Earned Premium and the Passage of Coverage

Introduction

Written premium records the premium associated with a policy transaction; earned premium answers a different question: how much of that premium is attributable to coverage already provided? For a policy in force partway through its term, some coverage time has passed and some has not. Under a straight-line time-proration assumption, the premium is allocated in proportion to the time elapsed.

This calculation is useful when measuring premium attributable to an experience period. It depends on the policy term and the premium for that term, not on when the policyholder pays. The method below is deliberately limited to unchanged premium and uniform earning over time; later adjustments to a policy’s premium require separate treatment.

Definition: Earned premium is the portion of premium attributable to insurance coverage already provided during a stated period. For a policy with written premium $WP$ over a term of length $T$, cumulative earned premium at elapsed coverage time $t$, under straight-line time proration, is $EP(t)=WP\times t/T$ when $0\leq t\leq T$. Before coverage starts, cumulative earned premium is zero; after the term ends, it is the full $WP$.

Calculating Earned Premium at a Valuation Date

Begin with the written premium applicable to the policy term. Next determine how much of that term has elapsed at the valuation date. Divide the elapsed coverage time by the full term length, using the same time unit for both quantities. Finally, multiply that fraction by the written premium.

EP(t)=WP\frac{t}{T},\qquad 0\leq t\leq T.

Here, $t$ is elapsed time since the policy’s effective date, and $T$ is the full length of the policy term. For a policy in force at the valuation date, the formula allocates only the premium attributable to coverage already provided. If the requested amount relates to a reporting window rather than all time since inception, calculate the premium earned within that window instead.

Worked Example: A Policy 90 Days Into Its Term

A hypothetical policy has written premium of $1,460 for a 365-day term. The policy is in force, and the valuation date is 90 days after coverage began. Find cumulative earned premium at that date and the premium earned during the 30-day window from elapsed day 60 to elapsed day 90. Assume uniform earning by day and no change in premium.

Method. Use straight-line time proration. The common trap is to use the 30-day reporting window to calculate cumulative earned premium, instead of using the full 90 days since inception.

1. Find the daily earning rate. The term is 365 days, so the premium attributable to each day is:

\frac{\$1{,}460}{365}=\$4\text{ per day}.

2. Calculate cumulative earned premium. By the valuation date, 90 days of coverage have been provided:

EP(90)=\$4\times 90=\$360.

3. Calculate premium earned in the reporting window. The window covers days 61 through 90, or 30 days. Therefore:

EP_{\text{window}}=\$4\times 30=\$120.

Check. The cumulative amount through day 60 is $\$4\times60=\$240$. Adding the window amount gives $\$240+\$120=\$360$, which matches cumulative earned premium at day 90. All amounts are exact to the cent.

Why the Proration Formula Holds

Straight-line proration assumes that premium is earned at a constant rate per unit of coverage time. If the policy premium is $WP$ and the term has $T$ days, the assumed amount earned per day is $WP/T$. After $t$ days, that constant daily amount has been earned $t$ times.

\text{daily earning rate}=\frac{WP}{T}, \qquad EP(t)=t\frac{WP}{T}=WP\frac{t}{T}.

This is an allocation rule, not a claim about when losses will occur or when premium cash will be received. The calculation requires no interest rate, mortality assumption, or probability distribution: it is deterministic given the premium, term length, and elapsed coverage time.

The same reasoning applies to a part of a term. Let $t_1$ and $t_2$ be the start and end of a reporting window, measured as elapsed time from the policy’s effective date. The window’s overlap with the policy term is the portion of that window during which the policy provides coverage. Its earned premium is the daily earning rate multiplied by the overlap. For a window entirely within the term, overlap is $t_2-t_1$.

Worked Example: Premium Earned in a Reporting Window

A hypothetical 360-day policy has written premium of $1,080. At the start of an experience window, 60 days of the policy term have elapsed. At the end of the window, 150 days have elapsed. Calculate (a) cumulative earned premium at the window end and (b) earned premium during the window. Assume uniform earning by day and unchanged premium.

Method. First calculate cumulative earning through day 150. Then calculate the amount attributable to the 90 days between the two cutoffs. The common mistake is to report cumulative earned premium as the amount earned in the experience window.

1. Find the daily earning rate.

\frac{\$1{,}080}{360}=\$3\text{ per day}.

2. Find cumulative earned premium at the window end. At that point, 150 days of coverage have been provided:

EP(150)=\$3\times150=\$450.

3. Find the number of days in the window.

150-60=90\text{ days}.

4. Calculate window earned premium.

EP_{\text{window}}=\$3\times90=\$270.

