Paid and Incurred Bases in Method Selection

Why the Development Basis Matters

Bornhuetter-Ferguson and Cape Cod both combine observed claims experience with an expected ultimate loss estimate. But the observed experience can be paid losses or incurred losses, and the development pattern must match that choice. Paid losses emerge as cash is disbursed; incurred losses include paid amounts plus case reserves. Because these amounts develop differently, their reported proportions are not interchangeable.

This distinction also affects what a calculated reserve represents. An incurred-basis Bornhuetter-Ferguson calculation estimates IBNR beyond the incurred amount. A paid-basis calculation estimates all unpaid losses beyond paid losses; to isolate IBNR from that result, subtract case reserves. Cape Cod uses the same basis-matching principle when it estimates its loss ratio.

Definition: For accident period $i$, let $P_i$ be cumulative paid losses, $I_i$ cumulative incurred losses, and $C_i=I_i-P_i$ case reserves at the valuation date. Let $E_i$ denote expected ultimate losses, and let $q_i^B$ be the reported proportion on basis $B$, where $B$ is paid or incurred. If $F_i^B$ is the selected cumulative development factor to ultimate on that basis, then $q_i^B=1/F_i^B$, provided the factor and reported proportion refer to the same valuation age and ultimate.

The calculations below treat all loss amounts as nominal dollars at the valuation date and assume no discounting. Expected ultimate losses, observed losses, and development proportions are assumed to be aligned by accident period, coverage, and loss definition. The selected development factors are inputs; the focus here is how to use the resulting paid or incurred reported proportions.

Bornhuetter-Ferguson on Paid and Incurred Bases

The Bornhuetter-Ferguson (BF) method estimates the unreported share of expected ultimate losses using the selected basis's reported proportion. Write $\widehat{R}_i^B=E_i(1-q_i^B)$ for the BF amount not yet reported on basis $B$.

\begin{aligned} \text{Paid-basis total unpaid}_i &= E_i(1-q_i^P),\\ \text{Paid-basis IBNR}_i &= E_i(1-q_i^P)-C_i,\\ \text{Incurred-basis IBNR}_i &= E_i(1-q_i^I),\\ \text{Total unpaid}_i &= \text{IBNR}_i+C_i. \end{aligned}

On the paid basis, the estimate $E_i(1-q_i^P)$ is the amount expected to remain unpaid, including any case reserves already established. Subtracting $C_i$ gives IBNR, if the result is positive and the case reserve definition is consistent with the data. On the incurred basis, the observed incurred amount already includes case reserves, so $E_i(1-q_i^I)$ is IBNR beyond incurred losses. Total unpaid is then IBNR plus case reserves.

Derivation: If the ultimate loss is expected to be $E_i$, and the selected basis is expected to have reported proportion $q_i^B$, the unreported share of that expected ultimate is $1-q_i^B$. Thus the expected unreported amount is $E_i(1-q_i^B)$. On an incurred basis this amount is IBNR. On a paid basis it is total unpaid, because it is the amount not yet paid; known case reserves must be removed to find IBNR alone.

Worked Example: BF Estimates on Both Bases

A hypothetical book has two accident periods. Expected ultimate losses are based on premiums of $100,000 and $200,000 and an expected loss ratio of 60%. For period 1, paid losses are $24,000 and incurred losses are $40,000; the selected paid and incurred reported proportions are 0.40 and 0.50. For period 2, paid losses are $72,000 and incurred losses are $100,000; the corresponding proportions are 0.60 and 0.80. Calculate paid- and incurred-basis IBNR and total unpaid losses.

Method. Apply BF separately using each basis's reported proportion. First, expected ultimates are $100{,}000(0.60)=\$60{,}000$ and $200{,}000(0.60)=\$120{,}000$. The case reserves are $40{,}000-24{,}000=\$16{,}000$ and $100{,}000-72{,}000=\$28{,}000$, for total case reserves of $44,000.

Paid basis. Period 1 total unpaid is $60{,}000(1-0.40)=\$36{,}000$; its IBNR is $36{,}000-16{,}000=\$20{,}000$. Period 2 total unpaid is $120{,}000(1-0.60)=\$48{,}000$; its IBNR is $48{,}000-28{,}000=\$20{,}000$. Therefore paid-basis total unpaid is $84,000 and IBNR is $40,000.

Incurred basis. Period 1 IBNR is $60{,}000(1-0.50)=\$30{,}000$. Period 2 IBNR is $120{,}000(1-0.80)=\$24{,}000$. Total incurred-basis IBNR is $54,000, and total unpaid is $54{,}000+44{,}000=\$98{,}000$.

