Midterm Effective Dates and Prorated Earning

Why the Midterm Date Matters

A rate change can apply only to policies starting at renewal, or it can apply to coverage already in force. When it applies to in-force coverage, the premium for the remaining coverage changes at the stated effective date. To calculate earned premium correctly, separate the coverage provided before that date from the coverage provided after it. The first portion earns at the old rate level; the second earns at the new rate level.

This tutorial uses a stated pro rata earning assumption: premium is earned uniformly over the policy term. The calculations are deterministic; no interest rate, mortality assumption, or probability distribution is involved. The focus is the timing of the rate change and the allocation of premium to the coverage periods on either side of it.

Definition: Suppose a policy has a term of $T$ equal time units and a total-term premium $P$ at the old rate level. A midterm change takes effect after $c$ time units, where $0 \leq c \leq T$, and changes the rate level by a factor $k$. Under uniform pro rata earning, coverage before the change earns at the old rate level, and coverage after it earns at the new rate level. Thus the new-rate premium for a full term would be $kP$, but the factor $k$ applies only to the portion of the term after the effective date.

Splitting Earned Premium at the Effective Date

When calculating from policy inception through the end of the term, the pre-change portion is $c/T$ of the term and the post-change portion is $(T-c)/T$. The earned premium assigned to each rate level is therefore:

E_{\text{old}} = P\frac{c}{T}, \qquad E_{\text{new}} = kP\frac{T-c}{T}, \qquad E_{\text{total}} = E_{\text{old}} + E_{\text{new}}.

The portions add because, under this assumption, every part of the policy term is earned exactly once. If the experience period ends before the policy expires, calculate only the coverage falling within that experience period. For example, if it includes $a$ time units before the change and $b$ time units after it, the earned amount in that period is $P(a/T) + kP(b/T)$. The relevant amounts of time must be measured on the same basis as $T$, such as equal months or days.

Key condition: The split-rate calculation applies when the change is effective for in-force coverage at the stated midterm date. If the change applies only at renewal, the current policy remains at the old rate through its term; do not assign part of its earned premium to the new rate level.

Why the Pro Rata Formula Works

Under uniform earning, each equal unit of coverage time earns the same fraction of the premium applicable to that time. At the old rate, the premium per time unit is $P/T$. The first $c$ time units therefore earn $c(P/T)$. After the change, the applicable full-term premium is $kP$, so each remaining time unit earns $kP/T$. Multiplying by the $T-c$ remaining units gives the post-change amount.

\underbrace{c\frac{P}{T}}_{\text{old-rate coverage}} + \underbrace{(T-c)\frac{kP}{T}}_{\text{new-rate coverage}} = P\frac{c}{T}+kP\frac{T-c}{T}.

This is also consistent with the premium adjustment for the remaining coverage. The original total-term premium was $P$; the change adds or subtracts $(k-1)P$ for a full term at the changed rate. Since only $(T-c)/T$ of the term remains, the midterm adjustment is:

\Delta P = (k-1)P\frac{T-c}{T}.

If the policy runs to expiration and premiums are earned uniformly, total written premium after the midterm adjustment equals total earned premium over the entire term:

P+\Delta P = P+(k-1)P\frac{T-c}{T} = P\frac{c}{T}+kP\frac{T-c}{T}.

The equality is a useful check, not a claim that all written premium is earned immediately. Before expiration, some written premium remains unearned because part of the coverage has not yet been provided.

Worked Example: A Rate Increase Halfway Through an Annual Policy

A hypothetical insurer issues a 12-month policy with a total-term premium of $1,200. A 20% rate increase applies to in-force coverage after 6 months. Assume equal-month pro rata earning and that the policy runs to expiration. Find the earned premium at each rate level, the midterm adjustment, and total written premium after the adjustment.

Method and assumptions. Use $T=12$, $P=\$1{,}200$, $c=6$, and $k=1.20$. The change applies to the remaining six months, not to the first six months. There is no interest or stochastic model in this calculation.

