What an Accident-Year Row Represents
A loss triangle organizes insurance experience by an origin period and a development age. In an accident-year triangle, the origin period is the calendar year in which the covered loss occurred. The first task in constructing a row is therefore to identify the date of loss for each claim and assign the claim to the corresponding calendar year. A claim reported or paid later still belongs to the accident year of its loss.
This distinction matters because the dates associated with a claim answer different questions. The date of loss identifies when the insured event occurred; the report date identifies when the insurer was notified; and payment dates identify when amounts were paid. For accident-year assignment, it is the date of loss that determines the row. The loss measure placed in a cell—such as paid loss or incurred loss—is a separate choice.
For a claim \(c\) with date of loss \(DOL_c\), the assignment rule is:
Thus, a loss dated December 31 belongs to that calendar year, while a loss dated the following January 1 belongs to the next year. This boundary rule applies regardless of when the claim is reported or when any payment is made.
The row’s membership and the amounts shown for its claims should not be confused. Claim membership is based on dates of loss. The amount recorded for an included claim depends on the selected measure and valuation date. For example, a paid-loss row may include payments made by a specified valuation date, while an incurred-loss row may use incurred amounts as of that date. Those choices affect the amount observed, but do not change the accident-year assignment.
A Reliable Assignment Process
When preparing an accident-year row from claim records, use the same assignment rule for every claim before adding or summarizing amounts. The following process keeps the origin-period definition separate from later dates and accounting measures.
Use the recorded date on which the covered event occurred, following the data definition for the exercise.
Assign January 1 through December 31 to one accident year. In particular, do not move a December loss because it was reported in January.
Each claim is assigned to the row matching its date of loss. If summarizing claim counts, count each included claim once.
Aggregate only the amounts required by the problem, using its stated valuation date and loss basis.
Confirm that claims have not been shifted by report or payment date and that row totals reconcile to the included claim records.
Worked Examples
Worked Example: Claims Around the Calendar-Year Boundary
A hypothetical insurer is assembling paid losses by accident year, with all amounts measured through December 31, 2024. Its claim list is:
| Claim | Date of loss | Report date | Paid through valuation |
|---|---|---|---|
| A | December 31, 2022 | January 5, 2023 | $1,200 |
| B | January 1, 2023 | January 2, 2023 | $800 |
| C | June 30, 2023 | July 5, 2023 | $2,500 |
| D | December 31, 2023 | January 10, 2024 | $1,500 |
| E | January 1, 2024 | January 1, 2024 | $600 |
Step 1: Assign each claim by date of loss. Claim A belongs to 2022 because its loss occurred on December 31, 2022, even though it was reported in 2023. Claims B, C, and D belong to 2023. Claim E belongs to 2024.
Step 2: Add paid amounts within each accident year. For accident year 2022, the total is \(1{,}200\). For accident year 2023, the total is \(800+2{,}500+1{,}500=4{,}800\). For accident year 2024, the total is \(600\).
Step 3: Check the totals. The sum of the annual amounts is \(1{,}200+4{,}800+600=6{,}600\), which equals the sum of all five claim amounts. The claim counts are one for 2022, three for 2023, and one for 2024.
Method and trap avoided. This is date-of-loss assignment followed by aggregation of paid amounts. Assigning Claim A to 2023 because of its report date would incorrectly move $1,200 out of accident year 2022; the row is determined by occurrence, not notification.
Worked Example: Several Payments on One Claim
A hypothetical portfolio is summarized as paid loss through December 31, 2025. One claim with a November 20, 2023 date of loss has payments of $500 in January 2024 and $300 in June 2025. A second claim has a February 14, 2024 date of loss and payments of $700 in March 2024 and $200 in August 2025. A third claim has a December 30, 2024 date of loss and a $1,100 payment in February 2025. A fourth claim has a January 2, 2025 date of loss and a $250 payment in December 2025. Construct the annual paid-loss totals.
Step 1: Assign the claims. The first claim belongs to accident year 2023. The second and third belong to accident year 2024. The fourth belongs to accident year 2025.
Step 2: Sum all payments made by the valuation date for each claim. The 2023 claim has \(500+300=800\) paid. The first 2024 claim has \(700+200=900\) paid, and the second 2024 claim has \(1{,}100\) paid. The 2025 claim has \(250\) paid.
Step 3: Aggregate by accident year. Accident year 2023 has \(800\). Accident year 2024 has \(900+1{,}100=2{,}000\). Accident year 2025 has \(250\).
Step 4: Reconcile. The three row totals add to \(800+2{,}000+250=3{,}050\). Adding all individual payments gives \(500+300+700+200+1{,}100+250=3{,}050\), so the aggregation checks.
