Extension Method for Annual Policies

Why Annual Policies Need Policy-Level On-Leveling

An annual policy earns premium over its term, while a ratemaking experience period is usually a fixed calendar interval. When policies have staggered effective dates, the experience period can include earned premium from terms written at different rate levels. The extension method handles this by adjusting each term’s earned premium from its own rate level to the selected target level, then adding the adjusted amounts.

The essential unit of calculation is the annual policy term, not simply the calendar year in which premium is earned. A policy term generally retains the rate level applicable when that term begins. Thus, premium earned during the same calendar month can come from terms written at different levels. Treating all earned premium as if it were written at one level can produce an incorrect on-level amount.

Definition: Under the extension method for annual policies, on-level earned premium is calculated by applying the rate-level factor for each policy term to that term’s earned premium in the experience period, then summing the adjusted premiums. The factor converts the term’s actual rate level to the chosen target rate level.

Rate Levels and the Policy-Term Factor

Represent a rate level by an index relative to a base level. For example, indices of 1.00 and 1.08 represent the base level and a level 8% higher. If term $j$ was written at rate-level index $L_j$, and the selected target level has index $L_T$, its extension factor is:

F_j=\frac{L_T}{L_j}

If $EP_j$ is the premium earned from that term in the experience period, its on-level earned premium is $EP_jF_j$. For $n$ terms, the total is:

EP_{\text{on-level}}=\sum_{j=1}^{n}EP_j\frac{L_T}{L_j}

Use consistent rate-level indices: both $L_j$ and $L_T$ must express the premium rate for the same risks and coverage on the same index scale. When the target level exceeds the term’s level, its factor exceeds 1 and increases its earned premium; when the target is lower, its factor is less than 1. A term already at target level has factor 1.

This calculation assumes that the rate-level change is the relevant difference between the actual and target premiums. In particular, it does not attempt to adjust for changes in exposure, coverage, classification, or other policy characteristics. The examples below assume annual terms whose rates are set at the term’s effective date and remain applicable throughout that term. If rates change midterm, one factor for the entire term may not be appropriate; the premium would need to be separated according to the applicable rate periods.

Why the Factor Is a Ratio

The ratio follows from comparing premiums for the same exposure at two rate levels. Suppose an exposure has base-level premium $B$. At level $L_j$, its premium is $BL_j$; at target level $L_T$, its premium would be $BL_T$. Therefore:

\frac{\text{target premium}}{\text{actual premium}} =\frac{BL_T}{BL_j} =\frac{L_T}{L_j}

The unknown base premium cancels. Applying this ratio to earned premium is justified when the earned premium represents the same exposure under the term’s actual rate level. It produces the premium that this earned exposure would have generated at the target level, subject to the assumptions just stated.

Applying the Method to Staggered Annual Terms

A policy can contribute earned premium to an experience period even if it began before that period or expires before the period ends. First determine how much premium from each term is earned in the period. Then identify the rate level attached to that term, apply its target-to-actual factor, and sum. For a simple monthly pro-rata example, a policy with a 12-month term earns one-twelfth of its annual premium for each month in which coverage is in force.

Identify the policy terms.
Separate renewals into distinct annual terms when they begin at different rate levels.

Determine earned premium in the experience period.
Use the premium earned from each term, including any partial-term contribution. Do not substitute written premium unless it equals the earned premium for that period.

Assign each term’s actual rate level.
Use the level applicable to the annual term, rather than the level in force on the date its premium happens to be earned.

Apply the factor and sum.
Multiply each term’s earned premium by $L_T/L_j$, then add the term-level results.

Worked Example: Adjusting Earned Premium at Three Rate Levels

A hypothetical insurer has annual policies contributing earned premium to an experience period. Their rate-level indices are 1.00, 1.08, and 1.12. The respective earned premiums are $25,000, $43,200, and $56,000. Calculate on-level earned premium at target index 1.12.

Assumptions. Each amount is earned premium for an annual term at its stated index; the rate level remains fixed during the term. This is a deterministic premium calculation, so no interest rate, mortality assumption, probability distribution, or distribution parameters apply.

Method. Apply the extension factor $L_T/L_j$ separately to each term.

At level 1.00, the factor is $1.12/1.00=1.12$, so adjusted premium is $25{,}000(1.12)=\$28{,}000.00$.

At level 1.08, the factor is $1.12/1.08=28/27$, so adjusted premium is $43{,}200(28/27)=1{,}600(28)=\$44{,}800.00$.

At level 1.12, the factor is $1.12/1.12=1.00$, so adjusted premium is $56{,}000(1.00)=\$56{,}000.00$.

The total is $\$28{,}000.00+\$44{,}800.00+\$56{,}000.00=\boxed{\$128{,}800.00}$. As a check, the unadjusted total is $\$124{,}200.00$; raising the first two groups to the highest target level should increase it. This method avoids applying one factor to the combined premium, which would ignore the different starting levels.

Worked Example: Staggered Dates and Partial-Year Earning

A hypothetical book has three annual policies contributing premium to the calendar-year experience period January 1 through December 31. Policy A begins January 1 at rate level 1.00, with an annual premium of $1,200. Policy B begins April 1 at level 1.00, also with an annual premium of $1,200. Policy C begins October 1 at level 1.10, with an annual premium of $1,320. Assume monthly pro-rata earning. Calculate on-level earned premium at target level 1.10.

