Why Examine Residuals by Class?
A selected set of class factors is intended to differentiate expected losses among risks. After applying those factors, an actuary can compare the losses actually observed in each class with the losses the model predicted. This comparison asks a practical question: after allowing for the selected factors, which classes still appear underpredicted or overpredicted?
A portfolio-wide comparison can hide important differences. If one class has losses above prediction and another has losses below prediction, their deviations may partly offset in the total. Class-level residual diagnostics preserve that detail. They are a screening tool: a persistent, interpretable pattern may point to remaining classification or model error, but it is not, by itself, proof of the cause.
Building a Class-Level Comparison
A simple frequency-severity or pure-premium prediction can be written as exposure multiplied by an expected loss cost. If the selected class factors are multiplicative, predicted losses for a class are
where \(E_c\) is the class exposure on the chosen basis, \(L_0\) is the base expected loss cost per exposure, and \(r_{jc}\) is the selected factor for the class's level of classification variable \(j\). This formula assumes that the base loss cost and selected factors are appropriate for the same coverage, time period, exposure definition, and loss basis as the observations. It also assumes the selected factors are applied as specified; a model with a different structure requires predictions from that structure instead.
Before calculating a ratio, confirm that observed and predicted losses are comparable. For example, the observed amount might be developed or trended, limited to a specified coverage, or adjusted for a change in benefit terms. The prediction must reflect that same basis. Likewise, exposures must use the measure intended by the model. Comparing losses from one basis with predictions from another can create an apparent residual pattern that is only a data mismatch.
The interpretation is direct but limited. An \(O_c/P_c\) of 1.10 means observed losses are 10% greater than predicted losses for that class. It does not mean that every risk in the class is underpriced by 10%, nor does it identify why the difference arose. Random fluctuation, unusual large losses, an exposure-data issue, or a missing model feature could all contribute.
Worked Example: Applying One Selected Class Factor
A hypothetical insurer uses a base expected loss cost of $400 per exposure and selected factors of 0.80, 1.00, and 1.25 for three classes. In a common analysis period, the classes have 100, 80, and 40 exposures, respectively. Observed losses on the same basis are $35,200, $28,800, and $23,000. Calculate each class's predicted losses, residual, and observed-to-predicted ratio, then compare the total ratio. Use four decimals for ratios.
Method and assumptions. Use exposure multiplied by the base loss cost and the selected class factor. Assume the observed losses and exposure counts are complete and comparable with the predictions.
Class 1. Predicted losses are \(100\times \$400\times0.80=\$32{,}000\). The residual is \(\$35{,}200-\$32{,}000=\$3{,}200\). The ratio is \(\$35{,}200/\$32{,}000=1.1000\), or 10% above prediction.
Class 2. Predicted losses are \(80\times \$400\times1.00=\$32{,}000\). The residual is \(\$28{,}800-\$32{,}000=-\$3{,}200\). The ratio is \(\$28{,}800/\$32{,}000=0.9000\), or 10% below prediction.
Class 3. Predicted losses are \(40\times \$400\times1.25=\$20{,}000\). The residual is \(\$23{,}000-\$20{,}000=\$3{,}000\). The ratio is \(\$23{,}000/\$20{,}000=1.1500\), or 15% above prediction.
Total check. Total predicted losses are \(\$32{,}000+\$32{,}000+\$20{,}000=\$84{,}000\). Total observed losses are \(\$35{,}200+\$28{,}800+\$23{,}000=\$87{,}000\). The aggregate ratio is \(\$87{,}000/\$84{,}000=1.0357\). The overall result is only 3.57% above prediction, even though two classes are more substantially above prediction and one is below it.
Common mistake avoided. The aggregate ratio does not replace the class ratios. Reporting only 1.0357 would conceal the opposing class-level deviations that may warrant investigation.
Why the Aggregate Ratio Is Predicted-Loss Weighted
Class ratios should not generally be averaged with equal weights. The aggregate observed-to-predicted ratio follows from the definition: total observed losses divided by total predicted losses. Since \(O_c=P_c(O_c/P_c)\), summing across classes gives
Thus the aggregate ratio is a weighted average of the class ratios, with each class's predicted losses as its weight. Equivalently, because \(R_c=O_c-P_c\),
This identity shows why the ratio describes the residual relative to the predicted amount. It also shows why the same dollar residual can have different significance in classes with different predicted losses.
