Worked Paid Bornhuetter-Ferguson Triangle

Why a Paid Bornhuetter-Ferguson Triangle Needs Two Inputs

A paid-loss triangle shows cumulative amounts paid for each accident year as claims develop. Recent accident years are less mature: their observed paid losses represent only part of the eventual cost. The paid Bornhuetter-Ferguson (BF) method combines those observed payments with an independent estimate of the portion expected to remain unpaid. For each accident year, it therefore needs both an a priori expected ultimate loss and a selected paid emergence percentage at the observed development age.

The method is useful when paid experience is incomplete and the observed amount alone would be unstable as a basis for projecting the ultimate. The key distinction is that BF does not scale the observed paid amount up by a development factor. Instead, it adds an expected unpaid amount to the paid amount already observed.

Definition: For an accident year observed at development age $d$, let $P_d$ be cumulative paid loss to age $d$, $E$ the a priori expected ultimate loss, and $p_d$ the selected paid emergence percentage. The paid BF unpaid reserve estimate is $E(1-p_d)$, and the estimated ultimate loss is $P_d+E(1-p_d)$. The emergence percentage is the selected proportion of ultimate loss expected to have been paid by age $d$.

For a premium-based estimate, the a priori expected ultimate is commonly calculated as earned premium multiplied by a selected expected loss ratio. Premium and expected loss ratio must be on a consistent basis with the losses being estimated. If the expected ultimate has already been provided, there is no need to reconstruct it from premium.

Working assumptions: The selected emergence percentage is appropriate for the accident year and the paid-loss definition being used; the a priori expected ultimate is on the same coverage, valuation, and monetary basis as the paid triangle; and losses are estimated without discounting unless the problem explicitly requires discounting. The calculations do not add a separate estimate of future paid amounts such as claim expenses unless those amounts are included in the stated loss basis.

How the Paid BF Formula Follows from the Loss Split

At a valuation date, ultimate loss can be separated into the cumulative paid amount and the amount still unpaid. The paid emergence selection says that the expected unpaid share is $1-p_d$. Applying that share to the a priori expected ultimate gives the BF estimate of unpaid loss.

\begin{aligned} U &= P_d + \text{unpaid loss},\\ \widehat{\text{unpaid loss}}_{\mathrm{BF}} &= E(1-p_d),\\ \widehat{U}_{\mathrm{BF}} &= P_d + E(1-p_d). \end{aligned}

This construction leaves the observed paid amount in the ultimate estimate at its recorded value. Only the unpaid portion is estimated from the a priori expectation and emergence pattern. That is why it is generally incorrect to estimate the BF ultimate as $P_d/p_d$: that calculation projects the observed paid amount to ultimate, rather than adding expected unpaid loss.

Worked Example: One Accident Year at 12 Months

A hypothetical equipment insurer has earned premium of $1,250,000 for one accident year. Its selected expected loss ratio is 72%. At 12 months, cumulative paid losses are $310,000. The selected paid emergence percentage at 12 months is 35%. Estimate the paid BF unpaid reserve and ultimate loss. Assume the loss ratio and emergence selection apply to this accident year and that amounts are undiscounted.

Method: calculate the a priori expected ultimate. The expected ultimate is earned premium multiplied by the expected loss ratio:

E=\$1{,}250{,}000(0.72)=\$900{,}000.

Calculate the unpaid share. With 35% expected to have emerged as paid, the unpaid share is $1-0.35=0.65$, or 65%.

1-p_{12}=1-0.35=0.65.

Calculate the reserve and ultimate. The reserve estimate is $900,000 multiplied by 65%; the ultimate adds that reserve to the $310,000 already paid.

\begin{aligned} \widehat{\text{unpaid loss}}_{\mathrm{BF}}&=\$900{,}000(0.65)=\$585{,}000,\\ \widehat{U}_{\mathrm{BF}}&=\$310{,}000+\$585{,}000=\$895{,}000. \end{aligned}

Check. The estimated ultimate consists of $310,000 paid and $585,000 unpaid, which totals $895,000. The reserve is not 65% of the observed paid amount; it is 65% of the a priori expected ultimate. A common mistake would be to calculate $310,000 divided by 0.35, or approximately $885,714, which is a paid-loss development projection, not this BF calculation.

