Proxy Variables and Indirect Use of Protected Traits

Why Proxy Variables Require Careful Review

A rating plan may omit a protected trait and still use a variable associated with that trait. For example, a hypothetical insurer might use a narrow geographic category, an occupation label, or a purchasing characteristic. Such a variable can be a proxy when it carries information about a protected trait and is used, intentionally or in effect, as a substitute for that trait. Association raises a question for review; by itself, it does not establish intent, causation, illegality, or unfair discrimination.

A useful actuarial review therefore asks several separate questions. How strongly is the candidate variable associated with the protected trait in the relevant insured population? What rating purpose is claimed for the variable? Does it contribute meaningfully to estimating expected costs for the insurance coverage? Are suitable alternatives available, and how well do they perform? The answers should be considered together, in light of applicable law, regulation, and actuarial standards. The answer is not determined by a single correlation or cost statistic.

Definition: A protected trait is a characteristic that is protected from specified forms of discrimination under the applicable law or regulatory framework. A proxy variable is a rating variable that may serve as a substitute for, or convey substantial information about, a protected trait. A variable’s association with a protected trait is evidence relevant to review, but it is not by itself proof that the variable is a proxy in a legally prohibited sense.

Direct use and possible proxy use are different concerns. A rating plan that explicitly uses a protected trait raises the direct-use question addressed by applicable restrictions. A plan that excludes the trait but uses a strongly associated variable raises an indirect-use question. Even where the candidate variable predicts expected costs, that fact does not alone resolve whether its use is permitted or appropriate. Conversely, observing an association does not alone establish that a variable lacks a legitimate actuarial purpose.

A Structured Screening Framework

A proxy review is best treated as a sequence of evidence questions, not a mechanical pass-or-fail test. The analyst should define the protected trait and the candidate rating variable precisely, specify the population and coverage being reviewed, and identify the proposed use of the variable. Then the analyst can assess descriptive association, the variable’s purpose, expected-cost contribution, and alternatives.

Define the question.
State the protected trait, candidate variable, insurance coverage, population, and rating decision. A broad label such as “location” is not precise enough if the filed factor is based on a particular geographic unit.

Describe association.
Compare how the candidate variable is distributed across groups defined by the protected trait. Identify the data source, time period, sample limitations, and any missing observations.

Examine the stated purpose and cost contribution.
Ask what rating function the variable is intended to serve and whether credible, relevant evidence supports its contribution to expected costs for the coverage in question.

Consider alternatives and constraints.
Compare feasible alternatives on predictive usefulness, data quality, stability, explainability, operational cost, and applicable restrictions. Record what the available evidence does and does not establish.

For a simple descriptive screen, let $P$ denote a protected-trait group and let $Z$ denote a candidate rating category. If $Z=1$ indicates the category of interest, an association can be described by comparing $P(Z=1\mid P=A)$ with $P(Z=1\mid P=B)$. The difference is descriptive: it says how common the category is in each observed group. It does not, on its own, show why the difference exists or whether the rating factor is impermissible.

\Delta_Z=P(Z=1\mid P=A)-P(Z=1\mid P=B)

Expected-cost contribution is a separate question. Let $L$ be annual loss cost for the defined coverage. If a category has different expected costs from other categories, that may support a rating purpose, subject to the quality and relevance of the estimates and any governing restrictions. It does not erase the need to consider the association and alternatives. Do not confuse “predicts costs” with “is automatically suitable for use.”

Worked Examples

Worked Example: Association and Expected-Cost Relevance

A hypothetical insurer is reviewing whether to use an optional home-security category in an annual homeowners rating plan. The protected-trait groups are A and B. Among 200 policies in group A, 80 have the security category; among 200 policies in group B, 20 have it. For this illustration, treat the counts as complete hypothetical observations for the same coverage and period. Expected annual loss costs estimated for the category are $600 with the security feature and $900 without it. There is no discounting, mortality assumption, or parametric loss distribution; these supplied amounts are annual expected costs. Calculate the descriptive association and group-average costs, then explain what the calculations establish.

Method. First compare the category shares. Then use weighted averages to calculate the expected cost in each protected-trait group, using the category-specific expected costs and the observed category mix.

In group A, the share with the security category is $80/200=0.4000$, or 40%. In group B, it is $20/200=0.1000$, or 10%. Thus the difference is $0.4000-0.1000=0.3000$, or 30 percentage points.

Group A’s average expected cost is $0.4000(\$600)+0.6000(\$900)=\$240+\$540=\$780$. Group B’s average expected cost is $0.1000(\$600)+0.9000(\$900)=\$60+\$810=\$870$.

