Introduction
A premium pays for insurance coverage over a defined period. As that period passes, part of the premium is associated with coverage already provided, and part remains associated with coverage still to be provided. The unearned portion at a specified date is therefore a way to measure how much of the premium corresponds to the remaining coverage under a policy.
This calculation is straightforward when premium is allocated evenly over time. The essential steps are to identify the premium being allocated, determine the fraction of the policy period that remains, and apply that fraction consistently. The result is a premium allocation, not a measure of cash held or a prediction of future claims. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Calculating the Unearned Portion
For a policy with a fixed premium and a fixed coverage period, the basic pro rata method allocates premium uniformly over the period. If the coverage period has total duration \(T\), and \(t\) units of that duration have elapsed at the valuation date, then the remaining duration is \(T-t\). The remaining fraction is \((T-t)/T\).
Equivalently, the earned and unearned portions divide the premium:
The expression works because the assumed allocation rate is constant: each unit of coverage time accounts for \(P/T\) of premium. There are \(T-t\) units of coverage remaining, so the premium assigned to those units is \((P/T)(T-t)\), which is \(P(T-t)/T\). The same reasoning applies whether time is measured in months, days, or another consistent unit.
The date convention matters. The elapsed and remaining periods must refer to the same coverage boundaries and use the same time unit. For example, a calculation using elapsed days should use total policy days and remaining policy days, not a mixture of calendar months and days. At the start of coverage, no time has elapsed and the full premium is unearned under this allocation. At the end of coverage, no time remains and the unearned amount is zero.
Worked Calculations
Worked Example: Annual Policy Valued Halfway Through
A hypothetical equipment policy has a fixed premium of $1,200 for a 12-month coverage period. The valuation date is exactly six months after coverage begins. Calculate the unearned premium and check the result against the earned portion.
This is a deterministic pro rata allocation. The interest rate is 0%; mortality is not modeled; no probability distribution is assumed. The only model parameter is the 12-month policy term.
Step 1: Identify the elapsed and remaining terms. Total duration is \(T=12\) months, and elapsed duration is \(t=6\) months. Thus the remaining duration is \(12-6=6\) months.
Step 2: Calculate the remaining fraction.
Step 3: Apply the fraction to premium.
Step 4: Check against the earned portion. The elapsed fraction is also \(6/12=0.5\), so earned premium is \(\$1{,}200(0.5)=\$600\). The two portions sum to \(\$600+\$600=\$1{,}200\), the original premium.
The method is a pro rata time allocation. The common mistake it avoids is treating the full premium as earned simply because it was charged for the policy: under this allocation, half of the coverage period remains, so half the premium remains unearned.
Worked Example: A Day-Based Policy With an Explicit Term
A hypothetical seasonal property policy has a fixed premium of $900 for a 180-day coverage period. At the valuation date, 75 coverage days have elapsed. Calculate the unearned premium, using the stated day counts.
This is a deterministic pro rata allocation. The interest rate is 0%; mortality is not modeled; no probability distribution is assumed. The model parameters are a 180-day total term and 75 elapsed days.
Step 1: Find days of coverage remaining.
Step 2: Find the fraction remaining.
Step 3: Calculate unearned premium.
Step 4: Check by calculating earned premium. The elapsed fraction is \(75/180=5/12\), so earned premium is \(\$900(5/12)=\$75(5)=\$375\). The check is \(\$525+\$375=\$900\).
The method uses the stated coverage days, not an estimate based on the names of the calendar months. A common error is to subtract 75 from the premium or to use elapsed days in the remaining fraction. The correct remaining fraction is remaining days divided by total days, \(105/180\).
Worked Example: Aggregating Policies With Different Terms
At one valuation date, a hypothetical insurer reviews three policies with unchanged fixed premiums. Policy A has a $600 premium for 12 months, of which 9 months have elapsed. Policy B has a $1,200 premium for 6 months, of which 2 months have elapsed. Policy C has a $900 premium for 90 days, of which 30 days have elapsed. Calculate the total unearned premium and reconcile it to total premium.
