The Bornhuetter-Ferguson Estimate as Expected Emergence

Why Use Expected Emergence?

A projection for a recent accident period can be sensitive to how much loss has emerged so far. A pure development-factor projection scales observed losses according to a selected development pattern. The Bornhuetter-Ferguson method takes a different approach: it keeps the observed amount to date, then adds an estimate of the losses expected to emerge in the future.

The expected future amount is based on an expected ultimate loss estimate and a development pattern, rather than on extrapolating the current reported amount alone. This can be useful when an accident period is immature: a relatively small or large observed amount does not, by itself, determine the full projection.

Definition: At a valuation date, let $R$ be cumulative losses observed on a consistent reported or paid basis, $E$ the selected expected ultimate losses on that same basis, and $p$ the selected proportion of ultimate losses expected to have emerged by that date. The Bornhuetter-Ferguson estimated future emergence, or reserve, is $(1-p)E$. The Bornhuetter-Ferguson estimated ultimate is $$ U_{\mathrm{BF}}=R+(1-p)E. $$ Here $1-p$ is the expected unreported or unpaid proportion, as appropriate to the loss basis being used.

The estimate $E$ is selected independently of the current observed losses, or at least is not simply obtained by extrapolating those losses. It might be based on exposure and an expected loss ratio, or on another suitable prior expectation. The method requires that $E$, $R$, and $p$ refer to compatible loss definitions, accident periods, valuation dates, and development bases.

How the Estimate Is Built

The calculation has two pieces. The first is the amount already observed, $R$. The second is expected future emergence, $(1-p)E$. Adding them gives the estimated ultimate. The method does not replace actual observed losses with the expected reported amount $pE$.

Select an expected ultimate.
Choose $E$ on the same basis as the observed losses. For example, an exposure measure multiplied by an expected loss cost or loss ratio can provide an expected ultimate.

Select the emerged proportion.
Use $p$, the proportion expected to have emerged at the current development age under the selected pattern. Thus the expected future proportion is $1-p$.

Estimate future emergence.
Multiply the expected ultimate by the expected future proportion: $(1-p)E$.

Add observed losses.
Set the estimated ultimate equal to observed losses plus estimated future emergence: $U_{\mathrm{BF}}=R+(1-p)E$.

A short justification explains why the unreported proportion is applied to the expected ultimate. If the selected pattern says a proportion $p$ of ultimate losses has emerged, the corresponding expected future portion of an ultimate amount $E$ is $E-pE=(1-p)E$. Bornhuetter-Ferguson uses that expected future portion, then adds what has actually been observed. The expected reported amount $pE$ is not substituted for the actual $R$.

\text{BF reserve}=(1-p)E,\qquad \text{BF ultimate}=R+\text{BF reserve}.

For comparison, under a simple chain-ladder projection with the same selected emerged proportion $p>0$, the observed amount is scaled to ultimate as $U_{\mathrm{CL}}=R/p$. Its implied reserve is $R/p-R$. The two approaches use the development proportion differently: Bornhuetter-Ferguson applies the unreported proportion to $E$, whereas chain ladder applies the inverse emerged proportion to $R$. If $p=0$, this chain-ladder calculation is undefined; if $p=1$, Bornhuetter-Ferguson adds no future emergence.

Interpretation: Bornhuetter-Ferguson is a combination of observed experience to date and expected future emergence. It does not average a chain-ladder ultimate and an expected ultimate. Its formula adds $R$ to a portion of $E$, namely $(1-p)E$.

Worked Calculations

Worked Example: An Immature Reported-Loss Estimate

A hypothetical insurer reviews a recent accident period. Cumulative reported losses are $R=\$510{,}000$. The selected expected ultimate reported losses are $E=\$1{,}200{,}000$, and the selected development pattern indicates that $p=0.40$ of ultimate losses are expected to have been reported. Find the Bornhuetter-Ferguson reserve and ultimate, then compare them with the pure chain-ladder projection. No discounting or mortality assumption is used; this is a deterministic loss projection, and no probability distribution is specified.

Method. Apply the selected future proportion to expected ultimate losses, then add the actual reported amount. This avoids treating the observed reported amount as the sole basis for projecting the unreported portion.

Step 1: Find the expected future proportion.

$$ 1-p=1-0.40=0.60. $$

Step 2: Calculate the Bornhuetter-Ferguson reserve.

