Earned Premium from Multiple Policy Transactions

Why Multiple Transactions Require Separate Earning

A policy's premium may change after issuance. A midterm endorsement can increase or decrease the premium for the coverage that remains, and a cancellation ends the period over which premium can be earned. To measure earned premium correctly, treat the policy as a sequence of time intervals. Each premium component contributes only while it is in effect.

The method here assumes straight-line earning over time and pro rata treatment of endorsements and cancellation. It does not allow for short-rate cancellation, minimum-earned premium, fees, taxes, or nonuniform coverage. Those features require separate adjustments. No interest or mortality assumption is needed: this is a premium allocation over a stated coverage period, not a discounted cash-flow or life-contingency calculation.

Definition: Earned premium through a specified date is the portion of premium allocated to coverage provided up to that date. Under straight-line earning, premium is allocated in proportion to elapsed coverage time. An endorsement affects the premium only from its effective date. A pro rata cancellation stops earning at the cancellation date.

Represent the Policy as Time Intervals

Let a policy have a term of $N$ days, with time measured from inception: day 0 is the start of coverage and day $N$ is expiration. A date at elapsed time $c$ represents $c$ days of coverage. Thus, the interval from day 0 up to day $c$ contains $c$ days; the date at the end of an interval is not counted as an additional day.

Let $P_0$ be the premium for the original policy term. For an endorsement effective at elapsed day $t_j$, let $A_j$ be its annualized full-term premium change: the change that would apply over a full $N$-day term at the same daily rate. For example, an increase of $A_j$ changes the daily premium by $A_j/N$, starting at day $t_j$.

The annualized change $A_j$ is not generally the amount booked for the endorsement. Under pro rata treatment, the actual endorsement transaction at day $t_j$ is based on the remaining $N-t_j$ days. Its amount is $A_j(N-t_j)/N$. If the data instead give the actual booked endorsement amount, use that amount and its affected period directly.

Result: If the valuation date is $c$, where $0\leq c\leq N$, and each endorsement is pro rata, earned premium through $c$ is

EP(c)=P_0\frac{c}{N}+\sum_j A_j\frac{\max(0,c-t_j)}{N}.

Here, each endorsement date $t_j$ is within the policy term. A term ending at $N$ limits the amount of time any component can earn. If the supplied endorsement figure is the actual booked transaction amount $W_j$, rather than an annualized change, its earned contribution through $c$ is $W_j\max(0,\min(c,N)-t_j)/(N-t_j)$, for $t_j<N$.

The positive-part term $\max(0,c-t_j)$ ensures that an endorsement contributes nothing before it takes effect. When cancellation occurs at day $c$, the same calculation gives pro rata earned premium to cancellation. Premium for coverage after that date is unearned, subject to any applicable cancellation provisions.

Why the Formula Works

Under straight-line earning, a premium component that applies for $N$ days earns the same fraction as the fraction of those days that has elapsed. The original premium $P_0$ applies from day 0, so by day $c$ its earned fraction is $c/N$. An endorsement change $A_j$ applies only from day $t_j$. By day $c$, it has affected $\max(0,c-t_j)$ days. Its earned fraction of the full-term equivalent is therefore $\max(0,c-t_j)/N$. Adding the earned portions of all components gives the result.

The same reasoning explains why an actual booked endorsement amount $W_j$ uses a different denominator. That booked amount already covers only the remaining $N-t_j$ days. By day $c$, the fraction of that transaction earned is the elapsed part of its affected period divided by the total affected period, or $(c-t_j)/(N-t_j)$ when $t_j<c\leq N$. Dividing $W_j$ by $N$ instead would prorate the transaction as though it applied to the entire original term.

Worked Example: An Increase Effective Midterm

A hypothetical 360-day equipment policy has an initial term premium of $3,600. On day 90, an endorsement increases the annualized full-term premium by $720. Find earned premium through day 210 and reconcile it to the written and unearned amounts under pro rata earning.

Method. Divide the original and annualized endorsement premiums by 360 to obtain their daily amounts. Apply the increase only from day 90 onward.

The original daily premium is $3,600 / 360 = $10. The endorsement changes the daily premium by $720 / 360 = $2. The policy has 90 days at the original rate before the endorsement and 120 days at the increased rate through day 210.

EP(210)=90(\$10)+120(\$10+\$2)=\$900+\$1{,}440=\$2{,}340.

The endorsement is effective for 270 days of the term, so its booked pro rata amount is $720(270/360) = $540. Total written premium is $3,600 + $540 = $4,140. After day 210, 150 days remain. Unearned premium is $150($10) + $150($2) = $1,800, and $2,340 + $1,800 = $4,140.

Check and common mistake. The earned and unearned amounts sum to written premium. A mistaken calculation would apply the endorsement from inception: ($3,600 + $720)(210/360) = $2,520. That overstates earned premium because the endorsement did not take effect until day 90. The correct answer is $2,340.

Adding Endorsements and a Pro Rata Cancellation

For a policy with several transactions, each premium component has its own effective date and daily amount. Work through the intervals in order, or calculate each component's earned contribution separately. A decrease is entered as a negative adjustment; it reduces premium only during the period it affects. For a pro rata cancellation, stop each component's earning at the cancellation date.

Set the term and dates.
Record the term length $N$, each endorsement's effective day, and the valuation or cancellation day $c$.

