Fairness Constraints in Predictive Rate Models

Turning a Fairness Objective Into a Rate Rule

A fairness diagnostic identifies a pattern; it does not, by itself, specify what rates should be. To change a predictive rate model, an actuary must select a fairness objective, express it as a measurable constraint or adjustment, and examine the resulting premiums and predictions. These choices matter because objectives can conflict: rates that satisfy one group comparison may fit observed losses less well or produce a different aggregate premium.

There is no single fairness constraint that is appropriate in every insurance setting. The metric should be selected in context, with attention to the groups and outcomes being compared, the permitted rating variables, and applicable regulatory requirements. The examples below use hypothetical portfolios and defined objectives to show the mechanics; they do not establish a universal standard.

Definition: A fairness constraint is a restriction on model parameters or predicted rates that requires a specified fairness condition to hold. A post-fit adjustment changes predictions from an already fitted model to pursue a specified condition. In either case, the objective must name the groups, the quantity compared, and any weighting or tolerance used.

Let $r_i$ be the predicted rate for policy $i$, $e_i$ its exposure, and $g(i)$ its group. For a group $g$, an exposure-weighted average rate is $ \bar r_g=\sum_{i:g(i)=g}e_i r_i \big/ \sum_{i:g(i)=g}e_i $. For example, a constraint might limit the ratio of two group averages to $c$, or require the averages to be equal. This compares average rates, not individual risks, claim outcomes, or every other possible measure of fairness.

A constrained fit can be written generally as minimizing a chosen model loss, such as a likelihood deviance, subject to fairness restrictions. A post-fit adjustment instead begins with predictions $r_i$ and applies a rule, such as $r_i^*=q_{g(i)}r_i$, where $q_g$ is a group-specific multiplier. The rule should also state whether the total premium is preserved, whether the adjustments are capped, and how the resulting rates will be assessed.

Deriving a Simple Rate Constraint

A transparent way to study a rate-ratio constraint is to adjust group rates as little as possible under a specified squared-deviation criterion. This is a teaching device, not a claim that squared deviations are the only appropriate objective. Suppose two groups have initial rates $m_A,m_B$ and exposure weights $w_A,w_B$. Choose adjusted rates $a,b$ to minimize $w_A(a-m_A)^2+w_B(b-m_B)^2$, subject to $b\leq ca$, where $c\geq1$.

If the initial ratio $m_B/m_A$ exceeds $c$, the constraint binds at the minimum. Otherwise, the original rates already satisfy it and are the minimizer. In the binding case, substitute $b=ca$. Differentiating the resulting one-variable quadratic gives $w_A(a-m_A)+c\,w_B(ca-m_B)=0$. Thus:

a=\frac{w_A m_A+c w_B m_B}{w_A+c^2w_B}, \qquad b=ca.

The calculation shows why the weights matter: a group with greater weight has greater influence on the adjusted rates under this particular objective. The constraint guarantees only the stated rate-ratio condition. It does not guarantee calibration, adequate premiums, or another fairness metric.

Worked Example: Capping a Group Rate Ratio

Method and assumptions. A hypothetical insurer has two groups, each with 100 exposure units. Its initial indicated loss-cost rates are $100 for group A and $160 for group B. It wants the group B-to-A rate ratio to be no greater than 1.20, while minimizing the exposure-weighted squared change in rates. For context, assume claims have fixed severity of $1,000 and Poisson frequency. The initial rates correspond to frequency parameters of 0.10 and 0.16 claims per exposure; the separately estimated expected loss costs are $110 and $170, equivalent to frequency parameters of 0.11 and 0.17. No interest discounting or mortality assumption is used.

The initial ratio is $160/100=1.60$, so the constraint binds. With equal weights, the objective is proportional to $(a-100)^2+(b-160)^2$, and the boundary condition is $b=1.20a$. Substituting gives:

$$ (a-100)^2+(1.20a-160)^2 $$

Differentiating and setting the derivative to zero yields $(a-100)+1.20(1.20a-160)=0$. Therefore, $2.44a=292$, $a=119.6721$, and $b=1.20(119.6721)=143.6066$. Rounded to the nearest cent, the constrained rates are $119.67 and $143.61. Their ratio is 1.20, as required.

