Comparison of offset and ratio weighted regressions in tweedie models with application to mid-term cancellations

In property and casualty insurance, particularly in automobile insurance, risk exposure is commonly assumed to be proportional to the duration of coverage. This assumption leads to two standard estimation strategies: the ratio approach, which normalizes the response variable (e.g., claim cost or premium) by the exposure, and the offset approach, which incorporates a transformation of the exposure (typically its logarithm) as a fixed regressor in the mean structure of the model. Although both approaches rely on the same proportionality assumption, they are not equivalent when the response variable follows a Tweedie distribution, a framework widely used in insurance analytics. In this paper, we show that each approach can be implemented independently and yields a consistent estimator of the true mean parameter vector. We then show that the offset approach is asymptotically more efficient than the ratio approach, a result established both theoretically and through simulation studies. However, when evaluated from the perspective of portfolio-level financial balance, the ratio approach exhibits superior performance, particularly in the presence of heterogeneous or truncated exposures arising from mid-term policy cancellations. These theoretical results are illustrated through an empirical analysis of an automobile insurance portfolio with a high cancellation rate, highlighting the practical implications of model choice for premium estimation under variable exposure conditions.