Chain-ladder IBNR, a late-reported large loss and the reported combined ratio
An analyst in the reserving cell of an engineering insurer is finalising gross IBNR at 31 March 2025 from the cumulative paid-claims triangle below (Rs lakh, development months 12 / 24 / 36 / 48):
- Accident year 2021: 100, 140, 168, 182
- Accident year 2022: 110, 154, 185, -
- Accident year 2023: 120, 168, -, -
- Accident year 2024: 130, -, -, -
The chief actuary wants a volume-weighted chain-ladder basis, no tail factor and no discounting. The CFO notes three things: accident year 2023 contains one offshore construction loss of Rs 45 lakh that was reported only in the 36-month period; the claims team will settle files materially faster from FY26 under a new TPA arrangement; and the line's earned premium is Rs 2,400 lakh, with incurred losses of Rs 1,310 lakh (before IBNR) and expenses of Rs 610 lakh.
1. Derive the volume-weighted age-to-age development factors, the projected ultimate for each accident year and the total IBNR.
Read the worked answer
Development factors (sum of later column divided by sum of earlier column, over the years where both are observed):
- 12 to 24 months: (140 + 154 + 168) / (100 + 110 + 120) = 462 / 330 = 1.4000
- 24 to 36 months: (168 + 185) / (140 + 154) = 353 / 294 = 1.2007
- 36 to 48 months: 182 / 168 = 1.0833
Projected ultimates (latest cumulative x product of remaining factors):
- AY2021: 182.00 (fully developed); IBNR 0.00
- AY2022: 185 x 1.0833 = 200.41; IBNR 15.41
- AY2023: 168 x 1.2007 x 1.0833 = 218.52; IBNR 50.52
- AY2024: 130 x 1.4000 x 1.2007 x 1.0833 = 236.74; IBNR 106.74
Total gross IBNR = 15.41 + 50.52 + 106.74 = about Rs 172.67 lakh.
Two arithmetic checks worth doing under exam conditions: the factor for AY2024 is the full product 1.4000 x 1.2007 x 1.0833 = 1.8211, and 130 x 1.8211 = 236.74; and each ultimate must exceed the latest paid figure, otherwise a factor has been inverted.
2. Critique the chain-ladder result for this line and state the adjustments you would make before the figure goes to the board.
Read the worked answer
The arithmetic is sound; the assumptions are not.
The Rs 45 lakh late-reported loss. A claim that appears only in the 36-month column of AY2023 inflates the 24-to-36 factor for that year and understates the 12-to-24 factor, because it is absent from the 24-month figure. Chain-ladder spreads that distortion across all accident years. The loss should be removed from AY2023's 36-month figure, the factors recomputed, and the Rs 45 lakh added back as a separately loaded, case-specific item. Better still, treat it as evidence that the 24-month data is incomplete and apply a loading to the recent years.
Faster settlement. Chain-ladder assumes a stable reporting and settlement pattern. A new TPA compressing settlement times will pull payments forward, raising the observed development factors in the near term and making the historical pattern overstate the future one. Either wait for credible data or blend the chain-ladder with a Bornhuetter-Ferguson or expected-loss-ratio method for the immature years where the triangle carries almost no information - AY2024 here is 100% dependent on factors.
No tail, no discounting. Both are conservative, so the Rs 172.67 lakh figure is, if anything, slightly understated at the tail but overstated if settlement speeds up. Report the figure with a range and disclose the assumptions; do not present a point estimate as certainty.
3. Using the data given, compute the reported and the IBNR-inclusive combined ratio, and explain what the gap tells the CFO.
Read the worked answer
As reported (before IBNR): (1,310 + 610) / 2,400 = 1,920 / 2,400 = 80.0%. Loss ratio 54.6%, expense ratio 25.4%.
Including IBNR: incurred losses become 1,310 + 172.67 = 1,482.67. (1,482.67 + 610) / 2,400 = 2,092.67 / 2,400 = 87.2%. Loss ratio 61.8%.
The 7.2-point gap is the reserving charge the CFO has not yet booked. The message is that the line is profitable but less profitable than the management report suggests, and that the entire cushion sits in IBNR: an 80% ratio looking comfortable becomes an 87% ratio once the accident years are fully developed, which is still under 100% but leaves little room for a single large loss on a Rs 2,400 lakh premium base. Also note the asymmetry - the 80% figure uses only paid and case reserves, so it will always flatter the account in a growing line where recent accident years carry the largest IBNR.