Comparing Rating Profiles
A rating table assigns factors to characteristics such as territory, vehicle, use, and experience. When several risks share some characteristics but differ on others, a useful comparison must do more than list their premiums: it should identify which factors drive the difference and express the result in dollars or as a percentage. The calculations below use one hypothetical factor table and one common base rate throughout.
Assume each factor applies multiplicatively to the base rate, the factors are used exactly as shown, and the table contains no interaction adjustments or other premium components. The resulting amount is an indicated annual rate per unit of exposure, before any items outside this simplified table. These assumptions matter: the comparisons describe what this specified rating formula indicates, not a claim that the factors are appropriate for every product.
One Factor Table and a Common Base
For all examples, let the common base rate be $600 per annual exposure unit. The hypothetical rating plan uses the factors below. A factor of 1.00 leaves the rate unchanged; a factor above 1.00 increases it, and a factor below 1.00 decreases it.
| Characteristic | Level | Factor |
|---|---|---|
| Territory | Metro | 1.20 |
| Territory | Small town | 0.95 |
| Territory | Rural | 0.80 |
| Vehicle | Compact | 0.90 |
| Vehicle | Standard | 1.00 |
| Vehicle | Performance | 1.25 |
| Use | Pleasure | 0.85 |
| Use | Commute | 1.10 |
| Use | Business | 1.30 |
| Experience | Established | 0.80 |
| Experience | Neutral | 1.00 |
| Experience | New | 1.25 |
If profile \(j\) has a selected factor \(f_{jk}\) for each of the four characteristics, its indicated rate \(R_j\) is
The product is essential: each selected factor acts on the amount produced by the other factors. To compare two profiles, calculate each rate at full precision, then subtract in the requested direction. If reporting a percentage, state which rate is the denominator; “percent higher” and “percent lower” depend on the comparison base.
Worked Comparisons
Worked Example: Two Profiles With Two Shared Characteristics
A hypothetical annual policy is rated using the table above. Profile A is Metro, Standard, Commute, Established. Profile B is Rural, Performance, Commute, Established. Calculate both indicated rates, the dollar difference \(R_B-R_A\), and the percentage by which B is higher or lower than A.
Method. Select one factor for each characteristic, multiply by the common base, and compare. Commute and Established are shared, so they can also be cancelled when forming the rate ratio.
For A, the selected factors are 1.20, 1.00, 1.10, and 0.80:
For B, the factors are 0.80, 1.25, 1.10, and 0.80:
The requested signed difference is \(R_B-R_A=\$528.00-\$633.60=-\$105.60\). Thus B is $105.60 lower than A. For the relative difference, A is the stated denominator:
Therefore, B is 16.67% lower than A, rounded to two decimal places. As a check, the shared factors give the same ratio directly: \(R_B/R_A=(0.80/1.20)(1.25/1.00)=5/6\), so B’s rate is five-sixths of A’s.
Common mistake avoided: Do not add the changes in territory and vehicle factors as if they were percentage-point adjustments. The factors multiply; their combined ratio is \((0.80)(1.25)/(1.20)(1.00)=5/6\), not the result of adding factor differences.
Worked Example: Ranking Three Profiles With Different Combinations
Three hypothetical risks have these profiles: A is Metro, Compact, Pleasure, New; B is Small town, Standard, Commute, Established; C is Rural, Performance, Business, Neutral. Calculate each indicated rate, rank them, and find the dollar gap between the highest and lowest rates.
Method. Compute each profile independently from the same base and factor table. Do not compare only one characteristic: every selected factor contributes to the product.
For A, the factor product is \(1.20(0.90)=1.08\), then \(1.08(0.85)=0.918\), then \(0.918(1.25)=1.1475\). Hence:
For B, the successive products are \(0.95(1.00)=0.95\), \(0.95(1.10)=1.045\), and \(1.045(0.80)=0.836\). Hence:
For C, the successive products are \(0.80(1.25)=1.00\), \(1.00(1.30)=1.30\), and \(1.30(1.00)=1.30\). Hence:
The ranking from highest to lowest is C, A, B. The high-to-low dollar gap is \(R_C-R_B=\$780.00-\$501.60=\$278.40\). A final check is that the respective products, 1.30, 1.1475, and 0.836, have the same ranking as their rates because all rates use the positive common base of $600.
Common mistake avoided: Do not infer the ranking from the single largest factor in each profile. A’s New factor is 1.25, but its other selections produce an overall factor of 1.1475; C’s combined factor is 1.30. The entire product, not one component, determines the indicated rate.
