Chain-ladder or Bornhuetter-Ferguson: choosing a reserving method per risk group

Chain-ladder and Bornhuetter-Ferguson are the two methods almost every non-life reserving actuary runs first, and they disagree most where the claims outstanding primarily are: on the youngest accident years of a long-tail risk group. This article explains what each method assumes, works both methods through the same paid triangle we used in the IBNR article, shows the arithmetic that makes them differ, and lists the conditions under which each one breaks. It ends with a decision table per type of risk group and a discussion of reserve uncertainty. The calculation is the one that TP Tool runs on your own risk groups, with chain-ladder and Bornhuetter-Ferguson side by side, only small enough to check by hand.

The chain-ladder method

Chain-ladder assumes that every accident year develops in the same proportions as the years before it. Cumulative paid or incurred claims are laid out by accident year and development year, and for each step to the next development year an age-to-age factor is taken from the historical columns. Let $C_{i,j}$ denote cumulative claims for accident year $i$ at development year $j$, and let $\mathcal{I}_j$ be the accident years with observations at both $j$ and $j+1$. The volume-weighted age-to-age factor is:

For accident year $i$, let $j_i$ be its latest observed development year and $J$ the last development year before the tail. The cumulative development factor from $j_i$ to ultimate is the product of the remaining age-to-age factors and the selected tail factor. The projected ultimate is:

Two things sit outside the triangle. The first is the tail. If the oldest accident year is still paying, the last observed factor is not 1.000 and a tail factor has to be fitted, usually by extrapolating the observed factors with a curve or by borrowing a tail from a longer market triangle. The second is judgement over the factors: an actuary looks at the individual age-to-age ratios, excludes a diagonal distorted by a one-off event, and may use the last three or five years rather than the whole history if the settlement speed has changed.

The IFoA Claims Reserving Manual lists it among the familiar techniques for projecting paid or incurred claims. Friedland’s CAS text, Estimating Unpaid Claims Using Basic Techniques, calls it the development technique and states the assumption plainly: a year will develop as the older years did. Everything chain-ladder gets wrong follows from that assumption not holding.

The Bornhuetter-Ferguson method

Bornhuetter and Ferguson introduced the method in their 1972 paper, “The Actuary and IBNR”, published in the Proceedings of the Casualty Actuarial Society. It addressed reserves where the data alone are not credible such as reinsurance. The idea is to stop multiplying the paid claims of a young year and instead ask how much of the expected claims has not yet emerged.

The method needs three inputs per accident year: earned premium $\mathrm{EP}_i$, an a priori expected loss ratio $\mathrm{ELR}_i$, and the same development pattern chain-ladder uses, now read as a percentage. Let $F_i$ be the cumulative development factor from the latest observed development age $j_i$ to ultimate. The developed and unpaid proportions are $1/F_i$ and $1-1/F_i$, respectively. If $F_i=1.98$, about 50.5% is developed and 49.5% remains unpaid. The Bornhuetter-Ferguson estimates are:

The paid to date enters only as an addition. If the year has had one unusually large early payment, the unpaid estimate does not change. The ultimate rises by the excess payment, not by the excess payment times the cumulative factor. That is the point of the method and also its exposure, because the answer is only as good as the a priori loss ratio. A 2016 CAS Working Party paper on the initial expected loss ratio surveyed how practitioners choose it. The common sources are the pricing loss ratio, the plan, and older accident years adjusted for rate changes, claims inflation and any change in reinsurance.

Both methods on the same triangle

Here is the cumulative paid triangle from the IBNR article, one homogeneous risk group, EUR thousand. The age-to-age factors are the volume weighted ones from that article: 1.500 from development year 1 to 2, 1.200 from 2 to 3, 1.100 from 3 to 4, and a tail of 1.000.

Applying the remaining factors to each latest diagonal gives the chain-ladder ultimates:

Now Bornhuetter-Ferguson for the same year. Assume the risk group earned a premium of 3,300 in 2025 and the a priori loss ratio for that year is 75%, taken from the pricing basis.

