Worked Bornhuetter-Ferguson Portfolio Problem

Purpose and Setup

The Bornhuetter-Ferguson (BF) method combines observed losses with an a priori estimate of expected losses. It is useful when an accident period is immature or its observed experience is too sparse to support a stable projection by itself. The method does not simply apply a development factor to the observed amount: it adds the expected portion of ultimate losses that has not yet emerged.

A portfolio calculation requires a consistent expected-loss basis, a development pattern appropriate to the selected data basis, and a clearly identified maturity for each accident period. The calculations below use hypothetical portfolios. Amounts are stated in thousands of dollars unless noted otherwise.

Definition: For accident period \(i\), let \(E_i\) be its exposure measure (for example, premium), \(r_i\) its selected expected loss ratio, and \(U_i^{(0)}=E_i r_i\) its a priori expected ultimate loss. Let \(F_i\) be the cumulative development factor from the period's current maturity to ultimate on the chosen paid or incurred basis. The implied emerged proportion is \(q_i=1/F_i\), and the implied unreported or unpaid proportion is \(1-q_i\), provided the factor represents development from the stated maturity to ultimate.

The exposure measure and expected loss ratio must be on compatible bases. For example, if exposure is premium in thousands of dollars and the expected loss ratio is 0.65, their product is expected ultimate losses in thousands of dollars. A paid development factor must be used with paid data; an incurred development factor must be used with incurred data.

The Bornhuetter-Ferguson Calculation

On a paid basis, the BF unpaid reserve is the a priori expected ultimate multiplied by the proportion expected to remain unpaid. The BF ultimate is observed cumulative paid losses plus that reserve. This paid-basis reserve is total unpaid loss, not necessarily pure IBNR: it can include amounts for claims already reported but not yet paid.

R_i^{\text{paid}}=U_i^{(0)}\left(1-\frac{1}{F_i^{\text{paid}}}\right), \qquad U_i^{\text{BF}}=P_i+R_i^{\text{paid}}.

Here \(P_i\) is cumulative paid loss to date. On an incurred basis, the same structure estimates IBNR as the expected ultimate multiplied by the proportion not yet reported in incurred losses. The BF ultimate is incurred loss to date plus that IBNR.

\operatorname{IBNR}_i^{\text{BF}} =U_i^{(0)}\left(1-\frac{1}{F_i^{\text{inc}}}\right), \qquad U_i^{\text{BF}}=I_i+\operatorname{IBNR}_i^{\text{BF}},

where \(I_i\) is cumulative incurred loss to date. If cumulative paid losses \(P_i\) are also available, total unpaid loss is \(U_i^{\text{BF}}-P_i\). On an incurred basis, the difference \(I_i-P_i\) is case outstanding; therefore total unpaid equals estimated IBNR plus case outstanding.

Why the formula holds: A cumulative development factor \(F_i\) means that the ultimate amount is projected as \(F_i\) times the amount currently emerged under the selected development pattern. Thus the emerged amount is \(1/F_i\) of ultimate, and the portion not yet emerged is \(1-1/F_i\). BF applies that remaining proportion to the a priori expected ultimate \(U_i^{(0)}\), rather than applying it to the observed amount. Adding the resulting expected remainder to observed losses gives the BF ultimate.

Worked Portfolio Calculations

Worked Example: Paid-Basis BF for Two Accident Periods

A hypothetical insurer has two accident periods. Premium, selected expected loss ratio, current cumulative paid losses, and paid cumulative development factors are shown below. Calculate the BF unpaid reserve and BF ultimate by period and in total. All amounts are in thousands of dollars.

Accident period	Premium	Expected loss ratio	Paid to date	Paid factor to ultimate
2024	1,000	0.65	260	2.00
2025	900	0.65	400	1.25

Method. Calculate expected ultimate losses from premium and the selected loss ratio. Convert each paid factor to an unpaid proportion, then add the BF reserve to paid-to-date losses.