(1-p)E=0.60(\$1{,}200{,}000)=\$720{,}000.

Step 3: Calculate the Bornhuetter-Ferguson ultimate.

U_{\mathrm{BF}}=\$510{,}000+\$720{,}000=\$1{,}230{,}000.

Step 4: Compare with chain ladder. Since $p=0.40$, the corresponding development factor to ultimate is $1/p=2.5$. Thus

U_{\mathrm{CL}}=\frac{\$510{,}000}{0.40}=\$1{,}275{,}000, \qquad \text{CL reserve}=\$1{,}275{,}000-\$510{,}000=\$765{,}000.

The Bornhuetter-Ferguson ultimate is $\$45{,}000$ lower than the chain-ladder ultimate, and its reserve is also $\$45{,}000$ lower. As a check, $\$510{,}000+\$720{,}000=\$1{,}230{,}000$. The common mistake avoided here is multiplying observed losses by $1-p$; the expected future portion is calculated from $E$, not from $R$.

Worked Example: Building Expected Ultimate Losses From Premium

A hypothetical specialty cover has earned premium of $\$800{,}000$. The selected expected loss ratio is $65\%$, so the expected ultimate losses are based on premium multiplied by that ratio. At the valuation date, reported losses are $\$420{,}000$, and $75\%$ of ultimate losses are expected to have been reported. Calculate the Bornhuetter-Ferguson reserve and ultimate and compare the ultimate with chain ladder. No discounting or mortality assumption is used, and the calculation is deterministic rather than based on a specified loss distribution.

Method. First calculate $E$ from premium and the selected loss ratio. Then apply the expected unreported proportion to $E$, not to premium alone.

Step 1: Calculate expected ultimate losses.

E=\$800{,}000(0.65)=\$520{,}000.

Step 2: Find the expected unreported proportion and reserve.

1-p=1-0.75=0.25, \qquad \text{BF reserve}=0.25(\$520{,}000)=\$130{,}000.

Step 3: Add reported losses to obtain the ultimate.

U_{\mathrm{BF}}=\$420{,}000+\$130{,}000=\$550{,}000.

Step 4: Calculate the chain-ladder comparison.

U_{\mathrm{CL}}=\frac{\$420{,}000}{0.75}=\$560{,}000, \qquad \text{CL reserve}=\$560{,}000-\$420{,}000=\$140{,}000.

The chain-ladder ultimate exceeds the Bornhuetter-Ferguson ultimate by $\$10{,}000$. The check is that the expected reported amount under the selected expectation is $0.75(\$520{,}000)=\$390{,}000$, while the observed amount is $\$420{,}000$; the BF estimate retains that observed amount and adds the expected $\$130{,}000$ future portion. The mistake avoided is replacing the actual $\$420{,}000$ with the expected reported amount of $\$390{,}000$.

Worked Example: Combining Two Accident Periods

For a hypothetical book, a reserving analyst has two accident periods, with amounts stated in thousands of dollars. For period A, $R_A=610$, $E_A=900$, and $p_A=0.65$. For period B, $R_B=320$, $E_B=1{,}100$, and $p_B=0.30$. Calculate the Bornhuetter-Ferguson reserve and ultimate for each period and in total, then compare the total ultimate with a separate chain-ladder calculation for each period. No discounting or mortality assumption is used; no probability distribution is specified.

Method. Calculate each period separately because its expected ultimate and emerged proportion differ. Add period-level estimates only after completing those calculations.

Period A. The expected future proportion is $1-0.65=0.35$. Therefore,

\text{BF reserve}_A=0.35(900)=315, \qquad U_{\mathrm{BF},A}=610+315=925.

The chain-ladder ultimate and reserve are

U_{\mathrm{CL},A}=\frac{610}{0.65}=938.461538\ldots, \qquad \text{CL reserve}_A=938.461538\ldots-610=328.461538\ldots.

Period B. The expected future proportion is $1-0.30=0.70$. Therefore,

\text{BF reserve}_B=0.70(1{,}100)=770, \qquad U_{\mathrm{BF},B}=320+770=1{,}090.

The chain-ladder ultimate and reserve are

U_{\mathrm{CL},B}=\frac{320}{0.30}=1{,}066.666667\ldots, \qquad \text{CL reserve}_B=1{,}066.666667\ldots-320=746.666667\ldots.

