Earned and Written Exposures in Pure Premium Analysis

Why Exposure Timing Matters

A pure premium is a loss amount expressed per exposure unit. The meaning of that rate depends on the denominator: the exposure should represent the risk underlying the losses being measured. A policy can be written in one period but provide coverage in another. Consequently, the number of units written during a period need not equal the number of units exposed to loss during that period.

This distinction matters whenever an experience period is defined by calendar dates. Losses assigned to the period in which accidents occur arise from coverage in force during those dates, including coverage on policies written earlier. Conversely, a policy written near the end of the period may contribute little exposure before the period ends, even though its full exposure unit appears in the written total.

Definition: Written exposure is the amount of exposure associated with business written during a specified period, according to the insurer’s exposure-counting rules. Earned exposure is the amount of exposure corresponding to the coverage actually provided during a specified period. For a time-based exposure unit, one full unit is earned when one unit of risk receives one full exposure period of coverage. Written exposure is associated with when the business is written; earned exposure is associated with when coverage is in force.

An exposure unit can be a policy-year, vehicle-year, payroll unit, or another measure appropriate to the insurance coverage. The exposure base must be specified before counting. For example, a policy covering two vehicles for a year represents two vehicle-years, not one policy-year, if the selected base is vehicle-years. This tutorial focuses on when the exposure is counted, not on selecting the exposure base.

How Time Earns Exposure

For a time-based exposure unit with uniform coverage, the earned portion is the covered part of the exposure period. If one annual policy is in force for six months, it earns one-half of a policy-year under a twelve-month convention. If an exposure base changes during the period, calculate exposure for each interval over which the base is constant, then add those amounts.

\text{Earned exposure} = \sum_j \left(\text{exposure units in force during interval }j\right) \left(\frac{\text{covered duration in interval }j}{\text{duration of one exposure period}}\right)

The calculation follows directly from what an earned unit measures. Divide the covered duration by the duration required for one full unit. That ratio is the fraction of a unit supplied by the coverage. For example, six months of coverage under a twelve-month convention supplies $6/12=0.5$ policy-years. When several units or intervals are involved, add their fractional units. Equivalently, if $N(t)$ is the number of exposure units in force at time $t$, earned exposure over a period is the accumulated in-force units over that period, expressed in the chosen time unit.

E_{\mathrm{earned}} = \int_{\text{experience period}} N(t)\,dt \qquad \text{when }t\text{ is measured in exposure-period units}

For monthly examples below, a year is treated as twelve equal months. Actual systems may use daily, monthly, or other prescribed earning conventions. Use the convention specified for the data; do not silently substitute a different day-count rule. If coverage is not uniform over time, simple pro-rata earning may not describe the exposure appropriately. The calculation must then follow the exposure definition and applicable rating or accounting rules.

Worked Example: Calendar-Year Written and Earned Policy-Years

A hypothetical insurer writes 12 annual policies effective January 1, 6 effective April 1, and 6 effective October 1 of the same calendar year. All policies remain in force through year-end, and each policy is one policy-year of exposure for a full twelve months. During that calendar year, calendar-year losses are $360,000. Calculate written and earned exposure for the year, and the illustrative calendar-year loss per earned exposure.

Method. Count all policies written in the calendar year for written exposure. For earned exposure, count only the fraction of each policy’s twelve-month coverage period that falls within that calendar year. No interest rate, mortality assumption, or probability distribution is used; this is a deterministic exposure calculation.

Written exposure. Each of the 12, 6, and 6 policies is written in the year, so written exposure is $12+6+6=24$ policy exposures.

Earned exposure from January 1 policies. Each of the 12 policies is in force for all twelve months of the calendar year. Their earned exposure is $12(12/12)=12$ policy-years.

Earned exposure from April 1 policies. Each of the 6 policies is in force for nine months in the calendar year. Their earned exposure is $6(9/12)=4.5$ policy-years.

Earned exposure from October 1 policies. Each of the 6 policies is in force for three months in the calendar year. Their earned exposure is $6(3/12)=1.5$ policy-years.

Total earned exposure is $12+4.5+1.5=18$ policy-years. The illustrative loss per earned exposure is $$360{,}000/18=$20{,}000$ per policy-year. As a check, 24 written policies have an average of $18/24=0.75$ of a policy-year earned in the calendar year, which is consistent with the effective dates.

