Purpose and Inputs
The Bornhuetter-Ferguson (BF) method combines an observed loss amount with an estimate of losses that have not yet emerged. It is useful when the observed experience is incomplete: the method does not simply develop the observed amount to ultimate, but uses the a priori expected ultimate to estimate the portion still unreported. This tutorial focuses on calculating and checking the resulting ultimate losses and reserves.
For each valuation unit, such as an accident year, the calculation requires three inputs on a consistent basis: observed losses to date, an expected ultimate loss estimate, and the selected percentage of ultimate losses expected to have emerged by the valuation date. The expected ultimate is assumed to be available from an appropriate prior estimate; for example, it may be based on exposures and an expected loss cost or on premium and an expected loss ratio. The inputs must use consistent loss definitions, coverage, currency, and valuation basis.
The emergence percentage \(q\) is the share expected to have emerged, not the share still to emerge. Therefore the complement \(1-q\) is used to calculate the reserve. The observed amount \(R\) is then added to that reserve to obtain the BF ultimate.
Why the Formula Reconciles
The calculation follows from splitting the a priori expected ultimate into an expected emerged portion and an expected unreported portion. The selected emergence pattern assigns the fraction \(q\) to the emerged portion and \(1-q\) to the portion not yet emerged. BF uses actual observed losses for the first portion, while retaining the expected amount for the second.
The expected unreported share is \((1-q)U^{\mathrm{exp}}\). The expected emerged share is \(qU^{\mathrm{exp}}\).
The observed amount \(R\), rather than \(qU^{\mathrm{exp}}\), represents the emerged portion in the BF estimate.
The ultimate is \(R+(1-q)U^{\mathrm{exp}}\), and the reserve is the second term. Consequently, ultimate minus observed losses equals the reserve.
This identity is both a derivation and a practical check: a calculated BF ultimate and reserve must reconcile exactly, apart from any displayed rounding. The method does not force the ultimate to equal the a priori expected ultimate. If observed losses differ from the expected emerged amount, the BF ultimate shifts by that difference:
Thus, observed losses above the expected emerged amount raise the BF ultimate above the a priori estimate; observed losses below it lower the BF ultimate. The reserve remains determined by the a priori expected ultimate and the unreported share.
Worked Calculations
Worked Example: One Accident Year With an Unexpected Observed Amount
A hypothetical property coverage has an a priori expected ultimate loss of $1,200,000 for one accident year. At the valuation date, $805,000 of losses have been observed. The selected emergence percentage is 65%. Calculate the BF reserve and ultimate, then check the reconciliation.
Method. Apply the BF expected-unreported-share method. The expected ultimate is $1,200,000, and \(q=0.65\), so the unreported share is \(1-0.65=0.35\).
Reserve calculation. The expected unreported amount is:
Ultimate calculation. Add the observed losses to the reserve:
Reconciliation check. The estimated ultimate less observed losses is \(1{,}225{,}000-805{,}000=420{,}000\), equal to the calculated reserve. The answer is internally consistent. The ultimate exceeds the a priori expected ultimate by $25,000 because observed losses exceed the expected emerged amount of \(0.65(1{,}200{,}000)=780{,}000\) by $25,000.
Common mistake avoided. The reserve is not calculated as the difference between the expected ultimate and observed losses. That difference is \(1{,}200{,}000-805{,}000=395{,}000\), which is not the BF reserve. The BF reserve uses the unreported share of the expected ultimate, giving $420,000.
Worked Example: Aggregating Estimates Across Accident Years
A hypothetical insurer reviews three accident years. Loss amounts are in thousands of dollars. For each year, the observed amount, expected ultimate, and selected emergence percentage are shown. Calculate the reserve and ultimate by year and in total.
| Accident year | Observed losses \(R\) | Expected ultimate \(U^{\mathrm{exp}}\) | Emergence \(q\) |
|---|---|---|---|
| Year A | 760 | 900 | 80% |
| Year B | 690 | 1,200 | 55% |
| Year C | 410 | 1,500 | 25% |
Year A. The unreported share is \(1-0.80=0.20\). The reserve is \(0.20(900)=180\), and the ultimate is \(760+180=940\).
Year B. The unreported share is \(1-0.55=0.45\). The reserve is \(0.45(1{,}200)=540\), and the ultimate is \(690+540=1{,}230\).
Year C. The unreported share is \(1-0.25=0.75\). The reserve is \(0.75(1{,}500)=1{,}125\), and the ultimate is \(410+1{,}125=1{,}535\).
Aggregate. Add the observed losses, reserves, and ultimates separately:
Reconciliation check. \(1{,}860+1{,}845=3{,}705\), so aggregate ultimate equals aggregate observed losses plus aggregate reserve. The three expected ultimates sum to \(900+1{,}200+1{,}500=3{,}600\). The aggregate BF ultimate is $105 thousand above that total because the observed losses exceed the total expected emerged losses by the same amount: \(1{,}860-[0.80(900)+0.55(1{,}200)+0.25(1{,}500)]=1{,}860-1{,}755=105\).
