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CAS Exam 5 · Premium On-Leveling Methods · Tutorial 114 of 989

Renewal-Only and Midterm Rate Changes

Distinguish renewal-only changes from midterm changes and correctly on-level earned premium when policies have experienced different rate levels.

Intermediate 10 min read

What You'll Learn

  • Distinguish changes that apply at renewal from changes that apply to in-force policies midterm.
  • Identify the rate level applicable to each earned-premium segment.
  • Extend segment premium from its current rate level to a common target level.
  • Calculate and interpret aggregate on-level factors using premium weights.
  • Recognize why a change’s implementation rule matters to an on-level calculation.
  • Avoid treating all premium as if it were written at the latest rate level.

Why the Implementation Rule Matters

A rate change does not necessarily apply to every policy on the same date. A renewal-only change generally affects a policy when it renews; an unrenewed policy continues at its existing level. A midterm change applies to affected in-force policies on its effective date, even if their policy terms have not yet renewed. Consequently, earned premium from one experience period can represent multiple rate levels.

On-leveling adjusts premium to a common target rate level. The essential step in this topic is not just knowing the change percentage: it is assigning the correct current rate level to each premium segment before adjusting it. A rate change’s implementation rule determines which policies, and which portions of earned premium, are at the old or new level.

Definition: A rate level is a numerical index describing how a rate compares with a chosen base level. The base level is commonly indexed as 1.00. A change of \(r\) percent multiplies the previous level by \(1+r\); sequential changes compound. On-level premium is the premium adjusted to a specified target level. For this tutorial, a premium segment is the earned premium attributable to exposure at one particular rate level.

Renewal-Only and Midterm Changes

For a renewal-only change, the renewal date determines when a policy enters the new level. Policies that have not renewed remain at the old level. Thus a single renewal-only change can leave a portfolio with earned premium at both levels during the same experience period.

For a midterm change that applies to all affected in-force policies, the effective date—not each policy’s renewal date—determines when the new level applies. Earned premium attributable to coverage before that date is at the previous level; premium attributable to coverage after it is at the changed level. If the change is a proportional increase of \(r\), the new level is the old level multiplied by \(1+r\). A midterm change may therefore create old- and new-level segments even within the same policy term.

The calculation below takes the earned-premium amount at each rate level as given. It does not determine how much premium was earned before or after a midterm effective date; the earning allocation must already be available or supplied. The key is to use the level that applied to each segment, rather than to classify all premium by the policy’s current or renewal status.

Formula: Suppose segment \(j\) has earned premium \(P_j\) at rate level \(L_j\), and the target level is \(L_T\). Under a proportional rate-level change, its on-level premium is \(P_j(L_T/L_j)\). For multiple segments, add the adjusted premiums. The aggregate on-level factor is total on-level premium divided by total actual earned premium.
$$ P_{\mathrm{on\text{-}level}}=\sum_j P_j\frac{L_T}{L_j}, \qquad F_{\mathrm{on\text{-}level}} =\frac{\sum_j P_j(L_T/L_j)}{\sum_j P_j}. $$

Use the same target level for every segment. If a segment is already at that level, its ratio is 1 and its premium is unchanged. An aggregate factor is a premium-weighted average of the segment factors; it is not generally the unweighted average of the rate-level ratios.

Why the Segment Formula Works

The formula follows from the proportional relationship between premium and rate level. Assume that each segment’s exposure and rating characteristics are held fixed, and that the rate-level change scales its premium proportionally. Let \(B_j\) denote the premium that segment \(j\) would have at the base level. At level \(L_j\), its premium is \(P_j=B_jL_j\). At the target level, the same exposure would produce \(B_jL_T\). Substituting \(B_j=P_j/L_j\) gives the segment adjustment:

$$ P_j^{T}=B_jL_T =\frac{P_j}{L_j}L_T =P_j\frac{L_T}{L_j}. $$

Adding these segment amounts gives total on-level premium. Dividing that total by actual earned premium gives the aggregate factor. This derivation assumes the premium difference is due to the rate-level change being adjusted. If exposure, coverage, classification, or other premium components also change, those differences should not be mistaken for a rate-level change; the calculation may need to control for them separately.

Worked Example: Renewal-Only Increase with Two Rate Levels

A hypothetical insurer has a 5% renewal-only increase, moving the rate-level index from 1.00 to 1.05. During the experience period, earned premium is $42,000 from policies not yet renewed and $31,500 from policies renewed into the increase. Calculate the on-level premium and factor at target level 1.05.

Method. Treat the two amounts as separate segments. The unrenewed policies remain at level 1.00; the renewed policies are at 1.05.

The $42,000 segment must be extended from 1.00 to 1.05:

$$ \$42{,}000\left(\frac{1.05}{1.00}\right)=\$44{,}100. $$

The $31,500 segment is already at the target level:

$$ \$31{,}500\left(\frac{1.05}{1.05}\right)=\$31{,}500. $$

Therefore, on-level premium is $44,100 + $31,500 = $75,600. Actual earned premium is $42,000 + $31,500 = $73,500, so the aggregate factor is $75,600 / $73,500 = 1.028571… = 1.0286 to four decimal places.

