The Bornhuetter-Ferguson method: the chain ladder with a second opinion

CM2 · Financial Engineering & Loss Reserving · 12 min read

1The problem it was invented to solve

The chain ladder takes the claims an accident year has paid so far and multiplies them up. For an old accident year that is sound: almost everything is known and the factor is close to 1. For the newest accident year it is uncomfortable, because the whole reserve rests on one early figure multiplied by the largest factor in the table.

And that early figure is the least reliable number on the triangle. Claims paid in the first few months of an accident year reflect how fast the claims department happened to be working, which large losses happened to settle early, and whether a reporting system changed. None of that tells you much about what the year will eventually cost.

The Bornhuetter-Ferguson method's answer is to stop asking the data to answer the question on its own. An insurer knows something else about that accident year before any claim is paid at all: how much premium it charged, and what loss ratio it was expecting when it set the price. That is a genuine, independent estimate of the eventual cost, and the method blends it with the triangle.

2The idea in one line

Take the chain ladder's view of how claims emerge over time, and the premium's view of how much there is in total.

That division of labour is the whole method, and it is the reason it works. The development pattern is estimated from several accident years pooled together, so it is the better-supported part of the chain ladder. The level for a single recent year is the poorly supported part. So keep the pattern and replace the level.

3The method

Three inputs, two of which come from outside the triangle:

• Earned premium for each accident year;
• an expected ultimate loss ratio, from pricing, from the business plan, or from a market benchmark;
• the cumulative development factors, from a chain ladder on the triangle.

Then, for each accident year:

In words: work out what the year was expected to cost in total, take the fraction of it that has not developed yet, and call that the reserve. Then add it to what has actually been paid.

The last line is where the method is often misread, so it is worth stating plainly. The claims actually paid are kept in full — they are facts, not estimates. Only the part that has not yet emerged is taken from the premium-based expectation. The method never overrides what is known; it only fills in what is not.

4The same method seen as a credibility weighting

There is a second way of writing Bornhuetter-Ferguson that makes what it is doing obvious. Let z = 1/F, the proportion of the accident year already developed. Then:

In words: the Bornhuetter-Ferguson ultimate is a weighted average of the chain ladder's answer and the premium's answer, and the weight on the chain ladder is exactly the proportion of the claims that have already come in.

This form is algebraically identical to the one above, and proving it takes one line: the chain ladder ultimate is C × F, so z × C × F = C, leaving C + (1 − z) × the expected ultimate, which is the first form. But it is worth knowing in its own right for three reasons.

It explains the behaviour at both ends. For a fully developed accident year z = 1 and the method returns the chain ladder's answer exactly — it cannot do anything else, because there is nothing left to estimate. For an accident year with nothing paid at all, z = 0 and the method returns the premium-based estimate, which is the only information there is.

It explains the weighting. The method trusts the data in proportion to how much of it there is. That is the definition of a credibility estimate, and this is the standard example of one in the reserving half of the syllabus.

And it gives a free check on any calculation, which is used in the worked example below.

5Reading the gap between the two methods

When a question gives you both reserves, the difference is the answer to "comment on your results", and it has exactly one source: whether the data is running above or below the expected loss ratio.

• If the chain ladder is projecting higher loss ratios than the expected one, Bornhuetter-Ferguson gives a lower reserve, because it is partly replacing the data's pessimism with the pricing assumption.
• If the data is running better than expected, Bornhuetter-Ferguson gives a higher reserve, for the same reason in reverse.

That symmetry is also the warning. Bornhuetter-Ferguson is only as good as the loss ratio fed into it. An optimistic loss ratio produces an optimistic reserve and does so quietly, because the method's arithmetic is unaffected by whether the assumption is any good. Where the chain ladder's weakness is that it believes the data too much, Bornhuetter-Ferguson's is that it believes the business plan too much. A reserving actuary runs both, compares the implied loss ratios, and investigates the difference rather than picking a winner.

6What it is better and worse at

Better: far more stable for recent accident years, because the reserve does not scale with an early paid figure; usable for an accident year with no claims at all, where the chain ladder simply cannot produce a number; and it makes the actuary's assumption explicit, as a stated loss ratio that a reader can challenge, rather than implicit in a factor.

Worse: it needs premium data, which may not be available by accident year or may not be comparable across years if rates have moved; it needs a loss ratio that someone has to justify; and it inherits every one of the chain ladder's assumptions about the development pattern, since that is still where the factors come from. It fixes the level, not the pattern.

One practical point that a question may reach for. If rates have risen sharply, premiums per unit of exposure have risen too, so last year's loss ratio is not this year's, and using one figure across all accident years will distort the newest. The repair is to use a loss ratio per accident year, adjusted for rate changes, and to say that is what you have done.

A worked example

The same portfolio as the chain ladder page, reprinted so this page stands alone. Cumulative claims paid, in ₹ lakh, fully run off by the end of development year 3:

The triangle, the premiums and the loss ratio are all invented for this example. They are not a real insurer's experience.

The expected ultimate loss ratio is 72% for every accident year. Claims paid to date total 764 + 789 + 742 + 560 = ₹2,855 lakh.

