Why Use Exposure Records?
Historical premium totals may show how much premium was earned, but they do not always reveal the rate level that applied to each policy or portion of a policy. If policies have different territories, classes, deductibles, or effective dates, a single total can conceal important differences. When exposure records and rating inputs are available, the extension method estimates the premium those exposures would produce at a selected current rate level.
The method is useful when the actuary can identify the exposure units and rate inputs but cannot reliably determine each item of historical premium from the aggregate total alone. It is a premium re-rating calculation, not a loss adjustment or a forecast of future exposure. The choice of experience period and target rate level should be consistent with the earlier material on choosing the experience period and target level.
Calculation Method
For each record \(i\), let \(e_i\) be the earned exposure and \(r_i^{\mathrm{cur}}\) be the premium per full exposure at the selected current rates, using that record’s current rating characteristics. If the rate is built from a base rate and multiplicative rating factors, then \(r_i^{\mathrm{cur}}\) is the base rate multiplied by the applicable current factors. The extension premium is the sum of each rate per exposure multiplied by its earned exposure.
If the historical earned premium for the same experience is \(EP^{\mathrm{hist}}\), the on-level factor is
An on-level factor above 1 indicates that the current-rate premium for these exposures exceeds the historical earned premium; a factor below 1 indicates the reverse. Applying the factor to the historical premium brings that total to the extension-premium level, provided the premium being adjusted corresponds to the same exposure base.
Confirm what one exposure unit means, the earned amount of each record, and which records belong in the experience period.
Use the selected current base rate and the current rating factors that apply to that record. Do not apply a factor twice if the provided rate already includes it.
Multiply each current rate per full exposure by the record’s earned exposure, then add the record-level amounts.
Divide the extension premium by historical earned premium for the same experience and verify that the direction of the ratio matches the intended adjustment.
Why the Formula Works
A rate per full exposure is the premium for one complete exposure unit at the specified rate level. If a record earns only a fraction \(e_i\) of an exposure, its premium at that rate level is \(e_i r_i^{\mathrm{cur}}\), assuming premium varies proportionally with exposure. The total premium for a collection of records is therefore the sum of those record-level premiums:
The historical earned premium is the premium at the rate levels actually reflected in the experience. Dividing the premium calculated at current rates by that historical total gives the multiplicative change needed to move the historical total to the current-rate level. In particular, \(EP^{\mathrm{hist}}\times OLF=EP^{\mathrm{cur}}\). This reasoning assumes comparable exposure bases, accurate exposure and rating records, and premium that is proportional to exposure. A fixed policy fee, minimum premium, or other nonproportional charge should be handled separately rather than treated as if it were a per-exposure rate.
Worked Example: Re-Rating Two Exposure Records
A hypothetical insurer uses a current base rate of $1,000 per full exposure. Record A has 0.20 earned exposure, a territory factor of 1.20, and an equipment factor of 0.90. Record B has 0.30 earned exposure, a territory factor of 0.80, and an equipment factor of 1.10. Historical earned premium for the two records is $684. Find the extension premium and on-level factor.
Method. Calculate each record’s current premium per full exposure by multiplying the current base rate by its current factors. Then multiply by its earned exposure. This avoids treating a fractional exposure as if it were a full year.
For record A, the current rate per full exposure is \(1{,}000\times1.20\times0.90=1{,}080\). Its extension premium is \(1{,}080\times0.20=\$216\).
For record B, the current rate per full exposure is \(1{,}000\times0.80\times1.10=880\). Its extension premium is \(880\times0.30=\$264\).
The total extension premium is \(216+264=\$480\). The on-level factor is \(480/684=0.701754\ldots\), or 0.7018 to four decimal places.
Check. Multiplying the historical premium by the unrounded factor gives \(684\times(480/684)=\$480\), the extension premium. The common mistake this calculation avoids is omitting the earned exposure: adding the two full-exposure rates, $1,080 and $880, would not be the premium for these records.
Worked Example: Combining Segments at Different Historical Multipliers
A hypothetical policy has a current annual rate of $1,200. During the experience period it earns 0.25 exposure at a historical multiplier of 0.80 and then 0.50 exposure at a historical multiplier of 0.90. Assume the current rate is the same for both segments and the historical multipliers are relative to that current rate. Calculate total extension premium, historical earned premium, and the on-level factor.
Method. At current rates, each segment is priced at the current annual rate times its exposure. Historical earned premium uses the corresponding historical multiplier as well.
The total exposure is \(0.25+0.50=0.75\). Thus extension premium is \(1{,}200\times0.75=\$900\).
For the first segment, historical earned premium is \(1{,}200\times0.25\times0.80=\$240\). For the second segment, it is \(1{,}200\times0.50\times0.90=\$540\). Total historical earned premium is \(240+540=\$780\).
