Workshop: Diagnosing Disparity in a Rate Plan

Diagnosing a Disparity Requires More Than a Difference

A rate plan can produce different average premiums for two groups without revealing why they differ. The gap may reflect differences in the plan’s rating factors, different distributions of risk characteristics within each group, coverage or exposure differences, or random variation in the observed data. A useful audit calculates the gap, examines its composition, quantifies uncertainty where the data permit, and identifies evidence that would distinguish among plausible explanations. This workshop-style approach is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: A subgroup premium disparity is a measured difference in premium outcomes between specified groups over a defined population, period, and coverage basis. A crude disparity compares each group’s average directly. A standardized disparity compares group outcomes after applying the same selected distribution of relevant characteristics to both groups. Neither measure, by itself, establishes the cause of a difference or determines whether a rate plan is appropriate.

Before calculating, define the comparison: name the groups, the premium measure, the exposure basis, the period, and the coverage being compared. Earned premium divided by earned exposure is generally more meaningful than a comparison of unadjusted premium totals when group sizes differ. Loss costs can add context, but should be compared on a consistent basis and are not automatically equivalent to indicated rates.

Start With Comparable Measures

For subgroup $g$, let $P_g$ be aggregate premium and $e_g$ be earned exposure units. The exposure-weighted average premium per unit is $P_g/e_g$. If $L_g$ denotes losses on the same exposure and coverage basis, $L_g/e_g$ is the corresponding average loss cost. These summaries describe observed averages; they do not control for differences in group composition or establish the adequacy of the premium.

Worked Example: Comparing Premium and Loss-Cost Summaries

A hypothetical insurer reviews two groups of policies for one fully earned year. Both groups have the same coverage definition and exposure unit. Group A has 200 earned exposures, aggregate earned premium of $180,000, and losses of $132,000. Group B has 100 earned exposures, aggregate earned premium of $120,000, and losses of $90,000. Calculate the average premium, average loss cost, and loss ratio for each group, then compare the gaps.

Method and assumptions. Divide each aggregate by earned exposures to obtain per-exposure averages. For this exercise, assume the reported losses are on the same period and coverage basis as the premium and include all loss components intended for this comparison. The data are hypothetical and are not adjusted for expenses, development, or credibility.

Group A average premium. $ \$180{,}000/200=\$900.00$ per exposure.

Group B average premium. $ \$120{,}000/100=\$1{,}200.00$ per exposure. The crude premium difference, B minus A, is $ \$1{,}200-\$900=\$300.00$.

Group A average loss cost. $ \$132{,}000/200=\$660.00$ per exposure.

Group B average loss cost. $ \$90{,}000/100=\$900.00$ per exposure. The loss-cost difference, B minus A, is $ \$900-\$660=\$240.00$.

Loss ratios. Group A has $ \$132{,}000/\$180{,}000=0.7333$, or 73.33%. Group B has $ \$90{,}000/\$120{,}000=0.7500$, or 75.00%.

The calculations show a $300.00 premium gap and a $240.00 loss-cost gap, while the loss ratios differ by 1.67 percentage points. These summaries do not show whether the remaining premium difference reflects expenses, risk characteristics, plan structure, or noise. Common mistake avoided: comparing aggregate premiums ($180,000 and $120,000) as if they were per-risk rates would confuse portfolio size with average price.

Check Whether Composition Changes the Comparison

A subgroup average is a mixture of the averages for its component segments. If groups have different proportions in segments with different premium levels, crude averages can differ even when the within-segment relationship is smaller or points in another direction. Standardization addresses one part of this problem by applying common weights to both groups. The chosen segments and weights should be relevant to the question and supported by the available data.

Definition: If group $g$ has segment rate $r_{gj}$ in segment $j$, and the selected common weights $w_j$ satisfy $\sum_j w_j=1$, its standardized average is $\sum_j w_jr_{gj}$. The standardized difference between groups A and B is $\sum_j w_j(r_{Aj}-r_{Bj})$. This comparison describes differences under the chosen common mix; it does not prove that the segmentation is complete or that the standardized measure is the only relevant fairness measure.

Worked Example: Separating a Mix Effect From Within-Segment Differences

A hypothetical plan has two segments, urban and rural. Group A is 80% urban and 20% rural; Group B is 20% urban and 80% rural. The respective segment average premiums are $1,300 urban and $700 rural for A, and $1,200 urban and $600 rural for B. Calculate the crude averages, standardize to a 50% urban and 50% rural mix, and decompose the crude difference using Group B’s segment rates as the reference.

