Common Bornhuetter-Ferguson Calculation Errors

Why Bornhuetter-Ferguson Errors Are Easy to Make

A Bornhuetter-Ferguson (BF) estimate combines observed losses with an a priori estimate of losses that have not yet emerged. The calculation is compact, but each input has a distinct role: observed losses supply the emerged portion, while the a priori expected ultimate supplies the estimate of the unreported portion. Confusing those roles can materially distort an ultimate loss or reserve.

The most useful diagnostic is to write down, before calculating, which loss measure is observed, which emergence pattern applies to it, what proportion remains unreported, and what premium basis supports the a priori estimate. The checks in this tutorial are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $\widehat U^{\,a}$ be the a priori expected ultimate loss for an accident year, $C$ the observed cumulative loss measure, and $p$ the expected proportion of ultimate loss emerged at the valuation age for that same measure. The unreported proportion is $q=1-p$. The BF ultimate estimate is $$ \widehat U_{\mathrm{BF}}=C+\widehat U^{\,a}q. $$ For a paid method, $C$ is cumulative paid loss and $p$ comes from a paid emergence pattern. For an incurred method, $C$ is cumulative incurred loss and $p$ comes from an incurred emergence pattern. The estimated unpaid reserve is $\widehat U_{\mathrm{BF}}$ less cumulative paid loss.

The word same matters in both cases. The observed measure and the emergence pattern must describe the same basis. Incurred loss generally includes paid loss plus case outstanding loss, so an incurred BF estimate and a paid BF estimate do not use interchangeable observed amounts or patterns.

Why the Unreported Share Is Used

The BF estimate does not apply the emergence percentage to the a priori ultimate as the amount to add. The emergence percentage identifies the portion expected already to have emerged; the complement identifies the portion expected not yet to have emerged. Since $C$ is already the observed emerged amount, BF adds an estimate only for the portion still unreported.

q=1-p,\qquad \widehat U_{\mathrm{BF}}=C+\widehat U^{\,a}(1-p)

This formula can be justified by partitioning ultimate losses into emerged and unreported components. The observed $C$ replaces an estimate of the emerged component. The a priori ultimate multiplied by $q$ estimates the remaining component. Adding those two components gives the BF ultimate. It would count the emerged portion twice to add $\widehat U^{\,a}p$ to $C$.

Identify the observed loss basis.
Label the observation as paid or incurred; do not treat those cumulative amounts as equivalent.

Select the matching emergence proportion.
Use $p_{\text{paid}}$ with paid losses or $p_{\text{incurred}}$ with incurred losses.

Calculate the unreported share.
Set $q=1-p$, using the matching $p$.

Calculate and label each result.
Compute $\widehat U_{\mathrm{BF}}=C+\widehat U^{\,a}q$, then calculate the unpaid reserve as $\widehat U_{\mathrm{BF}}$ less paid losses.

Worked Calculations and Their Error Checks

Worked Example: Using the Unreported Share

A hypothetical insurer estimates an a priori ultimate of $1,200,000 for an accident year. At the valuation date, the incurred emergence pattern indicates that 60% of ultimate incurred losses should have emerged. Actual incurred losses are $720,000, and cumulative paid losses are $650,000. Calculate the incurred BF ultimate and unpaid reserve. Then identify the error in using the emerged percentage as the BF complement.

Method. This is an incurred BF estimate, so use the incurred observation and its incurred emergence percentage. The emerged proportion is $p=0.60$, so the unreported proportion is $$ q=1-0.60=0.40. $$

The a priori estimate of unreported losses is $$ \$1{,}200{,}000(0.40)=\$480{,}000. $$ Add this amount to observed incurred losses: $$ \widehat U_{\mathrm{BF}}=\$720{,}000+\$480{,}000=\$1{,}200{,}000. $$ The unpaid reserve is the BF ultimate less cumulative paid losses: $$ \widehat R_{\mathrm{BF}}=\$1{,}200{,}000-\$650{,}000=\$550{,}000. $$

Check. The estimate consists of $720,000 incurred and $480,000 estimated unreported, totaling $1,200,000. The reserve also equals $70,000 of incurred-but-unpaid loss ($\$720{,}000-\$650{,}000$) plus $480,000 of estimated unreported loss, or $550,000.

Common mistake avoided. Using 60% as the complement gives $\$720{,}000+\$1{,}200{,}000(0.60)=\$1{,}440{,}000$, which counts the emerged share of the a priori ultimate as additional loss. The complement is 40%, not 60%.

