Why Trace the Pathways?
A rating variable may predict losses accurately and still raise questions about how that predictive relationship arose. For example, a location attribute might capture differences in physical hazards, but it might also reflect historical barriers that shaped where people could live and what resources were available. A useful fairness review therefore asks more than whether a variable is associated with loss: it asks what pathways could produce that association and what those pathways imply for the proposed rating rationale.
A causal diagram makes these questions explicit. It is not proof that a pathway exists, and it cannot settle a regulatory or ethical judgment by itself. It is a structured way to state hypotheses about how variables relate, distinguish competing explanations, and identify what evidence would help evaluate them. The discussion here is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Use \(G\) for a protected trait or group, \(R\) for a proposed rating variable, and \(Y\) for a loss outcome. Other nodes can represent exposure, a physical hazard, access to resources, or historical conditions. A simple predictive model describes how \(Y\) varies with \(R\). A causal diagram adds hypotheses about why the variables vary together. Those are different claims: accurate prediction from \(R\) does not alone show that \(R\) is a fair or appropriate basis for differentiating rates.
Reading a Pathway Without Overclaiming
Three pathway patterns are especially useful in a fairness review. They are analytical categories, not automatic verdicts about a rating variable.
- Risk or exposure pathway: \(R \to Y\), or \(R \leftarrow E \to Y\), where \(E\) is a relevant exposure. The rating variable may measure a feature of the insured exposure or a physical condition associated with loss.
- Proxy pathway: \(G \to C \to R \to Y\), where \(C\) represents social or institutional conditions. Here \(R\) may carry information about group-related conditions as well as, or instead of, an independently justified risk feature.
- Mediated pathway: \(G \to A \to R \to Y\), where \(A\) represents an intervening condition. The variable \(R\) may be genuinely predictive of loss while also lying on a pathway through which inequitable conditions affect outcomes.
A variable can participate in more than one pattern. For example, a geographic category can measure a real physical hazard and also reflect structural conditions that influence residential location. The diagram should represent the hypotheses under review, not force a complex situation into a single label.
The direction of an arrow matters. If a rating variable is measured before the loss, it can be a predictor of that loss, but temporal ordering alone does not establish that it causes the loss. A common cause may influence both \(R\) and \(Y\). For instance, an unrecorded exposure characteristic might affect a rating feature and claims. Conversely, an observed association between \(G\) and \(R\) does not, on its own, establish the social process that produced it.
Worked Examples: Tracing the Implications
Worked Example: Annual Mileage as an Exposure Measure
A hypothetical personal-auto insurer considers annual mileage as a rating variable. Assume claim count \(N\), conditional on annual mileage \(M\), follows a Poisson distribution with mean \(\lambda(M)=0.04+0.012M\), where \(M\) is measured in thousands of miles. Each claim has fixed severity of $500. Assume these relationships apply to both of two hypothetical groups, and that their average mileages are 5 and 10 thousand miles. Calculate expected annual loss at each mileage and describe what the diagram supports.
Method and diagram. The model specifies \(M \to N \to Y\): mileage is associated with expected claim count, which determines expected loss under the fixed-severity assumption. For a Poisson claim count, \(E[N\mid M]=\lambda(M)\), so \(E[Y\mid M]=500\lambda(M)\). This is a model-based predictive relationship; the assumptions do not establish that every difference in observed mileage is free of broader fairness concerns.
At 5 thousand miles, \(\lambda(5)=0.04+0.012(5)=0.10\) claims per year. Therefore, \(E[Y\mid M=5]=500(0.10)=\$50.00\).
At 10 thousand miles, \(\lambda(10)=0.04+0.012(10)=0.16\) claims per year. Therefore, \(E[Y\mid M=10]=500(0.16)=\$80.00\).
The expected-loss difference is \(\$80.00-\$50.00=\$30.00\) per year. The model gives a direct exposure-related explanation for this difference. It does not establish why the groups have different average mileages or whether mileage is the only relevant pathway. Common mistake avoided: treating the group difference in mileage as proof that all resulting rate differences are automatically fair. The calculation supports a specified risk relationship, not a complete fairness conclusion.
