Bornhuetter-Ferguson Method

Study guide: Exam 5

Bornhuetter-Ferguson Method (BF) estimates ultimate losses by adding to what has actually been reported an estimate of what has not — computed from an a priori expectation and the percentage still unreported.

U_{\text{BF}} = C + \left(1 - \tfrac{1}{\text{CDF}}\right) \times U_{0}

\text{IBNR} = \left(1 - \tfrac{1}{\text{CDF}}\right) \times \text{ELR} \times \text{EP}

• C is reported (or paid) losses to date, U_0 = \text{ELR} \times \text{EP} the a priori expectation, and 1 - 1/\text{CDF} the percentage unreported from the development pattern.
• The key structural property: actual emergence is taken at face value and only the unemerged portion is estimated. The chain ladder instead scales the whole diagonal, so a distorted diagonal is multiplied by the full CDF. BF's answer moves dollar-for-dollar with a reporting anomaly; the chain ladder's moves by CDF-times.
• BF is a credibility weighting in disguise: it equals Z \, U_{\text{CL}} + (1-Z)\,U_0 with Z = 1/\text{CDF}. The more of the year that has emerged, the more the answer relies on the data — automatically, without a separate credibility judgment.
• It is therefore the method of choice at immature maturities and for long-tail lines, converging to the chain ladder as the cohort matures.
• The a priori is the weak point. A stale ELR that ignores rate changes and trend produces a systematically biased reserve — and the bias persists year after year without ever showing up as instability. Where no reliable external ELR exists, the Cape Cod technique derives one from the data instead.
• BF does not respond to bad news. If emergence runs far ahead of the a priori, BF holds most of its estimate anchored — which is exactly right when the emergence is noise and exactly wrong when it is signal. Comparing BF against chain ladder each year, and investigating the gap, is how that is managed.

BF Against Chain Ladder AY 2023: earned premium \$2{,}000{,}000, ELR 65\%, reported losses \$600{,}000, \text{CDF}_{12} = 1.896.

Compute the BF ultimate and compare with the chain ladder.

Answer

\begin{align*}
U_0 &= 0.65 \times \$2{,}000{,}000 = \$1{,}300{,}000 \\[4pt]
\% \text{ unreported} &= 1 - \frac{1}{1.896} = 47.3\% \\[4pt]
\text{IBNR} &= 0.473 \times \$1{,}300{,}000 = \$614{,}900 \\[4pt]
U_{\text{BF}} &= \$600{,}000 + \$614{,}900 = \$1{,}214{,}900
\end{align*}

Chain ladder for comparison:

\$600{,}000 \times 1.896 = \$1{,}137{,}600

BF is \$77{,}300 higher, because emergence so far (\$600K against an expected \$1{,}300{,}000 \times 0.527 = \$685{,}000) is running behind the a priori and BF does not extrapolate the shortfall.

Verifying the credibility form, with Z = 1/1.896 = 0.527:

0.527(\$1{,}137{,}600) + 0.473(\$1{,}300{,}000) = \$1{,}214{,}900 \;\checkmark

Why BF Is Safer at an Immature Age Two accident years, both with \$14{,}000{,}000 of earned premium and a 65\% a priori. At 12 months (\text{CDF} = 3.20, so 31.25\% reported):

• AY A reported \$3{,}000{,}000 — right on expectation.
• AY B reported \$4{,}500{,}000 — inflated by one \$1.5M shock claim.

Compare the methods' responses.

Answer

\begin{align*}
\text{CL}_A &= \$3{,}000{,}000 \times 3.20 = \$9{,}600{,}000 \\
\text{CL}_B &= \$4{,}500{,}000 \times 3.20 = \$14{,}400{,}000 \\[6pt]
\text{BF}_A &= \$3{,}000{,}000 + 0.6875(\$9{,}100{,}000) = \$9{,}256{,}000 \\
\text{BF}_B &= \$4{,}500{,}000 + 0.6875(\$9{,}100{,}000) = \$10{,}756{,}000
\end{align*}

The \$1.5M reporting anomaly moves the chain ladder by \$4{,}800{,}000 and BF by exactly \$1{,}500{,}000 — the amount of the claim itself.

That is the whole argument for BF at immature ages. The chain ladder's proportionality assumption says a year that reports more early will report more throughout; when the extra reporting is one large claim rather than a faster pattern, that assumption manufactures three extra dollars of reserve for every real one.

(Neither answer is complete for AY B: the shock claim should be removed and reserved separately, with a provision for further large claims added back — see Large Loss.)