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CAS Exam 5 · Exposure Bases and Policy Data · Tutorial 26 of 989

Partial-Year Experience Windows

Calculate the covered time and exposure units that fall inside a partial-year experience window, including when coverage changes or ends during the window.

Beginner 9 min read

What You'll Learn

  • Define an experience window using explicit start and end dates.
  • Calculate the overlap between a coverage interval and an experience window.
  • Convert overlap days into exposure using a stated day-count convention.
  • Aggregate exposure across policies and across periods with different exposure amounts.
  • Avoid errors from counting dates outside the window or including an endpoint twice.

Why Partial-Year Windows Need a Boundary Calculation

An experience period does not always begin on a policy effective date or end on a policy expiration date. A policy may provide coverage before the period starts, continue after it ends, or cover only part of the period. To measure exposure for that experience period, include only the coverage time that lies within its boundaries.

The key operation is an interval overlap. First identify the coverage dates and the experience-window dates. Then determine the dates common to both intervals. Convert only that overlap into exposure. This approach applies whether the exposure basis is one policy unit, several insured items, or another quantity that is constant over the relevant interval.

Definition: An experience window is the specified date interval for which exposure is being measured. A coverage interval is the period during which a particular policy or coverage record provides insurance. The exposure assigned to the window is based on the portion of the coverage interval that overlaps the window.

Representing Dates as Intervals

For calculation, use half-open intervals: include the start date and exclude the end date. Write a coverage interval as \([s,e)\) and an experience window as \([A,B)\). Thus, coverage is active on date \(s\), but not on date \(e\). This convention lets adjacent intervals meet at a boundary without counting that boundary twice.

For a constant exposure amount, the overlapping time is the number of elapsed days from the later start date to the earlier end date. If the earlier end is not after the later start, there is no overlap. Under an Actual/365 convention, divide the overlap days by 365 to obtain policy-year exposure per unit of exposure basis. The day-count convention must be stated and used consistently; another specified convention may produce a different result.

$$ D_{\text{overlap}}=\max\left(0,\min(e,B)-\max(s,A)\right) $$

In this expression, date subtraction gives elapsed calendar days. If the exposure amount is \(q\) units throughout the overlapping interval, then its exposure is \(qD_{\text{overlap}}/365\) under Actual/365. If \(q\) changes during the window, split the record into intervals on which \(q\) is constant, calculate each overlap separately, and add the resulting exposures.

Calculation rule: Use the later of the two start dates and the earlier of the two end dates. Count the elapsed days between them only if the earlier end is after the later start. For a constant exposure amount, multiply those days by the amount and divide by the stated day-count denominator.

Worked Calculations

Worked Example: One Policy Crossing Both Window Boundaries

A hypothetical one-vehicle policy covers \([2025\text{-}02\text{-}01,2026\text{-}02\text{-}01)\). Measure its exposure in the experience window \([2025\text{-}06\text{-}15,2025\text{-}11\text{-}15)\). Use Actual/365. What is the exposure in vehicle-years?

Method. Find the intersection of the policy interval and the experience window. The policy starts before the window and ends after it, so the overlap is the whole window: June 15 through November 15, with November 15 excluded.

Count elapsed days. From June 15 to July 1 is 16 days; July contributes 31 days, August 31, September 30, October 31, and November 1 to November 15 contributes 14 days. Therefore,

$$ D_{\text{overlap}}=16+31+31+30+31+14=153\text{ days}. $$

There is one vehicle throughout the overlap, so \(q=1\). The exposure is

$$ 1\times\frac{153}{365}=0.419178\ldots\text{ vehicle-years}. $$

Rounded to four decimal places, the answer is 0.4192 vehicle-years. As a check, the window is shorter than half a 365-day year, and 153 days is less than 182.5 days. The result is therefore less than 0.5 vehicle-years. The common mistake is to assign the entire policy term, or a full vehicle-year, even though only the window overlap belongs in this experience period.

Worked Example: Two Policies With Different Boundary Overlaps

A hypothetical insurer measures exposure in the window \([2025\text{-}08\text{-}15,2026\text{-}02\text{-}15)\), using Actual/365. Policy A provides one unit of coverage from July 1, 2025, to July 1, 2026. Policy B provides one unit from December 1, 2024, to December 1, 2025. Calculate the combined exposure in policy-years.

Method. Calculate each policy's intersection with the window, then add the policy-day exposures. Policy A covers the entire window. The elapsed days from August 15, 2025, to February 15, 2026, are 184: the successive monthly intervals have lengths 31, 30, 31, 30, 31, and 31 days.

$$ D_A=31+30+31+30+31+31=184\text{ days}. $$

Policy B expires on December 1, 2025, before the window ends. Its overlap is August 15 to December 1. August 15 to September 1 is 17 elapsed days, followed by 30 days in September and 31 in October and 30 in November:

$$ D_B=17+30+31+30=108\text{ days}. $$

Each policy has one unit of exposure over its own overlap. The combined exposure is

$$ \frac{184}{365}+\frac{108}{365} =\frac{292}{365} =0.8000\text{ policy-years}. $$

The result is exactly 0.8000 policy-years. As a check, Policy A contributes the full 184 days in the window, while Policy B contributes only the 108 days before its expiration; neither policy contributes time outside the window. A common mistake is to give Policy B the entire window exposure, despite its December 1 expiration.

