Why Earned Exposure Depends on the Valuation Date
Written exposure records the amount of coverage associated with a policy when it is written. That amount does not necessarily represent coverage already provided. If a policy has been active for only part of its term, only part of its written exposure has elapsed by a particular valuation date. Measuring that elapsed portion gives earned exposure.
The distinction matters when compiling exposure for a period of experience. A policy effective on January 1 has elapsed more coverage by June 15 than an otherwise identical policy effective on May 15. Adding the policies’ full written exposures at June 15 would ignore that difference. The calculation requires both the exposure amount and the time elapsed within the coverage period.
The uniform-accrual assumption is appropriate when the exposure amount is constant throughout the term and the exposure is assigned in proportion to elapsed coverage time. It is a working assumption for the calculations here, not a claim that every coverage’s underlying risk is identical on every day. If exposure changes during a policy term, a single fixed amount multiplied by a time fraction may not represent the coverage accurately.
Date Boundaries and Elapsed Days
Use a consistent boundary convention before counting days. In this tutorial, coverage starts at the beginning of its effective date, and the valuation date is also treated as the beginning of that date. Thus, “elapsed by June 15” means coverage elapsed from the policy’s effective-date boundary through the start of June 15. The valuation date itself is not counted as an additional full day.
For example, from the start of May 15 to the start of June 15, 31 days elapse: 17 days from May 15 to the June 1 boundary, then 14 days from June 1 to the start of June 15. These are elapsed intervals between date boundaries. Do not count both endpoints as additional days.
For one policy, let \(W\) be its written exposure, \(n\) its full term in days, and \(k(t)\) the number of days from the effective-date boundary to valuation date \(t\). For a date before coverage begins, use zero elapsed days. For a date after the term ends, use the full term. Equivalently, the number of elapsed coverage days is the actual elapsed time limited to the interval from zero through \(n\).
Here \(E(t)\) denotes earned exposure at date \(t\). If \(t\) is on or before the effective-date boundary, \(E(t)=0\). If \(t\) is on or after the expiration boundary, \(E(t)=W\). Between those boundaries, the earned amount is the written exposure multiplied by the elapsed fraction of the term.
Why the Elapsed-Time Formula Works
Suppose a policy has written exposure \(W\) and a term of \(n\) days. Under uniform accrual, each day accounts for \(W/n\) units of exposure. After \(k\) days, the exposure corresponding to elapsed coverage is the daily amount multiplied by the number of elapsed days:
This also checks the boundary cases. At the start, \(k=0\), so earned exposure is zero. At the end, \(k=n\), so earned exposure is \(W(n/n)=W\). The formula therefore assigns no elapsed exposure before coverage starts, the full written amount after the entire term has elapsed, and the corresponding fraction between those boundaries.
Record the written exposure, effective date, expiration date, and term length in days for each policy.
Use the same convention for every policy. Here, the valuation date is the start of that date.
Count the days from the effective-date boundary to the valuation-date boundary, using zero before inception and no more than the full term after expiration.
Multiply each written exposure by its elapsed-days fraction, then add the policy-level earned amounts if a portfolio total is required.
Worked Calculations
Worked Example: One Policy at a Midterm Valuation
A hypothetical policy provides a constant exposure of 1.25 units over the period from April 1, 2025, to April 1, 2026. The policy term is specified as 365 days. Calculate its earned exposure at the start of October 1, 2025. Use uniform accrual over the term.
Method. Count the elapsed days from the April 1 boundary to the October 1 boundary, then multiply the written exposure by the elapsed fraction of the 365-day term.
Elapsed days. The date intervals are April 1 to May 1 (30 days), May 1 to June 1 (31), June 1 to July 1 (30), July 1 to August 1 (31), August 1 to September 1 (31), and September 1 to October 1 (30). Therefore, \(k=30+31+30+31+31+30=183\) days.
Elapsed fraction. \(183/365=0.50136986\ldots\).
Earned exposure. \(1.25(183/365)=228.75/365=0.62671232\ldots\). Rounded to four decimal places, the earned exposure is 0.6267 units.
Check. The valuation date falls within the term, and \(0.6267\) is less than the written exposure of 1.25 units. The unrounded amount is approximately half of the written exposure because 183 days is slightly more than half of 365 days.
Common mistake avoided. Counting both April 1 and October 1 as full elapsed days would add a boundary day. The calculation uses the difference between date boundaries, which is 183 days.
Worked Example: Different Effective Dates in a Portfolio
A hypothetical portfolio has three policies, all with 365-day terms and constant exposure amounts. Policy A runs from January 1, 2025, to January 1, 2026, and has 1.2 exposure units written. Policy B runs from May 15, 2025, to May 15, 2026, and has 0.8 units written. Policy C has 2.0 units written and is scheduled to run from June 20, 2025, to June 20, 2026. Calculate total earned exposure at the start of June 15, 2025, assuming uniform accrual.
Policy A. From January 1 to June 15, the elapsed intervals are January (31 days), February (28), March (31), April (30), May (31), and June 1 to June 15 (14). Thus, \(k_A=31+28+31+30+31+14=165\) days. Its earned exposure is \(1.2(165/365)=198/365=0.54246575\ldots\).
Policy B. From May 15 to June 1, 17 days have elapsed, followed by 14 days from June 1 to the start of June 15, for a total of \(k_B=17+14=31\) days. Its earned exposure is \(0.8(31/365)=24.8/365=0.06794520\ldots\).
