Rating Factors and Interactions in Insurance GLMs¶

Definition¶

A rating factor is a covariate used in an insurance generalized linear model (GLM) to distinguish expected insured risk. A categorical factor with $k$ levels can be represented by $k-1$ indicator variables relative to a reference level when an intercept is included. An interaction allows the effect of one factor to depend on another.

Interpretation¶

Under a log link, additive predictor terms become multiplicative rate effects. An interaction changes a rate ratio across groups rather than simply adding another independent multiplier.

Each cell is the rate $\lambda=\exp(\beta_0+\beta_AA+\beta_BB+\beta_{AB}AB)$ for one combination of the binary factors. Moving from $A=0$ to $A=1$ multiplies the rate by $\mathrm{e}^{\beta_A}=1.2$ when $B=0$ $0$ , but by $\mathrm{e}^{\beta_A+\beta_{AB}}=1.2\times1.5=1.8$ when $B=1$ $1$ , the values in the example. Without the interaction term the two arrows would carry the same multiplier.

Move $\beta_A$ , $\beta_B$ and $\beta_{AB}$ : the effect of $A$ is $e^{\beta_A}$ when $B=0$ and $e^{\beta_A+\beta_{AB}}$ when $B=1$ $1$ . Set $\beta_{AB}=0$ and the effect of $A$ no longer depends on $B$ .

Poisson regression supplies the log-link interpretation of predictor coefficients as multiplicative count effects. With exposure offset in claim-frequency models, $\log e_i$ is separated from those rating effects so that the coefficients compare claim rates at equal exposure.

Key Results¶

In $\log\lambda=\beta_0+\beta_AA+\beta_BB+\beta_{AB}AB$ $β_{A B} A B$ , with binary $A,B$ , the effect of $A$ is $\mathrm{e}^{\beta_A+\beta_{AB}B}$ .
Including all category indicators plus an intercept makes the design redundant: the indicators sum to the intercept column, so the coefficients are not identifiable unless a constraint is imposed, such as dropping a reference level.
Reference categories determine coefficient interpretation but do not change fitted means under an equivalent encoding.
A fitted association does not establish a causal effect of a rating factor.

Examples¶

If $\beta_A=\log1.2$ and $\beta_{AB}=\log1.5$ $lo g 1.5$ , the rate ratio for $A$ is 1.2 when $B=0$ $0$ , and 1.8 when $B=1$ $1$ .

Prerequisites¶

Information Criteria for Risk Models

Sources¶

Risk Insurance · Lecture 6

Aliases: categorical rating factors, multiplicative rating effects, rating factor, rating variable, tariff variable Keywords: dummy coding, reference level, interaction term, rate ratio, k - 1 indicator variables, identifiability Symbols: beta, lambda

Last updated: 17 October 2026 21:53