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Example board · Actuarial science

The chain-ladder method

A revision board made in MySummaries for short-term actuarial mathematics, shown as the app shows it in study mode. Updated 27 September 2026.

Highlights: the idea a number to know what to do, or the formula a trap

Short-Term Actuarial Mathematics The chain-ladder methodStudy
Loss reservingThe chain-ladder method10 sections · 3 columns
Chain ladder on a paid triangle ($000)
Accident year12 mo24 mo36 mo48 mo60 moCDFUltimateReserve
20219081,7312,1062,2482,2881.0002,2880
20229821,7942,2212,3782,4211.0182,42143
20231,0421,8442,3032,4622,5061.0882,506203
20241,0492,0052,4762,6472,6951.3442,695690
20251,1332,0982,5912,7702,8202.4892,8201,687
Age-to-age1.8521.2351.0691.0181.000 tail
Total12,7302,623
What it estimates
  • Ultimate: what an accident year's claims finally cost, once every claim is settled
  • Paid to date = the latest diagonal
  • Reserve = ultimate − paid to date: case reserves + IBNR
Which average?
  • 12–24 link ratios: 1.906, 1.827, 1.770, 1.911
  • All years: simple 1.854, volume-weighted 1.852
  • Latest 3 years, volume-weighted: 1.836
  • Volume-weighted is sum over sum, so big years count for more
  • Use the average the question names: all years, latest 3, excluding high and low
Paid or incurred?
  • Incurred (reported) = paid + case reserves
  • IBNR = ultimate − incurred, not ultimate − paid
  • Paid: slower and more leveraged, but case-reserve changes don't touch it
  • Incurred: develops faster, but moves with case-reserve adequacy
Bornhuetter–Ferguson, next
  • For immature (green) years, lean on an expected loss ratio (ELR) instead
  • Ult = paid + ELR × premium × (1 − 1/CDF)
  • 2025: 1−1/2.489=1 - 1/2.489 = 59.8% still to pay
  • As a year matures, 1/CDF → 1 and BF converges on the chain ladder
Known and projected
A loss-development triangle with accident periods down the left and development periods across the top. The observed upper-left portion is coloured yellow and divided into development cells, with a representative cell labeled xᵢⱼ. The unobserved lower-right portion is left white and identified as forecast using average age-to-age factors; curved arrows indicate development from one period to the next.
Picture: Figure 2 from Greg Taylor, “Loss Reserving Models: Granular and Machine Learning Forms”, Risks 7(3) 82 (2019). CC BY 4.0.
Step by step
  1. Cumulative paid in a triangle: accident years down, ages across
  2. Link ratios → average them → select a factor for each age
  3. Choose a tail for development past 60 months: 1.000 here
  4. Multiply from each age to ultimate: the CDF
  5. Ultimate = latest diagonal × CDF; applying the factors in turn fills the italic cells
  6. Reserve = ultimate − paid to date, summed over the years
The formulas
  • Ci,jC_{i,j}: cumulative paid, accident year ii, age jj
  • f^j=∑iCi,j+1/∑iCi,j\hat f_j = \sum_i C_{i,j+1} \big/ \sum_i C_{i,j}
  • Both sums over the same years: leave out the year with no Ci,j+1C_{i,j+1} yet
  • CDFj=f^j f^j+1⋯f^J−1×tail\mathrm{CDF}_j = \hat f_j\,\hat f_{j+1} \cdots \hat f_{J-1} \times \text{tail}
  • 1/CDFj1/\mathrm{CDF}_j = share of ultimate paid by age jj
  • Ultimate =Ci,latest×CDF= C_{i,\text{latest}} \times \mathrm{CDF}
Worked example: 2025
  1. f^12=7,374/3,981=1.852\hat f_{12} = 7{,}374 / 3{,}981 = 1.852 (2021–2024 only)
  2. CDF: 1.852×1.235×1.069×1.018=2.4891.852 \times 1.235 \times 1.069 \times 1.018 = 2.489
  3. Ultimate: 1,133×2.489=1{,}133 \times 2.489 = 2,820
  4. Reserve: 2,820−1,133=2{,}820 - 1{,}133 = 1,687 → 60% still unpaid
Assumptions and traps
  • Key assumption: the future develops like the past
  • Changes in claims handling, settlement speed, inflation or mix break it
  • The latest year is heavily leveraged: × 2.489 amplifies any noise in its 1,133
  • A tail of 1.000 says nothing develops after 60 months

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