From Claim Records to a Triangle
A loss development triangle is an organized summary of claim experience by origin period and development age. Building one correctly starts with the records, not with a triangle-shaped spreadsheet: each claim or transaction must be assigned to the right accident year, valuation date, and loss measure. A date assigned to the wrong period can produce a neat-looking triangle whose totals do not describe the intended experience.
This tutorial uses annual accident years and year-end valuations for illustration. The same construction principles apply to quarterly or monthly periods, provided the period definitions and valuation cutoffs are applied consistently. All amounts in the worked examples are hypothetical.
For example, an accident in 2022 belongs to accident year 2022, even if it is reported in 2023. With calendar-year accident periods and year-end valuations, a payment made during 2022 is assigned to age 1 (often labelled 12 months), a payment during 2023 to age 2 (24 months), and a payment during 2024 to age 3 (36 months). In practice, follow the specified valuation and development-period definitions, rather than assuming that labels such as “12 months” settle every boundary-date question.
A paid triangle begins with transaction amounts assigned to development periods. Cumulative paid amounts are then formed by adding the incremental amounts through each age:
An incurred triangle requires a valuation snapshot. For each claim, take cumulative paid and case outstanding as of the relevant valuation date, calculate incurred for that claim, and then aggregate claims within the accident-year and development-age cell. Do not add successive snapshots of a case reserve as though each were a payment.
Constructing Paid Triangles
The key distinction is between an incremental triangle and a cumulative triangle. The incremental cell contains payments assigned to that development period alone; the cumulative cell contains all payments through that age. A source file of payment transactions naturally supports incremental aggregation, while cumulative amounts are calculated from those increments.
Specify the covered losses, accident-period definition, valuation dates, and any transaction-date cutoff. Exclude transactions after the applicable cutoff.
Use the date that defines the chosen origin, such as the date of accident for an accident-year triangle. A report date or payment date does not replace the accident date.
For paid triangles, use the payment date and the stated valuation-period convention to place each transaction in its development age.
Sum included payments by accident year and development age, calculate cumulative amounts, and reconcile the included total to the source records.
The cumulative formula follows directly from how the cells are defined. The development periods from 1 through \(j\) are disjoint: each included payment belongs to exactly one of them. Their union is the set of payments made by the valuation at age \(j\). Adding the incremental totals for those periods therefore counts each payment once and produces the cumulative total through age \(j\). This construction assumes that the incremental amounts use the same coverage, loss basis, currency, and transaction cutoff across the periods.
Worked Example: Building Incremental and Cumulative Paid Triangles
A hypothetical insurer extracts payments for claims from accident years 2022–2024, valued at December 31, 2024. All amounts are dollars, and the extract contains only payments dated on or before the valuation date. The relevant records are: accident-year 2022 claim A, with payments of $1,200 in 2022, $800 in 2023, and $500 in 2024; accident-year 2022 claim B, with payments of $700 in 2022 and $300 in 2024; accident-year 2023 claim C, with payments of $900 in 2023 and $400 in 2024; and accident-year 2024 claim D, with a payment of $1,100 in 2024. Construct both triangles.
Method: assign by accident year and payment year, aggregate increments, then take running totals. A payment in the accident year is age 1; a payment in the following calendar year is age 2. Thus, claim A contributes $1,200, $800, and $500 to accident year 2022 at ages 1, 2, and 3. Claim B contributes $700 at age 1 and $300 at age 3.
For accident year 2022, the incremental amounts are: age 1, \(1{,}200+700=1{,}900\); age 2, \(800\); and age 3, \(500+300=800\). The cumulative amounts are: age 1, \(1{,}900\); age 2, \(1{,}900+800=2{,}700\); and age 3, \(2{,}700+800=3{,}500\).
For accident year 2023, the increments are $900 at age 1 and $400 at age 2. The cumulative amounts are $900 at age 1 and \(900+400=1{,}300\) at age 2. Accident year 2024 has $1,100 at age 1. Its later ages have not yet been observed at the valuation date.
| Accident year | Incremental age 1 | Incremental age 2 | Incremental age 3 |
|---|---|---|---|
| 2022 | $1,900 | $800 | $800 |
| 2023 | $900 | $400 | — |
| 2024 | $1,100 | — | — |
| Accident year | Cumulative age 1 | Cumulative age 2 | Cumulative age 3 |
|---|---|---|---|
| 2022 | $1,900 | $2,700 | $3,500 |
| 2023 | $900 | $1,300 | — |
| 2024 | $1,100 | — | — |
Check. The 2022 source payments sum to \(1{,}200+800+500+700+300=3{,}500\), equal to the final observed cumulative cell. The 2023 payments sum to \(900+400=1{,}300\), and the 2024 payments sum to $1,100. Across accident years, included payments total \(3{,}500+1{,}300+1{,}100=5{,}900\), also equal to the sum of the incremental cells.
