Comparing Methods on the Same Loss Triangle

Comparing Two Methods on a Common Basis

Bornhuetter-Ferguson (BF) and Cape Cod both use an estimate of the proportion of ultimate loss that is still unpaid or unreported. Their key difference is how they obtain the loss ratio used to estimate that remainder. BF uses a selected expected loss ratio; Cape Cod estimates a common loss ratio from the observed losses and the maturity-adjusted exposures in the data. Comparing the methods on the same loss basis makes that difference visible.

For a paid-loss analysis, the observed amounts below are cumulative paid losses, and each development proportion is the proportion of ultimate paid losses already paid. For an incurred-loss analysis, use cumulative incurred amounts and consistent incurred development proportions instead. The examples use hypothetical portfolios and treat each stated proportion as extending to ultimate, with no separate tail adjustment.

Definition: For accident period $i$, let $E_i$ be exposure, $p_i$ the proportion of ultimate losses represented by the observed cumulative loss measure, $C_i$ the observed cumulative losses, and $r$ the selected or estimated loss ratio. The unobserved proportion is $1-p_i$. BF estimates the reserve as $E_i r(1-p_i)$. Cape Cod first estimates $r$ as $\sum_i C_i/\sum_i E_i p_i$, then uses that estimated ratio in the reserve formula.

The reserve amounts are estimates of total unpaid losses on the chosen basis, not merely losses on claims that have not yet been reported. For example, on a paid basis, the unpaid share can include future payments on claims that are already known. Keep the loss amounts, development proportions, and reserve interpretation consistent throughout a calculation.

The Calculations and Their Rationale

For BF, the expected ultimate loss for period $i$ is $E_i r$. The estimated portion not yet represented in the observed cumulative amount is therefore $E_i r(1-p_i)$. The observed amount $C_i$ is not multiplied by the unpaid proportion to get the BF reserve; it is used when estimating ultimate loss as $C_i+$ reserve.

\widehat{R}^{\mathrm{BF}}_i=E_i r_{\mathrm{selected}}(1-p_i), \qquad \widehat{U}^{\mathrm{BF}}_i=C_i+\widehat{R}^{\mathrm{BF}}_i

For Cape Cod, the common loss-ratio assumption can be expressed as $C_i$ being approximately $E_i r p_i$: exposure times the common expected loss ratio times the proportion already observed. Summing across periods gives $\sum_i C_i$ approximately equal to $r\sum_i E_i p_i$. Solving this relationship for $r$ yields the Cape Cod loss ratio. The denominator is maturity-adjusted exposure, not total exposure.

\widehat{r}_{\mathrm{CC}} = \frac{\sum_i C_i}{\sum_i E_i p_i}, \qquad \widehat{R}^{\mathrm{CC}}_i = E_i(1-p_i)\widehat{r}_{\mathrm{CC}}

This derivation explains why Cape Cod weights each period’s contribution to the loss-ratio estimate by its maturity. A period with only 40% of ultimate losses observed contributes $0.4E_i$ to the denominator, not all of $E_i$. Both methods then allocate their estimated unpaid losses in proportion to exposure multiplied by the unpaid share, but their loss ratios can differ.

Check the basis.
Confirm that cumulative losses and development proportions are both paid-based or both incurred-based, and that the proportions represent the same development definition.

Calculate Cape Cod’s denominator.
For each period, calculate $E_i p_i$, then sum these maturity-adjusted exposures.

Calculate both ratios.
Use the selected expected loss ratio for BF and divide total observed losses by maturity-adjusted exposure for Cape Cod.

Allocate reserves by period.
Multiply each method’s loss ratio by $E_i(1-p_i)$, then sum the period estimates to obtain the portfolio reserve.

Worked Comparison on a Three-Period Portfolio

The table gives hypothetical exposures, cumulative paid losses, and paid development proportions. Use a selected BF expected loss ratio of 70%. The comparison is meaningful because both methods use the same paid-loss basis and the same development proportions.

Accident period	Exposure $E_i$	Paid proportion $p_i$	Cumulative paid loss $C_i$
1	1,000	0.50	$420
2	1,200	0.75	$760
3	800	0.90	$600

Worked Example: BF and Cape Cod on the Same Paid-Loss Data

Method. Calculate each reserve using the same exposures and paid proportions. BF uses the selected 70% loss ratio; Cape Cod estimates its loss ratio from total cumulative paid losses and maturity-adjusted exposure. Report reserve amounts to the nearest cent and the ratios to four decimal places as percentages.

Step 1: Find the unobserved exposure for each period. These values are $E_i(1-p_i)$: $1{,}000(0.50)=500$, $1{,}200(0.25)=300$, and $800(0.10)=80$. The total is $500+300+80=880$.

