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One Rate Change with the Parallelogram Method

Calculate how much of a single rate change is reflected in earned premium during an experience period, and convert that average level to a target level.

Intermediate 10 min read

What You'll Learn

  • State the assumptions behind the one-change parallelogram method
  • Calculate the earned share at the new rate level from the dates
  • Derive the average earned rate level for an experience period
  • Convert the average level to an on-level factor
  • Recognize why the share of new business is not the share of earned exposure

Why the Effective Date Is Not the Whole Story

A rate change affects policies according to when their premiums are earned, not simply when the change takes effect. For annual policies, a policy written at the new rate does not immediately contribute a full year of earned premium at that level. Its premium earns over the policy term. The parallelogram method represents this gradual transition and estimates the average rate level underlying earned premium for a selected experience period.

This tutorial considers one rate change effective during the experience period. The method is useful when policy writings and exposure are sufficiently uniform for a geometric approximation to represent the earned premium pattern. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: The average earned rate level is the average rate level reflected in earned premium during an experience period, weighted by the exposure earning premium. The on-level factor converts premium at that average level to premium at a specified target rate level.

Model and Assumptions

Let the experience period run from time \(a\) to time \(b\), measured in years, and let a single rate change take effect at time \(c\), where \(a<c<b\). Assume annual policies, a constant volume of exposures earning throughout the period, and policy effective dates distributed uniformly over time. Assume the rate change applies to policies as they are written or renewed; it does not change the rate on a policy already in force. Also assume \(b-c\leq 1\), so the end of the experience period is no more than one annual policy term after the change.

Let \(R_{\text{old}}\) and \(R_{\text{new}}\) denote rate-level indices before and after the change. For example, if the old level is indexed as 1.00 and the new rate represents a 12% increase, then \(R_{\text{old}}=1.00\) and \(R_{\text{new}}=1.12\). The indices need not be expressed relative to 1.00, provided both use the same scale.

Under the assumptions above, the share of exposure earning at the new level rises linearly from zero on the effective date to one at the end of a year. Before the effective date, none of the exposure has the new level. Consequently, the earned share at the new level at time \(t\) is \(t-c\) for \(c\leq t\leq b\). The condition \(b-c\leq 1\) ensures that this share has not yet reached its maximum of one by the period end.

Deriving the Average Earned Level

The fraction of exposure at the new level is zero at time \(c\), and it increases at a constant rate because policies are written uniformly. At time \(t\), the new-level exposure fraction is \(t-c\). Averaging that fraction over the experience period gives the proportion of earned exposure at the new level:

$$ f_{\text{new}} =\frac{1}{b-a}\int_a^b \max(t-c,0)\,dt =\frac{1}{b-a}\int_c^b(t-c)\,dt =\frac{(b-c)^2}{2(b-a)}. $$

The integral is the area of a triangle: its base is \(b-c\), its height is \(b-c\), and its area is \((b-c)^2/2\). Dividing by the full experience-period duration \(b-a\) converts that area to an average fraction. Since the rate level is \(R_{\text{old}}\) on the remaining exposure and \(R_{\text{new}}\) on the new-level exposure, the average earned rate level is

$$ R_{\text{avg}} =(1-f_{\text{new}})R_{\text{old}}+f_{\text{new}}R_{\text{new}} =R_{\text{old}}+(R_{\text{new}}-R_{\text{old}})f_{\text{new}}. $$

If \(P_{\text{earned}}\) is earned premium at this average level and \(R_{\text{target}}\) is the target rate level, the on-level factor and adjusted premium are

$$ \text{On-level factor}=\frac{R_{\text{target}}}{R_{\text{avg}}}, \qquad P_{\text{on-level}}=P_{\text{earned}}\frac{R_{\text{target}}}{R_{\text{avg}}}. $$

The factor is a ratio, not the percentage change in rates. It is greater than one when the target level exceeds the average earned level, and less than one when the target is lower.

