Why Classification Changes Need Their Own Support
A filing may support an overall rate change and also propose new classification relativities. The overall indication does not, by itself, show why one class should move more or less than another. A classification exhibit should let a reviewer trace the proposal from the experience by class through indicated factors, selected factors, and the resulting premium change.
The essential comparison uses the same exposure distribution and rate basis for the current and proposed classifications. This isolates the effect of the factor change. If the distribution of risks changes between periods, a separate mix effect may also change premium, but it should not be mistaken for a rate change caused by the revised relativities.
What a Support Exhibit Should Show
A useful exhibit gives enough information to reproduce the calculation and understand the choice. At a minimum, identify the experience period and exposure unit, the class definitions, the loss and expense basis used, and the adjustments already reflected in the experience. Then show the volume of exposure by class, the resulting experience measure, the indicated factors, the selected factors, and the premium comparison at current and proposed rates.
Exposure volumes matter for two distinct reasons. First, they indicate how much experience supports each class result. Second, they determine the weight each class has in the aggregate rate-level calculation. A class with a high relativity but few exposures may have less effect on total premium than a larger class with a smaller relativity change.
- State whether exposure means earned car-years, policy-years, payroll, or another unit, and use it consistently across classes.
- Show losses on the stated, consistently adjusted basis. If the exhibit uses losses already developed or trended for a rate indication, say so.
- Show current and proposed factors alongside the resulting class rates or premiums, so the direction and size of each change are visible.
- Reconcile the class totals to the aggregate premium comparison, and state whether the calculation holds exposure mix fixed.
The calculation below treats the displayed losses as already adjusted to a common basis. It does not redo the development, trend, or credibility selections that may have produced that basis. Those choices should be explained in the supporting filing material.
From Class Experience to Indicated Relativities
For a simple experience-based illustration, let \(E_j\) be exposures in class \(j\), and let \(L_j\) be the corresponding adjusted losses. The class pure premium is \(L_j/E_j\). If the reference pure premium is the overall experience pure premium, an indicated relativity is the class pure premium divided by that reference. This is one transparent approach; a filing should state its actual reference and method.
This unadjusted calculation is appropriate only when the experience measures are comparable across classes and no additional adjustments are needed for the purpose of the exhibit. In practice, the selected experience basis, credibility, and other assumptions must be appropriate to the filing. A calculated factor is not automatically a selected factor.
Worked Example: Calculating Indicated Relativities From Class Experience
A hypothetical insurer has three classes. The displayed losses are already adjusted to a common basis. Class A has 1,000 earned exposure units and $240,000 of losses; Class B has 600 units and $180,000; Class C has 400 units and $180,000. Calculate the overall pure premium and each indicated relativity using the overall pure premium as the reference.
Method. Calculate the class pure premiums, then divide each by the exposure-weighted overall pure premium. This method avoids treating the unweighted average of class results as the book average.
Step 1: Total the exposures and losses. Total exposure is \(1{,}000+600+400=2{,}000\) units. Total loss is \($240{,}000+$180{,}000+$180{,}000=$600{,}000\).
Step 2: Calculate the overall pure premium. The result is \($600{,}000/2{,}000=$300\) per exposure unit.
Step 3: Calculate class pure premiums. Class A is \($240{,}000/1{,}000=$240\). Class B is \($180{,}000/600=$300\). Class C is \($180{,}000/400=$450\).
Step 4: Divide by the overall pure premium. The indicated relativities are A: \($240/$300=0.8000\); B: \($300/$300=1.0000\); and C: \($450/$300=1.5000\).
Check. The exposure-weighted average relativity is \((1{,}000(0.8000)+600(1.0000)+400(1.5000))/2{,}000=(800+600+600)/2{,}000=1.0000\). The factors reconcile to the chosen overall reference. The common mistake avoided is averaging the three class pure premiums equally, despite their different exposure volumes.
