Why Paid and Incurred BF Estimates Must Be Separated
Bornhuetter-Ferguson (BF) estimates combine an a priori expected ultimate loss with observed experience. The method uses the expected ultimate to estimate the portion that has not yet emerged, while retaining the observed amount for the portion that has emerged. Paid and incurred losses do not emerge on the same schedule: paid losses reflect cash payments, while incurred losses also include case estimates for reported claims. Each measure therefore requires its own emergence pattern.
This distinction affects both the calculation and the meaning of the result. A paid BF estimate projects total future payments. An incurred BF estimate adds estimated IBNR to incurred losses; subtracting paid losses from that incurred ultimate gives total unpaid liability. These quantities should not be given the same label merely because both are called “reserves” in informal discussion.
The emergence pattern must match the observed measure: use \(q^P\) with paid losses and \(q^I\) with incurred losses. If development is provided as an age-to-ultimate factor rather than an emergence percentage, the factor must first be converted to an emerged proportion.
From Development Factors to Emergence Percentages
A cumulative age-to-ultimate factor describes the relationship between the estimated ultimate and the loss measure observed at the current age. If \(F^P\) is the paid age-to-ultimate factor, then the paid emergence percentage is \(q^P=1/F^P\). Similarly, if \(F^I\) is the incurred age-to-ultimate factor, then \(q^I=1/F^I\). In each case the un-emerged share is \(1-1/F\), using the factor for that measure.
For example, an incurred factor of 1.6 implies that incurred losses have emerged to \(1/1.6=0.625\), or 62.5% of ultimate under the selected pattern. The un-emerged share is \(1-0.625=0.375\), or 37.5%. The factor itself is not the unreported percentage.
Worked Example: Converting an Incurred Factor and Reconciling Liability
A hypothetical insurer selects an expected ultimate loss of $2,400,000 for an accident period. At the evaluation date, incurred loss is $1,100,000 and paid loss is $820,000. The incurred age-to-ultimate factor is 1.6. The paid emergence pattern indicates that 55% of ultimate paid loss is expected to have emerged. Calculate incurred BF IBNR, the incurred BF ultimate, and total unpaid liability. Also calculate the paid BF reserve and ultimate using the paid pattern.
Method. Convert the incurred factor into an incurred emergence percentage, apply its un-emerged share to the expected ultimate, and then distinguish IBNR from total unpaid liability. Use the given paid emergence percentage separately for the paid calculation.
Incurred emergence and IBNR. The incurred emerged share is \(1/1.6=0.625\). Therefore the un-emerged incurred share is \(1-0.625=0.375\). Incurred BF IBNR is:
The incurred BF ultimate is incurred loss plus IBNR:
Total unpaid liability is incurred BF ultimate less paid loss:
As a check, case outstanding is \(1{,}100{,}000-820{,}000=280{,}000\). Adding case outstanding and IBNR gives \(280{,}000+900{,}000=1{,}180{,}000\), the same total unpaid liability.
Paid estimate. The paid un-emerged share is \(1-0.55=0.45\). The paid BF reserve is \(2{,}400{,}000 \times 0.45=1{,}080{,}000\), and the paid BF ultimate is \(820{,}000+1{,}080{,}000=1{,}900{,}000\). The paid and incurred BF estimates differ because they use different emergence patterns and observed amounts; they are not required to produce identical ultimates.
Why the BF Formulas Take This Form
The core justification is a partition of expected ultimate loss into an emerged portion and an un-emerged portion. Under the selected pattern for a loss measure, \(q\) is the fraction expected to have emerged by the evaluation date, so \(1-q\) is the fraction expected to remain. BF uses the observed amount for the emerged part and the a priori expected ultimate for the un-emerged part.