Step 2: Calculate the BF reserves. Multiply each unobserved exposure by the selected 70% ratio: period 1, $500(0.70)=\$350.00$; period 2, $300(0.70)=\$210.00$; period 3, $80(0.70)=\$56.00$. The BF portfolio reserve is $\$350.00+\$210.00+\$56.00=\$616.00$.

Step 3: Estimate the Cape Cod ratio. The total cumulative paid loss is $\$420+\$760+\$600=\$1{,}780$. Maturity-adjusted exposure is $1{,}000(0.50)+1{,}200(0.75)+800(0.90)=500+900+720=2{,}120$. Thus the Cape Cod loss ratio is $1{,}780/2{,}120=0.8396226$, or 83.9623%.

Step 4: Calculate the Cape Cod reserves. Period 1: $500(0.8396226)=\$419.81$. Period 2: $300(0.8396226)=\$251.89$. Period 3: $80(0.8396226)=\$67.17$. The portfolio reserve is $\$419.81+\$251.89+\$67.17=\$738.87$.

Check and interpretation. Cape Cod exceeds BF by $\$738.87-\$616.00=\$122.87$, because its estimated ratio, 83.9623%, exceeds BF’s selected 70%. The Cape Cod reserve also reconciles to its estimated ultimate losses: total exposure is 3,000, so estimated ultimate losses are $3{,}000(0.8396226)=\$2{,}518.87$; subtracting observed paid losses of $\$1{,}780$ gives $\$738.87$. The common mistake avoided here is dividing the observed losses by total exposure rather than maturity-adjusted exposure.

What Drives the Difference?

The same $E_i(1-p_i)$ values allocate reserves under both methods. Therefore, when the inputs are identical, the method-level difference in each period is the difference in loss ratios multiplied by that period’s unobserved exposure. The portfolio difference follows directly:

\widehat{R}^{\mathrm{CC}}-\widehat{R}^{\mathrm{BF}} = \left(\widehat{r}_{\mathrm{CC}}-r_{\mathrm{selected}}\right) \sum_i E_i(1-p_i)

This identity follows by summing the two period reserve formulas: each is its loss ratio multiplied by $E_i(1-p_i)$. Subtracting one sum from the other factors out the difference in ratios. Thus, a higher Cape Cod ratio produces a higher portfolio reserve when any exposure remains unobserved; if all periods are fully mature, the reserve is zero under both formulas regardless of the ratio.

The allocation across periods depends on the maturity pattern. A period with a larger $E_i(1-p_i)$ receives a larger share under either method. Cape Cod’s ratio also depends on the data across all periods: each period contributes observed losses to the numerator and maturity-adjusted exposure to the denominator. Consequently, unusual experience in one period can move the ratio applied to every period.

Worked Example: A Different Cape Cod Ratio Changes the Portfolio Reserve

A hypothetical two-period portfolio has exposures of 400 and 600, paid proportions of 0.60 and 0.80, and cumulative paid losses of $180 and $390. Use a 75% selected BF ratio. Calculate the Cape Cod ratio and both portfolio reserves, rounding reserve amounts to cents and the ratios to four decimal places as percentages.

Step 1: Calculate maturity-adjusted and unobserved exposures. The maturity-adjusted exposures are $400(0.60)=240$ and $600(0.80)=480$, totaling 720. The unobserved exposures are $400(0.40)=160$ and $600(0.20)=120$, totaling 280.

Step 2: Calculate the Cape Cod ratio. Total observed losses are $\$180+\$390=\$570$. Hence $\widehat{r}_{\mathrm{CC}}=570/720=0.7916667$, or 79.1667%.

Step 3: Calculate BF. The period reserves are $160(0.75)=\$120.00$ and $120(0.75)=\$90.00$. The BF reserve is $\$210.00$.

Step 4: Calculate Cape Cod. The period reserves are $160(0.7916667)=\$126.67$ and $120(0.7916667)=\$95.00$. The Cape Cod reserve is $\$221.67$. Its excess over BF is $\$11.67$, which checks against $(0.7916667-0.75)(280)=\$11.67$, subject to rounding.

The comparison is not a test of which method is correct: it identifies how replacing a selected ratio with a ratio fitted to the observed portfolio changes the estimates. The common error to avoid is comparing ratios alone without checking how much unobserved exposure those ratios are applied to.

Worked Example: Equal Maturity and Reserve Allocation

Three hypothetical accident periods have exposures of 500, 1,000, and 1,500. Each has a paid proportion of 0.80; cumulative paid losses are $300, $500, and $700. Use a 60% BF ratio. Calculate both ratios and show how the common maturity pattern affects the reserve allocation. Report dollars to cents and percentages to four decimal places.

