Paid Cape Cod Estimation

Purpose of Paid Cape Cod Estimation

A paid loss triangle records cumulative claim payments by accident year and development age. For a recent accident year, the observed cumulative paid amount is only part of the eventual total. Paid Cape Cod estimation uses those observed payments together with a paid emergence pattern and an exposure base, typically earned premium, to estimate ultimate losses and the unpaid portion.

The central weighting idea is that a dollar of premium from an immature year contributes less expected paid loss to the data than a dollar from a mature year. The method therefore does not compare total paid losses with total premium as if every year had the same maturity. It first adjusts each year's premium for the share of ultimate loss expected to have been paid by the valuation date, then estimates a common expected loss ratio from the pooled experience. This is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let \(P_i\) be the selected earned-premium exposure for accident year \(i\), \(L_i\) its cumulative paid losses at the valuation date, and \(p_i\) the cumulative paid emergence percentage at that year’s development age. The percentage \(p_i\) is the expected fraction of ultimate losses already paid, with \(0 < p_i \leq 1\). The paid Cape Cod exposure base is \(\sum_i P_i p_i\). The paid Cape Cod expected loss ratio is \(\widehat{ELR}_{CC}=\frac{\sum_i L_i}{\sum_i P_i p_i}\). The resulting ultimate loss estimate for year \(i\) is \(\widehat{U}_i=P_i\widehat{ELR}_{CC}\), and the unpaid paid-loss liability is \(\widehat{R}_i=\widehat{U}_i-L_i\), or equivalently \(P_i\widehat{ELR}_{CC}(1-p_i)\) when the observed paid amount is consistent with the emergence assumption.

Model, Inputs, and Calculation

The basic model assumes that each accident year has the same expected ultimate loss ratio on the chosen premium base. In symbols, the expected ultimate loss for year \(i\) is \(P_i ELR\). It also assumes that the selected paid emergence percentage \(p_i\) applies to that year and that paid losses, premium and emergence percentages use consistent valuation, coverage and gross-or-net bases. Premium should represent the exposure for the loss period being estimated; for example, earned premium is generally more directly aligned with accident-year losses than written premium.

Under these assumptions, expected cumulative paid losses for year \(i\) are \(P_i ELR p_i\). Summing across years gives an expected pooled paid amount of \(ELR\sum_i P_i p_i\). Equating that expected amount to the observed pooled paid losses yields the Cape Cod loss-ratio estimate. The factor \(p_i\) makes the exposure base maturity-adjusted: premium from a year with only 30% of losses expected to be paid contributes 30% of its full premium to the denominator.

E[L_i]=P_i ELR p_i \qquad\Longrightarrow\qquad \widehat{ELR}_{CC} =\frac{\sum_i L_i}{\sum_i P_i p_i}

Once the common loss ratio is estimated, the ultimate estimate for each year is its own premium multiplied by that loss ratio. The liability is the estimated ultimate less the amount already paid. This is a paid-loss liability estimate; it does not, by itself, include future loss-adjustment expenses unless the loss amounts and assumptions have explicitly been defined to include them.

Align the inputs.
For each accident year, specify earned premium, cumulative paid losses at the valuation date, and the paid emergence percentage for that year’s maturity.

Build the adjusted exposure base.
Calculate \(P_i p_i\) for each year and sum these amounts. This is not generally the same as total premium multiplied by an unweighted average emergence percentage.

Estimate the common loss ratio.
Divide the sum of cumulative paid losses by the adjusted exposure base.

Estimate ultimate and unpaid losses.
For each year, calculate \(P_i\widehat{ELR}_{CC}\), then subtract that year’s cumulative paid losses. Sum the year-level unpaid estimates for the total liability.

Why the Exposure Adjustment Works

The denominator follows directly from the assumed relationship between ultimate losses and paid emergence. If the expected ultimate loss ratio is \(ELR\), a year with premium \(P_i\) has expected ultimate losses \(P_i ELR\). If the expected fraction paid by the valuation date is \(p_i\), the expected cumulative paid amount is that ultimate amount multiplied by \(p_i\). Therefore, for a set of years, the paid-loss observations are expected to represent the loss ratio times the sum of maturity-adjusted premium exposures.

Solving that pooled relationship for the loss ratio gives the estimator. The adjustment is important: without it, a recent year’s relatively low paid losses would be compared with its full premium, making it appear to have a lower loss ratio simply because less time has elapsed for claims to be paid.

\sum_i E[L_i] =\sum_i P_i ELR p_i =ELR\sum_i P_i p_i \quad\Longrightarrow\quad ELR=\frac{\sum_i E[L_i]}{\sum_i P_i p_i}

This derivation explains the estimator under the stated model; it does not guarantee that the estimate is suitable for every portfolio. Differences in expected loss ratios among years, an unsuitable emergence pattern, changes in claim payment practices, or inconsistent premium and loss bases can affect the result. A selected common loss ratio is an assumption used to pool experience, not proof that every year truly has identical risk.