Common mistake avoided. Dividing by 24 written exposures gives $$360{,}000/24=$15{,}000$, but that treats every policy written during the year as if it supplied a full year of coverage during that year. It did not. For losses assigned to this calendar year, the earned denominator reflects the coverage time within that calendar year.

Written Periods and Coverage Periods Are Different Views

Written exposure answers a production-timing question: how much exposure was attached to business written during the period? Earned exposure answers a coverage-timing question: how much exposure was provided during the period? Neither measure is inherently wrong. The appropriate measure depends on how the losses and experience period are defined.

For losses organized by accident date, a calendar or accident period collects claims arising from accidents in that period. Coverage for those accidents may come from policies written in earlier periods as well as policies newly written during the period. Earned exposure is therefore the natural time-based measure of the covered risk in force during the same period. In contrast, a policy-year view groups business by policy effective dates and follows that cohort of policies; its exposure and loss definitions must be interpreted on that basis.

A policy written near year-end illustrates the distinction. It contributes a full policy to a written count for the year under a count-at-issuance convention, but only a small fraction of its annual exposure is earned in that calendar year. A policy written before the year may contribute no written exposure in the current year while still earning coverage during it. Thus written exposure can be useful for tracking new business, but it does not by itself measure the amount of coverage provided during the period.

Worked Example: An October Policy Crossing Calendar Years

A hypothetical insurer issues 100 identical annual policies effective October 1. Each policy provides one policy-year of coverage from October 1 through September 30. Assume all remain in force for their full terms. Calculate written exposure in the issue calendar year and earned exposure in the issue year and the following calendar year. No interest rate, mortality assumption, or probability distribution is involved; use a twelve-month exposure period and uniform pro-rata earning.

Written exposure in the issue year. All 100 policies are written between January 1 and December 31, so the written exposure is 100 policies.

Earned exposure in the issue year. Each policy is in force for three months, October through December. Each earns $3/12=0.25$ policy-years. Across 100 policies, earned exposure is $100(0.25)=25$ policy-years.

Earned exposure in the following year. Each policy is in force for nine months, January through September. Each earns $9/12=0.75$ policy-years. Across 100 policies, earned exposure is $100(0.75)=75$ policy-years.

The check is $25+75=100$ earned policy-years across the two calendar years, equal to the 100 full annual policy terms supplied. The written count in the following calendar year is zero for this cohort, even though it earns 75 policy-years there. This demonstrates why a calendar-period loss measure should not be paired automatically with the written count from that same calendar year.

Common mistake avoided. Assigning all 100 policy-years to the issue year because all 100 policies were written then confuses the transaction date with the coverage dates. Only three months of each policy’s coverage fall in the issue year.

Changing Exposure Units During Coverage

The same timing principle applies when the number of insured units changes within a policy. An endorsement that adds a vehicle, location, or other exposure unit changes the amount of risk in force from its effective date. For earned exposure, divide the timeline into intervals with a constant exposure count, calculate the exposure for each interval, and sum. The policy count alone may hide important changes in the number of units covered.

Worked Example: Vehicle-Years with Midyear Changes

A hypothetical commercial auto policy covers two vehicles from January 1 through December 31. A third vehicle is added effective July 1 and remains covered through year-end. A second policy covers four vehicles from April 1 through September 30. Calculate the total earned vehicle-years during the calendar year. If calendar-year losses for these risks are $90,000, calculate the illustrative loss per earned vehicle-year. Treat a year as twelve equal months. No interest rate, mortality assumption, or probability distribution is used.

First policy, January through June. Two vehicles are covered for six months: $2(6/12)=1$ vehicle-year.

First policy, July through December. Three vehicles are covered for six months: $3(6/12)=1.5$ vehicle-years.

The first policy therefore earns $1+1.5=2.5$ vehicle-years.

Second policy, April through September. Four vehicles are covered for six months: $4(6/12)=2$ vehicle-years.

Total earned exposure is $2.5+2=4.5$ vehicle-years. The illustrative loss per earned vehicle-year is $$90{,}000/4.5=$20{,}000$. As a check, the first policy contributes 2.5 vehicle-years from its actual time-varying vehicle count, and the second contributes 2 vehicle-years; their sum equals 4.5.