Worked Example: A Location Variable and Historical Conditions
A hypothetical property insurer considers a binary location feature \(R\): high-hazard area or lower-hazard area. Assume expected annual claim frequency is 0.10 in the high-hazard area and 0.04 in the lower-hazard area, with fixed claim severity of $1,000. In a hypothetical portfolio, 80% of Group A policies and 20% of Group B policies are in the high-hazard area. Calculate the group-average expected losses. Then explain how a diagram changes the interpretation of the location variable.
Method and diagram. First calculate the expected loss within each location category as claim frequency times fixed severity. The proposed risk pathway is \(R \to Y\). A separate possible pathway is \(G \to C \to R\), where \(C\) represents historical or institutional conditions affecting residential location. A further cause, such as local physical hazard, could affect both \(R\) and \(Y\). These are hypotheses to examine, not facts established by the portfolio figures.
For the high-hazard area, expected loss is \(0.10(\$1{,}000)=\$100.00\). For the lower-hazard area, it is \(0.04(\$1{,}000)=\$40.00\).
For Group A, the weighted expected loss is \(0.80(\$100.00)+0.20(\$40.00)=\$80.00+\$8.00=\$88.00\).
For Group B, it is \(0.20(\$100.00)+0.80(\$40.00)=\$20.00+\$32.00=\$52.00\). The difference is \(\$88.00-\$52.00=\$36.00\) per policy year under these assumptions.
These calculations show how different location distributions produce different group-average expected losses in this hypothetical portfolio. They do not show why the groups have different location distributions. A fairness review would distinguish evidence for an independently relevant physical hazard from evidence that historical conditions helped determine residential location. Common mistake avoided: inferring from the loss difference alone that the location category is either necessarily justified or necessarily unjustified.
Worked Example: A Predictive Variable on a Mediated Pathway
A hypothetical insurer examines vehicle condition, classified as poor or good. Assume claim frequency is 0.12 for poor-condition vehicles and 0.04 for good-condition vehicles, with fixed severity of $1,000. In a hypothetical dataset, 60% of Group A and 20% of Group B vehicles are in poor condition. Assume, for this calculation, that the frequencies apply equally across the groups within each condition category. Calculate each group’s expected loss and the difference associated with the condition mix. What can and cannot be concluded?
Method and diagram. Consider the hypothesized pathway \(G \to A \to R \to Y\), where \(A\) is access to maintenance and \(R\) is vehicle condition. The assumed conditional claim frequencies provide a predictive relationship between \(R\) and \(Y\). To quantify the difference associated with the groups’ condition mix, weight the expected losses at each condition by the group’s condition distribution.
Poor condition has expected loss \(0.12(\$1{,}000)=\$120.00\); good condition has expected loss \(0.04(\$1{,}000)=\$40.00\).
Group A’s expected loss is \(0.60(\$120.00)+0.40(\$40.00)=\$72.00+\$16.00=\$88.00\). Group B’s is \(0.20(\$120.00)+0.80(\$40.00)=\$24.00+\$32.00=\$56.00\). The difference is \(\$32.00\).
The same difference follows by comparing the poor-condition shares: \((0.60-0.20)(\$120.00-\$40.00)=0.40(\$80.00)=\$32.00\). This is a decomposition of the stated weighted averages, not proof that group membership causally changes vehicle condition or that maintenance access is the true mechanism. It also does not decide whether condition is an appropriate rating basis. Common mistake avoided: treating strong prediction by vehicle condition as sufficient evidence that every pathway producing condition differences is acceptable.
Why Standardization Works—and What It Does Not Prove
The calculation in the third example uses a general weighted-average identity. Suppose \(R\) has categories \(r\), and within each group the conditional expected outcome is \(m(r)\). If that conditional expectation is the same in both groups, their difference in average outcome comes from the different distributions of \(R\). The identity is:
To see why, express each group’s mean by partitioning over the categories of \(R\):
Under the stated condition \(E[Y\mid R=r,G=g]=m(r)\), substitute \(m(r)\) into both group means and subtract. The resulting expression is the identity above. In the vehicle example, the difference in the poor-condition shares is 0.40, the remaining share difference is -0.40, and the conditional expected losses are $120.00 and $40.00. Thus \(0.40(\$120.00)-0.40(\$40.00)=\$32.00\).
