Introduction
A reserve estimate is not selected just because it is larger, smaller, or produced by a familiar formula. The actuary must judge whether the data and assumptions behind the estimate fit the portfolio being reserved. Bornhuetter-Ferguson (BF) relies on an expected loss ratio chosen independently of the current reported experience. Cape Cod (CC) estimates a common loss ratio from that experience, adjusted for how much of each period is expected to have been reported.
Both methods combine reported losses with an estimate of what remains unreported. Their results can be close while relying on different evidence, or far apart because one of the assumptions is unsuitable. The task is to identify that distinction, make a defensible selection, and explain how the selected reserve is allocated across accident periods. The methods are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
What the Two Estimates Rely On
For BF, the expected ultimate loss for a period is \(L EP_i\). The expected amount still unreported is therefore the expected ultimate multiplied by the unreported proportion, \(1-p_i\). BF adds that amount to reported losses to obtain an ultimate estimate. The external loss ratio should reflect relevant prospective expectations, rather than being selected merely to make the result agree with the observed losses.
Cape Cod instead estimates a common loss ratio from the observed losses across the selected periods. Its denominator is the premium corresponding to the portion of losses expected to have been reported, not total premium. It then applies that estimated ratio to each period’s unreported premium to estimate the reserve. This pooled estimate is most persuasive when periods are reasonably comparable and the premium and reporting-proportion data are dependable.
Here \(\widehat{R}_i\) denotes the estimated reserve and \(\widehat{U}_i\) the estimated ultimate for period \(i\). The displayed Cape Cod formula uses a common loss ratio for the selected periods. It is not appropriate to pool periods blindly if their exposure bases, coverage, loss definitions, or expected reporting patterns are inconsistent.
A Structured Selection Process
A useful selection is an argument about evidence, not a rule that one method is always preferable. First confirm that the data inputs are comparable. Then consider whether a credible independent expected loss ratio is available, whether the experience is mature enough to be informative, and whether the portfolio has been stable enough for a pooled Cape Cod ratio to represent the periods in question.
Confirm that premiums, reported losses, and \(p_i\) use consistent coverage, valuation, and loss definitions. Investigate exposure changes or mismatches before interpreting a method difference.
Ask whether \(L\) is supported by relevant pricing, underwriting, or other actuarial analysis, and whether it reflects the periods being estimated. If it is weak or stale, BF may be difficult to defend.
For Cape Cod, judge whether the periods share a sufficiently stable risk mix and loss environment. A pooled ratio can be misleading when the periods are not comparable or when reported data are unreliable.
Review each period’s reserve as well as the portfolio total. State which assumptions explain the difference and why the selected allocation is plausible.
Maturity affects how much current experience contributes to the two approaches. BF gives the unreported portion the external expected loss ratio; Cape Cod uses the pooled experience-derived ratio for that portion. Neither method makes immature data fully informative. In particular, a small \(p_i\) means a relatively large portion of that period’s estimate depends on assumptions rather than losses already reported.
Worked Comparisons and Decisions
Worked Example: Selecting a Method for a Stable Portfolio
A hypothetical insurer has three accident periods, with premiums and losses in thousands of dollars. The portfolio has had stable coverage and risk mix. Reporting proportions are considered reliable, but the available external expected loss ratio of 0.65 is based on an older analysis. Compare the estimates and select a method.
Step 1: Calculate the Cape Cod denominator and ratio. The reported-adjusted premiums are \(1{,}000(0.80)=800\), \(1{,}200(0.50)=600\), and \(800(0.25)=200\), all in thousands. Their sum is \(800+600+200=1{,}600\). Total reported losses are \(500+420+150=1{,}070\). Therefore,
Step 2: Calculate Cape Cod reserves by period. The unreported premiums are \(1{,}000(0.20)=200\), \(1{,}200(0.50)=600\), and \(800(0.75)=600\). Applying the estimated ratio gives reserves of \(200(0.66875)=133.75\), \(600(0.66875)=401.25\), and \(600(0.66875)=401.25\). The total Cape Cod reserve is \(133.75+401.25+401.25=936.25\) thousand.
Step 3: Calculate BF reserves for comparison. The same unreported premiums multiplied by the external ratio 0.65 give \(200(0.65)=130\), \(600(0.65)=390\), and \(600(0.65)=390\). The total BF reserve is \(130+390+390=910\) thousand. Cape Cod is higher by \(936.25-910=26.25\) thousand.
Selection and explanation. Given the stated stable portfolio and reliable reporting proportions, Cape Cod is defensible because it uses pooled experience to update a possibly stale external ratio. The selection is conditional on the comparability of the periods; stability alone does not prove the pooled ratio is right. The accident-period Cape Cod reserves are 133.75, 401.25, and 401.25 thousand, so the less mature second and third periods account for most of the estimate. The common mistake avoided is choosing the method solely because its total is higher or lower.
Check. The three reserves sum to 936.25 thousand. Adding reported losses gives estimated ultimates of \(633.75\), \(821.25\), and \(551.25\) thousand, totaling \(2{,}006.25\) thousand. This equals total reported losses \(1{,}070\) plus the reserve \(936.25\).
Worked Example: Correcting a Cape Cod Denominator
Two hypothetical accident periods each have premium 100. Their reported proportions are 0.50 and 0.75, and reported losses are 35 and 66. Calculate Cape Cod and BF reserves, then compare them. Monetary amounts are in the stated units.
Step 1: Estimate the Cape Cod loss ratio. The reported-adjusted premium is \(100(0.50)+100(0.75)=50+75=125\). Total reported losses are \(35+66=101\). Thus,
Step 2: Calculate the Cape Cod reserve. The unreported premiums are \(100(1-0.50)=50\) and \(100(1-0.75)=25\), totaling 75. Therefore the reserve is \(75(0.808)=60.6\).
