Why the Assumptions Matter
Bornhuetter-Ferguson (BF) and Cape Cod both estimate ultimate losses by combining reported experience with an estimate of losses not yet reported. They differ in where the expected loss level comes from. BF uses an expected loss ratio selected outside the current loss experience, while Cape Cod estimates a common loss ratio from reported losses and the amount of exposure expected to have emerged. The distinction matters when the risk level has changed, or when the assumed reporting pattern does not fit the business being estimated.
The comparison is meaningful only when the inputs are on consistent bases. In the examples below, premiums and losses are on the same valuation and coverage basis, and all dollar amounts are in thousands unless stated otherwise. The reported proportion is the fraction of ultimate losses expected to be reported by the valuation date. It must match the loss basis being analyzed.
These formulas represent different assumptions, not just different arithmetic. BF treats \(e\) as the appropriate expected loss ratio for the period being estimated, based on information other than the current reported losses being adjusted. Cape Cod treats the loss ratio estimated from the included experience as representative of the periods to which it is applied. Both methods rely on \(p_j\) being an appropriate estimate of loss emergence for the relevant periods and loss basis.
What Each Method Assumes
| Question | Bornhuetter-Ferguson | Cape Cod |
|---|---|---|
| Where does the expected loss level come from? | An externally selected expected loss ratio \(e\), applied to the period's exposure | A ratio estimated from reported losses divided by expected reported exposure |
| How is current reported experience used? | Reported losses are retained as observed; expected losses are used for the unreported portion | Reported losses contribute to estimating the loss ratio, which is then applied to exposure |
| What is assumed about risk level? | The selected ratio is appropriate for the period's risk, pricing, and exposure mix | The included experience supports a representative common ratio for the periods being estimated |
| What is assumed about emergence? | The reported proportion correctly measures the fraction emerged; the complement \(1-p_j\) is still unreported | The same type of reporting-pattern assumption is needed both to estimate the ratio and to apply it to exposure |
In BF, the expected loss ratio can reflect a considered change in rates, underwriting, limits, or mix, provided the exposure and ratio are aligned. In Cape Cod, the ratio is experience-derived: the method uses the loss experience, adjusted for expected emergence, to set the level. That can be useful when the included experience is relevant to the periods being estimated, but it can be misleading if risk levels differ materially across those periods.
Neither method makes an inaccurate reporting pattern harmless. If \(p_j\) is too high, the expected unreported fraction \(1-p_j\) is too small in BF. In Cape Cod, an incorrect \(p_j\) also changes the denominator used to estimate the loss ratio. A wrong emergence assumption can therefore affect both the estimated risk level and the reserve.
Why the Cape Cod Ratio Has This Form
Suppose that, for each included period, expected reported losses are proportional to exposure, the common expected loss ratio \(\theta\), and the reported proportion. In symbols, the model for the total reported losses is
where \(\varepsilon_j\) represents the difference between actual reported losses and their expected values under this model. If the errors have mean zero, then the total reported losses have expected value \(\theta\sum_j P_jp_j\). Solving the aggregate expected-value relationship for \(\theta\) gives the Cape Cod ratio:
The denominator is not total premium. It is the sum of premium weighted by the proportion expected to be reported. Dividing by this denominator adjusts the observed losses for the fact that some losses are still unreported. This justification depends on the common-ratio assumption being reasonable for the included periods; if their underlying risk levels differ, the aggregate ratio may not describe any one period well.
BF follows a different decomposition. If \(P_je\) is expected ultimate loss and \(p_j\) is the expected reported fraction, then expected unreported loss is \(P_je(1-p_j)\). Adding that expected unreported amount to actual reported loss gives:
This construction avoids replacing the actual reported amount with the expected reported amount. It uses actual experience for the emerged portion and the selected expectation only for the not-yet-emerged portion. It is not appropriate to use this formula with mismatched inputs—for example, a paid-loss reporting proportion with an incurred-loss amount.
Worked Example: Comparing the Estimates on a Shared Basis
A hypothetical two-period portfolio has the following premium, reported losses, and reported proportions. Amounts are in thousands. An externally selected BF expected loss ratio is \(e=0.75\). Calculate the BF and Cape Cod ultimate losses and reserves. Use the same reported proportions for both methods.
| Period | \(P_j\) | \(L_j\) | \(p_j\) |
|---|---|---|---|
| 1 | 1,000 | 260 | 0.40 |
| 2 | 1,200 | 630 | 0.70 |
Method: Bornhuetter-Ferguson. For period 1, the expected unreported fraction is \(1-0.40=0.60\). Expected unreported losses are \(1{,}000(0.75)(0.60)=450\), so the ultimate estimate is \(260+450=710\). The reserve is \(710-260=450\).