Before adjustment, expected premium is $100(100)+100(160)=$26,000$. After adjustment, it is $100(119.6721)+100(143.6066)=$26,327.87$. The estimated expected losses are $100(110)+100(170)=$28,000$, so the adjusted expected premium-to-loss ratio is $26,327.87/28,000=0.9403$, or 94.03%. The constraint improves neither adequacy automatically nor necessarily the loss-cost fit: here the specified adjustment still leaves expected premium below expected loss.

Common mistake avoided: Simply setting group B's rate to $1.20(160)=$192$ applies the cap to the original group B rate instead of comparing the adjusted group rates. It would increase the disparity rather than limit it. The constraint is $b\leq1.20a$, so both adjusted rates must be considered.

Post-Fit Adjustments and Aggregate Premium

A post-fit adjustment can be useful when the existing model is retained but its group averages do not meet a selected objective. For a simple equal-average-rate objective, one possible adjustment is to set both group averages to a common value. If total modeled premium must be preserved, that common rate is the original exposure-weighted average:

r^*=\frac{\sum_g E_g\bar r_g}{\sum_g E_g},

where $E_g$ is the group's exposure. This rule makes the group averages equal and preserves aggregate modeled premium. It does not preserve each group's premium, and it does not ensure that the adjusted predictions are equally well calibrated to group losses.

Worked Example: Equalizing Group Averages Without Changing Total Premium

Method and assumptions. A hypothetical portfolio has 60 exposure units in group A and 40 in group B. The fitted mean rates are $100 and $150, respectively. The selected objective is equal group-average rates, with total modeled premium held fixed. For a distributional illustration, assume fixed claim severity of $1,000 and Poisson claim frequency; the fitted frequency parameters are 0.10 and 0.15 per exposure. Estimated expected loss costs are $110 and $130, corresponding to frequency parameters 0.11 and 0.13. No discounting or mortality is used.

The original modeled premium is $60(100)+40(150)=$6,000+$6,000=$12,000$. There are $60+40=100$ exposure units, so the common rate that preserves this total is $12,000/100=$120$. The adjustment raises group A's average by $20, or 20%, and reduces group B's average by $30, or 20%. Adjusted modeled premium is $60(120)+40(120)=$7,200+$4,800=$12,000$: the total is unchanged.

Expected losses are $60(110)=$6,600$ for A and $40(130)=$5,200$ for B, or $11,800 in total. Aggregate expected premium exceeds expected loss by $200 both before and after the adjustment. However, group-level premium less expected loss changes. Before adjustment, the differences are $6,000-6,600=-$600$ for A and $6,000-5,200=+$800$ for B. After adjustment, they are $7,200-6,600=+$600$ and $4,800-5,200=-$400$.

For one explicitly defined accuracy summary, compare each group's average rate with its expected loss cost. The exposure-weighted sum of squared differences is initially $60(-10)^2+40(20)^2=22,000$, in squared dollars per exposure, and after adjustment it is $60(10)^2+40(-10)^2=10,000$. This particular group-average summary improves, but it does not establish that individual-level predictive accuracy improves: that would require individual predictions and an appropriate individual-level evaluation.

Common mistake avoided: Equal group averages do not mean equal group losses or equal group profit. Here the adjusted means are both $120, while the expected loss costs remain $110 and $130. The objective is achieved, but group calibration differs.

Constraining a Log-Link Model

A fairness condition can also be incorporated into model fitting. For a Poisson frequency model with log link, a simple two-group specification is $\log(\lambda_i)=\alpha+\beta I(g(i)=B)$, where $\lambda_i$ is expected claim frequency per exposure, $\alpha$ is the group A log-frequency, and $I(g(i)=B)$ equals 1 for group B and 0 otherwise. If severity is fixed and identical across the groups, the ratio of their expected loss-cost rates is $e^\beta$. A cap $c$ on that ratio is equivalent to $\beta\leq\log(c)$.

In a richer model, the comparison must be defined carefully. For example, raw group averages may reflect different distributions of other rating variables. One can instead compare predictions standardized to a common distribution of those variables. The chosen definition must be stated; a coefficient restriction and a standardized-average restriction are not generally interchangeable.

Worked Example: Capping a Log-Link Group Effect and Recalibrating

Method and assumptions. A hypothetical Poisson frequency model has 100 exposure units in each group, fixed claim severity of $1,000, and initial expected loss-cost rates of $80 for A and $120 for B. Thus the frequency parameters are 0.08 and 0.12, and the initial rate ratio is 1.50. The selected constraint caps the ratio at 1.25. After imposing it, the intercept is recalibrated to preserve the initial $20,000 total modeled premium. Estimated expected loss costs are $90 for A and $130 for B, corresponding to frequency parameters 0.09 and 0.13. There is no discounting or mortality assumption.

