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Mack 1993: chain-ladder reserves and standard errors of the Taylor–Ashe triangle

ToolLoss Reserving
SourceMack, T. (1993). "Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates." ASTIN Bulletin, Vol. 23, No. 2, pp. 213–225. Tables 1–3.
LevelProfessional

Problem

A 10 × 10 cumulative paid-claims run-off triangle (Taylor and Ashe 1983, accident years 1 to 10) is given. Estimate the age-to-age development factors, the variance parameters σ_k², the reserve of every accident year and the total, and the standard error of each reserve by Mack's distribution-free formulas.

Inputs to Enter

  • Cumulative triangle, 10 accident years × 10 development ages (loaded by default; first row 357,848 … 3,901,463, last row 344,014)
  • Volume-weighted averages over all years, no tail factor
Open the Loss Reserving tool (loaded with this example)

Answers

  • f_k = 3.49, 1.75, 1.46, 1.174, 1.104, 1.086, 1.054, 1.077, 1.018
  • σ_k²/1000 = 160, 37.7, 42.0, 15.2, 13.7, 8.19, 0.447, 1.15, 0.447 (the paper prints 0.477 for k = 9, a misprint; Mack's own rule gives 0.447)
  • Reserves (thousands) by accident year 2–10: 95, 470, 710, 985, 1,419, 2,178, 3,920, 4,279, 4,626; total 18,681 (tool 18,680,856)
  • Standard error as % of the reserve: 80, 26, 19, 27, 29, 26, 22, 23, 29; total 13 % (tool 2,447,095, equal to R's MackChainLadder(GenIns))
  • Total ultimate 53,038,946; the ODP GLM on the same triangle gives the same reserves with dispersion φ = 52.6 thousand (ChainLadder glmReserve)

Notes

The tool's default inputs are this triangle; press Submit to reproduce it. The Mack standard errors are given for the volume-weighted ladder without a tail factor.

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