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

Boucher Jean-Philippe, Coulibaly Raïssa

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.

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

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.

Paper Structure

This paper contains 34 sections, 2 theorems, 53 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Examples for claim counts
  3. Structure of the paper
  4. GLM with Tweedie-distributed Response Variable
  5. Exponential Family Form of Tweedie Distribution
  6. Notations
  7. Assumptions and properties
  8. Log-likelihood Function
  9. Iterative Algorithm
  10. Remarks
  11. The Offset and the Ratio Approaches
  12. Offset Approach
  13. Ratio Approach
  14. Link between Offset and Ratio Approaches
  15. Weights Analysis
  16. ...and 19 more sections

Key Result

Proposition 1

According to white1982maximum, if the Kullback--Leibler divergence between the true density $g$ and the density $f$ (assumed under either the offset or the ratio approach) exists, then the corresponding estimator of the true parameter vector is consistent under both approaches. More formally, if then where $\bm{\widehat{\beta}}^{O}$ and $\bm{\widehat{\beta}}^{R}$ denote the estimators of $\bm{\b

Figures (9)

  • Figure 1: Comparison of the Weight Parameters in the Ratio and Offset Approaches for Different Risk Exposures ($t$)
  • Figure 2: Financial balance density for $n = 1{,}000$, $5{,}000$, $20{,}000$, and $50{,}000$ under the offset and ratio approaches.
  • Figure 3: Individual observed gaps: single heterogeneous realization (left) and across multiple heterogeneous realizations (right).
  • Figure 4: Distribution of risk exposures
  • Figure 5: Loss cost references by risk exposures
  • ...and 4 more figures

Theorems & Definitions (10)

  • Example 1
  • proof
  • Example 2
  • proof
  • Definition 2.1.1
  • Definition 2.4.1
  • Proposition 1
  • proof
  • Proposition 2
  • proof