On Duration Effects in Non-Life Insurance Pricing

In this short note we discuss duration effects on the consistency of mean parameter and dispersion parameter estimators in exponential dispersion families (EDFs). Particular focus is on the situation corresponding to the standard generalised linear model assumptions where both the mean and variance, conditional on duration, are linear functions in terms of duration. We derive simple convergence results that highlight consequences when the linear conditional moment assumption is not satisfied. These results illustrate that: 1) the resulting mean estimators always have a relevant asymptotic interpretation in terms of the duration adjusted actuarially fair premium; 2) deviance based estimators of the dispersion parameter in an EDF should be avoided in favour of Pearson estimators; 3) when the linear moment assumption is not satisfied, the natural plug-in estimator of the variance of the response variable will result in spurious over-dispersion. A simple way to circumvent the issues in 3) is to estimate the variance of the response directly, without using specific duration based moment assumptions