Method and assumptions. Treat the stated proportions as exposure weights and assume both groups’ segment premiums use the same coverage and exposure basis. First calculate each group’s observed mixture; then calculate both standardized averages with the same 50/50 weights.

Crude average for A. Urban contribution: $0.80(\$1{,}300)=\$1{,}040$. Rural contribution: $0.20(\$700)=\$140$. Thus A’s average is $\$1{,}040+\$140=\$1{,}180$.

Crude average for B. Urban contribution: $0.20(\$1{,}200)=\$240$. Rural contribution: $0.80(\$600)=\$480$. Thus B’s average is $\$240+\$480=\$720$. The crude difference, A minus B, is $\$1{,}180-\$720=\$460$.

Standardized average for A. $0.50(\$1{,}300)+0.50(\$700)=\$650+\$350=\$1{,}000$.

Standardized average for B. $0.50(\$1{,}200)+0.50(\$600)=\$600+\$300=\$900$. The standardized difference is $\$1{,}000-\$900=\$100$.

The $460 crude difference is not the same as the $100 standardized difference. To make the components explicit, add and subtract the average that would result from applying A’s weights to B’s rates:

\begin{aligned} \sum_j w_{Aj}r_{Aj}-\sum_j w_{Bj}r_{Bj} &=\sum_j w_{Aj}(r_{Aj}-r_{Bj})\\ &\quad+\sum_j (w_{Aj}-w_{Bj})r_{Bj}. \end{aligned}

The first term is the within-segment component at A’s weights: $0.80(\$100)+0.20(\$100)=\$100$. The second is the composition component using B’s rates: $(0.80-0.20)(\$1{,}200)+(0.20-0.80)(\$600)=\$720-\$360=\$360$. They sum to $\$100+\$360=\$460$, matching the crude difference.

This identity holds because the right side expands to $\sum_j w_{Aj}r_{Aj}-\sum_j w_{Aj}r_{Bj}+\sum_j w_{Aj}r_{Bj}-\sum_j w_{Bj}r_{Bj}$; the middle terms cancel. The decomposition depends on the chosen reference rates and weights. Common mistake avoided: calling the entire crude gap a within-segment difference ignores the $360 composition component in this reference decomposition.

Quantify Sampling Uncertainty

Observed subgroup averages are estimates when they are calculated from samples or from a finite period of experience. A confidence interval can show how uncertain a difference in sample means is under stated statistical assumptions. It does not measure every source of uncertainty in a ratemaking analysis: changes in exposure mix, model misspecification, data limitations, and future trend may require separate assessment.

Approximate confidence interval: For two independent samples with sample means $\bar X_A,\bar X_B$, sample variances $s_A^2,s_B^2$, and sample sizes $n_A,n_B$, an approximate 95% interval for the difference in population means is $$ (\bar X_A-\bar X_B)\ \pm\ 1.96\sqrt{\frac{s_A^2}{n_A}+\frac{s_B^2}{n_B}}. $$ This normal approximation assumes independent observations within and between groups and sufficiently well-behaved sampling distributions of the means. If observations are clustered, dependent, or highly skewed with limited sample sizes, this standard error and interval may not be appropriate; an analysis accounting for the dependence or distribution is needed.

Worked Example: Does a Sample Premium Gap Clearly Differ From Zero?

In a hypothetical review, 300 independent risks in Group A have sample mean annual premium $1,250 and sample variance $ \$40{,}000$. Another 300 independent risks in Group B have sample mean $1,200 and sample variance $ \$75{,}000$. Estimate the difference A minus B and calculate an approximate 95% confidence interval. Assume the normal approximation above is appropriate.

Method and assumptions. Use the difference of sample means and the independent-samples standard error. Variances are in dollars squared; the resulting standard error and interval endpoints are in dollars.

Estimated difference. $\$1{,}250-\$1{,}200=\$50.00$.

Variance of the estimated difference. $\$40{,}000/300=\$133.3333$, and $\$75{,}000/300=\$250.0000$. Their sum is $\$383.3333$ in squared-dollar units.

Standard error. $\sqrt{383.3333}=\$19.5789$, approximately $19.58.

Margin of error. $1.96(\$19.5789)=\$38.3746$.

Interval. The lower endpoint is $\$50.00-\$38.3746=\$11.6254$, and the upper endpoint is $\$50.00+\$38.3746=\$88.3746$. Rounded to cents only at the final step, the approximate interval is $[\$11.63,\$88.37]$.

