Why Staggered Policy Dates Matter
When annual policies take effect on different dates, a rate change can apply to some policies before it applies to others. For a renewal-effective change, a policy generally keeps the rate level in effect when its term begins; a later renewal may receive the changed rates. As a result, an experience period can contain earned premium from policies at different rate levels, even if the period is only one calendar year.
The extension method handles this by using each policy’s effective date to identify its actual rate level, then adjusting its earned premium to a common target level. The earned amounts matter: a policy that contributes more earned premium has more influence on the aggregate factor. This tutorial focuses on annual policies and a single rate change implemented at different policy effective dates.
From Effective Dates to an Aggregate Factor
The effective date is the date on which the policy’s term begins. For a renewal-effective rate change, compare that date with the date on which the change applies to that policy. A policy effective before the change applies is assigned the old level; a policy effective on or after its applicable change date is assigned the new level. When policies receive the change at different renewal dates, make this assignment policy by policy rather than classifying the whole calendar year as “old” or “new.”
The earned premium is the portion of a policy’s premium attributable to coverage earned during the experience period. If a policy earns uniformly by month and \(m_i\) of its 12 months fall in the period, then its earned premium is its annual premium multiplied by \(m_i/12\). The applicable premium and the earned fraction answer different questions: the effective date identifies the rate level, while the overlap with the experience period identifies how much premium is earned.
To see why the aggregate formula holds, take one policy at actual level \(L_i\). Under the assumption that the target change scales its premium proportionally, its premium at the target level is \(EP_i(L_T/L_i)\). Adding this amount over all policies gives the total premium that the same earned business would produce at the target level. Dividing by the actual total earned premium gives the single factor that converts the actual total to that target-level total:
The result is a weighted average of the policy-level adjustment factors \(L_T/L_i\), with weights \(EP_i/\sum_j EP_j\). These weights sum to one. Thus the aggregate factor is pulled toward the adjustment factor for policies that contribute more earned premium.
State which earned premium is being adjusted and the rate level to which it will be brought.
Use the policy effective date and the applicable implementation date for the rate change.
Use the portion of the annual term earned in the experience period, along with the premium at that policy’s actual level.
Multiply each earned premium by its target-to-actual level ratio, sum the results, and divide by actual total earned premium.
Worked Examples
Worked Example: Two Staggered Renewal Cohorts
A hypothetical insurer has a renewal-effective rate increase of 8%. Policies renewing before their individual implementation dates remain at level 1.00; policies renewing on or after those dates receive level 1.08. In one experience period, policies at level 1.00 contribute $40,000 of earned premium and policies at level 1.08 contribute $60,000. Calculate the factor to bring the total to level 1.08.
Method. Apply the extension method by cohort. The target level is \(L_T=1.08\). The old-level cohort needs the ratio \(1.08/1.00=1.08\); the new-level cohort needs the ratio \(1.08/1.08=1.00\).
Re-rate the old-level earned premium: $$ \$40{,}000(1.08)=\$43{,}200. $$ The new-level earned premium remains: $$ \$60{,}000(1.00)=\$60{,}000. $$ The total on-level earned premium is \(\$43{,}200+\$60{,}000=\$103{,}200\). Actual total earned premium is \(\$40{,}000+\$60{,}000=\$100{,}000\). Therefore, $$ F=\frac{\$103{,}200}{\$100{,}000}=1.0320. $$
Check. The old-level share is \(40\%\), so the factor must be \(40\%(1.08)+60\%(1.00)=1.032\), matching the direct calculation. The common mistake this method avoids is assigning the full 8% increase to all earned premium, which would incorrectly give a factor of 1.0800.
Worked Example: Calculate Earned Amounts Before Weighting
A hypothetical book uses monthly pro rata earning, with 12 equal months in each annual policy term. The target level is 1.08. Policy A has an annual premium of $1,200 at level 1.00 and contributes six earned months to the experience period. Policy B has an annual premium of $1,296 at level 1.08 and contributes six earned months. Policy C has an annual premium of $900 at level 1.08 and contributes four earned months. Find the extension factor.
Method. First calculate each policy’s earned premium. Then adjust each amount from its actual level to the target level. The six months for Policy A mean half its annual premium is earned; they do not mean that its rate level should be changed.
Policy A’s earned premium is \(\$1{,}200(6/12)=\$600\). Its adjustment ratio is \(1.08/1.00=1.08\), so its on-level earned premium is \(\$600(1.08)=\$648\).
Policy B’s earned premium is \(\$1{,}296(6/12)=\$648\). Its adjustment ratio is \(1.08/1.08=1.00\), so its on-level earned premium is \(\$648\).
Policy C’s earned premium is \(\$900(4/12)=\$300\). Its adjustment ratio is also \(1.00\), so its on-level earned premium is \(\$300\).
