Renewal-Effective Changes on the Parallelogram

Why Renewal-Effective Changes Ramp Up Gradually

A rate change that applies only when policies renew does not reach every policy on its announced effective date. Policies already in force retain their existing rates until renewal, while newly renewed policies begin using the changed rates. As time passes, a larger share of the earned exposure is therefore at the new rate. The parallelogram method represents this transition as a ramp.

The key calculation is not simply the percentage change in the rate level. It is the average share of exposure at the new rate during the experience period. That share determines the average earned rate level to use when bringing historical premium to a target level. This tutorial considers one rate change, policies with a common term, and uniformly distributed renewal dates.

Definition: Let $T$ be the policy term, measured in the same time units as $t$, and let $t=0$ be the date the rate change becomes effective for renewing policies. The new-rate share $q(t)$ is the share of earned exposure at time $t$ that is subject to the changed rate. Under the standard parallelogram assumptions, $$ q(t)= \begin{cases} 0, & t\leq 0,\\ t/T, & 0<t<T,\\ 1, & t\geq T. \end{cases} $$

Before the effective date, no policies have renewed into the new rate level, so the share is zero. During the following policy term, the share rises linearly: at time $t$, the fraction of renewal dates that have occurred since the change is $t/T$. Once a full term has passed, all policies have had an opportunity to renew, so the share is one. The exact treatment of a renewal on the effective date does not affect the average; an instant has zero duration in the exposure calculation.

Average New-Rate Share and Earned Rate Level

Suppose the old rate level is indexed to 1 and the rate change is $r$, expressed as a signed decimal. For example, a 12% increase has $r=0.12$, while an 8% decrease has $r=-0.08$. At time $t$, the average rate level is $1+r q(t)$: the share $q(t)$ receives the changed rate, and the remaining share retains the old rate.

For an experience period from time $a$ to time $b$, with $b>a$, average the new-rate share over the period. This assumes earned exposure is uniform through calendar time and does not vary in a way that changes the weighting of the share.

\bar q=\frac{1}{b-a}\int_a^b q(t)\,dt, \qquad \bar L=1+r\bar q

Here $\bar q$ is the average new-rate share and $\bar L$ is the average earned rate level relative to the old level. If the target is the full changed level $1+r$, the factor applied to historical earned premium is:

F_{\text{on-level}}=\frac{1+r}{\bar L}, \qquad P_{\text{on-level}}=P_{\text{historical}}F_{\text{on-level}}.

The factor adjusts for the rate level already reflected in the historical premium. It is not generally equal to $1+r$, because some historical earned premium may already reflect the change. For a different target level, replace $1+r$ in the numerator by that target, expressed on the same index scale.

Why the Averaging Formula Works

The parallelogram follows from the renewal-date distribution. Assume policies have term $T$, their renewal dates are uniformly distributed through a full term, and the changed rate takes effect at renewal. At time $t$ after the change, where $0<t<T$, the fraction of renewal dates that have occurred since time zero is $t/T$. Those policies have renewed at the changed rate; the rest have not. Thus $q(t)=t/T$ during the ramp.

To find the average rate level over the experience period, divide the accumulated rate-level exposure by the period’s total exposure. At each time $t$, the rate level is $1+r q(t)$. Under uniform earning, integration over time gives:

\bar L =\frac{1}{b-a}\int_a^b \bigl(1+r q(t)\bigr)\,dt =1+r\left(\frac{1}{b-a}\int_a^b q(t)\,dt\right) =1+r\bar q.

The second equality holds because the integral of the constant 1 over $[a,b]$ is $b-a$. This also explains why the experience period’s position relative to the ramp matters: the average share is the area under $q(t)$, divided by the period length.

A Reliable Calculation Sequence

Set the time origin and units.
Measure $t=0$ at the renewal-effective date. Use the same units for the policy term and every experience-period boundary.

Mark the ramp boundaries.
For term $T$, the ramp runs from 0 to $T$. Before 0 the share is zero; after $T$ it is one.

Find the area within the experience period.
Split the period at 0 and $T$ wherever needed. Use a triangle or trapezoid for the linear part and a rectangle for a full-rate plateau.

Calculate the average level and factor.
Divide the area by the experience-period length to get $\bar q$; then calculate $\bar L$ and the target-to-average factor.

Apply and check.
Multiply historical earned premium by the factor. Check that an increase produces a factor above 1 when the target is the full new level and the historical average is between the old and new levels.

Worked Applications

Worked Example: A Period That Includes the Ramp and Plateau

A hypothetical insurer has annual policies with renewal dates uniformly distributed through the year. A 15% rate increase becomes effective at renewal at time zero. The experience period runs from month 6 through month 18, and historical earned premium is $1,150,000. Find the average new-rate share, the on-level factor to the full new rate, and the on-level premium.

Method and area. The term is $T=12$ months. From month 6 to month 12, the share rises from $6/12=0.5$ to 1. The area is a trapezoid:

A_{\text{ramp}}=6\left(\frac{0.5+1}{2}\right)=4.5

From month 12 to month 18, the share is 1 for 6 months, so the plateau area is $6(1)=6$. The experience period is 12 months long. Therefore:

\bar q=\frac{4.5+6}{12}=\frac{10.5}{12}=0.875

Average level and factor. With $r=0.15$, the average earned level is:

\bar L=1+0.15(0.875)=1.13125

The target level is $1.15$, so:

F_{\text{on-level}}=\frac{1.15}{1.13125} =\frac{184}{181} \approx 1.016574586

On-level premium. Applying the factor gives:

P_{\text{on-level}}=\$1{,}150{,}000\left(\frac{184}{181}\right) =\frac{\$211{,}600{,}000}{181} \approx \$1{,}169{,}060.77

The check is that the factor is slightly above 1: some premium is already at the new level, so the factor is smaller than 1.15. The common mistake is to average the ramp and plateau incorrectly. The ramp area is 4.5, not 3; using a 0.75 share would understate the rate level already earned and overstate the on-level factor.

