Economic Modelling · Run-off triangles

Run-off Triangles and Claims Data Basics Explained

Updated 11 October 2026 · Fact-checked

A run-off triangle is a table of claims data. Rows are origin periods (when claims arose), columns are development periods (time since origin). Cells hold incremental or cumulative paid amounts, incurred amounts or claim counts. Only past cells are known, giving a triangle. You set it up by grouping each claim by origin and development period.

Understand Run-off Triangles and Claims Data Basics

A general insurer does not know its final claims cost on the day a policy year ends. Claims are reported late, settled slowly and revised along the way. To estimate the final cost, actuaries track how claims for each period grow over time. The run-off triangle is the standard table for this.

The rows are origin periods. These are the periods in which the claims arose. Depending on the basis, this can be accident year, underwriting year or reporting year. The columns are development periods: the time elapsed since the start of the origin period, such as year 0, year 1, year 2. Each cell shows the amount for that origin period at that development period.

The table is a triangle because only past data exists. The earliest origin period has the most development periods observed. The latest origin period has only one. The cells fall along diagonals, and each diagonal is one calendar period (the period in which the payment was actually made or the position valued). This matters because inflation and changes in claims handling act along calendar diagonals.

There are two layouts. Incremental data gives the amount arising in each development period alone. Cumulative data gives the running total up to that development period. Cumulative = sum of incremental across the row. Most reserving methods, such as the chain ladder, work on cumulative data.

There are also different data types. Paid claims are the amounts actually paid. Incurred claims are paid amounts plus case estimates (the claims handlers' estimates of outstanding amounts on reported claims). Claim numbers (counts of reported or settled claims) are often used alongside amounts. Paid data is objective but slow to develop. Incurred data is more up to date but depends on how prudent the case estimates are, which can change over time.

Key rules to remember

Cumulative from incremental: C(i, j) = X(i, 0) + X(i, 1) + ... + X(i, j); i is the origin period, j the development period, X the incremental amount. Cumulative values never fall for paid data.
Incremental from cumulative: X(i, j) = C(i, j) − C(i, j−1), with X(i, 0) = C(i, 0); Differences along each row. Do not difference down a column.
Incurred claims: Incurred = Paid + Case estimates of outstanding claims; Excludes IBNR, which is the reserve for claims not yet reported plus allowance for further development.
Calendar period of a cell: Calendar period = origin period + development period; With both numbered from 0. Cells on the same diagonal share a calendar period.
Number of known cells in a run-off triangle: n(n + 1) ÷ 2; For n origin periods and n development periods. For n = 4 this gives 10 known cells.
Outstanding reserve (basic idea): Reserve = Ultimate cost − Paid to date; The triangle gives the paid to date. Reserving methods estimate the ultimate.

How to solve Run-off Triangles and Claims Data Basics questions

Use this order for any question that asks you to build, convert or interpret a claims triangle.

1Identify the basis of the origin period (accident, underwriting or reporting) and the length of the periods, such as years or quarters.
2Identify the data type: paid, incurred or claim numbers. Note whether the figures are incremental or cumulative.
3Place each claim or amount in the correct row (origin) and column (development period). Count development from 0 unless the question says otherwise.
4Convert between incremental and cumulative along the rows if the question needs it. Check each row total against the latest cumulative figure.
5Mark the latest diagonal. This is the current position and the paid or incurred to date for each origin period.
6Answer the question asked: build the table, read a value, compare paid and incurred, or comment on features such as late settlement or changing case estimates.
7State your assumptions, such as no inflation adjustment, no change in claims handling and all figures in the same units.

Quickest way: Row-by-row differencing and diagonal check

When to use it: Use when you are given one form of the triangle and need the other, or must spot an error in the data.

Work one row at a time from left to right. Add to go to cumulative, subtract to go to incremental.
Check that the last cumulative figure in each row equals the sum of that row's incremental figures.
For paid data, check that no cumulative figure falls. A fall signals an error or a recovery such as salvage or subrogation.
Shade the latest diagonal and check the number of known cells matches n(n + 1) ÷ 2.
Write units and the data type beside the table so you do not mix them.

