Paid Bornhuetter-Ferguson Reserves

Paid Bornhuetter-Ferguson: What Is Being Estimated?

For an accident period that is not fully developed, paid losses are only the amounts already disbursed. A paid Bornhuetter-Ferguson (BF) estimate combines those observed payments with an expected estimate of the portion of ultimate losses that remains unpaid. The method is useful when the observed paid amount is not, by itself, considered a sufficiently stable basis for projecting all future payments.

The method requires two inputs for each accident period: an expected ultimate loss amount, selected independently of the current paid amount, and a paid proportion at the period’s current age. The paid proportion describes the share of ultimate losses expected to have been paid by that age. Its complement is the expected unpaid proportion. The expected unpaid amount is the BF reserve estimate; adding it to paid losses gives the BF ultimate estimate.

Definition: Let $P$ be cumulative paid losses to the valuation date, $E$ the expected ultimate losses, and $q$ the expected proportion of ultimate losses still unpaid at the valuation date. The paid Bornhuetter-Ferguson estimate of unpaid losses is $R^{BF}=E q$. The corresponding ultimate estimate is $U^{BF}=P+E q$. If $p$ is the expected paid proportion, then $q=1-p$, so $R^{BF}=E(1-p)$.

Here, “reserve” means the estimated total amount of losses that remains unpaid, not an amount already paid. The expected ultimate $E$ is an input to the calculation; it is not automatically equal to the final BF ultimate. The BF ultimate combines actual paid losses to date with the expected amount still unpaid.

Choosing and Interpreting the Paid Proportion

The paid proportion must correspond to the accident period’s age and the development basis used for the losses. It can be supplied directly as a percentage paid or obtained as the reciprocal of a selected paid age-to-ultimate development factor. If $L$ is that factor, then $L$ represents ultimate paid losses divided by current cumulative paid losses, and $p=1/L$. The unpaid proportion is $q=1-1/L$.

For example, a paid age-to-ultimate factor of 2.00 implies that 50% of ultimate losses are expected to be paid by the current age. It does not imply that 200% is paid, or that 2.00 should be used directly as the unpaid proportion. A selected factor and the proportion it implies must refer to the same valuation age and loss definition as the paid amount.

The expected ultimate and paid development proportion also need compatible bases. The exposure period, coverage, loss definition, and valuation basis should not be mixed casually. The method’s arithmetic is simple, but an inconsistent expected ultimate or development pattern can make the result misleading.

Why the Formula Uses Expected Unpaid Losses

The BF reserve formula follows from the meaning of the unpaid proportion. If $p$ is the expected fraction of ultimate losses already paid, the expected fraction not yet paid is $1-p$. Applying that fraction to the expected ultimate gives the expected unpaid amount. The observed paid amount is not subtracted from expected ultimate to form the BF reserve; it is added afterward to calculate the BF ultimate.

$$ q=1-p $$

R^{BF}=E q=E(1-p)

U^{BF}=P+R^{BF}=P+E(1-p)

This formulation also explains the method’s response to maturity. When a small fraction is expected to remain unpaid, the estimate adds only a small portion of expected ultimate losses to actual paid losses. When a large fraction remains unpaid, more of the expected ultimate is included in the reserve.

If $L$ is a paid age-to-ultimate factor, then $p=1/L$. Substitution gives $R^{BF}=E(1-1/L)$. This conversion is valid when $L$ is specifically the factor from the current age to ultimate on the paid-loss basis being used. A factor between two intermediate ages cannot be treated as an age-to-ultimate factor without first obtaining the appropriate cumulative factor.

R^{BF}=E\left(1-\frac{1}{L}\right), \qquad U^{BF}=P+E\left(1-\frac{1}{L}\right)

Worked Example: Estimating a Paid Reserve and Ultimate

A hypothetical accident period has expected ultimate losses of $700,000. At the valuation date, cumulative paid losses are $360,000, and the selected paid emergence pattern indicates that 60% of ultimate losses are expected to have been paid. Case reserves for reported claims total $110,000. Calculate the BF unpaid estimate, BF ultimate, and implied IBNR.

Method. Use the expected unpaid proportion to estimate the total unpaid amount, then reconcile that estimate to the supplied case reserves.

