Bornhuetter-Ferguson: the intuition, then the formula

Chain ladder has a flaw you can feel: for a young accident year, it takes one thin sliver of experience and multiplies it into the whole answer. One early large claim, one late-reported batch, and the projection lurches. Bornhuetter-Ferguson (BF) is the profession’s answer, and it’s more elegant than its name.

The idea in one sentence

Trust your prior expectation for what hasn’t emerged yet; trust actual experience for what has.

The formula

Ultimate (BF) = Paid to date  +  Expected ultimate × (1 − 1/CDF)

where CDF is the cumulative development factor to ultimate, and expected ultimate comes from a prior, usually premium × an expected loss ratio (ELR).

The term $(1 - 1/CDF)$ is just “the fraction of losses we believe hasn’t shown up yet.” BF fills only that unreported portion with the prior, it never re-scales what’s already on the table.

Same numbers, new answer

From the chain ladder lesson: accident year 2025 has paid 1,500 at 12 months, and the CDF to 36m is 1.775 × 1.15 = 2.041. Suppose 2025’s premium was 4,000 with a 70% expected loss ratio → prior ultimate = 2,800.

• Unreported fraction: 1 − 1/2.041 = 0.510
• BF ultimate: 1,500 + 2,800 × 0.510 = 1,500 + 1,428 = 2,928

Chain ladder said 3,062; BF says 2,928. Now imagine 2025’s paid figure had been 1,900 instead (one early jumbo claim): chain ladder rockets to 3,878, while BF moves only to 3,328. That damping is the whole point: actual experience gets added dollar-for-dollar, not multiplied.

The quiet truth

BF is a credibility weighting in disguise: young years lean on the prior, mature years converge to chain ladder automatically as 1/CDF → 1. Which means the method’s honesty depends entirely on the honesty of your ELR, an optimistic prior quietly buries deterioration. Ask any reserving actuary about that meeting.

Practice both on triangles where they disagree, and write down why you chose what you chose. The methods map walks through when each one earns its place.