Segmentation Choices and Fairness Consequences

Why Segmentation Choices Matter

A rating plan groups risks and assigns different prices to those groups. A broad classification may produce stable averages but conceal real differences in expected cost. A highly segmented plan can better reflect supported differences, yet it can also make premiums depend on thin data or on variables that raise fairness concerns. The choice is therefore not simply whether a variable improves fit: the actuary must also ask what evidence supports the distinctions and what their consequences are.

This tutorial focuses on how grouping, rating relativities, and interactions can change subgroup premiums. It also considers how to distinguish a potentially meaningful cost difference from a noisy estimate. This is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: A segmentation choice determines which risks are grouped together and which rating variables distinguish them. A rating relativity is a multiplicative factor applied to a reference rate for a rating class. An interaction occurs when the effect of one variable on indicated cost depends on the value of another variable.

From Grouped Experience to Premium Relativities

For a group with total losses $L$ and exposure $E$, the observed pure premium is $L/E$, provided losses and exposures are measured on a consistent basis. It estimates average loss cost per exposure for that group; it is not automatically a final rate. Expenses, contingencies, credibility procedures, and other adjustments may also be relevant to a filed or implemented rate.

If two groups have indicated pure premiums $P_A$ and $P_B$, their relative cost indication is $P_A/P_B$. A rating plan can express that ratio as a relativity, often with one class chosen as the reference. The relativity describes the modeled cost relationship; the dollar premium also depends on the base rate and the rest of the rating plan.

Key result: A finer split changes indicated subgroup premiums when the subgroup cost estimates differ. Whether the split is defensible depends not only on the difference in estimates, but also on their credibility, stability, relevance to expected cost, and fairness consequences.

Worked Example: Comparing a Pooled Rate With Two Subgroup Rates

A hypothetical insurer observes two policy groups with the same coverage and exposure definition. Group A has 1,000 earned exposures and 20 claims; Group B has 1,000 earned exposures and 30 claims. Assume each claim has a fixed amount of $5,000, so claim counts follow independent Poisson distributions and severity is nonrandom. Calculate the pooled and subgroup pure premiums, then assess how precisely the subgroup difference is estimated.

Method. Under the stated assumptions, total loss is claim count multiplied by $5,000. The pure premium is total loss divided by exposure.

Group A has losses of $20\times\$5{,}000=\$100{,}000$, so its pure premium is $\$100{,}000/1{,}000=\$100$ per exposure. Group B has losses of $30\times\$5{,}000=\$150{,}000$, so its pure premium is $\$150{,}000/1{,}000=\$150$ per exposure.

The pooled pure premium is

\frac{\$100{,}000+\$150{,}000}{1{,}000+1{,}000} =\frac{\$250{,}000}{2{,}000} =\$125.

With the pooled rate as reference, Group A’s relativity is $\$100/\$125=0.80$, and Group B’s is $\$150/\$125=1.20$. Thus the split indicates premiums 20% below and 20% above the pooled amount, respectively.

To gauge process uncertainty, the Poisson standard error of a claim frequency estimate $N/E$ is $\sqrt{N}/E$. Multiplying by the fixed severity converts it to a pure-premium standard error. For A this is $(\sqrt{20}/1{,}000)\times\$5{,}000\approx\$22.36$; for B it is $(\sqrt{30}/1{,}000)\times\$5{,}000\approx\$27.39$. Independence gives a standard error for the difference of

\sqrt{(\$22.3607)^2+(\$27.3861)^2} =\sqrt{500+750} =\sqrt{1{,}250} \approx \$35.36.

The estimated difference is $\$150-\$100=\$50$, about $50/35.36=1.41$ standard errors. This is a modest signal under the stated process model, not compelling evidence by itself that the groups should always receive different rates. The calculation excludes uncertainty in severity and any uncertainty from selecting the groups or model.

Check. The exposure-weighted subgroup premiums reproduce the pooled amount: $(1{,}000\times\$100+1{,}000\times\$150)/2{,}000=\$125$. The method avoids treating the observed $50 difference as certain merely because it can be calculated.

Interactions Change the Meaning of a Relativity

A multiplicative rating model without an interaction applies the same factor for a variable across every level of other modeled variables. For example, a territory factor of 1.4 means a 40% increase in each usage class, if the model contains no territory-by-usage interaction. An interaction allows that relationship to vary by class.

