Extension On-Leveling for Loss Ratio Indications

Why Extension On-Leveling Uses Earned Exposures

Historical earned premium reflects the rate levels in force when the policies were written. To compare that premium with a selected current rate level, the extension method restates each part of the historical exposure at that selected level. The key information is the actual distribution of earned exposures by the rate level that applied to them, determined from the rate effective date and the policy effective dates.

This method is useful when the earned exposures can be separated into cohorts that received different rates. It does not require assuming that policies were written uniformly over time. A policy effective before a renewal-based rate change generally retains its old rate for its term; a policy effective on or after the change date receives the new rate. The policy’s effective date, not the calendar date on which each day of coverage was earned, determines its rate cohort in that situation.

Definition: Extension on-leveling restates earned premium at a selected rate level by applying the rate-level change to each historical earned-exposure cohort. If $E_j$ is the earned exposure in cohort $j$, $r_j$ is its rate level, and $r_c$ is the selected current rate level, then, when premium scales proportionally with the rate level, the on-level premium is

P_{\text{on}}=\sum_j E_j q_j r_c

Here $q_j$ is the premium per exposure unit at the reference rate level for cohort $j$. If the cohorts have the same underlying premium per exposure $q$, this simplifies to $P_{\text{on}}=q r_c\sum_j E_j$. The observed premium for cohort $j$ is $P_j=E_jq_jr_j$, so an equivalent calculation, when $P_j$ is available and the rate-level scaling applies, is

P_{\text{on}}=\sum_j P_j\frac{r_c}{r_j}, \qquad F_{\text{on}}=\frac{P_{\text{on}}}{P_{\text{actual}}}

The $r_j$ and $r_c$ values must use a consistent scale. They may be rate-level indices, such as 1.00, 1.08, and 1.14, or actual premium rates expressed in the same units. A rate-level ratio $r_c/r_j$ is a factor, not a percentage-point change.

Why the Formula Holds

The extension method follows directly from restating each cohort’s exposure at the chosen rate level. For one cohort, its actual premium is exposure multiplied by the premium rate applicable to that cohort: $P_j=E_jq_jr_j$. At the selected rate level, the same exposure and underlying risk characteristics would produce $E_jq_jr_c$. Substituting $P_j$ for $E_jq_jr_j$ gives

E_jq_jr_c = (E_jq_jr_j)\frac{r_c}{r_j} = P_j\frac{r_c}{r_j}.

Summing over cohorts produces the on-level premium formula. The adjustment does not change the exposure or the cohort’s underlying risk characteristics; it changes only the rate level used to value the premium. The formula assumes that premium scales proportionally with the rate-level change and that the cohort premiums and exposures have been assigned consistently. It should not be applied blindly when a change alters coverage, classification, or another premium component rather than just the rate level.

Once the on-level premium is calculated, the on-level loss ratio is the losses for the same experience period divided by that premium:

\text{On-level loss ratio} = \frac{\text{losses for the experience period}}{P_{\text{on}}}.

For this calculation, losses are held fixed while premium is restated to the selected rate level. Other adjustments to losses or the experience period are outside this calculation; do not add an adjustment unless the indication setup calls for it.

A Practical Calculation Sequence

Set the rate-change rule.
Record the change’s effective date and whether it applies based on policy effective date, renewal date, or another specified transaction date.

Assign earned exposure to rate cohorts.
Use the applicable date rule to identify the rate level that produced each cohort’s earned premium. Keep the experience-period definition consistent across premium and losses.

Rerate the cohorts.
Apply the ratio of selected current rate level to cohort rate level, or calculate each cohort’s premium directly from its exposure at the selected rate level.

Aggregate and calculate the loss ratio.
Sum the rerated cohort premiums, then divide the experience-period losses by the resulting on-level premium.

