What a Rate-Change Review Compares
A proposed rate change can move premiums for two distinct reasons. The overall rate level may change for the portfolio, while the relativities assigned to particular groups may also change. A fairness review should separate these effects before interpreting the resulting premium differences. Otherwise, a portfolio-wide increase can be mistaken for a group-specific change, or a shift in portfolio mix can be mistaken for a change in the rates themselves.
The comparison is meaningful only when the old and proposed premiums refer to the same coverage, exposure basis, rating period, and relevant insured profile. A difference caused by changed limits, deductibles, or other policy terms is not, by itself, a rate change. The examples below use hypothetical insurance products and data.
Separating Rate Level from Relativities
In a simple multiplicative rating structure, a premium for a comparable risk in group \(g\) can be represented as a common base premium \(B\) multiplied by a group relativity \(r_g\). If \(B_0,r_{0,g}\) are the old values and \(B_1,r_{1,g}\) are the proposed values, then:
This is an exact multiplicative decomposition when the premiums use the same rating structure and comparable terms. The first factor captures the overall level movement; the second captures how the group’s price position changes relative to the base. A group’s total premium change is not generally the arithmetic sum of these two percentage changes.
For a multiplicative decomposition, the percentage changes combine through multiplication. If the base rises 10% and a group’s relativity falls 5%, the total factor is \(1.10\times0.95=1.045\), or a 4.5% premium increase—not a 5% increase. Relativities may also be normalized differently between rate plans. In that case, comparing raw relativity numbers can be misleading; compare actual premiums for matched risks or first put the relativities on a compatible reference basis.
Worked Example: Isolating a Group’s Relativity Change
A hypothetical insurer is reviewing rates for two vehicle-use groups. The old base premium is $600. The old relativities are 0.80 for Group A and 1.20 for Group B. The proposed base premium is $660, with proposed relativities of 0.75 and 1.25. Assume coverage and all other rating characteristics are unchanged. Calculate each group’s old and proposed premium and separate the overall level effect from the relativity effect.
Method. Calculate premiums as base times relativity, then factor each group’s total change into the base factor and the relativity factor.
Step 1: Overall level factor. The base changes from $600 to $660, so the overall factor is \(660/600=1.10\), an increase of 10%.
Step 2: Group A. Its old premium is \(600(0.80)=\$480.00\). Its proposed premium is \(660(0.75)=\$495.00\). Thus its total factor is \(495/480=1.03125\), or a 3.125% increase. The relativity factor is \(0.75/0.80=0.9375\), a 6.25% decrease. The decomposition checks: \(1.10(0.9375)=1.03125\).
Step 3: Group B. Its old premium is \(600(1.20)=\$720.00\). Its proposed premium is \(660(1.25)=\$825.00\). Its total factor is \(825/720=1.145833\ldots\), or approximately a 14.58% increase. Its relativity factor is \(1.25/1.20=1.041666\ldots\), an increase of approximately 4.17%. The decomposition checks: \(1.10(1.25/1.20)=1.145833\ldots\).
Check and common mistake avoided. With equal weights, the old average relativity is \((0.80+1.20)/2=1.00\), and the proposed average is \((0.75+1.25)/2=1.00\). The base therefore produces a 10% portfolio increase on this two-group mix, while the group movements differ. It would be wrong to add the 10% level change to Group A’s \(-6.25\%\) relativity change and report 3.75%; the exact combined increase is 3.125% because the factors multiply.
Hold the Portfolio Mix Constant
An average premium can change even when the rates applied to every group are held fixed, simply because the proportions of insureds in the groups change. To assess the rate effect, apply old and proposed premiums to the same fixed group weights. This is a standardized comparison, not a claim that one particular weighting is the only appropriate fairness measure. The weights should match the review question and be disclosed.
If old and proposed average premiums instead use different weights, the difference includes both rate changes and composition changes. One transparent approach is to calculate the proposed average first using the old weights, isolating the rate effect at the old mix. Then compare the proposed average under the new weights with that standardized proposed average to quantify the composition effect along that chosen path.
Worked Example: Separating Rate Effects from a Mix Shift
A hypothetical home-insurance portfolio has two occupancy groups. Old premiums are $500 for Group 1 and $1,000 for Group 2; proposed premiums are $550 and $900, respectively. The old portfolio is 80% Group 1 and 20% Group 2, while the proposed portfolio is 50% in each group. Calculate the observed old and proposed averages, then separate rate and mix effects using the old mix as the standard.
Method. Compute weighted averages, first using the actual weights in each period and then applying the old weights to both rate plans.
Step 1: Old average at the old mix. \(0.80(500)+0.20(1{,}000)=400+200=\$600\).
Step 2: Proposed average at the new mix. \(0.50(550)+0.50(900)=275+450=\$725\). The unstandardized comparison rises by \(\$725-\$600=\$125\), or \(125/600=20.8333\ldots\%\).
Step 3: Proposed average at the old mix. \(0.80(550)+0.20(900)=440+180=\$620\). With mix held fixed, the rate effect is \(\$620-\$600=\$20\), or \(20/600=3.3333\ldots\%\).
Step 4: Mix effect on this path. The proposed average at the new mix exceeds the proposed average at the old mix by \(\$725-\$620=\$105\). The decomposition checks: \(\$20+\$105=\$125\). This attribution uses the old mix as the reference; using a different order for changing rates and weights can allocate an interaction differently.
Check and common mistake avoided. Group 1’s premium rises 10%, and Group 2’s falls 10%. Reporting the 20.83% observed average increase as the rate change would wrongly attribute the shift toward the higher-premium group to the rate plan. The standardized rate effect is 3.33% at the stated old mix. Neither average alone establishes whether the changes are justified or fair; it answers a specific comparison question.
