loss development triangle

Imagine a spreadsheet where each row is a year in which accidents happened, and each column is how much you had paid (or recorded) on those accidents 12 months later, 24 months later, 36 months later, and so on. The oldest accident years are almost fully paid, so their rows stretch all the way across. The newest year has only had time for its first column. Because old rows are long and new rows are short, the filled-in data forms a triangle — the loss development triangle, the single most important picture in reserving.

Precisely, a triangle arranges cumulative losses by accident (or report) year down the rows and by development age across the columns. Each diagonal is one calendar year of experience: the most recent diagonal is what you know as of today. The whole point of reserving is to fill in the empty lower-right corner — to project each immature row forward to its ultimate value. For example, if accident year 2023 has paid 6 million at age 24 months, and history says losses at 24 months are typically 80 percent of ultimate, you project 2023's ultimate at about 7.5 million and reserve the difference. The triangle turns a messy stream of payments into a clean grid where development patterns become visible.

Triangles matter because almost every reserving method — chain ladder, Bornhuetter-Ferguson, Cape Cod, Mack — reads its assumptions directly off them. They also expose problems: a kink in the diagonals can reveal a change in claims handling, a speed-up in payments, or a one-off catastrophe. A crucial caveat: a triangle assumes the future will develop like the past. When the mix of business, the legal environment, inflation, or claims practices shift, the historical pattern misleads, and a mechanical projection can be badly wrong.