With group B represented by an indicator, the initial coefficient difference is $\beta=\log(120/80)=\log(1.50)$. The constraint is $\beta\leq\log(1.25)$, so at the constrained boundary the rates have the ratio $r_B/r_A=1.25$. Preserving the original total premium requires $100r_A+100r_B=$20,000$, or $r_A+r_B=$200$. Substituting $r_B=1.25r_A$ gives $2.25r_A=200$, so $r_A=$88.8889$ and $r_B=$111.1111$. Rounded to the nearest cent, these are $88.89 and $111.11; their ratio is 1.25 and their total modeled premium is $20,000 before rounding.

Expected losses are $100(90)=$9,000$ for A and $100(130)=$13,000$ for B, totaling $22,000. The aggregate premium-to-expected-loss ratio is $20,000/22,000=0.9091$, or 90.91%. Before the constraint, each group was underpredicted by $1,000: A's modeled $8,000 versus $9,000 expected loss, and B's $12,000 versus $13,000. After adjustment, A is underpredicted by $111.11 and B by $1,888.89. The total shortfall remains $2,000, but its distribution between groups changes.

Common mistake avoided: Capping the group coefficient does not, by itself, preserve the aggregate premium. The intercept recalibration is a separate step. Without it, the constrained rates would depend on how the fitting procedure handles the intercept and would not be justified as aggregate-premium preserving.

Assessing the Tradeoffs

A constraint is meaningful only alongside its consequences. After fitting or adjustment, compare the results with the unconstrained model using clearly specified measures. Depending on the objective and data, these can include:

Fairness outcome: Did the stated group-rate comparison meet its target? Report the exact groups, weighting, and measure.
Adequacy: Compare indicated or modeled premium with expected losses overall and by relevant group. A preserved total does not imply adequate premiums.
Predictive accuracy: Recalculate an appropriate loss, deviance, or calibration measure on validation data. Do not infer individual-level accuracy from group averages alone.
Rate impact: Show the distribution of increases and decreases, not only the change in a group mean.

A constraint may be imposed during fitting, applied after fitting, or used together with a separate recalibration. These procedures can yield different predictions even when they target a similar group summary. In every case, disclose the objective, the fitting or adjustment rule, the weights used, and the measures used to evaluate the result. A constraint on average rates is not automatically a constraint on claim outcomes, access, or any other fairness definition.

Key takeaway: Define the fairness quantity first, encode it as an explicit parameter restriction or adjustment, and then remeasure adequacy, predictive performance, and group outcomes. Meeting the selected constraint is one result to report—not proof that every relevant objective has been met.

Check Your Understanding

For each hypothetical scenario, identify the stated objective and check the arithmetic before interpreting the result.

A portfolio has equal exposures in groups A and B, with initial rates $90 and $150. Under the exposure-weighted squared-change objective, cap the B-to-A ratio at 1.20. Find the constrained rates and total premium for 80 exposures in each group. (Answer: $110.66 and $132.79; total premium $19,475.41.)
A portfolio has 50 exposures at an average rate of $80 and 150 exposures at $140. Apply equal group-average rates while preserving total modeled premium. What is the common rate? (Answer: $125.00; total modeled premium remains $25,000.)
A log-link frequency model has group rates $70 and $105. A constraint limits the second-to-first rate ratio to 1.30. What is the largest permitted group coefficient difference? (Answer: $\log(1.30)$, approximately 0.2624.)
Two groups each have 100 exposures and constrained rates of $88.89 and $111.11. Their expected loss costs are $90 and $130. Find aggregate modeled premium, aggregate expected loss, and the premium-to-loss ratio using the unrounded rates $88.8889 and $111.1111. (Answer: $20,000; $22,000; 0.9091, or 90.91%.)
A proposed adjustment makes two group-average rates equal and preserves aggregate premium. What additional evidence is needed before concluding that individual predictive accuracy is unchanged? (Answer: Individual-level predictions and an appropriate individual-level evaluation on suitable validation data; equal group means and preserved aggregate premium do not establish individual-level accuracy.)

What You'll Learn