1. 1The development pattern, from a chain ladder

   Volume-weighted factors, each using only the accident years with a figure in both columns:

   f_0→1 = 2,008 / 1,340 = 1.4985075
   f_1→2 = 1,517 / 1,266 = 1.1982622
   f_2→3 = 764 / 728 = 1.0494505

   Chained into cumulative factors to ultimate, and inverted to give the proportion developed:

   Development year	F	Proportion developed, 1/F	Still to develop, 1 − 1/F
   0	1.8843986	0.5306733	0.4693267
   1	1.2575170	0.7952179	0.2047821
   2	1.0494505	0.9528796	0.0471204
   3	1.0000000	1.0000000	0.0000000

2. 2The expected ultimates from the premiums

   2022: 0.72 × 1,000 = 720
   2023: 0.72 × 1,100 = 792
   2024: 0.72 × 1,250 = 900
   2025: 0.72 × 1,400 = 1,008

3. 3The reserves

   Each one is the expected ultimate multiplied by the proportion still to develop.

   Accident year	Expected ultimate	1 − 1/F	BF reserve	Paid	BF ultimate
   2022	720	0.0000000	0.00	764	764.00
   2023	792	0.0471204	37.32	789	826.32
   2024	900	0.2047821	184.30	742	926.30
   2025	1,008	0.4693267	473.08	560	1,033.08
   Total	—	—	694.70	2,855	3,549.70

   The Bornhuetter-Ferguson reserve is ₹694.70 lakh. To four decimal places the components are 37.3194, 184.3039 and 473.0813, totalling 694.7046.

4. 4The second route: the credibility form

   This is a genuinely independent calculation, because it goes through the chain ladder ultimates rather than through the undeveloped proportions. The chain ladder ultimates are the paid figures multiplied by F:

   2023: 789 × 1.0494505 = 828.0165
   2024: 742 × 1.2575170 = 933.0776
   2025: 560 × 1.8843986 = 1,055.2632

   Now weight each against the premium-based expected ultimate, with weight z = 1/F:

   2023: 0.9528796 × 828.0165 + 0.0471204 × 792 = 826.3194
   2024: 0.7952179 × 933.0776 + 0.2047821 × 900 = 926.3039
   2025: 0.5306733 × 1,055.2632 + 0.4693267 × 1,008 = 1,033.0813

   The same three ultimates, to every decimal place shown. Subtracting the paid claims gives back 37.3194, 184.3039 and 473.0813, and a total reserve of 694.7046. Two routes, identical.

   And the closing arithmetic check: total ultimate minus total paid is 3,549.7046 − 2,855 = 694.7046.

5. 5Reading the weights

   The weights are the interesting part of the answer and are worth a sentence in any exam response.

   • 2022 is fully developed, so the weight on the data is 100% and the premium plays no part at all.
   • 2023 is 95.3% developed, so the reserve is almost entirely the chain ladder's.
   • 2024 is 79.5% developed — the data still leads, but a fifth of the answer now comes from the premium.
   • 2025 is only 53.1% developed, so nearly half of that year's ultimate comes from the pricing assumption rather than from the single paid figure.

   That last line is the method earning its keep. On the chain ladder, the ₹495.26 lakh reserve for 2025 rested entirely on ₹560 lakh of early payments. Here, 46.9% of the year's ultimate comes from the premium instead, and the reserve is ₹473.08 lakh — a figure that will move much less if that one early payment figure turns out to be unrepresentative.

6. 6Comparing the two methods

   Accident year	Chain ladder reserve	Bornhuetter-Ferguson reserve	Difference
   2022	0.00	0.00	0.00
   2023	39.02	37.32	−1.70
   2024	191.08	184.30	−6.77
   2025	495.26	473.08	−22.18
   Total	725.36	694.70	−30.65

   Bornhuetter-Ferguson is ₹30.65 lakh lower, about 4.2% below the chain ladder, and the whole of the difference has one cause. Divide the chain ladder ultimates by the premiums and the loss ratios the data implies are 76.4%, 75.3%, 74.6% and 75.4% — an overall 75.4% against the 72% the pricing assumed. The data is running about three and a half points worse than the business plan, so wherever the premium-based estimate is given weight it pulls the reserve down, and it is given most weight in the newest year, which is where the gap is largest.

   That is the comment the question is looking for, and notice that it is not a verdict. Two readings are available and the arithmetic cannot choose between them: either the pricing was optimistic, in which case the chain ladder is right and the loss ratio needs revising for next year as well as for this reserve; or the early development on recent years is running fast for a reason that will not persist, in which case Bornhuetter-Ferguson is right. The useful answer names both possibilities and says what evidence would settle it — here, the fact that the implied loss ratios are stable across all four accident years rather than drifting points towards the first explanation, and towards taking the chain ladder figure seriously.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

• Utility theory and risk aversion: non-satiation, A(w) and R(w)
• Measures of investment risk: variance, semi-variance, VaR and Tail VaR
• CAPM and the security market line: beta and the market price of risk
• Ito's lemma explained term by term, and what it does to a share price
• The chain ladder method: estimating outstanding claims from a run-off triangle
• Binomial option pricing: where the risk-neutral probability comes from
• The Black-Scholes formula and put-call parity, read term by term
• Ruin theory: the adjustment coefficient and Lundberg bound explained
• Mean-variance portfolio theory: the efficient frontier and why diversification works
• The term structure: spot rates, forward rates and par yields
• Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
• The Merton model: company equity as an option, and where a credit spread comes from
• The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
• The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
• The average cost per claim method: developing numbers and severity separately
• Valuing a maturity guarantee by simulation, and checking it against the formula
• The Efficient Markets Hypothesis: the three forms, and how each one is tested
• Single-index and multifactor models: getting a covariance matrix out of a few betas
• Excess-of-loss reinsurance: the retained claim, and choosing the retention
• State price deflators: pricing with real-world probabilities
• What makes a risk insurable, and what to do when it almost is
• Two-state and intensity credit models: backing a default rate out of a bond price
• The martingale representation theorem, and why hedging is possible at all

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.