Policy C. The effective date, June 20, is after the valuation-date boundary. No coverage has elapsed by June 15, so its earned exposure is zero, even though the policy has a stated written exposure.
Portfolio total. Add the unrounded policy amounts: \(198/365+24.8/365+0=222.8/365=0.61041095\ldots\). Rounded to four decimal places, total earned exposure is 0.6104 units.
Check. The portfolio has 4.0 written exposure units in total, so an earned amount of 0.6104 units is within the possible range from zero to 4.0. Policy A contributes more than Policy B because it has both a larger written amount and more elapsed days.
Common mistake avoided. A common error is to treat a future-effective policy as fully earned because its exposure is included in the portfolio’s written amount. Earned exposure at a date reflects elapsed coverage; Policy C contributes zero at this valuation date.
Worked Example: A Policy Past Expiration and Two Policies Still in Force
At the start of October 1, 2025, an analyst reviews three hypothetical policies with constant exposure amounts and 365-day terms. Policy A ran from August 1, 2024, to August 1, 2025, with 0.5 units written. Policy B runs from January 1, 2025, to January 1, 2026, with 2.0 units written. Policy C runs from September 15, 2025, to September 15, 2026, with 0.5 units written. Calculate the total earned exposure using uniform accrual.
Policy A. Its expiration boundary, August 1, 2025, is before the valuation-date boundary. The full 365-day term has elapsed, so earned exposure is \(0.5(365/365)=0.5\) units.
Policy B. From January 1 to October 1, the elapsed intervals are January (31), February (28), March (31), April (30), May (31), June (30), July (31), August (31), and September (30). Therefore, \(k_B=273\) days. Earned exposure is \(2.0(273/365)=546/365=1.49589041\ldots\) units.
Policy C. From September 15 to October 1, 16 days elapse. Earned exposure is \(0.5(16/365)=8/365=0.02191780\ldots\) units.
Total. Add the amounts before rounding: \(0.5+546/365+8/365=0.5+554/365=2.01780821\ldots\). Rounded to four decimal places, total earned exposure is 2.0178 units.
Check. Policy A contributes its full 0.5 units because its term has ended. Policies B and C contribute less than their respective written amounts because neither has reached expiration. The total is below the portfolio’s 3.0 written units.
Common mistake avoided. Do not continue counting elapsed days beyond a policy’s expiration boundary. Policy A contributes 365 elapsed days, not the longer calendar interval from August 1, 2024, to October 1, 2025.
Interpreting and Checking Earned Exposure
For a portfolio, calculate earned exposure policy by policy when the written amounts or effective dates differ. If policies share the same term length and valuation date, their elapsed fractions may differ because their effective dates differ. The portfolio total is the sum of those policy-level amounts, not the total written exposure multiplied by one average fraction unless that average is correctly weighted by the written exposures.
A useful reasonableness check is to compare each policy’s earned amount with its written amount. With nonnegative exposure and the stated uniform-accrual assumption, earned exposure must be at least zero and no greater than written exposure. It is zero before coverage begins, between zero and the written amount during the term, and equal to the written amount once the full term has elapsed.
Keep the units consistent. If \(W\) is measured in vehicle exposure units, then \(E(t)\) is also measured in vehicle exposure units; the ratio \(k/n\) has no units. Do not confuse earned exposure with earned premium: premium is a dollar amount, while exposure measures the amount of coverage on the selected exposure basis.
Common Errors
- Counting calendar dates inclusively. Elapsed days are measured from one date boundary to another. Do not add a day merely because a boundary date appears in the wording.
- Using one elapsed fraction for every policy. Policies with different effective dates generally have different elapsed fractions at the same valuation date.
- Giving a future-effective policy positive earned exposure. Before the effective-date boundary, elapsed coverage and earned exposure are zero.
- Allowing elapsed days to exceed the term. Once the expiration boundary is reached, the elapsed coverage used in the calculation is the full term, not the number of calendar days since inception.
- Rounding each policy too early. Retain unrounded policy-level amounts when aggregating, then round the final total to the requested precision.
Check Your Understanding
For each question, use the stated date-boundary convention and assume uniform accrual over the policy term. Round final exposure amounts to four decimal places where needed.
- A policy has 0.75 exposure units written for a 365-day term from January 1, 2025, to January 1, 2026. What is its earned exposure at the start of April 1, 2025? (Answer: \(0.75(90/365)=0.1849\) exposure units.)
- A policy has 1.4 units written for a 365-day term beginning May 15, 2025. How much is earned at the start of June 15, 2025? (Answer: \(1.4(31/365)=0.1189\) exposure units.)
- A policy with 2.3 units written is scheduled to begin July 1, 2025. What is its earned exposure at the start of June 20, 2025? (Answer: 0 exposure units.)
- A policy has 2.5 units written for a 365-day term from October 1, 2025, to October 1, 2026. How much is earned at the start of January 1, 2026? (Answer: \(2.5(92/365)=0.6301\) exposure units.)
- At the start of August 1, 2025, Policy A has 1.5 units written for a 365-day term from June 1, 2024, to June 1, 2025. Policy B has 0.9 units written for a 365-day term from July 15, 2025, to July 15, 2026. What is their combined earned exposure? (Answer: \(1.5+0.9(17/365)=1.5419\) exposure units.)