The paid proportion is $p=0.60$, so the unpaid proportion is $q=1-0.60=0.40$.

R^{BF}=E q=\$700{,}000(0.40)=\$280{,}000

The BF ultimate is the observed paid amount plus the estimated unpaid amount:

U^{BF}=P+R^{BF}=\$360{,}000+\$280{,}000=\$640{,}000

The implied IBNR is the BF ultimate less incurred losses. In this calculation, incurred losses are paid losses plus the given case reserves:

\text{Incurred}=\$360{,}000+\$110{,}000=\$470{,}000

\text{Implied IBNR}=\$640{,}000-\$470{,}000=\$170{,}000

Check. The estimated unpaid amount is $280,000, which equals $110,000 of case reserves plus $170,000 of implied IBNR. Adding the $360,000 already paid gives the $640,000 BF ultimate. The common mistake avoided here is calling the entire $280,000 reserve IBNR: it is the total unpaid estimate, including the case reserves.

Separating the Total Unpaid Estimate

Paid BF estimates total unpaid losses from paid data and the expected ultimate. Paid data alone do not identify how much of that unpaid estimate is already represented by case reserves. If case reserves are supplied on a compatible basis, they can be used to calculate an implied IBNR component.

Definition: If $C$ is the outstanding case-reserve amount at the valuation date, incurred losses are $P+C$. The implied IBNR under paid BF is $U^{BF}-(P+C)$, equivalently $R^{BF}-C$. Thus the BF total unpaid estimate consists of case reserves $C$ plus implied IBNR, provided the amounts use consistent valuation and loss bases.

This is an arithmetic reconciliation, not a separate estimate of case reserves. The case reserve amount is an input, and the residual is the implied IBNR. If the residual is negative, do not silently replace it with zero and claim the original calculation was positive. A negative result indicates that the supplied case reserves exceed the BF total unpaid estimate; review the inputs, bases, or intended interpretation.

Worked Example: Using a Paid Development Factor

A hypothetical accident period has $240,000 in cumulative paid losses. The selected paid age-to-ultimate factor is 2.50, expected ultimate losses are $500,000, and outstanding case reserves are $180,000. Calculate the implied paid proportion, BF reserve, BF ultimate, and implied IBNR. Report dollar amounts to the cent.

Method. Convert the age-to-ultimate factor to a paid proportion before applying the BF formula. Since the factor is ultimate divided by current paid losses, the paid proportion is its reciprocal:

p=\frac{1}{2.50}=0.40, \qquad q=1-0.40=0.60

The BF reserve is the expected ultimate multiplied by the unpaid proportion:

R^{BF}=\$500{,}000(0.60)=\$300{,}000.00

Add current paid losses to obtain the BF ultimate:

U^{BF}=\$240{,}000+\$300{,}000.00=\$540{,}000.00

Incurred losses are paid losses plus case reserves:

P+C=\$240{,}000+\$180{,}000=\$420{,}000

Therefore, the implied IBNR is:

\text{Implied IBNR}=\$540{,}000-\$420{,}000=\$120{,}000.00

Check. The $300,000.00 BF reserve equals $180,000 of case reserves plus $120,000 of implied IBNR. Paid losses of $240,000 plus the reserve of $300,000.00 equal the $540,000.00 ultimate. The common mistake avoided is using 2.50 as the paid proportion; the correct proportion is $1/2.50$.

Applying Paid BF Across Accident Periods

For multiple accident periods, calculate the BF unpaid estimate separately for each period using that period’s expected ultimate and paid proportion. Then sum the period-level estimates. If the periods’ case reserves are also supplied on the same basis, sum them separately and reconcile the aggregate implied IBNR.

A single average paid percentage should not be substituted unless it is appropriately weighted and represents the relevant periods’ development. Each period may have a different maturity and therefore a different unpaid proportion. The following example shows the period-by-period calculation and the aggregate check.

Worked Example: Combining Two Accident Periods

A hypothetical insurer is estimating unpaid losses for two accident periods. All amounts below are in thousands of dollars. For Period A, expected ultimate losses are 800, the paid proportion is 75%, cumulative paid losses are 510, and case reserves are 125. For Period B, expected ultimate losses are 600, the paid proportion is 40%, cumulative paid losses are 210, and case reserves are 140. Calculate the BF reserve, BF ultimate, and implied IBNR by period and in total.