Check. Only the old-level segment increases; the target-level segment stays fixed. The on-level total must therefore exceed actual premium, as it does. The common mistake is to leave the $42,000 unchanged because it is already earned premium; the correction is to multiply it by 1.05/1.00.

Worked Example: Midterm Increase Applying to In-Force Policies

A hypothetical portfolio receives an 8% midterm increase that applies to all affected in-force policies on its effective date. The experience-period earned premium is $48,000 for coverage before the effective date at level 1.00 and $72,000 for coverage afterward at level 1.08. Find the on-level premium and factor at target level 1.08.

Method. Assign the pre-effective-date segment to level 1.00 and the post-effective-date segment to level 1.08. The fact that a policy may not have renewed does not keep its post-effective-date coverage at the old level when the change applies midterm.

Adjust the old-level amount and leave the new-level amount unchanged:

$$ \$48{,}000\left(\frac{1.08}{1.00}\right)=\$51{,}840, \qquad \$72{,}000\left(\frac{1.08}{1.08}\right)=\$72{,}000. $$

On-level premium is $51,840 + $72,000 = $123,840. Actual earned premium is $48,000 + $72,000 = $120,000. Thus the aggregate factor is $123,840 / $120,000 = 1.0320.

Check. The post-effective-date segment is already at the target and is unchanged. The factor is less than 1.08 because only the pre-effective-date portion needs the full 8% adjustment.

Worked Example: Assigning Levels After Sequential Changes

A hypothetical portfolio first receives a 6% renewal-only increase, from level 1.00 to 1.06. Later, a 4% midterm increase applies to all in-force policies. During an experience period, earned premium is $20,000 at level 1.00, $30,000 at level 1.06, and $25,000 at the level after both increases. Find on-level premium and the factor at the level after both changes.

Method. Compound sequential changes to find the target level. The level after both increases is \(1.06(1.04)=1.1024\). The three segments are already identified by their applicable levels; in particular, the middle segment is at 1.06, not at the final level.

Adjust each segment:

$$ \$20{,}000\left(\frac{1.1024}{1.00}\right)=\$22{,}048, $$
$$ \$30{,}000\left(\frac{1.1024}{1.06}\right) =\$30{,}000(1.04)=\$31{,}200, $$
$$ \$25{,}000\left(\frac{1.1024}{1.1024}\right)=\$25{,}000. $$

On-level premium is $22,048 + $31,200 + $25,000 = $78,248. Actual earned premium is $20,000 + $30,000 + $25,000 = $75,000. The aggregate factor is $78,248 / $75,000 = 1.043306… = 1.0433 to four decimal places.

Check. The level 1.06 segment needs only the later 4% increase to reach 1.1024; multiplying it by the full 10.24% increase from level 1.00 would overstate its adjustment. The segment already at 1.1024 needs no adjustment.

Common Mistakes and Practical Checks

The most consequential error is assigning a rate level based only on the date a policy was written or renewed. For renewal-only changes, policies that have not renewed remain at the old level. For a midterm change applying to all in-force policies, coverage after the effective date is at the changed level even if the policy has not renewed. The applicable implementation rule must come before the arithmetic.

Another frequent mistake is applying the full increase to every segment. For each segment, use the ratio of target level to its own level. A segment already at target has a ratio of 1; a segment at an intermediate level gets only the adjustment from that intermediate level to target.

Key takeaway: Separate earned premium by the rate level that actually applied to it. Renewal-only changes follow renewal status; midterm changes follow their effective date for affected in-force policies. Then adjust each segment by target level divided by its current level and sum.

As a final check, confirm that the segment premiums add to actual earned premium and that the adjusted segments add to on-level premium. If the target is above every segment’s current level, no segment should decrease; if some segment is already at target, it should remain unchanged. These checks help catch both misclassification and arithmetic errors.

Check Your Understanding

For each hypothetical situation, identify the applicable levels before calculating.

  1. A 6% renewal-only increase moves the index from 1.00 to 1.06. Earned premium is $50,000 from policies not yet renewed and $20,000 from renewed policies. Find on-level premium and the factor at 1.06. (Answer: on-level premium = $73,000; factor = 1.0429.)
  2. A 9% midterm increase applies to all in-force policies. Earned premium is $36,000 before its effective date at level 1.00 and $54,000 afterward at level 1.09. Find on-level premium and the factor at 1.09. (Answer: on-level premium = $93,240; factor = 1.0360.)
  3. A portfolio has earned premium of $40,000 at level 1.00, $25,000 at level 1.05, and $35,000 at level 1.1025. Find on-level premium and the aggregate factor at target 1.1025. (Answer: on-level premium = $105,350; factor = 1.0535.)
  4. At target level 1.04, earned premium is $18,000 at level 0.96 and $42,000 at level 1.04. Calculate the on-level premium and factor. (Answer: on-level premium = $61,500; factor = 1.0250.)
  5. A 5% renewal-only change moves a portfolio from 1.00 to 1.05. Earned premium is $24,000 at 1.00 and $16,000 at 1.05. A student leaves the $24,000 unchanged and reports on-level premium of $40,000. Correct the result at target 1.05. (Answer: on-level premium = $41,200; factor = 1.0300.)

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