For paid losses, the observed amount is \(P\) and the relevant un-emerged share is \(1-q^P\). Thus the projected future payments are \(E(1-q^P)\), and the paid BF ultimate is \(P+E(1-q^P)\). For incurred losses, the observed amount is \(I\) and the relevant un-emerged share is \(1-q^I\). Thus incurred BF IBNR is \(E(1-q^I)\), and the incurred BF ultimate is \(I+E(1-q^I)\). The calculation does not multiply the observed amount by a development factor; the factor is used to determine the expected un-emerged share of \(E\).
For a paid calculation, use cumulative paid loss. For an incurred calculation, use cumulative incurred loss.
Use the paid emergence pattern with paid loss and the incurred emergence pattern with incurred loss.
If given an age-to-ultimate factor \(F\), calculate emergence as \(1/F\), then calculate the un-emerged share as \(1-1/F\).
For paid BF, the un-emerged expected amount is the paid reserve. For incurred BF, it is incurred IBNR.
Paid BF reserve is total projected unpaid payment. Incurred BF IBNR excludes case outstanding; add case outstanding, or subtract paid loss from incurred BF ultimate, to obtain total unpaid liability.
Interpreting the Two Estimates
Paid BF and incurred BF answer related but distinct questions. The paid calculation estimates future payments by adding an expected unpaid-payment amount to cash already paid. This unpaid-payment amount can include future payments on claims already reported as well as payments on claims not yet reported. It is therefore not generally the same as incurred IBNR.
Incurred BF IBNR estimates the portion of ultimate loss not yet represented in incurred loss under the selected incurred pattern. Since incurred loss already includes case outstanding, incurred BF IBNR is not the total amount still to be paid. If case outstanding is positive, total unpaid liability exceeds IBNR by that case outstanding amount.
These calculations assume that \(E\), observed losses, and development patterns are on a consistent basis for the same accident period and evaluation date. The selected pattern must represent the emergence of the measure being projected. The basic formulas are not appropriate without adjustment if, for example, the expected ultimate is on a different loss basis from the observed data or the selected pattern does not match the evaluation age. In those situations, the inputs need to be reconciled before applying BF.
Worked Example: Applying Different Paid and Incurred Patterns
For a hypothetical accident period, the a priori expected ultimate is $1,500,000. Paid loss is $250,000 and incurred loss is $900,000. The selected paid emergence percentage is 30%, while the selected incurred emergence percentage is 70%. Calculate paid BF reserve and ultimate, incurred BF IBNR and ultimate, and incurred-basis total unpaid liability.
Paid BF. Paid un-emerged share is \(1-0.30=0.70\). The paid BF reserve is \(1{,}500{,}000 \times 0.70=1{,}050{,}000\). The paid BF ultimate is \(250{,}000+1{,}050{,}000=1{,}300{,}000\).
Incurred BF. Incurred un-emerged share is \(1-0.70=0.30\). Incurred BF IBNR is \(1{,}500{,}000 \times 0.30=450{,}000\). The incurred BF ultimate is \(900{,}000+450{,}000=1{,}350{,}000\).
Total unpaid liability. Case outstanding is \(900{,}000-250{,}000=650{,}000\). Total unpaid liability is \(650{,}000+450{,}000=1{,}100{,}000\). Subtracting paid loss from the incurred BF ultimate gives the same check: \(1{,}350{,}000-250{,}000=1{,}100{,}000\).
The paid reserve of $1,050,000 is not the incurred IBNR of $450,000. The first estimates all future payments under the paid pattern; the second estimates unincurred loss under the incurred pattern. The common mistake this example avoids is treating both values as interchangeable “IBNR.”
Worked Example: Using Age-to-Ultimate Factors on Both Bases
A hypothetical specialty insurer has expected ultimate loss of $3,600,000, paid loss of $1,400,000, and incurred loss of $2,700,000. The paid age-to-ultimate factor is 1.8 and the incurred age-to-ultimate factor is 1.2. Calculate the paid BF reserve and ultimate, the incurred BF IBNR and ultimate, and total unpaid liability.