Each policy is treated as a deterministic pro rata allocation. The interest rate is 0%; mortality is not modeled; no probability distribution is assumed. The model parameters are the stated premiums, terms, and elapsed periods.
Step 1: Calculate Policy A. Three of 12 months remain, so the remaining fraction is \(3/12=1/4\). Its unearned premium is \(\$600(1/4)=\$150\).
Step 2: Calculate Policy B. Four of 6 months remain, so the remaining fraction is \(4/6=2/3\). Its unearned premium is \(\$1{,}200(2/3)=\$800\).
Step 3: Calculate Policy C. Sixty of 90 days remain, so the remaining fraction is \(60/90=2/3\). Its unearned premium is \(\$900(2/3)=\$600\).
Step 4: Add the policy amounts.
Step 5: Reconcile to total premium. Total premium is \(\$600+\$1{,}200+\$900=\$2{,}700\). Earned premium by policy is \(\$600(9/12)=\$450\), \(\$1{,}200(2/6)=\$400\), and \(\$900(30/90)=\$300\), for total earned premium of \(\$1{,}150\). Therefore, \(\$1{,}550+\$1{,}150=\$2{,}700\).
The method calculates each policy’s remaining fraction before summing. A common mistake is to apply one unweighted average remaining fraction to all premiums. These policies have different terms and different amounts of coverage elapsed, so their unearned amounts must be calculated individually and then added.
Interpreting the Result
Unearned premium measures premium assigned to the future part of the coverage period, not the amount of premium that has not yet been paid. The calculation is based on premium and coverage time. Whether a policyholder has paid an invoice, or when an insurer received cash, does not change the time-based allocation itself.
Likewise, unearned premium is not a forecast of claims on the remaining coverage. A policy could have substantial unearned premium even if no claims have occurred, or claims could already have occurred while part of the policy term remains. The allocation answers a narrower question: under the stated earning basis, what portion of premium corresponds to coverage after the valuation date?
Use the premium applicable to the coverage being measured and establish the period over which it is allocated.
Determine elapsed and remaining coverage using consistent boundaries and units.
Divide remaining time by total coverage time under a uniform pro rata assumption.
Multiply premium by the remaining fraction, then verify that earned plus unearned premium equals the premium being allocated.
Common Errors and Limits of the Simple Method
- Using elapsed time instead of remaining time. The elapsed fraction produces earned premium. Unearned premium uses the remaining fraction.
- Mixing time units. If total duration is measured in days, remaining duration must also be measured in days. Do not combine a day count with an unadjusted month fraction.
- Applying the formula to the wrong premium amount. The calculation allocates the premium for the coverage under consideration. A premium amount that does not correspond to that coverage period cannot be allocated correctly by choosing a convenient fraction.
- Calling the result a cash balance or refund. It is an allocation to unexpired coverage. It does not, by itself, state how much cash has been collected or what amount would be returned on cancellation.
- Assuming uniform earning when the coverage pattern is not uniform. The formula relies on even allocation over time. If the policy’s applicable earning basis is not uniform, the simple fraction does not represent that basis.
Check Your Understanding
For each problem, use the stated pro rata time allocation and check that the calculation uses remaining coverage.
- A hypothetical 12-month policy has a fixed premium of $840. Five months have elapsed at the valuation date. What is the unearned premium? (Answer: $490)
- A hypothetical policy has a fixed premium of $1,800 for 240 days. Ninety days have elapsed. Calculate unearned premium and earned premium. (Answer: Unearned premium $1,125; earned premium $675)
- At a valuation date, Policy A has a $1,500 premium for 12 months with 8 months elapsed. Policy B has a $720 premium for 6 months with 1 month elapsed. What is their combined unearned premium? (Answer: $1,100)
- A hypothetical policy has a fixed premium of $960 for 12 months. At the valuation date, 10 months have elapsed. Calculate both portions and verify the total. (Answer: Unearned premium $160; earned premium $800; total $960)
- Policy A has a $1,080 premium for a 360-day term, with 240 days elapsed. Policy B has a $750 premium for a 150-day term, with 60 days elapsed. Find total unearned premium across both policies. (Answer: $810)