Tutorials › CAS Exam 8 - Advanced Ratemaking › Matching Losses to Earned Exposure

CAS Exam 8 · Experience Rating Plan Design · Tutorial 500 of 989

Matching Losses to Earned Exposure

Build and check a period-by-period schedule so that losses and earned exposures describe the same risk, coverage, and dates.

Intermediate 10 min read

What You'll Learn

  • Define the experience-period boundaries and the risk units included in a matching schedule
  • Assign losses to periods by their covered occurrence dates
  • Accumulate exposure only for the matching risk and time intervals
  • Reconcile period totals and identify experience outside the selected window
  • Diagnose how unmatched losses or exposure can distort a loss-per-exposure measure

Why Matching Matters

An experience-rating schedule is only meaningful when its losses and exposures describe the same underlying experience. A loss belongs with exposure for the risk that generated it, under the relevant coverage, during the time the exposure was earned. A calendar-year loss total divided by exposure from a different year may produce a number, but it does not describe a valid period-specific loss rate.

The schedule is a recordkeeping and measurement step, not a credibility formula. It organizes experience before a plan uses it in a rating calculation. Start by fixing the experience window and identifying the risk unit and exposure basis. Then assign each covered loss and each earned exposure increment to the same period, and reconcile the resulting totals.

Definition: A matching schedule is a period-by-period record that assigns each eligible loss and each earned exposure amount to the same defined risk, coverage, and time interval. A loss is assigned according to the date and rule specified for the experience measure; in the examples below, that date is the covered occurrence date. Exposure is assigned according to when coverage and the associated exposure actually apply.

A risk unit must be identifiable in both records. Depending on the plan and exposure basis, it might be an insured account, location, vehicle, or payroll exposure for a specified class. The unit should not change silently between the loss record and the exposure record. Where the selected basis is measured in units such as vehicle-years or payroll per thousand dollars, calculate each period’s exposure in those units before aggregating.

The schedule should also state the period boundaries precisely. For example, if the experience window runs from January 1 through December 31, a covered occurrence on December 31 is in the window, while one on the following January 1 is outside it. Do not infer the period from when a claim was reported or paid when the schedule’s assignment rule is based on occurrence date.

A Practical Construction Sequence

1
Set the boundaries.
Write the start and end dates for every experience period, including whether each boundary date is included.
2
Identify the matching risk and coverage.
Use consistent identifiers so a loss and its supporting exposure refer to the same insured risk and relevant coverage.
3
Assign loss records.
Apply the stated date rule to each covered loss, place it in one period, and exclude records outside the window or outside the defined risk.
4
Earn exposure by time interval.
Allocate exposure to the period in which it applies. For partial periods, use the selected exposure basis and the corresponding earned amount, rather than treating a partial term as a full term.
5
Reconcile and review.
Sum each period’s losses and exposures separately, check the grand totals against the source records, and investigate unmatched or duplicated entries.

Worked Schedules

Worked Example: Fleet Exposure Across Two Experience Years

A hypothetical delivery firm’s experience window is January 1, 2024 through December 31, 2025. Consider only its covered auto-liability experience. The selected exposure basis is vehicle-years, earned monthly: each covered vehicle contributes one-twelfth of a vehicle-year for each month in force. Vehicle A is covered throughout 2024, vehicle B from April 1, 2024 through March 31, 2025, and vehicle C from April 1 through December 31, 2025. Covered occurrence losses are $18,400 on March 15, 2024; $7,600 on December 20, 2024; and $11,000 on February 2, 2025. A $900 occurrence on January 4, 2026 is outside the window.

Method. Divide the window into calendar-year periods. Assign each covered occurrence to its calendar year, and calculate vehicle-years from the months each vehicle was covered in that year.

2024 exposure. Vehicle A contributes \(12/12=1\) vehicle-year. Vehicle B contributes \(9/12=0.75\) vehicle-year, from April through December. Vehicle C contributes none. Thus, 2024 exposure is \(1+0.75=1.75\) vehicle-years.

2024 losses. Both 2024 occurrences fall in this period, so losses are \($18{,}400+$7{,}600=$26{,}000\).

