Paid and incurred triangles
The two views of the same claims book, one by actual payments, the other by payments plus case reserves, whose divergence reveals reserving practice as much as risk.
Gerhard Quarg and Thomas Mack published Munich chain ladder in 2004, projecting paid and incurred together instead of separately. The paid triangle holds only cash that has left the company, so nothing arguable, but it develops slowly and gives little signal on recent years. The incurred triangle adds case reserves set by claims handlers on top of payments. So it reacts fast, but it embeds their judgment and distorts as soon as case reserving guidance changes. An actuary always projects both and compares the resulting ultimates: their convergence is the first consistency check of a closing. When the incurred triangle returns a clearly higher ultimate, the most frequent explanation is case reserve strengthening rather than deteriorating risk. The reverse signals loosening. The paid-to-incurred ratio by accident year and development period settles it. It is also the variable Munich chain ladder uses to reconcile the two views.
Review of a commercial liability book as of June 30, 2026. The paid-to-incurred ratio at 24 months falls from 0.41 on accident years 2019 to 2021 to 0.29 on 2022 and 2023. Investigation shows a case reserve strengthening instruction issued in March 2022, not a deterioration in claims: the incurred ultimate is cut by 6M EUR, the paid ultimate is left unchanged.
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