Basic Chain Ladder
Basic Chain Ladder
The deterministic chain ladder estimates how a cumulative triangle will develop if historical age-to-age patterns continue.
1. Start with cumulative observations
Let be cumulative paid loss for origin year at development age .
| AY | Dev 0 | Dev 1 | Dev 2 | Dev 3 |
|---|---|---|---|---|
| 2020 | 100 | 180 | 240 | 280 |
| 2021 | 120 | 220 | 300 | — |
| 2022 | 140 | 260 | — | — |
| 2023 | 160 | — | — | — |
Units are £m. The triangle is cumulative, paid, nominal, gross of reinsurance, and valued at year-end 2023.
2. Estimate age-to-age factors
The usual volume-weighted factor is a ratio of column sums over rows for which both ages are observed:
For the example:
| Transition | Calculation | Factor | Data supporting it |
|---|---|---|---|
| 0 → 1 | 1.8333 | 3 origin years | |
| 1 → 2 | 1.3500 | 2 origin years | |
| 2 → 3 | 1.1667 | 1 origin year |
Volume weighting gives larger cells more influence. It is not automatically superior to an arithmetic or credibility-weighted selection; it is a modelling choice whose sensitivity should be shown.
3. Convert link ratios into cumulative development factors
The cumulative development factor (CDF) from age to ultimate age is
where is the implied percentage paid by age .
| Latest age | CDF to Dev 3 | Implied paid proportion |
|---|---|---|
| 0 | 34.63% | |
| 1 | 63.49% | |
| 2 | 85.71% | |
| 3 | 100% |
A “100%” at Dev 3 is an assumption of this example. Real long-tailed business generally requires a selected tail factor beyond the observed triangle.
4. Project ultimate loss and reserve
If the latest observed value for origin year is ,
| AY | Latest age | Latest paid | CDF | Ultimate | Reserve |
|---|---|---|---|---|---|
| 2020 | 3 | 280.0 | 1.0000 | 280.0 | 0.0 |
| 2021 | 2 | 300.0 | 1.1667 | 350.0 | 50.0 |
| 2022 | 1 | 260.0 | 1.5750 | 409.5 | 149.5 |
| 2023 | 0 | 160.0 | 2.8875 | 462.0 | 302.0 |
| Total | 1,000.0 | 1,501.5 | 501.5 |
Why the youngest year dominates
AY 2023 supplies only £160m of observed paid loss yet contributes £302m of reserve. It is both the least mature and the most leveraged by assumptions: a 5% change in its selected CDF changes its ultimate by about £23.1m. This is why reserve review should focus on influence, not merely on the number of rows.
5. Reproduce the calculation
Chain-ladder calculation and factor sensitivity
6. State the assumptions as testable claims
The chain ladder does not merely assume “the past repeats.” It requires more precise claims:
| Assumption | Observable warning sign | Response |
|---|---|---|
| comparable origin years | exposure or mix changes sharply | on-level, segment, or use an exposure-based method |
| stable development by age | link ratios trend by origin year | investigate operations, inflation, or model calendar effects |
| no unmodelled calendar shock | residuals align on diagonals | isolate shock or use a model with calendar terms |
| adequate volume | one cell controls a factor | use credibility, benchmarks, or a range |
| latest cumulative value is meaningful | large recoveries/reopenings | reconcile and model the mechanism |
| selected tail is adequate | oldest years still develop | fit or benchmark a tail factor |
7. Three quick diagnostics
Leave-one-origin-year-out
Re-estimate each factor after removing one eligible origin year. Large movement identifies influential observations; it does not automatically justify deleting them.
Backtest a historical diagonal
Pretend the latest diagonal was unavailable, project it using only older data, and compare projection with what actually emerged. Repeat across diagonals where possible.
Compare paid and incurred
Agreement is not proof, but divergence can reveal settlement-speed changes or case-reserving shifts. Define incurred consistently before comparing.
Practice
- If is unchanged but rises from 1.35 to 1.42, calculate AY 2022 ultimate and reserve.
- Why is the Dev 2→3 factor especially uncertain here?
- A £20m payment is moved from Dev 1 to Dev 0 without changing ultimate loss. Predict the direction of the early link ratios.
Answers
- Ultimate ; reserve .
- Only AY 2020 supplies a fully observed pair, so process variation and that year's peculiarities are inseparable from the selected factor.
- Dev 0 rises and Dev 1 cumulative is unchanged, so the 0→1 ratio falls. The change is payment timing, not necessarily ultimate severity.