Why Explanations Need More Than a Factor Label
A rating factor table can show that a factor is 1.15, but that number alone does not tell a consumer or regulator what the factor means. A useful explanation identifies the comparison behind the number, translates it into a premium effect, describes the evidence supporting its use, and states what the evidence does not establish. Without those details, an accurate number can still be misunderstood.
This tutorial focuses on explaining factors in an existing hypothetical rating plan. It does not treat a factor as proof that a customer’s behavior caused a loss, nor does it replace the required analysis of a proposed new variable. These explanations are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Explain the Comparison Before the Number
In a multiplicative rating plan, a premium can be represented as a base amount multiplied by rating factors. For an illustration with two factors,
Here, \(B\) is the base premium and \(r_1\) and \(r_2\) are the selected relativities. A factor of 1.15 means 15% above its own reference level, with the other components of this calculation held fixed. It does not mean that the total premium is 15% higher than every other customer’s premium. Other factors may differ, and the comparison depends on the rating plan’s reference levels.
For a change from factor \(r_{\text{old}}\) to \(r_{\text{new}}\), holding the base and all other factors fixed, the premium ratio is \(r_{\text{new}}/r_{\text{old}}\). Therefore the percentage change from the old premium is that ratio minus 1, not necessarily the difference between the two factor values. The dollar change is the old premium multiplied by the ratio minus 1.
Why this holds: Write the old premium as \(P_{\text{old}}=C r_{\text{old}}\), where \(C\) is the product of the base and every unchanged factor. The new premium is \(P_{\text{new}}=C r_{\text{new}}\). Dividing cancels \(C\), giving \(P_{\text{new}}/P_{\text{old}}=r_{\text{new}}/r_{\text{old}}\). Subtracting the old premium gives the dollar-change formula. The result requires all other inputs and the rating-plan structure to remain unchanged.
Worked Example: Translating a Relativity Change into Dollars
Method and assumption. A hypothetical policy has a base premium of $720. Its territory factor is 1.15 and its use factor is 0.90. The territory factor is revised to 1.25, with the base, use factor, and all other rating inputs unchanged. Calculate the old premium, new premium, dollar change, and percentage change. This fixed-input comparison avoids treating a factor-point difference as the percentage change in premium.
Old premium. Multiply the base by both old factors: \(720 \times 1.15 = 828.00\), then \(828.00 \times 0.90 = \$745.20\).
New premium. Replace only the territory factor: \(720 \times 1.25 = 900.00\), then \(900.00 \times 0.90 = \$810.00\).
Dollar change. \(810.00-745.20=\$64.80\).
Percentage change. The factor ratio is \(1.25/1.15=1.0869565\). Subtracting 1 gives 0.0869565, or 8.6957%. The same result follows from \(\$64.80/\$745.20=0.0869565\). To the nearest cent for premiums and four decimal places for the percentage, the new premium is $810.00, an increase of $64.80 or 8.6957%. The check is that multiplying $745.20 by 1.0869565 gives approximately $810.00.
Common mistake avoided: The factor rose by 0.10, but that is not a 10% premium increase from the old rating. The appropriate ratio is \(1.25/1.15\), because the old factor was already above 1.00.
Describe Evidence Without Overclaiming
A defensible explanation describes how the factor was evaluated, rather than presenting a factor value as self-explanatory. Depending on the rating plan and available analysis, the supporting record may identify the data period and population, the outcome being modeled, how exposures and losses were aligned, the model specification, and relevant validation or fairness diagnostics. The explanation should distinguish results actually observed or measured from conclusions that the analysis cannot establish.
A factor estimated while accounting for other variables is ordinarily a conditional association in that model: it compares the model’s indicated results at specified values of the other variables. That is not automatically a causal effect. Even if the model predicts differences in expected loss, the factor alone does not show that changing a policyholder’s characteristic would change that person’s losses. A clear explanation says what was compared and does not substitute causal language such as “causes” or “saves” unless the evidence and design support that claim.