Step 3: Calculate the BF reserve. With expected loss ratio \(L=0.60\), the reserve is \(0.60(50+25)=45\). Cape Cod is higher by \(60.6-45=15.6\).
Common trap and correction. Using 175 as the Cape Cod denominator produces \(101/175\), but 175 is not the sum of reported-adjusted premiums. The correct denominator is \(50+75=125\), since each premium is multiplied by its own reported proportion. The corrected Cape Cod ratio is 0.8080, the reserve is 60.6, and the difference from BF is 15.6—not 27.1429.
Selection comment. This calculation establishes that the estimates differ, not which one is better. If the external 0.60 ratio is independently well supported and the pooled experience is distorted or non-comparable, BF may be preferable. If the periods are comparable and the external ratio is weak, Cape Cod may be more supportable. The method choice needs evidence beyond the arithmetic.
Worked Example: Choosing BF After a Portfolio Change
A hypothetical book has undergone a material underwriting and risk-mix change. The new, independently reviewed expected loss ratio is 0.72. The older accident-period experience is not considered representative of the current mix, although the reported proportions are credible. Premiums and losses are in thousands of dollars. Compare the two methods and select an estimate for the periods.
Step 1: Calculate Cape Cod for comparison. Reported-adjusted premiums are \(600(0.75)=450\) and \(900(0.40)=360\), totaling 810. Reported losses total \(310+360=670\). Thus \(\widehat{L}^{CC}=670/810=0.8271605\) (shown to seven decimal places).
Step 2: Calculate the unreported premiums. These are \(600(0.25)=150\) and \(900(0.60)=540\), totaling 690.
Step 3: Calculate the reserves. Cape Cod gives \(150(670/810)=124.0741\) and \(540(670/810)=446.6667\), totaling \(570.7407\) thousand. BF gives \(150(0.72)=108\) and \(540(0.72)=388.8\), totaling \(496.8\) thousand. The Cape Cod estimate exceeds BF by \(570.7407-496.8=73.9407\) thousand.
Selection and explanation. BF is the more defensible choice under the stated facts because its expected loss ratio reflects the changed prospective portfolio, while the Cape Cod ratio pools experience from a different risk mix. The accident-period BF reserves are 108 and 388.8 thousand. The second period’s reserve is larger because its unreported premium is larger; that allocation follows the method’s common expected loss ratio, not a claim that its reported losses are less reliable.
Check. The BF reserves sum to \(108+388.8=496.8\) thousand. A wrong approach would treat the larger Cape Cod result as automatically prudent or automatically correct. The comparison identifies an assumption difference; the stated portfolio change supplies the reason to prefer BF.
Communicating the Selection
A useful reserve communication separates the estimate from the reason for selecting it. State the selected method and total, identify the assumptions that matter, and describe the accident-period allocation. If the competing method differs materially, give the difference and explain what evidence drives it. Avoid presenting a single point estimate as if it were independent of reporting patterns, premium quality, or the chosen expected loss ratio.
| Evidence about the portfolio | Selection implication | Point to communicate |
|---|---|---|
| A credible prospective expected loss ratio; material portfolio change or non-comparable experience | BF may be more supportable | The external ratio and the unreported portion drive the estimate |
| Stable portfolio; reliable premium and reporting data; no strong external ratio | Cape Cod may be more supportable | The experience-derived ratio is pooled using reported-adjusted premium |
| Weak reporting proportions, inconsistent bases, or mismatched exposures | Neither result is automatically reliable | Resolve the data issue or qualify the estimates before selecting |
The Cape Cod denominator is a particularly important audit check: it is \(\sum_i EP_i p_i\), while the reserve exposure is \(\sum_i EP_i(1-p_i)\). These quantities serve different purposes and should not be interchanged. In BF, the externally selected loss ratio is applied to the unreported exposure, not to all premium when calculating the reserve.
Check Your Understanding
For each problem, use the stated loss and premium units. For numerical comparisons, retain unrounded values through the calculation and round only the final answer to four decimal places when needed.
- A period has premium 750, a reported proportion of 0.60, and an independently selected expected loss ratio of 0.64. Calculate its BF reserve and state the unreported premium. (Answer: Unreported premium \(=300\); BF reserve \(=192\).)
- Two periods have premiums 200 and 500, reported proportions 0.50 and 0.80, and reported losses 70 and 260. Calculate the Cape Cod ratio and total reserve. (Answer: Ratio \(=330/(100+400)=0.6600\); reserve \(=200(0.6600)=132\).)
- Two periods have premiums 300 and 600, reported proportions 0.50 and 0.75, and reported losses 100 and 300. Compare the Cape Cod reserve with the BF reserve using \(L=0.60\). (Answer: Cape Cod ratio \(=400/600=0.6667\); Cape Cod reserve \(=300(0.6667)=200\); BF reserve \(=0.60(150+150)=180\); Cape Cod is higher by \(20\).)
- Two accident periods have premiums 400 and 800, reported proportions 0.25 and 0.75, and reported losses 50 and 360. Calculate the Cape Cod ratio, total reserve, and allocation by period. (Answer: Ratio \(=410/(100+600)=0.5857143\); reserves \(=300(0.5857143)=175.7143\) and \(200(0.5857143)=117.1429\); total \(=292.8571\).)
- A portfolio has stable coverage but a recent underwriting change; an independently reviewed expected loss ratio reflects the new mix, while the experience pooled for Cape Cod mostly predates the change. Which method is generally easier to justify, and what assumption should be highlighted? (Answer: BF is generally easier to justify; highlight that the external expected loss ratio is credible and appropriate for the changed portfolio.)