For 2024, the a priori expected ultimate is \(1{,}000(0.65)=650\). The paid proportion emerged is \(1/2.00=0.50\), so the unpaid proportion is \(1-0.50=0.50\). The BF unpaid reserve is \(650(0.50)=325\), and the BF ultimate is \(260+325=585\).

For 2025, the a priori expected ultimate is \(900(0.65)=585\). The paid proportion emerged is \(1/1.25=0.80\), so the unpaid proportion is \(1-0.80=0.20\). The BF unpaid reserve is \(585(0.20)=117\), and the BF ultimate is \(400+117=517\).

The portfolio paid amount is \(260+400=660\), the BF unpaid reserve is \(325+117=442\), and the BF ultimate is \(585+517=1{,}102\). The check \(660+442=1{,}102\) reconciles. The method avoids treating the paid development factor itself as an unpaid percentage.

Worked Example: Incurred-Basis IBNR and Total Unpaid Losses

A hypothetical insurer uses an expected loss ratio of 0.75 for two accident periods. Incurred development factors are selected for the current incurred maturity. Calculate the expected ultimate, BF IBNR, BF ultimate, and total unpaid loss for each period. All amounts are in thousands of dollars.

Accident period	Premium	Incurred to date	Paid to date	Incurred factor to ultimate
2023	800	420	320	1.50
2024	600	300	240	1.25

Method. Use the incurred factors to estimate the not-yet-incurred proportion. Add IBNR to incurred-to-date losses to obtain the BF ultimate, then subtract paid-to-date losses to obtain total unpaid.

For 2023, expected ultimate losses are \(800(0.75)=600\). The emerged proportion is \(1/1.50=2/3\), so the unreported proportion is \(1/3\). BF IBNR is \(600(1/3)=200\). The BF ultimate is \(420+200=620\), and total unpaid is \(620-320=300\). Case outstanding is \(420-320=100\), so the unpaid check is \(200+100=300\).

For 2024, expected ultimate losses are \(600(0.75)=450\). The emerged proportion is \(1/1.25=0.80\), so the unreported proportion is \(0.20\). BF IBNR is \(450(0.20)=90\). The BF ultimate is \(300+90=390\), and total unpaid is \(390-240=150\). Case outstanding is \(300-240=60\), so \(90+60=150\).

In total, expected ultimate losses are \(600+450=1{,}050\), incurred losses are \(420+300=720\), and BF IBNR is \(200+90=290\). Thus BF ultimate is \(720+290=1{,}010\). Paid losses are \(320+240=560\), so total unpaid is \(1{,}010-560=450\). The separate reconciliation is IBNR \(290\) plus case outstanding \(100+60=160\), also \(450\).

Worked Example: Building Cumulative Factors Before Applying BF

A hypothetical insurer has selected incurred age-to-age factors of 1.40 from 12 to 24 months and 1.25 from 24 months to ultimate. Its 2024 accident period is at 24 months; the 2025 and 2026 periods are at 12 months. Premium is 500 for 2024 and 700 for each later period, with an expected loss ratio of 0.60. Incurred-to-date losses are 390, 260, and 280, respectively; paid-to-date losses are 300, 200, and 180. Calculate the BF ultimates, IBNR, and total unpaid losses.

Method. First multiply the age-to-age factors needed to reach ultimate. Then calculate expected ultimate losses and the BF unreported proportion for each period.

At 24 months, the cumulative factor to ultimate is \(1.25\). At 12 months, it is \(1.40(1.25)=1.75\). The 2024 expected ultimate is \(500(0.60)=300\); its unreported proportion is \(1-1/1.25=0.20\). Its IBNR is \(300(0.20)=60\), its BF ultimate is \(390+60=450\), and its total unpaid is \(450-300=150\).

For 2025, expected ultimate is \(700(0.60)=420\). Its unreported proportion is \(1-1/1.75=1-4/7=3/7\). IBNR is \(420(3/7)=180\), BF ultimate is \(260+180=440\), and total unpaid is \(440-200=240\).