Reading Patterns Without Overinterpreting Them
A residual table is most useful when the comparisons are organized around a meaningful question. For example, the analyst might inspect ratios by one classification variable, then by another, or by combinations of levels where sufficient experience exists. If one level is consistently above 1 across several relevant groupings, that is a possible signal that its expected losses are not fully captured. If deviations appear only in one small cell, random variation or unusual claims may be a more plausible explanation than a broad class effect.
Residual patterns also depend on the data and model structure. A factor applied to a broad class may predict its average adequately while missing differences among subgroups. Conversely, an apparent pattern may disappear once the data are checked for incorrect class assignments, exposure errors, or losses recorded on a different basis. A diagnostic is evidence to investigate, not a mechanical instruction to change a factor.
Specify the loss measure, exposure measure, coverage, and period. Apply the same basis to observed and predicted amounts.
Apply the selected factors to the relevant base expected loss cost and exposures, using the model's stated structure.
For each class, report \(O_c-P_c\) and \(O_c/P_c\). State whether the class is above or below prediction.
Look for consistent differences across related groups, but consider data quality, exposure volume, large losses, and random fluctuation before drawing conclusions.
Worked Example: Looking for a Pattern Across Two Variables
A hypothetical fleet program applies an age factor and a use factor to a base expected loss cost of $500 per exposure. The selected age factors are 1.40 for novice operators and 0.80 for experienced operators. The use factors are 1.20 for business use and 0.90 for pleasure use. The table gives exposures and observed losses. Calculate predicted losses and observed-to-predicted ratios for each combination, then summarize by age group and use.
| Combination | Exposures | Observed losses |
|---|---|---|
| Novice, business | 20 | $20,160 |
| Novice, pleasure | 50 | $31,500 |
| Experienced, business | 30 | $12,960 |
| Experienced, pleasure | 100 | $32,400 |
Method and assumptions. Multiply exposure by $500 and both applicable factors. Assume the combination factors are multiplicative and observed losses use the same basis as the predictions.
Novice, business. Prediction: \(20\times \$500\times1.40\times1.20=\$16{,}800\). Ratio: \(\$20{,}160/\$16{,}800=1.2000\).
Novice, pleasure. Prediction: \(50\times \$500\times1.40\times0.90=\$31{,}500\). Ratio: \(\$31{,}500/\$31{,}500=1.0000\).
Experienced, business. Prediction: \(30\times \$500\times0.80\times1.20=\$14{,}400\). Ratio: \(\$12{,}960/\$14{,}400=0.9000\).
Experienced, pleasure. Prediction: \(100\times \$500\times0.80\times0.90=\$36{,}000\). Ratio: \(\$32{,}400/\$36{,}000=0.9000\).
Summarize by age. Novice predictions total \(\$16{,}800+\$31{,}500=\$48{,}300\); observed losses total \(\$20{,}160+\$31{,}500=\$51{,}660\). The ratio is \(\$51{,}660/\$48{,}300=1.0696\). Experienced predictions total \(\$14{,}400+\$36{,}000=\$50{,}400\); observed losses total \(\$12{,}960+\$32{,}400=\$45{,}360\). The ratio is \(\$45{,}360/\$50{,}400=0.9000\).
Summarize by use. Business predictions total \(\$16{,}800+\$14{,}400=\$31{,}200\), with observed losses of \(\$20{,}160+\$12{,}960=\$33{,}120\), for a ratio of 1.0615. Pleasure predictions total \(\$31{,}500+\$36{,}000=\$67{,}500\), with observed losses of \(\$31{,}500+\$32{,}400=\$63{,}900\), for a ratio of 0.9467.
Interpretation and check. The largest cell ratio is novice business at 1.2000, while both experienced cells are at 0.9000. This pattern may motivate a closer review of the selected factors or the underlying data, but does not establish the cause. Total predicted losses are \(\$98{,}700\), and total observed losses are \(\$97{,}020\), giving an overall ratio of 0.9830. The near-one aggregate therefore masks meaningful differences among cells. The common mistake avoided is to infer that all classes fit well just because the overall ratio is close to 1.
