A Single-Period Bornhuetter-Ferguson Calculation

Purpose of a Single-Period Bornhuetter-Ferguson Estimate

The Bornhuetter-Ferguson method estimates losses that have not yet emerged for an accident period by applying an expected loss amount to the portion of losses expected to remain unreported or unpaid. It uses both an a priori expectation and observed losses: reported experience informs the part already emerged, while the expected loss amount informs the part still to emerge. This can make the estimate less sensitive than a direct projection of reported losses when the period is immature.

This tutorial focuses on one accident period. The expected loss ratio and exposure or premium base must describe the same business, coverage, and loss basis as the reported losses. The age-to-ultimate factor must likewise match the reported or paid measure being used. These alignments are essential: a correct calculation with mismatched inputs is not a meaningful estimate.

Definition: For one accident period, let $E$ be its appropriate exposure or premium measure, let $r$ be the selected expected loss ratio, let $L$ be the age-to-ultimate factor for the selected loss measure, and let $R$ be the losses reported or otherwise emerged to date on that same measure. The expected ultimate loss amount is $E r$. The estimated percentage reported is $1/L$, and the estimated percentage remaining is $1-1/L$. The Bornhuetter-Ferguson unpaid estimate is $$ U_{\mathrm{BF}}=E r\left(1-\frac{1}{L}\right), $$ and the estimated ultimate is $$ \widehat{U}_{\mathrm{ultimate}}=R+U_{\mathrm{BF}}. $$

Here, $L$ is a cumulative age-to-ultimate factor, not the percentage reported. For example, if $L=1.25$, the estimated percentage reported is $1/1.25=0.80$, or 80%, and the percentage remaining is 20%. A factor greater than one therefore corresponds to a reported percentage less than one.

The word “unpaid” depends on the basis of the calculation. With paid losses and paid development factors, the remaining amount estimates losses not yet paid. With incurred or reported losses and a matching development basis, it estimates the amount needed to bring that measure to ultimate. It is not necessarily identical to IBNR: on an incurred basis, some future development can arise from claims already known. In all cases, use one consistent loss basis throughout.

Why the Formula Uses the Remaining Percentage

The method does not project the reported amount by multiplying it by $L$. Instead, it estimates the portion still to emerge from the expected ultimate amount. The age-to-ultimate factor supplies the proportion of the ultimate expected to have emerged by the evaluation date.

Convert the factor to an emerged proportion.
By definition, an age-to-ultimate factor is ultimate divided by the amount at the current age. Thus the current amount is $1/L$ of ultimate, so the estimated percentage reported is $1/L$.

Find the proportion still outstanding.
The reported and remaining proportions add to one. Therefore, the remaining proportion is $1-1/L$.

Apply that proportion to expected ultimate losses.
If the expected ultimate amount is $E r$, the expected amount not yet emerged is $E r(1-1/L)$. Adding the observed amount $R$ gives the estimated ultimate.

This is the core justification: the factor determines the fraction of the expected amount still to emerge, and the method applies that fraction to the expected ultimate amount. The observed amount is then added to the estimate of future emergence. The calculation assumes that the expected loss amount and the selected development pattern are appropriate for this accident period and evaluation date.

Worked Calculations

Worked Example: One Immature Accident Period

A hypothetical insurer has earned premium of $800,000 for one accident period. A selected expected loss ratio is 65%. On the applicable reported-loss basis, the age-to-ultimate factor is 1.25, and $310,000 has been reported. Estimate unpaid losses and ultimate losses. Assume the premium and expected loss ratio are aligned with the reported-loss basis, and that the selected factor is appropriate for this period.

Method. Calculate expected ultimate losses, convert the factor to the percentage remaining, and apply that percentage to expected ultimate losses. The common mistake to avoid is applying the reported percentage to the expected amount when estimating the unpaid portion.

Step 1: Expected ultimate losses. The expected loss ratio is 65%, or 0.65. Therefore, $$ E r=\$800{,}000(0.65)=\$520{,}000. $$

Step 2: Percentage reported and percentage remaining. $$ \text{Percentage reported}=\frac{1}{1.25}=0.80, \qquad \text{Percentage remaining}=1-0.80=0.20. $$

Step 3: Bornhuetter-Ferguson unpaid estimate. $$ U_{\mathrm{BF}}=\$520{,}000(0.20)=\$104{,}000. $$

Step 4: Estimated ultimate. Add reported losses to the estimated amount remaining: $$ \widehat{U}_{\mathrm{ultimate}}=\$310{,}000+\$104{,}000=\$414{,}000. $$

Check. The expected amount not yet reported is 20% of $520,000, which is $104,000. The ultimate estimate reconciles to reported plus unpaid: $310,000 + $104,000 = $414,000. The calculation uses the remaining percentage, not the 80% reported percentage.

Worked Example: A Different Age-to-Ultimate Factor

For a hypothetical accident period, earned premium is $1,200,000 and the expected loss ratio is 72%. The appropriate age-to-ultimate factor for the loss measure is 1.60. Reported losses to date are $470,000. Calculate the estimated percentage remaining, unpaid losses, and ultimate losses. Assume all inputs refer to the same coverage and loss basis.

Method. Use the reciprocal of the age-to-ultimate factor for the reported percentage, then use its complement for the unpaid estimate. A common mistake is to interpret 1.60 as 160% reported; a cumulative factor is not itself a percentage.

Step 1: Expected ultimate losses. $$ E r=\$1{,}200{,}000(0.72)=\$864{,}000. $$

Step 2: Convert the factor to percentages. $$ \text{Percentage reported}=\frac{1}{1.60}=0.625, \qquad \text{Percentage remaining}=1-0.625=0.375. $$ The period is estimated to be 62.5% reported and 37.5% remaining.