Method. Calculate each period’s unpaid proportion, BF reserve, and ultimate independently; then aggregate the amounts.

For Period A, the unpaid proportion is $1-0.75=0.25$. Thus:

R_A^{BF}=800(0.25)=200,\qquad U_A^{BF}=510+200=710

Period A incurred losses are $510+125=635$. Its implied IBNR is $710-635=75$. The check is $125+75=200$, equal to the BF reserve.

For Period B, the unpaid proportion is $1-0.40=0.60$. Thus:

R_B^{BF}=600(0.60)=360,\qquad U_B^{BF}=210+360=570

Period B incurred losses are $210+140=350$. Its implied IBNR is $570-350=220$. The check is $140+220=360$, equal to the BF reserve.

Now aggregate the period-level amounts. Total paid losses are $510+210=720$; total BF reserve is $200+360=560$; and total BF ultimate is $710+570=1{,}280$. Total case reserves are $125+140=265$, so total implied IBNR is $75+220=295$.

\text{Total incurred}=720+265=985,\qquad \text{Total implied IBNR}=1{,}280-985=295

Check. Total case reserves plus implied IBNR are $265+295=560$, equal to the total BF reserve. Total paid losses plus the reserve are $720+560=1{,}280$, equal to the sum of the period ultimates. The common mistake avoided is applying one period’s paid percentage to both periods.

Common Errors and Practical Checks

Confusing paid and unpaid proportions. If 65% is paid, the BF reserve uses the remaining 35%, not 65%.
Using the development factor as a percentage. For an age-to-ultimate paid factor of 2.00, the implied paid proportion is 50%, not 200%.
Subtracting paid losses from expected ultimate to get the BF reserve. The BF reserve is expected ultimate multiplied by the unpaid proportion. The BF ultimate is then paid losses plus that reserve.
Calling total unpaid losses IBNR. If case reserves are available, total unpaid losses include both case reserves and implied IBNR. Subtract case reserves to obtain the implied IBNR component.
Mixing incompatible inputs. The expected ultimate, paid losses, development proportion, and case reserves must be on consistent accident-period, valuation, and loss bases.

A useful final check is to verify both identities: BF ultimate equals paid losses plus BF reserve, and BF reserve equals case reserves plus implied IBNR when those components are being reconciled. A failed check usually signals an arithmetic error or a mismatch in definitions.

Key takeaway: Paid Bornhuetter-Ferguson estimates total unpaid losses as expected ultimate losses multiplied by the expected unpaid proportion. Add that reserve to paid losses for the BF ultimate. When compatible case reserves are available, subtract them from the total unpaid estimate to obtain implied IBNR.

Check Your Understanding

For each problem, calculate the requested amounts using the paid Bornhuetter-Ferguson method.

A hypothetical accident period has expected ultimate losses of $900,000, paid losses of $520,000, a paid proportion of 70%, and case reserves of $90,000. Find the BF reserve, BF ultimate, and implied IBNR. (Answer: BF reserve $270,000; BF ultimate $790,000; implied IBNR $180,000.)
A hypothetical period has expected ultimate losses of $640,000, paid losses of $320,000, a paid age-to-ultimate factor of 2.00, and case reserves of $100,000. Find the paid proportion, BF reserve, BF ultimate, and implied IBNR. (Answer: paid proportion 50%; BF reserve $320,000; BF ultimate $640,000; implied IBNR $220,000.)
Two hypothetical periods have the following data, in thousands: Period A has expected ultimate 900, paid proportion 60%, paid losses 420, and case reserves 150; Period B has expected ultimate 500, paid proportion 40%, paid losses 180, and case reserves 120. Find the total BF reserve, total BF ultimate, and total implied IBNR. (Answer: total BF reserve 660; total BF ultimate 1,260; total implied IBNR 390, all in thousands.)
A hypothetical period has expected ultimate losses of $800,000, paid losses of $400,000, a paid proportion of 75%, and case reserves of $250,000. Calculate the BF reserve, BF ultimate, and implied IBNR, and interpret the sign of IBNR. (Answer: BF reserve $200,000; BF ultimate $600,000; implied IBNR -$50,000; case reserves exceed the BF total unpaid estimate by $50,000.)

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