Common mistake avoided. Counting one exposure for each policy would produce only two policy counts, not 4.5 vehicle-years. It ignores both the selected exposure base and the dates when the insured vehicle counts changed.

Choosing the Measure for a Specified Experience Period

When a problem specifies losses from a calendar or accident period, first ask what the period means. If losses are assigned to accidents occurring during that period, the exposure measure should represent the risk in force during those dates. For a time-based exposure base, that is generally earned exposure. Written exposure is appropriate when the question explicitly concerns business written in the period or when the loss experience is defined on a corresponding written-business basis.

This distinction is a timing principle, not a claim that every analysis uses calendar-year earned exposure. Other adjustments may be needed to make loss and exposure definitions comparable. For example, a change in the exposure base or a mismatch in the way claims are assigned can affect the analysis. The essential first step here is to identify which coverage period the exposure denominator represents; do not infer that from the date a policy was issued.

Identify the loss period.
Determine whether losses are grouped by accident date, policy effective date, or another specified basis.

Identify the exposure unit.
State whether one unit means a policy-year, vehicle-year, or another exposure base.

Locate coverage in time.
Use effective dates, expiration dates, cancellations, and exposure-changing endorsements to determine units in force during the period.

Select and calculate the denominator.
For period losses arising from coverage in force during the period, calculate earned exposure over those dates. Use written exposure only when the analysis calls for business written in the period.

Common Errors and Practical Checks

Using policy count instead of the specified exposure base. One policy can cover multiple vehicles or locations. Count the units required by the exposure definition.
Putting a full annual exposure in the issue year. A policy written during the year may provide only part of its coverage in that calendar year. Earn the covered fraction in each period.
Ignoring policies written earlier. Such policies can still be in force and earn exposure in the current period, even if they add nothing to the current written count.
Using the same label for different timing bases. Always qualify exposure as written or earned and state the period to which it applies.
Applying pro-rata time when exposure is not uniform. Simple duration fractions require uniform earning over time and a defined exposure period. Follow the specified method when those assumptions do not hold.

A useful arithmetic check is to compare the result with the amount of coverage that could actually have been supplied. A policy that is in force for three months under a twelve-month convention cannot earn more than $3/12$ of one policy-year. For a cohort of full-term policies spanning several calendar periods, earned amounts across those periods should sum to the cohort’s total full-term exposure, subject to the stated counting convention.

Key takeaway: Written exposure records exposure associated with business written during a period; earned exposure measures coverage provided during a period. For losses defined by accidents in a specified period, use earned exposure for the risks in force during that period, unless the problem defines a different matching basis.

Check Your Understanding

For each problem, identify the relevant exposure timing and calculate the requested amount. Assume uniform pro-rata earning and twelve equal months unless stated otherwise.

A hypothetical insurer writes 40 annual policies effective July 1, all remaining in force through year-end. Find written exposure and earned exposure for that calendar year. (Answer: 40 written policies; 20 earned policy-years.)
Fifty annual policies are effective November 1 and remain in force for twelve months. Find the cohort’s earned policy-years in the issue year and the following calendar year. (Answer: $50(2/12)=8.3333$ policy-years in the issue year and $50(10/12)=41.6667$ in the following year; 50 in total.)
A policy covers three machines from January 1 through December 31. One additional machine is added July 1 and remains covered through year-end. Find earned machine-years. (Answer: $3(12/12)+1(6/12)=3.5$ machine-years.)
A calendar-year accident analysis has $126,000 of losses. During that year, 30 annual policies are in force for the full year and 20 more are in force for six months each. Find earned exposure and illustrative losses per earned policy-year. (Answer: $30+20(6/12)=40$ policy-years; $3,150 per earned policy-year.)
An insurer writes 80 annual policies effective October 1, all still in force through the next September 30. A candidate uses 80 written policies as the denominator for losses from the following calendar year. Find the cohort’s earned exposure in that following year, and explain why it differs from the written count for that year. (Answer: 60 earned policy-years, because each policy supplies nine months of coverage in the following calendar year; the cohort contributes no written policies in that year.)

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