The two methods land 49 apart on the same year. The gap is small because the assumptions happen to agree: chain-ladder’s 2,574 implies a loss ratio of $2{,}574/3{,}300=78\%$, against the 75% assumed, and the methods differ by that disagreement scaled by the unpaid share. With the 900 of 2025 case reserves used in the IBNR article, the implied IBNR after case reserves is:

The difference shows when the data move. Suppose the 2025 paid figure had been 1,500 instead of 1,300 because one large claim settled early. Chain-ladder becomes $1{,}500 \times 1.98=2{,}970$, an increase of 396 from an extra 200 of payments. Bornhuetter-Ferguson becomes $1{,}500+1{,}225=2{,}725$, an increase of exactly 200. Now hold payments at 1,300 and move the loss ratio instead: at 65% the Bornhuetter-Ferguson ultimate is approximately 2,362. At 85% it is approximately 2,688, but chain-ladder stays at 2,574 either way. Each method is stable against the input the other one is sensitive to.

For the more mature 2024 year the two are similar. With a premium of 2,900 and the same 75% loss ratio, Bornhuetter-Ferguson gives $1{,}650 + 2{,}900 \times 0.75 \times (1-1/1.32) \approx 2{,}177$, against chain-ladder’s 2,178. Once about three quarters of the ultimate has emerged, there is less room for the a priori view to influence the estimate, so method choice matters most for the youngest years. The formulas and inputs above reproduce the figures directly.

Where each method fails

Chain-ladder is most vulnerable on immature years. The 2025 row is about 50.5% developed, so its factor of 1.98 nearly doubles the first diagonal. On a liability group the leverage can be larger still. It also relies on a stable development pattern. Changes in case-reserving practice can shift incurred development, while changes in settlement speed can shift paid development. Applying factors selected across different regimes can bias the projection in either direction. The actuary should reconcile the triangle to changes in claims handling, case reserving, inflation and large-loss experience before selecting factors. Friedland devotes a chapter to reading the triangle for exactly these shifts.

Bornhuetter-Ferguson is deliberately less responsive to a young year’s emerging experience, so a stable unpaid estimate does not validate its prior. An a priori loss ratio from an earlier pricing or underwriting regime may be inappropriate if expected gross claim costs, premium rates or portfolio mix have changed. A reinsurance programme change alone does not change the gross claims obligation under Solvency II: gross claims provisions are calculated without deducting reinsurance recoverables, which are valued separately. The change matters to a gross Bornhuetter-Ferguson prior only if the prior is net of reinsurance or the programme change accompanies changes to coverage, underwriting or portfolio mix that affect gross claims. Use a prior and premium or exposure basis consistent with the gross claim triangle. The method also inherits the development pattern from the triangle, so a distorted pattern can bias both methods, even if its effect on an immature Bornhuetter-Ferguson estimate is dampened.

Large losses hurt chain-ladder directly and Bornhuetter-Ferguson only through the paid to date. The usual practice is to remove individual large claims from the triangle, reserve them separately, and run the methods on the attritional remainder. Sparse triangles, common in captives and niche lines, defeat chain-ladder because a factor taken from one or two cells is not a factor at all, so those groups run on Bornhuetter-Ferguson with a borrowed pattern or on an expected loss ratio alone.

Quantifying reserve uncertainty

The point estimates differ by 49, and the estimated unpaid amounts differ by the same amount because both methods use the same 1,300 paid to date. That difference alone does not show whether either estimate is materially more reliable. Mack’s 1993 paper in the ASTIN Bulletin derives a chain-ladder reserve prediction standard error under assumptions about the mean and variance of development and the independence of accident years. This standard error is conditional on the chain-ladder model and does not measure uncertainty in the Bornhuetter-Ferguson prior or provide a direct test of the difference between these two estimates. Mack’s standard error does not require a full distributional assumption, but converting it into an error band or interval does. Calculating the standard error, including Mack’s extrapolation for sparse development-age data, is beyond the scope of this article. Assessing Bornhuetter-Ferguson uncertainty would separately require sensitivity analysis or a model for its prior and development pattern.

For a stochastic alternative focused on incurred triangles, the over-dispersed Skellam reserving model uses a chain-ladder-compatible multiplicative mean structure, allowing its expected incremental cell values to be calibrated cell by cell to a chain-ladder projection. Unlike an over-dispersed Poisson model, it allows both negative observed increments and negative expected incremental cell means. Note that is a distinct model and not a distribution-free extension of Mack. Its cell-independence and other model assumptions, calibration and fit still need to be assessed for the given risk group.