Worked Applications

Worked Example: One Increase and Two Effective-Date Cohorts

A hypothetical insurer has a renewal-based rate increase of 12% effective August 1. In the experience period, 600 earned exposure units came from policies effective before August 1, and 400 came from policies effective on or after that date. The annual premium at the old rate is $100 per exposure unit. Losses for the experience period are $78,400. Calculate the on-level premium at the new rate level, the on-level factor, and the on-level loss ratio.

Method. Apply the rate-level change to each cohort separately. The pre-August 1 cohort is at rate level 1.00; the later cohort is already at the new rate level 1.12. The selected current rate level is 1.12.

Actual premium by cohort. The old-rate cohort has premium $600\times \$100=\$60{,}000$. Its premium at the new rate level is $600\times \$100\times1.12=\$67{,}200$. The later cohort has actual and on-level premium $400\times \$100\times1.12=\$44{,}800$.

On-level premium. Summing the rerated cohort premiums gives $\$67{,}200+\$44{,}800=\$112{,}000$. Actual premium is $\$60{,}000+\$44{,}800=\$104{,}800$, so the on-level factor is $\$112{,}000/\$104{,}800=1.068702\ldots$, or 1.0687 to four decimal places.

On-level loss ratio. Divide the unchanged experience-period losses by on-level premium: $\$78{,}400/\$112{,}000=0.70$, or 70.00%.

Check and common mistake. The rerated premium exceeds actual premium because some exposures received the increase only after the effective date. Applying 1.12 to all actual premium would give $\$104{,}800\times1.12=\$117{,}376$, which incorrectly increases the premium for the later cohort a second time.

Worked Example: Several Rate Levels and a Current-Level Loss Ratio

A hypothetical book has earned exposures at three historical rate levels. There are 250 exposures at level 1.00, 300 at level 1.08, and 450 at level 1.14. The premium at level 1.00 is $80 per exposure unit, and the selected current rate level is 1.14. Experience-period losses are $68,400. Calculate actual premium, on-level premium, the on-level factor, and the on-level loss ratio.

Method. Extend each cohort’s exposures to level 1.14. The premium per exposure at that selected level is $\$80\times1.14=\$91.20$.

Actual premium. At level 1.00, premium is $250\times\$80=\$20{,}000$. At level 1.08, it is $300\times\$80\times1.08=\$25{,}920$. At level 1.14, it is $450\times\$80\times1.14=\$41{,}040$. Total actual premium is $\$20{,}000+\$25{,}920+\$41{,}040=\$86{,}960$.

On-level premium. Total earned exposures are $250+300+450=1{,}000$. At $91.20 per exposure, on-level premium is $1{,}000\times\$91.20=\$91{,}200$. Equivalently, the three cohort premiums at level 1.14 are $\$22{,}800$, $\$27{,}360$, and $\$41{,}040$, which also sum to $\$91{,}200$.

On-level factor and loss ratio. The factor is $\$91{,}200/\$86{,}960=1.048757\ldots$, or 1.0488 to four decimal places. The on-level loss ratio is $\$68{,}400/\$91{,}200=0.75$, or 75.00%.

Check and common mistake. The level-1.14 cohort is already at the selected level, so its premium remains $41,040. Multiplying total actual premium by 1.14 would treat the historical level 1.00 as though it were the current level and would also fail to distinguish the three cohorts.

Worked Example: On-Leveling From Cohort Premiums

A hypothetical insurer has cohort premium records but not an exposure count by cohort. Earned premium is $76,000 at rate level 0.95 and $54,000 at rate level 1.00. The selected current level is 1.06, and experience-period losses are $95,000. Calculate on-level premium, the on-level factor, and the on-level loss ratio.

Method. Use the premium-based form of the extension formula, multiplying each cohort’s premium by the ratio of the selected level to its own level.

Rerate each cohort. For the first cohort, the factor is $1.06/0.95=1.115789\ldots$, so on-level premium is $\$76{,}000\times(1.06/0.95)=\$84{,}800$. For the second cohort, the factor is $1.06/1.00=1.06$, so on-level premium is $\$54{,}000\times1.06=\$57{,}240$.

