Why Variable Selection Matters
A classification variable divides risks using information available for rating, such as building construction, vehicle use, or an insured’s location. The goal is not to create as many groups as possible. It is to identify characteristics that help distinguish expected insurance cost in a way that can be measured consistently and used appropriately.
A variable can appear promising because groups have different observed losses, yet that difference may reflect exposure volume, another characteristic, random fluctuation, or inconsistent data. Conversely, a useful variable may be difficult to apply reliably or may not be suitable for rating. Screening therefore requires more than asking whether two groups have different averages.
What Makes a Variable Useful?
A useful variable should help distinguish expected cost for the coverage being rated. The relevant comparison is not simply the total losses in each group: larger groups naturally tend to have more losses. A basic screening measure is adjusted loss per unit of exposure.
The denominator matters because it puts losses on a comparable scale. For example, a group with twice as many policy-years might have about twice the total losses without having a higher expected cost per policy-year. The estimated loss cost is most informative when the loss and exposure data represent the same coverage, period, and population, and when adjustments for development, trend, or other changes have been handled consistently.
A classification variable need not cause losses to be predictive. It may instead be associated with an underlying characteristic that affects losses. The practical question is whether the variable provides a reliable and appropriate distinction in expected cost, not whether changing the variable itself would change the risk.
Worked Example: Screening a Building Protection Variable
A hypothetical insurer is considering whether monitored fire protection is useful for rating a commercial property policy. After placing losses on a common basis, the review has 800 policy-years with monitored protection and $1,440,000 of adjusted losses, and 1,200 policy-years without it and $1,800,000 of adjusted losses. Calculate the two loss costs and compare them. State what the comparison does and does not establish.
Method. Use adjusted loss per policy-year for each group. This comparison corrects for the different exposure volumes; it does not control for differences in other risk characteristics.
Protected group. $$ \widehat{m}_{P}=\frac{\$1{,}440{,}000}{800\text{ policy-years}}=\$1{,}800\text{ per policy-year}. $$
Unprotected group. $$ \widehat{m}_{U}=\frac{\$1{,}800{,}000}{1{,}200\text{ policy-years}}=\$1{,}500\text{ per policy-year}. $$
The protected group’s estimated loss cost is $300 higher per policy-year. Relative to the unprotected group, the ratio is \(\$1{,}800/\$1{,}500=1.20\), or 20% higher. The arithmetic is checked by multiplying each rate by its exposure: \(800\times\$1{,}800=\$1{,}440{,}000\), and \(1{,}200\times\$1{,}500=\$1{,}800{,}000\).
This result is a screening signal, not evidence that protection increases losses or that the variable should be adopted as stated. The protected properties might differ in occupancy, size, location, or other risk characteristics. The comparison also does not show whether the result is stable or whether protection status is recorded accurately. Those issues require further evaluation.
Common mistake avoided: Comparing total losses and concluding that the protected group costs less because its total, $1.44 million, is below $1.80 million. That comparison ignores the unequal exposure counts and gives the wrong impression about per-exposure cost.
Separate Predictive Signal from Portfolio Mix
A candidate variable can be related to expected cost while a crude comparison suggests the opposite. This can happen when the groups have different mixes of other important characteristics. A useful screening approach is to compare like with like on relevant dimensions, or to standardize the comparison to a common mix.
For a standardized comparison, choose common weights \(w_z\) for the relevant groups \(z\), with \(w_z\geq 0\) and \(\sum_z w_z=1\). If \(m_{c,z}\) is the loss cost for candidate class \(c\) within group \(z\), then the standardized comparison is $$ m_c^{\mathrm{std}}=\sum_z w_z m_{c,z}. $$ This is a weighted average of within-group costs. Its interpretation depends on the selected weights and on whether the chosen groups capture the mix differences relevant to the question. It does not replace a broader review of the variable.