For two binary indicators $x$ and $z$, a log-link model for expected cost per exposure can be written as

\log(\mu_{xz})=\beta_0+\beta_1x+\beta_2z+\beta_{12}xz, \qquad \mu_{xz}=\exp(\beta_0+\beta_1x+\beta_2z+\beta_{12}xz).

Here $\mu_{xz}$ is expected cost per exposure. The interaction multiplier is $\exp(\beta_{12})$. If it equals one, the model has no interaction. If it differs from one, the joint effect is not obtained simply by multiplying the two main-effect relativities.

Why the interaction is a ratio of ratios: In the model above, the effect of changing $x$ from 0 to 1 when $z=0$ is $\mu_{10}/\mu_{00}=\exp(\beta_1)$. When $z=1$, it is $\mu_{11}/\mu_{01}=\exp(\beta_1+\beta_{12})$. Dividing the second ratio by the first gives $\exp(\beta_{12})$. The interaction therefore measures how much the $x$-relativity changes across the two values of $z$.

Worked Example: An Interaction Changes the Joint-Class Premium

A hypothetical portfolio has four equally exposed cells: Zone A and personal use; Zone B and personal use; Zone A and commercial use; Zone B and commercial use. Suppose supported indicated costs for the four cells are, respectively, $500, $700, $800, and $1,400 per exposure. Take Zone A personal use as the reference. Calculate the main-effect relativities, the interaction, and the implication of omitting it.

Method. Divide each cell indication by the $500 reference cost. The Zone B factor among personal-use risks is $\$700/\$500=1.40$. The commercial-use factor in Zone A is $\$800/\$500=1.60$. Without an interaction, the joint prediction for Zone B commercial use is

\$500\times1.40\times1.60=\$1{,}120.

The indicated joint cost is $1,400, so the interaction multiplier is $\$1{,}400/\$1{,}120=1.25$. In this hypothetical illustration, the joint effect is 25% higher than the product of the two main effects. The four relativities to the reference are 1.00, 1.40, 1.60, and 2.80.

If the interaction is omitted, the model predicts total cost of $\$500+\$700+\$800+\$1{,}120=\$3{,}120$ across one exposure in each cell. The stated indications total $\$500+\$700+\$800+\$1{,}400=\$3{,}400$. The difference is $280 across the four exposures. A global rate-level adjustment could change all four predictions, but would not remove the omitted model’s incorrect relative relationship between the cells.

Check. The implied Zone B factor for commercial use is $\$1{,}400/\$800=1.75$; dividing by the personal-use Zone B factor of 1.40 gives $1.75/1.40=1.25$, the same interaction multiplier. The method avoids assuming that main effects combine multiplicatively when the cell evidence indicates otherwise.

Credibility and the Cost of Finer Groups

A subdivision can reveal useful differences, but smaller cells usually contain less experience. Random variation can then produce apparently large relativities that are unstable from one period to the next. Confidence in a difference should reflect the amount and quality of data, not just the point estimates.

One simple diagnostic is to compare the estimated difference with its standard error under a stated process model, as in the first example. A credibility blend offers another way to temper thin experience. For illustration, let $P_g$ be a group’s observed pure premium, $P_0$ an a priori expected premium, and $Z_g$ a credibility weight between zero and one. The indicated blended premium is $Z_gP_g+(1-Z_g)P_0$. The method and its parameters must be justified; a chosen credibility formula is not a substitute for examining whether a rating distinction is appropriate.

Worked Example: Blending a Thin Subgroup Indication

A hypothetical class has two subgroups. Subgroup A has 100 exposures and 1 claim; subgroup B has 1,900 exposures and 39 claims. Assume independent Poisson claim counts and fixed claim severity of $5,000. For this illustration only, use $Z=n/(n+K)$ with $K=400$ exposures and an a priori pure premium $P_0=\$100$ for each subgroup. Calculate observed and blended indications.

Method. Calculate claim frequency times fixed severity for each group. Then apply the supplied credibility weights; $K$ and $P_0$ are hypothetical inputs, not universal constants.

For A, the observed frequency is $1/100=0.0100$, so the observed pure premium is $0.0100\times\$5{,}000=\$50$. Its credibility is $Z_A=100/(100+400)=0.20$. The blended indication is

0.20(\$50)+0.80(\$100)=\$10+\$80=\$90.