For 2026, expected ultimate is also \(420\), and the unreported proportion is \(3/7\). IBNR is \(420(3/7)=180\), BF ultimate is \(280+180=460\), and total unpaid is \(460-180=280\).

Portfolio IBNR is \(60+180+180=420\), and incurred to date totals \(390+260+280=930\). Therefore portfolio BF ultimate is \(930+420=1{,}350\), which also equals \(450+440+460\). Paid to date totals \(300+200+180=680\), so total unpaid is \(1{,}350-680=670\). Case outstanding is \((390-300)+(260-200)+(280-180)=90+60+100=250\); adding IBNR gives \(250+420=670\), as required.

Common Errors and Interpretation

The most important practical check is that the factor, observed loss, and quantity being estimated all use the same basis and maturity. Do not use paid factors with incurred-to-date amounts, or treat an age-to-age factor as though it were already a factor from the current age to ultimate. Multiply successive age-to-age factors when a cumulative factor is required.

A common factor error is to treat \(F-1\) as the unpaid proportion. For the 2024 period in the first example, this mistake gives \(650(2.00-1)=650\) as the reserve. But a factor of 2.00 means paid-to-date is half of ultimate, so the expected unpaid proportion is \(1-1/2=0.50\), and the BF unpaid reserve is \(650(0.50)=325\). The erroneous reserve doubles the correct result.

Another important distinction is the meaning of “unpaid.” On a paid basis, BF unpaid is ultimate less paid and can include both case outstanding and IBNR. On an incurred basis, BF IBNR is ultimate less incurred; total unpaid still requires subtracting paid from ultimate. Calling incurred-basis IBNR “total unpaid” would omit the case outstanding.

Key takeaway: For each accident period, calculate expected ultimate losses, convert the appropriate cumulative factor into an unreported proportion, and add the resulting BF amount to losses observed on the same basis. Reconcile portfolio totals both by summing accident periods and by comparing ultimate with paid or incurred losses.

Check Your Understanding

For each hypothetical problem, use the stated development basis and report amounts in the units given.

A paid-basis period has premium of 400 thousand dollars, an expected loss ratio of 0.70, paid losses of 150 thousand dollars, and a paid factor to ultimate of 1.40. Find the BF unpaid reserve and BF ultimate. (Answer: reserve = 80 thousand dollars; ultimate = 230 thousand dollars.)
An incurred-basis period has expected ultimate losses of 400 thousand dollars, incurred losses of 290 thousand dollars, paid losses of 230 thousand dollars, and an incurred factor to ultimate of 2.00. Find IBNR and total unpaid. (Answer: IBNR = 200 thousand dollars; total unpaid = 260 thousand dollars.)
Selected incurred age-to-age factors are 1.25 from 12 to 24 months and 1.20 from 24 months to ultimate. A period at 12 months has expected ultimate losses of 600 thousand dollars and incurred losses of 200 thousand dollars. Find its cumulative factor to ultimate, BF IBNR, and BF ultimate. (Answer: factor = 1.50; IBNR = 200 thousand dollars; ultimate = 400 thousand dollars.)
Two paid-basis periods have premium and paid data, respectively, of 300 and 100 thousand dollars, and 500 and 250 thousand dollars. Both have an expected loss ratio of 0.60; their paid factors to ultimate are 1.50 and 1.25. Find each BF reserve and the portfolio BF ultimate. (Answer: reserves = 60 and 60 thousand dollars; portfolio ultimate = 470 thousand dollars.)
An incurred-basis period has expected ultimate losses of 500 thousand dollars, incurred losses of 360 thousand dollars, paid losses of 290 thousand dollars, and an incurred factor to ultimate of 1.25. Find IBNR, BF ultimate, case outstanding, and total unpaid, and verify the reconciliation. (Answer: IBNR = 100 thousand dollars; ultimate = 460 thousand dollars; case outstanding = 70 thousand dollars; total unpaid = 170 thousand dollars, equal to IBNR plus case outstanding.)

What You'll Learn