Actual earned premium totals \(\$600+\$648+\$300=\$1{,}548\). On-level earned premium totals \(\$648+\$648+\$300=\$1{,}596\). Thus, $$ F=\frac{\$1{,}596}{\$1{,}548}=1.03100775\ldots \approx 1.0310. $$
Check. The only increase is on Policy A’s $600 of earned premium, which adds \(8\%(\$600)=\$48\). The resulting total is \(\$1{,}548+\$48=\$1{,}596\), as calculated. The common mistake this method avoids is multiplying the annual premium by the rate adjustment without first limiting it to the months earned in the experience period.
Worked Example: Assigning Rate Levels by Policy Effective Date
A hypothetical insurer implements a 6% renewal-effective increase for each policy when that policy renews on or after its assigned implementation date. Its experience period is January through December 2025, and earning is uniform by month. Policy A has a September 1, 2024 effective date, an annual premium of $1,200 at level 1.00, and eight months in the experience period. Policy B has an April 1, 2025 effective date, an annual premium of $1,060.80 at level 1.06, and nine months in the period. Policy C has a July 1, 2024 effective date, an annual premium of $1,500 at level 1.00, and six months in the period. Policy D has an October 1, 2025 effective date, an annual premium of $1,590 at level 1.06, and three months in the period. Calculate the factor to level 1.06.
Method. Use policy-level effective dates to identify the stated actual levels, calculate earned premium from the months in the experience period, and re-rate only the level 1.00 amounts.
Policy A contributes \(\$1{,}200(8/12)=\$800\) at level 1.00. Policy B contributes \(\$1{,}060.80(9/12)=\$795.60\) at level 1.06. Policy C contributes \(\$1{,}500(6/12)=\$750\) at level 1.00. Policy D contributes \(\$1{,}590(3/12)=\$397.50\) at level 1.06.
The actual earned premium total is \(\$800+\$795.60+\$750+\$397.50=\$2{,}743.10\). The two old-level policies contribute \(\$800+\$750=\$1{,}550\), which must be multiplied by \(1.06/1.00=1.06\). Their on-level amount is \(\$1{,}550(1.06)=\$1{,}643\). The level 1.06 policies are already at target and contribute \(\$795.60+\$397.50=\$1{,}193.10\). Therefore, the total on-level earned premium is \(\$1{,}643+\$1{,}193.10=\$2{,}836.10\), and $$ F=\frac{\$2{,}836.10}{\$2{,}743.10} =1.03390325\ldots \approx 1.0339. $$
Check. The old-level share of actual earned premium is \(\$1{,}550/\$2{,}743.10\). Increasing only that share by 6% gives an aggregate increase of \(0.06(\$1{,}550/\$2{,}743.10)=0.03390325\ldots\), agreeing with the factor. The common mistake this method avoids is treating all 12 months as if every policy were at the same level simply because the experience period is a calendar year.
Assumptions and Common Exam Traps
The extension calculation assumes that changing the rate level scales the premium being adjusted by the stated ratio. It also assumes the policy’s exposure and risk characteristics are otherwise held fixed for this comparison. If a policy’s classification, coverage, limit, or other rating inputs change, its premium may not be represented accurately by one common rate-level ratio; the appropriate calculation must then account for those changes separately.
The key distinction is between when a rate level applies and how much premium is earned. The policy effective date determines the applicable level for a renewal-effective change. The portion of the policy term within the experience period determines its earned amount. A policy can therefore be at the new rate level while contributing only a small fraction of its annual premium to the experience period.
A frequent trap is to average policy-level factors without weights. For instance, if $10,000 of earned premium is at level 1.00 and $90,000 is at level 1.08, with level 1.08 as the target, the policy-level factors are 1.08 and 1.00. Their simple average is 1.0400, but the correct factor is $$ \frac{\$10{,}000(1.08)+\$90{,}000(1.00)}{\$100{,}000}=1.0080. $$ The simple average gives equal influence to two groups that contribute very different amounts of earned premium. Extension factors must use earned-premium weights, not an unweighted average of rate changes or cohort factors.
Check Your Understanding
For each problem, identify the actual levels and earned-premium weights before calculating the factor.
- Earned premium is $25,000 at level 1.00 and $75,000 at level 1.12. Find the factor to level 1.12. (Answer: 1.0300)
- A 5% increase has produced levels 1.00 and 1.05. There is $18,000 of earned premium at level 1.00 and $42,000 at level 1.05. Find the factor to level 1.05. (Answer: 1.0150)
- Earned premium is $12,000 at level 1.00, $20,000 at level 1.04, and $8,000 at level 1.10. Find the factor to level 1.10. Round the factor to four decimal places. (Answer: 1.0588)
- Policies earn uniformly by month. An annual policy at level 1.00 has an annual premium of $1,800 and contributes four months to the experience period. An annual policy at level 1.10 has an annual premium of $2,400 and contributes six months. Find the factor to level 1.10, rounded to four decimal places. (Answer: 1.0333)
- A hypothetical book has $20,000 of earned premium at level 1.00 and $30,000 at level 1.08. The target level is 1.1124. Calculate the factor using the target-to-actual ratio for each group, and round to four decimal places. (Answer: 1.0630)