For B, the observed frequency is $39/1{,}900\approx0.0205263$, giving an observed pure premium of $(39/1{,}900)\times\$5{,}000\approx\$102.63$. Its credibility is $Z_B=1{,}900/(1{,}900+400)=1{,}900/2{,}300\approx0.8261$. Using unrounded inputs, the blended indication is

\frac{1{,}900}{2{,}300}\left(\frac{39}{1{,}900}\times\$5{,}000\right) +\frac{400}{2{,}300}(\$100) \approx \$102.17.

The raw estimates differ by about $\$102.63-\$50=\$52.63$, while the illustrative blended indications differ by about $12.17. The blend gives the thin subgroup less weight on its own experience. It does not prove that the subgroups should be combined or that the chosen $K$ is appropriate; those judgments require support.

Check. A’s blended amount lies between $50 and $100, closer to $100 because its weight is 0.20. B’s blended amount lies between about $102.63 and $100, closer to its experience because its weight is about 0.8261. The calculation avoids assigning full weight to a single claim in a small cell.

Assessing Fairness Consequences Alongside Cost Evidence

Segmentation has distributional consequences: when one group’s relativity rises, other groups may pay a smaller share of the total indicated premium, depending on how the rate level and relativities are calibrated. A factor can be predictive and still deserve careful scrutiny if it separates groups in ways that create substantial burdens or acts as a proxy for a protected trait. The earlier material on protected traits, proxy variables, causal pathways, and fairness metrics provides the relevant concepts for that assessment.

Cost evidence is necessary, but it is not the only consideration. A defensible review asks whether the variable has a sound relationship to expected insured cost, whether its measurement is reliable and consistently available, whether the subgroup estimates are sufficiently credible, and whether a simpler grouping captures the supported pattern. It also examines the direction and magnitude of premium changes across groups and whether the distinctions can be explained clearly. These questions help identify when additional detail is useful and when it mainly adds instability or opaque differentiation.

Practical takeaway: Do not choose segmentation solely because it improves an in-sample fit. Compare the simpler and more detailed indications, inspect subgroup credibility and stability, quantify how relativities alter premiums, and document why the distinctions are relevant and appropriate.

Common Mistakes

Confusing fit with justification. A model can fit observed losses more closely after adding groups, even if the new cell estimates are noisy or the variable lacks a sound cost rationale.
Multiplying main effects despite evidence of an interaction. In the second example, the main-effect calculation gives $1,120 for the joint class, while the indicated cost is $1,400. The correction is to assess the joint cell and, if supported, include the 1.25 interaction multiplier.
Ignoring exposure weights. The pooled pure premium is total loss divided by total exposure, not an unweighted average of subgroup rates when exposures differ.
Relying on a point estimate alone. A large estimated difference in a very small group can be highly uncertain. Show the data volume and quantify or otherwise assess uncertainty.
Treating credibility as an automatic fairness test. Shrinking a factor toward a benchmark addresses one form of estimation uncertainty; it does not establish that the variable or resulting premium distinctions are appropriate.

Check Your Understanding

Work each problem before reading its final answer.

A hypothetical class has 600 exposures and $72,000 of losses; a second has 900 exposures and $99,000 of losses. Calculate both pure premiums, the pooled pure premium, and the first class’s relativity to the pooled amount. (Answer: $120 and $110; pooled pure premium $114; first-class relativity $120/114\approx1.0526$.)
In a log-link model, the effect of a factor is 1.30 when $z=0$ and 1.56 when $z=1$. What is the interaction multiplier? (Answer: $1.56/1.30=1.20$.)
Two equally exposed cells have indicated costs of $400 and $600. A third cell, with both rating characteristics, has indicated cost $960. Using the first cell as reference, calculate the two main effects and the interaction multiplier. (Answer: Main effects 1.50 and 1.60; their product predicts $960, so the interaction multiplier is 1.00.)
A subgroup has 80 exposures, 2 claims, and fixed severity $4,000. Under a Poisson frequency assumption, calculate its observed pure premium and process standard error. (Answer: Pure premium $2/80\times\$4{,}000=\$100$; standard error $(\sqrt{2}/80)\times\$4{,}000\approx\$70.71$.)
A subgroup’s observed pure premium is $140, its a priori premium is $100, and its specified credibility is 0.25. Calculate the blended indication and explain what the blend does. (Answer: $0.25(\$140)+0.75(\$100)=\$110$; it gives 25% weight to subgroup experience and 75% to the a priori amount.)

What You'll Learn