Check and trap avoided. Paid losses total $96,000, so the paid-basis ultimate is $96{,}000+84{,}000=\$180{,}000$. Incurred losses total $140,000, so the incurred-basis ultimate is $140{,}000+54{,}000=\$194{,}000$; equivalently, $96{,}000+98{,}000=\$194{,}000$. The estimates need not match because the two bases use different reported proportions. A common error is to add case reserves to the paid-basis total unpaid a second time; case reserves are already included in total unpaid.

Cape Cod Requires Basis-Matched Calibration

Cape Cod estimates an expected loss ratio from observed losses and exposure, giving greater weight to periods with more expected reported experience. For a selected basis $B$, let $X_i^B$ be paid losses if $B=P$, or incurred losses if $B=I$. Let $Z_i$ be the exposure measure, such as premium. With compatible definitions and valuation dates, the Cape Cod loss ratio is:

\widehat{\mathrm{ELR}}_B = \frac{\sum_i X_i^B}{\sum_i Z_iq_i^B}.

The denominator is the exposure weighted by the reported proportion on the same basis used for the observed-loss numerator. The estimated ultimate for period $i$ is $Z_i\widehat{\mathrm{ELR}}_B$. On a paid basis, estimated total unpaid is that ultimate times $1-q_i^P$; subtract case reserves to isolate IBNR. On an incurred basis, estimated IBNR is the ultimate times $1-q_i^I$, and total unpaid adds case reserves.

Choose one basis.
Use paid losses with paid reported proportions, or incurred losses with incurred reported proportions.

Calibrate the Cape Cod loss ratio.
Divide the selected-basis observed losses by exposure weighted by the selected-basis reported proportions.

Estimate unpaid losses.
Multiply each period's expected ultimate by its unreported proportion on that same basis.

Reconcile the components.
For paid basis, subtract case reserves to obtain IBNR. For incurred basis, add case reserves to IBNR to obtain total unpaid.

Worked Example: Cape Cod Loss Ratios by Basis

A hypothetical portfolio has two periods with premiums of $100,000 and $200,000. Period paid losses are $20,000 and $60,000, and incurred losses are $30,000 and $90,000. Paid reported proportions are 0.40 and 0.60; incurred reported proportions are 0.50 and 0.75. Calculate the Cape Cod estimates on each basis, including IBNR and total unpaid. Assume the incurred amounts include case reserves.

Paid-basis calibration. Total paid losses are $20{,}000+60{,}000=\$80{,}000$. The exposure-weighted reported amount is $100{,}000(0.40)+200{,}000(0.60)=40{,}000+120{,}000=\$160{,}000$. Thus the paid-basis loss ratio is $80{,}000/160{,}000=0.50$, or 50%.

Expected ultimates are $100{,}000(0.50)=\$50{,}000$ and $200{,}000(0.50)=\$100{,}000$. Paid-basis total unpaid is $50{,}000(0.60)+100{,}000(0.40)=30{,}000+40{,}000=\$70{,}000$. Case reserves total $(30{,}000-20{,}000)+(90{,}000-60{,}000)=\$40{,}000$, so IBNR is $70{,}000-40{,}000=\$30{,}000$.

Incurred-basis calibration. Total incurred losses are $30{,}000+90{,}000=\$120{,}000$. The weighted exposure is $100{,}000(0.50)+200{,}000(0.75)=50{,}000+150{,}000=\$200{,}000$. The incurred-basis loss ratio is $120{,}000/200{,}000=0.60$, or 60%.

Expected ultimates are $60,000 and $120,000. Incurred-basis IBNR is $60{,}000(0.50)+120{,}000(0.25)=30{,}000+30{,}000=\$60{,}000$. Total unpaid is $60{,}000+40{,}000=\$100{,}000$.

Check and trap avoided. Paid-basis ultimate is $80{,}000+70{,}000=\$150{,}000$. Incurred-basis ultimate is $120{,}000+60{,}000=\$180{,}000$, and also $80{,}000+100{,}000=\$180{,}000$. The paid-basis $70,000 is already total unpaid; adding case reserves to it would incorrectly report $110,000. Case reserves are subtracted from paid-basis total unpaid to find IBNR, not added to that total.

Interpreting Differences Between Bases

A paid- and incurred-basis estimate can differ even when both calculations are internally consistent. Paid development may be slower because payments lag reporting and case reserving. Incurred development can be faster at early ages, but it may also reflect changes in case-reserving practices. Consequently, the reported proportions differ, changing both BF's unreported share and Cape Cod's exposure-weighted loss ratio.

