Why Compare Cape Cod and Bornhuetter-Ferguson?
Cape Cod and Bornhuetter-Ferguson (BF) both estimate unpaid losses by combining observed claims with an estimate of losses not yet emerged. Their key difference is the source of the expected ultimate loss: Cape Cod estimates a common loss ratio from pooled experience, while BF takes an a priori expected loss as an input. The distinction matters because two calculations using the same reported losses and development assumptions can produce different reserves if their expected-loss assumptions differ.
To keep the comparison precise, use the following notation. For accident year \(i\), let \(P_i\) be earned premium on a consistent, on-level basis; \(L_i\) the cumulative loss observed at the valuation date; and \(w_i\) the proportion of ultimate losses expected to have emerged by that date. Let \(E_i\) denote a supplied a priori expected ultimate loss for BF. All amounts in the examples are hypothetical.
The formulas below assume that \(L_i\), \(P_i\), and the expected losses use the same loss basis, coverage, and valuation conventions, and that the selected \(w_i\) is appropriate for the data. Cape Cod also assumes a common expected loss ratio across the experience years being pooled. BF does not need a common loss ratio if a separate expected loss \(E_i\) is supplied for each year. Neither formula should be applied mechanically when its inputs are not comparable or the selected emergence proportions are unsuitable.
The Two Expected-Loss Inputs
In BF, the a priori expected ultimate loss is specified before using that accident year's observed loss in the reserve formula. It may be built from an expected loss ratio and premium, or from an exposure-based expected loss cost. Once \(E_i\) is selected, BF estimates the unreported portion as \(E_i(1-w_i)\).
Cape Cod instead estimates a common expected loss ratio, \(\widehat r_{CC}\), from the pooled experience. The method compares total observed losses with the premium-weighted amount of expected emergence. Its expected ultimate loss basis for year \(i\) is \(P_i\widehat r_{CC}\). The Cape Cod reserve is the unreported fraction of that expected loss, not the whole expected ultimate loss.
For BF, use the corresponding supplied expected ultimate loss \(E_i\):
In both methods, the reserve is an estimate of future emergence and the ultimate estimate is observed loss plus that reserve. In particular, \(P_i\widehat r_{CC}\) is Cape Cod's expected-loss basis; it is not, by itself, the Cape Cod ultimate estimate.
Worked Example: Calculate Both Methods from the Same Experience
A hypothetical portfolio has earned premiums of \(1{,}000\) and \(2{,}000\) thousand dollars for two accident years. Their development proportions are \(0.40\) and \(0.80\), and observed cumulative losses are \(180\) and \(1{,}120\) thousand dollars. A separate BF analysis supplies expected ultimate losses of \(600\) and \(1{,}200\) thousand dollars. Calculate the Cape Cod and BF reserves and ultimate losses.
Method: Cape Cod. First calculate the pooled observed losses: \(180+1{,}120=1{,}300\) thousand dollars. The maturity-adjusted premium denominator is \(1{,}000(0.40)+2{,}000(0.80)=400+1{,}600=2{,}000\) thousand dollars. Therefore, \(\widehat r_{CC}=1{,}300/2{,}000=0.65\).
For the first year, the expected unreported fraction is \(1-0.40=0.60\), so its reserve is \(1{,}000(0.65)(0.60)=390\) thousand dollars. Its ultimate estimate is \(180+390=570\) thousand dollars.
For the second year, the expected unreported fraction is \(1-0.80=0.20\), so its reserve is \(2{,}000(0.65)(0.20)=260\) thousand dollars. Its ultimate estimate is \(1{,}120+260=1{,}380\) thousand dollars. The total Cape Cod reserve is \(390+260=650\) thousand dollars; the total ultimate estimate is \(570+1{,}380=1{,}950\) thousand dollars.
Method: BF. Apply the supplied expected ultimate losses rather than estimating a loss ratio. The first reserve is \(600(1-0.40)=360\) thousand dollars, giving ultimate loss \(180+360=540\). The second reserve is \(1{,}200(1-0.80)=240\) thousand dollars, giving ultimate loss \(1{,}120+240=1{,}360\). The total BF reserve is \(600\) thousand dollars, and the total BF ultimate estimate is \(1{,}900\) thousand dollars.
Check and common mistake avoided. For each method, total ultimate equals total observed loss plus total reserve: Cape Cod gives \(1{,}300+650=1{,}950\), and BF gives \(1{,}300+600=1{,}900\). Do not report \(P_i\widehat r_{CC}\) as the Cape Cod ultimate estimate: it is the expected-loss basis used to calculate the unreported portion.
