Worked Cape Cod Portfolio Problem

Introduction

A Cape Cod estimate uses the portfolio's observed losses to calibrate an expected loss ratio, while allowing for the fact that accident periods are at different stages of development. The calculation is simple only if the exposure, reported-loss amounts, and reported proportions all use a consistent basis. Mixing, for example, paid losses with incurred-loss maturity proportions can produce a precise-looking answer that has no coherent interpretation.

This tutorial works through a multi-period portfolio calculation: convert development factors to reported proportions, calibrate the exposure-weighted loss ratio, estimate ultimate losses by accident period, and calculate unpaid amounts. All scenarios and data below are hypothetical. The deterministic calculations use no interest or mortality assumption and posit no probability distribution for losses; the selected reported proportions and exposure amounts are inputs. This treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For accident period $i$, let $E_i$ be exposure on the selected basis, $C_i$ cumulative reported losses at the valuation date on that same basis, and $p_i$ the selected proportion of ultimate losses expected to be reported by that date. Under the Cape Cod model, a common portfolio loss ratio $\lambda$ gives expected ultimate losses $E_i\lambda$. The calibrated estimate is $$ \widehat{\lambda} = \frac{\sum_i C_i}{\sum_i E_i p_i}, \qquad \widehat{U}_i=E_i\widehat{\lambda}. $$

Set the Basis Before Calculating

The reported proportion $p_i$ measures the expected share of ultimate losses already reported. If a selected age-to-ultimate loss development factor for the relevant basis is $f_i$, then $$ p_i=\frac{1}{f_i}. $$ For example, a factor of $2.00$ implies a reported proportion of $0.50$. A factor of $1.25$ implies a reported proportion of $0.80$. The factor must be appropriate to the same loss measure used for $C_i$: a paid development factor belongs with cumulative paid losses, and an incurred development factor belongs with cumulative incurred losses.

The exposure $E_i$ must also be on the basis to which the loss ratio applies. It may be a premium or another suitable exposure measure, but it cannot be switched partway through the calculation. In particular, the denominator is not simply total exposure and is not total exposure multiplied by one unweighted average maturity. Each period contributes $E_i p_i$.

Fix the valuation and measurement basis.
Identify the valuation date, cumulative reported-loss measure, and matching exposure for every accident period.

Determine each reported proportion.
Use selected proportions directly, or take the reciprocal of the matching age-to-ultimate development factor.

Calibrate the portfolio loss ratio.
Divide the total cumulative reported losses by the sum of exposure multiplied by reported proportion.

Calculate accident-period estimates.
Multiply each exposure by the calibrated loss ratio to obtain expected ultimate losses; subtract reported losses for actual-based unpaid estimates.

Why the Calibration Formula Holds

Suppose the same loss ratio $\lambda$ applies to every accident period. The expected ultimate for period $i$ is $E_i\lambda$, so the expected amount reported by the valuation date is $E_i\lambda p_i$. Summing across periods gives total expected reported losses $\lambda\sum_i E_i p_i$. Cape Cod chooses $\lambda$ so this expected total matches the observed total reported losses. Therefore,

\lambda\sum_i E_i p_i=\sum_i C_i \quad\Longrightarrow\quad \widehat{\lambda}=\frac{\sum_i C_i}{\sum_i E_i p_i}.

This is a portfolio calibration: it matches the totals, not necessarily the reported amount in each accident period separately. That distinction explains why an accident-period reserve can be negative even when the portfolio reserve is positive.

Worked Cape Cod Portfolio Calculations

Worked Example: Three Accident Periods on One Basis

A hypothetical insurer has three accident periods, all measured at the same valuation date using cumulative incurred losses and premium exposure in thousands of dollars. The selected incurred age-to-ultimate factors are $2.50,\ 1.428571,$ and $1.111111$, giving reported proportions $0.40,\ 0.70,$ and $0.90$, respectively. Premium exposures are $1{,}000,\ 800,$ and $600$, and cumulative reported incurred losses are $250,\ 350,$ and $300$. Estimate the Cape Cod loss ratio, each ultimate, and unpaid losses.

Method and maturity inputs. Take reciprocals of the factors: $1/2.50=0.40$, $1/1.428571\approx0.70$, and $1/1.111111\approx0.90$. The stated proportions are used as $0.40,\ 0.70,\ 0.90$. All monetary figures remain in thousands.

Calibration denominator. Weight each exposure by its own reported proportion:

$$ 1{,}000(0.40)+800(0.70)+600(0.90) =400+560+540 =1{,}500. $$

Calibrated loss ratio. Total reported losses are $250+350+300=900$. Thus

\widehat{\lambda}=\frac{900}{1{,}500}=0.60.

