Loss development triangle (chain ladder)
A table crossing accident years with the development of payments, the basis for projecting ultimate losses and for reserving.
Thomas Mack, 1993, ASTIN Bulletin: the chain ladder, until then a plain calculation procedure, gets a distribution-free stochastic model and a standard error formula for the reserves it produces. The loss development triangle is the table it works on. Rows are accident years, columns are development years (the time elapsed since occurrence), and each cell holds the cumulative amount paid or incurred at that stage. It is triangular because recent cohorts have little observed development while older ones are almost fully run off. The whole job is to fill in the missing part. The chain ladder computes age-to-age factors from one development year to the next, applies them to recent cohorts and derives the ultimate loss of each, hence the reserves to set up. On a mature, stable line it holds up. On cyber, the history is short and development patterns shift with the threat, so the computed factors are worth little without a judgmental adjustment.
Textbook case: an actuary sees on the triangles that a line's losses are 90 percent paid after three years. They apply that pattern to recent cohorts to estimate their ultimate loss. On cyber the triangles are too unstable to rely on alone, so expert judgment has to be added.
loss triangle, development triangle, chain ladder, claims triangle
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