Worked Example: Isolating a Difference When Most Factors Match
Two hypothetical risks are both Metro, Business, and New. One has a Standard vehicle and the other a Performance vehicle. Find each rate and the dollar and percentage increase for the Performance profile relative to the Standard profile.
Method. Calculate the common multiplier from the base rate and shared factors, then apply each vehicle factor. This isolates the effect of the differing characteristic while retaining the scale created by all shared characteristics.
The common multiplier is \(600(1.20)(1.30)(1.25)\). First, \(1.20(1.30)=1.56\); then \(1.56(1.25)=1.95\); thus the common multiplier is \(600(1.95)=\$1,170.00\). The Standard profile uses a factor of 1.00, so its rate is \(1,170(1.00)=\$1,170.00\). The Performance profile uses 1.25, so its rate is \(1,170(1.25)=\$1,462.50\).
The dollar increase is \(\$1,462.50-\$1,170.00=\$292.50\). Relative to the Standard rate, the increase is:
The result also follows from the vehicle factors: \(1.25/1.00=1.25\), a 25% increase. The dollar check is \(25\%\) of the common multiplier, or \(0.25(1,170)=\$292.50\).
Common mistake avoided: Applying 25% to the $600 base would give $150, which is not the rate difference here. The vehicle factor changes a rate that already includes the Metro, Business, and New factors; its increase is 25% of $1,170, not 25% of the unadjusted base.
Why Shared Factors Cancel
Cancellation is a convenient way to compare rates, but it follows directly from the multiplicative model. Suppose two profiles have the same selected factors for some characteristics. Let \(C\) be the product of those shared factors, let \(F_A\) and \(F_B\) be the products of the remaining factors for A and B, and let \(B\) be their common positive base rate. Then \(R_A=BCF_A\) and \(R_B=BCF_B\). Dividing gives:
The base and the shared factors cancel in a ratio. For a dollar difference, subtract instead of divide:
This identity explains why a change in one factor can create different dollar changes for profiles with different shared multipliers. The same relative change in a factor applies to each affected rate, but the dollar impact is scaled by the rate before that factor is applied. Cancellation is valid only for factors that are actually identical in the two profiles; if a factor differs, it belongs in the relevant product.
Record the rating level selected for every characteristic for each risk.
Use the same base and multiply all selected factors for each profile.
For a dollar difference, name the subtraction order. For a percentage, name the denominator.
Cancel only identical factors in a ratio, or factor them out of a dollar difference.
Interpretation and Exam Traps
A comparison is clearest when it reports both the numerical result and its direction. “The rates differ by $105.60” does not identify which profile is higher; “Profile B is $105.60, or 16.67%, lower than Profile A” does. For a percentage comparison, the chosen denominator should match the wording. If B is 16.67% lower than A, that does not mean A is 16.67% higher than B: the latter percentage would use B as its denominator.
The calculations also depend on consistent units. Here every rate is annual and per the same exposure unit, so subtracting the rates is meaningful. If profiles had different exposure bases or different periods, those amounts would need to be brought to a comparable basis before interpreting their difference. This tutorial assumes the stated base rate and factors are already applicable to each profile; it does not assess how those factors were selected.
Check Your Understanding
Use the common $600 base rate and factor table above. Round final dollar amounts to the nearest cent and percentages, if requested, to two decimal places.
- A profile is Metro, Compact, Commute, Established. Calculate its indicated rate. (Answer: $570.24)
- Compare a Small town, Performance, Pleasure, Neutral profile with a Rural, Standard, Pleasure, Neutral profile. Which is higher, and by how many dollars? (Answer: The Small town profile is higher by $197.63.)
- Compare Metro, Standard, Business, Established with Small town, Compact, Business, Established. Calculate the dollar difference, with the Metro profile first in the subtraction. (Answer: $215.28; the Metro profile is higher.)
- For a Rural, Performance, New profile, calculate the change in rate when use changes from Pleasure to Commute. Give the dollar increase and the percentage increase relative to Pleasure. (Answer: $187.50 increase, or 29.41%.)
- Rank these profiles from highest to lowest and give the high-to-low dollar gap: Metro, Performance, Pleasure, Established; Rural, Compact, Business, New; Small town, Standard, Commute, Neutral. (Answer: Rural/Compact/Business/New at $702.00, Small town/Standard/Commute/Neutral at $627.00, then Metro/Performance/Pleasure/Established at $612.00; the gap is $90.00.)