Why an Incurred Cape Cod Calculation Is Weighted
An incurred triangle records cumulative losses by accident year and development age. At a valuation date, older accident years have generally had more time to develop than younger ones. A direct comparison of incurred losses with total premium would therefore mix years at different stages of emergence. The incurred Cape Cod method accounts for that difference by weighting each year's premium by the proportion of its ultimate losses expected to have emerged.
The method uses selected age-to-ultimate factors, the latest incurred amount available for each accident year, and earned premium on a consistent basis. It then estimates one expected loss ratio across the included years. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
“Incurred” here means cumulative paid losses plus case outstanding losses, using the same loss components for every accident year. The factors \(f_i\) are cumulative age-to-ultimate factors for the respective evaluation ages. They must be on the same basis as the incurred triangle; for example, factors that include a tail should be paired with incurred amounts and a valuation basis that use that tail consistently.
From Triangle to Cape Cod Inputs
Use the latest diagonal of the triangle: for each accident year, take its most recent cumulative incurred amount and record the corresponding development age. Then pair that amount with the appropriate age-to-ultimate factor. For a factor of \(f_i\), the estimated emerged proportion is \(w_i=1/f_i\), while the proportion still unreported is \(1-w_i\).
For this calculation, premium should be earned premium aligned with the losses being estimated. It should be on a consistent rate and exposure basis across the accident years. The method assumes the selected emergence weights adequately represent how incurred losses emerge. It also assumes the included years can reasonably share a common expected loss ratio. If these conditions are not credible, the resulting estimate may be misleading.
For each accident year, identify its latest cumulative incurred loss and the development age at which it was evaluated.
For each year, take the reciprocal of its selected age-to-ultimate factor: \(w_i=1/f_i\).
Divide total latest incurred losses by total premium weighted by the emerged proportions.
Apply the common expected loss ratio to premium and the un-emerged proportion. Add that unpaid amount to current incurred losses to obtain ultimate losses.
Why the Weighted Expected Loss Ratio Works
Suppose accident year \(i\) has ultimate losses equal to premium times a common expected loss ratio \(L\), and suppose the selected emergence weight \(w_i\) is the proportion of those ultimate losses expected to be incurred by the evaluation age. Then the expected incurred amount at that age is \(P_iLw_i\). Across the included years, the model therefore gives:
Solving this relationship for \(L\) gives \(\widehat{L}=\sum_i I_i/\sum_i P_iw_i\). The denominator is not total premium: each year's premium contributes in proportion to the amount of its ultimate losses expected to have emerged. Once \(\widehat{L}\) is estimated, the un-emerged share of expected ultimate losses is \(P_i\widehat{L}(1-w_i)\). Adding that amount to the incurred losses already observed produces the Cape Cod ultimate estimate.
Worked Example: Completing an Incurred Triangle
A hypothetical insurer evaluates a line at 12, 24, and 36 months. Its cumulative incurred triangle and earned premiums are shown below. Selected age-to-ultimate factors are 2.00 at 12 months, 1.25 at 24 months, and 1.00 at 36 months. Assume no further development after 36 months. Calculate the Cape Cod expected loss ratio, each accident year's ultimate losses, and total unpaid reserve. Amounts are in thousands of dollars.
| Accident year | Incurred at 12 months | Incurred at 24 months | Incurred at 36 months | Earned premium |
|---|---|---|---|---|
| Year 1 | 300 | 480 | 600 | 1,000 |
| Year 2 | 360 | 576 | — | 1,200 |
| Year 3 | 420 | — | — | 1,400 |
Method: incurred Cape Cod. First take the latest diagonal: Year 1 has incurred losses of 600 at 36 months, Year 2 has 576 at 24 months, and Year 3 has 420 at 12 months. Their respective factors are 1.00, 1.25, and 2.00.
Step 1: Calculate emergence weights. The weights are \(1/1.00=1.00\) for Year 1, \(1/1.25=0.80\) for Year 2, and \(1/2.00=0.50\) for Year 3.
Step 2: Weight premium and total the incurred losses. The weighted premiums are \(1{,}000(1.00)=1{,}000\), \(1{,}200(0.80)=960\), and \(1{,}400(0.50)=700\). Their sum is \(2{,}660\). The incurred total is \(600+576+420=1{,}596\).
Step 3: Estimate the expected loss ratio. Divide incurred losses by weighted premium: \(1{,}596/2{,}660=0.600\), or 60.0%.
Step 4: Calculate unpaid reserve and ultimate by year. Year 1 has no estimated un-emerged losses: \(1{,}000(0.600)(1-1.00)=0\), so its ultimate is \(600+0=600\). Year 2 reserve is \(1{,}200(0.600)(1-0.80)=144\), giving ultimate \(576+144=720\). Year 3 reserve is \(1{,}400(0.600)(1-0.50)=420\), giving ultimate \(420+420=840\).
Check. Total unpaid reserve is \(0+144+420=564\). Total ultimate losses are \(600+720+840=2{,}160\), which equals total incurred \(1{,}596\) plus reserve \(564\). The method avoids treating each accident year's current incurred loss as though it were already ultimate.
Reading the Ultimate Estimate Correctly
Cape Cod does not generally set each accident year's ultimate equal to its current incurred amount divided by its emergence weight. It estimates a common expected loss ratio from all the weighted experience, then adds expected un-emerged losses to each year's observed incurred amount. As a result, a year’s estimated ultimate can differ from its premium times the common expected loss ratio: its own observed incurred amount still contributes directly to the estimate.