Why the Cape Cod Ratio Has This Form
The Cape Cod ratio is a pooled estimate, not an ordinary average of accident-year loss ratios. Its denominator accounts for how much premium-weighted expected loss has emerged by the valuation date.
To justify the formula, suppose each year has the same expected loss ratio \(r\), and the expected ultimate loss for year \(i\) is \(P_i r\). Under the selected emergence proportion, expected observed loss is \(w_iP_ir\). Summing over the pooled years gives the expected total observed loss:
Solving this relationship for \(r\), using the observed pooled loss total in place of its expectation, gives the method-of-moments estimate \(\widehat r_{CC}=\sum_i L_i/\sum_iP_iw_i\). Thus, a less mature year contributes less to the denominator than an equally premium-sized, more mature year. The formula relies on a common ratio for the pooled experience; if that assumption is unsuitable, pooling may not produce a meaningful common indication.
The reserve formulas follow from the same emergence assumption. If expected ultimate loss is \(E_i\), the expected emerged portion is \(w_iE_i\), leaving \(E_i-w_iE_i=(1-w_i)E_i\) unreported. For Cape Cod, substitute \(E_i=P_i\widehat r_{CC}\); for BF, retain the supplied \(E_i\).
Worked Example: Recalculate a Cape Cod Ratio with Unequal Maturity
A hypothetical book has earned premiums of \(700\) and \(1{,}300\) thousand dollars, development proportions of \(0.50\) and \(0.75\), and observed losses of \(100\) and \(650\) thousand dollars. Calculate the Cape Cod ratio, each reserve, and the total reserve.
Method: maturity-adjusted pooled experience. Total observed loss is \(100+650=750\) thousand dollars. The denominator is \(700(0.50)+1{,}300(0.75)=350+975=1{,}325\) thousand dollars. Therefore, \[ \widehat r_{CC}=\frac{750}{1{,}325}=0.5660377\ldots, \] or \(0.5660\) to four decimal places.
The first year's unreported fraction is \(1-0.50=0.50\). Its reserve is \(700(0.5660377\ldots)(0.50)=198.1132\ldots\) thousand dollars.
The second year's unreported fraction is \(1-0.75=0.25\). Its reserve is \(1{,}300(0.5660377\ldots)(0.25)=183.9623\ldots\) thousand dollars. The total reserve is \(198.1132\ldots+183.9623\ldots=382.0755\ldots\), or \(382.08\) thousand dollars to the nearest cent.
Check and common mistake avoided. Equivalently, the reserve is \([700(0.50)+1{,}300(0.25)](750/1{,}325)=675(750/1{,}325)=382.08\) thousand dollars. A common error is to divide by total premium, obtaining \(750/(700+1{,}300)=0.3750\). That denominator ignores maturity and is not the Cape Cod denominator. The correct maturity-adjusted denominator is \(1{,}325\), not \(2{,}000\).
When the Methods Agree—and When They Do Not
If BF is supplied with \(E_i=P_i\widehat r_{CC}\) for every year, its reserve equals the Cape Cod reserve year by year: \(E_i(1-w_i)=P_i\widehat r_{CC}(1-w_i)\). Their ultimate estimates then agree as well, because both add the same reserve to \(L_i\). This is an algebraic equivalence conditional on using the same expected-loss basis and development proportions.
More often, BF expected losses reflect information not encoded in the pooled Cape Cod ratio, such as different exposure-based expected loss costs by accident year. In that case, the methods can differ. Cape Cod's common ratio pulls the expected-loss basis toward a shared level, while BF preserves the year-specific a priori expectations. Neither method is automatically preferable: the selection depends on the relevance of the supplied expectations and the comparability of the pooled experience.
Worked Example: Different Year-Specific Expectations Change the BF Result
A hypothetical portfolio has earned premiums of \(1{,}500\) and \(2{,}500\) thousand dollars, development proportions of \(0.60\) and \(0.30\), and observed losses of \(540\) and \(450\) thousand dollars. A separate BF analysis supplies expected ultimate losses of \(1{,}000\) and \(1{,}250\) thousand dollars. Calculate both methods and explain the difference.
