Worked Gamma Model for Claim Severities

Why a Gamma Severity GLM?

Claim severities are positive amounts, and their variability often increases as their expected value increases. A Gamma generalized linear model (GLM) represents this pattern by modeling a positive conditional mean and making the conditional variance proportional to the square of that mean. With a log link, changes in rating characteristics act multiplicatively on expected severity, which makes fitted coefficients interpretable as severity relativities.

The response must first be defined consistently. In this tutorial, each observation is one claim, and the response $Y$ is the positive amount for that claim under a specified severity target. The examples use complete, uncensored claim amounts and equal observation weights. They do not include claims selected by a deductible or censored at a limit; those sampling and response definitions require a likelihood consistent with the observed data.

Definition: A Gamma severity GLM with a log link specifies $Y_j\mid \mathbf{x}_j$ as Gamma distributed with conditional mean $\mu_j>0$ and conditional variance $\phi\mu_j^2$, where $\phi>0$ is the dispersion parameter. The linear predictor is $$ \log(\mu_j)=\eta_j=\mathbf{x}_j^\mathsf{T}\boldsymbol{\beta}, $$ so $\mu_j=\exp(\eta_j)$. Claims are assumed conditionally independent given their predictors, and the model uses one common dispersion parameter unless a different dispersion structure is specified.

One standard Gamma parameterization uses shape $\alpha=1/\phi$ and scale $\theta_j=\phi\mu_j$. Under this parameterization, the conditional mean is $\alpha\theta_j=\mu_j$, and the conditional variance is $\alpha\theta_j^2=\phi\mu_j^2$. The Gamma distribution is positive and continuous, so it is not suitable for a response that can equal zero without an additional modeling treatment.

Fitting and Interpreting a Small Model

For a categorical predictor, a coefficient is a difference in log mean relative to the chosen reference level. Exponentiating the coefficient converts that difference into a ratio of expected severities. With more than one predictor, the linear predictor adds the relevant coefficients, while the expected mean is the product of their exponentiated effects.

Worked Example: Fit a Two-Factor Gamma Severity Model

A hypothetical insurer models complete claim severities in dollars using territory and vehicle class. Territory A and standard class are the reference levels. The response for each row is the individual claim's positive severity amount, with no deductible selection or limit censoring. The 12 observed claims are:

Territory	Vehicle class	Three claim severities
A	Standard	$900, $1,000, $1,100
A	High	$1,350, $1,500, $1,650
B	Standard	$1,080, $1,200, $1,320
B	High	$1,620, $1,800, $1,980

Fit a model with an intercept, territory indicator $T=1$ for B, class indicator $H=1$ for High, and their interaction:

\log(\mu)=\beta_0+\beta_TT+\beta_HH+\beta_{TH}TH.

Step 1: Calculate the four sample means. The means by cell are:

\begin{aligned} \bar{Y}_{A,\mathrm{Standard}}&=(900+1{,}000+1{,}100)/3=1{,}000,\\ \bar{Y}_{A,\mathrm{High}}&=(1{,}350+1{,}500+1{,}650)/3=1{,}500,\\ \bar{Y}_{B,\mathrm{Standard}}&=(1{,}080+1{,}200+1{,}320)/3=1{,}200,\\ \bar{Y}_{B,\mathrm{High}}&=(1{,}620+1{,}800+1{,}980)/3=1{,}800. \end{aligned}

Step 2: Convert the cell means into coefficients. The reference mean is $\exp(\widehat{\beta}_0)=1{,}000$, so $\widehat{\beta}_0=\log(1{,}000)=6.907755$. The territory effect is the log ratio $1{,}200/1{,}000=1.2$, and the class effect is the log ratio $1{,}500/1{,}000=1.5$. The B-High cell has mean $1{,}800$, exactly the product $1{,}000(1.2)(1.5)$, so the interaction coefficient is zero.

\widehat{\beta}_T=\log(1.2)=0.182322,\qquad \widehat{\beta}_H=\log(1.5)=0.405465,\qquad \widehat{\beta}_{TH}=0.

Thus the fitted model is $\log(\widehat{\mu})=6.907755+0.182322T+0.405465H$. Its fitted cell means are $1,000, $1,200, $1,500, and $1,800, matching the four observed cell means.

