Why Triangle Totals Depend on the Grouping
A loss triangle can be totaled in more than one direction. Summing across an origin-period row answers how much loss is recorded for that origin period across the development ages shown. Summing along a calendar-year diagonal answers how much loss activity is recorded in that calendar year across the origin periods represented. These totals use the same cells but answer different questions.
The distinction is especially important when the triangle contains incremental amounts. Each incremental cell belongs to one origin period and one calendar year, so its position determines which row total and which diagonal total include it. A total is meaningful only after the grouping has been identified.
For example, the 2042 diagonal contains the age-1 cell for origin year 2042, the age-2 cell for origin year 2041, and the age-3 cell for origin year 2040, if those cells are present. By contrast, the 2040 origin-period total contains the age-1, age-2, and later cells in the 2040 row. The diagonal crosses rows; the origin-period total stays within one row.
Locating the Cells Before Adding
For an annual triangle, development age \(j\) places an origin year \(i\) in calendar year \(i+j-1\). Equivalently, for a specified calendar year \(k\), the cell from origin year \(i\) is at development age \(k-i+1\). The condition is useful in either direction: start with a cell and determine its calendar year, or start with a calendar year and identify its contributing cells.
A reliable calculation begins by writing down the requested grouping and listing the cells that belong to it. Then add only those cells, using incremental amounts for calendar-year activity. If the question requests an origin-period amount from cumulative data, use the last cumulative value available for that row as the total through that age; do not add the cumulative values together.
Decide whether the question asks for a calendar-year diagonal or an origin-period row.
For an annual diagonal, use \(i+j-1=k\). For an origin-period total, stay in the specified row.
Sum incremental values to measure activity in a calendar year. For a cumulative row total, use its latest stated cumulative value, which already includes earlier increments.
Confirm that every included cell belongs to the requested grouping and that no cell has been counted twice.
Worked Example: Comparing Row Totals With Diagonal Totals
A hypothetical annual incremental paid-loss triangle, in thousands of dollars, contains the following values. The listed cells are the observed cells for this example.
| Origin year | Age 1 | Age 2 | Age 3 |
|---|---|---|---|
| 2040 | 40 | 15 | 5 |
| 2041 | 50 | 20 | — |
| 2042 | 60 | — | — |
Find the origin-period total for each row and the calendar-year total for 2042. Also check the total of all listed incremental amounts by adding the row totals and then by adding the diagonal totals.
Method. First calculate each row total. For 2040, \(40+15+5=60\) thousand dollars. For 2041, the listed row total is \(50+20=70\) thousand dollars. For 2042, it is \(60\) thousand dollars. These are totals of the incremental amounts shown in each origin row.
For calendar year 2042, apply \(i+j-1=2042\). The matching cells are origin 2040 at age 3, origin 2041 at age 2, and origin 2042 at age 1. Their incremental amounts are \(5\), \(20\), and \(60\), respectively. Therefore, the 2042 diagonal total is \(5+20+60=85\) thousand dollars.
To check all listed amounts using diagonals, the 2040 diagonal contains only \(40\), for a total of \(40\). The 2041 diagonal contains \(15+50=65\). The 2042 diagonal is \(85\). Thus the diagonal totals sum to \(40+65+85=190\) thousand dollars. The row totals also sum to \(60+70+60=190\) thousand dollars.
Check. Both ways of grouping the same listed incremental cells give \(190\) thousand dollars overall. The 2042 diagonal total, \(85\), is not the 2042 origin-period row total, \(60\): the diagonal includes amounts from three different origin years.
Common mistake avoided: Adding every cell in the 2042 origin-year row to answer the 2042 calendar-year question would give \(60\), not \(85\). That calculation uses the right year label but the wrong grouping.
Why the Two Ways of Summing Reconcile
A cell in an annual incremental triangle has one origin-period row and one calendar-year diagonal. It is therefore counted once in the sum across rows and once in the sum across diagonals. This gives a useful check when both sets of totals cover the same observed cells.
The equality follows by regrouping a finite sum: the middle expression lists each included cell once. Grouping those cells by row produces the row totals; grouping the same cells by calendar year produces the diagonal totals. This check does not imply that an individual row total equals an individual diagonal total. It also does not establish that unlisted or unobserved cells have value zero; the equality applies to the same set of cells on both sides.
Worked Example: Finding a Diagonal From Incremental and Cumulative Values
A hypothetical triangle has these incremental paid losses, in thousands of dollars: origin 2050 has \(30,12,8\) at ages 1, 2, and 3; origin 2051 has \(40,15\) at ages 1 and 2; origin 2052 has \(50\) at age 1. Find the 2051 diagonal total. Then determine what happens if the cumulative cells on that diagonal are added instead.
Method. The 2051 diagonal consists of cells for which \(i+j-1=2051\). These are origin 2050 at age 2 and origin 2051 at age 1. The relevant incremental amounts are \(12\) and \(40\), so the 2051 calendar-year total is \(12+40=52\) thousand dollars.
To form the cumulative values, add increments within each row. For origin 2050, the cumulative values are \(30\), \(30+12=42\), and \(30+12+8=50\). For origin 2051, they are \(40\) and \(40+15=55\). Thus the cumulative cells located on the 2051 diagonal are \(42\) and \(40\). Adding them gives \(42+40=82\), which is not the 2051 incremental calendar-year total.
The reason is that the cumulative value \(42\) at origin 2050, age 2, includes both the 2050 age-1 increment of \(30\) and the 2051 calendar-year increment of \(12\). The 2050 amount belongs to an earlier calendar year, so the cumulative diagonal sum includes activity outside 2051. The 2051 diagonal total must be calculated from incremental amounts, or by extracting the increments from cumulative values.