Deriving the Aggregate Rate-Level Effect
To measure the effect of changed factors, compare the premium that the same exposures would produce under proposed rates with the premium they produce under current rates. Let \(r_{0j}\) be the base rate applicable to class \(j\), and let \(f_{cj}\) and \(f_{pj}\) be its current and proposed relativities. With exposure \(E_j\), current and proposed premiums are respectively \(E_jr_{0j}f_{cj}\) and \(E_jr_{0j}f_{pj}\). Therefore, the aggregate rate-level effect is:
The justification follows directly from the premium calculation: each class contributes exposure multiplied by its rate. Summing those contributions gives total premium at each rate level; dividing proposed premium by current premium gives the proportional change. Subtracting 1 expresses that ratio as a change. Thus the denominator is current premium, not exposure alone, and the method holds the exposure mix and all non-changing rate components constant.
If the base rate is the same for every class, it cancels from the ratio. If base rates differ, the calculation must retain them or use each class's current premium as its weight. A simple exposure-weighted average of factor changes is generally not the premium change.
Worked Example: Measuring the Rate-Level Effect of Selected Factors
Use the three classes in the prior example, with 1,000, 600, and 400 exposures. A hypothetical $500 base rate applies to all classes. Current relativities are 0.90, 1.00, and 1.40; selected relativities are 0.80, 1.00, and 1.50. Calculate the aggregate effect and each class's rate change.
Method. Calculate current and proposed premium using the same exposures and base rate. The selected factors equal the indicated factors from the experience calculation, but the premium comparison uses the stated current factors as its baseline.
Step 1: Calculate current premium by class. A: \(1{,}000\times$500\times0.90=$450{,}000\). B: \(600\times$500\times1.00=$300{,}000\). C: \(400\times$500\times1.40=$280{,}000\). Total current premium is \($450{,}000+$300{,}000+$280{,}000=$1{,}030{,}000\).
Step 2: Calculate proposed premium by class. A: \(1{,}000\times$500\times0.80=$400{,}000\). B: \(600\times$500\times1.00=$300{,}000\). C: \(400\times$500\times1.50=$300{,}000\). Total proposed premium is \($400{,}000+$300{,}000+$300{,}000=$1{,}000{,}000\).
Step 3: Calculate the aggregate effect. The change is \($1{,}000{,}000/$1{,}030{,}000-1=-0.029126\), or a 2.913% decrease, rounded to three decimal places.
Step 4: Check class changes. A's rate changes from \($500(0.90)=$450\) to \($500(0.80)=$400\), a decrease of \(400/450-1=-11.111\%\). B is unchanged. C changes from \($700\) to \($750\), an increase of \(750/700-1=7.143\%\).
Check. The class premium changes are \(-$50{,}000+$0+$20{,}000=-$30{,}000\), equal to \($1{,}000{,}000-$1{,}030{,}000\). The common mistake is to average class percentage changes or factor values without premium weights. That would not reproduce the \(-2.913\%\) book change.
Separating Relativity Changes From Overall Level
A proposed set of classification factors can change both the distribution of premium among classes and the aggregate premium. Sometimes the intent is to revise the distribution while keeping the overall level unchanged. In that case, normalize the proposed factors using current premium weights. This is different from simply making the unweighted average of the factors equal to 1.
Worked Example: Normalizing Proposed Class Multipliers
A hypothetical book has 500, 300, and 200 exposures in classes A, B, and C. Current rates are $400, $600, and $800 per exposure unit, giving current class premiums of $200,000, $180,000, and $160,000. Preliminary proposed multipliers to those rates are 0.90, 1.10, and 1.30. Normalize the multipliers so the change is premium-neutral on this exposure distribution, then verify the total.
Method. Find the preliminary proposed premium, then divide current total premium by that amount. Multiply every preliminary factor by this common normalization factor.
Step 1: Calculate current total premium. The class amounts sum to \($200{,}000+$180{,}000+$160{,}000=$540{,}000\).
Step 2: Calculate preliminary proposed premium. A contributes \($200{,}000(0.90)=$180{,}000\); B contributes \($180{,}000(1.10)=$198{,}000\); C contributes \($160{,}000(1.30)=$208{,}000\). The total is \($586{,}000\), an unnormalized increase of \($586{,}000/$540{,}000-1=8.519\%\), rounded to three decimal places.
