Purpose of an Incurred Bornhuetter-Ferguson Triangle
An incurred Bornhuetter-Ferguson (BF) estimate combines two kinds of information: the incurred losses observed so far and an a priori estimate of ultimate losses. Incurred losses include amounts paid and case reserves established for reported claims. The method adds an estimate of the losses that have not yet emerged in the incurred triangle; it does not develop the entire observed incurred amount to ultimate.
The calculation is made separately for each accident year at its current development age. That matters in a triangle: a recent year may have only 12 months of incurred experience, while an older year has substantially more. The selected incurred emergence pattern determines what share of expected ultimate losses remains unreported at each age.
The factors and expected losses must be on a consistent basis. For example, the expected ultimate should correspond to the same accident year, coverage, loss definition, and monetary basis as the incurred triangle. In the examples below, all loss amounts are stated in thousands of dollars unless otherwise noted.
From Incurred Development Factors to Emergence
A cumulative incurred development factor describes how much larger ultimate incurred loss is expected to be than incurred loss at a given age. Under the selected development pattern, if \(F_t^I=2\), then incurred losses at age \(t\) represent one-half of ultimate. More generally, the emerged proportion is the reciprocal of the factor, not the factor itself:
The factors may be selected directly from development experience or built from age-to-age factors. If the selected age-to-age factors are \(f_{t,t+1}^I\), \(f_{t+1,t+2}^I\), and so on, multiply the relevant factors to obtain \(F_t^I\), the cumulative factor from age \(t\) to ultimate. The inverse of that cumulative factor is the portion expected to have emerged by age \(t\).
This construction uses the selected emergence pattern for incurred losses, including both payments and case-reserve estimates. It is not the same as applying a paid emergence pattern to incurred amounts. Nor does it assert that observed incurred losses will develop exactly according to the expected pattern; the pattern is used to estimate the unreported share of the a priori ultimate.
Worked Incurred BF Triangle
Worked Example: Three Accident Years at Different Ages
A hypothetical insurer selects these cumulative incurred development factors: 2.00 from 12 months to ultimate, \(4/3\) from 24 months to ultimate, and \(10/9\) from 36 months to ultimate. The a priori expected ultimates are 1,400 for accident year 2025, 1,200 for accident year 2024, and 1,000 for accident year 2023. At the valuation date, the incurred triangle contains 500 for 2025 at 12 months, 900 for 2024 at 24 months, and 980 for 2023 at 36 months. Calculate each BF ultimate and reserve, then reconcile the totals.
Method: incurred Bornhuetter-Ferguson. First convert each cumulative factor to an emerged proportion and an unreported proportion. Then add expected unreported losses to the observed incurred amount.
| Accident year | Age | Incurred to date \(C_t\) | Expected ultimate \(E\) | \(F_t^I\) | \(p_t\) | \(q_t\) |
|---|---|---|---|---|---|---|
| 2025 | 12 months | 500 | 1,400 | 2.00 | 50% | 50% |
| 2024 | 24 months | 900 | 1,200 | \(4/3\) | 75% | 25% |
| 2023 | 36 months | 980 | 1,000 | \(10/9\) | 90% | 10% |
Accident year 2025. The emerged share is \(1/2.00=0.50\), so the unreported share is \(1-0.50=0.50\). Expected unreported losses are \(1{,}400(0.50)=700\). Thus the BF ultimate is \(500+700=1{,}200\), and the reserve is \(1{,}200-500=700\).
Accident year 2024. The emerged share is \(1/(4/3)=3/4=0.75\), so the unreported share is \(1-0.75=0.25\). Expected unreported losses are \(1{,}200(0.25)=300\). The BF ultimate is \(900+300=1{,}200\), and the reserve is \(1{,}200-900=300\).
Accident year 2023. The emerged share is \(1/(10/9)=9/10=0.90\), leaving an unreported share of \(0.10\). Expected unreported losses are \(1{,}000(0.10)=100\). The BF ultimate is \(980+100=1{,}080\), and the reserve is \(1{,}080-980=100\).