Worked Example: A Crude Comparison Reverses Within-Group Evidence
A hypothetical insurer is screening annual mileage for a vehicle coverage. Adjusted losses and exposures are separated by territory because urban and rural risks have different observed loss costs. In the urban territory, low-mileage risks have 900 policy-years and $2,700,000 of losses; high-mileage risks have 100 policy-years and $400,000. In the rural territory, low-mileage risks have 100 policy-years and $100,000 of losses; high-mileage risks have 900 policy-years and $1,080,000. Compare the crude results with results standardized to equal urban and rural weights.
Method. First calculate each loss cost within territory. Then calculate crude loss costs by combining each mileage group’s total losses and exposures. Finally, average the two within-territory costs using weights of 50% each for both mileage groups. This standardization is a diagnostic comparison, not a final rating calculation.
Within urban territory. Low mileage: \(\$2{,}700{,}000/900=\$3{,}000\) per policy-year. High mileage: \(\$400{,}000/100=\$4{,}000\) per policy-year.
Within rural territory. Low mileage: \(\$100{,}000/100=\$1{,}000\) per policy-year. High mileage: \(\$1{,}080{,}000/900=\$1{,}200\) per policy-year. Within both territories, the high-mileage group has the higher loss cost.
Crude low-mileage result. Total losses are \(\$2{,}700{,}000+\$100{,}000=\$2{,}800{,}000\). Total exposure is \(900+100=1{,}000\) policy-years. The crude loss cost is \(\$2{,}800{,}000/1{,}000=\$2{,}800\).
Crude high-mileage result. Total losses are \(\$400{,}000+\$1{,}080{,}000=\$1{,}480{,}000\). Total exposure is \(100+900=1{,}000\) policy-years. The crude loss cost is \(\$1{,}480{,}000/1{,}000=\$1{,}480\). Crude data therefore make high mileage appear less costly.
Equal-territory standardization. Low mileage: \(0.5(\$3{,}000)+0.5(\$1{,}000)=\$2{,}000\). High mileage: \(0.5(\$4{,}000)+0.5(\$1{,}200)=\$2{,}600\). With the same territory weights for each mileage group, the standardized high-mileage cost is $600 higher, or \(\$2{,}600/\$2{,}000=1.30\) times the low-mileage cost.
Check. The crude result is consistent with the portfolio composition: 90% of low-mileage exposure is urban, compared with only 10% of high-mileage exposure. The standardized calculation removes that particular difference in territory mix; it does not prove that mileage is independently predictive after accounting for every relevant factor.
Common mistake avoided: Selecting or rejecting mileage based only on the crude rates. The reversal shows why the mix of other risks must be examined before interpreting a group difference.
Measurability, Stability, and Practical Suitability
Predictive evidence is only one part of variable selection. A rating variable also needs a clear definition that can be applied consistently. Ask whether the information is available at the time a quote is produced, whether its source is dependable, how missing or conflicting records are handled, and whether the classification can be explained and administered.
Stability has more than one meaning. First, the variable’s value may change for a particular risk over time. Some changes are legitimate, such as a change in vehicle use; others may reflect noisy measurement or arbitrary threshold crossings. Second, the relationship between the variable and expected cost may change over time. A variable can be measured reliably yet become a weaker predictor as circumstances change. Both kinds of stability matter.
A variable should also be reviewed for appropriate use. Consider whether it is legally permitted, whether its use is consistent with applicable rules, whether it creates unintended incentives, and whether its distinctions are proportionate to the evidence. A proxy can raise suitability concerns even when it is statistically associated with losses. Statistical predictive power alone does not resolve those questions.
Worked Example: Testing an Odometer-Based Variable
A hypothetical insurer is considering annual mileage bands for a personal vehicle product. Among 600 renewal records, 570 have a mileage value that can be verified. In a separate audit of 300 records with verified values, 18 differ from a repeat measurement by more than the insurer’s stated tolerance. Among 2,000 vehicles observed at consecutive renewals, 360 move to a different mileage band. Assess these figures as screening evidence.
Method. Calculate the verified-record rate, the audit disagreement rate, and the band-transition rate. These measures describe data quality and movement; none is a direct measure of predictive value.