Common mistake avoided: adding each incremental cell to the previous cumulative cell is correct; treating an incremental amount as already cumulative would double-count earlier payments. Also, the dash for accident year 2023 at age 3 is not a zero: that age was not yet observable at the valuation date.
Constructing an Incurred Triangle
An incurred triangle is built from claim-level values at each valuation, not by summing every reserve movement recorded during the year. For claim \(c\) at a valuation, incurred equals that claim’s cumulative paid amount plus its case outstanding amount. Aggregate those claim-level incurred values by accident year and development age.
Here \(c\in(i,j)\) means that claim \(c\) belongs to accident year \(i\) and is included in the snapshot for development age \(j\). The formula requires paid and case outstanding to be measured at the same valuation and on the same loss basis. It does not combine paid loss with a reserve basis that includes different expense components.
Worked Example: Aggregating Claim Snapshots Into Incurred Cells
A hypothetical insurer has two claims in accident year 2022. At ages 1, 2, and 3, claim A has cumulative paid of $1,000, $1,500, and $1,800, and case outstanding of $2,000, $1,200, and $0. Claim B has cumulative paid of $600, $600, and $900, and case outstanding of $1,400, $900, and $300. Calculate the accident-year incurred cells.
Method: calculate incurred for each claim at each snapshot, then add within each cell. At age 1, claim A incurred is \(1{,}000+2{,}000=3{,}000\), and claim B incurred is \(600+1{,}400=2{,}000\). The accident-year cell is \(3{,}000+2{,}000=5{,}000\).
At age 2, claim A incurred is \(1{,}500+1{,}200=2{,}700\), and claim B incurred is \(600+900=1{,}500\). The cell is \(2{,}700+1{,}500=4{,}200\).
At age 3, claim A incurred is \(1{,}800+0=1{,}800\), and claim B incurred is \(900+300=1{,}200\). The cell is \(1{,}800+1{,}200=3{,}000\). Therefore, the cumulative incurred row is $5,000, $4,200, and $3,000.
Check. At each age, the total incurred equals total cumulative paid plus total case outstanding: age 1, \(1{,}600+3{,}400=5{,}000\); age 2, \(2{,}100+2{,}100=4{,}200\); age 3, \(2{,}700+300=3{,}000\). The checks match the claim-by-claim calculation.
Common mistake avoided: adding the three case-reserve snapshots for claim A, \(2{,}000+1{,}200+0\), would not give its age-3 outstanding amount. The age-3 incurred value uses only the age-3 snapshot: cumulative paid at age 3 plus case outstanding at age 3.
Date, Cell, and Total Checks
A finished triangle should be checked against both its design and its source records. The checks are not a substitute for assigning records correctly; they help expose inconsistencies after the assignments have been made.
- Check the accident-year assignment. Confirm that each claim uses the date defining the origin period. For an accident-year triangle, a later report date does not move the claim to a later accident year.
- Check the valuation cutoff. Include only transactions permitted by the stated cutoff. A transaction dated after the valuation date does not belong in the valued triangle, even if it appears in a later extract.
- Check the development-age mapping. Recalculate the valuation year or date associated with each cell. Confirm the boundary convention for transactions on period-end dates.
- Check blanks and zeros. A structurally unobserved age is left unfilled, not set to zero. An observed cell with no payments may be recorded as zero if the claim population and data definition show that the cell was observed.
- Reconcile totals. Sum included transaction amounts and compare them with the corresponding total of incremental paid cells. For incurred, check each cell against its paid and case-outstanding components.
A total reconciliation alone cannot prove that every record is assigned to the right cell: two offsetting date errors could preserve the overall total. Pair the total check with claim-level spot checks and checks of period boundaries. Where claim records contain duplicate transactions or inconsistent status snapshots, resolve those issues before aggregation; otherwise, the triangle may faithfully sum an invalid input file.