Working Across Accident Years and Development Ages

A triangle applies the same calculation separately to each accident year, using the paid amount at that year’s latest observed age and the emergence selection for that age. The a priori expected losses may differ by year because earned premium or the expected loss ratio differs. Keep the triangle’s paid amounts, expected losses, reserves, and ultimates on one consistent unit; the example below uses thousands of dollars throughout.

Worked Example: Completing a Three-Year Paid BF Triangle

A hypothetical insurer has the following cumulative paid-loss triangle, with amounts in $000. The latest available ages are 36 months for AY 2022, 24 months for AY 2023, and 12 months for AY 2024.

Accident year	Paid at 12 months	Paid at 24 months	Paid at 36 months
AY 2022	540	930	1,200
AY 2023	500	900	—
AY 2024	620	—	—

Selected paid emergence percentages are 40% at 12 months, 65% at 24 months, and 80% at 36 months. Earned premium and expected loss ratios are: AY 2022, $2,000 thousand and 60%; AY 2023, $1,800 thousand and 65%; AY 2024, $1,600 thousand and 62.5%. Estimate each year’s expected ultimate, unpaid reserve, and BF ultimate. Assume the selections are appropriate for these losses, and use undiscounted amounts.

Step 1: Calculate a priori expected ultimates.

\begin{aligned} E_{2022}&=2{,}000(0.60)=1{,}200\text{ thousand},\\ E_{2023}&=1{,}800(0.65)=1{,}170\text{ thousand},\\ E_{2024}&=1{,}600(0.625)=1{,}000\text{ thousand}. \end{aligned}

Step 2: Match each accident year to its observed age and emergence. AY 2022 is observed at 36 months, so $p=0.80$ and the unpaid share is 0.20. AY 2023 is observed at 24 months, so $p=0.65$ and the unpaid share is 0.35. AY 2024 is observed at 12 months, so $p=0.40$ and the unpaid share is 0.60.

Step 3: Calculate reserve and ultimate for each row. Apply each unpaid share to that accident year’s expected ultimate, then add the observed paid amount.

\begin{aligned} \text{AY 2022 reserve}&=1{,}200(0.20)=240,\\ \text{AY 2022 ultimate}&=1{,}200+240=1{,}440;\\[4pt] \text{AY 2023 reserve}&=1{,}170(0.35)=409.5,\\ \text{AY 2023 ultimate}&=900+409.5=1{,}309.5;\\[4pt] \text{AY 2024 reserve}&=1{,}000(0.60)=600,\\ \text{AY 2024 ultimate}&=620+600=1{,}220. \end{aligned}

All amounts in those calculations are $000. The completed results are:

Accident year	Observed age	Paid to date ($000)	Expected ultimate ($000)	BF reserve ($000)	BF ultimate ($000)
AY 2022	36 months	1,200	1,200	240	1,440
AY 2023	24 months	900	1,170	409.5	1,309.5
AY 2024	12 months	620	1,000	600	1,220
Total	—	2,720	3,370	1,249.5	3,969.5

Check. The three reserves total $240+409.5+600=1{,}249.5$ thousand. Adding the observed paid total gives $2{,}720+1{,}249.5=3{,}969.5$ thousand, equal to the sum of the row ultimates. The year-specific emergence was essential: assigning the 12-month percentage to AY 2022, for example, would use the wrong maturity for that row.

When the Input Is a Development Factor

Paid emergence may be given directly as a percentage or inferred from selected cumulative paid development factors. A cumulative paid development factor from age $d$ to ultimate is the selected ratio of ultimate paid losses to paid losses at age $d$. If that factor is $F_d$, the corresponding emergence percentage is its reciprocal, $p_d=1/F_d$, provided the factor is defined on that cumulative-to-ultimate basis.

Worked Example: Converting a Cumulative Factor to Emergence

A hypothetical specialty insurer has paid $760,000 at 24 months for an accident year. Its a priori expected ultimate loss is $1,500,000. Selected cumulative paid development factors are 1.25 from 24 to 36 months and 1.20 from 36 months to ultimate. Estimate the paid BF reserve and ultimate. Assume these are cumulative paid factors on a consistent loss basis.

Method: multiply the development factors to obtain the 24-to-ultimate factor. A dollar paid at 24 months is projected through both subsequent development intervals:

F_{24}=1.25(1.20)=1.50.

