Chain Ladder versus Bornhuetter-Ferguson for an Immature Liability Accident Year
IBNR reserves for a commercial public liability portfolio are being set at 31 March. Cumulative paid claims, in Rs lakh, by accident year and development period from the start of the accident year are:
| Accident year | 12 months | 24 months | 36 months | 48 months |
|---|---|---|---|---|
| 2021 | 1,200 | 1,800 | 2,050 | 2,180 |
| 2022 | 1,320 | 1,980 | 2,270 | - |
| 2023 | 1,450 | 2,150 | - | - |
| 2024 | 1,600 | - | - | - |
The selected tail beyond 48 months is 1.03. Case reserves held by the claims department are: AY2021 Rs 120 lakh, AY2022 Rs 260 lakh, AY2023 Rs 430 lakh and AY2024 Rs 520 lakh. Earned premium for AY2024 was Rs 4,800 lakh and the pricing expected loss ratio for this class is 68%. Two years ago the claims department changed its case reserving philosophy and now sets case estimates earlier and more fully than before.
1. Derive volume-weighted age-to-age factors and cumulative development factors, project an ultimate for each accident year, and separate the total unpaid amount into case reserves and pure IBNR.
Read the worked answer
Age-to-age factors (volume weighted):
- 12-24: (1,800 + 1,980 + 2,150) / (1,200 + 1,320 + 1,450) = 5,930/3,970 = 1.4937
- 24-36: (2,050 + 2,270) / (1,800 + 1,980) = 4,320/3,780 = 1.1429
- 36-48: 2,180/2,050 = 1.0634
- 48-ultimate (tail): 1.0300
Cumulative development factors to ultimate:
- from 12 months: 1.4937 x 1.1429 x 1.0634 x 1.03 = 1.8699
- from 24 months: 1.1429 x 1.0634 x 1.03 = 1.2519
- from 36 months: 1.0634 x 1.03 = 1.0953
- from 48 months: 1.0300
Projected ultimates (Rs lakh): AY2021 = 2,180 x 1.0300 = 2,245.4; AY2022 = 2,270 x 1.0953 = 2,486.3; AY2023 = 2,150 x 1.2519 = 2,691.6; AY2024 = 1,600 x 1.8699 = 2,991.8. Total = 10,415.1.
Unpaid (ultimate less paid): 65.4 + 216.3 + 541.6 + 1,391.8 = 2,215.1 lakh.
Split: case reserves total 120 + 260 + 430 + 520 = 1,330 lakh; pure IBNR = 2,215.1 - 1,330 = 885.1 lakh. The split matters because the case component is the claims department's estimate and the IBNR component is the actuary's; the balance sheet and any reserve review will treat them differently.
2. Apply a Bornhuetter-Ferguson estimate to AY2024 and compare it with the chain ladder result. Which would you book and why?
Read the worked answer
BF calculation for AY2024. Percentage reported at 12 months = 1/1.8699 = 53.5%. Expected ultimate = 68% x 4,800 = Rs 3,264 lakh. BF ultimate = reported + expected ultimate x (1 - % reported) = 1,600 + 3,264 x 0.465 = 1,600 + 1,517.8 = Rs 3,117.8 lakh. The chain ladder gives Rs 2,991.8 lakh, an implied loss ratio of 62.3% against the pricing expectation of 68%.
The gap is Rs 126 lakh, or 4.2% of the AY2024 ultimate. It is not a rounding difference; it tells you the two methods disagree about the underlying loss ratio.
Which to book. AY2024 is 12 months old and 46.5% of its ultimate is still unreported. The chain ladder projection is therefore almost entirely an assumption borrowed from prior accident years, and those years were reserved under the old case philosophy. A move to fuller, earlier case estimates converts what would have been IBNR into case reserves and accelerates paid development as cases are settled, which can distort the paid-based factors in either direction.
Recommendation: use BF or a credibility blend for AY2024, and chain ladder for the mature years where its own data dominates. BF anchors the immature year to the pricing loss ratio, so a single noisy factor cannot swing the reserve by tens of lakh. Document the blend and test sensitivity: at a 60% expected loss ratio the BF ultimate falls to Rs 2,878 lakh, below chain ladder, so the choice between methods is itself worth about Rs 240 lakh.
3. The claims manager argues that AY2023 is under-reserved because case reserves of Rs 430 lakh are below the chain ladder unpaid of Rs 541.6 lakh, while AY2021 is over-reserved for the same reason. Evaluate this argument and list the sensitivities you would disclose.
Read the worked answer
The argument confuses two different things. Chain ladder unpaid is total unpaid - claims already reported but not yet paid, plus claims incurred but not reported at all, plus further development on reported claims. Case reserves cover only the first component. For a long-tail liability class, unpaid will almost always exceed case reserves in early development, so case below chain ladder unpaid is the normal condition, not evidence of deficiency. For AY2023 the shortfall is Rs 111.6 lakh of IBNR, which is plausible at 24 months. For AY2021 the case of Rs 120 lakh exceeding the chain ladder unpaid of Rs 65.4 lakh means the claims department expects negative development on known claims, i.e. redundant case estimates; that is a claims-quality signal, not an actuarial one, and should be raised with the claims manager rather than absorbed silently into the reserve.
Sensitivities to disclose:
- Tail. The 1.03 factor adds 3% to every ultimate; for AY2024 that is Rs 90 lakh. Doubling it to 1.06 adds a further Rs 90 lakh. This is the single assumption with the least data behind it.
- Change in case philosophy. Test whether the age-to-age factors are stable across the pre- and post-change cohorts.
- Inflation. Paid-based methods implicitly assume future claim inflation equals the historical average. A court-award or medical inflation step-up is not captured.
- Large claims. Any single claim above Rs 100 lakh should be projected individually, not by a factor.
- Discounting. Report the undiscounted figure and the discounted figure separately; do not let discounting mask an adequacy problem.