Aggregate and calculate. On-level premium is $\$84{,}800+\$57{,}240=\$142{,}040$. Actual premium is $\$76{,}000+\$54{,}000=\$130{,}000$, so the on-level factor is $\$142{,}040/\$130{,}000=1.092615\ldots$, or 1.0926 to four decimal places. The on-level loss ratio is $\$95{,}000/\$142{,}040=0.668825\ldots$, or 66.88% to two decimal places.

Check and common mistake. Each cohort is converted directly to level 1.06. Using 1.06 as the adjustment factor for both cohorts would leave the level-1.00 cohort correctly adjusted but would understate the adjustment for the level-0.95 cohort.

Conditions, Interpretation, and Exam Traps

The extension result depends on assigning the earned exposures or premiums to the rate levels that actually applied. If a change is effective at renewal, a policy that began before the change date is not automatically assigned the new rate merely because some of its coverage was earned after that date. Conversely, a policy effective on the change date is assigned according to the stated rule, which is often that the change applies on or after the effective date. Read that boundary condition carefully.

The exposure-based and premium-based forms are equivalent when the premiums are consistent with the stated rate levels and the premium scales proportionally with those levels. If only aggregate premium is available, the historical distribution across rate levels still matters. Applying a single factor to the aggregate is justified only when that factor correctly represents the premium-weighted adjustment for the underlying cohorts.

Common errors include:

Using the calendar earning date instead of the policy rating date. Follow the specified effective-date rule for the rate change.
Adjusting already-current premium again. A cohort at the selected level has adjustment factor $r_c/r_j=1$.
Using an unweighted average of rate levels. Cohorts with different exposure or premium volumes do not contribute equally just because there are several of them.
Reversing the ratio. To move premium from historical level $r_j$ to current level $r_c$, multiply by $r_c/r_j$, not $r_j/r_c$.
Changing the losses along with premium. In this on-level calculation, the numerator remains the losses for the defined experience period; the denominator is the premium restated to the selected rate level.

Key takeaway: Assign each earned exposure or cohort premium to its actual rate level, convert it to the selected rate level, and aggregate before calculating the loss ratio. The on-level factor is premium-weighted through the cohort calculation; it is not automatically the rate change applied to all historical premium.

Check Your Understanding

For each problem, identify the applicable cohorts before doing the arithmetic. Round dollar amounts to the cent where needed and percentages to two decimal places.

A hypothetical book has 200 earned exposures at rate level 1.00 and 300 at rate level 1.10. The base premium is $50 per exposure and losses are $20,625. Calculate actual premium, on-level premium at level 1.10, and the on-level loss ratio. (Answer: actual premium $26,500; on-level premium $27,500; loss ratio 75.00%)
Earned premium is $42,000 at rate level 0.90 and $66,000 at rate level 1.10. Calculate on-level premium at level 1.21. (Answer: $129,066.67)
A hypothetical class has 150 earned exposures at level 0.96 and 250 at level 1.04. The base premium is $120 per exposure. Losses are $40,320. Calculate on-level premium at level 1.12 and the on-level loss ratio. (Answer: on-level premium $53,760; loss ratio 75.00%)
Earned premium is $30,000 at level 1.00 and $45,000 at level 1.05. At current level 1.10, losses are $60,107.14. Calculate on-level premium and the on-level loss ratio. (Answer: on-level premium $80,142.86; loss ratio 75.00%)
A rate increase to level 1.08 takes effect for policies effective on or after October 1. There are 500 earned exposures from policies effective before October 1 at level 1.00 and 300 exposures from policies effective on or after October 1 at level 1.08. Base premium is $70 per exposure. Losses are $57,456. What are the on-level premium and loss ratio at level 1.08? (Answer: on-level premium $60,480; loss ratio 95.00%)

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