1
Set the time interval.
Express the experience-period start \(a\), end \(b\), and rate-change date \(c\) in consistent year units.
2
Find the new-level earned share.
For one annual-policy change within the period, calculate \(f_{\text{new}}=(b-c)^2/[2(b-a)]\).
3
Calculate the average level.
Weight the old and new rate-level indices by \(1-f_{\text{new}}\) and \(f_{\text{new}}\).
4
Convert to the target.
Divide the target level by the average earned level; multiply earned premium by that factor if an adjusted premium is required.

Worked Applications

Worked Example: A Midyear Increase in a Calendar-Year Experience Period

A hypothetical insurer writes annual policies uniformly throughout the year. A 12% rate increase takes effect July 1, 2025. Calculate the average earned rate level for calendar year 2025, the factor to the new rate level, and the on-level premium if earned premium is $203,000.

Method and dates. Use the one-change parallelogram method. Set \(a=\) January 1, 2025, \(b=\) January 1, 2026, and \(c=\) July 1, 2025. In years, \(b-a=1\) and \(b-c=0.5\). The old and new indices are 1.00 and 1.12.

New-level earned share.

$$ f_{\text{new}}=\frac{(0.5)^2}{2(1)} =\frac{0.25}{2} =0.125. $$

Thus 12.5% of the period's earned exposure is at the new level. The average earned rate level is

$$ R_{\text{avg}} =1.00+(1.12-1.00)(0.125) =1.00+0.12(0.125) =1.015. $$

The target is the new level, 1.12, so the on-level factor is

$$ \frac{1.12}{1.015} =\frac{224}{203} =1.103448\ldots \approx 1.1034. $$

Multiplying the earned premium by the unrounded ratio gives

$$ \$203{,}000\left(\frac{224}{203}\right)=\$224{,}000. $$

The result checks: the factor converts premium at level 1.015 to level 1.12, since \(1.015(224/203)=1.12\). The common mistake avoided here is treating all policies written after July 1 as though they contributed a full year's earned premium at the new level.

Worked Example: A Decrease During a July-to-June Experience Period

A hypothetical insurer has annual policies and uniform writings. Its experience period is July 1, 2024, through June 30, 2025. A 8% rate decrease takes effect October 1, 2024. Earned premium for the period is $391,000 at the average earned level. Find the average level and premium on-leveled to the new rate level.

Method and dates. The experience period is one year, so \(b-a=1\). The change occurs three months after the period begins and nine months before it ends, so \(b-c=9/12=0.75\) years. The new index is \(1-0.08=0.92\), while the old index is 1.00.

New-level earned share.

$$ f_{\text{new}}=\frac{(0.75)^2}{2(1)} =\frac{0.5625}{2} =0.28125. $$

The average earned level is therefore

$$ R_{\text{avg}} =1.00+(0.92-1.00)(0.28125) =1.00-0.08(0.28125) =0.9775. $$

The factor to the new level is

$$ \frac{0.92}{0.9775} =\frac{368}{391} =0.941176\ldots \approx 0.9412. $$

Using the exact ratio, the on-level premium is

$$ \$391{,}000\left(\frac{368}{391}\right)=\$368{,}000. $$

The calculation checks because the earned premium is reduced when moving from average level 0.9775 to target level 0.92. A decrease does not mean using the increase formula with a positive change: the signed difference \(R_{\text{new}}-R_{\text{old}}\) must be negative.

Worked Example: A Change Early in the Experience Year

A hypothetical annual-policy portfolio has uniform writings and stable exposure volume. The experience period is calendar year 2025, and a 15% increase takes effect March 1, 2025. Earned premium is $505,000 at the average level. Calculate the average earned level and the factor and premium at the new level.

Method and dates. The period duration is one year. March 1 is two months after the start, leaving ten months, or \(10/12=5/6\) of a year, between the change and the end. The old and new levels are 1.00 and 1.15.

