Why Unequal Policy Terms Need Careful On-Leveling
On-leveling adjusts historical premium to a common rate level so premium can be compared consistently with losses and other experience measures. With policies of unequal term lengths, a policy’s written premium cannot be treated as though the same fraction was earned for every policy. The earned share depends on both how much of the term has elapsed and the length of that term.
The extension method handles this by calculating earned premium for each applicable policy-term portion, then extending that portion from its original rate level to the selected target level. A six-month term, a twelve-month term, and an eighteen-month term may each earn premium at different rates per month, even when their premiums appear similar. The calculation must preserve those differences.
The Extension Calculation
A rate-level index represents the relative premium level in effect for a policy term. For example, if the base level is assigned index 1.00, an index of 1.08 represents a level 8% above that base. A policy term is associated with the rate level used to determine its written premium. A later rate change does not by itself change the premium already written for that term. If a policy renews at a new level, treat the renewal as a new term with its own written premium, term length, and rate-level index.
Assume premium is earned uniformly over the policy term, using the same time unit for \(e_j\) and \(T_j\). The extension calculation is:
The aggregate on-level factor is the total on-level premium divided by total earned premium:
The factor is therefore weighted by earned premium, not by policy count or by the number of months in a term. A policy-term portion with more earned premium has more influence on the aggregate factor.
Why the Weighted Factor Formula Holds
The justification follows directly from the definition of extending a premium amount. Each earned premium amount is already stated at its own rate level. To express that amount at the target level, multiply by the ratio of the target index to the original index. Adding these adjusted amounts gives the portfolio’s on-level premium.
Uniform pro rata earning gives \(EP_j=W_j(e_j/T_j)\). This step accounts for the portion of each term that belongs to the experience period.
The relative premium change from level \(L_j\) to \(L_T\) is \(L_T/L_j\), so the adjusted amount is \(EP_j(L_T/L_j)\).
Summing over all portions gives \(\sum_j EP_j(L_T/L_j)\). Dividing by \(\sum_j EP_j\) expresses the result as a single aggregate factor.
This calculation assumes that written premium earns uniformly through each term, the stated premium corresponds to the stated rate-level index, and the experience-period earned portions are measured consistently. It does not apply without modification if earning is nonuniform, premium is not allocated pro rata, or a policy has a premium change within a term that must be separately identified. In those situations, the premium must be split into appropriate portions before extending it.
Worked Example: Three Different Term Lengths
A hypothetical portfolio has three policy-term portions in an experience period. Assume uniform pro rata earning and a target rate-level index of 1.10. Calculate the total earned premium, total on-level premium, and aggregate on-level factor.
| Portion | Term length \(T_j\) | Written premium \(W_j\) | Earned time \(e_j\) | Rate level \(L_j\) |
|---|---|---|---|---|
| A | 6 months | $600 | 4 months | 1.00 |
| B | 12 months | $1,320 | 9 months | 1.05 |
| C | 18 months | $1,980 | 12 months | 1.10 |
Portion A. Its earned fraction is \(4/6\), so \(EP_A=\$600(4/6)=\$400\). Extend from 1.00 to 1.10: \(OLP_A=\$400(1.10/1.00)=\$440\).
Portion B. Its earned fraction is \(9/12\), so \(EP_B=\$1{,}320(9/12)=\$990\). Extend from 1.05 to 1.10: \(OLP_B=\$990(1.10/1.05)=\$1{,}037.142857\ldots\).
Portion C. Its earned fraction is \(12/18\), so \(EP_C=\$1{,}980(12/18)=\$1{,}320\). It is already at the target level, so \(OLP_C=\$1{,}320(1.10/1.10)=\$1{,}320\).
Total earned premium is \(\$400+\$990+\$1{,}320=\$2{,}710\). Total on-level premium is \(\$440+\$1{,}037.142857\ldots+\$1{,}320=\$2{,}797.142857\ldots\). Thus the aggregate factor is \(\$2{,}797.142857\ldots/\$2{,}710=1.032155\ldots\). Rounded to the nearest cent for premium and to four decimal places for the factor, the results are $2,797.14 and 1.0322. The factor is above 1.00 because two portions are below the target level.
Method and trap avoided: This is an earned-premium-weighted extension. Applying the three rate-level factors equally would incorrectly give equal weight to portions with different earned premium.