Identify the premium basis.
For each transaction, determine whether the stated amount is an annualized full-term change or the actual booked amount for the remaining term.

Earn each component over its active interval.
Use elapsed days in that component's coverage interval, not elapsed days since policy inception for every transaction.

Reconcile the result.
Compare written premium with earned and unearned premium, applying the same pro rata assumptions to both.

Worked Example: Two Endorsements Before Cancellation

A hypothetical 365-day policy has an initial premium of $4,380. On day 100, an endorsement increases the annualized full-term premium by $730. On day 220, a second endorsement decreases it by $365. The policy is cancelled pro rata on day 280. Find earned premium, the booked endorsement amounts, and unearned premium.

Method. The original premium applies for all 280 covered days. The first change applies from day 100 to cancellation, or 180 days; the second applies from day 220 to cancellation, or 60 days.

The daily amounts are $4,380/365 = $12 for the original premium, $730/365 = $2 for the first change, and $-$365/365 = -$1$ for the second change. Therefore:

EP(280)=280(\$12)+180(\$2)+60(-\$1) =\$3{,}360+\$360-\$60=\$3{,}660.

The first endorsement is effective for 265 remaining days, so its booked amount is $730(265/365) = $530. The second is effective for 145 remaining days, so its booked amount is $-$365(145/365)=-$145$. Total written premium is $4,380 + $530 - $145 = $4,765.

For a component-by-component check, the original premium has 85 unexpired days: 85($12) = $1,020 unearned. The first change has 85 unexpired affected days: 85($2) = $170 unearned. The second change has 85 unexpired affected days: 85(-$1) = -$85 unearned. Net unearned premium is $1,020 + $170 - $85 = $1,105. The reconciliation is $3,660 + $1,105 = $4,765.

Common mistake avoided. Treating $730 and $-$365$ as if they were the endorsement transaction amounts would overstate the changes booked: those figures are annualized full-term changes. The actual pro rata transactions are $530 and $-$145$. The cancellation also does not erase the earned effect of an endorsement; it stops that effect from earning after day 280.

Worked Example: A Decrease Followed by Cancellation

A hypothetical 360-day policy has an initial premium of $5,400. An endorsement effective on day 120 decreases the annualized full-term premium by $900. The policy is cancelled pro rata on day 240. Calculate earned premium, the endorsement transaction amount, and unearned premium.

Method. The original daily premium is $5,400/360 = $15. The endorsement reduces the daily amount by $900/360 = $2.50, so the premium rate is $15 for the first 120 days and $12.50 for the next 120 days.

EP(240)=120(\$15)+120(\$12.50) =\$1{,}800+\$1{,}500=\$3{,}300.

The decrease applies over the 240 days remaining when it takes effect, so the booked endorsement amount is $-\$900(240/360)=-\$600$. Written premium is $5,400 - $600 = $4,800. There are 120 days left after cancellation. The original component has $1,800 unearned, while the decrease component contributes $-120($2.50)=-$300$ to unearned premium. Net unearned premium is $1,500, and $3,300 + $1,500 = $4,800.

Check and common mistake. The endorsement's negative premium effect must be included in both the earned and unearned periods. Ignoring it after cancellation would incorrectly report $1,800 unearned. The correct net unearned amount is $1,500.

Common Traps and Practical Checks

Transaction records can show either a premium change on an annualized basis or the actual amount booked for the remaining policy term. The distinction determines the denominator. If an annualized change is $A_j$, divide by the full term $N$ to obtain its daily effect. If the recorded amount is $W_j$, and it applies from day $t_j$ to expiration, allocate it over $N-t_j$ days. Do not prorate a booked transaction as though it covered the original full term.

A second frequent error is applying all premium changes from policy inception. Endorsements have effective dates; before that date, the original premium basis remains in force. A third is using the cancellation date as though the policy earned premium after coverage ended. Under the pro rata assumption, the cancellation date ends the earning interval. Short-rate cancellation or other contractual provisions can change the amount retained and are outside the straight-line calculation shown here.

For a final check, confirm that the endorsement's sign is correct, that every effective date is applied to the right interval, and that earned premium plus unearned premium reconciles to written premium. This calculation measures the premium allocated to coverage through the chosen date; assigning that premium to a particular experience period is a separate question. The treatment here is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical policy, assume straight-line earning, pro rata endorsements, and pro rata cancellation where stated. Treat endorsement figures explicitly identified as annualized changes as full-term equivalents.

A 300-day policy has an initial premium of $3,000. An annualized increase of $600 takes effect on day 60. Find earned premium through day 150. (Answer: $1,680)
A 365-day policy has an initial premium of $5,475 and an annualized increase of $365 effective on day 125. It is cancelled pro rata on day 245. Find earned premium. (Answer: $3,795)
A 360-day policy has an initial premium of $7,200. An annualized decrease of $720 takes effect on day 90, followed by an annualized increase of $360 on day 240. Cancellation is pro rata on day 300. Find earned premium. (Answer: $5,640)
A 360-day policy has an endorsement increase of $900 effective on day 90. Find the booked pro rata endorsement amount, without considering other transactions. (Answer: $675)
A 360-day policy has an initial premium of $7,200 and an annualized increase of $900 effective on day 90. It is cancelled pro rata on day 270. Find both earned and unearned premium. (Answer: earned $5,850; unearned $2,025)

What You'll Learn