Worked Example: Matching Paid and Incurred Patterns

For a hypothetical accident year, the a priori expected ultimate is $1,000,000. Cumulative paid losses are $500,000 and cumulative incurred losses are $700,000. At the valuation age, the paid emergence pattern indicates 50% emerged, while the incurred emergence pattern indicates 70% emerged. Calculate each BF ultimate and the corresponding unpaid reserve. Also calculate the incorrect incurred ultimate that results from using the paid emergence pattern.

Paid method. Use paid losses with $p_{\text{paid}}=0.50$. Then $q_{\text{paid}}=1-0.50=0.50$, and $$ \widehat U_{\mathrm{BF,paid}} =\$500{,}000+\$1{,}000{,}000(0.50) =\$1{,}000{,}000. $$ The paid-method reserve is $\$1{,}000{,}000-\$500{,}000=\$500{,}000$.

Incurred method. Use incurred losses with $p_{\text{incurred}}=0.70$. Then $q_{\text{incurred}}=1-0.70=0.30$, and $$ \widehat U_{\mathrm{BF,incurred}} =\$700{,}000+\$1{,}000{,}000(0.30) =\$1{,}000{,}000. $$ The incurred-method reserve is $\$1{,}000{,}000-\$500{,}000=\$500{,}000$. The reserve subtracts paid losses, even though the BF calculation used incurred losses.

Pattern-mismatch error. If the paid pattern is incorrectly applied to incurred losses, the result is $$ \$700{,}000+\$1{,}000{,}000(1-0.50)=\$1{,}200{,}000. $$ That calculation pairs an incurred observation with a paid emergence proportion and produces a $200,000 higher ultimate than either correctly matched calculation.

Check. The paid and incurred estimates agree here because the hypothetical data and emergence proportions are internally consistent: $500,000 is 50% of $1,000,000, and $700,000 is 70% of $1,000,000. In practice, the estimates need not agree exactly.

Worked Example: Separating IBNR from the Total Unpaid Reserve

A hypothetical insurer has $800,000 of a priori expected ultimate incurred loss for an accident year. Incurred losses are $560,000, the matching incurred emergence percentage is 70%, and paid losses are $500,000. Calculate the BF ultimate, estimated unreported increment, and total unpaid reserve. Show how adding the unreported increment twice creates an error.

The unreported proportion is $q=1-0.70=0.30$. The BF estimate of the unreported increment is $$ \$800{,}000(0.30)=\$240{,}000. $$ Therefore, $$ \widehat U_{\mathrm{BF}}=\$560{,}000+\$240{,}000=\$800{,}000. $$ The total unpaid reserve is $$ \widehat R_{\mathrm{BF}}=\$800{,}000-\$500{,}000=\$300{,}000. $$

The $240,000 increment is the estimated unreported amount added to incurred loss to form the ultimate. It is not the whole unpaid reserve: the $60,000 difference between incurred and paid losses is also unpaid. Equivalently, the reserve is $(\$560{,}000-\$500{,}000)+\$240{,}000=\$300{,}000$.

Common mistake avoided. Adding the $240,000 increment again after calculating the reserve gives $\$300{,}000+\$240{,}000=\$540{,}000$. That double counts the BF unreported increment, which was already included in the ultimate before paid losses were subtracted.

Check. The unpaid reserve comprises $60,000 already incurred but unpaid plus $240,000 estimated unreported: $\$60{,}000+\$240{,}000=\$300{,}000$.

Worked Example: Keeping the Premium Basis Consistent

A hypothetical portfolio has $2,000,000 of earned premium at historical rates. A factor of 1.10 converts that premium to the selected current rate level. The selected a priori loss ratio is 65% on that current rate level. Incurred losses are $900,000, paid losses are $820,000, and the matching incurred emergence percentage is 60%. Calculate the BF ultimate and unpaid reserve. Then show the result of incorrectly applying the a priori loss ratio to unadjusted premium.

