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Munich chain ladder

Coupled chain ladder that corrects the paid and the incurred triangle at the same time using their ratio, narrowing the gap between the ultimates the two triangles produce separately.

Definition

A paid triangle and an incurred triangle describe the same book and should converge to the same ultimate. Run separately they almost never do, and the actuary arbitrates by hand. Gerhard Quarg and Thomas Mack showed in 2004 that the gap is not noise: when an accident year's paid-to-incurred ratio sits above its average, future paid development is systematically lower and future incurred development higher. Munich chain ladder exploits that correlation by correcting each development factor by a term proportional to the paid-to-incurred residual. The two projections then converge mechanically. The method needs two consistent triangles and a stable definition of incurred: if case reserving practice changed mid-period, the correlation it measures is the correlation of the practice change, not of the risk.

Its 5 cross-references, woven across the glossary's 18 categories
Munich chain ladder: Actuarial science glossary illustration, AlgoPolis
Example

Motor bodily injury book as of December 31, 2025. Chain ladder on paid returns a 112M EUR ultimate, on incurred 127M EUR, a 15M EUR gap that three successive closings had not resolved. Munich chain ladder brings the two projections to 119M and 121M EUR, and the 2M EUR residual gap is of the same order as the estimation error of each projection.

Read next
Actuarial scienceChain ladderReserving method that projects each accident year to its ultimate cost by applying the development factors observed on older accident years.Read the entry →Actuarial sciencePaid and incurred trianglesThe 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.Read the entry →Actuarial scienceMack methodStochastic model that puts a confidence interval around chain ladder without assuming a distribution, by estimating the mean squared error of prediction of the reserve.Read the entry →
Sources
  • Quarg and Mack, "Munich Chain Ladder: A Reserving Method that Reduces the Gap between IBNR Projections Based on Paid Losses and IBNR Projections Based on Incurred Losses", Blätter der DGVFM, vol. 26, 2004, pp. 597-630
  • Quarg and Mack, Blätter der DGVFM, vol. 26, 2004 (no prediction error given for Munich chain ladder); Liu and Verrall, "Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims", Casualty Actuarial Society
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