New-level earned share.

$$ f_{\text{new}} =\frac{(5/6)^2}{2(1)} =\frac{25}{72} \approx 0.347222. $$

The average level is

$$ R_{\text{avg}} =1.00+(1.15-1.00)\frac{25}{72} =1+\frac{3}{20}\frac{25}{72} =1+\frac{5}{96} =\frac{101}{96} \approx 1.052083. $$

The on-level factor is

$$ \frac{1.15}{101/96} =\frac{23}{20}\frac{96}{101} =\frac{552}{505} \approx 1.0931. $$

The premium at the new level is

$$ \$505{,}000\left(\frac{552}{505}\right)=\$552{,}000. $$

The check is direct: the premium ratio \(552/505\) equals the rate-level ratio \(1.15/(101/96)\). The computation retains exact fractions until the displayed rounded factor.

Interpretation and Common Traps

The parallelogram is a model of earned premium transition, not a count of policies written at each rate. For the July 1 change in the first example, half the calendar year's writings occur after the effective date, but those policies earn only part of their annual premium during that same calendar year. The triangular area gives a new-level earned share of 12.5%, not 50%.

For instance, incorrectly using 50% in that example would produce an average level of \(1+0.12(0.50)=1.06\), and a factor of \(1.12/1.06\approx1.0566\). The correct average is 1.015 and the correct factor is approximately 1.1034. The incorrect approach substantially understates the adjustment because it confuses the share of writings with the share of earned exposure.

The formula also has specific limits. It assumes annual policy terms, uniform writings, a stable exposure base, and one rate change within the period. It is not the appropriate direct calculation when writings are materially seasonal, the exposure mix changes, policy terms differ, or there are multiple rate changes. Nor should the triangular formula be extended past a full year after the change without accounting for the fact that the new-level share cannot exceed one. Those situations require a model that represents the actual timing and exposure pattern.

A final exam trap is reversing the factor. To adjust premium from the average earned level to a target level, use target divided by average. The reverse ratio answers a different question: it converts from target level back to average level. Always identify the source and destination rate levels before forming the ratio.

Key takeaway: For one rate change within an experience period, annual policies, uniform writings, and at most one year from change to period end, the new-level earned share is the triangle area \((b-c)^2/2\) divided by the period duration \(b-a\). Use that share to find the average level, then divide the target level by the average to obtain the on-level factor.

Check Your Understanding

For each hypothetical annual-policy portfolio, apply the stated assumptions of uniform writings and stable exposure. Retain precision in intermediate calculations.

  1. A 10% increase takes effect April 1 during a January-to-December experience year. Find the new-level earned share, average earned level, and factor to the new level. (Answer: \(f_{\text{new}}=0.28125\), average level \(=1.028125\), factor \(=1.0699\).)
  2. A 12% decrease takes effect January 1 during a July-to-June experience year. Find the new-level earned share, average level, and factor to the new level. (Answer: \(f_{\text{new}}=0.125\), average level \(=0.985\), factor \(=0.8934\).)
  3. An 8% increase takes effect October 1 during a calendar-year experience period. Calculate the average level and the factor to level 1.08. (Answer: \(f_{\text{new}}=0.03125\), average level \(=1.0025\), factor \(=1.0773\).)
  4. An 18-month experience period has a 20% increase effective three months before the end. Under the stated annual-policy assumptions, calculate the new-level earned share, average level, and factor to the new level. (Answer: \(f_{\text{new}}=0.0208333\), average level \(=1.0041667\), factor \(=1.1950\).)
  5. A 6% increase takes effect May 1 during a calendar-year experience period. A colleague says the new-level earned share is \(8/12\), based on the time from May 1 to year-end. Correct the earned share, average level, and factor to level 1.06. (Answer: \(f_{\text{new}}=0.2222222\), average level \(=1.0133333\), factor \(=1.0461\).)

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