2025 exposure. Vehicle A contributes none. Vehicle B contributes \(3/12=0.25\) vehicle-year, from January through March. Vehicle C contributes \(9/12=0.75\) vehicle-year, from April through December. Thus, 2025 exposure is \(0.25+0.75=1.00\) vehicle-year.

2025 losses. The February occurrence is in this period, giving \($11{,}000\). The January 4, 2026 occurrence is outside the experience window and is not included.

Experience periodVehicle AVehicle BVehicle CTotal exposureMatched losses
20241.000.750.001.75 vehicle-years$26,000
20250.000.250.751.00 vehicle-year$11,000

As a check, exposure totals \(1.75+1.00=2.75\) vehicle-years, equal to the 12 months for vehicle A, 12 months for B, and 9 months for C within the window: \(12/12+12/12+9/12=2.75\). Included losses total \($26{,}000+$11{,}000=$37{,}000\), equal to the three occurrences dated within the window.

If a simple loss-per-exposure summary is desired, the 2024 figure is \($26{,}000/1.75=$14{,}857.14\) per vehicle-year, and the 2025 figure is \($11{,}000/1.00=$11{,}000.00\) per vehicle-year. These are descriptive calculations, not a claim that the periods have identical risk characteristics.

Worked Example: Matching Site-Level Losses to Machine-Months

A hypothetical property account has a July 1, 2025 through June 30, 2026 experience window. For this schedule, the risk unit is a covered machine at a named site, the selected coverage is property coverage, and exposure is measured in machine-months. Machine A at Site North is covered from July 1 through December 31, 2025. Machine B at Site South is covered from October 1, 2025 through June 30, 2026. Covered occurrences are $12,000 at Site North on August 14, 2025, $8,000 at Site South on November 2, 2025, and $5,000 at Site South on February 10, 2026. A $4,000 occurrence at Site South on July 3, 2026 is outside the window.

Method. Use three intervals that follow the exposure changes: July–September, October–December, and January–June. Count covered machine-months within each interval, then match only occurrences at the relevant covered site and within that interval.

July–September 2025. Machine A contributes 3 machine-months; Machine B is not yet covered. Exposure is 3 machine-months. The August occurrence adds $12,000 in losses.

October–December 2025. Both machines are covered for 3 months, so exposure is \(3+3=6\) machine-months. The November occurrence adds $8,000. The North machine’s coverage remains part of exposure, even though this interval has no North loss.

January–June 2026. Only Machine B is covered, for 6 months. Exposure is 6 machine-months, and the February occurrence adds $5,000. The July 3 occurrence is after the window and adds no loss to this schedule.

PeriodNorth machine-monthsSouth machine-monthsTotal exposureMatched losses
July–September 2025303$12,000
October–December 2025336$8,000
January–June 2026066$5,000

The exposure check is \(3+6+6=15\) machine-months, which also equals 6 months for Machine A plus 9 months for Machine B. Included losses total \($12{,}000+$8{,}000+$5{,}000=$25{,}000\). The excluded July occurrence is not added merely because it concerns an insured machine: its occurrence date lies outside the selected experience window.

Why the Matching Rule Supports a Meaningful Summary

Suppose a schedule has several periods indexed by \(j\). Let \(L_j\) be the losses assigned to period \(j\), and \(E_j\) its earned exposure. If \(L_j/E_j\) is calculated, it compares losses with the exposure that generated them. When periods have different underlying loss rates, the combined rate is an exposure-weighted average of their rates.

$$ \frac{\sum_j L_j}{\sum_j E_j} = \frac{\sum_j E_j(L_j/E_j)}{\sum_j E_j}, \qquad E_j>0. $$

The equality follows because \(E_j(L_j/E_j)=L_j\) for every period with positive exposure. Summing those equalities gives \(\sum_j L_j\), and dividing by \(\sum_j E_j\) gives the combined rate. Thus the combined result weights each period’s rate by its share of total exposure; it is not generally the simple arithmetic average of the period rates.

This interpretation depends on matching. If a loss from one period is placed over exposure from another, the numerator and denominator no longer describe the same experience. Even if the total dollar losses and total exposure across all years are unchanged, individual period rates can shift, and the combined rate may also be wrong if the mismatched records alter the selected window or risk set.

