Incurred Bornhuetter-Ferguson IBNR

Why Use an Incurred Bornhuetter-Ferguson Estimate?

An incurred loss amount combines payments made to date with case reserves for reported claims. It therefore reflects more information than paid losses alone, but it is still an incomplete view of an accident period’s ultimate losses. The incurred Bornhuetter-Ferguson (BF) method estimates the portion not yet recognized in incurred losses using an expected ultimate and an incurred reported proportion.

The key distinction is between IBNR and total unpaid losses. The BF IBNR amount is added to incurred losses to estimate ultimate losses. Total unpaid losses, however, are the estimated ultimate less payments already made, so they include both IBNR and the case reserves already included in incurred losses.

Definition: For an accident period, let $E$ be expected ultimate losses, $p$ the incurred reported proportion at the valuation date, $I$ incurred losses, and $P$ paid losses. The incurred BF estimate of IBNR is $E(1-p)$. The BF ultimate estimate is $I+E(1-p)$. If incurred losses consist of paid losses plus case reserves, then case reserves are $I-P$, and total unpaid losses are the BF ultimate estimate less $P$.

The incurred reported proportion describes the share of ultimate losses expected to be represented in incurred losses by the valuation date under the selected incurred development pattern. If an incurred cumulative development factor $f$ is given, where $f$ is the ratio of ultimate losses to incurred losses at the selected age, then $p=1/f$. A development factor is generally greater than or equal to one for a pattern with positive future development; it is not itself a percentage reported.

Formula and Interpretation

Under the BF method, the expected ultimate $E$ supplies the basis for estimating the part not yet recognized in incurred losses. The method does not estimate IBNR by simply projecting the observed incurred amount. Instead, it applies the unreported proportion to the expected ultimate:

\text{BF IBNR}=E(1-p)

The BF ultimate estimate retains the actual incurred amount to date, then adds the expected amount not yet reported in incurred losses:

\widehat{U}_{\mathrm{BF}}=I+E(1-p)

In this tutorial, “BF IBNR” means the BF estimate of losses not yet recognized in incurred losses. Depending on the claim-reserve definitions and development pattern, the remaining development can reflect more than claims that have not yet been reported; it can also include future changes to estimates on reported claims. The formula should be interpreted consistently with the incurred data and reporting pattern being used.

Why the Formula Holds

The justification is a partition of the expected ultimate into the part expected to be recognized by the valuation date and the part not yet recognized. If $p$ is the proportion reported, the expected reported component of $E$ is $pE$, leaving $(1-p)E$ unreported. BF uses the actual incurred amount $I$ for the recognized component and the expected amount for the unrecognized component.

$$ E=pE+(1-p)E $$

Replacing the expected reported component $pE$ with actual incurred losses $I$ gives the BF ultimate estimate:

\widehat{U}_{\mathrm{BF}}=I+(1-p)E

This construction explains why the BF IBNR amount is not generally equal to $E-I$. The method adds the expected unreported portion to incurred losses; it does not force the resulting ultimate to equal the expected ultimate. The actual incurred experience can be above or below $pE$.

Worked Calculations

Worked Example: IBNR and Total Unpaid for One Accident Period

A hypothetical insurer has expected ultimate losses of $1,200,000 for an accident period. At the valuation date, 75% of ultimate losses are expected to be represented in incurred losses. Incurred losses are $840,000, of which $600,000 has been paid. Calculate BF IBNR, the BF ultimate estimate, case reserves, and total unpaid losses.

Method. Apply the incurred BF formula to the expected ultimate and unreported proportion, then reconcile the ultimate estimate to paid losses.

Step 1: Find the unreported proportion. The reported proportion is $p=0.75$, so the remaining proportion is $1-p=1-0.75=0.25$.

Step 2: Calculate BF IBNR.

\text{BF IBNR} =1{,}200{,}000(0.25) =300{,}000

Step 3: Calculate the BF ultimate estimate.

\widehat{U}_{\mathrm{BF}} =840{,}000+300{,}000 =1{,}140{,}000

Step 4: Calculate case reserves and total unpaid losses. Case reserves are incurred less paid:

$$ I-P=840{,}000-600{,}000=240{,}000 $$

Total unpaid losses are ultimate less paid:

\widehat{U}_{\mathrm{BF}}-P =1{,}140{,}000-600{,}000 =540{,}000

Check. Case reserves plus BF IBNR equal total unpaid losses: $240{,}000+300{,}000=540{,}000$. The common mistake this avoids is calling the $300,000 IBNR estimate the total unpaid amount; it excludes the $240,000 of case reserves.

Worked Example: Combining Two Accident Periods

For two hypothetical accident periods, an analyst has the following estimates. Period A has expected ultimate losses of $900,000, an incurred reported proportion of 60%, incurred losses of $500,000, and paid losses of $350,000. Period B has expected ultimate losses of $1,500,000, an incurred reported proportion of 80%, incurred losses of $1,250,000, and paid losses of $1,000,000. Calculate each period’s BF IBNR and total unpaid losses, then calculate the combined amounts.

Method. Calculate BF IBNR separately for each period because each has its own expected ultimate and reported proportion. Then calculate ultimate less paid for total unpaid losses.

Period A. The unreported proportion is $1-0.60=0.40$. Thus:

\text{IBNR}_A=900{,}000(0.40)=360{,}000

Its BF ultimate estimate is $500{,}000+360{,}000=860{,}000$. Its case reserves are $500{,}000-350{,}000=150{,}000$, so total unpaid losses are $860{,}000-350{,}000=510{,}000$. The check is $150{,}000+360{,}000=510{,}000$.

