Exposure Offset in Claim-Frequency Models¶

Definition¶

For positive exposure $e_i$ and claim rate $\lambda_i=\exp(\boldsymbol{x}_i^{\top}\boldsymbol{\beta})$ $exp (x_{i}^{⊤} β)$ , a Poisson frequency model specifies $N_i\mid \boldsymbol{x}_i,e_i\sim\operatorname{Pois}(e_i\lambda_i)$ $Pois (e_{i} λ_{i})$ . Therefore $\log\mathbb{E}[N_i\mid \boldsymbol{x}_i,e_i]=\log e_i+\boldsymbol{x}_i^{\top}\boldsymbol{\beta}$ $x_{i}^{⊤} β$ . The known term $\log e_i$ is an offset, with coefficient fixed at one.

Interpretation¶

A group with twice as much insured time should have twice the expected claim count if its underlying claim rate is unchanged. The offset lets rating coefficients describe rate differences.

Expected claim count against exposure on log–log axes: a line of slope 1 through (10, 2) and (100, 20) for rate 0.2, where ten times the exposure gives ten times the count.

The figure draws the Examples case with rate $\lambda_i=0.2$ per policy-year. On log scales, $\log\mathbb{E}[N_i]=\log e_i+\boldsymbol{x}_i^{\top}\boldsymbol{\beta}$ is a straight line whose slope in $\log e_i$ is fixed at 1 by the offset, while the rating term $\boldsymbol{x}_i^{\top}\boldsymbol{\beta}=\log 0.2$ only sets its height. Multiplying exposure by 10, from 10 to 100, multiplies the expected count by 10, from 2 to 20, so the fitted rate stays 0.2.

Poisson regression relates the logarithm of an expected count to rating predictors. Exposure changes that count mean from $\lambda_i$ to $e_i\lambda_i$ , so its logarithm adds the known offset $\log e_i$ without estimating an extra coefficient.

Key Results¶

Expected count is $e_i\lambda_i$ ; fitted rate is count mean divided by $e_i$ .
An offset differs from an estimated covariate coefficient.
Exposure must be positive and measured in a consistent unit.
A zero-exposure record requires separate treatment; $\log0$ is undefined.

Examples¶

At rate 0.2 per policy-year, exposures 10 and 100 have expected counts 2 and 20. Both have the same fitted rate.

Prerequisites¶

Poisson Regression

Sources¶

Risk Insurance · Lecture 6

Aliases: log exposure offset, Poisson regression with exposure, exposure offset, offset term Keywords: log e_i, exposure in policy-years, claim rate per unit exposure, coefficient fixed at one Symbols: beta, lambda

Last updated: 14 October 2026 08:21