Worked Example: A Shorter Policy Term

A hypothetical equipment policy has a six-month term, with renewal dates uniformly distributed through each term. A 12% increase applies at renewal from time zero. The experience period is months 3 through 9, and historical earned premium is $840,000. Calculate the average share and premium at the full new rate.

Method and area. Here $T=6$. Between months 3 and 6, the share increases from $3/6=0.5$ to 1. The trapezoid area is:

A_{\text{ramp}}=3\left(\frac{0.5+1}{2}\right)=2.25

Between months 6 and 9, the share is fully at the new rate, giving area $3(1)=3$. The period length is 6 months, so:

\bar q=\frac{2.25+3}{6}=0.875

The average rate level is $1+0.12(0.875)=1.105$. The target-to-average factor is:

F_{\text{on-level}}=\frac{1.12}{1.105} =\frac{224}{221} \approx 1.013574661

Thus the premium at the full new level is:

\$840{,}000\left(\frac{224}{221}\right) =\frac{\$188{,}160{,}000}{221} \approx \$851{,}402.71

The calculation checks because the experience period contains both a ramp segment and a plateau, and the average share is below 1. A common mistake is to use 12 months as the ramp length simply because the policy year is a familiar unit. The relevant term here is six months, so the ramp is complete at month 6.

Worked Example: A Decrease with Pre-Change Experience

A hypothetical service contract has a 12-month term and uniformly distributed renewal dates. A 8% rate decrease applies at renewal at time zero. An experience period runs from month $-6$ through month 6, and historical earned premium is $990,000. Find the average earned level and the premium at the full decreased rate.

Method and area. Before month zero, the new-rate share is zero. From month zero through month 6, the share rises linearly from 0 to $6/12=0.5$. The triangular area is:

A_{\text{ramp}}=\frac{1}{2}(6)(0.5)=1.5

There is no plateau portion in the experience period. Its total length is 12 months, so:

\bar q=\frac{1.5}{12}=0.125

For a decrease, $r=-0.08$. The average level is $1+(-0.08)(0.125)=0.99$, and the target is $1-0.08=0.92$. Therefore:

F_{\text{on-level}}=\frac{0.92}{0.99} =\frac{92}{99} \approx 0.929292929

Applying the factor gives:

\$990{,}000\left(\frac{92}{99}\right)=\$920{,}000.00

The result is below the historical premium, as expected when moving premium from an average level of 0.99 to a target of 0.92. The common mistake is to assign the new level to all experience after the effective date. Only a fraction of the policies have renewed by month 6; the share then is 0.5, not 1.

Assumptions and Common Traps

The geometric ramp is useful when the model’s assumptions fit the premium being adjusted. In particular, the calculation assumes one rate change; a common policy term; renewal dates spread uniformly through that term; the changed rate applies at renewal rather than midterm; and exposure earns uniformly through calendar time. If renewal dates are concentrated, terms differ, exposure is seasonal, or rate changes apply midterm, the simple ramp may not represent the correct premium weights. The rate change should also apply consistently to the business being averaged.

A renewal-effective change must not be treated as an immediate change to every policy. That would make the share jump from 0 to 1 at time zero, which describes a different implementation. Nor should the ramp be extended beyond one term: after $T$, the new-rate share is 1. To calculate an experience-period average that crosses either boundary, split the period into the applicable zero-share, ramp, and full-share sections before finding the area.

Mixing up share and rate level. The share $\bar q$ is not the average rate level. Convert it using $\bar L=1+r\bar q$.
Ignoring the period length. The area under the share curve must be divided by $b-a$, not by the policy term unless those lengths happen to be equal.
Using the wrong sign. Represent a decrease with negative $r$; do not subtract the change a second time.
Using the wrong target. For the full new level, the numerator is $1+r$, while the denominator is the average level actually reflected in the historical premium.

Key takeaway: For a renewal-effective change, the new-rate share rises linearly from zero to one over a policy term. Find the area under that share curve within the experience period, divide by the period length, convert the average share to an average rate level, and then compare that level with the target.

Check Your Understanding

For each problem, use the renewal-effective ramp and state the average share before applying any premium factor.

A hypothetical insurer has 12-month policies and uniformly distributed renewal dates. A 10% increase applies at renewal from time zero. The experience period is months 0 through 12, and historical earned premium is $420,000. Find the average share and the premium at the full new level. (Answer: $\bar q=0.5000$; on-level premium = $440,000.00.)
A 12-month policy has a 12% renewal-effective increase. The experience period runs from month 9 through month 15, and historical premium is $890,000. Find the average share and on-level premium. (Answer: $\bar q=0.9375$; on-level premium = $896,000.00.)
A 12-month policy has an 8% renewal-effective increase. The experience period is month $-3$ through month 9, and historical earned premium is $818,000. Find the average share and premium at the full new level. (Answer: $\bar q=0.28125$; on-level premium = $864,000.00.)
A six-month policy has a 10% renewal-effective decrease. The experience period runs from month 2 through month 8, and historical earned premium is $830,000. Find the average share and premium at the full decreased level. (Answer: $\bar q=7/9\approx0.7778$; on-level premium = $810,000.00.)
A 12-month policy has a 20% renewal-effective increase. The experience period runs from month 4 through month 16, and historical earned premium is $1,040,000. Find the average share and premium at the full new level. (Answer: $\bar q=7/9\approx0.7778$; on-level premium = $1,080,000.00.)

What You'll Learn