Worked Example: One Common Valuation Age

A hypothetical insurer is estimating paid losses for three accident years. All amounts below are in millions of dollars. The earned premiums are \(2.00\), \(3.00\) and \(4.00\); the cumulative paid losses are \(0.60\), \(1.35\) and \(2.16\); and the selected paid emergence percentages are \(50\%\), \(75\%\) and \(90\%\), respectively. Estimate the Cape Cod loss ratio, each year’s ultimate losses and total unpaid paid-loss liability.

Method. Use the paid Cape Cod estimator, adjusting each premium by its own emergence percentage. The common valuation date does not imply a common maturity: the selected percentages differ by year.

Step 1: Calculate the adjusted exposure base. The three adjusted exposures are \(2.00(0.50)=1.00\), \(3.00(0.75)=2.25\), and \(4.00(0.90)=3.60\) million. Their sum is \(1.00+2.25+3.60=6.85\) million.

Step 2: Pool the paid losses and estimate the loss ratio. Total paid losses are \(0.60+1.35+2.16=4.11\) million. Thus \(\widehat{ELR}_{CC}=4.11/6.85=0.60\), or \(60\%\).

Step 3: Estimate ultimate losses and unpaid amounts. The ultimate estimates are \(2.00(0.60)=1.20\), \(3.00(0.60)=1.80\), and \(4.00(0.60)=2.40\) million. Subtracting paid losses gives unpaid amounts of \(1.20-0.60=0.60\), \(1.80-1.35=0.45\), and \(2.40-2.16=0.24\) million.

Check. Total ultimate losses are \(1.20+1.80+2.40=5.40\) million. Subtracting total paid losses, \(5.40-4.11=1.29\) million, agrees with \(0.60+0.45+0.24=1.29\) million of estimated unpaid losses.

Common mistake avoided. Dividing \(4.11\) by total premium \(9.00\) would produce \(45.7\%\), not the Cape Cod estimate. That calculation treats all premium as fully emerged; the correct denominator is the maturity-adjusted \(6.85\) million.

Worked Example: Different Maturities and a Premium-Weighted Base

For a hypothetical specialty coverage, three accident years have earned premiums of \(1.50\), \(2.50\) and \(3.50\) million dollars. Their cumulative paid losses are \(0.288\), \(0.960\) and \(1.792\) million. Their paid emergence percentages are \(30\%\), \(60\%\) and \(80\%\), respectively. Calculate the common Cape Cod loss ratio and total unpaid paid-loss liability.

Method. Use each year’s own paid emergence percentage when constructing the premium-weighted exposure base. The premium amounts are unequal, so an unweighted average of the percentages would not preserve the pooled expected-paid-loss relationship.

Step 1: Calculate adjusted exposures. They are \(1.50(0.30)=0.45\), \(2.50(0.60)=1.50\), and \(3.50(0.80)=2.80\) million. The denominator is \(0.45+1.50+2.80=4.75\) million.

Step 2: Calculate the loss ratio. Total paid losses equal \(0.288+0.960+1.792=3.040\) million. Therefore, \(\widehat{ELR}_{CC}=3.040/4.75=0.64\), or \(64\%\).

Step 3: Project ultimate losses and unpaid amounts. Ultimate losses are \(1.50(0.64)=0.960\), \(2.50(0.64)=1.600\), and \(3.50(0.64)=2.240\) million. The respective unpaid estimates are \(0.960-0.288=0.672\), \(1.600-0.960=0.640\), and \(2.240-1.792=0.448\) million.

Check. The unpaid total is \(0.672+0.640+0.448=1.760\) million. The ultimate total is \(0.960+1.600+2.240=4.800\) million, and \(4.800-3.040=1.760\) million, confirming the result.

Common mistake avoided. Applying one average emergence percentage to total premium can give the wrong denominator when premium amounts differ. The calculation uses \(\sum_i P_i p_i\), not total premium multiplied by an unweighted average of \(30\%\), \(60\%\) and \(80\%\).

When Development Factors Are Supplied

A paid emergence percentage is sometimes provided directly. In other problems, the input is a cumulative paid age-to-ultimate development factor. Under the usual multiplicative relationship, if \(F_i\) is the factor from the current maturity to ultimate, then \(F_i=1/p_i\), so \(p_i=1/F_i\). The factor and the emergence percentage point in opposite directions: a larger age-to-ultimate factor means a smaller share has emerged.

Use the reciprocal conversion only when the supplied factor has that meaning and uses a compatible paid-loss basis. If a problem directly gives emergence percentages, do not invert them. Nor should a factor be treated as a percentage to multiply by premium; that would increase the adjusted exposure for an immature year rather than reduce it.

Worked Example: Converting Paid Development Factors

A hypothetical book has earned premiums of \(1{,}000\), \(2{,}000\) and \(3{,}000\) thousand dollars. The age-to-ultimate paid development factors are \(4.00\), \(2.00\) and \(1.25\). Cumulative paid losses are \(125\), \(500\) and \(1{,}200\) thousand dollars. Estimate the paid Cape Cod loss ratio and unpaid liability.

Method. Convert each age-to-ultimate factor to its cumulative paid emergence percentage, then use the paid Cape Cod formula. Because the factors are defined as ultimate divided by current paid, the conversion is the reciprocal.

