Straight-Line Earning of Policy Premium

Introduction

A policy premium buys coverage over a stated period, but an insurer may need to measure how much of that premium relates to coverage already provided at a particular date. The premium paid for a full policy term is not necessarily all earned on the effective date. Under straight-line earning, the premium is allocated evenly over the coverage period, so the fraction earned is the fraction of the term that has elapsed.

This method is appropriate when the policy has a constant premium and coverage is treated as uniform over time. It is a time-based allocation, not a prediction of how many claims will occur or how much they will cost. The key calculation is simple, but the result depends on using the correct policy term and measuring elapsed coverage consistently.

Definition: At a valuation date, earned premium is the portion of a policy's premium allocated to coverage provided by that date. Unearned premium is the portion allocated to the remaining coverage period. Under straight-line earning for a constant premium and uniform coverage, premium is allocated in direct proportion to time.

The Straight-Line Formula

Let $P$ be the constant premium for a policy's full term, $T$ be the total term measured in a consistent time unit, and $t$ be the portion of the term that has elapsed by the valuation date. Assume the policy is in force over the interval being measured, coverage accrues uniformly throughout the term, and no premium-changing transaction alters the full-term premium. For a valuation date within the term, $0 \le t \le T$, the straight-line allocation is:

\text{Earned premium}=P\frac{t}{T}

\text{Unearned premium}=P\frac{T-t}{T} =P\left(1-\frac{t}{T}\right)

The time units for $t$ and $T$ must agree. If the term is measured in months, elapsed time must also be measured in months. If a valuation date is before coverage begins, the earned amount is zero and the unearned amount is $P$. Once the policy term is complete, the earned amount is $P$ and the unearned amount is zero. The formulas above apply between those endpoints.

Why the Formula Works

Straight-line earning assigns the same amount of premium to each equal unit of coverage time. The allocation rate is therefore the full premium divided by the full term. Multiplying that rate by the elapsed term gives the amount assigned to coverage already provided.

\text{Premium allocated per unit of time}=\frac{P}{T}

\text{Earned premium} =\frac{P}{T}\,t =P\frac{t}{T}

The unearned portion is the premium allocated to the remaining time, $T-t$. It can be calculated directly, or as the difference between the full premium and the earned amount. In either case, the two portions reconcile to the original premium:

P\frac{t}{T}+P\frac{T-t}{T} =P\frac{T}{T} =P

This derivation depends on uniform allocation over time. If coverage or the premium is not uniform, a straight-line time fraction may not represent the appropriate allocation. This tutorial uses the straight-line assumption for the full policy term and does not calculate alternative earning patterns.

Identify the full-term premium and term.
Use the premium for the policy's stated coverage term and express the term in a clear time unit.

Measure elapsed coverage.
Determine how much of that same term has elapsed by the valuation date.

Allocate the premium.
Multiply the full premium by the elapsed-time fraction to obtain earned premium.

Calculate and check unearned premium.
Subtract earned premium from the full premium, then verify that the two amounts sum to the full premium.

Worked Calculations

Worked Example: A Six-Month Valuation of an Annual Policy

A hypothetical commercial property policy has a constant premium of $1,200 for 12 months of uniform coverage. At the valuation date, exactly 6 months of coverage have elapsed. Calculate earned and unearned premium. This is a deterministic time-allocation calculation; no interest rate, mortality assumption, or probability distribution is used.

Method and step 1: Set the full term and elapsed time. The full-term premium is $P=\$1{,}200$, the policy term is $T=12$ months, and elapsed coverage is $t=6$ months. Both time quantities are in months.

Step 2: Find the elapsed fraction.

\frac{t}{T}=\frac{6}{12}=0.5

Step 3: Calculate earned premium.

\$1{,}200(0.5)=\$600

Step 4: Calculate unearned premium and check. The remaining term is $12-6=6$ months, so the unearned fraction is $6/12=0.5$.

\text{Unearned premium}=\$1{,}200(0.5)=\$600

The check is $\$600+\$600=\$1{,}200$, equal to the full premium. A common mistake is to treat the 6 elapsed months as 6 percent of the term; the correct fraction is $6/12$, or 50 percent. The straight-line result is $600 earned and $600 unearned.

Worked Example: A Partial Term That Is Not a Whole Number of Years

A hypothetical equipment policy provides 18 months of uniform coverage for a constant full-term premium of $2,700. At the valuation date, 7.5 months have elapsed. Find the earned and unearned amounts. This is deterministic; it uses no interest rate, mortality assumption, or probability distribution.

Method and step 1: Record the term and elapsed time in the same unit. Here $T=18$ months and $t=7.5$ months. The elapsed fraction is:

\frac{t}{T}=\frac{7.5}{18}=\frac{5}{12}

Step 2: Allocate the premium to elapsed coverage.

