Introduction
A triangle row is a record of how the experience for one origin period appears at successive development ages. Reading the row from left to right means moving from an earlier valuation of that period’s losses to a later valuation. The period being observed stays fixed; the cells are not new origin periods.
This distinction matters when interpreting a loss triangle. A value at a later development age is a later observation of the same origin-period experience, not a separate year’s experience. The meaning of the change depends on the triangle basis: cumulative paid amounts, incremental paid amounts, and incurred amounts each describe the row differently.
Keep the Origin Period Fixed
A useful way to read a triangle is to name both the row and the column for every cell. Let \(T_{i,j}\) denote the cell for origin period \(i\) at development age \(j\). For example, \(T_{2021,24}\) is the value for origin period 2021 at development age 24 months. Moving from \(T_{2021,12}\) to \(T_{2021,24}\) changes the development age, not the origin period.
In an annual triangle, the development ages might be 12, 24, and 36 months. Under a stated year-end valuation convention, these ages correspond to successive valuation dates for a particular origin period. The precise date convention must be read from the triangle’s definitions; do not infer dates from the age labels alone if the convention is unknown.
| Origin period | Development age | Meaning of the cell |
|---|---|---|
| 2021 | 12 months | Origin-period 2021 experience observed at age 12 months |
| 2021 | 24 months | The same origin-period experience observed at age 24 months |
| 2021 | 36 months | The same origin-period experience observed at age 36 months |
The row does not imply that every claim occurred at the same time or that every amount was known immediately. It organizes the observed results for one origin period according to how developed they were at each valuation. A later cell can reflect additional payments, updated estimates, or both, depending on the triangle’s measure.
Worked Example: Mapping a Cumulative Paid Row to Valuations
A hypothetical annual accident-year triangle uses year-end valuations and reports cumulative paid losses in thousands of dollars. For accident year 2021, the row is 180 at 12 months, 310 at 24 months, 390 at 36 months, and 430 at 48 months. Identify the successive observations and calculate the paid amounts added in each development interval.
Method. Keep accident year 2021 fixed, then read the development ages in order. Under the stated year-end convention, the ages correspond to December 31, 2021; December 31, 2022; December 31, 2023; and December 31, 2024.
The cumulative paid amount at 12 months is 180 thousand dollars. At 24 months, the cumulative amount for that same accident year is 310 thousand dollars. The interval amount from 12 to 24 months is therefore \(310-180=130\) thousand dollars.
From 24 to 36 months, the amount added is \(390-310=80\) thousand dollars. From 36 to 48 months, it is \(430-390=40\) thousand dollars.
The row’s interval amounts are therefore 180, 130, 80, and 40 thousand dollars, where the first amount is paid through 12 months and each later amount is paid during the interval after the previous age. The check is \(180+130+80+40=430\), which agrees with the cumulative paid value at 48 months.
Interpretation. All four cumulative observations concern accident year 2021. The value 430 at 48 months is not a separate 2024 accident year; it is the accident-year 2021 cumulative paid amount observed at a later valuation.
What Changes Along a Row?
The interpretation of a row comparison depends on the loss measure. In a cumulative paid triangle, each cell is the total paid to date for the origin period, as of that development age. In an incremental paid triangle, each cell is the amount paid during a particular development interval. A cumulative incurred triangle records the incurred amount at each valuation, so a later value can be higher or lower than an earlier value as estimates are revised.
This relationship assumes that the cumulative and incremental values use the same origin-period definition, coverage, currency, and paid-loss basis, and that the incremental intervals are consecutive and non-overlapping. The first incremental value is the amount paid from the start of the origin period through the first stated age. If those conditions do not hold, subtraction may not produce a meaningful interval amount.
The justification follows directly from the definition of cumulative paid loss. The total paid through age \(j\) consists of the amounts paid in each interval up to and including age \(j\). The total through age \(j-1\) includes all those same earlier intervals except the interval ending at \(j\). Subtracting cancels the shared intervals and leaves only \(I_{i,j}\).
For paid losses, cumulative values generally do not decrease when they record positive payments and no recoveries or other offsets. Incurred values are different. Incurred loss includes paid amounts and estimates of amounts still outstanding; when an estimate changes, a later incurred value can be lower than an earlier one. A decrease in incurred values is therefore not, by itself, evidence that the row has changed origin periods.
Worked Example: Interpreting a Decrease in an Incurred Row
A hypothetical report-year triangle records cumulative incurred losses in thousands of dollars. For report year 2023, the values are 250 at 12 months, 220 at 24 months, and 240 at 36 months. State what the row represents and calculate the changes between successive observations.
Method. Keep report year 2023 fixed. The three values are incurred observations for that report year at increasing development ages; they are not three report years.
From 12 to 24 months, the change is \(220-250=-30\) thousand dollars. The incurred amount decreased by 30 thousand dollars. From 24 to 36 months, the change is \(240-220=20\) thousand dollars. The incurred amount increased by 20 thousand dollars.