Step 3: Estimate unpaid losses. $$ U_{\mathrm{BF}}=\$864{,}000(0.375)=\$324{,}000. $$

Step 4: Estimate ultimate losses. $$ \widehat{U}_{\mathrm{ultimate}}=\$470{,}000+\$324{,}000=\$794{,}000. $$

Check. The estimated remaining amount is 37.5% of $864,000, or $324,000. Adding this to the $470,000 reported gives $794,000. Although reported losses differ from the expected amount already emerged, this method retains the observed reported amount and adds expected future emergence.

Worked Example: Keeping the Loss Basis Consistent

A hypothetical insurer uses an expected loss ratio of 60% on $900,000 of earned premium. On the incurred-loss basis being evaluated, the applicable age-to-ultimate factor is 2.50, and current incurred losses are $250,000. Estimate the remaining amount and ultimate incurred losses. Assume the expected loss ratio, premium, incurred amount, and development factor all refer to the same coverage and valuation basis.

Method. Apply the factor to the incurred measure and use current incurred losses as $R$. The method does not mix paid losses with an incurred factor or substitute a different reported amount. The common mistake is to calculate the complement of the factor itself rather than first taking its reciprocal.

Step 1: Expected ultimate losses. $$ E r=\$900{,}000(0.60)=\$540{,}000. $$

Step 2: Find the percentage remaining. $$ \text{Percentage reported}=\frac{1}{2.50}=0.40, \qquad \text{Percentage remaining}=1-0.40=0.60. $$

Step 3: Estimate the amount remaining. $$ U_{\mathrm{BF}}=\$540{,}000(0.60)=\$324{,}000. $$

Step 4: Estimate ultimate incurred losses. $$ \widehat{U}_{\mathrm{ultimate}}=\$250{,}000+\$324{,}000=\$574{,}000. $$

Check and interpretation. The unpaid or remaining estimate is 60% of the $540,000 expected ultimate amount. The ultimate estimate reconciles as $250,000 current incurred plus $324,000 remaining, equaling $574,000. Because the input is incurred losses, the remaining estimate represents development to ultimate on that basis; it should not automatically be labelled purely unreported claims.

Common Errors and Practical Checks

A single-period calculation is short, but several errors can change the result materially. Before accepting an estimate, verify the factor interpretation, the expected loss amount, and the basis of each input.

Using the factor as a percentage. If $L=1.25$, the reported percentage is 80%, not 125%. Take the reciprocal first.
Applying the wrong proportion. The unpaid estimate uses $1-1/L$, not $1/L$. The latter is the proportion already reported.
Multiplying reported losses by the factor. That is a direct development projection, not the Bornhuetter-Ferguson calculation. The Bornhuetter-Ferguson remaining amount is based on expected ultimate losses.
Mixing bases. A paid development factor does not automatically apply to incurred losses. Similarly, an expected loss ratio on one premium or coverage basis cannot be combined without adjustment with losses on another.
Calling every remaining amount IBNR. The appropriate interpretation depends on whether the underlying measure is paid, reported, or incurred. State what the calculation estimates on the chosen basis.

A useful arithmetic check is to confirm that the reported and remaining percentages sum to 100%, and that estimated ultimate losses equal observed losses plus the estimated remaining amount. A useful modelling check is to ask whether the expected loss ratio and development factor are reasonable for the specific period, coverage, and evaluation date. The arithmetic cannot repair an inappropriate assumption.

Key takeaway: For one accident period, calculate expected ultimate losses as exposure or premium times the aligned expected loss ratio. Convert the age-to-ultimate factor $L$ into a remaining percentage $1-1/L$, apply it to expected ultimate losses, and add the result to the observed losses on the same basis.

This single-period method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. In an exam calculation, show the reciprocal conversion explicitly: it makes clear whether the percentage used is reported or remaining and helps expose the most common factor-interpretation error.

Check Your Understanding

For each hypothetical one-period calculation, state the remaining percentage, Bornhuetter-Ferguson unpaid estimate, and estimated ultimate losses as requested. Assume each stated expected loss ratio and development factor is appropriate for the stated loss basis.

A period has earned premium of $500,000, an expected loss ratio of 68%, an age-to-ultimate factor of 1.25, and $220,000 reported losses. Find unpaid and ultimate losses. (Answer: Remaining percentage 20%; unpaid $68,000; ultimate $288,000.)
A hypothetical portfolio period has premium of $750,000, expected loss ratio 64%, age-to-ultimate factor 1.333333, and reported losses of $390,000. Use the factor as $4/3$. Find unpaid and ultimate losses. (Answer: Remaining percentage 25%; unpaid $120,000; ultimate $510,000.)
A period has an exposure measure of $1,100,000, expected loss ratio 55%, age-to-ultimate factor 2.00, and current losses of $280,000 on the matching basis. Find unpaid and ultimate losses. (Answer: Remaining percentage 50%; unpaid $302,500; ultimate $582,500.)
Expected ultimate losses are $900,000. The applicable factor is 1.666667, treated as $5/3$, and reported losses are $420,000. Find the percentage remaining, unpaid estimate, and ultimate. (Answer: Remaining percentage 40%; unpaid $360,000; ultimate $780,000.)
Earned premium is $600,000, the expected loss ratio is 70%, reported losses are $310,000, and the matching age-to-ultimate factor is 1.40. Find the remaining percentage, unpaid estimate, and ultimate; do not use the factor as the reported percentage. (Answer: Remaining percentage $2/7$, or 28.5714%; unpaid $120,000; ultimate $430,000.)

What You'll Learn