Why Reconciliation Matters
A reserving estimate is often calculated by accident period and then summarized for a portfolio. Those totals should agree with the sum of the period-level estimates. If they do not, the discrepancy may signal a wrong development proportion, an inconsistent exposure measure, or an aggregation error rather than a substantive feature of the methods.
This tutorial focuses on reconciling accident-period expected losses, reported losses, ultimates, and unpaid amounts under Bornhuetter-Ferguson and Cape Cod. All monetary amounts in the worked examples are stated in thousands of dollars. The loss amounts, exposures, and development proportions are hypothetical.
Reconciling Bornhuetter-Ferguson Results
Under Bornhuetter-Ferguson, the reported amount is supplemented by the expected portion of ultimate losses that has not yet been reported. Thus the accident-period ultimate is \(R_i+E_i(1-p_i)\), and the unpaid amount is the added expected unreported portion. The expected amount \(E_i\) and reported proportion \(p_i\) must refer to the same accident period and valuation basis.
At the portfolio level, add the period amounts. The total ultimate is the sum of reported losses plus the sum of the expected unreported amounts; the total unpaid amount is the sum of the accident-period unpaid amounts. This is an additive reconciliation, not an average of period ultimates or reserve ratios.
Worked Example: Reconciling Bornhuetter-Ferguson by Accident Period
A hypothetical liability portfolio has three accident periods. The expected ultimate losses, reported amounts, and reported proportions are as follows, with monetary amounts in thousands. Calculate each period's Bornhuetter-Ferguson ultimate and unpaid amount, then reconcile the portfolio totals.
| Period | \(E_i\) | \(R_i\) | \(p_i\) |
|---|---|---|---|
| 1 | 1,000 | 520 | 0.50 |
| 2 | 800 | 610 | 0.75 |
| 3 | 600 | 140 | 0.25 |
Method. For each period, calculate \(Q_i^{\mathrm{BF}}=E_i(1-p_i)\), then add that amount to reported losses to obtain the ultimate.
Period 1. The expected unreported amount is \(1{,}000(1-0.50)=500\). The ultimate is \(520+500=1{,}020\), and the unpaid amount is \(1{,}020-520=500\).
Period 2. The expected unreported amount is \(800(1-0.75)=200\). The ultimate is \(610+200=810\), and the unpaid amount is \(810-610=200\).
Period 3. The expected unreported amount is \(600(1-0.25)=450\). The ultimate is \(140+450=590\), and the unpaid amount is \(590-140=450\).
Portfolio reconciliation. Total reported losses are \(520+610+140=1{,}270\). Total unpaid is \(500+200+450=1{,}150\), so portfolio ultimate is \(1{,}270+1{,}150=2{,}420\). Directly summing period ultimates gives \(1{,}020+810+590=2{,}420\), and \(2{,}420-1{,}270=1{,}150\). The totals agree. The common mistake avoided here is applying one portfolio reported proportion to all three expected amounts when the period-specific proportions differ.
Reconciling Cape Cod Results
Cape Cod estimates a common loss ratio from the portfolio's reported losses and the exposure-weighted amount expected to be reported. In this setup, the estimated loss ratio is total reported losses divided by \(\sum_i w_i p_i\). The resulting loss ratio is then applied to each period's exposure to estimate its ultimate. This calculation requires the exposure measure and reported losses to be on compatible bases.
The period-level estimates reconcile because their sum is the common estimated loss ratio multiplied by total exposure. Subtracting total reported losses then gives total unpaid. The denominator is not total exposure alone: it is exposure weighted by the period-specific reported proportion.
Worked Example: Reconciling Cape Cod Losses and Unpaid Amounts
A hypothetical property portfolio has three accident periods. Exposure is earned premium in thousands of dollars; reported losses are also in thousands. Using the stated reported proportions, calculate the Cape Cod loss ratio, each accident period's ultimate and unpaid amount, and the portfolio totals.
| Period | Exposure \(w_i\) | Reported \(R_i\) | Reported proportion \(p_i\) |
|---|---|---|---|
| 1 | 2,000 | 500 | 0.40 |
| 2 | 1,500 | 700 | 0.60 |
| 3 | 1,000 | 180 | 0.20 |
Step 1: Calculate the denominator. The exposure-weighted reported amounts are \(2{,}000(0.40)=800\), \(1{,}500(0.60)=900\), and \(1{,}000(0.20)=200\). Their sum is \(800+900+200=1{,}900\).
Step 2: Estimate the common loss ratio. Total reported losses are \(500+700+180=1{,}380\). Therefore \(\widehat{\mathrm{LR}}_{\mathrm{CC}}=1{,}380/1{,}900=0.726315789\ldots\), or \(72.63\%\) when rounded to two decimal places.
Step 3: Calculate period ultimates. Before rounding, the period estimates are \(2{,}000(1{,}380/1{,}900)=1{,}452.631579\ldots\), \(1{,}500(1{,}380/1{,}900)=1{,}089.473684\ldots\), and \(1{,}000(1{,}380/1{,}900)=726.315789\ldots\). Rounded to two decimal places, these are 1,452.63, 1,089.47, and 726.32.