Paid pattern. Paid emergence is \(1/1.8=0.5556\) to four decimal places. The un-emerged share is \(1-1/1.8=0.4444\) to four decimal places. Using the unrounded fraction \(4/9\), the paid reserve is \(3{,}600{,}000 \times 4/9=1{,}600{,}000\). The paid BF ultimate is \(1{,}400{,}000+1{,}600{,}000=3{,}000{,}000\).
Incurred pattern. Incurred emergence is \(1/1.2=0.8333\) to four decimal places, so the un-emerged share is \(1-1/1.2=1/6\). Incurred BF IBNR is \(3{,}600{,}000 \times 1/6=600{,}000\). The incurred BF ultimate is \(2{,}700{,}000+600{,}000=3{,}300{,}000\).
Total unpaid liability. Case outstanding is \(2{,}700{,}000-1{,}400{,}000=1{,}300{,}000\). Therefore total unpaid liability is \(1{,}300{,}000+600{,}000=1{,}900{,}000\). The direct check is \(3{,}300{,}000-1{,}400{,}000=1{,}900{,}000\).
The paid and incurred factors cannot be switched: doing so would change the un-emerged share applied to \(E\). This method uses each factor only with its corresponding observed loss measure.
Common Mistakes and Exam Checks
A frequent error is to confuse an emergence percentage with its complement. If an incurred age-to-ultimate factor is 1.6, the emerged share is \(1/1.6=0.625\), not the un-emerged share. The un-emerged share is \(1-0.625=0.375\). For expected ultimate loss of $2,400,000, incurred BF IBNR is therefore \(2{,}400{,}000 \times 0.375=\$900{,}000\), not \(2{,}400{,}000 \times 0.625=\$1{,}500{,}000\).
Another error is to report incurred BF IBNR as total unpaid liability. In the same example, if incurred loss is $1,100,000 and paid loss is $820,000, incurred BF ultimate is $1,100,000+$900,000=$2,000,000. Total unpaid liability is $2,000,000-$820,000=$1,180,000. The $900,000 IBNR estimate alone omits $280,000 of case outstanding.
- Match the basis: paid observations require a paid pattern; incurred observations require an incurred pattern.
- Take the complement: BF uses the un-emerged share of expected ultimate, not the emerged share.
- Label the result precisely: paid BF reserve, incurred BF IBNR, and total unpaid liability are different quantities.
- Check the reconciliation: incurred-basis total unpaid must equal incurred BF ultimate minus paid loss, and also equal case outstanding plus incurred BF IBNR.
Check Your Understanding
For each problem, use the pattern corresponding to the loss measure and distinguish the requested BF amount from total unpaid liability.
- A hypothetical insurer has expected ultimate loss of $900,000, paid loss of $310,000, and a paid emergence percentage of 40%. Find the paid BF reserve and paid BF ultimate. (Answer: paid BF reserve $540,000; paid BF ultimate $850,000.)
- Expected ultimate loss is $2,000,000, incurred loss is $1,250,000, and paid loss is $1,000,000. The incurred age-to-ultimate factor is 2.0. Find incurred BF IBNR and total unpaid liability. (Answer: incurred BF IBNR $1,000,000; total unpaid liability $1,250,000.)
- Expected ultimate loss is $800,000, paid loss is $350,000, and the paid age-to-ultimate factor is 1.25. Find the paid emergence percentage, paid BF reserve, and paid BF ultimate. (Answer: paid emergence 80%; paid BF reserve $160,000; paid BF ultimate $510,000.)
- A hypothetical accident period has expected ultimate loss of $4,500,000, paid loss of $2,100,000, and incurred loss of $3,700,000. Paid emergence is 60% and incurred emergence is 80%. Find the paid BF reserve and ultimate, incurred BF IBNR, incurred BF ultimate, and incurred-basis total unpaid liability. (Answer: paid BF reserve $1,800,000; paid BF ultimate $3,900,000; incurred BF IBNR $900,000; incurred BF ultimate $4,600,000; total unpaid liability $2,500,000.)