Old-rate portion. Six of the twelve months earn at the old rate:

E_{\text{old}} = \$1{,}200\frac{6}{12} = \$600.

New-rate portion. The full-term premium at the new rate would be $1.20(\$1{,}200)=\$1{,}440$. The remaining six months earn half of that amount:

E_{\text{new}} = \$1{,}440\frac{6}{12} = \$720.

Total earned premium is $\$600+\$720=\$1{,}320$. The midterm adjustment is the 20% increase applied to the remaining half of the original premium:

\Delta P = (1.20-1)(\$1{,}200)\frac{6}{12} = \$120.

Thus total written premium is $\$1{,}200+\$120=\$1{,}320$, matching total earned premium at expiration. Common mistake avoided: applying the increase to all 12 months would produce $\$1{,}440$, incorrectly repricing coverage provided before the effective date. The correct result is $1,320.

Partial Experience Periods and Unequal Portions

A midterm date need not divide either the policy term or the experience period in half. Identify the amount of covered time in each rate segment that falls inside the period being measured, then apply the corresponding rate-level premium to each segment. Do not use the policy’s full remaining term if the experience period ends earlier.

For clarity, suppose the policy term is measured from time 0 to $T$, the change occurs at time $c$, and the experience period covers times $a$ to $b$, with $0 \leq a \leq b \leq T$. The old-rate overlap is the length of the experience period before $c$; the new-rate overlap is its length after $c$. The earned amount in that experience period is the old-rate premium per unit time times the old overlap, plus the new-rate premium per unit time times the new overlap.

Worked Example: A Rate Decrease With an Experience Period Ending Before Expiration

A hypothetical six-month policy has a total-term premium of $900. After two months, a 20% rate decrease applies to in-force coverage. Calculate premium earned from inception through the end of month 5, the midterm written-premium adjustment, and unearned premium at that point. Assume each month earns uniformly and the policy remains in force through month 6.

Method and assumptions. Use $T=6$, $P=\$900$, $c=2$, $k=0.80$. The experience period includes the first two months at the old rate and the next three months at the new rate. No interest or probability distribution is involved.

Earned at the old rate. The old rate applies to the first two months:

E_{\text{old}} = \$900\frac{2}{6} = \$300.

Earned at the new rate. The full-term premium at the reduced rate is $0.80(\$900)=\$720$. Three months at that rate earn:

E_{\text{new}} = \$720\frac{3}{6} = \$360.

Earned premium through the end of month 5 is $\$300+\$360=\$660$. The midterm endorsement covers the four months remaining after the change, so its adjustment is:

\Delta P = (0.80-1)(\$900)\frac{4}{6} = -\$120.

Total written premium is $\$900-\$120=\$780$. The final month is still unearned at the end of month 5. Its premium at the new rate is $\$720/6=\$120$, so unearned premium is $120. The check is $\$660+\$120=\$780$, equal to written premium. Common mistake avoided: counting all four post-change months as earned would give $\$780$ earned by month 5, although only three post-change months have elapsed.

Several Policies Affected on the Same Date

For a group of policies, the calculation is performed policy by policy and the resulting earned amounts are summed. Policies can have different inception dates, so a single calendar-date change can occur at different points in their respective terms. The effective date must be located within each policy’s coverage period before its pre-change and post-change portions are calculated.

Worked Example: Two Staggered Policies With a Common Midterm Change

Two hypothetical 12-month policies are subject to a 15% increase effective January 1. Policy A began September 1 and has an old-rate total-term premium of $2,400. Policy B began November 1 and also has an old-rate total-term premium of $2,400. Calculate total earned premium from each policy’s inception through June 30, and reconcile the total to written and unearned premium. Assume month-boundary dates and equal-month pro rata earning.