Triangle totals and check. The total incurred amount is \(500+900+980=2{,}380\). Total expected unreported losses are \(700+300+100=1{,}100\), so total BF ultimate is \(2{,}380+1{,}100=3{,}480\). The three reserves total 1,100, and \(3{,}480-2{,}380=1{,}100\), which confirms the reconciliation.
Common mistake avoided: For accident year 2024, using the 75% emerged share as the unreported share would produce an erroneous reserve of \(1{,}200(0.75)=900\) and ultimate of \(900+900=1{,}800\). The correction is to use the complement: \(q_t=1-0.75=0.25\), giving reserve 300 and ultimate 1,200.
Building the A Priori Ultimate from Premium
When the expected ultimate is not supplied directly, a common input is earned premium multiplied by a selected a priori loss ratio. This provides \(E\) for the incurred BF calculation; the observed incurred amount is still added separately. The loss ratio and premium must be selected on a basis consistent with the losses being estimated.
Worked Example: Premium-Based Expectations at Two Maturities
A hypothetical specialty insurer estimates ultimate losses using earned premium and selected a priori loss ratios. Accident year 2026 has earned premium of 1,600 and an expected loss ratio of 62.5%; its incurred losses at 12 months are 460. Accident year 2025 has earned premium of 1,200 and an expected loss ratio of 75%; its incurred losses at 24 months are 720. Use cumulative incurred factors of 2.00 at 12 months and \(4/3\) at 24 months. Calculate the expected ultimate, BF ultimate, and reserve for each year.
Step 1: Calculate each a priori expected ultimate. For accident year 2026, \(E=1{,}600(0.625)=1{,}000\). For accident year 2025, \(E=1{,}200(0.75)=900\).
Step 2: Calculate unreported proportions. At 12 months, \(p_{12}=1/2=0.50\), so \(q_{12}=0.50\). At 24 months, \(p_{24}=1/(4/3)=0.75\), so \(q_{24}=0.25\).
Step 3: Calculate BF estimates. For 2026, expected unreported losses are \(1{,}000(0.50)=500\). The ultimate is \(460+500=960\), and the reserve is \(960-460=500\). For 2025, expected unreported losses are \(900(0.25)=225\). The ultimate is \(720+225=945\), and the reserve is \(945-720=225\).
Check: The total expected ultimate is \(1{,}000+900=1{,}900\), while the total BF ultimate is \(960+945=1{,}905\). This difference is consistent with the method: the observed incurred amounts do not have to equal the expected emerged amounts. Total incurred is \(460+720=1{,}180\), and total reserve is \(500+225=725\); \(1{,}180+725=1{,}905\).
Common mistake avoided: Do not calculate the reserve as the expected ultimate less the observed incurred amount. That would give \(1{,}000-460=540\) for 2026, rather than the BF reserve of 500. The BF reserve is the expected ultimate multiplied by the unreported share; the observed incurred amount is added to that reserve to obtain the BF ultimate.
Using Age-to-Age Factors
The selected factor may be presented as a series of age-to-age factors rather than a cumulative factor to ultimate. In that case, multiply the factors along the path from the triangle's current age to ultimate. Do not take the reciprocal of a single link factor unless that link itself runs from the current age all the way to ultimate.
Worked Example: Converting Selected Links Before Applying BF
For a hypothetical liability coverage, the selected incurred age-to-age factors are 1.40 from 12 to 24 months and 1.25 from 24 months to ultimate. An accident year has incurred losses of 520 at 12 months and an a priori expected ultimate of 700. A second accident year has incurred losses of 760 at 24 months and an expected ultimate of 1,100. Calculate both BF estimates and the combined reserve.
Step 1: Obtain cumulative factors. From 12 months to ultimate, the cumulative factor is \(1.40(1.25)=1.75\). From 24 months to ultimate, it is 1.25.
Step 2: Convert to unreported proportions. At 12 months, \(p_{12}=1/1.75=4/7\), so \(q_{12}=1-4/7=3/7\). At 24 months, \(p_{24}=1/1.25=0.80\), so \(q_{24}=0.20\).