Assumptions. Each policy’s rate level is fixed for its annual term; there are no other premium adjustments. Premium is earned evenly by month. No interest rate, mortality assumption, probability distribution, or distribution parameters apply.

Policy A is in force for all 12 months of the experience period. Its earned premium is $12/12(\$1{,}200)=\$1{,}200.00$. Its factor is $1.10/1.00=1.10$, giving on-level earned premium of $\$1{,}200.00(1.10)=\$1{,}320.00$.

Policy B is in force from April through December, or 9 months. Its earned premium is $9/12(\$1{,}200)=\$900.00$. Its factor is also 1.10, giving $\$900.00(1.10)=\$990.00$.

Policy C is in force from October through December, or 3 months. Its earned premium is $3/12(\$1{,}320)=\$330.00$. Because its rate level is already 1.10, its factor is $1.10/1.10=1.00$, giving $\$330.00$.

The on-level total is $\$1{,}320.00+\$990.00+\$330.00=\boxed{\$2{,}640.00}$. The unadjusted earned premium is $\$1{,}200.00+\$900.00+\$330.00=\$2{,}430.00$, and the adjustment adds $\$210.00$. The calculation uses each term’s start-date rate level, not the rate level in force on every day its premium is earned.

Worked Example: Compounding Sequential Rate Changes

A hypothetical insurer has three annual-policy groups. The first group was written at index 1.00 and contributed $23,100 of earned premium. A 5% rate change then produced index 1.05; policies at this level contributed $33,600. A subsequent 10% increase produced the target level, and policies at that level contributed $46,200. Calculate total on-level earned premium at the latest level.

Assumptions. The increases apply sequentially to the rate level, and each annual term retains its level. Earned premium amounts are given. No interest rate, mortality assumption, probability distribution, or distribution parameters apply.

First calculate the target index by compounding: $1.05(1.10)=1.155$. The factor for the first group is $1.155/1.00=1.155$, giving $\$23{,}100(1.155)=\$26{,}680.50$.

The factor for the second group is $1.155/1.05=1.10$, giving $\$33{,}600(1.10)=\$36{,}960.00$. The latest-level group has factor $1.155/1.155=1.00$, giving $\$46{,}200.00$.

Total on-level earned premium is $\$26{,}680.50+\$36{,}960.00+\$46{,}200.00=\boxed{\$109{,}840.50}$.

Common mistake and correction. Adding 5% and 10% and using index 1.15 treats sequential changes as if both were measured from the original level. That gives a wrong first-group adjustment of $\$23{,}100(1.15)=\$26{,}565.00$, and a wrong second-group factor of $1.15/1.05$, giving $\$36{,}800.00$. The resulting total, $\$109{,}565.00$, is $275.50 too low. The correct target is $1.05(1.10)=1.155$, because the second increase applies to the already increased level.

Interpretation and Common Checks

The method can also be expressed as a weighted average factor, but the weights must be earned premiums. Specifically, total on-level earned premium equals total actual earned premium multiplied by the earned-premium-weighted average of the term factors. An unweighted average treats a small premium group as equally important as a large one and generally gives the wrong result.

EP_{\text{on-level}} =\left(\sum_j EP_j\right) \left(\frac{\sum_j EP_jF_j}{\sum_j EP_j}\right)

Before finalizing a calculation, check that each earned premium is assigned to the correct annual term, every factor uses the same target level, and an unchanged level has factor 1. Also verify that the direction is sensible: moving a group from a lower level to a higher target should increase its premium. Keep full precision through the calculation and round only the final dollar result to the cent.

The extension method is useful because it directly adjusts the premium actually earned from the policy terms in the experience period. Its accuracy depends on having a defensible earned-premium allocation and correctly identifying the rate level for each term. A calendar-year label alone does not identify that level: staggered policy effective dates mean terms contributing to one year’s earned premium can have different rate histories.

Check Your Understanding

For each problem, determine the on-level earned premium at the stated target level. Round dollar answers to the cent when needed.

A hypothetical portfolio has $18,000 of earned premium at level 1.00 and $24,000 at level 1.08. What is the on-level total at 1.08? (Answer: $43,440.00)
Earned premiums are $10,000 at level 1.00, $21,000 at level 1.05, and $32,000 at level 1.155. Find the on-level total at target level 1.155. (Answer: $66,650.00)
A policy with a $960 annual premium at level 1.00 begins February 1, 2025. Another with a $1,152 annual premium at level 1.20 begins September 1, 2025. Under monthly pro-rata earning, calculate their combined on-level earned premium for calendar year 2025 at target level 1.20. (Answer: $1,440.00)
A book has $12,500 of earned premium at level 1.04 and $8,000 at level 1.12. Calculate the total at target level 1.12. Why would applying the unweighted average of the two factors be incorrect? (Answer: $21,461.54; the factors must be weighted by earned premium, not averaged equally.)
A 6% increase changes an index of 1.00 to 1.06, followed by a 5% increase. Earned premiums are $10,000 at 1.00, $22,260 at 1.06, and $30,000 at the latest level. Calculate the on-level total at the latest level. (Answer: $64,503.00)

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