For period 2, the expected unreported fraction is \(1-0.70=0.30\). Expected unreported losses are \(1{,}200(0.75)(0.30)=270\), so the ultimate estimate is \(630+270=900\). The reserve is \(900-630=270\). Thus aggregate BF ultimate losses are \(710+900=1{,}610\), and aggregate reserve is \(450+270=720\).
Method: Cape Cod. The expected reported exposure is \(1{,}000(0.40)+1{,}200(0.70)=400+840=1{,}240\). Total reported losses are \(260+630=890\). Therefore,
The Cape Cod ultimate for period 1 is \(1{,}000(890/1{,}240)=717.7419\ldots\), and its reserve is \(717.7419\ldots-260=457.7419\ldots\). For period 2, the ultimate is \(1{,}200(890/1{,}240)=861.2903\ldots\), and the reserve is \(861.2903\ldots-630=231.2903\ldots\). The aggregate ultimate is \(1{,}579.0323\ldots\), and the aggregate reserve is \(689.0323\ldots\). Rounded to two decimals, Cape Cod gives ultimate losses of \(1{,}579.03\) and a reserve of \(689.03\).
The Cape Cod ratio, about \(0.7177\), is below the BF selection of \(0.75\), so Cape Cod produces a lower aggregate ultimate here. This is a consequence of the experience-derived ratio, not a general ordering between the methods. The calculation also checks: Cape Cod aggregate ultimate \(1{,}579.0323\ldots\) less aggregate reported losses \(890\) equals reserve \(689.0323\ldots\).
Common mistake avoided: Using total premium \(2{,}200\) as the Cape Cod denominator would give \(890/2{,}200\), which does not adjust for emergence. The correct denominator is \(1{,}240=\sum_jP_jp_j\).
Risk-Level Assumptions in Practice
A selected BF ratio can be preferable as a representation of the current risk level when the periods being estimated differ from the historical experience used to calibrate Cape Cod. This does not make BF automatically correct: its selected ratio still has to reflect the actual exposure basis and expected loss level. Conversely, Cape Cod can usefully derive a level from experience when the included periods are sufficiently representative and a common ratio is defensible.
The following example isolates a risk-mix change. It assumes the current portfolio has a higher expected loss ratio than the older experience. The ratio for BF is selected to reflect that current risk; the Cape Cod calculation pools the older and current experience.
Worked Example: A Current Period with a Higher Risk Level
A hypothetical older period has premium \(1{,}000\), reported losses \(640\), and reported proportion \(0.80\). A current period has premium \(500\), reported losses \(90\), and reported proportion \(0.20\). All amounts are in thousands. For the current period, an external analysis supports a BF expected loss ratio of \(1.00\). Calculate the current-period BF and Cape Cod estimates, using both periods to derive the Cape Cod ratio.
Bornhuetter-Ferguson. The current period's unreported fraction is \(1-0.20=0.80\). Its expected unreported losses are \(500(1.00)(0.80)=400\). Thus current ultimate losses are \(90+400=490\), and the reserve is \(490-90=400\).
Cape Cod. The older period contributes \(1{,}000(0.80)=800\) of expected reported exposure. The current period contributes \(500(0.20)=100\). Total reported losses are \(640+90=730\), and total expected reported exposure is \(800+100=900\). Therefore, \(\hat e_{CC}=730/900=0.811111\ldots\). Applying this ratio to current premium gives current ultimate losses of \(500(730/900)=405.5556\ldots\), and a reserve of \(405.5556\ldots-90=315.5556\ldots\). Rounded to two decimals, the Cape Cod current-period ultimate is \(405.56\) and the reserve is \(315.56\).
The methods differ because the BF selection reflects a higher current risk level, while the pooled Cape Cod ratio is influenced by the older period. This interpretation depends on the stated scenario: a different external selection or a different experience mix could change the comparison. As a check, each reserve equals its ultimate estimate less the current reported losses of \(90\).
Common mistake avoided: Applying the older period's reported loss ratio \(640/1{,}000=0.64\) directly to current premium ignores emergence. Cape Cod adjusts for emergence in its denominator; the pooled ratio is \(730/900\), not \(730/1{,}500\) or \(640/1{,}000\).