Observed averages can also differ from a model’s adjusted comparison because groups may have different mixes of other rating characteristics. A standardized comparison makes that dependence visible by applying a common distribution of those characteristics to each group. The chosen common distribution matters, so it should be stated. The following simplified calculation illustrates the communication issue; it is not a substitute for a full model analysis.
Worked Example: Explaining an Adjusted Comparison When Group Mixes Differ
Method and assumptions. A hypothetical insurer reviews loss per exposure for policies with and without a specified safety feature. Its experience is split between two territories. Assume the table’s exposure and loss totals are correctly aligned for the same period, and use an equal 50%/50% territory mix as a common comparison. Calculate territory averages, the standardized averages, and their ratio. This method avoids describing the unadjusted overall difference as the factor’s adjusted effect.
| Territory | Feature | Exposures | Losses |
|---|---|---|---|
| Low-cost | Yes | 80 | $32,000 |
| Low-cost | No | 20 | $10,000 |
| High-cost | Yes | 20 | $16,000 |
| High-cost | No | 80 | $72,000 |
Step 1: Calculate each territory average. In the low-cost territory, the average with the feature is \(\$32{,}000/80=\$400\); without it, \(\$10{,}000/20=\$500\). In the high-cost territory, the corresponding averages are \(\$16{,}000/20=\$800\) and \(\$72{,}000/80=\$900\).
Step 2: Apply the common mix. With an equal territory mix, the standardized average for feature-yes policies is \(0.50(\$400)+0.50(\$800)=\$600\). For feature-no policies it is \(0.50(\$500)+0.50(\$900)=\$700\).
Step 3: Compare the standardized results. The ratio is \(\$600/\$700=0.8571429\). On this simplified comparison, the feature-yes group has a standardized average 14.2857% lower than the feature-no group. The unadjusted totals happen to be $48,000 over 100 exposures, or $480, for the yes group, and $82,000 over 100 exposures, or $820, for the no group. Those totals use very different territory mixes, so their ratio, \(480/820=0.5854\), is not the standardized comparison.
A responsible summary would say that the standardized comparison is about 0.8571 under the stated equal-territory mix, in this simplified hypothetical experience. It would not claim that the feature itself caused a 14.2857% loss reduction. The calculation does not establish causality, uncertainty, or how the factor performs after accounting for other variables. Those limitations must be considered before describing a selected rating factor.
Common mistake avoided: Comparing the raw averages of $480 and $820 ignores the different territory distributions. Standardizing makes the comparison explicit; it does not, by itself, prove a causal explanation.
State Uncertainty and Scope for Each Audience
Regulators generally need enough detail to reproduce and assess the actuarial reasoning: the factor definition, reference level, model role, relevant data scope, validation and fairness results, and material caveats. A consumer explanation is usually shorter and should translate the factor into understandable premium terms. In either setting, clarity requires separating a model result from a promise about an individual outcome.
If an estimate is uncertain, state what the uncertainty measure describes. For example, an interval around an estimated relativity concerns uncertainty in that estimate under the stated model and assumptions; it is not a range guaranteed to contain an individual policyholder’s future losses or premium. Disclose material limits such as a restricted observation period, limited data for a subgroup, sensitivity to modeling choices, or an assumption about future conditions when relevant. Avoid presenting a rounded factor as more precise than its evidence supports.
Worked Example: Explaining a Factor with an Uncertainty Interval
Method and assumptions. A hypothetical model gives a factor of 1.08 for a category relative to a reference factor of 1.00. The uncertainty interval for the estimated factor is 0.98 to 1.19. A policy’s base premium is $1,000, and its other unchanged factors multiply to 0.85. Assume the interval describes uncertainty in the estimated factor under the model, not an individual loss range. Calculate the point premium, the premium range obtained by varying only this factor over the interval, and a careful interpretation.
Point estimate. The amount before applying this factor is \(1{,}000 \times 0.85=\$850.00\). The point premium is \(\$850.00 \times 1.08=\$918.00\), which is $68.00 above the $850.00 reference-factor premium. The relative increase is \(1.08/1.00-1=8\%\).