Prioritize Segments for Further Review
A useful review brings together several pieces of evidence rather than treating a single percentage as a verdict. Examine group-level premium changes, the overall level change, exposure counts or weights, the relevant benchmark or expected cost basis, and the rationale for the change. Check whether the group definitions and underlying risk profile are comparable between periods. A segment with a large change may merit investigation even if its exposure is limited; limited data can make the estimate uncertain, not automatically unimportant.
Where an indicated benchmark premium is available, comparing proposed premiums with that benchmark can help locate differences that need explanation. The benchmark must be on a consistent basis—for example, it should reflect the same coverage and expense treatment. A deviation is a review signal, not proof of unfairness: the benchmark may omit legitimate factors, and a premium-to-benchmark comparison does not resolve legal or regulatory requirements. Group averages can also conceal variation within a group, so a favorable average does not rule out concerns for relevant subgroups.
Worked Example: Flagging a Segment Without Treating the Flag as a Finding
A hypothetical insurer proposes new premiums for three equipment-protection segments. The exposure counts, average premiums, and indicated benchmark premiums are shown below. Assume the average premiums are quoted for identical coverage, and the benchmark uses the same expense basis. Calculate each segment’s premium change, the portfolio-weighted change, and proposed premium divided by benchmark. Identify which segment should receive focused follow-up.
| Segment | Exposures | Old premium | Proposed premium | Indicated benchmark |
|---|---|---|---|---|
| Cedar | 400 | $900 | $990 | $950 |
| Birch | 60 | $1,200 | $1,560 | $1,300 |
| Alder | 500 | $700 | $665 | $700 |
Method. Calculate segment-level changes, weight old and proposed premiums by exposures for the portfolio comparison, and compare proposed premiums with the stated benchmark.
Step 1: Segment changes. Cedar changes by \((990-900)/900=0.10\), or +10%. Birch changes by \((1{,}560-1{,}200)/1{,}200=0.30\), or +30%. Alder changes by \((665-700)/700=-0.05\), or -5%.
Step 2: Portfolio-weighted change. The old premium total is \(400(900)+60(1{,}200)+500(700)=360{,}000+72{,}000+350{,}000=\$782{,}000\). The proposed total is \(400(990)+60(1{,}560)+500(665)=396{,}000+93{,}600+332{,}500=\$822{,}100\). The weighted change is \((822{,}100-782{,}000)/782{,}000=0.0512788\ldots\), or approximately +5.13%.
Step 3: Proposed premium relative to benchmark. Cedar’s ratio is \(990/950=1.0421\), or 104.21%. Birch’s is \(1{,}560/1{,}300=1.2000\), or 120.00%. Alder’s is \(665/700=0.9500\), or 95.00%.
Check and common mistake avoided. Birch merits focused follow-up: it has the largest increase and the largest proposed-to-benchmark ratio, while its 60 exposures are fewer than the other groups. That does not establish that Birch is unfairly rated. Review the change’s actuarial basis, group definition, and uncertainty before drawing a conclusion. A common mistake would be to use the +5.13% portfolio average to dismiss Birch’s +30% change; the overall average does not describe each segment.
Review Discipline and Common Traps
A proposed-rate review is strongest when the comparison is explicit and reproducible. State the unit of comparison, the weights used, and whether premiums are actual, average, or standardized. Keep the portfolio-level result distinct from group-level results, and explain which segments need more evidence. A change in a premium relationship is an important observation, but it is not itself a complete fairness assessment or a legal conclusion. The appropriate interpretation depends on the purpose of the review, the relevant groups and metrics, the rating context, and applicable regulatory requirements.
- Adding percentage changes. In a multiplicative plan, combine factors by multiplication. A 10% base increase and a 5% relativity decrease give \(1.10(0.95)=1.045\), a 4.5% increase.
- Comparing different mixes. Use consistent weights when the question concerns the rate effect; otherwise report that composition also changed.
- Comparing incompatible relativities. Normalization or reference-group changes can alter the displayed factors. Use matched premiums or a common basis.
- Treating a flag as a finding. A large change, benchmark deviation, or group disparity identifies a question for review. It does not by itself explain the cause or establish that a rate is unfair.
- Ignoring uncertainty and context. Exposure volume, credibility, data quality, and within-group variation affect how confidently a pattern can be interpreted.
Check Your Understanding
For each hypothetical review, distinguish the requested rate effect from other changes. Answers use the assumptions stated in each question.
- A proposed plan raises its base premium from $400 to $440. A group’s relativity changes from 1.40 to 1.25. For an otherwise unchanged risk, find the old and proposed premiums and the total percentage change. (Answer: old premium $560.00; proposed premium $550.00; decrease of approximately 1.79%.)
- Two groups have old premiums of $700 and $1,400, and proposed premiums of $770 and $1,260. The old mix is 75% Group 1 and 25% Group 2; the new mix is 50% each. Find the old average, the proposed average at the new mix, and the proposed average standardized to the old mix. (Answer: $875.00; $1,015.00; $892.50, respectively.)
- A rating review has three segments: 100 exposures change from $900 to $990, 20 change from $800 to $1,040, and 80 change from $600 to $570. Calculate the exposure-weighted old and proposed totals and their percentage change. (Answer: $154,000.00 old; $165,400.00 proposed; approximately +7.40%.)
- The overall level factor is 1.08, and a particular group’s proposed-to-old relativity factor is 0.88. Calculate the group’s total premium change factor and percentage change. (Answer: factor 0.9504, a 4.96% decrease.)
- A group’s premium rises 20% while the overall rate level rises 10%. Under the multiplicative decomposition, find the group’s relativity change factor and percentage change. (Answer: factor \(1.20/1.10=1.0909\ldots\), an increase of approximately 9.09%.)