Step 3: Estimate dispersion using the Pearson statistic. For each cell, the observations are $0.9m,m,1.1m$, where $m$ is that cell's fitted mean. The cell's Pearson contribution is $((-0.1m)^2+0^2+(0.1m)^2)/m^2=0.02$. There are four cells, so the total Pearson statistic is $4(0.02)=0.08$. Four coefficients have been estimated from 12 observations, leaving $12-4=8$ residual degrees of freedom. Therefore,

\widehat{\phi}_{P}=\frac{0.08}{8}=0.01.

The fitted model gives a territory B versus A severity relativity of $\exp(0.182322)=1.2$, and a High versus Standard relativity of $\exp(0.405465)=1.5$, holding the other predictor fixed. The residual-dispersion estimate is 0.01 under the specified Pearson method.

Why the Cell Means Give These Estimates

The cell-mean result in the example follows from the Gamma likelihood when the model has a separate fitted mean for each cell. It is not a rule that the mean of any arbitrary subset of claims must equal a fitted GLM mean: the predictor structure and likelihood determine the fit.

For a Gamma observation with mean $\mu$ and dispersion $\phi$, the log-likelihood contribution, apart from terms not depending on $\mu$, can be written as $(1/\phi)(-\log\mu-y/\mu)$. Under a log link, $\eta=\log\mu$, so differentiating with respect to $\eta$ gives the score contribution $(1/\phi)(y/\mu-1)$. For a cell with its own intercept, the likelihood score equation sets the sum of these contributions to zero:

\sum_{j=1}^{n}\left(\frac{y_j}{\mu}-1\right)=0 \quad\Longrightarrow\quad \frac{\sum_{j=1}^{n}y_j}{\mu}=n \quad\Longrightarrow\quad \widehat{\mu}=\frac{1}{n}\sum_{j=1}^{n}y_j.

The factor $1/\phi$ cancels from the score equation. In a saturated categorical model, each cell has its own fitted mean, so this argument applies cell by cell. The corresponding log-link coefficients are then obtained by taking logarithms of the fitted means and forming the contrasts implied by the reference coding.

Coefficients, Relativities, and Contrasts

A coefficient itself is on the log-mean scale, not the severity ratio scale. For a binary predictor with coefficient $\beta$, changing that predictor from 0 to 1 multiplies the conditional mean by $\exp(\beta)$, provided other model terms do not change. If the model includes an interaction, the effect of changing one predictor depends on the level of the interacting predictor.

Worked Example: Convert a Main Effect and Interaction to Relativities

Suppose a separate hypothetical fitted model has reference expected severity $900 and coefficients $\widehat{\beta}_T=\log(1.25)$, $\widehat{\beta}_H=\log(1.4)$, and $\widehat{\beta}_{TH}=\log(0.9)$, using the same indicator definitions as above. Find the expected severity for a B-High claim and the relativity of B versus A within the High class.

Step 1: Calculate the reference mean. By definition, the reference case has $T=0,H=0$, so its mean is the model's intercept-scale mean, $900.

Step 2: Combine the B-High effects. With both indicators equal to 1, the mean is multiplied by all three factors, including the interaction:

\widehat{\mu}_{B,\mathrm{High}} =900(1.25)(1.4)(0.9) =900(1.575) =\$1{,}417.50.

Step 3: Compare B with A while holding class at High. For B-High, the factors involving territory are $1.25$ and $0.9$. For A-High, the territory indicator and interaction are zero, so only the class factor applies. The within-High territory relativity is therefore $(1.25)(0.9)=1.125$. This agrees with the direct means: B-High has predicted mean $1,417.50, while A-High has mean $900(1.4)=\$1,260$, and $1{,}417.50/1{,}260=1.125$.

Check. The B-High prediction exceeds the A-High prediction by 12.5%, as the calculated within-High relativity indicates. Including the interaction is essential; applying the territory main effect alone would give the wrong contrast.

Claim-Level Prediction and Conditional Variability

A fitted claim-level mean is the model's estimate of the conditional average severity for claims with specified predictors. It is not a prediction that the next observed claim will equal that mean. Under the Gamma assumption, the claim amount varies around the mean, with variance determined by both the fitted mean and dispersion. The Pearson estimate of dispersion is commonly calculated as Pearson residual sum of squares divided by residual degrees of freedom; it is an estimate, not a fixed known parameter.