Worked Example: Applying a Cutoff and Distinguishing Zero From Missing
A hypothetical insurer values an annual paid triangle at December 31, 2024. Claim F has accident year 2023 and a $500 payment dated January 2, 2024. Claim G also has accident year 2023 and payments of $300 dated November 1, 2023, and $200 dated December 31, 2024. Claim H has accident year 2024 and a $400 payment dated June 1, 2024. Claim I has accident year 2024 but no payment by the valuation date. A further $90 payment for claim H is dated January 5, 2025. Build the observed incremental and cumulative cells through the valuation date.
Method: apply the cutoff first, then assign included payments by accident year and payment year. The $90 payment dated January 5, 2025 is after the cutoff, so it is excluded. For accident year 2023, claim G contributes $300 at age 1. Claim F’s January 2, 2024 payment and claim G’s December 31, 2024 payment both belong to age 2, contributing \(500+200=700\). Thus, the 2023 incremental cells are $300 and $700, and the cumulative cells are $300 and \(300+700=1{,}000\).
For accident year 2024, claim H contributes $400 at age 1 and claim I contributes no payment. Because age 1 is observed, its cell is $400; claim I does not add to it. Age 2 for accident year 2024 is unobserved at the December 31, 2024 valuation, so it is blank rather than zero.
| Accident year | Incremental age 1 | Incremental age 2 |
|---|---|---|
| 2023 | $300 | $700 |
| 2024 | $400 | — |
| Accident year | Cumulative age 1 | Cumulative age 2 |
|---|---|---|
| 2023 | $300 | $1,000 |
| 2024 | $400 | — |
Check. The included payments total \(300+500+200+400=1{,}400\). The incremental cells also total \(300+700+400=1{,}400\). The post-cutoff $90 is absent from both totals. Claim I has no payment but does not create a separate, later-age observation.
Common mistake avoided: placing claim F’s payment in accident year 2024 because it was paid in 2024 confuses payment year with accident year. It remains in accident year 2023 and is assigned to that year’s age-2 development period.
Common Construction Errors
- Mixing date fields. Accident date determines accident year in an accident-year triangle; payment date determines the paid development period. A report date is not interchangeable with either.
- Mixing incremental and cumulative values. Cumulative cells are running totals of the incremental cells, not a second copy of the payment transactions.
- Summing reserve changes instead of snapshots. Incurred at a valuation uses cumulative paid plus case outstanding at that valuation.
- Filling the unobserved triangle with zeros. A future development age has not been observed; zero is appropriate only for a cell that is observed and has no amount under the stated data definition.
- Reconciling to the wrong source total. Compare the triangle with records filtered to the same cutoff, coverage, and loss basis. A full extract may contain post-valuation transactions that should not be included.
Check Your Understanding
For each problem, state the relevant assignment or calculation and check that the result follows the stated cutoff and cell definitions.
- A hypothetical claim has accident date November 20, 2021, and payments of $250 in 2021 and $150 in 2022. Under annual accident years and year-end valuations, what are its accident year and incremental development-age amounts? (Answer: Accident year 2021; age 1 is $250 and age 2 is $150.)
- For one accident year, incremental paid amounts at ages 1, 2, and 3 are $400, $0, and $250. Give the cumulative amounts and explain whether the age-2 zero is observed or structurally missing. (Answer: Cumulative amounts are $400, $400, and $650; age 2 is an observed zero as specified.)
- At one valuation, two claims in the same accident year have cumulative paid and case outstanding of $900 and $600, and $300 and $700, respectively. What is the incurred cell, and what component check confirms it? (Answer: $2,500; total paid is $1,200 and total case outstanding is $1,300, which sum to $2,500.)
- A triangle is valued at December 31, 2025. An accident-year 2024 claim has payments of $500 in 2024, $200 on December 31, 2025, and $80 on January 1, 2026. What are its incremental amounts through the valuation date, and what is the included total? (Answer: Age 1 is $500, age 2 is $200, and the included total is $700; the $80 payment is after the cutoff.)
- For accident year 2022, observed incremental cells are $600 at age 1 and $350 at age 2. The source records filtered to the same scope and cutoff total $1,000. What discrepancy remains, and what should be checked first? (Answer: The triangle totals $950, so it is short by $50; first check for an omitted, misassigned, or incorrectly filtered transaction before treating the source total as reconciled.)