From a Selected Cape Cod Ratio to Accident-Year Expected Losses
A Cape Cod analysis uses experience across accident years to select an expected loss ratio. Once that ratio has been selected, the next calculation is year-specific: apply it to each accident year’s earned premium on the basis used for the selection. Keeping those steps separate matters. Development information helps determine the selected ratio, while the expected loss for an accident year is calculated from that ratio and that year’s premium.
This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. It focuses on calculating accident-year expected losses from a selected Cape Cod ratio, not on selecting the ratio or estimating unpaid claims.
What the Calculation Uses—and What It Does Not
The selected ratio is common to the accident years being analyzed when the Cape Cod selection assumes a common expected loss ratio. Each year’s premium remains its own: a larger earned premium produces a larger expected loss at the same ratio. If the analysis instead uses different selected ratios for different groups or years, use the applicable ratio for each premium; do not apply a common ratio where the selection does not support one.
The earned premium must be measured consistently with the ratio. For example, if the ratio is selected using on-level earned premium, apply it to on-level earned premium for each accident year. If a problem gives raw earned premium and a premium adjustment factor, first adjust the premium and then multiply by the selected ratio. Do not adjust the ratio and premium independently for the same change unless the stated method calls for both.
Reported proportions, also called development proportions, are relevant in the Cape Cod ratio selection. They describe the proportion of ultimate losses expected to be reported by the valuation date. They are not another multiplier in the accident-year expected loss formula. Once the ratio has been selected, multiplying \(P_i\widehat{ELR}_{CC}\) by a reported proportion would calculate only a reported-to-date portion under an additional assumption; it would not give the expected ultimate losses requested here.
Why Premium Times Ratio Gives the Expected Loss
A loss ratio is a ratio of loss dollars to premium dollars. If the selected expected loss ratio is \(\widehat{ELR}_{CC}\), then each dollar of relevant earned premium corresponds to \(\widehat{ELR}_{CC}\) dollars of expected ultimate loss under the selected assumption. For \(P_i\) dollars of earned premium, add that expected loss amount across the premium dollars:
More generally, the equation follows directly from the definition of a loss ratio. If expected ultimate losses equal the ratio times premium, then dividing both sides by premium recovers the ratio:
This calculation assumes that the selected ratio is appropriate for the accident year’s risk mix, coverage, loss basis, and premium basis. A consistent formula cannot correct an inappropriate ratio or mismatched premium. It also does not imply that actual ultimate losses will equal the expectation; it gives the expected amount under the selected assumption.
Calculating Accident-Year Amounts
Use the selected Cape Cod expected loss ratio as a decimal. For example, 67% is \(0.67\).
Use the earned premium basis consistent with the ratio. Apply any stated premium adjustment before multiplying.
Multiply each accident year’s applicable earned premium by the selected ratio. Keep the premiums and results associated with their own accident years.
Sum the accident-year expected losses and verify that the result equals the selected ratio times total applicable premium.
Worked Example: Applying a Common Ratio to Adjusted Premiums
A hypothetical liability portfolio has a selected Cape Cod expected loss ratio of 67%. Accident year 1 earned $4.8 million before a stated premium adjustment factor of 1.04. Accident year 2 earned $6.25 million before a factor of 0.96. Calculate the expected ultimate losses by year and in total. Assume the factors put each year’s premium on the basis consistent with the selected ratio.
Method. Adjust each year’s premium first, then multiply by the selected ratio.
For accident year 1, adjusted earned premium is \(4.8\times1.04=\$4.992\) million. Its expected losses are \(4.992\times0.67=\$3.34464\) million.
For accident year 2, adjusted earned premium is \(6.25\times0.96=\$6.000\) million. Its expected losses are \(6.000\times0.67=\$4.02000\) million.
The total adjusted premium is \(4.992+6.000=\$10.992\) million. Total expected losses are \(3.34464+4.02000=\$7.36464\) million. As a check, \(10.992\times0.67=\$7.36464\) million, the same total.
Answer: Accident year 1 has expected losses of $3.34464 million, accident year 2 has $4.02000 million, and the total is $7.36464 million. Common mistake avoided: applying the ratio to the unadjusted premiums, or adjusting the premium after calculating expected losses.
Worked Example: Development Proportions Are Not an Extra Multiplier
A hypothetical insurer has selected a Cape Cod expected loss ratio of 72% for three accident years. The earned premiums are $3.2 million, $4.5 million, and $5.1 million. Their reported proportions are 80%, 50%, and 25%, respectively. Calculate the expected ultimate losses by year and in total.
Method. Apply the selected ratio to each year’s earned premium. The reported proportions may have contributed to selecting the ratio, but they are not applied again to the expected ultimate losses.
Accident year 1: \(3.2\times0.72=\$2.304\) million.
Accident year 2: \(4.5\times0.72=\$3.240\) million.