Step 3: Determine the common normalization factor. It is \($540{,}000/$586{,}000=270/293\), or 0.921502 to six decimal places. The normalized multipliers are \(0.90(270/293)=243/293=0.8294\), \(1.10(270/293)=297/293=1.0137\), and \(1.30(270/293)=351/293=1.1980\), rounded to four decimal places.
Step 4: Verify the aggregate premium. Using the exact factors, the class premiums are \($200{,}000(243/293)=$165{,}870.31\), \($180{,}000(297/293)=$182{,}457.34\), and \($160{,}000(351/293)=$191{,}672.35\), each rounded to the nearest cent. Their rounded sum is $540,000.00. Before rounding, the numerator is \($48{,}600{,}000+$53{,}460{,}000+$56{,}160{,}000=$158{,}220{,}000\); dividing by 293 gives exactly $540,000.
Check. The normalized factors preserve the current aggregate premium on the stated mix, while changing the relative distribution across classes. Rounding the factors before the premium calculation can create a small residual, so the exhibit should disclose its rounding convention and reconcile using unrounded factors where possible. The common mistake avoided is normalizing by the simple average of 0.90, 1.10, and 1.30, which ignores the current premium weights.
Selection, Reconciliation, and Common Traps
An exhibit should distinguish the experience result from the proposed choice. If selected factors differ from indicated factors, state the reason and quantify the effect of the selections. A concise comparison table can make the audit trail visible:
| Class | Exposure volume | Indicated factor | Selected factor | Current premium | Proposed premium |
|---|---|---|---|---|---|
| A | Units and period | Calculated result | Proposed result | Using current rate | Using proposed rate |
| B | Units and period | Calculated result | Proposed result | Using current rate | Using proposed rate |
| Total | Sum of class volumes | Method stated | Method stated | Sum of rows | Sum of rows |
The exhibit should identify the base class or normalization convention, and whether displayed factors are rounded. It should also explain any changes in class definitions or exposure assignment that affect comparability. Those details let a reviewer distinguish a changed risk factor from a changed mix of business.
- Wrong: “The average class change is the overall rate effect.” Correction: Calculate total proposed premium divided by total current premium using the same exposures.
- Wrong: Use the number of policies as the weight when the stated exposure unit is car-years. Correction: Match the exposure measure to the rate basis.
- Wrong: Present indicated factors as though they were automatically selected. Correction: Display both, explain selection judgment, and calculate the premium effect from the selected factors.
- Wrong: Attribute a change in premium to revised relativities when the exposure mix also changed. Correction: Hold mix fixed for the rate-level comparison, and show any mix comparison separately.
Clear support for classification changes is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. In practice, the exhibit is also a communication tool: it should make the data basis, factor choices, and aggregate consequence understandable without requiring the reader to infer the calculation.
Check Your Understanding
For each problem, calculate with the stated exposure basis and retain unrounded values until the final answer.
- A hypothetical class has 400 exposure units and $96,000 of adjusted losses; a second has 300 units and $90,000. Using combined experience as the reference, calculate the overall pure premium and both indicated relativities. (Answer: Overall pure premium $265.71; relativities 0.9032 and 1.1290.)
- A book has 200 exposures in class A and 100 in B, with a $600 common base rate. Current factors are 1.00 and 1.20; proposed factors are 0.90 and 1.30. Calculate the aggregate rate-level effect. (Answer: Current premium $192,000; proposed premium $186,000; decrease 3.125%.)
- Current class premiums are $120,000 and $80,000. Proposed multipliers are 0.95 and 1.10. Calculate the unnormalized rate-level effect. (Answer: Proposed premium $202,000; increase 1.000%.)
- Using the premiums and multipliers in Question 3, find the common normalization factor and normalized multipliers for a premium-neutral change. (Answer: Normalization factor \(200{,}000/202{,}000=0.990099\); normalized multipliers 0.9406 and 1.0891.)
- A current book has $300,000 premium in class A and $200,000 in class B. Proposed factors relative to current rates are 0.88 and 1.18. Calculate proposed total premium and the aggregate effect. (Answer: Proposed total $500,000; rate-level effect 0.000%.)