Calculating Exposure For Annual Policies

Annual Policy Exposure: Written Versus Earned

An annual policy may be written on any date, but its full year of coverage does not necessarily fall within one calendar year. This distinction matters when exposure is summarized by calendar year: a policy can contribute one full unit of written exposure at inception while contributing only a fraction of a unit of earned exposure to that year's results.

The calculations here use a count exposure base: one unit of exposure for one covered policy. The same time-allocation idea applies when an exposure unit is defined differently, but the amount of the exposure unit must be identified before calculating. The earlier material on exposure bases explains why the chosen unit must match the coverage being measured.

Definition: For a policy with one exposure unit and a 12-month term, one policy-year is the exposure provided by that unit for the full annual term. Written exposure is the full exposure associated with a policy term when it is written. Earned exposure is the portion of that term's exposure corresponding to coverage that has elapsed by the measurement date or within the experience period.

For a standard annual policy with no change in coverage, writing the policy records one policy-year of written exposure at inception. The exposure earns over the policy term. If a fraction \(t\) of the annual term has elapsed, where \(0 \leq t \leq 1\), the policy has earned \(t\) policy-years and has \(1-t\) policy-years unearned. The assumption behind this proportional allocation is that exposure accrues uniformly over time and the exposure unit remains unchanged.

\text{Written exposure}=1,\qquad \text{Earned exposure}=t,\qquad \text{Unearned exposure}=1-t

When time is measured in whole months, an annual term contains 12 equal months, so \(t=m/12\), where \(m\) is the number of months of coverage elapsed. For example, six elapsed months correspond to \(6/12=0.5\) earned policy-years. When dates do not fall on month boundaries, use the applicable day-count or other time convention specified for the calculation; do not silently treat every month as having the same number of days.

Why Proportional Earning Works

The proportional formula follows from treating exposure as accumulating at a constant rate during the term. A full annual term contains 1 policy-year, so each equal fraction of the term contributes the same fraction of that unit. If \(m\) of 12 equal months have elapsed, each month contributes \(1/12\) of the annual exposure, and the total earned amount is the sum of the contributions from those \(m\) months.

\text{Earned exposure} =\underbrace{\frac{1}{12}+\frac{1}{12}+\cdots+\frac{1}{12}}_{m\text{ elapsed months}} =\frac{m}{12}

For multiple policies, exposures add across policies. If policy \(j\) has \(e_j\) exposure units for a full annual term and fraction \(t_j\) of its term has elapsed in the period being measured, its earned exposure is \(e_jt_j\). Therefore, total earned exposure is the sum of the individual earned amounts. For the count basis in the examples below, each policy has \(e_j=1\).

\text{Total earned exposure}=\sum_{j=1}^{n} e_jt_j

Worked Example: A Full Year of Coverage

A hypothetical insurer writes four unchanged annual policies, each with one policy exposure unit, effective January 1, 2026. Each policy expires at the start of January 1, 2027. Calculate the cohort's written exposure at inception and its earned exposure during calendar year 2026. Use 12 equal months for the year.

Step 1: Calculate written exposure. Each policy contributes one full annual policy-year when written. For four policies:

\text{Written exposure}=4 \times 1=4\text{ policy-years}

Step 2: Find the portion of each term earned in 2026. Each policy is in force for the full 12 months of the calendar year, so its elapsed fraction is \(12/12=1\).

\text{Earned exposure per policy}=\frac{12}{12}=1\text{ policy-year}

Step 3: Sum across the four policies.

\text{Earned exposure}=4 \times 1=4\text{ policy-years}

Check. All four annual terms are fully elapsed by the end of the calendar-year interval, so earned exposure equals written exposure: 4 policy-years. No exposure from these terms remains unearned at that point.

Partial Terms at a Valuation Date

For a valuation made partway through an annual term, calculate how much of the term has elapsed—not how many calendar years have started or how many policy records are present. Under the equal-month convention, divide elapsed months by 12. The remaining fraction is unearned. These amounts refer to the same policy term, so they must add to one full policy-year for a policy with one exposure unit.

Worked Example: Earned Exposure at a Midterm Valuation

A hypothetical annual policy with one exposure unit is effective October 1, 2025, and expires October 1, 2026. At the end of June 2026, calculate its written exposure, earned exposure to date, and unearned exposure. Treat the policy term as 12 equal months, and measure elapsed coverage from October 1 through the end of June.

Step 1: Record the written exposure. The policy was written for one annual term, so it contributes 1 policy-year of written exposure.

Step 2: Count elapsed months. Coverage has elapsed for October, November, and December 2025, followed by January through June 2026. That is \(3+6=9\) months.

\text{Earned exposure}=\frac{9}{12}=0.75\text{ policy-years}

Step 3: Calculate the unearned portion. Three months of the 12-month term remain, from July through September.