A decision table per risk group

The choice is made per homogeneous risk group and per accident year, not once for the company. Treat the table as a screening aid for candidate methods, not as a rule that a line of business or development age determines the method. A triangle’s appearance does not establish that a model’s assumptions hold.

For every row, first check data completeness, triangle consistency, gross or net basis, and risk-group homogeneity. Investigate calendar-year effects, claims inflation, changes in coverage, case-reserving and claims-handling practices, and large losses. Chain-ladder assumes the selected age-to-age relationships remain suitable for future development, so test factor selections, stability across accident years, and tail assumptions, then back-test as later diagonals emerge. Standard Bornhuetter-Ferguson applies the a priori ultimate to the un-emerged proportion and does not let a young year’s paid emergence correct a deficient prior. Validate the prior, premium or exposure basis, and development pattern on their respective merits, and stress alternative selections. If using Mack standard errors, test its assumptions—including the chain-ladder mean structure, accident-year independence, and variance proportional to cumulative claims. Such standard errors quantify uncertainty conditional on the model, but they do not measure bias from a wrong pattern, prior, or segmentation. Compare methods as diagnostics, quantify material uncertainty and document why the selected assumptions remain appropriate. If they fail, do not select a method solely because of the table.

How the choice reaches the Solvency II balance sheet

On this paid triangle, each method estimates undiscounted ultimate paid claims per accident year. The ultimate less paid to date gives an estimate of future payments, but it is not itself the Solvency II claims provision. For a Solvency II gross claims provision, the triangle and any Bornhuetter-Ferguson prior must be on a consistent gross basis. Article 77(2) of the Directive requires the best estimate to be calculated without deduction for reinsurance recoverables. Article 81 requires those recoverables to be valued separately. The illustrative triangle here is not labelled gross or net, so its arithmetic is a method comparison, not a Solvency II valuation; no gross-to-net or reinsurance bridge is shown. Article 36 of Delegated Regulation (EU) 2015/35 requires the best estimate for non-life obligations to be calculated separately for the premium provision and the provision for claims outstanding, and states that the claims provision relates to claim events that have already occurred, whether or not they have been reported. The ultimate minus the claims paid is the starting point of that claims provision.

The claims cash-flow projection must also include claims-handling expenses and appropriate inflation assumptions, and assess material effects from events not adequately captured in the data (ENIDs), rather than apply an unspecified generic allowance. Future cash flows are discounted at the applicable risk-free curve. The result is reported in template S.17.01. The triangles themselves, and the best estimate by accident year, go into template S.19.01, Non-life insurance claims, where a supervisor can see how each accident year’s estimate has moved between year ends. The net claims provision per line of business is also the volume measure for reserve risk in the SCR standard formula, so a method that overstates the youngest years costs capital as well as reserves.

Overall conclusion

The chain-ladder and Bornhuetter-Ferguson calculations estimate undiscounted paid-claims ultimates. Subtracting paid to date gives estimated unpaid claims, but not the full Solvency II claims provision or complete technical provisions. The example’s gross or net basis is unspecified, so it is a method illustration rather than an actual Solvency II valuation.

Article 36 requires separate premium and claims provisions. A paid-triangle ultimate-minus-paid projection is only a starting point for the claims provision. A full valuation would also require:

• A dated claims cash-flow projection, including claims-handling expenses and an appropriate treatment of claims inflation.
• A payment pattern and discounting on the applicable risk-free term structure.
• A gross-to-net reconciliation, with reinsurance recoverables valued separately and including the required counterparty-default adjustment.
• A separate premium provision for future claim events and the risk margin.
• Documented validation and sensitivity testing for the data, segmentation, development pattern and tail, Bornhuetter-Ferguson prior, and large-loss treatment.

Where this lands in the software

TP Tool makes the per-group choice a setting. In the Risk Group Designer the actuary defines the homogeneous risk groups the transactions are analysed in, and the Statistical Engine builds the run-off triangles for each group automatically from the claim and premium transactions. chain-ladder, Bornhuetter-Ferguson and fixed loss ratio estimates run side by side on the same triangle, with a choice of development factor selections and diagnostics that show how each method reacts to the data, so the sensitivity worked through above is on screen for every group and every accident year. Every intermediate figure stays visible for review, reserve uncertainty is assessed next to the central estimate, and the selected results flow into S.17.01, S.19.01 and the other technical provisions templates without a separate aggregation step.