Why Exposure Selection Matters
A pure premium indication depends on both the loss amount and the exposure volume used to measure it. An exposure denominator that is too large makes the calculated loss cost too small; one that is too small makes it too large. The central selection task is therefore not simply to choose every exposure record available. It is to choose the records that represent the same insured risk, coverage, and experience interval as the losses in the numerator.
This tutorial focuses on defining that eligible exposure volume. It does not project exposures to a future period or adjust losses for trend. Those are separate calculations. Here, assume the losses supplied for an example are already on the stated basis and that the exposure unit is defined.
For a set of eligible records, let \(L_j\) be the losses on the selected basis for record or period \(j\), and let \(E_j\) be its exposure volume. The pooled pure premium is:
The sum in the denominator must cover precisely the experience represented in the numerator. For example, losses from a six-month interval should not be divided by a full year of exposure for the same risks unless the loss data also represent that full year. The relevant correspondence is determined by the coverage and experience-period definitions; it should not be assumed from the policy start date or the date a report was extracted.
A Practical Selection Process
A disciplined selection process makes the intended basis explicit before calculating a rate. For each record, ask whether its coverage, exposure measure, and time interval are known, and whether its losses are included on the same basis. A complete record may be eligible; a partial record may also be eligible if the partial amount is measured accurately. An incomplete record should be reconciled, reconstructed from reliable detail, or excluded together with its corresponding losses.
Specify the dates, population, and coverage represented by the indication. State the exposure unit, such as vehicle-years or another defined risk unit.
Confirm which losses belong to that population, coverage, and interval. Use the same period convention for numerator and denominator, and document any cutoff applied to both.
Use the exposure amount actually earned or otherwise measured for the selected interval. Do not substitute a full-period amount for a partial period without support.
Reconstruct a missing amount only from reliable source data and consistent rules. If a loss or exposure cannot be matched, do not leave only one side of the pair in the calculation.
Sum eligible losses and eligible exposures separately, divide, and verify that the totals refer to the same selected experience.
Why Partial Exposure Can Be Included
A partial period is not inherently unusable. It contributes the exposure actually at risk if that amount can be measured and the corresponding losses are selected for the same interval. For a time-based exposure unit, the basic justification follows from a constant expected claim rate over the interval.
Suppose an insured risk has an expected claim frequency of \(\lambda\) claims per exposure-year, and it is covered for a fraction \(t\) of a year. Under the assumption that expected frequency is proportional to time at risk, its expected claim count for that interval is:
If there are \(n\) comparable risks each covered for the same fraction \(t\), total exposure is \(nt\) risk-years, and expected claims are \(n\lambda t\). Dividing gives:
Thus, a correctly measured partial exposure can be combined with full-period exposure without treating the partial risk as if it were covered for a full year. This justification relies on the exposure unit being proportional to the relevant risk opportunity and on the expected rate being stable over the interval. It may not be appropriate when exposure does not accrue in proportion to time, or when material seasonality or other changes make the partial interval unrepresentative. For a non-time-based exposure such as actual payroll, use the measured payroll for the selected interval rather than replacing it with an assumed fraction of annual payroll.
Worked Example: Combining Complete and Matched Partial Periods
A hypothetical insurer is calculating a pure premium per vehicle-year for a defined coverage. Its selected experience contains two complete periods and one partial period. Losses are already on the required basis. Period A has 10,000 vehicle-years and $4,800,000 of losses; period B has 8,000 vehicle-years and $4,160,000 of losses; and the selected six-month portion of period C has 3,000 vehicle-years and $1,710,000 of losses. A fourth data extract shows 2,000 vehicle-years, but its corresponding loss record is not complete. Calculate the indication using eligible records.
Method. Include only complete or accurately measured exposure whose losses correspond to the same selected experience. The six-month period C amount is stated as 3,000 vehicle-years for that six-month portion, not as a full-year amount. Exclude period D for now because its loss record is incomplete.
Step 1: Sum eligible exposure.
Step 2: Sum corresponding losses.
Step 3: Divide losses by exposure.
The result is $507.62 per vehicle-year, rounded to the nearest cent. As a check, the numerator includes only the losses from A, B, and the selected portion of C, and the denominator includes exactly their 21,000 vehicle-years. Period D contributes neither its exposure nor an unsupported loss amount.
Common mistake avoided: Adding D’s 2,000 vehicle-years while leaving its losses out would increase the denominator without adding the corresponding experience. That would calculate $10,670,000 / 23,000 = $463.91, which is not the rate for the matched records. The issue is not whether D’s exposure is real; it is whether its corresponding loss data are complete on the same basis.
Reconstructing or Excluding Incomplete Records
An incomplete extract does not always require permanent exclusion. If reliable source data can establish the missing exposure amount and the losses for that same interval are available, the record can be reconstructed using a consistent rule. Do not infer exposure from the loss amount: losses are outcomes, not a measure of time at risk. If reconstruction is not reliable, keeping the loss while omitting its exposure, or keeping the exposure while omitting its loss, generally changes the selected experience rather than merely cleaning the data.
