of determination, R2 (squared), the estimated proportion of total variation explained by the model. We introduce a novel coefficient of individual determination (CID), which excludes noise variance and is defined as the estimated fraction of total individual variation explained by the model. We argue that CID is a more relevant measure of explained variation than R2 (squared) for data with Poisson variation. We also generalize previously used estimates and tests of overdispersion and introduce new coefficients of individual explained and unexplained variance.

Application to a Swedish three year motor TPL data set reveals that only 0.5 percent of the total variation and 11 percent of the total individual variation is explained by a model with seven rating factors, including interaction between sex and age. Even though the amount of overdispersion is small (4.4 percent of the noise variance) it is still highly significant. The coefficient of variation of explained and unexplained individual variation is 29 percent and 81 percent respectively.

Keywords: claim frequency variation, coefficient of determination, coefficient of individual determination, unexplained individual variation, overdispersion, Poisson regression, rating factors.