Worked Example: The Term Length Determines the Earned Share
A hypothetical insurer writes a nine-month policy for $1,350 at rate level 1.08. Six months of the term are earned in the experience period. The target level is 1.12. Find the earned premium and on-level premium, rounding money to the nearest cent.
The earned fraction uses the actual nine-month term: \(6/9=2/3\). Therefore,
The extension factor is \(1.12/1.08=28/27=1.037037\ldots\). The on-level premium is
The check is that the factor exceeds 1.00 because the target level, 1.12, is higher than the original level, 1.08. The adjusted premium should therefore exceed the $900 earned premium.
Method and trap avoided: The earned fraction is six months divided by nine months, not six months divided by twelve. Treating the policy as annual would produce \( \$1{,}350(6/12)=\$675\) of earned premium and an incorrectly low on-level amount of \(\$700\).
Worked Example: A Renewal Creates a New Policy-Term Portion
A hypothetical policy has a six-month term with written premium $720 at rate level 1.00. It then renews for a twelve-month term with written premium $1,440 at rate level 1.10. The experience period includes all six months of the first term and six months of the renewal term. Calculate the on-level premium at target level 1.10.
For the first term, \(e_1/T_1=6/6=1\), so \(EP_1=\$720(6/6)=\$720\). Its extension factor is \(1.10/1.00=1.10\), giving \(OLP_1=\$720(1.10)=\$792\).
For the renewal term, \(e_2/T_2=6/12=1/2\), so \(EP_2=\$1{,}440(6/12)=\$720\). This term is already at the target level, so \(OLP_2=\$720(1.10/1.10)=\$720\).
The total earned premium is \(\$720+\$720=\$1{,}440\), and the total on-level premium is \(\$792+\$720=\$1{,}512\). The aggregate factor is \(\$1{,}512/\$1{,}440=1.05\). As a check, the two term portions have equal earned premium, so their factors 1.10 and 1.00 average to 1.05 when weighted equally.
Method and trap avoided: The renewal is treated as a new term because it has a new written premium and rate level. Applying the renewal’s level to all earned premium would incorrectly adjust the first term as well.
Common Errors and Practical Checks
Unequal term calculations are especially vulnerable to using the right rate-level ratio with the wrong earned-premium weight. Keep the earning and extension steps distinct: first determine how much premium was earned from each term, and then adjust that amount to the target level.
- Using a standard term instead of the actual term. Calculate \(e_j/T_j\) using the policy’s stated term length. Do not assume every policy is annual.
- Weighting by policies rather than earned premium. The portfolio factor uses earned premium weights because the amount being extended is earned premium.
- Applying the current level to an entire policy history. Use the rate level attached to the written premium for each term. A renewal can have a different level from the term that preceded it.
- Mixing time units. If the term is measured in months, express earned time in months as well. If days are used, use a consistent day-count convention in both quantities.
- Reversing the ratio. To move from original level \(L_j\) to target \(L_T\), multiply by \(L_T/L_j\). If the target is higher, the factor must exceed 1.00.
The method matters because premium is the denominator in many experience comparisons. Incorrectly estimating the earned share for short or long terms can distort the on-level premium even when the rate-level ratios are correct. This extension approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, assume uniform pro rata earning unless stated otherwise. Round money to the nearest cent and factors to four decimal places.
- A nine-month policy has written premium $1,080 at level 0.96. Three months are earned, and the target level is 1.08. Find earned premium and on-level premium. (Answer: earned premium $360.00; on-level premium $405.00.)
- A fifteen-month policy has written premium $2,250 at level 1.05. Ten months are earned, and the target is 1.10. Find the on-level premium. (Answer: $1,571.43.)
- A six-month term has written premium $840 at level 1.00, and all six months are earned. It renews for twelve months at level 1.10 with written premium $1,320; six months of the renewal term are earned. At target level 1.10, find total earned premium, total on-level premium, and the aggregate factor. (Answer: $1,500.00; $1,584.00; 1.0560.)
- An eighteen-month policy has written premium $2,700 at level 1.08. Nine months are earned in the experience period, and the target is 1.134. Find earned premium and on-level premium. (Answer: earned premium $1,350.00; on-level premium $1,417.50.)
- At target level 1.10, portion A has a six-month term, written premium $600, three months earned, and original level 1.00. Portion B has a twelve-month term, written premium $1,440, nine months earned, and original level 1.20. Find total earned premium, total on-level premium, and the aggregate factor. (Answer: $1,380.00; $1,320.00; 0.9565.)