Economic Modelling · Run-off triangles

Bornhuetter-Ferguson Method Explained with Examples

Updated 11 October 2026 · Fact-checked

The Bornhuetter-Ferguson (BF) method estimates the reserve for each origin year as the a priori ultimate claims multiplied by the proportion still to be reported or paid. You then add the claims already incurred or paid. It blends prior belief with actual development, so recent years are more stable than under chain ladder.

Understand Bornhuetter-Ferguson Method

Run-off triangles show claims by origin year and development year. The chain ladder method projects each origin year using only its own claims to date. For a recent year, claims to date are small and can be unusually high or low. Chain ladder then scales that noise up into a large, unreliable ultimate.

The Bornhuetter-Ferguson method avoids this. You start with an a priori estimate of ultimate claims for each origin year. This usually comes from premium multiplied by an expected loss ratio, or from past experience, pricing or judgement. It does not use the claims data in the triangle for that year.

Next you use the triangle to find the development pattern. This is the proportion of ultimate claims that you expect to have emerged by each development year. With chain ladder factors, the proportion developed to date is 1 ÷ (product of the remaining development factors).

The BF reserve is the a priori ultimate multiplied by the proportion not yet developed. You assume future claims do not depend on claims to date. So a bad or good start does not change the expected future claims. The ultimate is claims to date plus this reserve.

BF suits recent origin years with little data, or classes with volatile claims. For old years, almost all claims have emerged, so BF and chain ladder give similar answers. The weakness of BF is that it depends on the a priori estimate. If that is wrong, the reserve is wrong, and the method ignores real signals in the data.

Key rules to remember

Proportion developed: p = 1 ÷ (f₁ × f₂ × ... × fₖ), where f₁ ... fₖ are the remaining development factors; This is the proportion of ultimate claims expected to have emerged to date. For the latest development year, p = 1 ÷ (product of all remaining factors to ultimate).
BF reserve: Reserve = A × (1 − p); A is the a priori ultimate claims for the origin year. (1 − p) is the proportion still to emerge.
BF ultimate: Ultimate = Claims to date + A × (1 − p); Use the same basis for claims to date and for the pattern: both cumulative paid, or both cumulative incurred.
A priori ultimate: A = Premium × expected loss ratio; Other sources are allowed. State the source and any inflation or exposure adjustment.
Chain ladder ultimate (comparison): Ultimate = Claims to date × f₁ × f₂ × ... × fₖ = Claims to date ÷ p; Chain ladder is fully driven by the data. BF replaces Claims to date ÷ p with a blend.

How to solve Bornhuetter-Ferguson Method questions

Follow this order for any BF question. Keep paid and incurred data separate.

1Check the triangle is cumulative. If it is incremental, accumulate it first.
2Calculate the development factors, for example the volume-weighted ones, as in the chain ladder method. Use any factors given.
3For each origin year, multiply the remaining factors and take the reciprocal. This gives the proportion developed, p.
4Find the a priori ultimate A for each origin year. Use the given loss ratio and premium, or the figure supplied.
5Calculate the reserve as A × (1 − p).
6If asked, add claims to date to get the BF ultimate.
7Sum the reserves across origin years for the total. Comment on the result and compare with chain ladder if asked, and state the assumptions.

Quickest way: Reserve from p and A

When to use it: Use when factors or the developed proportions are given and you need only the reserve.

Write p for each year straight from the pattern. If given cumulative factors to ultimate F, then p = 1 ÷ F.
Compute 1 − p. If F is the factor to ultimate, 1 − p = (F − 1) ÷ F.
Multiply by A. That is the reserve. You do not need claims to date.
Add the reserves for the total. Only add claims to date if the ultimate is asked for.