The 36-month incurred amount is 240 thousand dollars for report year 2023. It is 10 thousand dollars below the 12-month amount, since \(240-250=-10\). This net change agrees with the two interval changes: \(-30+20=-10\).
Interpretation. The row records successive incurred estimates for the same report-year experience. Incurred values can change when estimates of outstanding losses are revised, so the decrease is possible. These differences describe changes in the incurred measure, not amounts of cash paid during the intervals.
Check. The successive changes sum to the overall change: \(-30+20=-10=240-250\). The arithmetic is consistent.
A Short Row-Reading Procedure
Use a consistent sequence whenever a problem asks you to interpret one row. This keeps the origin period, development age, valuation, and measure from being confused.
Identify the origin period and its type, such as accident year or report year. Keep that period fixed throughout the comparison.
Determine which cell is earlier and which is later. Use the stated valuation convention before assigning calendar dates.
Check whether the row is cumulative paid, incremental paid, or incurred. The same numerical change does not have the same interpretation for every measure.
If calculating a change, verify that the cells use the same origin-period definition and loss basis. For cumulative paid values, subtract the earlier cumulative amount from the later one to find the interval amount.
Say which origin period was observed, at which ages, and what the difference means for that measure.
Worked Example: Converting Incremental Values Within One Row
A hypothetical accident-year triangle gives incremental paid losses, in thousands of dollars, for accident year 2022: 95 through 12 months, 70 from 12 to 24 months, and 35 from 24 to 36 months. Calculate the cumulative paid amount at each age and interpret the 36-month value.
Method. These are incremental amounts, so add the intervals in chronological order while keeping accident year 2022 fixed.
Through 12 months, cumulative paid is \(95\) thousand dollars. Through 24 months, it is \(95+70=165\) thousand dollars. Through 36 months, it is \(95+70+35=200\) thousand dollars.
The cumulative row is therefore 95, 165, and 200 thousand dollars at 12, 24, and 36 months, respectively. As a check, the consecutive cumulative differences are \(165-95=70\) and \(200-165=35\), matching the second and third incremental amounts.
Interpretation. At 36 months, 200 thousand dollars had been paid cumulatively for accident year 2022 under the hypothetical triangle’s stated basis. The 35 thousand dollars is the amount paid in the interval from 24 to 36 months, not a new origin-period total.
Common Mistake: Treating Cells as Separate Origin Periods
A frequent error is to read the development age as though it identified another origin period. For example, suppose a row labelled accident year 2021 has cumulative paid amounts of 180 at 12 months and 310 at 24 months. A wrong interpretation is: “180 belongs to 2021, and 310 belongs to 2022.” That changes the row’s meaning and loses the connection between the observations.
The correct reading is: “180 is the accident-year 2021 cumulative paid amount at 12 months; 310 is the accident-year 2021 cumulative paid amount at 24 months.” The interval paid amount is \(310-180=130\), not 310. The later age indicates a later valuation of the same origin period.
Another trap is to treat every difference as a payment. That interpretation applies to differences between cumulative paid amounts when the data have a consistent basis. A difference between incurred amounts is a change in the incurred measure, which can reflect estimate revisions and need not equal cash paid.
Check Your Understanding
For each problem, identify the row’s meaning before calculating or interpreting its values.
- A hypothetical accident-year cumulative paid row, in thousands of dollars, is 140 at 12 months, 205 at 24 months, and 260 at 36 months. Find the amounts paid from 12 to 24 months and from 24 to 36 months, and state the origin period represented by all three cells. (Answer: 65 thousand dollars and 55 thousand dollars; all cells represent the same accident year.)
- A hypothetical report-year incurred row is 410, 395, and 430 thousand dollars at 12, 24, and 36 months. Calculate each successive change and say whether the decrease from 12 to 24 months means that the origin period changed. (Answer: a decrease of 15 thousand dollars, then an increase of 35 thousand dollars; no, both cells remain for the same report year.)
- An incremental paid row for a hypothetical origin period has amounts of 72, 48, and 30 thousand dollars through 12 months, from 12 to 24 months, and from 24 to 36 months. Find the cumulative paid amounts at all three ages and check the final total using differences. (Answer: 72, 120, and 150 thousand dollars; the differences are 48 and 30 thousand dollars.)
- A hypothetical annual triangle states that origin period 2020 is valued at year-end and shows cells at 12, 24, and 36 months. Assign the valuation dates and describe what changes as you move from the first cell to the third. (Answer: December 31, 2020; December 31, 2021; and December 31, 2022; the valuation age changes while origin period 2020 stays fixed.)
- A cumulative paid row for one hypothetical origin period is 90, 160, 210, and 245 thousand dollars at 12, 24, 36, and 48 months. Find the paid amount from 24 to 48 months and the amount from 36 to 48 months. (Answer: 85 thousand dollars from 24 to 48 months and 35 thousand dollars from 36 to 48 months.)