Accident-period ultimates and actual-based unpaid estimates. Multiply each exposure by $0.60$, then subtract its observed reported loss:

\begin{aligned} \widehat{U}_1&=1{,}000(0.60)=600, &\widehat{R}_1&=600-250=350,\\ \widehat{U}_2&=800(0.60)=480, &\widehat{R}_2&=480-350=130,\\ \widehat{U}_3&=600(0.60)=360, &\widehat{R}_3&=360-300=60. \end{aligned}

Check. The estimated ultimates total $600+480+360=1{,}440$, and the actual-based unpaid estimates total $350+130+60=540$. The portfolio reported total is $900$, so $1{,}440-900=540$, as required.

The common mistake avoided here is using $0.70$, or another average maturity, for every accident period. The denominator must use the period-specific products $E_i p_i$.

Worked Example: A Portfolio With Unequal Maturities

A hypothetical portfolio is evaluated using cumulative paid losses, with premium exposures of $700$ and $1{,}000$ thousand dollars. The selected paid reported proportions are $0.50$ and $0.80$, and cumulative paid losses are $280$ and $720$ thousand dollars. Calculate the calibrated loss ratio, accident-period ultimates, actual-based unpaid estimates, and expected unpaid amounts implied by the model.

Calibration. The exposure-weighted denominator is $700(0.50)+1{,}000(0.80)=350+800=1{,}150$. Total reported paid losses are $280+720=1{,}000$. Hence

\widehat{\lambda}=\frac{1{,}000}{1{,}150} =\frac{20}{23} \approx0.8695652.

Expected ultimates. Using the unrounded ratio, the first ultimate is $700(20/23)=14{,}000/23=608.6957$ thousand dollars. The second is $1{,}000(20/23)=20{,}000/23=869.5652$ thousand dollars.

Actual-based unpaid estimates. Subtract the actual cumulative reported paid amounts:

\widehat{R}_1=608.6957-280=328.6957, \qquad \widehat{R}_2=869.5652-720=149.5652

Expected unpaid amounts. The model's expected unpaid amount is ultimate multiplied by the unreported proportion. For period 1 this is $608.6957(1-0.50)=304.3478$. For period 2 it is $869.5652(1-0.80)=173.9130$. Rounded to cents, the ultimates are $608.70$ and $869.57$ thousand dollars; actual-based unpaid estimates are $328.70$ and $149.57$; and expected unpaid amounts are $304.35$ and $173.91$.

Check. Model-expected reported losses are $608.6957(0.50)=304.3478$ and $869.5652(0.80)=695.6522$, totaling $1{,}000$, the observed total. Expected unpaid amounts total $478.2608$; actual-based unpaid estimates total $478.2609$, with the tiny difference due to displayed rounding. The two approaches agree in aggregate before rounding, but not by accident period.

The mistake avoided is setting the loss ratio to $1.00$ merely because reported losses total $1{,}000$. The correct denominator is $1{,}150$, not $1{,}000$ or the sum of exposures.

Worked Example: A Negative Accident-Period Reserve

A hypothetical two-period portfolio uses premium exposure and cumulative incurred losses, both in thousands of dollars. Exposures are $600$ and $800$; reported proportions are $0.60$ and $1.00$; and reported losses are $300$ and $700$. Find the calibrated loss ratio, ultimates, and unpaid amounts. State how to handle a negative period result.

Calibration. The denominator is $600(0.60)+800(1.00)=360+800=1{,}160$. Reported losses total $300+700=1{,}000$, so

\widehat{\lambda}=\frac{1{,}000}{1{,}160} =\frac{25}{29} \approx0.8620690.

Ultimates. The first estimate is $600(25/29)=15{,}000/29=517.2414$. The second is $800(25/29)=20{,}000/29=689.6552$, in thousands of dollars.

Actual-based unpaid estimates. Subtract the observed reported amounts: $517.2414-300=217.2414$ for period 1 and $689.6552-700=-10.3448$ for period 2. Since the second period is fully reported under the selected maturity assumption, its model-expected unpaid amount is $689.6552(1-1.00)=0$. The first period's model-expected unpaid amount is $517.2414(1-0.60)=206.8966$.

Rounded to cents, the ultimates are $517.24$ and $689.66$; actual-based unpaid estimates are $217.24$ and $-10.34$; and expected unpaid amounts are $206.90$ and $0.00$ thousand dollars.

Check and interpretation. The modeled reported amounts are $517.2414(0.60)=310.3448$ and $689.6552(1.00)=689.6552$, totaling $1{,}000$. Actual-based unpaid estimates sum to $206.8966$, equal to model-expected unpaid amounts in total. Do not silently replace the negative period estimate with zero: it follows from the supplied inputs. Investigating whether the inputs or assumptions are appropriate is a separate judgment.