Worked Example: Recovering Missing Exposure From Insured Days
A hypothetical portfolio has two complete groups on the same coverage basis: group 1 has 400 vehicle-years and $220,000 in losses, and group 2 has 250 vehicle-years and $162,500 in losses. A third group has $70,000 of losses for a known six-month interval, but its exposure total is missing from the extract. Source records confirm that 100 vehicles were insured for exactly six months each during that same interval. Calculate the pooled pure premium after reconstruction.
Step 1: Reconstruct group 3 exposure. Each of the 100 vehicles was covered for half a year, so:
The six-month loss amount is matched to these 50 vehicle-years, so group 3 can be included.
Step 2: Add all eligible exposures.
Step 3: Add all matching losses.
Step 4: Calculate and check.
Rounded to the nearest cent, the indication is $646.43 per vehicle-year. The reconstructed exposure is included because it comes from insured-day detail for the same 100 vehicles and six-month interval as the losses. If that detail were unavailable, the analyst should not guess the missing denominator. The affected experience should be investigated or excluded consistently, with the impact and potential selection bias considered.
Common mistake avoided: Counting the third group as 100 full vehicle-years would treat each six-month policy as though it were exposed for a full year. That would give 750 total vehicle-years and a misleading $452,500 / 750 = $603.33 per vehicle-year.
Partial Records and Cutoff Consistency
A partial record can arise because the experience window begins or ends during a policy term, because coverage starts or ends mid-period, or because only part of an exposure measure is available. These situations require different evidence. If the record gives actual exposure for the selected dates, use that amount. If the actual amount is not known, a time-based proration may be reasonable only when the underlying exposure accrues proportionally to time and the covered interval is established. Do not prorate a non-time-based exposure by elapsed days when the measured basis is available.
The loss selection must follow the same boundary. Suppose an indication uses exposures through June 30. Losses from July onward for those same policies should not be included simply because the policies began before June 30. Conversely, if losses are selected for a full policy year, using only the first six months of its earned exposure would mismatch the bases. In practice, the exact matching convention depends on the coverage and data definition. State the convention and apply it uniformly rather than mixing record dates.
Worked Example: Applying a Common Cutoff to Losses and Exposure
A hypothetical book has 1,200 vehicle-years and $720,000 of losses for a complete twelve-month interval. A separate extract for the next six months reports 500 vehicle-years and $325,000 of losses through June 30, with both amounts confirmed to cover only that six-month interval. A later extract contains another 180 vehicle-years and $90,000 of losses for July through September. The indication is defined to include experience only through June 30. Calculate the eligible pure premium.
Step 1: Apply the stated cutoff. Include the complete twelve-month interval and the matched January-through-June portion. Exclude the July-through-September losses and exposure because they fall after the cutoff.
Step 2: Sum eligible exposure.
Step 3: Sum eligible losses.
Step 4: Calculate.
The answer is $614.71 per vehicle-year, rounded to the nearest cent. The cutoff check is that neither the third-period losses nor its 180 vehicle-years are included. Adding only one of those amounts would violate the selected basis.
Common mistake avoided: Including the July-through-September exposure but not its losses would enlarge the denominator with experience beyond the stated cutoff. Including the losses but not the exposure would create the opposite mismatch. Neither is a valid application of the June 30 selection rule.
Common Selection Errors
- Using every available exposure record. Availability does not establish that a record belongs to the selected coverage or time window.
- Treating a partial period as a full period. A six-month risk contributes six months of time-based exposure, not twelve, unless the stated exposure amount already reflects the correct basis.
- Keeping only one side of an incomplete record. A loss without matched exposure, or exposure without the associated losses, may distort the ratio.
- Applying inconsistent cutoffs. The numerator and denominator must represent the same selected experience, even when source systems use different reporting dates.
- Prorating by time without checking the exposure definition. Time proration is suitable only when exposure is proportional to time at risk. Actual measured exposure should be used when available.
- Excluding records to improve the result. Eligibility should be determined by a documented rule applied consistently, not by whether a record raises or lowers the indication.
Check Your Understanding
For each problem, identify the eligible exposure and losses before calculating. Monetary answers are rounded to the nearest cent.
- A hypothetical insurer has 900 risk-years with $495,000 of matching losses and 300 risk-years with $180,000 of matching losses. A third extract has 200 risk-years, but its losses are still incomplete. What pure premium results from the records that can currently be matched? (Answer: $562.50 per risk-year)
- For a selected three-month interval, 240 units were continuously insured for the entire interval. The corresponding losses are $42,000. What time-based exposure and pure premium should be used? (Answer: 60 unit-years and $700.00 per unit-year)
- A complete group has 500 exposure units and $275,000 of losses. A second group has 80 policies insured for nine months each, with $54,000 of losses for those same nine months. Assuming time-proportional exposure, calculate the pooled pure premium. (Answer: 560 exposure units and $587.50 per exposure unit)
- An indication includes experience through September 30. It has 1,000 exposure units and $620,000 of losses through June 30, plus 400 exposure units and $260,000 of losses from July through September. What is the pure premium through September 30? (Answer: $628.57 per exposure unit)
- A data extract has 300 exposure units and $180,000 of losses for a full interval. A second, matched partial interval has 120 exposure units and $78,000 of losses. A third partial record has 70 exposure units, but its losses cannot be confirmed. What pure premium uses only the records with matched losses and exposure? (Answer: $614.29 per exposure unit)