Check. The category mix differs by 30 percentage points, and the resulting group-average expected costs differ by $\$780-\$870=-\$90$. Those results are consistent with the security category’s lower supplied expected cost. They do not establish that the category is legally acceptable, that the cost estimates are sufficiently credible, or that no better alternative exists.

Common mistake avoided. It would be wrong to treat the 30-point association as proof of prohibited proxy use, or to treat the lower category cost as automatic approval. The arithmetic describes association and expected-cost implications; suitability requires the broader review.

Worked Example: Checking Group Averages Before Interpreting a Factor

A hypothetical commercial insurer is reviewing a two-level equipment-use category. For 150 exposures in category A, expected annual loss costs are $1,200 for 60 exposures and $900 for 90 exposures. For 150 exposures in category B, they are $1,100 for 20 exposures and $900 for 130 exposures. A separate descriptive review finds that a protected-trait group is more prevalent in category A than in category B. Calculate the average cost in each category and the overall average. Treat the supplied exposure counts and costs as hypothetical, complete summaries on the same annual coverage basis; no mortality, interest, or parametric loss-distribution assumption is used.

Method. Calculate each category mean as total expected losses divided by exposures. Then calculate the overall mean using all exposures and losses, not an unweighted average of the two category means.

For category A, total expected losses are $60(\$1,200)+90(\$900)=\$72,000+\$81,000=\$153,000$. Its average is $\$153,000/150=\$1,020.00$.

For category B, total expected losses are $20(\$1,100)+130(\$900)=\$22,000+\$117,000=\$139,000$. Its average is $\$139,000/150=\$926.6667$, which is $\$926.67$ to the nearest cent.

Across both categories, total losses are $\$153,000+\$139,000=\$292,000$, and total exposures are $150+150=300$. The overall average is $\$292,000/300=\$973.3333$, or $\$973.33$ to the nearest cent.

As a descriptive comparison, the category averages are $\$1,020.00$ and $\$926.67$. Their difference is $\$93.33$ per exposure. This is evidence of different average costs in these summaries, but it does not by itself show that category membership is the cause of the difference or that the variable is suitable for rating. The separate association with a protected group makes the category relevant to proxy review; the reviewer still needs to assess data quality, purpose, alternatives, and applicable requirements.

Check. Because both categories contain 150 exposures, the overall mean is also $(\$1,020.00+\$926.6667)/2=\$973.3333$, matching the pooled calculation. A common incorrect calculation gives category B as $\$920$; the correct numerator is $\$139,000$, and $\$139,000/150=\$926.67$.

Common mistake avoided. Do not miscalculate a weighted mean or infer that a cost difference resolves the proxy question. The corrected category B average is essential to any subsequent interpretation.

Worked Example: Comparing a Candidate Variable With an Alternative

A hypothetical small-business insurer is considering a detailed vendor credential as a rating variable. In a review sample, 72 of 120 policies in protected-trait group A have the credential, compared with 24 of 120 in group B. The candidate category’s supplied expected annual costs are $840 with the credential and $960 without it. A broader job-task category is available as an alternative; its two expected-cost levels are $860 and $940. A hypothetical validation summary reports mean absolute prediction errors of $42 for the credential-based model and $45 for the job-task model, measured on the same coverage and validation sample. These are supplied summaries, not estimates from an assumed parametric distribution; no mortality or interest assumption applies. Compare the association, the category cost differences, and the reported error difference.

Method. Calculate the credential prevalence by protected-trait group, then compare the absolute cost separations and validation errors. Keep the measures distinct: association describes group composition, cost separation describes the supplied category means, and validation error summarizes predictive performance as reported.

Credential prevalence is $72/120=0.6000$, or 60%, in group A, and $24/120=0.2000$, or 20%, in group B. The prevalence difference is 40 percentage points.

For the credential variable, the supplied expected-cost difference is $\$960-\$840=\$120$. For the job-task alternative, it is $\$940-\$860=\$80$. The credential model’s reported mean absolute error is $\$45-\$42=\$3$ lower than the alternative’s, a $3/45=0.0667$, or 6.67%, reduction relative to the alternative’s error.

Interpretation. The credential is more unevenly distributed across the two groups in these observations, and its supplied category costs are more separated. Its reported validation error is also modestly lower. These summaries make the credential’s predictive contribution and association relevant to a review, but they do not show that the difference in predictive error is statistically or practically decisive. Nor do they establish that the alternative is adequate for all rating purposes. A defensible assessment would check comparable data and definitions, stability, operational feasibility, and applicable restrictions before drawing a conclusion.

Check. The 40-point prevalence difference, $120 versus $80 category cost separations, and $3 lower reported error are separately calculated quantities. None can be substituted for either of the others.