Check. The row totals are \(50\), \(55\), and \(50\), summing to \(155\). The diagonal totals are \(30\) for 2050, \(12+40=52\) for 2051, and \(8+15+50=73\) for 2052. They also sum to \(30+52+73=155\).
Common mistake avoided: A diagonal position tells which calendar-year grouping a cell occupies, but a cumulative cell still contains earlier development amounts. The position does not turn cumulative values into calendar-year increments.
When the Requested Diagonal Is Only Partly Specified
A requested diagonal may include a cell whose value is not supplied. In that case, calculate the subtotal from the known cells and state clearly that the full diagonal total cannot be determined from the given information. An unavailable value is not automatically a zero. This is a matter of keeping the total tied to the information actually provided.
Worked Example: Reporting a Known Diagonal Subtotal
A hypothetical incremental triangle gives origin 2045 values of \(30\) at age 1 and \(10\) at age 2; origin 2046 values of \(20\) at age 1 and \(5\) at age 2; and origin 2047 value \(15\) at age 1. The origin 2045 age-3 value is not supplied. What can be concluded about the 2047 diagonal? Also compare the answer with the sum of the age-2 cumulative cells for origins 2045 and 2046.
Method. For calendar year 2047, the age-1 cell from origin 2047 contributes \(15\), and the age-2 cell from origin 2046 contributes \(5\). The age-3 cell from origin 2045 also belongs on the 2047 diagonal, because \(2045+3-1=2047\), but its value is not given.
The known diagonal subtotal is therefore \(5+15=20\) thousand dollars. If the unprovided age-3 amount is denoted by \(x_{2045,3}\), the full diagonal total is \(20+x_{2045,3}\) thousand dollars. Since its value is unknown, the full diagonal total cannot be determined.
The age-2 cumulative value for origin 2045 is \(30+10=40\). For origin 2046, it is \(20+5=25\). Their sum is \(40+25=65\) thousand dollars. This is not the 2047 incremental diagonal total: the first cumulative value includes the origin 2045 age-1 increment, and the second includes the origin 2046 age-1 increment.
Check. The stated information supports a known 2047 diagonal subtotal of \(20\) thousand dollars, plus an unknown contribution from origin 2045 at age 3. It does not support a full diagonal total of \(20\), nor does the cumulative-cell sum of \(65\) answer the calendar-year activity question.
Common mistake avoided: Do not silently treat an unspecified diagonal cell as zero. Report the subtotal and identify the additional value needed to calculate the full total.
Interpretation and Exam Traps
The name of a requested total signals the direction of aggregation. A calendar-year total describes the incremental amounts assigned to one calendar year across origin periods. An origin-period total describes the incremental amounts within one row across development ages. Neither total should be substituted for the other merely because both use dollar amounts from the same triangle.
| Question asks for | Cells to group | Appropriate values |
|---|---|---|
| Calendar-year activity | One diagonal across origin periods | Incremental amounts |
| Origin-period amount over listed ages | One row across development ages | Incremental amounts summed within the row |
| Cumulative amount through an age for one origin period | One cumulative cell in that row | The stated cumulative value |
A useful final check is to ask whether every included cell has the correct origin year and development age. For an annual triangle, substitute them into \(i+j-1\). Then confirm that a calendar-year total is based on incremental values. If the question supplies only some diagonal cells, distinguish the known subtotal from the full total.
Check Your Understanding
For each problem, identify the requested grouping before calculating. All amounts are hypothetical and stated in thousands of dollars.
- Origin 2048 has incremental amounts \(22,11,4\) at ages 1, 2, and 3. Origin 2049 has \(30,14\) at ages 1 and 2, and origin 2050 has \(35\) at age 1. Find the 2049 calendar-year diagonal total and the 2049 origin-period row total. (Answer: 2049 diagonal \(=11+30=41\); 2049 origin-period total \(=30+14=44\) thousand dollars.)
- A hypothetical cumulative triangle has origin 2055 values \(18,29,37\) at ages 1, 2, and 3, and origin 2056 values \(25,39\) at ages 1 and 2. Find the 2056 incremental diagonal total. (Answer: \((29-18)+(25)=36\) thousand dollars.)
- Origin 2060 has increments \(16,9,6\) at ages 1, 2, and 3; origin 2061 has increments \(21,12\) at ages 1 and 2; origin 2062 has increment \(28\) at age 1. Calculate the 2062 diagonal total and check it against the sum of all row totals and all diagonal totals. (Answer: 2062 diagonal \(=6+12+28=46\); row totals sum to \(31+33+28=92\), and diagonal totals \(16+30+46\) also sum to \(92\) thousand dollars.)
- Origin 2070 has increments \(24\) and \(13\) at ages 1 and 2; origin 2071 has increments \(31\) and \(8\) at ages 1 and 2; origin 2072 has increment \(27\) at age 1. The origin 2070 age-3 value is not supplied. State the known subtotal for the 2072 diagonal and explain whether its full total is determined. (Answer: Known subtotal \(=8+27=35\) thousand dollars; the full total is not determined because origin 2070 at age 3 also belongs to that diagonal.)
- A cumulative triangle has origin 2080 values \(20,32,41\) at ages 1, 2, and 3; origin 2081 values \(26,40\) at ages 1 and 2; and origin 2082 value \(33\) at age 1. Find the 2081 incremental diagonal total, the 2081 origin-period incremental total, and the sum of the cumulative cells on the 2081 diagonal. Explain why the last value is not the diagonal activity. (Answer: Diagonal \(=(32-20)+26=38\); origin 2081 total \(=40\); cumulative diagonal-cell sum \(=32+26=58\) thousand dollars. The cumulative sum includes the earlier origin 2080 age-1 amount.)