Step 3: Calculate each accident year. At 12 months, expected unreported losses are \(700(3/7)=300\). The BF ultimate is \(520+300=820\), and the reserve is 300. At 24 months, expected unreported losses are \(1{,}100(0.20)=220\). The BF ultimate is \(760+220=980\), and the reserve is 220.
Check: Combined incurred losses are \(520+760=1{,}280\), and the combined reserve is \(300+220=520\). Combined ultimate is \(820+980=1{,}800\), and \(1{,}280+520=1{,}800\). The calculation uses the cumulative factor at each age, not just the 1.40 link for the 12-month observation.
Common mistake avoided: Taking \(1/1.40\) as the 12-month emerged share ignores the additional 24-month-to-ultimate development. The correct factor is \(1.40(1.25)=1.75\), giving an emerged share of \(4/7\) and an unreported share of \(3/7\).
Interpretation and Practical Checks
An incurred BF reserve is the estimated ultimate less incurred losses to date. Since incurred losses already include case reserves for reported claims, this reserve is not simply an estimate of claims that have never been reported. It also reflects expected future development in the incurred value of claims already known. The estimate is useful only when the selected incurred emergence pattern and the a priori expected losses are appropriate for the portfolio and valuation basis.
Before accepting a triangle calculation, check the following:
- Match the age and factor. Use the cumulative incurred factor from the accident year's actual evaluation age to ultimate.
- Use the complement for the reserve share. The unreported proportion is \(1-1/F_t^I\), not \(1/F_t^I\).
- Keep the units aligned. If expected losses are in thousands, incurred amounts and reserves must also be in thousands.
- Reconcile each row. For every accident year, \(U_t^{BF}=C_t+R_t^{BF}\).
- Reconcile the triangle. Total BF ultimate must equal total incurred to date plus total estimated reserves.
These checks catch many mechanical errors, but they do not validate the selected expected ultimate or the selected development pattern. Those remain actuarial assumptions. An incurred BF result is a structured combination of those assumptions and observed incurred experience, not a guarantee that the estimated ultimate will equal the eventual outcome.
Check Your Understanding
For each problem, calculate using incurred Bornhuetter-Ferguson and keep monetary amounts in the units stated.
- An accident year has expected ultimate losses of 800 thousand, incurred losses of 350 thousand at 12 months, and a cumulative incurred factor of 1.25 to ultimate. Calculate the unreported proportion, BF reserve, and BF ultimate. (Answer: unreported proportion \(=0.2000\), reserve \(=160\) thousand, ultimate \(=510\) thousand)
- At 24 months, a hypothetical accident year has incurred losses of 810 thousand and expected ultimate losses of 950 thousand. Its cumulative incurred factor to ultimate is 1.25. Calculate the BF reserve and ultimate, and state the emerged proportion. (Answer: emerged proportion \(=0.8000\), reserve \(=190\) thousand, ultimate \(=1{,}000\) thousand)
- The selected incurred links are 1.40 from 12 to 24 months and 1.25 from 24 months to ultimate. At 12 months, incurred losses are 520 thousand and expected ultimate losses are 700 thousand. Calculate the cumulative factor, unreported proportion, reserve, and BF ultimate. (Answer: cumulative factor \(=1.75\), unreported proportion \(=3/7\), reserve \(=300\) thousand, ultimate \(=820\) thousand)
- For one accident year, expected ultimate is 1,200 thousand, incurred loss at 12 months is 640 thousand, and the cumulative factor is 2.00. For another, expected ultimate is 900 thousand, incurred loss at 24 months is 760 thousand, and the cumulative factor is \(4/3\). Find each reserve and the combined ultimate. (Answer: reserves \(=600\) thousand and \(225\) thousand; combined ultimate \(=2{,}225\) thousand)
- A hypothetical portfolio has earned premium of 2,000 thousand, an a priori loss ratio of 55%, incurred losses of 990 thousand at an age where the cumulative incurred factor to ultimate is 1.10. Calculate expected ultimate, unreported proportion, BF reserve, and BF ultimate. (Answer: expected ultimate \(=1{,}100\) thousand, unreported proportion \(=1/11\), reserve \(=100\) thousand, ultimate \(=1{,}090\) thousand)