Period B. The unreported proportion is $1-0.80=0.20$. Thus:

\text{IBNR}_B=1{,}500{,}000(0.20)=300{,}000

Its BF ultimate estimate is $1{,}250{,}000+300{,}000=1{,}550{,}000$. Its case reserves are $1{,}250{,}000-1{,}000{,}000=250{,}000$, so total unpaid losses are $1{,}550{,}000-1{,}000{,}000=550{,}000$. The check is $250{,}000+300{,}000=550{,}000$.

Combined amounts. Add the period estimates:

\begin{aligned} \text{Combined IBNR}&=360{,}000+300{,}000=660{,}000,\\ \text{Combined case reserves}&=150{,}000+250{,}000=400{,}000,\\ \text{Combined total unpaid}&=510{,}000+550{,}000=1{,}060{,}000. \end{aligned}

Check. Combined IBNR plus combined case reserves is $660{,}000+400{,}000=1{,}060{,}000$, matching combined total unpaid losses. Applying one average reporting percentage to the combined expected ultimate would not generally give the same result; each period’s own proportion must be applied to its own expected ultimate.

Worked Example: Converting an Incurred Development Factor

A hypothetical portfolio has expected ultimate losses of $2,400,000 for an accident period. The selected incurred cumulative development factor at the current age is 1.25. Incurred losses are $1,680,000 and paid losses are $1,200,000. Calculate BF IBNR and total unpaid losses.

Method. Convert the development factor to a reported proportion before applying BF. By definition, the development factor is ultimate divided by the amount incurred at the current age, so the corresponding reported proportion is its reciprocal.

Step 1: Calculate the incurred reported proportion.

p=\frac{1}{1.25}=0.80

The unreported proportion is $1-0.80=0.20$.

Step 2: Calculate BF IBNR and the ultimate estimate.

\begin{aligned} \text{BF IBNR}&=2{,}400{,}000(0.20)=480{,}000,\\ \widehat{U}_{\mathrm{BF}}&=1{,}680{,}000+480{,}000=2{,}160{,}000. \end{aligned}

Step 3: Calculate total unpaid losses. Case reserves are $1{,}680{,}000-1{,}200{,}000=480{,}000$. Total unpaid losses are $2{,}160{,}000-1{,}200{,}000=960{,}000$.

Check and trap. The total unpaid amount also equals $480{,}000+480{,}000=960{,}000$. A common error is to use 1.25 as the reported proportion, producing $2{,}400{,}000(1-1.25)=-600{,}000$ of IBNR. That is not a valid interpretation: 1.25 is a development factor, not a proportion. Its reciprocal, 0.80, is the reported proportion.

Reconcile IBNR to Total Unpaid Losses

Assume incurred losses consist of paid losses plus case reserves, so $I=P+C$, where $C$ denotes case reserves. The BF ultimate estimate is $I+\text{IBNR}$. Subtracting paid losses gives:

\begin{aligned} \text{Total unpaid} &=\widehat{U}_{\mathrm{BF}}-P\\ &=I+\text{IBNR}-P\\ &=(I-P)+\text{IBNR}\\ &=C+\text{IBNR}. \end{aligned}

This identity is a useful check, not a second method for calculating BF IBNR. It shows why the valuation basis matters: incurred losses already include case reserves, whereas paid losses do not. If incurred data are defined differently—for example, if they include adjustments not represented by paid losses plus case reserves—the reconciliation must use the actual data definitions rather than assume $I=P+C$.

Common Exam Traps

Confusing a development factor with a proportion. If the incurred development factor is $f$, use $p=1/f$, then calculate IBNR as $E(1-p)$.
Using incurred losses in place of expected ultimate losses. The BF IBNR calculation uses $E$, not $I$. Incurred losses are added afterward to obtain the BF ultimate estimate.
Calling total unpaid losses IBNR. Total unpaid is BF ultimate less paid. It includes case reserves as well as the BF IBNR amount.
Combining periods before respecting their reported proportions. Calculate $E_j(1-p_j)$ by accident period $j$, then sum the results.
Mixing incompatible bases. Expected ultimate losses, incurred amounts, and the development pattern must use consistent loss definitions and valuation bases for the calculation to be meaningful.

Key takeaway: Incurred BF IBNR is expected ultimate losses multiplied by the proportion not yet represented in incurred losses. Add that IBNR to incurred losses for the BF ultimate estimate; subtract paid losses from the ultimate estimate for total unpaid losses.

The method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. In exam calculations, write down whether a supplied percentage is the reported proportion or the unreported proportion, and state whether the requested reserve is IBNR or total unpaid.

Check Your Understanding

For each problem, calculate the requested estimate and keep IBNR distinct from total unpaid losses.

A hypothetical accident period has expected ultimate losses of $800,000 and an incurred reported proportion of 65%. Calculate BF IBNR. (Answer: $280,000)
An incurred cumulative development factor is 1.60 and expected ultimate losses are $1,120,000. Calculate the incurred reported proportion and BF IBNR. (Answer: Reported proportion $0.625$; BF IBNR $420,000)
Expected ultimate losses are $1,000,000, the incurred reported proportion is 70%, incurred losses are $720,000, and paid losses are $510,000. Calculate BF IBNR, BF ultimate losses, and total unpaid losses. (Answer: IBNR $300,000; BF ultimate $1,020,000; total unpaid $510,000)
Two hypothetical periods have expected ultimate losses of $600,000 and $1,400,000, with incurred reported proportions of 50% and 80%, respectively. Calculate combined BF IBNR. (Answer: $580,000)
A period has expected ultimate losses of $2,000,000, an incurred reported proportion of 90%, incurred losses of $1,850,000, and paid losses of $1,500,000. Calculate BF IBNR, case reserves, and total unpaid losses, and verify the reconciliation. (Answer: IBNR $200,000; case reserves $350,000; total unpaid $550,000; $200,000+$350,000=$550,000)

What You'll Learn