Check. Cumulative earned premium at day 60 is $\$3\times60=\$180$. The difference between cumulative amounts is $\$450-\$180=\$270$, agreeing with the direct window calculation. Both answers are exact to the cent.

Combining Policies with Different Terms

When calculating earned premium for a group of policies, apply the proration to each policy using its own written premium, term length, and elapsed time. Sum the resulting earned amounts. Do not average the policies’ elapsed-time percentages unless the question calls for a particular weighted average; the direct total is the sum of policy-level earned premium.

Worked Example: Two Policies at a Common Valuation Date

At a common valuation date, two hypothetical policies are in force. Policy A has written premium of $1,200 for a 365-day term, with 120 days elapsed. Policy B has written premium of $840 for a 280-day term, with 70 days elapsed. Calculate the combined cumulative earned premium. Assume uniform earning by day and unchanged premium for each policy.

Method. Calculate earned premium separately for each policy and then add the results. The common trap is to use one policy’s term length for both or to apply the unweighted average of the elapsed-time percentages to total written premium.

1. Calculate Policy A’s daily earning rate and earned premium.

\frac{\$1{,}200}{365}=\$3.287671\ldots\text{ per day}, \qquad EP_A=\$1{,}200\frac{120}{365} =\$394.520547\ldots.

2. Calculate Policy B’s daily earning rate and earned premium.

\frac{\$840}{280}=\$3\text{ per day}, \qquad EP_B=\$3\times70=\$210.

3. Add the policy-level amounts. Keep full precision in the intermediate calculation and round the final combined amount to cents:

EP_{\text{total}}=\$394.520547\ldots+\$210 =\$604.520547\ldots\approx \$604.52.

Check. Policy A has earned less than its $1,200 written premium, and Policy B has earned less than its $840 written premium, as both are still in force partway through their terms. Their total, $604.52, is below their combined written premium of $2,040. The final amount is rounded to two decimal places.

Assumptions, Interpretation, and Exam Traps

Time proration is a straightforward way to measure earned premium, but it relies on an assumption: each unit of coverage time earns the same fraction of the term premium. This is often a practical convention for a basic calculation. It may be a poor representation when risk or coverage varies materially by time, such as when the insured exposure is strongly seasonal. In that case, a uniform daily allocation may not match the underlying pattern of coverage.

Earned premium is attributable to coverage provided. It is not automatically the amount of premium collected, nor does the calculation predict the losses that will occur. A policy may have earned premium even if the policyholder has not yet paid the corresponding amount. Conversely, the timing of a payment does not by itself change how much coverage time has passed.

For a window that partly falls outside the policy term, count only the time during which the policy is in force. For example, if a term ends before the reporting window ends, no further premium is earned under that term after the coverage ends. A window entirely before the policy begins has no overlap with the term and therefore has zero earned premium. The general calculation is:

EP_{\text{window}} = \frac{WP}{T} \times \text{number of coverage days in the overlap of the window and the term}.

A reliable check is to compare the answer with the limits for the policy. Cumulative earned premium should be between zero and the policy’s written premium, inclusive, under this method. Premium earned in a window should also be nonnegative and cannot exceed the premium attributable to the full term. If the calculated amount exceeds those limits, recheck the time unit, the term length, and whether cumulative and window amounts were confused.

Key takeaway: Under straight-line time proration, earned premium equals the policy’s written premium multiplied by the fraction of its term’s coverage time that has been provided. For a reporting window, use only the coverage time overlapping that window, and calculate each policy using its own term.

Check Your Understanding

For each hypothetical policy, assume unchanged written premium and uniform earning by day. Round money to the nearest cent where needed.

A policy has written premium of $730 for a 365-day term. How much premium has been earned after 146 days? (Answer: $292.00)
A 240-day policy has written premium of $960. At the start of a reporting window, 60 days have elapsed; at the end, 150 days have elapsed. Find cumulative earned premium at the end and premium earned during the window. (Answer: $600.00 cumulative; $360.00 during the window)
A 300-day policy has written premium of $1,500. An experience window runs from elapsed day 90 through elapsed day 210. Find the premium earned in that window and cumulative earned premium at its end. (Answer: $600.00 during the window; $1,050.00 cumulative)
At a valuation date, Policy A has written premium of $1,000 for 250 days, with 50 days elapsed. Policy B has written premium of $720 for 180 days, with 90 days elapsed. Find combined cumulative earned premium. (Answer: $560.00)
A policy has written premium of $900 for a 300-day term. A reporting window covers elapsed days 260 through 330. How much premium is earned in the window, accounting for the term ending at day 300? (Answer: $120.00)

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