The next example isolates the effect of the two proportions in a BF calculation. It also shows why estimated ultimate losses should be reconciled to the observed amount on the selected basis, rather than assumed to equal the expected ultimate input.

Worked Example: Reconciling a Single Period

For a hypothetical accident period, expected ultimate losses are $200,000. At the valuation date, paid losses are $50,000 and incurred losses are $80,000, including $30,000 of case reserves. The selected paid reported proportion is 0.40 and the incurred reported proportion is 0.50. Calculate BF IBNR, total unpaid, and ultimate on both bases.

Paid basis. Total unpaid is $200{,}000(1-0.40)=\$120{,}000$. Subtract the $30,000 case reserve to obtain IBNR of $90,000. The estimated ultimate is paid losses plus total unpaid: $50{,}000+120{,}000=\$170{,}000$.

Incurred basis. IBNR is $200{,}000(1-0.50)=\$100{,}000$. Total unpaid is $100{,}000+30{,}000=\$130{,}000$. The estimated ultimate is $80{,}000+100{,}000=\$180{,}000$, which also equals paid losses plus total unpaid: $50{,}000+130{,}000=\$180{,}000$.

Check and trap avoided. Both estimates use the same expected ultimate input, but the observed amounts and reported proportions differ, so the resulting ultimates need not equal $200,000 or each other. The reconciliation identities hold: total unpaid equals ultimate less paid losses; for the incurred estimate, $180{,}000-50{,}000=\$130{,}000$. Do not mistake the expected ultimate input for the method's final estimated ultimate.

Common Basis and Reconciliation Errors

Mixing paid and incurred inputs. A paid-loss numerator paired with incurred reported proportions does not produce a basis-consistent Cape Cod loss ratio.
Confusing IBNR with total unpaid. Incurred-basis IBNR is added to case reserves to obtain total unpaid. Paid-basis unreported losses are already total unpaid; subtract case reserves to obtain IBNR.
Adding case reserves twice. Check that total unpaid equals estimated ultimate less paid losses, not that amount plus case reserves.
Using a factor as a proportion. If a selected cumulative development factor to ultimate is 2.5, the reported proportion is $1/2.5=0.40$, not 2.5.
Overlooking an implausible paid-basis IBNR. If paid-basis total unpaid is less than case reserves, the implied IBNR is negative. Treat this as a diagnostic of potentially inconsistent inputs or assumptions, not as a reason to silently change the formula.

Key takeaway: Match observed losses, development factors, and reported proportions to one basis throughout each calculation. Then distinguish IBNR from total unpaid and verify that total unpaid equals estimated ultimate less paid losses.

Check Your Understanding

Work each problem using basis-matched proportions. Answers are rounded to the nearest cent where needed.

A hypothetical period has expected ultimate losses of $80,000, incurred losses of $50,000, case reserves of $15,000, and an incurred reported proportion of 0.75. Find BF IBNR and total unpaid. (Answer: IBNR $20,000; total unpaid $35,000.)
A hypothetical period has expected ultimate losses of $120,000, paid losses of $40,000, case reserves of $25,000, and a paid reported proportion of 0.50. Find paid-basis BF total unpaid, IBNR, and ultimate. (Answer: total unpaid $60,000; IBNR $35,000; ultimate $100,000.)
A two-period hypothetical portfolio has premiums of $80,000 and $120,000, paid losses of $20,000 and $60,000, and paid reported proportions of 0.50 and 0.75. Case reserves are $10,000 and $25,000. Find the paid-basis Cape Cod loss ratio, total unpaid, and IBNR. (Answer: loss ratio 61.5385%; total unpaid $43,076.92; IBNR $8,076.92.)
A hypothetical portfolio has premiums of $100,000 in each of two periods, incurred losses of $24,000 and $56,000, incurred reported proportions of 0.40 and 0.80, and paid losses of $15,000 and $40,000. Find the incurred-basis Cape Cod loss ratio, IBNR, and total unpaid. (Answer: loss ratio 66.6667%; IBNR $53,333.33; total unpaid $78,333.33.)
A hypothetical period has expected ultimate losses of $90,000, paid losses of $27,000, incurred losses of $55,000, and paid and incurred reported proportions of 0.30 and 0.60. Find BF IBNR and total unpaid on each basis. (Answer: paid basis—IBNR $35,000 and total unpaid $63,000; incurred basis—IBNR $36,000 and total unpaid $64,000.)

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