The on-level factor is \(900/780=1.153846\ldots\), or 1.1538 to four decimal places. As a check, \(780\times(900/780)=\$900\). The factor exceeds 1 because the current-rate premium is greater than the historical earned premium.
Common mistake avoided. Do not use the total exposure alone to calculate historical premium. The two segments had different historical multipliers, so their historical amounts must be calculated separately before summing.
Worked Example: Applying Multiple Current Rating Factors
A hypothetical insurer has three records with the current base rates and rating factors shown below. The stated exposures are earned exposures, and the historical earned premiums are amounts for the corresponding records. Find the total extension premium and on-level factor.
| Record | Base rate | Territory factor | Class factor | Earned exposure | Historical premium |
|---|---|---|---|---|---|
| A | $900 | 1.20 | 0.80 | 0.50 | $390 |
| B | $1,500 | 0.80 | 1.10 | 0.30 | $520 |
| C | $1,100 | 1.00 | 0.75 | 0.60 | $600 |
Method. Re-rate each record independently at the current base rate and factors, apply its earned exposure, and then sum. This preserves the rating mix in the exposure records rather than applying a single average factor to the whole portfolio.
Record A has a current rate per full exposure of \(900\times1.20\times0.80=\$864\), so its extension premium is \(864\times0.50=\$432\). Record B has a current rate per full exposure of \(1{,}500\times0.80\times1.10=\$1{,}320\), so its extension premium is \(1{,}320\times0.30=\$396\). Record C has a current rate per full exposure of \(1{,}100\times1.00\times0.75=\$825\), so its extension premium is \(825\times0.60=\$495\).
The extension premium is \(432+396+495=\$1{,}323\). Historical earned premium is \(390+520+600=\$1{,}510\). Therefore, the on-level factor is \(1{,}323/1{,}510=0.876821\ldots\), or 0.8768 to four decimal places.
Check. Applying the unrounded factor to $1,510 gives $1,323. A useful control is to verify both sums independently: one sum uses the re-rated exposure records, while the other uses the stated historical premiums.
Common Traps and Practical Limits
Exposure-level detail improves the calculation only when the records and rate inputs are interpreted consistently. In particular, distinguish a rate per full exposure from the premium for the earned portion. If a rate is annual and the record represents a fraction of a year, multiply by the fractional exposure. If the record already reports an earned premium or an exposure-adjusted amount, do not prorate it again.
- Using one average factor for unlike records. Apply each record’s current rating factors before summing. An average factor can be wrong when exposure amounts differ across rating groups.
- Mixing exposure bases. Confirm that the historical premium and extension premium cover the same policies, segments, and exposure period. Otherwise, their ratio does not isolate the rate-level difference.
- Reversing the ratio. The factor that converts historical premium to current-rate premium is extension premium divided by historical earned premium, not the reverse.
- Ignoring nonproportional charges. If premiums include fixed fees or minimums, a simple rate-times-exposure calculation may not reproduce them. Identify and treat such amounts consistently.
- Using inappropriate rating inputs. The factors must be those specified for the selected current rate level. Do not combine a current base rate with historical factors unless that is explicitly the intended target calculation.
The extension method with exposure records is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. Its central discipline is to calculate at the record level first and aggregate second; doing so makes the exposure base, rating mix, and rate-level adjustment visible and checkable.
Check Your Understanding
For each hypothetical scenario, calculate the requested amount and check that exposure and rate levels are applied consistently.
- A current rate is $800 per full exposure. One record has 0.25 earned exposure, a territory factor of 1.15, and a class factor of 0.80. Historical earned premium is $230. Find the extension premium and on-level factor. (Answer: extension premium $184; factor 0.8000.)
- A policy earns 0.25 exposure at historical multiplier 0.80 and 0.50 exposure at historical multiplier 0.90. Its current annual rate is $1,200. Find extension premium, historical earned premium, and the on-level factor. (Answer: $900; $780; 1.1538.)
- Two records have current rates per full exposure of $900 and $1,400, with earned exposures 0.40 and 0.25. Their historical earned premiums are $380 and $420. Find total extension premium and the on-level factor. (Answer: extension premium $710; factor 0.8875.)
- A record has a current annual base rate of $1,250, territory factor 1.08, class factor 0.92, deductible factor 0.85, and earned exposure 0.60. Historical earned premium is $680. Calculate its extension premium and on-level factor, rounding the factor to four decimals. (Answer: extension premium $633.42; factor 0.9315.)
- Three records have current base rates $900, $1,600, and $1,100; current factor products 0.90, 0.935, and 0.72; and earned exposures 0.50, 0.25, and 0.40, respectively. Historical earned premiums are $380, $420, and $350. Find the total extension premium and on-level factor, rounding the factor to four decimals. (Answer: extension premium $1,095.80; factor 0.9537.)