Convert the factor to emergence. A 24-month paid amount is 1.50 times the corresponding ultimate amount under the selected factor, so the paid emergence percentage is the reciprocal:

p_{24}=\frac{1}{1.50}=\frac{2}{3}=0.6667\text{ (to four decimal places)}.

The unpaid share is $1-\frac{2}{3}=\frac{1}{3}$. Therefore:

\begin{aligned} \widehat{\text{unpaid loss}}_{\mathrm{BF}} &=\$1{,}500{,}000\left(\frac{1}{3}\right)=\$500{,}000,\\ \widehat{U}_{\mathrm{BF}} &=\$760{,}000+\$500{,}000=\$1{,}260{,}000. \end{aligned}

Check. The ultimate is $760,000 paid plus $500,000 unpaid. The common mistake is to treat 1.50 as an emergence percentage, even though it is a development factor greater than one. Taking the reciprocal correctly gives emergence of two-thirds. The final amounts are in dollars to the cent; no rounding was needed.

Practical Checks and Exam Traps

A row-by-row calculation is straightforward, but a plausible-looking result can still be wrong if the development age, expected-loss basis, or meaning of a factor is misread. Before accepting a triangle result, verify the following:

Match age to emergence. Use the percentage for the row’s latest observed development age, not one common percentage for every accident year.
Use the unpaid share. If emergence is $p_d$, the BF reserve uses $1-p_d$, not $p_d$.
Use the expected ultimate for the reserve base. The unpaid share multiplies $E$, not the paid amount $P_d$.
Distinguish factors from percentages. A cumulative-to-ultimate development factor $F_d$ corresponds to emergence $1/F_d$, not $F_d$.
Reconcile totals. Across all accident years, total BF ultimate must equal total paid to date plus total BF unpaid reserve.
Keep the basis consistent. Do not combine an expected loss estimate for one coverage or loss definition with paid amounts and emergence selections for another.

The method’s main strength is also its defining feature: the observed paid experience determines the paid part of the estimate, while the expected loss and selected emergence determine the unpaid part. This makes the a priori expected ultimate and the maturity selection consequential inputs, especially for rows with a large unpaid share. The arithmetic should always preserve that separation.

Key takeaway: For each accident-year row, calculate the a priori expected ultimate, select the paid emergence percentage at that row’s observed age, multiply the expected ultimate by the remaining unpaid share, and add observed cumulative paid loss. Check that the resulting ultimate equals paid to date plus the estimated unpaid reserve.

Check Your Understanding

For each hypothetical case, calculate the requested paid BF result. Amounts are undiscounted unless stated otherwise.

An accident year has earned premium of $900,000, an expected loss ratio of 70%, paid losses of $280,000 at 12 months, and 12-month paid emergence of 40%. What are the expected ultimate, BF reserve, and BF ultimate? (Answer: expected ultimate $630,000; reserve $378,000; ultimate $658,000.)
A row has expected ultimate loss of $2,400 thousand, paid loss of $1,100 thousand, and selected emergence of 75% at its observed age. Calculate the reserve and BF ultimate. (Answer: reserve $600 thousand; ultimate $1,700 thousand.)
A paid triangle row has earned premium of $1,500 thousand and an expected loss ratio of 64%. It is observed at 24 months with $700 thousand paid; the selected 24-month emergence is 56%. Calculate the a priori expected ultimate, reserve, and ultimate. (Answer: expected ultimate $960 thousand; reserve $422.4 thousand; ultimate $1,122.4 thousand.)
At 18 months, paid loss is $840,000 and expected ultimate loss is $1,600,000. Selected cumulative paid factors are 1.10 from 18 to 30 months and 1.25 from 30 months to ultimate. Calculate the 18-month emergence, BF reserve, and BF ultimate. (Answer: emergence $1/1.375=0.7273$ to four decimals; reserve $436,363.64; ultimate $1,276,363.64.)
Two accident years have the following data: AY A has expected ultimate $1,300 thousand, paid $780 thousand, and emergence 60%; AY B has expected ultimate $1,500 thousand, paid $450 thousand, and emergence 30%. Calculate each BF reserve and ultimate, then check the portfolio totals. (Answer: AY A reserve $520 thousand and ultimate $1,300 thousand; AY B reserve $1,050 thousand and ultimate $1,500 thousand; total reserve $1,570 thousand and total ultimate $2,800 thousand.)

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