Purpose and Structure of an Adjusted Incurred Triangle
A Berquist-Sherman incurred adjustment puts case outstanding amounts on a consistent adequacy basis before incurred losses are used for development analysis. Once adjusted case outstanding amounts have been determined, the next task is to rebuild the incurred triangle correctly. This requires combining each adjusted outstanding balance with paid losses to date, while preserving the original paid amounts.
The triangle is organized by accident year and development age. Each observed cell represents cumulative experience for one accident year as of a particular valuation. A missing cell means that development age was not yet available at the valuation; it is not a zero.
Use the following notation for a single triangle cell: \(P\) is cumulative paid loss, \(R\) is reported case outstanding, \(R^*\) is adjusted case outstanding, \(I\) is reported incurred loss, and \(I^*\) is adjusted incurred loss. These are amounts at the same valuation and for the same accident year. The reported and adjusted incurred amounts are
The adjusted case outstanding amounts are treated here as inputs from the adequacy-adjustment process. The task is to use each one at its corresponding valuation, calculate adjusted incurred, and place the result in the proper triangle cell.
Rebuilding a Cell
The adjustment can also be expressed as a change to reported incurred. Subtract the reported incurred amount from adjusted incurred:
The paid amount cancels because it is identical in both calculations. Thus, the adjustment to incurred is exactly the change in case outstanding. This identity is useful as a check: an adjusted incurred amount must differ from reported incurred by the same amount as adjusted case outstanding differs from reported case outstanding.
Worked Example: Rebuilding Two Valuation Cells
For a hypothetical accident year, cumulative paid loss is $180,000 at 12 months and $310,000 at 24 months. Reported case outstanding is $120,000 and $80,000 at those ages, respectively. The adequacy adjustment has produced adjusted case outstanding of $150,000 at 12 months and $100,000 at 24 months. Calculate reported and adjusted incurred at each age, and verify that paid loss is unchanged.
Method. Calculate incurred separately at each valuation using cumulative paid plus the case outstanding amount applicable at that same valuation.
At 12 months: Reported incurred is \(180{,}000+120{,}000=300{,}000\). Adjusted incurred is \(180{,}000+150{,}000=330{,}000\). The adjustment is \(330{,}000-300{,}000=30{,}000\), which equals \(150{,}000-120{,}000=30{,}000\).
At 24 months: Reported incurred is \(310{,}000+80{,}000=390{,}000\). Adjusted incurred is \(310{,}000+100{,}000=410{,}000\). The adjustment is \(410{,}000-390{,}000=20{,}000\), which equals \(100{,}000-80{,}000=20{,}000\).
The paid amounts remain $180,000 and $310,000. The common mistake avoided here is adding the change in case outstanding to paid loss without first identifying the reported incurred amount; either route works only if the same valuation and the correct case outstanding amounts are used. The cell calculations reconcile: the changes in incurred match the changes in case outstanding.
From Cell Calculations to a Triangle
After calculating the cells, arrange them in a consistent triangle. Each row is an accident year; each column is a development age. Include a cell only when that accident year has reached the relevant age by the valuation date. The adjusted incurred amount for each observed cell is calculated from cumulative paid and adjusted case outstanding at that age.
The paid triangle used in the calculation is cumulative. If the source data give incremental paid amounts, first sum the increments from the start of the accident year through the relevant development age. By contrast, adjusted case outstanding is a balance at a valuation date. It is not a flow to be accumulated over development ages.
Confirm the accident year, development age, and valuation associated with each input amount.
If paid data are incremental, accumulate them through the cell’s development age. Do not change cumulative paid amounts that are already supplied.
For each observed cell, calculate adjusted incurred as cumulative paid plus adjusted case outstanding at that valuation.
Enter the adjusted incurred amount in the matching accident-year row and development-age column. Verify the paid component and, if reported amounts are available, the adjustment identity.
Worked Example: Completing a Cumulative Adjusted Incurred Triangle
A hypothetical insurer has the following cumulative paid losses and adjusted case outstanding amounts, in thousands of dollars. Rebuild the observed portion of the adjusted incurred triangle. Leave unavailable cells blank.
| Accident year | Age | Cumulative paid | Adjusted case outstanding |
|---|---|---|---|
| 2022 | 12 | 100 | 160 |
| 2022 | 24 | 220 | 90 |
| 2022 | 36 | 300 | 40 |
| 2023 | 12 | 120 | 180 |
| 2023 | 24 | 250 | 100 |
| 2024 | 12 | 140 | 210 |
Method. Add cumulative paid and adjusted case outstanding for each observed row of input. Amounts are in thousands, so the results are also in thousands.
For accident year 2022, the 12-month cell is \(100+160=260\). The 24-month cell is \(220+90=310\). The 36-month cell is \(300+40=340\).
For accident year 2023, the 12-month cell is \(120+180=300\), and the 24-month cell is \(250+100=350\). The 36-month cell is blank because it is not supplied. For accident year 2024, the 12-month cell is \(140+210=350\); the 24- and 36-month cells are not yet observed.
| Accident year | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2022 | 260 | 310 | 340 |
| 2023 | 300 | 350 | |
| 2024 | 350 |
The paid inputs appear in the calculations without adjustment. The common mistake avoided here is filling unavailable future cells with zero: those cells are unobserved, not known to have zero incurred losses. As a check, each displayed adjusted incurred cell equals its listed paid amount plus its listed adjusted case outstanding.