Common mistake avoided. Do not combine the three emergence percentages using a simple average and apply that percentage to the sum of expected ultimates. Each year has a different expected ultimate and emergence pattern, so its reserve is calculated separately before aggregation.
Worked Example: Building Expected Ultimates Before Applying BF
A hypothetical commercial segment has two groups of earned premium. Group 1 has $1,600,000 of earned premium and an expected loss ratio of 55%. Group 2 has $2,000,000 of earned premium and an expected loss ratio of 60%. The selected emergence percentages are 70% and 40%, respectively. Observed losses are $640,000 for Group 1 and $390,000 for Group 2. Calculate the combined BF reserve and ultimate.
Step 1: Calculate expected ultimates. For each group, expected ultimate equals earned premium multiplied by the expected loss ratio:
Step 2: Calculate reserves. Group 1 has an unreported share of \(1-0.70=0.30\), so its reserve is \(0.30(880{,}000)=264{,}000\). Group 2 has an unreported share of \(1-0.40=0.60\), so its reserve is \(0.60(1{,}200{,}000)=720{,}000\).
Step 3: Calculate group ultimates and combine. Group 1's ultimate is \(640{,}000+264{,}000=904{,}000\). Group 2's ultimate is \(390{,}000+720{,}000=1{,}110{,}000\). Therefore:
Reconciliation check. Total observed losses are \(640{,}000+390{,}000=1{,}030{,}000\), and \(1{,}030{,}000+984{,}000=2{,}014{,}000\). The aggregate ultimate reconciles. The method is applied at the group level because the emergence selections differ; the resulting reserves and ultimates can then be summed.
Common mistake avoided. Do not apply one group's expected loss ratio or emergence percentage to the combined premium or losses. Doing so would ignore the stated group-specific assumptions.
Calculation Checks and Interpretation
A disciplined calculation keeps the inputs and outputs distinct. The expected ultimate is an a priori estimate; it is not the final BF ultimate. The emergence percentage determines the fraction of that expected ultimate assigned to the unreported portion. Observed losses enter the ultimate directly, while the reserve is calculated from expected ultimate and the unreported share.
| Check | What to verify |
|---|---|
| Percentage | Use \(1-q\) for the unreported share; confirm that \(q\) is expressed as a decimal in calculations. |
| Basis | Observed losses, expected ultimate, and emergence selection refer to the same coverage and loss measure. |
| Reserve | For nonnegative expected ultimate and \(0\le q\le1\), the calculated reserve is nonnegative and no greater than the expected ultimate. |
| Reconciliation | Verify \(\widehat{U}_{BF}=R+\widehat{I}_{BF}\) for each unit and in aggregate. |
The method has intuitive boundary checks. If \(q=1\), the entire expected ultimate is assumed to have emerged, so the BF reserve is zero and the BF ultimate equals observed losses. If \(q=0\), none is assumed to have emerged, so the reserve equals the expected ultimate and the BF ultimate is observed losses plus that expected amount. These boundary cases are arithmetic checks, not a claim that either selection is suitable for a particular valuation.
A frequent exam error is to mistake the unreported portion of the expected ultimate for the entire ultimate. For example, if \(U^{\mathrm{exp}}=1{,}000\), \(q=0.60\), and \(R=620\), then the reserve is \(0.40(1{,}000)=400\), not the BF ultimate. The correct ultimate is \(620+400=1{,}020\). The incorrect calculation that reports 400 as the ultimate omits observed losses; the correct result passes the check \(1{,}020-620=400\).
Another error is to divide the reserve by the emergence percentage. The formula uses the selected unreported share of expected ultimate, \((1-q)U^{\mathrm{exp}}\), not an adjustment to the observed amount. Before calculating, label the three inputs and write the reserve formula. After calculating, add observed losses and verify the reconciliation. These checks are especially valuable when working across several accident years or segments, where a correct formula can still be applied to the wrong row or percentage.
Check Your Understanding
For each problem, calculate the requested quantity and use the BF reconciliation where applicable.
- A hypothetical liability portfolio has an expected ultimate of $800,000, observed losses of $650,000, and an emergence percentage of 75%. Find the BF reserve and ultimate. (Answer: reserve $200,000; ultimate $850,000.)
- A hypothetical accident year has expected ultimate losses of $1,250,000, an emergence percentage of 40%, and observed losses of $430,000. Calculate the reserve and ultimate. (Answer: reserve $750,000; ultimate $1,180,000.)
- A hypothetical segment has earned premium of $900,000, an expected loss ratio of 68%, observed losses of $280,000, and emergence of 50%. First calculate the expected ultimate, then the BF reserve and ultimate. (Answer: expected ultimate $612,000; reserve $306,000; ultimate $586,000.)
- For two hypothetical groups, Group A has expected ultimate $500,000, emergence 90%, and observed losses $470,000. Group B has expected ultimate $700,000, emergence 30%, and observed losses $180,000. Find the total reserve and ultimate. (Answer: total reserve $540,000; total ultimate $1,190,000.)
- A hypothetical book has an expected ultimate of $2,000,000 and observed losses of $1,100,000. The selected BF ultimate is $1,550,000. Find the BF reserve and the emergence percentage. (Answer: reserve $450,000; emergence percentage 77.5%.)