\text{Unearned exposure}=\frac{3}{12}=0.25\text{ policy-years}

Check. Earned plus unearned exposure is \(0.75+0.25=1.00\) policy-year, equal to the written exposure. The calculation assumes continuous coverage with no change to the policy's exposure unit.

A valuation date is a cutoff for measuring how much coverage has elapsed. It does not change the original written exposure. For example, the policy above remains one policy-year written whether the valuation occurs one month or eleven months into the term; only its earned and unearned portions change.

Policies with Staggered Effective Dates

When annual policies have different effective dates, calculate the elapsed fraction separately for each policy or group of policies with the same effective date. Then add the earned amounts. A common shortcut—multiplying the number of policies by one average fraction—works only if that fraction is correctly weighted for the policies' exposure amounts and elapsed terms. Calculating each cohort separately makes the timing explicit.

Worked Example: Staggered Policies in a Calendar Year

A hypothetical insurer writes six annual policies, each with one exposure unit. Two are effective January 1, 2026; three are effective April 1, 2026; and one is effective October 1, 2026. All terms are unchanged and last 12 months. Calculate the total written exposure at inception during 2026 and the exposure earned during the calendar-year interval from January 1, 2026, through January 1, 2027. Use 12 equal months.

Step 1: Calculate written exposure by effective-date group. Every policy contributes one policy-year when written.

\text{Written exposure}=(2+3+1)\times 1=6\text{ policy-years}

Step 2: Calculate the fraction earned by each group in 2026. Each January policy is covered for all 12 months of the interval, each April policy for nine months, and the October policy for three months.

t_{\text{Jan}}=\frac{12}{12}=1,\qquad t_{\text{Apr}}=\frac{9}{12}=0.75,\qquad t_{\text{Oct}}=\frac{3}{12}=0.25

Step 3: Multiply each fraction by the number of policies in that group.

\begin{aligned} \text{January group earned}&=2\times 1=2.00,\\ \text{April group earned}&=3\times 0.75=2.25,\\ \text{October group earned}&=1\times 0.25=0.25. \end{aligned}

Step 4: Add the group amounts.

\text{Total earned exposure}=2.00+2.25+0.25=4.50\text{ policy-years}

Check. The six written policy-years split into 4.50 earned and 1.50 unearned at the end of the interval: \(4.50+1.50=6.00\). The unearned amounts are \(3\times(3/12)=0.75\) for the April group and \(1\times(9/12)=0.75\) for the October group. The January group has no unearned exposure remaining at this cutoff.

Common Mistake: Treating Written Exposure as Earned Exposure

A frequent error is to count each annual policy as one earned policy-year as soon as it is written. Writing records the full term's exposure, but earning allocates that exposure as coverage elapses. For the staggered cohort above, the incorrect calculation would be:

\text{Incorrect earned exposure}=2+3+1=6\text{ policy-years}

That calculation incorrectly treats all six policies as if their full terms fell within 2026. The correction is to apply each group's elapsed fraction: \(2(12/12)+3(9/12)+1(3/12)=4.50\) earned policy-years. The other 1.50 policy-years are not earned within the stated calendar-year interval.

A second trap is to calculate the fraction using the calendar year number rather than the actual dates of coverage. A policy effective October 1 does not earn a full policy-year merely because it was written during that calendar year. Identify the measurement interval, count the coverage elapsed within that interval, and divide by the annual term only when the stated time convention supports that division.

Key takeaway: For an unchanged annual policy with one exposure unit, written exposure is one policy-year at inception. Earned exposure is the fraction of the annual term that has elapsed within the period being measured. For multiple policies, calculate by effective-date cohort and sum; confirm that earned plus unearned exposure reconciles to written exposure.

Check Your Understanding

For each question, use the stated equal-month convention and assume the annual policies remain unchanged and in force for their stated terms.

A hypothetical insurer writes five one-unit annual policies effective January 1, 2027. What are the cohort's written exposure and earned exposure during the 12-month term? (Answer: 5 written policy-years and 5 earned policy-years.)
One one-unit annual policy is effective May 1, 2026. At the end of August 2026, how much has it earned and how much remains unearned? (Answer: \(4/12=0.3333\) earned policy-years and \(8/12=0.6667\) unearned policy-years.)
Three one-unit annual policies are effective July 1, 2026. Calculate their total written exposure and their total exposure earned from July 1 through December 31, 2026, a six-month interval. (Answer: 3 written policy-years and \(3\times 6/12=1.5\) earned policy-years.)
A cohort has two one-unit annual policies effective January 1, 2027, and four effective October 1, 2027. For the calendar-year interval beginning January 1 and ending January 1, 2028, calculate written and earned exposure. (Answer: 6 written policy-years and \(2\times12/12+4\times3/12=3\) earned policy-years.)
At a valuation date, a cohort of eight one-unit annual policies has 5.25 policy-years earned. If all eight terms were written and no exposure unit changed, how much exposure is unearned? (Answer: \(8-5.25=2.75\) unearned policy-years.)

What You'll Learn