GLMs and Predictive Modeling in Pricing

Generalized linear models are the workhorse of modern rating plan construction: they let an actuary estimate a multiplicative rating structure — base rate times relativities — directly from policy-level data while correctly handling non-normal, skewed claims data and unequal exposures.

Why GLMs, and the Tweedie distribution

Insurance loss data is right-skewed with a point mass at zero (most policies have no claims). A GLM generalizes ordinary least squares by allowing (1) a distribution from the exponential family appropriate to the data's variance-mean relationship, and (2) a link function $g(\cdot)$ connecting the linear predictor to the mean:

$g(E[Y_i]) = \mathbf{x}_i^\top\boldsymbol\beta.$

For pure premium (losses per exposure), the Tweedie distribution with power parameter $1<p<2$ is standard: it is a Poisson-compound-Gamma mixture, meaning it naturally represents "frequency of claims, each with Gamma-distributed severity," and produces exactly the zero-inflated, right-skewed shape of aggregate loss data in one model, avoiding the need to fit frequency and severity separately (though a frequency/severity split, modeling counts with Poisson and severity with Gamma separately, remains common and often more interpretable, especially when different rating variables drive frequency vs. severity).

The Tweedie variance function is $\mathrm{Var}(Y_i) = \phi\,\mu_i^p$ — variance grows faster than the mean for $p>1$, matching insurance data's tendency for large losses to also be more volatile.

Log link and offsets

Nearly all pricing GLMs use the log link, $g(\mu)=\ln(\mu)$, because it produces a multiplicative rating structure — exactly the structure of a rating manual:

$\mu_i = \exp(\mathbf{x}_i^\top\boldsymbol\beta) = \exp(\beta_0)\times\exp(\beta_1 x_{i1})\times\exp(\beta_2 x_{i2})\times\cdots,$

so each categorical level's coefficient exponentiates directly into a relativity relative to a base level (the reference class, coefficient folded into $\beta_0$).

An offset enters the linear predictor with a coefficient fixed at 1 (not estimated), typically $\ln(\text{exposure})$, so the model predicts a rate per unit exposure while still using total exposure to weight the likelihood correctly — critical when policies have varying terms (e.g., a 6-month vs. 12-month policy). Offsets are also used to hold a previously-selected variable's effect fixed while fitting the remainder of the model (e.g., holding a known territory relativity fixed while re-fitting all other rating variables).

Extracting relativities

Given fitted coefficients, the relativity for level $j$ of a categorical variable versus its base level is simply $r_j = \exp(\hat\beta_j)$. For a continuous variable modeled linearly, a one-unit increase multiplies the mean by $\exp(\hat\beta)$; these are often converted back into a step or banded rating table for filing simplicity. A full multiplicative rate is then $\text{Rate} = \text{Base Rate}\times\prod_j r_j$ across all rating variables — a structure regulators recognize and can audit variable-by-variable, an important practical advantage of GLMs over black-box models.

Validation: lift charts and Gini

A fitted model is validated on holdout data (or by cross-validation) using:

• Lift charts: sort policies by predicted pure premium into deciles and plot actual vs. predicted average loss per decile — a well-fit model shows a monotonic, steep actual-loss gradient across deciles (strong "lift"), meaning the model successfully separates good risks from bad.
• Double lift charts: compare two competing models (e.g., current GLM vs. GBM challenger) by sorting on the ratio of their predictions, revealing which model better discriminates where they disagree.
• Normalized Gini coefficient: derived from the Lorenz-curve-like ordered Lorenz curve of losses vs. premium when policies are sorted by model score; a higher Gini indicates better discriminatory power, and it is now a standard single-number model comparison metric in CAS predictive modeling literature.

GBM vs. GLM

Gradient boosted trees (GBMs) and other machine-learning challengers typically achieve higher raw predictive lift than GLMs because they automatically capture non-linearities and interactions the GLM's linear predictor does not include unless explicitly specified. However, GLMs retain three practical advantages that keep them dominant for filed rating plans: (1) transparency — coefficients map directly to filed relativities regulators can review; (2) stability — smooth, monotonic, low-variance relativities that don't overfit sparse cells; (3) ease of implementation in legacy rating engines. A common workflow uses a GBM as an exploratory tool — to discover interactions and non-linear transformations (e.g., a spline for age) — which are then engineered as new features into the production GLM, capturing much of the GBM's lift while keeping an interpretable, filable model.

Interpretability and ASOP 56

ASOP No. 56 (Modeling) applies to any actuarial model, including predictive models, and requires the actuary to: understand the model sufficiently to rely on it (a "black box" model still requires the actuary to understand its inputs, assumptions, sensitivities, and limitations even without deriving its internals); evaluate whether it is fit for its intended purpose; assess reasonableness of output; and disclose reliance on other experts (e.g., a data science team) or software. This obligation applies with equal force whether the model is a GLM or a GBM — the standard is about the actuary's understanding and governance of the model in context, not about the specific algorithm used.

Worked mini-example

A frequency GLM with log link and Poisson error, offset $\ln(\text{car-years})$, has fitted coefficients (relative to a "Suburban, Sedan" base cell): $\hat\beta_0=-2.30$ (base cell log-frequency), $\hat\beta_{Urban}=0.22$, $\hat\beta_{SUV}=-0.05$. Base cell predicted frequency $=\exp(-2.30)=0.1003$ claims/car-year. An Urban SUV risk has predicted frequency $\exp(-2.30+0.22-0.05)=\exp(-2.13)=0.1188$, i.e., a combined relativity of $\exp(0.22)\times\exp(-0.05)=1.246\times0.951=1.185$ relative to the base — a 18.5% frequency load applied multiplicatively, exactly reproducible in a rating manual.

Pitfalls

• Omitting the exposure offset, which forces exposure to be treated as an ordinary predictor and distorts every other coefficient.
• Reading off relativities from a model with strong multicollinearity between correlated rating variables (e.g., age and years-licensed) without checking variance inflation.
• Treating high lift/Gini as sufficient justification alone — ASOP 56 requires broader review of stability, business reasonableness, and disparate-impact considerations, not just predictive accuracy.
• Overfitting sparse interaction cells in a GBM and porting them uncritically into a filed GLM without credibility-weighting or smoothing.

Exam relevance

GLM structure, Tweedie/Poisson/Gamma model families, and predictive model validation are core to the CAS Modern Actuarial Statistics (MAS-I/MAS-II) exams and CAS Exam 8; ASOP 56 is directly tested on the CAS Course on Professionalism.

Further reading

• Goldburd, Khare & Tevet, Generalized Linear Models for Insurance Rating, CAS Monograph 5.
• Anderson et al., A Practitioner's Guide to Generalized Linear Models, CAS.
• ASOP No. 56, Modeling, ASB.