Method: Cape Cod. Total observed loss is \(540+450=990\) thousand dollars. The denominator is \(1{,}500(0.60)+2{,}500(0.30)=900+750=1{,}650\) thousand dollars. Thus \(\widehat r_{CC}=990/1{,}650=0.60\).
The first Cape Cod reserve is \(1{,}500(0.60)(1-0.60)=360\) thousand dollars. The second is \(2{,}500(0.60)(1-0.30)=1{,}050\) thousand dollars. Total reserve is \(1{,}410\) thousand dollars, and total ultimate is \(990+1{,}410=2{,}400\) thousand dollars.
Method: BF. The first reserve is \(1{,}000(0.40)=400\) thousand dollars. The second is \(1{,}250(0.70)=875\) thousand dollars. Total reserve is \(1{,}275\) thousand dollars. The ultimate estimates are \(540+400=940\) and \(450+875=1{,}325\) thousand dollars, totaling \(2{,}265\) thousand dollars.
Check and common mistake avoided. The BF total satisfies \(990+1{,}275=2{,}265\). The methods differ by \(135\) thousand dollars in total ultimate because their expected-loss bases differ. In particular, the BF expectations imply ratios of \(1{,}000/1{,}500=0.6667\) and \(1{,}250/2{,}500=0.5000\), rather than Cape Cod's pooled \(0.60\). Do not replace those supplied BF expectations with the Cape Cod ratio unless the problem directs that selection.
Practical Checks and Exam Traps
A reliable comparison begins by labeling the input source before doing arithmetic. In BF, identify the supplied \(E_i\) and calculate \(E_i(1-w_i)\). In Cape Cod, first calculate the pooled ratio using \(\sum_iP_iw_i\), then calculate \(P_i\widehat r_{CC}(1-w_i)\). Keep unrounded values through the calculations and round only the final reported results.
- Using total premium in the Cape Cod denominator. The denominator is maturity-adjusted premium, \(\sum_iP_iw_i\), not \(\sum_iP_i\).
- Using the emerged proportion for the reserve. The unreported share is \(1-w_i\), not \(w_i\).
- Confusing expected ultimate loss with ultimate estimate. In either method, the ultimate estimate is observed loss plus reserve.
- Replacing a supplied BF expectation with a pooled ratio. That changes the method's input and can change the result.
- Mixing incompatible bases. Premium, observed losses, expected losses, and development proportions must correspond to a consistent set of coverage and valuation assumptions.
Check Your Understanding
For each problem, identify the expected-loss input and retain full precision until the final reported answer.
- A hypothetical portfolio has premiums of \(800\) and \(1{,}200\) thousand dollars, development proportions of \(0.25\) and \(0.75\), and observed losses of \(120\) and \(660\) thousand dollars. Calculate the Cape Cod ratio and total reserve. (Answer: \(\widehat r_{CC}=0.7091\); total reserve \(638.18\) thousand dollars.)
- For a hypothetical BF estimate, supplied expected ultimate losses are \(500\) and \(900\) thousand dollars. The development proportions are \(0.40\) and \(0.80\), and observed losses are \(210\) and \(680\) thousand dollars. Calculate total reserve and ultimate loss. (Answer: reserve \(480\) thousand dollars; ultimate loss \(1{,}370\) thousand dollars.)
- A Cape Cod calculation uses premiums of \(900\) and \(1{,}100\) thousand dollars, development proportions of \(0.50\) and \(0.80\), and observed losses of \(300\) and \(700\) thousand dollars. Calculate the indicated ratio and total reserve. (Answer: \(\widehat r_{CC}=0.7519\); total reserve \(503.76\) thousand dollars.)
- A hypothetical accident year has an age-to-ultimate factor of \(2.50\), supplied BF expected ultimate loss of \(750\) thousand dollars, and observed loss of \(260\) thousand dollars. Convert the factor to a development proportion, then calculate the reserve and ultimate. (Answer: \(w=0.40\); reserve \(450\) thousand dollars; ultimate \(710\) thousand dollars.)
- Two years have premiums of \(1{,}000\) and \(2{,}000\) thousand dollars, development proportions of \(0.50\) and \(0.25\), and observed losses of \(400\) and \(300\) thousand dollars. BF expected ultimate losses are \(800\) and \(2{,}400\) thousand dollars. Calculate the Cape Cod ratio and total reserve under each method. (Answer: \(\widehat r_{CC}=0.7000\); Cape Cod reserve \(1{,}400\) thousand dollars; BF reserve \(2{,}200\) thousand dollars.)