Run-off triangles
Claims reserving as the exam asks it: a cumulative triangle, development factors from the column sums, and three reserve estimates that only agree when the pricing assumption matches the data. Edit any cell and everything recomputes.
Cumulative claims triangle (edit any known cell)
| Origin year | Dev 1 | Dev 2 | Dev 3 | Dev 4 |
|---|---|---|---|---|
| 2023 | ||||
| 2024 | 3,807 | |||
| 2025 | 3,960 | 4,165 | ||
| 2026 | 3,852 | 4,282 | 4,504 | |
| Dev factor | 1.3698 | 1.1117 | 1.0517 |
White cells are your data; purple cells are the chain ladder projection, each one the cell to its left times the column's development factor. Factors are weighted averages of the observed link ratios (column sums, the standard basic chain ladder).
Reserves by method
| Origin year | Chain ladder | Expected loss ratio | Bornhuetter-Ferguson |
|---|---|---|---|
| 2023 | 0 | 26 | 0 |
| 2024 | 187 | 35 | 180 |
| 2025 | 603 | 433 | 578 |
| 2026 | 1,692 | 1,523 | 1,628 |
| Total | 2,482 | 2,017 | 2,386 |
Chain ladder trusts the data completely; the expected loss ratio method trusts the pricing assumption completely (reserve = ELR × premium − paid to date, floored at zero); Bornhuetter-Ferguson splits the difference, keeping actual experience for the developed share and the ELR ultimate for the share still to come. Watch the recent origin years: that is where the methods disagree most, and why BF exists.
Three methods, one disagreement
The chain ladder assumes the future develops like the past, so a distorted latest diagonal (one big claim, a change in settlement speed) is grossed up all the way to ultimate. The expected loss ratio method ignores the claims data entirely, which is all you can do when an origin year is too green to trust. Bornhuetter-Ferguson blends them in exactly the proportion the triangle has developed, which is why it is the default answer for the most recent origin years. Try inflating the newest year's single known cell and watch the chain ladder panic while BF barely moves — and the expected loss ratio reserve not move at all, because it never looks at the claims data in the first place.
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Illustrative figures. For education only, not financial or reserving advice.