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

TL;DR

This paper compares offset and ratio weighted regressions in Tweedie models for insurance data, showing their theoretical properties, efficiency differences, and practical implications in portfolios with high cancellations.

Contribution

It demonstrates that both approaches are consistent estimators, with the offset approach being more efficient, and highlights the ratio approach's advantages in portfolio-level financial balance.

Findings

Offset approach is asymptotically more efficient than ratio approach.

Both methods provide consistent estimators under the proportionality assumption.

Ratio approach performs better in portfolios with heterogeneous or truncated exposures.

Abstract

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.