The Bornhuetter-Ferguson Estimate

Purpose and Structure of the Bornhuetter-Ferguson Estimate

For an immature accident period, observed losses do not yet include all claims that will ultimately be paid. A projection based only on observed losses can therefore be sensitive to the period’s early experience. The Bornhuetter-Ferguson method addresses this by estimating the un-emerged portion from an expected ultimate loss amount, while retaining the observed losses to date.

The method requires three pieces of information on a consistent basis: observed cumulative losses, an expected ultimate loss estimate, and the proportion of ultimate losses not yet emerged. The expected ultimate is set independently of the current observed loss amount; the method does not simply apply a development factor to the observed losses.

Definition: For an accident period, let $O$ be observed cumulative losses to date, $E$ be the expected ultimate loss, and $q$ be the proportion of ultimate losses expected to have emerged by the valuation date. The un-emerged proportion is $1-q$. The Bornhuetter-Ferguson ultimate loss estimate is $$ \widehat{U}_{BF}=O+E(1-q). $$ The second term, $E(1-q)$, is the Bornhuetter-Ferguson estimate of losses that have not yet emerged relative to the observed basis.

The amount $E(1-q)$ has a direct interpretation: it is the share of the independently selected expected ultimate that remains to emerge. For paid-to-date data, this amount is an estimate of unpaid losses, so the Bornhuetter-Ferguson unpaid liability is $E(1-q)$. For incurred-to-date data, it estimates the increment not yet included in incurred losses, usually called an IBNR estimate. It is not, by itself, the total unpaid liability when some claims have already been reported and have case reserves.

From Emergence to the Un-Emerged Share

An emergence percentage is the proportion of ultimate losses expected to be present in the observed data at the valuation age. If a development pattern is expressed as a cumulative loss development factor $LDF$ to ultimate, the corresponding emergence proportion is $q=1/LDF$. The un-emerged proportion is therefore $1-1/LDF$. This conversion is valid when the selected factor and the observed losses refer to the same loss measure and valuation age.

q=\frac{1}{LDF}, \qquad 1-q=1-\frac{1}{LDF}.

For example, a factor of $1.25$ means that ultimate losses are estimated to be 1.25 times the observed amount under a development-based projection. Its reciprocal, $0.80$, is the implied emerged share; the un-emerged share is $0.20$. The factor itself is not a percentage and must not be used as the un-emerged share.

Worked Example: Paid Data with an Emergence Percentage

A hypothetical specialty insurer has paid $ \$910{,}000 $ for an accident period. The selected expected ultimate loss is $ \$1{,}200{,}000 $, and the paid emergence percentage at the valuation date is 70%. Estimate ultimate losses and unpaid losses using Bornhuetter-Ferguson. Assume all amounts are on the same loss and valuation basis, with no discounting.

Step 1: Find the un-emerged proportion. The emerged share is $q=0.70$, so the un-emerged share is $1-q=1-0.70=0.30$.

Step 2: Estimate unpaid losses. Apply the un-emerged share to expected ultimate losses: $ \$1{,}200{,}000 \times 0.30 = \$360{,}000 $.

Step 3: Estimate ultimate losses. Add observed paid losses to the estimated unpaid amount: $ \$910{,}000 + \$360{,}000 = \$1{,}270{,}000 $.

Check. Estimated ultimate less paid-to-date is $ \$1{,}270{,}000-\$910{,}000=\$360{,}000 $, equal to the estimated unpaid amount. The method is Bornhuetter-Ferguson; it avoids letting the observed paid amount determine the estimate of losses not yet emerged.

Why the Formula Holds

The formula follows from an accounting decomposition of ultimate losses. Ultimate losses consist of losses already observed plus losses not yet observed. Under the Bornhuetter-Ferguson assumption, the latter component is estimated as the expected ultimate amount multiplied by the proportion still un-emerged.

Decompose the ultimate.
Write ultimate losses as observed losses plus the future, un-emerged portion: $U=O+U_{\text{un-emerged}}$.

Estimate the un-emerged portion.
Use the expected ultimate $E$ and un-emerged share $1-q$: $\widehat{U}_{\text{un-emerged}}=E(1-q)$.