Step 1: Calculate the a priori expected ultimate. The premium basis is current rate level, so first calculate the corresponding premium: $$ \$2{,}000{,}000(1.10)=\$2{,}200{,}000. $$ Then apply the 65% a priori loss ratio: $$ \widehat U^{\,a}=\$2{,}200{,}000(0.65)=\$1{,}430{,}000. $$

Step 2: Calculate the incurred BF estimate. The unreported proportion is $q=1-0.60=0.40$, so the expected unreported amount is $$ \$1{,}430{,}000(0.40)=\$572{,}000. $$ Thus, $$ \widehat U_{\mathrm{BF}}=\$900{,}000+\$572{,}000=\$1{,}472{,}000. $$

Step 3: Calculate the unpaid reserve. $$ \widehat R_{\mathrm{BF}}=\$1{,}472{,}000-\$820{,}000=\$652{,}000. $$

Premium-basis error. Applying 65% to unadjusted premium gives $\$2{,}000{,}000(0.65)=\$1{,}300{,}000$, an a priori ultimate that does not match the stated current-rate-level loss ratio. Its resulting BF ultimate would be $\$900{,}000+\$1{,}300{,}000(0.40)=\$1{,}420{,}000$, and its reserve would be $\$1{,}420{,}000-\$820{,}000=\$600{,}000$. Both are $52,000 below the consistent-basis results.

Check. The difference is explained by the $130,000 difference between the two a priori ultimates multiplied by the 40% unreported share: $(\$1{,}430{,}000-\$1{,}300{,}000)(0.40)=\$52{,}000$.

A Practical Audit Before You Report a Result

BF inputs should be consistent not only within the emergence calculation, but also across the a priori estimate. If the a priori ultimate is obtained from premium multiplied by an a priori loss ratio, the premium period and basis must be appropriate for that loss ratio. For example, a loss ratio selected for current rate-level premium should not be multiplied by premium that has not been brought to that basis. Likewise, confirm that the premium and expected loss refer to the exposure period being estimated. Do not apply a rate-level adjustment twice if the premium has already been adjusted.

The following checks help expose errors before they flow into a reserve selection:

Complement check: Confirm that $p+q=1$, and use $q$, not $p$, for expected unreported losses.
Pattern check: Match paid observations to paid emergence and incurred observations to incurred emergence.
Quantity check: Label the BF increment, BF ultimate, and unpaid reserve separately. The increment is not automatically the total reserve.
Subtraction check: Calculate an unpaid reserve as BF ultimate less paid losses, not less incurred losses.
Basis check: Verify that the premium basis is consistent with the a priori loss ratio and the accident year being estimated.

A final reasonableness check is to reconstruct the result from its components. For a paid method, the ultimate is paid loss plus estimated unreported loss. For an incurred method, the reserve is incurred-but-unpaid loss plus estimated unreported loss. These identities do not guarantee that the assumptions are appropriate, but they can reveal arithmetic or classification errors.

Key takeaway: In a BF calculation, observed losses represent the emerged component and the a priori expected ultimate supplies the unreported component. Use the matching emergence pattern, add the unreported estimate only once, subtract paid losses to obtain the unpaid reserve, and keep the a priori premium and loss ratio on a consistent basis.

Check Your Understanding

For each problem, identify the relevant BF inputs and check whether the requested result is an ultimate or an unpaid reserve.

A hypothetical insurer has an a priori expected ultimate of $900,000. Paid losses are $390,000, and the paid emergence pattern indicates 45% emerged. Calculate the paid BF ultimate and unpaid reserve. (Answer: Ultimate $885,000; unpaid reserve $495,000.)
A hypothetical accident year has an a priori expected ultimate of $1,200,000, incurred losses of $930,000, paid losses of $800,000, and an incurred emergence percentage of 75%. Calculate the incurred BF ultimate and unpaid reserve. (Answer: Ultimate $1,230,000; unpaid reserve $430,000.)
A hypothetical insurer has expected ultimate incurred losses of $750,000, incurred losses of $480,000, paid losses of $420,000, and a matching incurred emergence percentage of 60%. Calculate the BF ultimate and total unpaid reserve. (Answer: Ultimate $780,000; unpaid reserve $360,000.)
Historical earned premium is $1,500,000. A factor of 1.08 converts it to the selected rate level, and the a priori loss ratio on that basis is 64%. Paid losses are $310,000, and the paid emergence percentage is 40%. Calculate the paid BF ultimate and unpaid reserve. (Answer: A priori expected ultimate $1,036,800; BF ultimate $932,080; unpaid reserve $622,080.)
A hypothetical accident year has an a priori expected ultimate of $2,000,000, incurred losses of $1,400,000, paid losses of $1,100,000, incurred emergence of 70%, and paid emergence of 55%. Calculate both correctly matched BF ultimates and the incurred BF reserve. What incurred ultimate results from incorrectly using the paid pattern? (Answer: Paid BF ultimate $2,000,000; incurred BF ultimate $2,000,000; incurred BF reserve $900,000; mismatched incurred ultimate $2,300,000.)

What You'll Learn