Total. The BF reserves total $315+770=1{,}085$, and the BF ultimates total $925+1{,}090=2{,}015$. The observed total is $610+320=930$, so the check is $930+1{,}085=2{,}015$. The chain-ladder ultimates total $938.461538\ldots+1{,}066.666667\ldots=2{,}005.128205\ldots$. In thousands, rounded to three decimals at the final step, the BF ultimate is $2{,}015.000$, compared with $2{,}005.128$ for chain ladder. The key trap avoided is applying one common emerged proportion to both periods despite their different development maturity.

Assumptions, Use, and Common Traps

The arithmetic is simple; the judgment lies in whether the inputs are suitable. The selected expected ultimate should be reasonable for the accident period and measured on the same basis as $R$. The selected $p$ should correspond to the applicable development age and loss measure. If $R$ is cumulative reported losses, the pattern used for $p$ should describe reported emergence; if $R$ is cumulative paid losses, use a compatible paid pattern. Mixing bases can produce an internally inconsistent estimate.

Bornhuetter-Ferguson is especially useful when little experience has emerged and the observed amount may be volatile. Its future emergence component remains tied to the expected ultimate. As more losses emerge, the observed amount has a larger role in the estimate, while the expected future portion falls as $p$ increases. This does not guarantee a more accurate result: a poor expected ultimate or an inappropriate development pattern can still lead to a poor estimate.

Wrong base for expected emergence. The reserve is $(1-p)E$, not $(1-p)R$.
Subtracting observed losses from expected ultimate for the reserve. $E-R$ is not the BF reserve. The BF reserve is the expected future emergence, $(1-p)E$; the BF ultimate is then $R+(1-p)E$.
Confusing the BF ultimate with the expected ultimate. In general, $U_{\mathrm{BF}}$ need not equal $E$, because the observed amount $R$ replaces the expected emerged portion $pE$.
Using incompatible development patterns. A reported-loss proportion should not be applied mechanically to paid losses, or vice versa.
Treating chain ladder and BF as interchangeable. Chain ladder projects from observed losses; BF adds expected future emergence to observed losses. Their estimates coincide only for particular input values.

Key takeaway: Bornhuetter-Ferguson estimates ultimate losses as observed losses plus the expected unreported portion of an independently selected expected ultimate. Keep the observed amount, expected ultimate, and development proportion on a consistent basis.

Check Your Understanding

For each problem, use the stated proportion as the fraction expected to have emerged by the valuation date. Amounts are hypothetical, and no discounting or mortality assumption is required.

A hypothetical property portfolio has expected ultimate losses of $\$640{,}000$, reported losses of $\$390{,}000$, and an expected reported proportion of $60\%$. Find the BF reserve and ultimate. (Answer: reserve $\$256{,}000$; ultimate $\$646{,}000$)
A hypothetical insurer has earned premium of $\$1{,}000{,}000$, an expected loss ratio of $125\%$, reported losses of $\$980{,}000$, and an expected reported proportion of $80\%$. Find expected ultimate losses, the BF reserve and ultimate, and the chain-ladder ultimate. (Answer: expected ultimate $\$1{,}250{,}000$; BF reserve $\$250{,}000$; BF ultimate $\$1{,}230{,}000$; chain-ladder ultimate $\$1{,}225{,}000$)
For a hypothetical accident period, $E=720$ thousand dollars, $R=250$ thousand dollars, and $p=0.35$. Calculate the BF reserve and ultimate, then the chain-ladder ultimate. Round to three decimals in thousands. (Answer: BF reserve $468.000$, BF ultimate $718.000$, chain-ladder ultimate $714.286$)
Two hypothetical periods have inputs in thousands: period A has $E=400$, $R=330$, $p=0.80$; period B has $E=600$, $R=290$, $p=0.50$. Find each BF ultimate and the total BF reserve and ultimate. (Answer: period A ultimate $410$; period B ultimate $590$; total reserve $380$; total ultimate $1{,}000$)
A hypothetical account has expected ultimate losses of $\$500{,}000$, reported losses of $\$400{,}000$, and an expected reported proportion of $70\%$. Calculate both ultimates and explain which calculation uses the observed losses directly. (Answer: BF ultimate $\$550{,}000$; chain-ladder ultimate $\$571{,}428.57$; BF uses observed losses directly and adds expected future emergence)

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