Substitute the estimate.
Then $\widehat{U}_{BF}=O+\widehat{U}_{\text{un-emerged}}=O+E(1-q)$.

This derivation also explains a key feature of the method: the observed amount is added as observed, while only the expected ultimate is multiplied by the un-emerged share. Multiplying the observed amount by a factor to estimate the remaining losses would instead make the un-emerged estimate depend directly on the observed experience.

Paid, Incurred, and Unpaid Amounts

The meaning of “unpaid” depends on the data used. With paid losses, the observed amount is cumulative paid-to-date. Adding the Bornhuetter-Ferguson unpaid estimate to paid-to-date gives the ultimate estimate, and the difference between ultimate and paid is the unpaid liability.

With incurred losses, observed incurred losses generally include paid losses plus case reserves on reported claims. Applying an incurred emergence pattern to expected ultimate losses gives an estimate of the part not yet included in incurred losses. This is an IBNR increment relative to incurred losses. To estimate the total unpaid liability, subtract paid-to-date from the estimated ultimate; that total includes both case outstanding and the IBNR increment.

Worked Example: Two Accident Periods at Different Maturities

A hypothetical insurer estimates paid losses for two accident periods. For Period A, expected ultimate losses are $ \$2{,}000{,}000 $, paid-to-date losses are $ \$1{,}500{,}000 $, and paid emergence is 75%. For Period B, expected ultimate losses are $ \$1{,}200{,}000 $, paid-to-date losses are $ \$360{,}000 $, and paid emergence is 30%. Estimate the Bornhuetter-Ferguson ultimate and unpaid losses for each period and in total. Assume consistent loss definitions and no discounting.

Period A. The un-emerged share is $1-0.75=0.25$. Estimated unpaid losses are $ \$2{,}000{,}000 \times 0.25=\$500{,}000 $. Thus estimated ultimate losses are $ \$1{,}500{,}000+\$500{,}000=\$2{,}000{,}000 $.

Period B. The un-emerged share is $1-0.30=0.70$. Estimated unpaid losses are $ \$1{,}200{,}000 \times 0.70=\$840{,}000 $. Thus estimated ultimate losses are $ \$360{,}000+\$840{,}000=\$1{,}200{,}000 $.

Total. Paid-to-date losses are $ \$1{,}500{,}000+\$360{,}000=\$1{,}860{,}000 $. Estimated unpaid losses are $ \$500{,}000+\$840{,}000=\$1{,}340{,}000 $. The total estimated ultimate is $ \$1{,}860{,}000+\$1{,}340{,}000=\$3{,}200{,}000 $, which also equals $ \$2{,}000{,}000+\$1{,}200{,}000 $, the sum of the two expected ultimates.

Check. For each period, ultimate less paid equals estimated unpaid. The method avoids applying one maturity percentage across accident periods that have different emergence patterns.

Worked Example: Incurred IBNR Versus Total Unpaid Liability

For a hypothetical accident period, incurred losses are $ \$620{,}000 $, paid losses are $ \$500{,}000 $, expected ultimate losses are $ \$800{,}000 $, and the selected incurred emergence percentage is 80%. Estimate the Bornhuetter-Ferguson ultimate, the IBNR increment relative to incurred losses, and the total unpaid liability. Assume all amounts are on a consistent basis and no discounting.

Step 1: Calculate the un-emerged share. It is $1-0.80=0.20$.

Step 2: Estimate the IBNR increment. Multiply expected ultimate by the un-emerged share: $ \$800{,}000 \times 0.20=\$160{,}000 $.

Step 3: Estimate ultimate losses. Add the incurred amount to that increment: $ \$620{,}000+\$160{,}000=\$780{,}000 $.

Step 4: Calculate total unpaid liability. Subtract paid-to-date losses from estimated ultimate: $ \$780{,}000-\$500{,}000=\$280{,}000 $. The difference between incurred and paid is case outstanding: $ \$620{,}000-\$500{,}000=\$120{,}000 $. Thus total unpaid is $ \$120{,}000+\$160{,}000=\$280{,}000 $.

