The loss development triangle is the single most important artifact in property and casualty reserving. Almost every method you will see in a reserve report starts from one. If you can read a triangle with confidence, you can ask the right questions about the report that comes out of it.
What the shape means
A triangle has accident years running down the left side and development periods running across the top. Each cell shows losses, either paid or reported, evaluated at the end of a given development period for a given accident year. The reason the shape is a triangle, not a rectangle, is simple: the 2024 accident year cannot have 48 months of development until late 2028.
The most recent evaluation date sits on the diagonal. The diagonal is what you have. Everything below the diagonal is the future. Everything a reserve method produces for the triangle is an estimate of those empty cells and the tail beyond them.
Two triangles, not one
A report will usually show at least two triangles side by side: one for paid losses and one for reported losses (paid plus case reserves). They tell you different things.
- Paid triangle. The most objective data. Paid losses do not depend on the adjuster’s opinion. But the paid pattern for long-tailed lines can be slow and volatile.
- Reported triangle. Includes the adjuster’s case estimate. Reacts faster to changes in severity and frequency. But it is sensitive to case reserving philosophy and adequacy, which can drift.
Whenever the paid triangle and the reported triangle tell different stories, you have a puzzle worth asking about. It is one of the clearest diagnostic signals in reserving.
Rows, columns, diagonals
Reading a triangle is mostly about pattern recognition on rows, columns, and diagonals.
- Rows. Each row is one accident year’s development. You are looking for a smooth, stable development pattern. A sharp jump or drop in a row relative to its neighbors is worth understanding.
- Columns. Each column is one development period across all accident years. You are looking for the age-to-age factors to be stable from one year to the next. If the 12-to-24 factor drifts year over year, something about recent experience has changed.
- Diagonals. Each diagonal is one calendar year of actual claim activity across accident years. Diagonals pick up calendar-year phenomena: a claims system change, a settlement initiative, reserving strengthening, inflation shocks. Calendar-year effects are the hardest to see in a routine chain ladder and the easiest to miss.
Age-to-age factors, in English
Below the triangle you will usually see a row of development factors: 1.45, 1.18, 1.08, 1.03, and so on. These are age-to-age factors, each computed as the ratio of one column to the previous column for a given accident year, then averaged across years in some way.
The usual averaging choices are:
- All-year simple average. Equal weight to every year.
- Volume-weighted average (also called dollar-weighted). Larger accident years count more. This is the most common default in software.
- Three-year, five-year, or ex-high-ex-low. Attempts to down-weight old or anomalous years.
None of these is inherently right. A good reserving report tells you which average was selected for each period and why. A report that does not tell you is asking you to trust the default in the software.
The tail
The bottom-right cell of the triangle is almost never the end of the story. For long-tailed lines there are payments still to come long after the last observed development period. The estimate for those payments is the tail factor, and it is where most of the subjective judgment in a reserve review lives.
When reviewing a report, look for how the tail was selected: from industry curves, from a fitted curve extrapolating the observed pattern, from a judgmental pick, or from external benchmarking. Ask what the answer changes by if the tail factor changes by a reasonable amount. If a small change in the tail moves the indicated reserve by ten percent, that is important to know.
Diagnostic questions
Before you accept the projection that comes out of a triangle, ask:
- Is the development pattern stable from year to year in the columns you care about?
- Do the paid and reported triangles tell the same story, and if not, why not?
- Are there visible diagonals? If yes, what calendar-year event is driving them?
- How was the tail factor selected, and how sensitive is the answer to it?
- Are there accident years with too few data points to trust? If so, how are they handled?
A reserving actuary will have an answer to each of these. If the answers are clear and specific, the triangle is doing its job. If not, the number at the bottom is not the number you should be relying on.