Method and assumptions. The rate factor is $k=1.15$, making each policy’s full-term premium at the new rate $\$2{,}400(1.15)=\$2{,}760$. From September 1 through December 31, Policy A has four old-rate months; from November 1 through December 31, Policy B has two. Each policy earns six months at the new rate from January through June.

Policy A. Its old-rate earned premium is $\$2{,}400(4/12)=\$800$. Its new-rate earned premium is $\$2{,}760(6/12)=\$1{,}380$. Total earned through June 30 is $\$800+\$1{,}380=\$2{,}180$.

Policy B. Its old-rate earned premium is $\$2{,}400(2/12)=\$400$. Its new-rate earned premium is $\$2{,}760(6/12)=\$1{,}380$. Total earned through June 30 is $\$400+\$1{,}380=\$1{,}780$.

Across both policies, earned premium is $\$2{,}180+\$1{,}780=\$3{,}960$. Policy A has eight months remaining on January 1, so its endorsement is $(0.15)(\$2{,}400)(8/12)=\$240$. Policy B has ten months remaining, so its endorsement is $(0.15)(\$2{,}400)(10/12)=\$300$. Total written premium is $\$2{,}400+\$2{,}400+\$240+\$300=\$5{,}340$.

At June 30, Policy A has two unearned months, or $\$2{,}760(2/12)=\$460$. Policy B has four unearned months, or $\$2{,}760(4/12)=\$920$. Total unearned premium is $\$1{,}380$, and the check is $\$3{,}960+\$1{,}380=\$5{,}340$. Common mistake avoided: using the same pre-change fraction for both policies simply because the calendar effective date is common. Their policy inception dates differ, so their term fractions differ.

Common Traps

Applying the new rate retroactively. A change effective midterm applies to coverage from the effective date onward, not to coverage already provided.
Confusing the full-term new premium with the endorsement. The full-term premium at the new rate is $kP$; the endorsement changes only the premium for the remaining fraction of the policy term.
Using the remaining term instead of elapsed experience time. When the experience period ends before expiration, count only the coverage provided through that ending date.
Mixing time units. If $T$ is measured in months, measure $c$ and each overlap in months too. For day-based pro rata earning, use consistent day counts for the full term and each segment.
Ignoring policy-specific dates. For a portfolio, a common calendar change date may fall at different points in different policy terms.

Key takeaway: Under pro rata earning, split the covered time at the effective date, apply the old rate to the earlier portion and the new rate to the later portion, and include only time inside the experience period. When the policy runs to expiration, the sum of the two earned portions should equal written premium after the prorated midterm adjustment.

Check Your Understanding

For each hypothetical policy, assume uniform pro rata earning, no interest, and no cancellation unless stated. Check that your time fractions use the policy term as the denominator.

A 12-month policy has a total-term premium of $960. After three months, a 25% increase applies to in-force coverage. Find the old-rate and new-rate earned premium at expiration and the endorsement amount. (Answer: Old-rate earned $240; new-rate earned $900; endorsement $180.)
A six-month policy has a total-term premium of $900. A 20% decrease takes effect after two months. Find earned premium through the end of month 5 and the unearned premium then. (Answer: Earned premium $660; unearned premium $120.)
A 12-month policy has a total-term premium of $1,500. A 12% increase applies after four months. Find earned premium through the end of month 9, split by rate level. (Answer: Old-rate earned $500; new-rate earned $700; total $1,200.)
Two 12-month policies each have an old-rate total-term premium of $1,200 and $1,800, respectively. A 10% increase applies after five months on the first policy and after eight months on the second. Find their combined earned premium at expiration. (Answer: First policy $1,270; second policy $1,860; combined $3,130.)
A 12-month policy has an old-rate total-term premium of $2,400. An 8% increase applies after seven months. Calculate earned premium for policy months 5 through 10, and the total written premium after the midterm endorsement. (Answer: Earned in months 5–7 at old rate $600 and in months 8–10 at new rate $648, for $1,248 total; written premium $2,480.)

What You'll Learn