This arithmetic describes how much of the observed mean difference is associated with the category mix under the assumptions. It does not establish that changing \(R\) would change loss, identify the cause of different distributions of \(R\), or determine whether a rate based on \(R\) is fair. A causal interpretation requires a credible causal structure and appropriate assumptions about confounding, measurement, and the population being considered.
A Practical Review of a Proposed Pathway
A disciplined pathway review helps keep actuarial evidence and fairness reasoning distinct while allowing both to inform the decision. State the proposed rationale precisely, draw the hypothesized arrows, and identify evidence that could distinguish the plausible explanations. For example, if the rationale is a physical hazard, ask whether the variable measures that hazard or merely stands in for a broader social pattern. If the rationale is a behavior or condition, consider whether access or constraints influence that variable.
Define \(Y\) and \(R\) in the actual rating context. Specify the loss measure, observation period, and unit of exposure so that the proposed relationship is interpretable.
Include relevant exposures, physical conditions, group membership where appropriate to the analysis, and historical or institutional conditions. Mark uncertain arrows as hypotheses requiring evidence.
Ask whether \(R\) reflects an exposure, a physical hazard, an intervening condition, or a proxy for other conditions. A pathway can contain both a loss-related mechanism and a fairness concern.
Predictive validation can support the claim that \(R\) forecasts \(Y\). Further evidence and applicable standards are needed to assess the pathway and the appropriateness of using \(R\) in a rate.
A DAG also helps expose missing information. If an unmeasured factor could affect both a proposed rating variable and losses, an observed association may not isolate the relationship the analyst intends to use. If a variable lies between a group-related condition and loss, controlling for it changes the question being answered: it describes outcomes conditional on that variable, rather than the overall difference. Neither a diagram nor a statistical adjustment decides which question is the right fairness question.
Common Traps
- Prediction is not a complete justification. A variable can forecast loss well while its pathway still warrants examination.
- An arrow is a hypothesis, not a finding. Drawings should not be presented as proven mechanisms unless supported by evidence.
- Do not label every association a proxy pathway. Identify a plausible route connecting the group-related condition, the rating variable, and loss; explain what evidence supports it.
- Do not treat a mediator as automatically acceptable or unacceptable. Its predictive value and its position in a pathway are separate considerations.
- Do not interpret a weighted-average decomposition as a causal effect. The calculation describes differences under its assumptions; causal conclusions require additional support.
The central discipline is to state what the analysis establishes and what it leaves unresolved. A rate filing or actuarial rationale may need to explain predictive evidence, the proposed risk mechanism, limitations, and relevant practical or regulatory constraints. A transparent causal diagram can organize that explanation, but it cannot substitute for the evidence or judgment the question requires.
Check Your Understanding
For each scenario, identify the relevant pathway or calculation and distinguish what is supported from what remains unproven.
- A hypothetical insurer models annual claim frequency as 0.03 plus 0.01 times annual use measured in thousands of hours. Expected claim severity is $800. Calculate expected loss at 4 thousand hours and 9 thousand hours. (Answer: $56.00 and $96.00, respectively; the difference is $40.00 under the model, but this alone does not resolve fairness questions about use differences.)
- A hypothetical rating category has expected losses of $150 in category 1 and $50 in category 2. Group A has 70% in category 1, and Group B has 30%. Assuming the conditional expected losses are the same by group, calculate their average expected losses and difference. (Answer: Group A $120.00; Group B $80.00; difference $40.00.)
- A proposed predictor is associated with claims and is also influenced by access to a service that differs across groups. Give the pathway pattern and one conclusion that predictive evidence alone cannot establish. (Answer: A possible mediated pathway is \(G \to A \to R \to Y\); prediction alone cannot establish that the pathway is fair or that \(R\) is an appropriate rating basis.)
- In a hypothetical portfolio, a factor has conditional expected losses of $90 and $30 in its two categories. The first category’s share is 50% for Group A and 25% for Group B. Assuming the conditional expected losses are identical across groups, calculate the difference attributable to the category mix. (Answer: \((0.50-0.25)(\$90-\$30)=\$15.00\).)
- A location variable predicts losses. A diagram includes both a local physical hazard causing losses and historical conditions affecting where groups live. What does the diagram help clarify, and what does it not establish? (Answer: It separates a possible physical-risk pathway from a possible historical pathway affecting location; the diagram alone establishes neither pathway as true nor the fairness of using the variable.)