Common mistake avoided. Do not claim that the variable with the lowest error must be used, or that a smaller group association automatically makes an alternative preferable. The decision requires context and a complete comparison, not a ranking on one statistic.

Why the Weighted-Average Formula Holds

A group’s mean expected cost is a weighted average because each category contributes its expected cost in proportion to its share of the group. Suppose a group has $n_1$ exposures in category 1 and $n_0$ in category 0, with expected costs $\mu_1$ and $\mu_0$. The total expected cost is $n_1\mu_1+n_0\mu_0$. Dividing by the group’s total exposure count $n_1+n_0$ gives the group mean. Writing the exposure shares as $w_1=n_1/(n_1+n_0)$ and $w_0=n_0/(n_1+n_0)$ yields the weighted-average form.

\bar{\mu} =\frac{n_1\mu_1+n_0\mu_0}{n_1+n_0} =w_1\mu_1+w_0\mu_0, \qquad w_1+w_0=1

This identity explains why a variable can be associated with different group averages even when the category-specific expected costs are the same across groups: the groups may have different category weights. It is an arithmetic decomposition, not a causal explanation of why the weights or costs differ.

Interpreting Evidence and Avoiding Overreach

A careful proxy review separates what is observed from what is inferred. A cross-tabulation can show a difference in prevalence. Cost summaries can show different estimated expected costs across rating categories. Validation evidence can indicate predictive usefulness under the stated design. None of these alone establishes intent, a causal mechanism, the legal status of the variable, or whether its use is justified under the governing standard.

The strength of any conclusion depends on the evidence. Relevant considerations include whether the protected-trait information is reliable and lawfully available for the review; whether the rating and audit populations align; whether there are missing or misclassified observations; whether expected-cost estimates are sufficiently credible; and whether definitions are consistent across groups. Sparse data can make apparent differences unstable. A result for one coverage, time period, or population should not be generalized automatically to another.

Alternative variables should be compared on a like-for-like basis. An alternative with a lower observed association may be less useful, less stable, or unavailable for operational use; an alternative with similar predictive performance may merit closer consideration. The comparison should document the intended purpose, evidence, limitations, and tradeoffs rather than declaring an alternative superior from a single number.

Association is not proof. A strong relationship warrants scrutiny but does not establish prohibited use on its own.
Predictive value is not automatic permission. Evidence of expected-cost contribution is relevant, but does not override applicable restrictions or complete the review.
Absence of a direct input is not the end of the inquiry. An excluded protected trait may still be associated with variables used by the rating plan.
Compare alternatives fairly. Use comparable definitions, populations, coverage periods, and validation criteria where possible.
State uncertainty. Small samples, incomplete records, and model limitations affect how strongly results can be interpreted.

Key takeaway: A sound proxy screen considers association, stated rating purpose, expected-cost contribution, alternatives, data limitations, and applicable constraints together. It treats each calculation as evidence for a defined question—not as a self-sufficient verdict.

Check Your Understanding

For each problem, distinguish the numerical result from the conclusion it can support. All scenarios are hypothetical.

In a sample, 45 of 90 policies in group A and 24 of 160 in group B have a proposed rating category. Calculate the category prevalence in each group and the difference in percentage points. What does this calculation alone establish? (Answer: 50% in group A, 15% in group B, a 35-percentage-point difference; it establishes descriptive association in the sample, not intent or a legal conclusion.)
A category is present for 35% of a protected-trait group and 15% of another group. Expected annual costs are $700 with the category and $900 without it. Calculate each group’s average expected cost under these mixes. (Answer: first group, $0.35(\$700)+0.65(\$900)=\$830$; second group, $0.15(\$700)+0.85(\$900)=\$870$.)
A rating category has 40 exposures at an expected cost of $1,050 and 60 exposures at $850. Calculate the category’s overall average expected cost. Assume the summaries are on the same annual coverage basis and no discounting is applied. (Answer: \([40(\$1,050)+60(\$850)]/100=(\$42,000+\$51,000)/100=\$930.)
A proposed variable has a 25-percentage-point prevalence difference across two groups and a reported validation error of $38. An available alternative has a 10-point prevalence difference and a reported validation error of $40 on the same validation sample. What is the error increase for the alternative, and why is that not by itself a final decision? (Answer: the alternative’s error is $2 higher; the result alone does not resolve the variable’s purpose, the practical significance of the error difference, applicable constraints, data quality, or other alternatives.)
A candidate variable is strongly associated with a protected trait, but its supplied category costs are identical. Name two additional issues an analyst should investigate before reaching a conclusion about its use. (Answer: for example, verify the reliability and relevance of the cost estimates and assess feasible alternatives; also examine data quality, the stated rating purpose, and applicable legal or regulatory restrictions.)

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