Actuarial Science & Loss Reserving
Bornhuetter-Ferguson vs Chain-Ladder IBNR Reserve Formula
Compare Chain-Ladder IBNR = Paid * (CDF - 1) against Bornhuetter-Ferguson IBNR = EP * ELR * (1 - 1 / CDF) to stabilize immature accident years.
Prevent green-year Chain-Ladder swings by anchoring unreported % to a-priori ELR
Pure Chain-Ladder (CL) projects Incurred-But-Not-Reported (IBNR) reserves as Paid_Loss * (CDF - 1), multiplying cumulative losses to date by Age-to-Ultimate Cumulative Development Factor CDF >= 1. On an immature accident year where CDF = 5.00 (only 1 / 5 = 20% developed!), Chain-Ladder amplifies early claim noise 5-fold — and if zero claims have been paid yet, Chain-Ladder books ZERO reserve! Bornhuetter and Ferguson (1972) replace that unstable projection on the undeveloped fraction (1 - 1 / CDF) with Earned Premium * Expected Loss Ratio: IBNR_BF = EP * ELR * (1 - 1 / CDF), making BF Ultimate Loss Paid + IBNR_BF an exact credibility blend (1/CDF) * Ult_CL + (1 - 1/CDF) * (EP * ELR)!
Audit three accident years in a shared reserving workbook: Mature AY 2024 (EP = 10,000, ELR = 0.65, Paid = 5,400, CDF = 1.25 -> CL IBNR = 1,350 vs BF IBNR = 1,300), Mid AY 2025 (EP = 12,000, ELR = 0.65, Paid = 4,500, CDF = 2.00 -> CL = 4,500 vs BF = 3,900), and Immature AY 2026 (EP = 15,000, ELR = 0.65, quiet early Paid = 900, CDF = 5.00 -> CL under-reserves at 3,600 vs BF = 7,800, raising total portfolio IBNR from 9,450 to 13,000!). In a shared Arte Sheets reserving review, a peer analyst with the suggester role can submit a cell suggestion on G4 and anchor a whole-cell comment @mentioning the signing actuary (owner), who tests setting D4 = 0 before accepting.
| Accident Year (ELR = 65%) | Earned Prem / Paid / CDF | CL IBNR vs BF IBNR |
|---|---|---|
| AY 2024 (80% Developed) | 10,000 / 5,400 / 1.25 | 1,350.00 vs 1,300.00 |
| AY 2025 (50% Developed) | 12,000 / 4,500 / 2.00 | 4,500.00 vs 3,900.00 |
| AY 2026 (20% Developed) | 15,000 / 900 / 5.00 | 3,600.00 vs 7,800.00 |
| Total Portfolio IBNR | 37,000 / 10,800 / — | 9,450.00 vs 13,000.00 |
Bornhuetter-Ferguson Accident-Year IBNR Reserve EP * ELR * (1 - 1 / CDF) Formula
Validate Earned Premium EP > 0, ELR > 0, Paid >= 0, and CDF >= 1 (when CDF = 1.00 on a closed mature year, 1 - 1/1 = 0 cleanly books 0 IBNR).
=LET(ep_y, B4, elr_y, C4, paid_y, D4, cdf_y, E4, IF(OR(ep_y <= 0, elr_y <= 0, paid_y < 0, cdf_y < 1), "Invalid Reserve Input (Require CDF >= 1)", ROUND(ep_y * elr_y * (1 - 1 / cdf_y), 2)))Chain-Ladder Accident-Year IBNR Paid * (CDF - 1) and Portfolio Total Formula
Compare Chain-Ladder row IBNR side-by-side against Bornhuetter-Ferguson and sum across accident years.
=IF(OR(D4 < 0, E4 < 1), "Invalid Reserve Input (Require CDF >= 1)", ROUND(D4 * (E4 - 1), 2))Try the example
Use the original values or change them to test your understanding.