Interval endpoints. At 0.98, the premium is \(\$850.00 \times 0.98=\$833.00\). At 1.19, it is \(\$850.00 \times 1.19=\$1{,}011.50\). The interval corresponds to a factor-based premium difference from the reference of \(-\$17.00\) to \(+\$161.50\). These amounts are to the nearest cent.
A careful explanation would report the 1.08 point factor and its 0.98-to-1.19 interval, and state that this interval reflects uncertainty in the factor estimate under the specified model. It would not imply that every policy in the category will experience an 8% change in losses, or that the interval is an individual premium prediction interval. This avoids treating uncertainty in an estimated factor as certainty about a particular policyholder.
A Practical Structure for a Clear Explanation
A concise explanation can be organized as a sequence. The order helps prevent a reader from seeing a factor value before learning what comparison it represents.
Define the category or value being discussed and identify the comparison level represented by 1.00.
Say which other rating inputs are held fixed and whether the result is an unadjusted average, a standardized comparison, or a model-based conditional relativity.
Use the applicable old-to-new factor ratio for a change, or compare the factor with 1.00 for a reference-level comparison. Give a dollar illustration with its assumptions.
Identify the relevant experience, methods, and checks actually used, without implying that any one diagnostic proves every aspect of the factor’s appropriateness.
Describe uncertainty and scope, and distinguish predictive association from causal effect where that distinction matters.
This structure also helps keep a consumer-facing explanation readable. A short explanation need not reproduce a technical filing, but it should not omit a caveat that changes the meaning of the premium comparison. A regulator-facing explanation can provide more detail while retaining the same definitions and comparison.
Common Explanation Errors
- Calling a factor difference a premium percentage. A move from 1.15 to 1.25 is not a 10% increase from the old factor level. The percentage change is \(1.25/1.15-1\), when all other terms are fixed.
- Leaving the reference implicit. A factor of 1.08 is meaningful only relative to the factor’s defined reference level and within the specified rating plan.
- Using causal language for a predictive association. A model-based difference, even after adjustment, does not alone show that changing the characteristic would change losses.
- Presenting an unadjusted comparison as adjusted evidence. If groups have different mixes of other characteristics, raw averages may not represent a like-for-like comparison.
- Omitting uncertainty and scope. A point estimate without its relevant limitations can sound more certain or broadly applicable than the analysis warrants.
Check Your Understanding
For each problem, state the comparison assumptions before interpreting the result.
- A hypothetical policy has a $800 base premium and factors of 0.75 and 1.10. The second factor changes from 1.10 to 1.20, with all else fixed. Find the old and new premiums, dollar change, and percentage change. (Answer: old premium $660.00; new premium $720.00; increase $60.00, or 9.0909%.)
- In a hypothetical standardization, feature-yes averages are $400 in class A and $900 in class B; feature-no averages are $500 in class A and $1,000 in class B. Use a 50%/50% class mix to calculate both standardized averages and their ratio. (Answer: yes $650; no $750; ratio 0.8667, so the yes average is 13.33% lower under this mix.)
- A hypothetical model gives a factor of 1.04 with an interval from 0.94 to 1.12. The premium subtotal before this factor is $1,250. Calculate the point premium and endpoint premiums, and explain what the interval does not represent. (Answer: point premium $1,300.00; endpoints $1,175.00 and $1,400.00; it is not an individual policyholder’s future-loss or guaranteed premium range.)
- A model-based factor is 0.92 relative to 1.00 after accounting for other specified variables. Write one careful sentence explaining the comparison without making a causal claim. (Answer: “Under the specified model and holding its other inputs fixed, this category has an indicated relativity of 0.92 compared with the 1.00 reference category; this conditional comparison does not by itself establish a causal effect.”)
- A hypothetical policy has a $900 base premium and two unchanged factors, 0.80 and 1.10. A third factor changes from 0.95 to 1.05. Find the old premium, new premium, dollar change, and percentage change. (Answer: old premium $752.40; new premium $831.60; increase $79.20, or 10.5263%.)