Step 4: Calculate period unpaid amounts. Subtract reported losses: \(1{,}452.631579\ldots-500=952.631579\ldots\); \(1{,}089.473684\ldots-700=389.473684\ldots\); and \(726.315789\ldots-180=546.315789\ldots\). Rounded to two decimal places, these are 952.63, 389.47, and 546.32.
Check the portfolio. Total exposure is \(2{,}000+1{,}500+1{,}000=4{,}500\), so total ultimate is \(4{,}500(1{,}380/1{,}900)=3{,}268.421053\ldots\), or 3,268.42. Total unpaid is \(3{,}268.421053\ldots-1{,}380=1{,}888.421053\ldots\), or 1,888.42. The period unpaid amounts sum to \(952.631579\ldots+389.473684\ldots+546.315789\ldots=1{,}888.421052\ldots\), which rounds to 1,888.42. The method avoids the error of dividing reported losses by total exposure without accounting for different maturity proportions.
Aggregation Errors and Consistent Weighting
A common aggregation error is to average accident-period proportions without considering the amounts to which they apply. For Bornhuetter-Ferguson, if a portfolio calculation uses one combined reported proportion with total expected losses, that proportion must be weighted by expected ultimate losses: \(\sum_i E_i p_i/\sum_i E_i\). An unweighted average gives equal influence to periods with very different expected losses and generally does not reproduce the sum of the period estimates.
For Cape Cod, the corresponding weighting is built into the denominator \(\sum_i w_i p_i\). A simple average of the reported proportions is not a substitute. In either method, calculate using unrounded values and round only after the totals reconcile; independently rounded period entries can differ from a rounded portfolio figure by a cent or a small displayed unit.
Worked Example: Finding a Misweighted Bornhuetter-Ferguson Portfolio
A hypothetical two-period portfolio has expected ultimate losses of 1,000 and 3,000, reported amounts of 250 and 2,400, and reported proportions of 0.20 and 0.80. Amounts are in thousands. Calculate the period-level results, then test whether an unweighted average reported proportion reproduces the portfolio ultimate.
Period-level method. For period 1, unpaid is \(1{,}000(1-0.20)=800\), so ultimate is \(250+800=1{,}050\). For period 2, unpaid is \(3{,}000(1-0.80)=600\), so ultimate is \(2{,}400+600=3{,}000\). The portfolio ultimate is \(1{,}050+3{,}000=4{,}050\), and the portfolio unpaid amount is \(800+600=1{,}400\).
Incorrect unweighted calculation. The unweighted average reported proportion is \((0.20+0.80)/2=0.50\). Applying it to total expected losses gives \(4{,}000(1-0.50)=2{,}000\) of expected unpaid losses. Adding total reported \(250+2{,}400=2{,}650\) gives an incorrect portfolio ultimate of \(4{,}650\).
Correction. The appropriate combined reported proportion is weighted by expected ultimate: \((1{,}000(0.20)+3{,}000(0.80))/4{,}000=(200+2{,}400)/4{,}000=0.65\). Expected unpaid is \(4{,}000(1-0.65)=1{,}400\), and ultimate is \(2{,}650+1{,}400=4{,}050\), agreeing with the period-level sum. The trap is treating periods as equally weighted when their expected losses differ.
Practical Reconciliation Checks
A reliable review follows the same sequence for either method: calculate each period on a consistent basis, add reported amounts and unpaid amounts separately, and verify that total ultimate equals total reported plus total unpaid. For Cape Cod, also verify that the portfolio ultimate equals the common loss ratio multiplied by total exposure. For Bornhuetter-Ferguson, verify that portfolio unpaid equals \(\sum_i E_i(1-p_i)\).
Investigate a mismatch before interpreting it. Common causes include using a reported proportion from a different valuation date, pairing exposure and loss measures that are not on compatible bases, applying a period's proportion to another period's expected losses, or rounding intermediate values. These checks confirm arithmetic and aggregation; they do not by themselves validate the selected assumptions or inputs.
Check Your Understanding
For each problem, calculate the requested period or portfolio quantities before checking the final answer. Monetary amounts are in thousands unless stated otherwise.
- A Bornhuetter-Ferguson period has expected ultimate losses of 900, reported losses of 520, and a reported proportion of 0.60. Find ultimate and unpaid losses. (Answer: Ultimate 880; unpaid 360.)
- Two Bornhuetter-Ferguson periods have \((E_i,R_i,p_i)=(400,150,0.25)\) and \((600,500,0.75)\). Find portfolio reported, unpaid, and ultimate losses. (Answer: Reported 650; unpaid 450; ultimate 1,100.)
- A Cape Cod portfolio has exposures 1,200 and 800, reported losses 400 and 300, and reported proportions 0.50 and 0.25. Find the estimated loss ratio, total ultimate, and total unpaid. (Answer: Loss ratio 0.875; total ultimate 1,750; total unpaid 1,050.)
- A two-period Cape Cod portfolio has exposures 1,000 and 3,000, reported losses 250 and 2,000, and reported proportions 0.20 and 0.80. Find the loss ratio and total unpaid amount, rounded to two decimal places. (Answer: Loss ratio 0.8654, or 86.54%; total unpaid 1,211.54.)