Worked Example: Predict a Claim Mean and Its Gamma Variability

Use the fitted model from the first example for a hypothetical B-High claim. Calculate its conditional mean, variance, standard deviation, and Gamma shape and scale, using the unrounded dispersion estimate.

Step 1: Calculate the conditional mean. The fitted mean for B-High is the reference mean times both relativities: $1{,}000(1.2)(1.5)=\$1{,}800.00$.

Step 2: Calculate conditional variance and standard deviation. With $\widehat{\phi}=0.01$, the estimated conditional variance is $0.01(1{,}800)^2=0.01(3{,}240{,}000)=\$32{,}400.00$ squared. Its square root is $\$180.00$.

Step 3: Express the fitted Gamma distribution. Using shape $1/\widehat{\phi}$ and scale $\widehat{\phi}\widehat{\mu}$, the shape is $1/0.01=100$, and the scale is $0.01(1{,}800)=\$18.00$. The mean check is $100(18)=\$1{,}800$; the variance check is $100(18)^2=\$32{,}400$. Thus, under this fitted conditional distribution, the claim severity is modeled as Gamma with shape 100 and scale $18.

The result describes uncertainty in an individual claim amount conditional on its modeled characteristics. It does not include uncertainty in the estimated coefficients or dispersion.

Common Mistakes and Model Checks

Using a coefficient as a relativity. If a fitted coefficient is $0.182322$, the relativity is $\exp(0.182322)=1.2$, not 0.182322.
Adding relativities instead of multiplying them. Under a log link, two active main effects with factors 1.2 and 1.5 produce a combined factor $1.2(1.5)=1.8$, not $1+0.2+0.5=1.7$. Include any applicable interaction factor as well.
Using the wrong residual degrees of freedom. Pearson dispersion is not the Pearson statistic divided by the number of claims. In the first example, the denominator is $12-4=8$, accounting for the four estimated coefficients.
Confusing a conditional mean with an observed claim amount. The prediction $\$1{,}800$ is an expected severity; it does not imply that each B-High claim will be $1,800.
Ignoring how the response was observed. A complete claim severity, a deductible-selected amount, and a limit-censored observation do not necessarily have the same likelihood. A standard complete-observation Gamma fit should not be applied without checking that its response and sampling assumptions match the data.

Key takeaway: For a Gamma severity GLM with a log link, $\exp(\mathbf{x}^{\mathsf T}\boldsymbol{\beta})$ is the conditional mean and exponentiated coefficient contrasts are severity relativities. The dispersion controls conditional variability through $\operatorname{Var}(Y\mid\mathbf{x})=\phi\mu^2$; it does not change the meaning of the fitted mean.

A useful validation check is to compare fitted means with observed averages across relevant predictor groups and to inspect Pearson residuals for systematic patterns or unusually large values. A sensible coefficient interpretation does not by itself establish that the Gamma variance assumption or the predictor structure is adequate. The model remains conditional on the chosen response definition, claim sample, and predictors, and should be interpreted within those limits. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

Work each problem using the stated response, link, and dispersion convention.

A one-factor Gamma log-link model has a reference fitted mean of $700 and a fitted category coefficient of $\log(1.3)$. What is the category's fitted mean and its relativity to the reference? (Answer: mean $910.00; relativity 1.300)
A one-mean Gamma fit has four complete claim severities: $400, $600, $450, and $550. The fitted mean is $500. Estimate dispersion using the Pearson statistic divided by residual degrees of freedom. (Answer: $\widehat{\phi}_{P}=0.0333$, approximately)
A log-link model has a coefficient of $-0.223144$ for a binary indicator, with no interaction. If the reference mean is $1,250, find the indicator's relativity and fitted mean. (Answer: relativity 0.8000; fitted mean $1,000.00)
A fitted claim-level Gamma model has mean $2,000 and dispersion 0.02. Calculate its conditional variance, standard deviation, shape, and scale using shape $1/\phi$ and scale $\phi\mu$. (Answer: variance $80,000.00 squared; standard deviation $282.84; shape 50; scale $40.00)
A reference mean is $1,600. For a particular class, two active main effects have relativities 1.1 and 1.25, while their interaction has relativity 0.8. Find the class's predicted mean and explain why the interaction must be included. (Answer: $1,760.00; multiply all applicable factors, $1{,}600(1.1)(1.25)(0.8)$)

What You'll Learn