Worked Example: Exposure Amount Changes Before Cancellation

A hypothetical commercial policy covers two insured units from January 1 through May 1, 2025. An endorsement effective May 1 increases the covered amount to three units. Coverage is cancelled effective September 1, 2025. Calculate exposure in the window \([2025\text{-}03\text{-}01,2025\text{-}10\text{-}01)\), using Actual/365.

Method. Split the coverage at the endorsement date, where the exposure amount changes, and at the cancellation date, where coverage stops. Use the half-open intervals \([2025\text{-}01\text{-}01,2025\text{-}05\text{-}01)\) at two units and \([2025\text{-}05\text{-}01,2025\text{-}09\text{-}01)\) at three units. No exposure is provided after September 1.

For the first interval, the overlap with the window is March 1 to May 1: 31 days in March and 30 in April, for 61 days. Its exposure is \(2\times61=122\) unit-days. For the second interval, the overlap is May 1 to September 1: 31 days in May, 30 in June, 31 in July, and 31 in August, for 123 days. Its exposure is \(3\times123=369\) unit-days.

$$ \text{Total exposure} =\frac{122+369}{365} =\frac{491}{365} =1.345205\ldots\text{ unit-years}. $$

Rounded to four decimal places, the answer is 1.3452 unit-years. The check is \(491\) unit-days in total: \(122\) before the endorsement change and \(369\) after it. The common mistake is to apply three units to the entire window, including dates before the endorsement, or to include September after cancellation.

Why the Overlap Formula Works

The coverage interval and the experience window each impose two restrictions on an included date. A date must be on or after both start dates, so the first possible included date is the later start, \(\max(s,A)\). It must also be before both end dates, so the last possible endpoint is the earlier end, \(\min(e,B)\). The dates satisfying both restrictions therefore form the interval from \(\max(s,A)\) to \(\min(e,B)\).

If that earlier endpoint is after the later start, the elapsed time between them is the overlap length. If it is equal to or before the later start, no positive-duration interval satisfies both restrictions, so the overlap is zero. This gives the overlap formula directly. Multiplying overlap days by a constant exposure amount counts the amount for every covered day; division by 365 converts unit-days into unit-years under Actual/365.

1
Specify the window and coverage dates.
Write both intervals with explicit start and end dates, using the same endpoint convention.
2
Find the overlap.
Take the later start and earlier end. If the earlier end is not after the later start, assign zero overlap days.
3
Apply the exposure amount and day-count rule.
Multiply overlap days by the amount in force, divide by the stated denominator, and sum across intervals or policies.

Boundary Choices and Common Traps

A clear endpoint convention is important even though a single instant has no duration. With half-open intervals, one coverage interval can end on May 1 and a replacement interval can start on May 1 without a gap or duplicate day. This convention also makes an effective cancellation date unambiguous: if coverage ends effective September 1, the interval ending on that date does not include September 1.

  • Using the policy term instead of the window overlap. A policy may be active for a full year but contribute only a fraction of a year to a shorter experience window.
  • Extending exposure past an expiration or cancellation. The window ending later does not make coverage continue later.
  • Applying one exposure amount across a change. Split at an endorsement or other effective change, and use the amount applicable to each segment.
  • Counting both intervals at a shared boundary. Use one consistent start-inclusive, end-exclusive convention rather than adding an extra day to each segment.
  • Mixing day-count conventions. State the denominator and apply it consistently. Actual/365 uses actual elapsed calendar days divided by 365; do not silently substitute a month-based approximation.

Partial-window exposure matters because the experience period is the basis for deciding which coverage time belongs in a particular measurement. If dates outside the window are included, exposure is assigned to the wrong period; if dates inside it are omitted, the period's exposure is understated. The overlap calculation keeps the assignment tied to the stated boundaries without requiring the window to align with policy terms.

Key takeaway: Exposure in a partial-year window is based on the intersection of the coverage interval and the window, not on the full policy term. Identify the overlap first, split whenever the exposure amount changes, and apply the chosen day-count convention only to included days.

Check Your Understanding

Use half-open intervals and Actual/365 for each problem. Round exposure answers to four decimal places.

  1. A one-unit policy covers January 1, 2025, to January 1, 2026. Find its exposure in the window April 1 to October 1, 2025. (Answer: 0.5014 policy-years)
  2. A one-unit policy covers November 1, 2024, to November 1, 2025. Find its exposure in the window September 1, 2025, to January 1, 2026. (Answer: 0.1671 policy-years)
  3. One one-unit record covers January 1 to July 1, 2025, and a second one-unit record covers March 1 to May 1, 2025. Find combined exposure in the window February 1 to June 1, 2025. (Answer: 0.4959 policy-years)
  4. A policy covers four units from January 1 to April 1, 2025, then two units from April 1 to July 1, 2025. Find exposure in the window March 1 to June 1, 2025. (Answer: 0.6740 unit-years)
  5. A one-unit policy covers October 1, 2025, to April 1, 2026. Find exposure in the window December 15, 2025, to March 15, 2026. (Answer: 0.2466 policy-years)

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