Under the assumptions, this interval is entirely above zero. That is evidence of a positive difference in these population mean premiums under the model, not an explanation of the difference or a conclusion about its fairness. Common mistake avoided: the point estimate is $50, not $10, and the interval endpoints must be found by subtracting from and adding to $50. Reversing that center or using a standard error as if it were the margin of error gives an incorrect interval.

Choose Follow-Up Analyses That Address the Evidence

An audit is most useful when each follow-up is tied to a plausible source of the observed pattern. A large crude gap that shrinks after standardization suggests that composition matters, but does not establish that composition is the only relevant factor. A gap that remains within several segments raises different questions: whether segment definitions and model controls are adequate, whether coverage and exposure are comparable, and whether the estimates are sufficiently stable.

Specify the estimand.
State the groups, period, coverage, exposure unit, outcome, and whether the question concerns average charged premium, indicated premium, or loss cost.

Reconcile the comparison basis.
Check that the groups use comparable exposure definitions, coverage limits, deductibles, term treatment, and premium basis before interpreting a difference.

Examine composition and segments.
Compare relevant characteristics and calculate standardized results using a stated common mix. Inspect segment-level results rather than relying only on a single aggregate.

Assess stability and uncertainty.
Review exposure volume, claim experience, variation, and dependence. Use uncertainty measures suited to the data, and avoid interpreting a noisy estimate as a settled pattern.

Form targeted next tests.
Depending on the pattern, test alternative standardization choices, examine relevant model variables and interactions, check data or coverage differences, and document what each analysis can and cannot establish.

These steps keep measurement separate from interpretation. For example, if standardized premiums remain different but loss costs do not, investigate the rate plan’s components and the basis of the loss-cost comparison. If both remain different, examine whether material risk characteristics or exposure conditions have been accounted for, and whether the differences are estimated with adequate precision. If confidence intervals are wide, additional or more carefully pooled experience may be more informative than a categorical conclusion.

Common Audit Traps

Comparing totals instead of rates. A larger group can have a larger aggregate premium simply because it has more exposure. Divide by a consistent exposure measure before comparing averages.
Treating a standardized result as definitive. Standardization controls only for the variables and weights used. It cannot resolve omitted characteristics, poor measurement, or a poorly chosen reference mix.
Ignoring uncertainty. A point estimate alone does not indicate how stable the observed difference is. Small or sparse subgroups need particular caution.
Overinterpreting a confidence interval. An interval addresses sampling uncertainty under its assumptions. It is not a test of causation, legality, or the full fairness of a plan.
Switching the comparison direction. State whether the difference is A minus B or B minus A, and keep that order in the calculation and interpretation.

Key takeaway: A defensible disparity review makes the comparison basis explicit, reports crude and appropriately standardized results, evaluates uncertainty, and proposes follow-up analyses matched to the observed pattern. A measured gap is a signal to investigate, not a standalone explanation.

Check Your Understanding

Work through each hypothetical audit question and check the final answer.

Group C has 120 earned exposures and $102,000 aggregate earned premium; Group D has 80 exposures and $76,000 premium. Calculate each average premium per exposure and the C-minus-D difference. (Answer: C = $850.00; D = $950.00; difference = -$100.00 per exposure.)
Two groups have urban/rural premiums of $1,100/$500 and $1,000/$600, respectively. Standardize both to 40% urban and 60% rural. What are the standardized averages and difference, first group minus second? (Answer: first = $740; second = $760; difference = -$20.)
Group A is 70% in a high-premium segment and 30% in a low-premium segment; Group B is 30% and 70%. Segment premiums are $1,400 and $800 for A, and $1,300 and $700 for B. Calculate both crude averages and the crude A-minus-B gap. (Answer: A = $1,220; B = $880; gap = $340.)
Independent groups each have 100 observations. Their sample means are $980 and $950, and their sample variances are $10,000 and $15,000. Using the stated normal approximation, calculate the standard error and approximate 95% interval for the first-minus-second difference. (Answer: difference = $30; standard error = $\sqrt{250}=\$15.81$; interval approximately $[-\$0.99,\$60.99]$.)
A crude premium gap is large, falls substantially after standardizing to a common mix, and the remaining within-segment estimates have wide intervals. Name two focused follow-up analyses and explain the immediate limitation on the conclusion. (Answer: check the segment weights and coverage/exposure comparability, and assess segment-level stability or obtain more experience; the crude gap is partly associated with composition, but the noisy residual does not support a precise conclusion about the remaining difference.)

What You'll Learn