Common mistakes in Run-off Triangles and Claims Data Basics

Mixing up origin period and calendar period.
Both are dates, and students read the rows as the year of payment.
Fix: Rows are when the claim arose. Calendar period is origin plus development, shown by the diagonals.
Differencing down the columns instead of along the rows.
Students are used to column subtraction in other tables.
Fix: Incremental and cumulative conversion is always within one origin row, across development periods.
Treating incurred claims as the same as the ultimate cost.
Incurred sounds like the final figure.
Fix: Incurred is paid plus case estimates only. It excludes IBNR and can still change as estimates are revised.
Saying paid and incurred triangles are equally reliable at early development periods.
Both look like claims cost.
Fix: Paid is objective but slow, so early values understate the ultimate. Incurred is more up to date but depends on case estimate strength.
Starting development periods at 1 when the question starts at 0, or the reverse.
Conventions differ between questions.
Fix: Read the headings. Label the first column exactly as given and keep it consistent.
Ignoring the basis of the origin period when comparing triangles.
Students assume all triangles use accident year.
Fix: State the basis. Underwriting-year and reporting-year triangles behave differently. On a reporting-year basis, the claims in a row are by definition already reported, so there is no IBNR for late reporting within that row, though further development on those claims is still possible.

Worked examples

Example 1

An insurer's incremental paid claims (₹ lakh) by accident year are: 2022: 40, 30, 10. 2023: 50, 36. 2024: 60. Development periods are years 0, 1, 2. Construct the cumulative paid triangle and state the paid to date for each accident year.

Show the solution

Row 2022: cumulative = 40, then 40 + 30 = 70, then 70 + 10 = 80.
Row 2023: cumulative = 50, then 50 + 36 = 86.
Row 2024: cumulative = 60.
The latest diagonal is 80 (2022, year 2), 86 (2023, year 1) and 60 (2024, year 0).
Check: the triangle has 3 × 4 ÷ 2 = 6 cells, and we have 3 + 2 + 1 = 6.

Answer: Cumulative paid (₹ lakh): 2022: 40, 70, 80. 2023: 50, 86. 2024: 60. Paid to date: ₹80 lakh, ₹86 lakh and ₹60 lakh respectively.

Example 2

For accident year 2023, cumulative paid claims at development years 0, 1 and 2 are ₹50 lakh, ₹86 lakh and ₹100 lakh. Case estimates for outstanding claims at the same dates are ₹70 lakh, ₹30 lakh and ₹12 lakh. Find the cumulative incurred claims at each date and the incremental incurred amounts. Comment briefly.

Show the solution

Incurred = paid + case estimates.
Year 0: 50 + 70 = 120.
Year 1: 86 + 30 = 116.
Year 2: 100 + 12 = 112.
Incremental incurred: year 0 = 120. Year 1 = 116 − 120 = −4. Year 2 = 112 − 116 = −4.
Incurred falls over time. This suggests the early case estimates may have been more than needed, that is, over-prudent. Other causes are possible, such as recoveries, claims closed without payment, or a change in case-estimate practice. One accident year is limited evidence, so you would want to see other years before concluding. Cumulative paid rises (50, 86, 100) and never falls, so paid data would not show this directly.

Answer: Cumulative incurred is ₹120 lakh, ₹116 lakh and ₹112 lakh. Incremental incurred is ₹120 lakh, −₹4 lakh and −₹4 lakh. The fall suggests early case estimates may have been over-prudent, but recoveries or a change in case-estimate practice could also explain it, and one accident year is limited evidence. Incurred data needs care when used for reserving. Cumulative paid (₹50 lakh, ₹86 lakh, ₹100 lakh) only rises, so it would not show this directly.

Exam tips

Write the headings on every triangle you draw: origin basis, development period, paid or incurred, incremental or cumulative.
Always check one row total after converting. It takes seconds and catches most arithmetic slips.
For comment questions, give a point on each of paid, incurred and claim numbers: speed, objectivity and what can distort them.
Name the assumptions the triangle needs, such as stable settlement pattern and no inflation change, because later method questions build on them.
In computer-based questions, keep the triangle in a clear grid with blank cells for future periods so formulas do not pick up zeros.

Practice questions from Run-off triangles

Run-off Triangles and Claims Data Basics: frequently asked questions

What is a run-off triangle in general insurance?

It is a table of claims data with origin periods as rows and development periods as columns. It shows how claims for each origin period have developed so far. Only past cells are filled, so the table forms a triangle.

What is the difference between paid and incurred claims triangles?

A paid triangle contains only amounts actually paid. An incurred triangle adds case estimates for outstanding reported claims. Incurred is more up to date, but it depends on how strong the case estimates are.

What is the difference between incremental and cumulative triangles?

Incremental shows the amount in each development period alone. Cumulative is the running total along each row. You convert between them by adding or subtracting along the rows.

Why is the data arranged by origin period?

Grouping by origin period lets you compare how similar blocks of business develop over time. This pattern is then used to estimate the future development of recent periods. It is the basis of reserving methods like the chain ladder.