\text{Earned premium} =\$2{,}700\left(\frac{5}{12}\right) =\$225(5) =\$1{,}125

Step 3: Allocate the premium to remaining coverage. The remaining term is $18-7.5=10.5$ months, so the unearned fraction is $10.5/18=7/12$.

\text{Unearned premium} =\$2{,}700\left(\frac{7}{12}\right) =\$225(7) =\$1{,}575

The check is $\$1{,}125+\$1{,}575=\$2{,}700$. The common mistake here is to divide by 12 because a year has 12 months. The denominator must be the policy's full term, which is 18 months, not a standard calendar year. The answer is $1,125 earned and $1,575 unearned.

Worked Example: Summing Earned Premium for Two Policies

At a hypothetical valuation date, an insurer reviews two in-force policies, each with a constant premium and uniform coverage. Policy A has a 12-month term, a premium of $1,440, and 5 months elapsed. Policy B has a 24-month term, a premium of $2,400, and 9 months elapsed. Calculate total earned premium, total unearned premium, and reconcile the result to total premium. No interest rate, mortality assumption, or probability distribution is used.

Method and step 1: Calculate Policy A. Its elapsed fraction is $5/12$, so:

\text{A earned}=\$1{,}440\left(\frac{5}{12}\right) =\$120(5)=\$600

The remaining fraction is $7/12$, giving:

\text{A unearned}=\$1{,}440\left(\frac{7}{12}\right) =\$120(7)=\$840

Step 2: Calculate Policy B. Its elapsed fraction is $9/24=3/8$, so:

\text{B earned}=\$2{,}400\left(\frac{3}{8}\right) =\$300(3)=\$900

The remaining fraction is $15/24=5/8$, giving:

\text{B unearned}=\$2{,}400\left(\frac{5}{8}\right) =\$300(5)=\$1{,}500

Step 3: Add each measure across the policies.

\text{Total earned}=\$600+\$900=\$1{,}500

\text{Total unearned}=\$840+\$1{,}500=\$2{,}340

Step 4: Reconcile. Total full-term premium is $\$1{,}440+\$2{,}400=\$3{,}840$, while total earned plus total unearned is $\$1{,}500+\$2{,}340=\$3{,}840$. The reconciliation holds. A common mistake is to combine the elapsed months across policies before calculating; each policy needs its own elapsed fraction because its term and premium differ. The portfolio totals are $1,500 earned and $2,340 unearned.

Interpretation and Common Traps

Under the stated assumptions, straight-line earning measures how the full-term premium is allocated between elapsed and remaining coverage time. The unearned amount is not automatically the amount a policyholder would receive as a refund. A refund calculation can depend on policy terms and transactions; the straight-line calculation here is only the earned-versus-unearned allocation for a policy with a constant full-term premium.

The main exam traps are errors in the fraction or in what the result represents:

Using the wrong denominator. Divide elapsed time by the policy's full term, not automatically by 12 months or by the number of months remaining.
Mixing time units. If $T$ is in months, $t$ must also be in months. Convert consistently before calculating.
Reversing earned and unearned. Elapsed time determines the earned fraction; remaining time determines the unearned fraction.
Ignoring reconciliation. For a full-term premium $P$, earned plus unearned should equal $P$. A mismatch often signals a time-fraction or arithmetic error.
Applying straight-line earning without its assumptions. The method assumes a constant premium and uniform coverage over the term. It does not by itself account for a change in premium or a nonuniform allocation of coverage.

Key takeaway: For constant premium and uniform coverage, multiply the full-term premium by the elapsed-term fraction to find earned premium. The remainder is unearned, and the two amounts must sum to the full-term premium.

Check Your Understanding

For each hypothetical policy, use straight-line earning and check that earned plus unearned premium equals the full-term premium.

A 12-month policy has a constant premium of $960. Three months have elapsed. Find earned and unearned premium. (Answer: $240 earned; $720 unearned.)
A 15-month policy has a constant premium of $2,250. Eight months have elapsed. Find the elapsed fraction, earned premium, and unearned premium. (Answer: Elapsed fraction $8/15$; $1,200 earned; $1,050 unearned.)
A two-year policy has a constant premium of $3,600. Fifteen months have elapsed. Calculate the earned amount using months for both elapsed and total term, then find unearned premium. (Answer: $2,250 earned; $1,350 unearned.)
At one valuation date, Policy A has a 10-month term, premium of $1,500, and 4 months elapsed. Policy B has a 14-month term, premium of $2,800, and 9 months elapsed. Find the portfolio's total earned and unearned premium and reconcile to total premium. (Answer: $2,400 earned; $1,900 unearned; total premium $4,300.)
A 12-month policy has a constant premium of $2,190. Exactly 5.5 months have elapsed. Calculate earned premium to the nearest cent and then calculate unearned premium. (Answer: $1,003.75 earned; $1,186.25 unearned.)

What You'll Learn