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 money is: on the youngest accident years of a long-tail risk group. This article explains what each method assumes, works both through the same four-year paid triangle we used in the IBNR article, shows the arithmetic that makes them differ by 49 on an ultimate of about 2,500, and lists the conditions under which each one breaks. It ends with a decision table per type of risk group and a note on the two methods that usually come next, Cape Cod and the Mack standard error. 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. The volume weighted version divides the sum of the later column by the sum of the earlier column, using only the rows that have both values. The product of the remaining factors is the cumulative development factor, and the latest diagonal multiplied by that factor is the ultimate.

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 published the method in 1972 in the Proceedings of the Casualty Actuarial Society, in a paper written for reserves where the data alone are not credible, such as reinsurance and umbrella liability. 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: the earned premium, an a priori expected loss ratio, and the same development pattern Chain-Ladder uses, now read as a percentage. If the cumulative development factor is 1.98, then 1 / 1.98 = 50.5 percent of the ultimate is expected to have been paid by now and 49.5 percent is still unpaid. The formula is:

Unpaid claims = premium × expected loss ratio × percentage unpaid

Ultimate = paid to date + unpaid claims

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 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.

Chain-Ladder gives the ultimates we already know: 1,980 for 2022, 2,376 for 2023, 2,178 for 2024, and for the youngest year 1,300 × 1.500 × 1.200 × 1.100 = 1,300 × 1.98 = 2,574. The unpaid amount for 2025 is 2,574 minus 1,300 = 1,274.

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 percent, taken from the pricing basis.

Expected ultimate claims = 3,300 × 0.75 = 2,475

Percentage paid after one development year = 1 / 1.98 = 50.5 percent, so 49.5 percent is unpaid

Unpaid claims = 2,475 × 0.495 = 1,225

Ultimate = 1,300 + 1,225 = 2,525

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 percent against the 75 percent we assumed, and the methods only differ by that disagreement scaled by the unpaid share. On the case reserves used in the IBNR article, 900 for 2025, the IBNR is 374 under Chain-Ladder and 325 under Bornhuetter-Ferguson.

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 × 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 the payments at 1,300 and move the loss ratio instead: at 65 percent the Bornhuetter-Ferguson ultimate is 2,362, at 85 percent it is 2,688, and 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 converge. With a premium of 2,900 and the same 75 percent, Bornhuetter-Ferguson gives 1,650 + 2,900 × 0.75 × 0.242 = 2,177 against Chain-Ladder’s 2,178. Once three quarters of a year has been paid there is little left for the a priori view to influence, so the method choice is a question about the youngest years only. A short Python script over the triangle reproduces every figure above.

Where each method fails

Chain-Ladder fails on immature years. The 2025 row is 50 percent developed, so a factor of 1.98 doubles whatever noise sits in the first diagonal, and on a liability group the leverage is larger still. It also fails when the pattern changes. If the claims department started setting case reserves earlier or more prudently two years ago, an incurred triangle shows faster early development and Chain-Ladder projects that speed onto the old, slower pattern, overstating the ultimates. The reverse happens when settlement slows, for example when a court backlog delays bodily injury claims. Friedland devotes a chapter to reading the triangle for exactly these shifts.

Bornhuetter-Ferguson fails when the a priori loss ratio is wrong, and it fails quietly, because the method will not let the data correct the assumption for a young year. A loss ratio carried over from a soft market into a hard one, or one that ignores a change in the reinsurance programme, produces an unpaid estimate that looks stable while it is simply insensitive. The method also inherits the development pattern from the triangle, so a distorted pattern hurts both methods, only less under Bornhuetter-Ferguson.

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.

Cape Cod and the Mack standard error as the next steps

Cape Cod, chapter 10 of Friedland’s text, keeps the Bornhuetter-Ferguson formula but estimates the loss ratio from the triangle instead of assuming it. The estimate is total claims to date divided by total used-up premium, where each year’s premium is multiplied by its percentage developed. On our triangle, with premiums of 2,600, 2,900, 2,900 and 3,300 for the four years, the used-up premium is 9,100, the claims to date are 7,090, and the Cape Cod loss ratio is 77.9 percent. Applied to 2025 it gives an ultimate of 2,573, close to Chain-Ladder, because the triangle’s own experience now drives both numbers. It suits a group where one a priori figure is hard to defend but the history is long enough to use.

The second step is uncertainty. Mack’s 1993 paper in the ASTIN Bulletin derives the standard error of the Chain-Ladder reserve for each accident year and in total without assuming a distribution, under three conditions: the expected next value is the current value times the factor, accident years are independent, and the variance of the next value is proportional to the current one. The result turns a single ultimate into a central estimate with an error band. If the standard error on the 2025 reserve is several hundred, 2,574 and 2,525 sit inside each other’s noise; if it is fifty, they do not.

A decision table per risk group

The choice is made per homogeneous risk group and per accident year, not once for the company. The table is the usual starting position before the diagnostics are read.

Whatever the row says, the method not selected is still shown next to the one that was, so the reviewer can see the gap.

How the choice reaches the Solvency II balance sheet

Whichever method wins, its output is an undiscounted ultimate per accident year. 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; claims handling expenses, discounting at the risk-free curve and an allowance for events not in the data complete it, and 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.

Where this lands in the software

TP Tool makes the per-group choice a setting rather than a spreadsheet. 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.