Derivation: Let the correct loss and exposure for period \(j\) be \(L_j\) and \(E_j\), and let the period’s loss per exposure be \(r_j=L_j/E_j\). Then \(L_j=r_jE_j\), so total losses are \(\sum_j r_jE_j\). Dividing by total matched exposure yields \(\sum_j r_jE_j/\sum_jE_j\), the exposure-weighted average. Replacing an \(E_j\) with exposure from a different period breaks this identity for the actual period rates.

Common Mistake: Matching by Claim Processing Date

A common error is to assign a loss to the year it was paid or processed even though the schedule specifies occurrence date. That can move a loss away from the exposure period in which the covered event happened. In the fleet example, suppose the $7,600 loss from December 20, 2024 is paid in 2025. Assigning it by payment year would incorrectly produce 2024 losses of $18,400 and 2025 losses of $18,600, while leaving exposures at 1.75 and 1.00 vehicle-years.

The correction is to follow the schedule’s assignment rule: the occurrence was in 2024, so 2024 losses are $26,000 and 2025 losses are $11,000. The example does not establish a universal rule for every plan’s loss record; it demonstrates why the date basis must be explicit and consistently applied.

Other checks help prevent the same type of mismatch:

  • Do not include an occurrence outside the experience window simply because it involves the same insured.
  • Do not count exposure for months before coverage begins or after it ends.
  • Do not combine exposures from a different risk unit merely because they share an account name.
  • Do not compare losses for one selected coverage with exposure that does not represent that coverage.
  • Do not silently change the exposure unit or the period boundary rule partway through the schedule.

A completed schedule should allow a reviewer to trace every included loss to a period and a defined risk, and every exposure amount to the time and risk it represents. Reconciliation is not a substitute for matching, but it can reveal missing records, duplicated amounts, or exposure that has been assigned to the wrong interval.

Check Your Understanding

For each problem, apply the stated date rule and exposure basis before calculating totals.

  1. A hypothetical policy’s experience window is January 1 through June 30, 2025. Four covered trucks are in force for all six months, and exposure is measured in truck-years earned monthly. Covered occurrences are $7,000 on June 30 and $3,000 on July 1. State the matched exposure and losses. (Answer: 2 truck-years and $7,000.)
  2. A hypothetical risk has calendar-year experience periods and payroll exposure measured in units of $1,000. Payroll is $360,000 in 2024 and $240,000 in 2025. Occurrences are $4,200 on December 31, 2024; $5,500 on January 1, 2025; $2,100 on December 31, 2025; and $8,000 on January 1, 2026. Give matched losses and exposures by year. (Answer: 2024: $4,200 and 360 exposure units; 2025: $7,600 and 240 exposure units; the 2026 occurrence is excluded.)
  3. A hypothetical equipment policy covers two machines from April 1 through September 30, 2025. Exposure is machine-months. Machine X is covered for all six months; Machine Y is covered only from July 1 through September 30. Covered occurrences are $2,400 for X on May 3 and $1,600 for Y on August 8. Give total exposure and losses for April–June and July–September. (Answer: April–June: 3 machine-months and $2,400; July–September: 6 machine-months and $1,600.)
  4. A hypothetical account has two experience periods, 2023 and 2024. Its matched exposures are 100 and 200 units, respectively. Occurrence losses are $9,000 in 2023 and $6,000 in 2024. Calculate each period’s loss per exposure and the combined loss per exposure. (Answer: 2023: $90 per unit; 2024: $30 per unit; combined: $15,000/300 = $50 per unit.)
  5. A hypothetical policy’s experience window is October 1, 2024 through March 31, 2025. One covered unit earns exposure at one unit-month per month. It is covered from October 1 through December 31, and a second unit joins on January 1 and remains through March 31. Occurrences are $3,000 on December 31, $4,000 on January 1, and $2,000 on April 1. Give exposure and losses for October–December and January–March, applying occurrence dates to the periods. (Answer: October–December: 3 unit-months and $3,000; January–March: 3 unit-months and $4,000; the April 1 occurrence is outside the window.)
Clinical Biostats