Purpose and Setup
The Cape Cod method estimates ultimate losses by combining observed loss experience with an expected loss pattern. On a paid basis, the method uses cumulative paid losses and the expected proportion of ultimate losses paid by each accident year’s evaluation age. This makes the development pattern essential: a small paid amount from a very immature year cannot be compared directly with a large paid amount from a mature year.
The calculation below treats all premiums and losses as being on the same basis and in thousands of dollars. Assume no discounting. For each accident year \(i\), let \(EP_i\) be earned premium, \(C_i\) the cumulative paid loss at the valuation date, \(L_i\) the selected age-to-ultimate paid loss development factor, and \(w_i\) the expected proportion of ultimate losses paid by that age.
The development factor \(L_i\) is cumulative and links paid losses at the current age to estimated ultimate paid losses. Thus, if \(L_i=2\), the selected pattern says that 50% of ultimate losses are expected to have been paid. The factor does not mean that the current paid amount is 200% of ultimate.
Why the Weighted Ratio Formula Works
The method assumes a common expected loss ratio across the included accident years. Under this assumption, expected ultimate losses for year \(i\) are \(EP_i\mathrm{ELR}\). If the expected proportion paid by the valuation age is \(w_i\), expected cumulative paid losses are \(EP_i\mathrm{ELR}w_i\). Summing those expected paid amounts gives the model’s expected total paid loss:
Equating this expected total to the observed total paid, \(\sum_i C_i\), and solving for the common ratio gives
The denominator is therefore not total premium. It is premium adjusted for how much loss is expected to have emerged as paid by the valuation date. Once the ratio is estimated, the expected ultimate loss for each year is premium times that ratio. Only the expected unpaid portion is added to the paid amount already observed.
Calculation Procedure
Use the latest available diagonal of the paid triangle at the common valuation date.
For each year, calculate \(w_i=1/L_i\), using the selected factor for that year’s maturity.
Divide the sum of observed paid losses by \(\sum_i EP_iw_i\).
For each year, multiply premium by the estimated ratio, then multiply expected ultimate by \(1-w_i\) to obtain unpaid losses.
Check that estimated ultimate equals observed paid plus estimated unpaid for each year, and in total.
Worked Paid-Triangle Calculation
Worked Example: Full Paid Cape Cod Triangle
A hypothetical commercial package has the cumulative paid triangle below, evaluated at the end of 2024. Amounts are in thousands of dollars. The selected age-to-ultimate paid factors are 4.00 at 12 months, 2.00 at 24 months, \(4/3\) at 36 months, and 1.00 at 48 months. Earned premiums are also shown in thousands. Estimate the Cape Cod expected loss ratio, expected losses, ultimate losses, and unpaid reserves.
| Accident year | 12-month paid | 24-month paid | 36-month paid | 48-month paid | Earned premium |
|---|---|---|---|---|---|
| 2021 | 300 | 600 | 900 | 1,200 | 2,000 |
| 2022 | 270 | 540 | 810 | — | 1,800 |
| 2023 | 240 | 480 | — | — | 1,600 |
| 2024 | 187.5 | — | — | — | 1,250 |
Step 1: Select each year’s current cumulative paid amount and age. The latest diagonal is 1,200 at 48 months for 2021, 810 at 36 months for 2022, 480 at 24 months for 2023, and 187.5 at 12 months for 2024. Their total is
Step 2: Convert each selected factor to a paid proportion. The proportions are \(1/1=1.00\) at 48 months, \(1/(4/3)=0.75\) at 36 months, \(1/2=0.50\) at 24 months, and \(1/4=0.25\) at 12 months.
Step 3: Calculate premium weighted by the proportions.
Step 4: Calculate the Cape Cod expected loss ratio.
Step 5: Estimate expected ultimate losses and unpaid losses. For 2021, expected ultimate loss is \(2{,}000(0.60)=1{,}200\), and unpaid loss is \(1{,}200(1-1.00)=0\). For 2022, expected ultimate loss is \(1{,}800(0.60)=1{,}080\), and unpaid loss is \(1{,}080(1-0.75)=270\). For 2023, expected ultimate loss is \(1{,}600(0.60)=960\), and unpaid loss is \(960(1-0.50)=480\). For 2024, expected ultimate loss is \(1{,}250(0.60)=750\), and unpaid loss is \(750(1-0.25)=562.5\).
| Accident year | Observed paid | Paid proportion \(w_i\) | Expected ultimate loss | Estimated unpaid | Estimated ultimate |
|---|---|---|---|---|---|
| 2021 | 1,200 | 1.00 | 1,200 | 0 | 1,200 |
| 2022 | 810 | 0.75 | 1,080 | 270 | 1,080 |
| 2023 | 480 | 0.50 | 960 | 480 | 960 |
| 2024 | 187.5 | 0.25 | 750 | 562.5 | 750 |
| Total | 2,677.5 | — | 3,990 | 1,312.5 | 3,990 |
Check. Total estimated ultimate is total observed paid plus total estimated unpaid: \(2{,}677.5+1{,}312.5=3{,}990\). The 2022–2024 accident-year ultimates are also equal to observed paid plus estimated unpaid in each row. All amounts remain in thousands of dollars. The method used is the paid Cape Cod method; the key trap avoided is applying the development factor itself as though it were the percentage paid.
Additional Worked Examples
Worked Example: Two Accident Years at Different Maturities
A hypothetical inland marine portfolio has two accident years. The first has earned premium of $900,000 and cumulative paid losses of $400,000 at 12 months; its selected paid factor to ultimate is 3.00. The second has earned premium of $1,500,000 and cumulative paid losses of $900,000 at 24 months; its selected factor is 1.50. Calculate the Cape Cod ratio and each year’s unpaid reserve. Work in thousands of dollars.