Check. Estimated ultimate less paid equals total unpaid, while the $ \$160{,}000 $ Bornhuetter-Ferguson increment is only the estimated IBNR relative to incurred losses. Confusing these two amounts would omit the $ \$120{,}000 $ already held in case reserves.

Assumptions, Use, and Common Exam Trap

The calculation is meaningful only when the inputs are aligned. The expected ultimate and observed losses must use compatible loss definitions, such as both including the same treatment of loss adjustment expense. The emergence percentage must correspond to the observation basis: a paid emergence pattern for paid data or an incurred emergence pattern for incurred data. Expected losses and observed amounts must also refer to the same accident period, coverage, and valuation basis.

The expected ultimate is an input to the method, not a result inferred from the observed amount. The method estimates the un-emerged portion from that expected amount. Its reliability therefore depends on both the expected ultimate selection and the selected emergence pattern. A wrong expected amount or a mismatched development pattern can materially change the result.

Common exam trap: A loss development factor is not the un-emerged percentage. If $LDF=1.25$, then the emerged proportion is $1/1.25=0.80$, and the un-emerged proportion is $1-0.80=0.20$.

Suppose expected ultimate losses are $ \$900{,}000 $, paid-to-date losses are $ \$640{,}000 $, and $LDF=1.25$. The incorrect calculation treats $1/LDF=0.80$ as the un-emerged share and estimates unpaid losses as $ \$900{,}000 \times 0.80=\$720{,}000 $, producing ultimate losses of $ \$1{,}360{,}000 $. The correction is to calculate the un-emerged share as $1-0.80=0.20$. Estimated unpaid losses are then $ \$900{,}000 \times 0.20=\$180{,}000 $, and estimated ultimate losses are $ \$640{,}000+\$180{,}000=\$820{,}000 $. The check is $ \$820{,}000-\$640{,}000=\$180{,}000 $.

Bornhuetter-Ferguson is a useful way to combine current experience and an expected estimate without extrapolating the observed amount to ultimate. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the un-emerged share before calculating. Keep paid, incurred, and total unpaid amounts distinct.

A hypothetical portfolio has expected ultimate paid losses of $ \$750{,}000 $, paid-to-date losses of $ \$525{,}000 $, and a paid emergence percentage of 70%. Find the Bornhuetter-Ferguson unpaid and ultimate estimates. (Answer: unpaid $ \$225{,}000 $; ultimate $ \$750{,}000 $)
A hypothetical accident period has expected ultimate losses of $ \$1{,}400{,}000 $, paid-to-date losses of $ \$770{,}000 $, and a paid-to-ultimate development factor of 1.60. Find the un-emerged share, unpaid estimate, and ultimate estimate. (Answer: un-emerged share 0.375; unpaid $ \$525{,}000 $; ultimate $ \$1{,}295{,}000 $)
For a hypothetical period, expected ultimate losses are $ \$900{,}000 $, incurred-to-date losses are $ \$540{,}000 $, incurred emergence is 60%, and paid-to-date losses are $ \$410{,}000 $. Find the IBNR increment, estimated ultimate, and total unpaid liability. (Answer: IBNR $ \$360{,}000 $; ultimate $ \$900{,}000 $; total unpaid $ \$490{,}000 $)
Two hypothetical periods have expected ultimates of $ \$600{,}000 $ and $ \$1{,}000{,}000 $, paid-to-date amounts of $ \$290{,}000 $ and $ \$830{,}000 $, and emergence percentages of 50% and 80%, respectively. Find the combined unpaid and ultimate estimates. (Answer: combined unpaid $ \$500{,}000 $; combined ultimate $ \$1{,}620{,}000 $)
A hypothetical period has expected ultimate losses of $ \$2{,}000{,}000 $, paid-to-date losses of $ \$1{,}100{,}000 $, and a paid-to-ultimate factor of 2.50. Find the correct un-emerged share, unpaid estimate, and ultimate estimate. (Answer: un-emerged share 0.60; unpaid $ \$1{,}200{,}000 $; ultimate $ \$2{,}300{,}000 $)

What You'll Learn