Unequal Class Sizes and a Common Averaging Trap
Classes often have different exposure volumes and expected losses. An arithmetic mean of class ratios gives a small class the same influence as a large class, which is not how total observed and predicted losses combine. The next example makes that distinction explicit.
Worked Example: Unequal Exposures and Offsetting Deviations
For a hypothetical product, the base expected loss cost is $500 per exposure. A moderate class has 120 exposures and factor 0.80; a high-risk class has 40 exposures and factor 1.50. Observed losses are $52,800 and $25,500, respectively. Calculate both class ratios and the aggregate ratio. Use four decimals for ratios.
Method and assumptions. Predict each class's losses as exposure times $500 times its selected factor. The two observed amounts are assumed comparable with these predictions.
Moderate class. Predicted losses are \(120\times \$500\times0.80=\$48{,}000\). The ratio is \(\$52{,}800/\$48{,}000=1.1000\).
High-risk class. Predicted losses are \(40\times \$500\times1.50=\$30{,}000\). The ratio is \(\$25{,}500/\$30{,}000=0.8500\).
Aggregate. Predicted losses total \(\$48{,}000+\$30{,}000=\$78{,}000\). Observed losses total \(\$52{,}800+\$25{,}500=\$78{,}300\). The aggregate ratio is \(\$78{,}300/\$78{,}000=1.0038\).
Common mistake avoided. The unweighted mean of class ratios is \((1.1000+0.8500)/2=0.9750\), which is not the aggregate ratio. The correct aggregate is weighted by predicted losses: \((\$48{,}000\times1.1000+\$30{,}000\times0.8500)/\$78{,}000=1.0038\). The near-one aggregate conceals the moderate class's 10% excess and the high-risk class's 15% shortfall.
Common Diagnostic Mistakes
- Using premium as the prediction denominator without checking its meaning. The defined diagnostic here compares observed losses with predicted losses. A loss ratio using premium is a different measure and depends on the premium basis.
- Comparing inconsistent amounts. Observed losses and predictions must refer to the same coverage, period, loss adjustments, and exposure basis.
- Interpreting a ratio without its size or context. A ratio from a small class can be volatile. Examine exposure volume and the underlying loss experience alongside the ratio.
- Treating an above-one ratio as proof of a wrong factor. Data errors, random variation, large losses, and omitted risk characteristics can also contribute.
- Relying only on the portfolio total. Opposing class residuals can offset. Report class-level comparisons when the diagnostic question concerns class performance.
Residual diagnostics are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus because they connect selected class factors to observed experience. Their value lies in making model fit visible at the class level while keeping the interpretation appropriately cautious.
Check Your Understanding
For each problem, calculate the requested ratio or interpretation using observed losses divided by predicted losses.
- A hypothetical class has 50 exposures, a base expected loss cost of $300, and a selected factor of 1.20. Its observed losses are $19,800. Calculate predicted losses and the class ratio. (Answer: Predicted losses are $18,000; the ratio is 1.1000.)
- Two hypothetical classes use a base loss cost of $400. Class A has 80 exposures, factor 0.90, and observed losses of $30,240. Class B has 40 exposures, factor 1.40, and observed losses of $20,160. Find each class ratio and the aggregate ratio. (Answer: Class A is 1.0500; Class B is 0.9000; the aggregate is \(50{,}400/51{,}200=0.9844\).)
- A class has predicted losses of $24,000 and observed losses of $21,600. State its signed residual and interpret its ratio. (Answer: Residual \(R=-\$2{,}400\); ratio \(0.9000\), so observed losses are 10% below prediction.)
- Three classes have predicted losses of $10,000, $30,000, and $60,000, with ratios of 1.20, 0.90, and 1.05. Calculate the aggregate ratio. (Answer: \((\$12{,}000+\$27{,}000+\$63{,}000)/\$100{,}000=1.0200\).)
- A class ratio is 1.25 in one period, but the class has few exposures and one unusually large claim. What is a defensible diagnostic conclusion? (Answer: Observed losses are 25% above prediction for that period, warranting review; the ratio alone does not establish that the class factor is wrong.)