Method. First invert the factors to get paid proportions: \(w_1=1/3\), and \(w_2=1/1.5=2/3\). The weighted premium denominator is
Observed paid losses total \(400+900=1{,}300\), so
Expected ultimate losses are \(900(1.000)=900\) and \(1{,}500(1.000)=1{,}500\). The estimated unpaid amounts are \(900(1-1/3)=600\) and \(1{,}500(1-2/3)=500\). Thus the estimated ultimates are \(400+600=1{,}000\) and \(900+500=1{,}400\), respectively. The total unpaid estimate is \(600+500=1{,}100\), and the total ultimate is \(1{,}300+1{,}100=2{,}400\).
Check and trap avoided. The paid numerator is the sum of the current cumulative paid amounts, not paid divided by premium separately for each year and then averaged. The resulting common ratio is 100.0%, and the year-specific unpaid amounts use the complements of the paid proportions.
Worked Example: Reconciling All Accident-Year Reserves
A hypothetical equipment insurer has three accident years. In thousands of dollars, the earned premiums are 500, 1,000, and 1,500; the current cumulative paid amounts are 75, 375, and 900. Their respective paid factors to ultimate are 5.00, 2.00, and 1.25. Calculate the common expected loss ratio, expected ultimate losses, unpaid estimates, and aggregate ultimate.
Method. The paid proportions are \(1/5=0.20\), \(1/2=0.50\), and \(1/1.25=0.80\). The premium-weighted denominator is
Observed paid losses total \(75+375+900=1{,}350\). Therefore,
Expected ultimate losses by year are \(500(0.75)=375\), \(1{,}000(0.75)=750\), and \(1{,}500(0.75)=1{,}125\). Estimated unpaid amounts are \(375(0.80)=300\), \(750(0.50)=375\), and \(1{,}125(0.20)=225\). Adding the current paid amounts gives ultimate estimates of \(75+300=375\), \(375+375=750\), and \(900+225=1{,}125\). Total unpaid is \(300+375+225=900\); aggregate ultimate is \(1{,}350+900=2{,}250\), equal to \(375+750+1{,}125=2{,}250\).
Check and trap avoided. The denominator is 1,800, not the total premium of 3,000: each premium is weighted by its expected paid proportion. The resulting ratio and every accident-year reserve reconcile to the aggregate.
Assumptions, Interpretation, and Common Errors
The calculation is meaningful only if its inputs and assumptions are coherent. The common expected loss ratio assumes that expected ultimate losses are proportional to earned premium across the accident years included. The selected paid development factors must represent the expected emergence of paid losses on the same loss basis as the triangle. Premium and paid loss amounts must be aligned by accident year and valued consistently. If these conditions are not reasonable—for example, if the years have materially different expected loss ratios or the selected paid pattern does not represent the portfolio—the single weighted ratio can be misleading.
A useful exam check is to reconstruct each ultimate in two ways. The Cape Cod expected ultimate is \(EP_i\widehat{\mathrm{ELR}}_{\mathrm{CC}}\). The paid-based reserve calculation gives \(C_i+\widehat{U}_i(1-w_i)\). Under the formula and assumptions above, these agree because the method implies \(C_i\) is the emerged portion, approximately \(\widehat{U}_iw_i\). They need not agree exactly for every accident year in observed data; the method estimates a common expected loss ratio from the aggregate paid experience, while retaining each year’s actual cumulative paid amount in the reserve calculation. For the final reserve, use observed paid plus expected unpaid, not expected ultimate alone.
- Using \(L_i\) instead of \(1/L_i\). A factor of 2.00 implies an expected paid proportion of 0.50, not 2.00.
- Using unweighted premium in the denominator. Cape Cod adjusts premium for expected loss emergence; total premium alone ignores the different maturity ages.
- Applying the paid proportion to the wrong quantity. Expected unpaid is expected ultimate times \(1-w_i\), not observed paid times \(1-w_i\).
- Adding expected ultimate to paid. That double counts the paid portion. The reserve is the unpaid portion; ultimate is paid plus reserve.
Check Your Understanding
For each problem, use the paid Cape Cod method and work in thousands of dollars unless stated otherwise. Round displayed ratios to four decimal places where needed.
- A single accident year has earned premium 800, cumulative paid loss 320, and a selected 24-month paid factor of 2.00. Find the Cape Cod ratio, expected ultimate loss, and unpaid estimate. (Answer: ELR \(=80.00\%\); expected ultimate \(=640\); unpaid \(=320\).)
- Earned premium is 1,200 and paid loss is 450 at 12 months; the selected factor is 4.00. Find the Cape Cod ratio and unpaid estimate. (Answer: ELR \(=150.00\%\); unpaid \(=1{,}350\).)
- Two years have \((EP,C,L)\) values \((1{,}000,200,4.00)\) and \((2{,}000,900,4/3)\), where \(C\) is current paid loss and \(L\) is the factor to ultimate. Find the common ratio and total estimated unpaid. (Answer: ELR \(=62.8571\%\); total unpaid \(=785.714\).)
- Two years have \((EP,C,L)\) values \((600,360,1.25)\) and \((900,270,3.00)\). Calculate the weighted denominator, Cape Cod ratio, and total unpaid estimate. (Answer: denominator \(=780\); ELR \(=80.7692\%\); total unpaid \(=581.538\).)
- Two years have \((EP,C,L)\) values \((400,100,4.00)\) and \((800,480,2.00)\). Calculate expected ultimate losses by year and aggregate unpaid. (Answer: ELR \(=116.00\%\); expected ultimates \(=464\) and \(928\); aggregate unpaid \(=812\).)