Accident year 3: \(5.1\times0.72=\$3.672\) million.
The total premium is \(3.2+4.5+5.1=\$12.8\) million. The total expected losses are \(2.304+3.240+3.672=\$9.216\) million. The aggregate check gives \(12.8\times0.72=\$9.216\) million.
For comparison, multiplying each expected ultimate amount by its reported proportion would produce amounts of \(2.304\times0.80=\$1.8432\) million, \(3.240\times0.50=\$1.6200\) million, and \(3.672\times0.25=\$0.9180\) million. Those are not the expected ultimate losses asked for; they are proportions of those amounts under the given reporting assumptions.
Answer: Expected ultimate losses are $2.304 million, $3.240 million, and $3.672 million, totaling $9.216 million. Common mistake avoided: multiplying by the reported proportion and thereby confusing an ultimate expectation with a reported-to-date amount.
Worked Example: Applying a Ratio After a Premium-Level Adjustment
A hypothetical property portfolio has a selected Cape Cod expected loss ratio of 58%. Accident year A has $2.75 million of earned premium and an adjustment factor of 1.04; accident year B has $3.60 million and a factor of 0.975. The factors convert the earned premiums to the level consistent with the selected ratio. Calculate expected losses by year, the total, and the implied aggregate expected loss ratio.
Method. Convert each premium to the stated level, calculate the expected losses using the selected ratio, and check the aggregate ratio.
For accident year A, adjusted premium is \(2.75\times1.04=\$2.86\) million. Expected losses are \(2.86\times0.58=\$1.6588\) million.
For accident year B, adjusted premium is \(3.60\times0.975=\$3.51\) million. Expected losses are \(3.51\times0.58=\$2.0358\) million.
Total adjusted premium is \(2.86+3.51=\$6.37\) million, and total expected losses are \(1.6588+2.0358=\$3.6946\) million. The implied aggregate expected loss ratio is \(3.6946/6.37=0.58\), or 58%.
Answer: Accident year A has expected losses of $1.6588 million; accident year B has $2.0358 million; the total is $3.6946 million and the implied ratio is 58%. Common mistake avoided: using inconsistent premium levels or reporting an aggregate ratio that does not reconcile to the selected ratio.
Common Errors and Practical Checks
A reliable calculation is short, but it depends on keeping the inputs and their roles straight. Before finalizing a result, check the following:
- Percent versus decimal: use \(0.72\), not \(72\), for a 72% ratio.
- Premium basis: make sure earned premium and the selected ratio are expressed on a consistent basis. Apply stated premium adjustments before calculating expected losses.
- Year alignment: multiply each accident year’s own premium by the applicable ratio. Do not apply one year’s premium or adjustment to another year.
- Development proportions: do not multiply the expected ultimate loss by a reported proportion when the requested amount is expected ultimate losses.
- Units: if premium is stated in millions, the result is in millions; convert to dollars only if requested.
- Aggregate reconciliation: with one common ratio and consistent premium basis, \(\sum_i\widehat{L}_{i,\mathrm{exp}}^{CC}=\widehat{ELR}_{CC}\sum_iP_i\).
The calculation’s purpose is to preserve the premium information by accident year while applying the selected common expected loss ratio. That makes the expected loss amounts responsive to year-specific premium volume without mistakenly treating development maturity as a second scaling factor. These amounts are an expected-loss allocation; further methods are needed to estimate unpaid claims from them.
Check Your Understanding
For each problem, calculate the expected losses on the stated premium basis. Treat each selected ratio as applicable to all years in its problem.
- A hypothetical specialty insurer selects a Cape Cod expected loss ratio of 64%. An accident year has $2.8 million in earned premium. What are its expected losses? (Answer: $1.792 million.)
- A selected ratio is 73%. An accident year has $4.2 million of earned premium before a factor of 1.05, which adjusts it to the selected ratio’s premium basis. Calculate its expected losses. (Answer: $3.2193 million.)
- A selected ratio of 61% applies to three accident years with earned premiums of $1.8 million, $2.4 million, and $3.3 million. Calculate each year’s expected losses and the total. (Answer: $1.098 million, $1.464 million, and $2.013 million; total $4.575 million.)
- A hypothetical portfolio has a selected ratio of 70%. Two years have earned premiums of $2.0 million and $5.0 million, with reported proportions of 90% and 40%. Calculate expected ultimate losses by year and in total. (Answer: $1.4 million and $3.5 million; total $4.9 million.)
- A selected ratio of 66% applies to a class with annual earned premium rates of $240 per exposure in year 1 and $255 per exposure in year 2. The years have 1,200 and 1,500 earned exposures, respectively. Calculate earned premium and expected losses for each year, then the total expected losses. (Answer: premiums are $288,000 and $382,500; expected losses are $190,080 and $252,450; total $442,530.)