This feature gives the calculation its balance between observed experience and expected unreported losses. A mature year has \(w_i\) near one, so it receives little additional expected development. An immature year has a smaller weight, so a larger share of its expected losses remains un-emerged. A zero reserve for a fully emerged year follows from the formula only when its weight is exactly one; it does not imply that every mature year must have zero unpaid claims in practice.
Worked Example: A Young Year Does Not Equal Premium Times the Ratio
A hypothetical portfolio has three accident years. The selected emergence weights are 0.40, \(2/3\), and 1.00. Earned premiums are 800, 1,000, and 1,200 thousand dollars; latest incurred losses are 200, 440, and 1,000 thousand dollars. Calculate the expected loss ratio and the Cape Cod ultimate and reserve for each year.
Method: exposure-weighted incurred Cape Cod. The weighted premiums are \(800(0.40)=320\), \(1{,}000(2/3)=666.667\), and \(1{,}200(1.00)=1{,}200\). The denominator is \(2{,}186.667\), and the incurred total is \(200+440+1{,}000=1{,}640\). Thus \(\widehat{L}=1{,}640/2{,}186.667=0.750\), or 75.0%.
For Year 1, the reserve is \(800(0.750)(1-0.40)=360\), and the ultimate is \(200+360=560\). For Year 2, the reserve is \(1{,}000(0.750)(1-2/3)=250\), and the ultimate is \(440+250=690\). For Year 3, the reserve is \(1{,}200(0.750)(1-1.00)=0\), and the ultimate is \(1{,}000\). Total reserve is \(360+250+0=610\), and total ultimate is \(560+690+1{,}000=2{,}250\).
Check and common mistake avoided. The reserve sum of 610 plus the incurred sum of 1,640 gives total ultimate losses of 2,250. The method does not force Year 1's ultimate to equal its expected ultimate of \(800(0.750)=600\); its estimate is 560 because the observed incurred amount is combined with expected un-emerged losses.
Worked Example: Use Cumulative Incurred, Not Paid Loss Alone
A hypothetical insurer reports the following latest values, in thousands of dollars. Year 1 has paid losses of 650 and case outstanding losses of 150; Year 2 has paid losses of 320 and case outstanding losses of 180; Year 3 has paid losses of 200 and case outstanding losses of 82. Earned premiums are 1,000, 1,200, and 900, and the selected emergence weights are 1.00, 0.75, and 0.40. Calculate the expected loss ratio and total reserve.
Method: incurred Cape Cod using paid plus case outstanding. The latest incurred amounts are \(650+150=800\), \(320+180=500\), and \(200+82=282\). The weighted premiums are \(1{,}000(1.00)=1{,}000\), \(1{,}200(0.75)=900\), and \(900(0.40)=360\), for a denominator of 2,260. Total incurred is \(800+500+282=1{,}582\).
The expected loss ratio is \(1{,}582/2{,}260=0.700\), or 70.0%. Year 1 reserve is \(1{,}000(0.700)(0)=0\). Year 2 reserve is \(1{,}200(0.700)(0.25)=210\). Year 3 reserve is \(900(0.700)(0.60)=378\). Total reserve is \(210+378=588\).
Check and common mistake avoided. Total ultimate is \(1{,}582+588=2{,}170\). Using only paid losses would substitute \(650+320+200=1{,}170\) for the incurred numerator and would omit case outstanding losses from the current incurred amounts. The calculation must use the incurred basis consistently.
Common Mistakes and Checks
- Using total premium as the denominator. The denominator is \(\sum_iP_iw_i\), not \(\sum_iP_i\). The latter treats every accident year as fully emerged.
- Using an age-to-ultimate factor as an emergence weight. If the factor is 2.00, the emerged proportion is \(1/2.00=0.50\), not 2.00.
- Using the wrong triangle diagonal. Each accident year’s incurred amount must be paired with its own latest evaluation age and corresponding factor.
- Adding the un-emerged proportion with the wrong sign. The expected unpaid share is \(1-w_i\), not \(w_i\). A more mature year should have a smaller estimated un-emerged share, all else equal.
- Mixing paid and incurred bases. If the selected factors and method are for incurred losses, use cumulative incurred losses, not paid losses alone.
For example, in the first worked example, dividing 1,596 by total unweighted premium of \(1{,}000+1{,}200+1{,}400=3{,}600\) gives 44.3%. That is the wrong calculation: it counts the immature Year 3 premium as though all of its expected losses had emerged. The correct weighted denominator is 2,660, yielding 60.0%. After calculation, check that total estimated ultimate losses equal total current incurred losses plus total estimated unpaid reserve.
Check Your Understanding
For each hypothetical portfolio, calculate the requested Cape Cod quantities using the supplied emergence weights.
- Two accident years have premiums of 500 and 800, weights of 0.50 and 0.80, and incurred losses of 120 and 480, respectively. Find the expected loss ratio and total reserve. (Answer: expected loss ratio 0.6742; total reserve $276.40.)
- Three years have premiums of 600, 900, and 1,000; weights of 0.25, 0.60, and 1.00; and incurred losses of 100, 450, and 700. Find the expected loss ratio and total reserve. (Answer: expected loss ratio 0.7396; total reserve $599.11.)
- A 24-month year has premium 500, weight 0.80, and incurred losses 270. A 12-month year has premium 700, weight 0.50, and incurred losses 240. Find the expected loss ratio and each year's reserve. (Answer: expected loss ratio 0.6800; reserves $68 and $238; total $306.)
- A young year has premium 900, weight 0.40, and incurred losses 360. A fully emerged year has premium 1,100, weight 1.00, and incurred losses 770. Find the expected loss ratio, the young year's ultimate, and total reserve. (Answer: expected loss ratio 0.7740; young-year ultimate $777.95; total reserve $417.95.)