Common mistakes in Bornhuetter-Ferguson Method

Using p as the proportion still to emerge
The factor 1 ÷ F looks like a reserve percentage.
Fix: p is the proportion already developed. The reserve uses (1 − p). Check that your reserve is smaller than A for any year with some development.
Multiplying the a priori ultimate by the development factor
Students mix BF with chain ladder and apply the factor to the wrong quantity.
Fix: Apply factors only to claims to date in chain ladder. In BF, use them only to get p.
Letting claims to date change the reserve
Students want to scale the reserve by how actual claims compare with expected.
Fix: BF assumes future claims are independent of claims to date. Claims to date enter only through the ultimate, not the reserve.
Mixing paid and incurred bases
The pattern comes from one triangle and the claims to date from another.
Fix: Use one basis throughout. The pattern and the claims to date must both be paid, or both be incurred.
Forgetting to adjust the a priori for inflation or exposure
The loss ratio is applied directly to premium from a different year or level.
Fix: State the basis. Check that the loss ratio matches the origin year's prices and exposure, and adjust if the question says so.
Reporting the ultimate when the reserve is asked
The two numbers sound alike.
Fix: BF reserve = A × (1 − p), which equals ultimate − claims to date on the chosen basis. On a paid basis this gives the total outstanding reserve. On an incurred basis it gives the IBNR. Underline what is asked before you start.

Worked examples

Example 1

For origin year 2024, premium is ₹50,00,000 and the expected loss ratio is 70%. Cumulative paid claims to date are ₹12,00,000. The remaining development factors to ultimate are 1.25, 1.10 and 1.05. Calculate the BF reserve and ultimate.

Show the solution

A = 50,00,000 × 0.70 = ₹35,00,000.
Product of factors = 1.25 × 1.10 × 1.05 = 1.44375.
p = 1 ÷ 1.44375 = 0.69264.
1 − p = 0.30736.
Reserve = 35,00,000 × 0.30736 = ₹10,75,758 (to the nearest rupee).
Ultimate = 12,00,000 + 10,75,758 = ₹22,75,758.

Answer: BF reserve ≈ ₹10,75,758 and BF ultimate ≈ ₹22,75,758.

Example 2

For a recent origin year, cumulative incurred claims to date are ₹8,00,000 and the cumulative factor to ultimate is 2.5. The a priori ultimate is ₹30,00,000, which you take to be a reasonable estimate. Compare the chain ladder and BF ultimates and explain the difference.

Show the solution

Chain ladder ultimate = 8,00,000 × 2.5 = ₹20,00,000.
p = 1 ÷ 2.5 = 0.4, so 1 − p = 0.6.
BF IBNR = 30,00,000 × 0.6 = ₹18,00,000. This is an IBNR, not the total outstanding reserve, because the claims to date are on an incurred basis.
BF ultimate = 8,00,000 + 18,00,000 = ₹26,00,000.
The difference is ₹6,00,000. Chain ladder gives ₹20,00,000 because it trusts the low claims to date and scales them up.
BF gives ₹26,00,000 because it uses the a priori view for the 60% still to emerge.

Answer: Chain ladder ultimate = ₹20,00,000. BF ultimate = ₹26,00,000, with a BF IBNR of ₹18,00,000. BF is higher because it relies on the a priori estimate for future claims, so the low claims to date do not carry forward. This holds only if the a priori of ₹30,00,000 is reasonable.

Exam tips

Write the formula Reserve = A × (1 − p) first. It shows the method and earns marks even if arithmetic slips.
If the question asks you to compare methods, say that chain ladder uses only the data and BF blends in the a priori. Comment on which is more stable for recent years.
State your assumptions: future claims are independent of claims to date, the pattern is stable, and the a priori is reasonable.
In written answers, comment on how reliable the a priori is and what you would check, such as loss ratio source and inflation.
In MCQs, check whether the answer is a reserve or an ultimate. Distractors often use p instead of 1 − p.

Practice questions from Run-off triangles

Bornhuetter-Ferguson Method: frequently asked questions

What is the difference between chain ladder and Bornhuetter-Ferguson?

Chain ladder projects each origin year from its own claims to date using development factors. BF uses an a priori ultimate and the development pattern to estimate only the future claims. BF is less affected by random swings in early claims.

When should I use the Bornhuetter-Ferguson method?

Use it for recent origin years where little has developed, or where claims are volatile. It works best when you have a credible a priori estimate. For older years, chain ladder and BF give similar results.

How do I calculate the BF reserve?

Find the proportion developed p from the development factors, then take 1 − p. Multiply by the a priori ultimate. That product is the reserve. Adding claims to date gives the ultimate.

Where does the a priori ultimate come from?

It is usually premium multiplied by an expected loss ratio. It can also come from pricing, past years adjusted for inflation, or expert judgement. It is independent of the claims in the triangle for that year.

What is the main weakness of the BF method?

It depends on the a priori estimate and ignores what the actual claims to date suggest. If the a priori is poor, the reserve will be biased even when the data clearly points the other way.