Step 1: Calculate the Cape Cod ratio. Total paid losses are $\$300+\$500+\$700=\$1{,}500$. Total maturity-adjusted exposure is $(500+1{,}000+1{,}500)(0.80)=3{,}000(0.80)=2{,}400$. Thus the Cape Cod ratio is $1{,}500/2{,}400=0.625$, or 62.5000%.

Step 2: Find unobserved exposures. The unobserved proportion is 0.20 for every period, so unobserved exposures are $500(0.20)=100$, $1{,}000(0.20)=200$, and $1{,}500(0.20)=300$, totaling 600.

Step 3: Calculate BF reserves. The period amounts are $100(0.60)=\$60.00$, $200(0.60)=\$120.00$, and $300(0.60)=\$180.00$, totaling $\$360.00$.

Step 4: Calculate Cape Cod reserves. The period amounts are $100(0.625)=\$62.50$, $200(0.625)=\$125.00$, and $300(0.625)=\$187.50$, totaling $\$375.00$. The difference is $\$15.00$, also given by $(0.625-0.60)(600)=\$15.00$. Because all periods have the same maturity, each method allocates its total reserve in proportion to exposure: one-sixth, one-third, and one-half, respectively.

This check illustrates both the ratio effect and the allocation effect. With equal proportions, the maturity-adjusted exposure denominator is 80% of total exposure, and unobserved exposure is 20% of total exposure. The period allocation is proportional to exposure, although the two methods still produce different totals when their ratios differ.

Common Exam Trap: Using Total Exposure in the Cape Cod Denominator

A frequent error is to divide total observed losses by total exposure, then call the result the Cape Cod ratio. That calculation ignores the fact that some periods have had more losses emerge than others. It will generally produce a different ratio and can materially change the estimated reserve.

\text{Incorrect: }\quad \frac{\sum_i C_i}{\sum_i E_i} \qquad\text{Correct: }\quad \widehat{r}_{\mathrm{CC}} = \frac{\sum_i C_i}{\sum_i E_i p_i}

For the three-period example above, the incorrect calculation is $\$1{,}780/3{,}000=59.3333\%$, instead of the correct 83.9623%. If that incorrect ratio were applied to the total unobserved exposure of 880, it would give a reserve of $\$522.13$, rather than the Cape Cod reserve of $\$738.87$. The correction is to calculate every $E_i p_i$ first and use their sum as the denominator.

Do not mix bases: paid losses require paid development proportions; incurred losses require incurred development proportions.
Do not substitute observed losses for the BF ratio: BF uses the selected expected loss ratio in its reserve formula.
Do not confuse reserve and ultimate: on a paid basis, Cape Cod ultimate is total exposure times its ratio, while its reserve is estimated ultimate less cumulative paid losses.
Do not use rounded ratios prematurely: retain unrounded values through the period calculations and round only the final reported amounts.

Key takeaway: On common data, BF and Cape Cod use the same maturity-weighted reserve allocation pattern, $E_i(1-p_i)$. Their estimates differ because BF supplies a selected loss ratio while Cape Cod estimates a common ratio from total observed losses divided by maturity-adjusted exposure. This comparison is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, keep the loss basis consistent, calculate the maturity-adjusted exposure before the Cape Cod ratio, and round reserves to cents.

A two-period hypothetical paid-loss portfolio has exposures of 700 and 900, paid proportions of 0.50 and 0.75, and cumulative paid losses of $280 and $630. Calculate the Cape Cod ratio and total Cape Cod reserve. (Answer: Cape Cod ratio = 88.7805%; total reserve = $510.49.)
A hypothetical incurred-loss portfolio has exposures of 800 and 500, incurred proportions of 0.25 and 0.80, and a selected BF expected loss ratio of 65%. Calculate each BF reserve and the total. (Answer: Period reserves = $390.00 and $65.00; total = $455.00.)
A two-period hypothetical paid-loss portfolio has exposures of 400 and 1,000, paid proportions of 0.50 and 0.90, and cumulative paid losses of $190 and $720. Calculate the Cape Cod ratio and reserve. Then calculate the BF reserve using a selected ratio of 70%. (Answer: Cape Cod ratio = 82.7273%; Cape Cod reserve = $248.18; BF reserve = $210.00.)
A hypothetical two-period paid-loss portfolio has exposures of 600 and 600, paid proportions of 0.50 and 0.75, and cumulative paid losses of $240 and $450. Compare the Cape Cod reserve with the BF reserve using a selected ratio of 80%. (Answer: Cape Cod ratio = 92.0000%; Cape Cod reserve = $414.00; BF reserve = $360.00.)
A hypothetical portfolio has exposures of 1,000 and 500, paid proportions of 0.40 and 0.80, and cumulative paid losses of $200 and $360. Calculate the Cape Cod ratio and each period’s Cape Cod reserve. (Answer: Cape Cod ratio = 70.0000%; period reserves = $420.00 and $70.00; total reserve = $490.00.)

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