The mistake avoided is flooring a negative accident-period reserve without stating a separate constraint. The Cape Cod calculation itself produces ultimate less reported, which can be negative for an individual period.

Unpaid Amounts and Portfolio Reconciliation

Two quantities are useful, and they answer different questions. The actual-based unpaid estimate for period $i$ is the Cape Cod ultimate less actual cumulative reported losses, $\widehat{U}_i-C_i$. The model-expected unpaid amount is the estimated ultimate multiplied by the estimated unreported proportion, $\widehat{U}_i(1-p_i)$. The first uses the actual reported amount for that period; the second uses the model's expected reporting pattern.

The calibrated loss ratio ensures that these approaches reconcile in total, provided all periods use the same inputs and no separate adjustments are introduced:

\begin{aligned} \sum_i \widehat{U}_i(1-p_i) &=\sum_i E_i\widehat{\lambda} -\widehat{\lambda}\sum_i E_i p_i\\ &=\sum_i E_i\widehat{\lambda}-\sum_i C_i\\ &=\sum_i(\widehat{U}_i-C_i). \end{aligned}

The second equality follows directly from the calibration formula. The accident-period amounts need not match individually, because the calibration matches only portfolio totals. This reconciliation is a valuable arithmetic check: if the two portfolio totals differ materially, revisit the denominator, reporting proportions, or loss and exposure bases.

Common Exam Traps

Using total exposure in the denominator. Use $\sum_i E_i p_i$, not $\sum_i E_i$.
Reversing the development factor. An age-to-ultimate factor of $f_i$ implies reported proportion $1/f_i$, not $f_i$.
Combining different bases. Match cumulative paid losses with paid maturity proportions, or cumulative incurred losses with incurred maturity proportions, and keep exposure consistent.
Confusing expected unpaid with actual-based unpaid. Calculate $\widehat{U}_i(1-p_i)$ and $\widehat{U}_i-C_i$ separately when asked for both.
Forgetting units. If exposures and losses are stated in thousands, the calculated ultimates and unpaid amounts are also in thousands.
Forcing every accident-period result to be positive. A negative period result can arise from the calibrated portfolio estimate; do not alter the formula without an explicit rule.

Key takeaway: Cape Cod calibrates one loss ratio by matching total observed reported losses to exposure-weighted expected reported losses. Calculate every $E_i p_i$ on a consistent basis, then distinguish each period's expected ultimate, expected unpaid amount, and ultimate less actual reported losses.

Check Your Understanding

For each hypothetical problem, retain full precision during calculation and round money to the nearest cent at the end. Amounts are in dollars unless stated otherwise.

A single period has exposure $500$, reported proportion $0.40$, and reported losses $100$. Find the calibrated loss ratio, ultimate, and actual-based unpaid estimate. (Answer: $\widehat{\lambda}=0.5000$; ultimate $250.00$; unpaid $150.00$.)
Two periods have exposures $900$ and $600$, reported proportions $0.50$ and $0.75$, and reported losses $200$ and $430$. Find the loss ratio, each ultimate, and each actual-based unpaid estimate. (Answer: $\widehat{\lambda}=0.7000$; ultimates $630.00,\ 420.00$; actual-based unpaid $430.00,\ -10.00$.)
A portfolio has two periods with exposures $400$ and $800$, reported proportions $0.25$ and $0.80$, and reported losses $90$ and $500$. Find the calibrated loss ratio, both ultimates, and both model-expected unpaid amounts. (Answer: $\widehat{\lambda}=590/740=0.7973$; ultimates $318.92,\ 637.84$; expected unpaid $239.19,\ 127.57$.)
Two periods have exposures $1{,}000$ and $500$, age-to-ultimate factors $2.50$ and $1.25$, and reported losses $240$ and $360$. Convert the factors to proportions, then calculate the loss ratio and actual-based unpaid estimates. (Answer: proportions $0.40,\ 0.80$; denominator $800$; $\widehat{\lambda}=0.7500$; ultimates $750.00,\ 375.00$; unpaid $510.00,\ 15.00$.)
Two periods have premium exposures $800$ and $1{,}200$, reported proportions $0.25$ and $0.75$, and reported losses $100$ and $600$. Calculate the portfolio loss ratio, accident-period actual-based unpaid estimates, and model-expected unpaid amounts. Check the aggregate. (Answer: denominator $1{,}100$; $\widehat{\lambda}=700/1{,}100=0.6364$; ultimates $509.09,\ 763.64$; actual-based unpaid $409.09,\ 163.64$; model-expected unpaid $381.82,\ 190.91$; both aggregate unpaid totals $572.73$.)

What You'll Learn