Chain Ladder, Bornhuetter–Ferguson and a Large Claim
An insurer is reserving for a general liability portfolio. Cumulative paid claims (£m) by accident year and development year are: AY2021: 12.0, 18.0, 22.0 at development years 1, 2 and 3; AY2022: 13.5, 19.5 at development years 1 and 2; AY2023: 15.0 at development year 1. The chain-ladder development factors estimated from the triangle are f1 = 1.45, f2 = 1.22 and f3 = 1.05. The claims team notes that AY2021 includes a single notified claim of £6m in development year 2 that is expected to develop to £9m; without it, the AY2021 paid pattern would be 12.0, 12.0, 14.0. The chief actuary must decide whether to use chain ladder, Bornhuetter–Ferguson with an expected loss ratio of 65%, or a tail-adjusted method. Earned premium is £36m for AY2021, £38m for AY2022 and £40m for AY2023.
1. Calculate the chain-ladder ultimate for AY2023 and the Bornhuetter–Ferguson ultimate using a 65% expected loss ratio. Which is more defensible and why?
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Chain ladder for AY2023 uses the latest paid of 15.0 and the three development factors: 15.0 × 1.45 × 1.22 × 1.05 = 15.0 × 1.85745 = 27.86. The Bornhuetter–Ferguson ultimate with a 65% expected loss ratio is 40 × 0.65 = 26.0. The BF estimate is lower by 1.86. For AY2023 there is only one development year, so the chain-ladder result relies entirely on factors estimated from AY2021 and AY2022, and the AY2021 pattern is distorted by a large claim. The CL method assumes the future development pattern is stable and that the latest paid is a reliable signal; with one observation, that is weak. BF uses an independent expected loss ratio and only the reported proportion of ultimate, so it is more stable for an immature year. I would recommend BF for AY2023, or a blend of BF and CL, provided the 65% loss ratio is supported by pricing and market data. I would also show sensitivity to the loss ratio and to the large-claim treatment.
2. Explain how the £6m large claim in AY2021 affects the development factors and how you would adjust the reserving triangle.
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The £6m claim inflates AY2021’s paid amount at development year 2 and therefore the estimated f2 factor, which is used to project AY2022 and AY2023. If f2 is estimated from a triangle containing the large claim, it will overstate the development expected for ordinary claims, and the chain-ladder reserve will be too high for the other years. The correct treatment is to separate the large claim: remove it from the triangle, estimate development factors on an ex-large basis, project the large claim separately to its expected ultimate of £9m, and add the two components. This preserves the ordinary development pattern. Alternatively, if the claim is not atypical of the portfolio, keep it in and accept the higher factor, but that should be a deliberate decision. I would also check whether the £9m estimate is case-reserving based, whether there are other large claims, and whether a tail factor is needed. The adjusted ultimate for AY2021 should be the ex-large projected ultimate plus 9.0, not simply the raw chain-ladder result.