Step 1: Convert factors to emergence percentages. The percentages are \(1/4.00=0.25\), \(1/2.00=0.50\), and \(1/1.25=0.80\).

Step 2: Calculate the adjusted exposure base. In thousands, the adjusted exposures are \(1{,}000(0.25)=250\), \(2{,}000(0.50)=1{,}000\), and \(3{,}000(0.80)=2{,}400\). The total denominator is \(250+1{,}000+2{,}400=3{,}650\).

Step 3: Estimate the loss ratio. Paid losses total \(125+500+1{,}200=1{,}825\) thousand. Therefore, \(\widehat{ELR}_{CC}=1{,}825/3{,}650=0.50\), or \(50\%\).

Step 4: Estimate ultimate and unpaid losses. Ultimate losses are \(1{,}000(0.50)=500\), \(2{,}000(0.50)=1{,}000\), and \(3{,}000(0.50)=1{,}500\) thousand. Subtracting paid losses gives unpaid amounts of \(375\), \(500\) and \(300\) thousand, for a total of \(1{,}175\) thousand.

Check. Total ultimate losses are \(3{,}000\) thousand, and \(3{,}000-1{,}825=1{,}175\) thousand, equal to the sum of the three unpaid estimates.

Common mistake avoided. Using the factors \(4.00\), \(2.00\) and \(1.25\) as emergence percentages would reverse their meaning. The correct emergence percentages are their reciprocals: \(25\%\), \(50\%\) and \(80\%\).

Interpreting and Checking the Liability

A useful check is that the total estimated unpaid amount equals total estimated ultimate losses less total observed paid losses. At the year level, the basic calculation is \(\widehat{R}_i=\widehat{U}_i-L_i\). If the model inputs are internally consistent so that \(L_i=P_i\widehat{ELR}_{CC}p_i\), this is also \(P_i\widehat{ELR}_{CC}(1-p_i)\). In actual estimates, observed losses need not equal that fitted amount for each year, so use ultimate less actual paid for the year-level liability rather than forcing the alternative expression to hold exactly.

Paid Cape Cod is sensitive to the inputs that define its pooled relationship. The premium base must match the loss experience, each maturity must be paired with the appropriate paid emergence percentage, and the cumulative paid amounts must be measured at the same valuation date. Results can be distorted if, for example, one year’s losses are net of recoveries while its expected emergence pattern is on a gross basis.

Do not omit the maturity adjustment. The denominator is \(\sum_i P_i p_i\), not \(\sum_i P_i\).
Do not average emergence percentages without premium weights. The correct weights arise from the individual \(P_i p_i\) terms.
Do not confuse emergence with a development factor. An age-to-ultimate factor is generally inverted to obtain the emerged share.
Do not confuse ultimate with unpaid. Ultimate includes losses already paid; unpaid is ultimate less cumulative paid losses.
Do not mix measurement bases. Premium, losses and emergence assumptions must correspond in period, coverage and gross-or-net treatment.

The resulting unpaid estimate is a projection under the selected common loss-ratio and paid-emergence assumptions. It is not an observed amount, and it should be interpreted with those assumptions in view. A numerical result may be calculated correctly yet still be inappropriate if the selected premium base or emergence percentages do not represent the portfolio.

Check Your Understanding

For each problem, use the stated premium and paid emergence assumptions. Amounts are in millions unless noted otherwise.

A single accident year has earned premium of \(2.00\), cumulative paid losses of \(0.300\), and paid emergence of \(40\%\). Find the paid Cape Cod loss ratio, ultimate losses and unpaid amount. (Answer: \(37.5\%\); ultimate \(0.750\); unpaid \(0.450\).)
Three years have premiums \(1.00\), \(2.00\) and \(3.00\), emergence percentages \(25\%\), \(50\%\) and \(75\%\), and paid losses \(0.100\), \(0.400\) and \(0.900\). Find the common loss ratio and total unpaid liability. (Answer: \(40\%\); unpaid \(1.000\).)
Premiums are \(400\), \(800\) and \(1{,}200\) thousand; age-to-ultimate paid factors are \(2.50\), \(2.00\) and \(1.25\); and paid losses are \(64\), \(160\) and \(384\) thousand. Convert the factors, then estimate the loss ratio and total unpaid liability. (Answer: emergence \(40\%\), \(50\%\), \(80\%\); loss ratio \(40\%\); unpaid \(352\) thousand.)
Three years have premiums \(1.00\), \(3.00\) and \(2.00\), emergence percentages \(20\%\), \(60\%\) and \(100\%\), and paid losses \(0.100\), \(0.900\) and \(0.800\). Calculate the Cape Cod loss ratio and total unpaid liability. (Answer: \(45\%\); unpaid \(0.900\).)
A portfolio has premiums of \(1.00\) and \(4.00\), emergence percentages of \(20\%\) and \(80\%\), and paid losses of \(0.100\) and \(1.600\). Calculate the correct loss ratio and unpaid liability. Then calculate the loss ratio that results from incorrectly using the unweighted average emergence percentage with total premium. (Answer: correct loss ratio \(50\%\), correct unpaid liability \(1.200\); incorrect loss ratio \(68\%\).)

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