Incremental Paid Data and Changing Case Balances
A frequent source of error is confusing cumulative amounts with incremental amounts. Cumulative paid at 24 months already includes payments made during the first 12 months. Incremental paid at 24 months, by contrast, refers only to payments made during the 12-to-24-month interval. Convert incremental paid to cumulative paid before adding the adjusted outstanding balance.
Adjusted case outstanding is a balance at each valuation, not an amount paid during the interval. It should not be summed across ages. It is also possible for adjusted incurred to decrease from one age to the next: a decline in adjusted case outstanding may exceed the increase in cumulative paid. That result does not mean paid losses decreased; paid amounts must remain unchanged and cumulative paid cannot decrease.
Worked Example: Converting Incremental Paid Before Adding Case Outstanding
For a hypothetical accident year, incremental paid losses at 12, 24, and 36 months are $80,000, $90,000, and $70,000. Adjusted case outstanding at those valuations is $220,000, $140,000, and $60,000. A second accident year has incremental paid losses of $100,000 at 12 months and $110,000 at 24 months, with adjusted case outstanding of $260,000 and $170,000. Convert the paid amounts to cumulative amounts and build the observed adjusted incurred triangle, in thousands.
First accident year. Cumulative paid is 80 at 12 months, \(80+90=170\) at 24 months, and \(80+90+70=240\) at 36 months. Adjusted incurred is therefore \(80+220=300\), \(170+140=310\), and \(240+60=300\).
Second accident year. Cumulative paid is 100 at 12 months and \(100+110=210\) at 24 months. Adjusted incurred is \(100+260=360\) and \(210+170=380\). The 36-month cell is unavailable.
| Accident year | 12 months | 24 months | 36 months |
|---|---|---|---|
| First accident year | 300 | 310 | 300 |
| Second accident year | 360 | 380 |
For the first accident year, adjusted incurred falls from 310 to 300 between 24 and 36 months even though cumulative paid rises from 170 to 240. The adjusted case outstanding balance falls by 80, while paid rises by 70, so adjusted incurred changes by \(70-80=-10\). The common mistake avoided is treating the 36-month case balance as an incremental payment or forcing incurred to increase. The calculations preserve cumulative paid and use each case balance only at its own valuation.
Checks and Common Traps
Before using the rebuilt triangle for development analysis, check that each amount is attached to the correct accident year and age, and that the paid basis is consistent across cells. In particular:
- Keep the paid component unchanged. Use the supplied paid amounts, converting incremental to cumulative only when necessary.
- Use the adjusted case balance for that valuation. Do not carry a case outstanding amount from one age into another unless the data explicitly specify it for that cell.
- Do not accumulate case outstanding across ages. Each amount is a balance as of its own valuation date.
- Leave unobserved cells blank. A future age is unavailable, not a zero-loss observation.
- Do not impose monotonicity on adjusted incurred. Cumulative paid is nondecreasing, but adjusted incurred can decrease when the adjusted case balance falls sufficiently.
If reported case outstanding and reported incurred are available, an additional cell check is \(I^*-I=R^*-R\). If only paid and adjusted case outstanding are supplied, check directly that every adjusted incurred cell is their sum. This construction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus; the focus here is the triangle itself, not selecting development factors or projecting ultimate losses.
Check Your Understanding
For each problem, show the cumulative paid basis before forming adjusted incurred. Amounts are in thousands unless stated otherwise.
- A cell has cumulative paid of 245 and adjusted case outstanding of 135. Calculate adjusted incurred. (Answer: 380 thousand)
- At 12 and 24 months, cumulative paid is 90 and 210, while adjusted case outstanding is 170 and 95. Calculate the two adjusted incurred cells. (Answer: 260 thousand at 12 months; 305 thousand at 24 months)
- Incremental paid at 12, 24, and 36 months is 60, 85, and 55. Adjusted case outstanding at those ages is 190, 120, and 45. Calculate cumulative paid and adjusted incurred at each age. (Answer: Cumulative paid is 60, 145, and 200; adjusted incurred is 250, 265, and 245 thousand)
- A cell has cumulative paid of 500, reported case outstanding of 140, and adjusted case outstanding of 112. Calculate reported and adjusted incurred, then check the change. (Answer: Reported incurred is 640; adjusted incurred is 612; the change is \(-28\) thousand, equal to \(112-140\))
- A triangle has these inputs: accident year 2021 has cumulative paid of 130 and adjusted case outstanding of 190 at 12 months, and cumulative paid of 250 and adjusted case outstanding of 80 at 24 months. Accident year 2022 has cumulative paid of 150 and adjusted case outstanding of 210 at 12 months. Rebuild the observed adjusted incurred triangle and show unavailable cells as blank. (Answer: Accident year 2021: 320 at 12 months and 330 at 24 months; accident year 2022: 360 at 12 months; later cells are blank)