Method and trap avoided. Payments are first combined within each claim and then assigned to the claim’s accident year. Putting the $500 payment in 2024 because that is when it was paid would mix payment-year grouping with accident-year grouping. Both payments on the 2023 loss remain in the 2023 accident-year total.
Worked Example: Building One Incurred-Loss Row
A hypothetical insurer wants the accident-year 2024 incurred-loss total as of December 31, 2025. Its records show: Claim J, loss dated February 29, 2024, incurred amount $300; Claim K, loss dated December 31, 2024, reported January 2, 2025, incurred amount $1,400; Claim L, loss dated January 1, 2025, incurred amount $2,000; and Claim M, loss dated August 15, 2024, incurred amount $900. The given incurred amounts are already measured at the stated valuation date.
Step 1: Select claims by date of loss. Claims J, K, and M have dates of loss in 2024, so they belong to the 2024 row. Claim L has a date of loss in 2025, so it is excluded from that row.
Step 2: Add the selected incurred amounts. The 2024 total is \(300+1{,}400+900=2{,}600\).
Step 3: Check the boundary and total. Claim K is included despite being reported in 2025 because the loss occurred on December 31, 2024. Claim L is excluded despite its amount being known at the valuation date because its loss occurred on January 1, 2025. The three included claims sum to $2,600.
Method and trap avoided. The row is constructed by selecting claims on the incurred-loss basis stated in the problem, but selection into the row still uses the date of loss. Excluding Claim K by report year would produce \(300+900=1{,}200\), which is not the 2024 accident-year total.
Why the Grouping Rule Works
The construction can be expressed as an indicator-weighted sum. Suppose the claim records under consideration are \(c=1,\ldots,n\), and let \(L_c\) be the selected loss amount for claim \(c\), measured on a consistent basis and at a common valuation date. Define an indicator \(I(c,y)\) that equals 1 if claim \(c\)'s date of loss falls in calendar year \(y\), and 0 otherwise. Then the total for accident year \(y\) is:
This formula follows directly from the row definition. If the loss occurred in year \(y\), then \(I(c,y)=1\), so the claim’s selected loss amount is included. If it occurred in any other year, then \(I(c,y)=0\), so its contribution to row \(y\) is zero. The date of report and the dates of payment do not appear in the indicator because they do not define accident-year membership.
The formula assumes the claim list and \(L_c\) values are defined consistently: the same coverage and loss basis are used, each claim is represented according to the data’s claim-unit convention, and amounts are measured at the stated valuation. If the available records are incomplete, the calculated total describes the records included under that data rule; it does not establish that no other losses occurred in the year. A date missing or ambiguous in the source data must be resolved under the stated data convention rather than silently assigned by report or payment date.
Common Mistakes and Practical Checks
- Using report year instead of accident year. A late-reported December loss remains in the year in which it occurred.
- Using payment year. A payment made after the accident year does not shift the claim into the year of payment.
- Misreading a boundary date. December 31 belongs to the year ending that day; January 1 belongs to the new year.
- Changing the loss basis while adding. Do not combine paid amounts for some claims with incurred amounts for others unless the problem explicitly defines such a measure.
- Confusing row membership with what was known at an earlier valuation. A loss can belong conceptually to an accident year even if it was not reported by a particular valuation date. A triangle cell at that valuation can only reflect information included under its data and valuation rules.
A useful reconciliation is to compare the total of the accident-year amounts with the total of the same selected claim amounts across the full data set. If the totals differ, investigate omitted claims, duplicate records, inconsistent valuation bases, or a date-based assignment error. The reconciliation checks the arithmetic and grouping; it does not replace the date-of-loss rule.
Check Your Understanding
For each problem, identify the relevant accident-year row before calculating any requested total.
- A covered loss occurs on December 31, 2021, is reported on January 4, 2022, and has $750 paid by the valuation date. Which accident year contains the claim, and how much does it contribute to that row’s paid total? (Answer: Accident year 2021; $750.)
- Two claims have loss dates of December 30, 2022, and January 1, 2023, with incurred amounts of $400 and $900, respectively. What is the incurred total for accident year 2022? (Answer: $400.)
- A claim has a March 2023 loss date and payments of $250 in 2023 and $450 in 2024. For a paid-loss summary through December 31, 2024, which accident-year row receives the payments, and what is its total contribution? (Answer: Accident year 2023; $700.)
- A claim register has three losses: May 5, 2024, incurred $600; December 31, 2024, incurred $1,100 and reported in 2025; and January 1, 2025, incurred $800. What is the accident-year 2024 incurred total? (Answer: $1,700.)
- Through a stated valuation date, a hypothetical insurer records paid amounts of $300 and $500 for two claims with loss dates of December 31, 2023, and January 1, 2024, respectively. Both claims were reported in 2024. Give the paid total for each of accident years 2023 and 2024. (Answer: Accident year 2023: $300; accident year 2024: $500.)