Verified-record rate. \(570/600=0.95\), so 95% of the renewal records have verified mileage. The missing or unverified share is \(600-570=30\) records, or \(30/600=5\%\).
Audit disagreement rate. \(18/300=0.06\), or 6% of audited records exceed the stated tolerance. The other \(300-18=282\) records, or 94%, do not exceed it.
Band-transition rate. \(360/2{,}000=0.18\), or 18% of vehicles move to a different band between observations. The remaining 1,640, or 82%, stay in the same band.
These results suggest that mileage is available for most, but not all, renewals and that some measurements may disagree. The transition rate alone does not establish instability: a vehicle may genuinely have changed use, and a small change in measured mileage may cross a band boundary. The insurer should distinguish genuine changes from measurement error, assess how missing values would be handled, and test whether loss-cost differences persist across periods. The variable should not be accepted merely because 95% of records are verified.
Common mistake avoided: Treating the 18% transition rate as proof that the variable is unreliable. A change in classification can reflect a real change in the risk, so the reason for movement matters.
A Structured Evaluation
A disciplined review makes the decision easier to explain and less vulnerable to misleading evidence. The following sequence keeps predictive performance separate from data and implementation concerns.
Specify what is measured, its units, the source, the classification rules, and when the information is available.
Use consistent exposure definitions and loss adjustments. Do not compare raw totals across groups of different sizes.
Compare loss costs and examine whether differences remain meaningful when relevant portfolio-mix differences are considered.
Measure missingness, audit disagreement, changes in classification, and persistence of the observed relationship over time.
Consider administration, customer impact, applicable rules, and whether the distinction can be supported and applied consistently.
The order is useful because it prevents an attractive loss pattern from ending the analysis prematurely. A variable with apparent predictive value can fail because it is unavailable at quote time, poorly defined, unstable, or inappropriate for use. A variable with reliable data can also add little if its apparent effect disappears when the portfolio mix is examined.
Common Exam Traps
- Using total losses as the comparison. Divide by a consistent exposure measure before comparing groups.
- Confusing association with causation. A predictive relationship does not show that the classification characteristic itself causes the losses.
- Ignoring mix differences. Different distributions of territory or other characteristics can distort a crude comparison.
- Calling all movement instability. A changed class may reflect a genuine change in the insured risk, not faulty measurement.
- Equating statistical usefulness with suitability. Availability, reliability, operational use, and applicable constraints also matter.
Check Your Understanding
For each problem, distinguish the arithmetic result from what it does and does not establish about the candidate variable.
- A hypothetical property group has $450,000 of adjusted losses over 300 policy-years. A second group has $540,000 over 360 policy-years. Calculate each loss cost and state whether these figures alone show differentiation. (Answer: Both are $1,500 per policy-year; these figures show no difference in the observed loss costs but do not establish that the variable has no predictive value in other data or after further analysis.)
- A proposed rating field is verified for 570 of 600 policies. In an independent audit, 18 of 300 records exceed a measurement tolerance. Calculate the verification and disagreement rates. (Answer: 95.00% verified; 6.00% exceed the audit tolerance.)
- Within two territories, the low-use group has loss costs of $2,400 and $1,200, while the high-use group has loss costs of $3,000 and $1,500, respectively. Standardize both groups using equal territory weights. (Answer: Low use, $1,800; high use, $2,250; the standardized high-use cost is 1.25 times the low-use cost.)
- A candidate variable has estimated relative costs of 1.18, 1.22, and 1.20 in three successive review periods. Calculate the range and give one interpretation and one limitation. (Answer: The range is 0.04; the estimates are close across the periods, which is encouraging evidence of temporal stability, but this alone does not establish data quality or appropriateness for rating.)
- A risk characteristic appears to separate observed loss costs, but it is unavailable when most customers request a quote and is recorded differently by two data sources. Name two suitability concerns and one next step. (Answer: Quote-time availability and inconsistent measurement are concerns; define a consistent source and rule, then assess whether the information can be obtained reliably when needed.)