Loss Emergence Is a Separate Assumption
A ratio selection addresses the expected loss level; \(p_j\) addresses how much of that loss has emerged. The two choices should not be conflated. In BF, the expected unreported loss is proportional to \(1-p_j\). In Cape Cod, \(p_j\) changes the experience-derived ratio itself, because it changes the expected reported exposure in the denominator.
For example, if reported losses are modest because a period is very immature, Cape Cod may infer a large loss ratio when the expected reported proportion is small. That result is not necessarily evidence that the underlying risk is high; it may reflect a small emergence proportion, actual random variation, or a misspecified pattern. BF does not derive the loss ratio from the current reported losses, but its expected unreported amount still depends directly on the assumed complement \(1-p_j\).
Worked Example: How the Reporting Pattern Changes the Estimate
A hypothetical period has premium \(2{,}000\) and reported losses \(500\), both in thousands. BF uses an external expected loss ratio of \(0.60\). Compare BF and Cape Cod estimates under two alternative assumptions about the reported proportion, \(p=0.25\) and \(p=0.50\). Hold the observed premium and reported losses fixed to isolate the effect of the emergence assumption.
If \(p=0.25\): BF expected unreported losses are \(2{,}000(0.60)(1-0.25)=2{,}000(0.60)(0.75)=900\). BF ultimate losses are \(500+900=1{,}400\), with reserve \(900\). Cape Cod's ratio is \(500/[2{,}000(0.25)]=500/500=1.00\). Its ultimate is \(2{,}000(1.00)=2{,}000\), and its reserve is \(2{,}000-500=1{,}500\).
If \(p=0.50\): BF expected unreported losses are \(2{,}000(0.60)(0.50)=600\). BF ultimate losses are \(500+600=1{,}100\), with reserve \(600\). Cape Cod's ratio is \(500/[2{,}000(0.50)]=500/1{,}000=0.50\). Its ultimate is \(2{,}000(0.50)=1{,}000\), and its reserve is \(1{,}000-500=500\).
Changing only \(p\) has a substantial effect, especially on the Cape Cod ratio: the denominator doubles when \(p\) increases from \(0.25\) to \(0.50\). These are alternative model assumptions, not two estimates of the same known reporting pattern. In practice the reporting proportion should be selected for the relevant loss basis and period. In both cases, the check is that the reserve equals ultimate less the reported \(500\).
Common mistake avoided: Keeping the Cape Cod ratio fixed while changing \(p\) would ignore the fact that \(p\) is part of its denominator. For BF, the external ratio remains fixed in this example, but the expected unreported fraction changes.
Common Exam Traps
- Using the wrong Cape Cod denominator. The denominator is \(\sum_jP_jp_j\), not total premium, and each period's premium must be weighted by its own reported proportion.
- Treating the two ratios as interchangeable. BF's \(e\) is selected externally for the period; Cape Cod's \(\hat e_{CC}\) is calculated from the included reported losses and expected reported exposure.
- Applying the expected loss ratio to the whole premium and then adding all reported losses. In BF, only the expected unreported part \(P_je(1-p_j)\) is added to actual reported losses. Adding \(P_je\) would count expected reported losses on top of the actual reported amount.
- Assuming the same risk mix without checking. Cape Cod's pooled ratio is most interpretable when the included experience is relevant to the periods receiving that ratio. A systematic change in risk level can make the pooled ratio inappropriate for a particular period.
- Mixing paid and incurred bases. Reported losses, the emergence proportion, and the exposure basis must be aligned. A reported proportion for one loss measure cannot be substituted casually for another.
Check Your Understanding
For each problem, state the method or assumption being tested and retain the units given.
- A hypothetical period has premium \(800\) thousand, reported losses \(180\) thousand, a reported proportion of \(0.40\), and a selected BF expected loss ratio of \(0.65\). Calculate expected unreported losses, ultimate losses, and reserve. (Answer: Expected unreported losses \(312\) thousand; ultimate \(492\) thousand; reserve \(312\) thousand.)
- Two periods have, respectively, premium \(600\) thousand, reported losses \(210\) thousand, and \(p=0.50\); and premium \(900\) thousand, reported losses \(540\) thousand, and \(p=0.80\). Calculate the Cape Cod ratio, aggregate ultimate, and aggregate reserve. (Answer: Cape Cod ratio \(750/1{,}020=0.7353\); aggregate ultimate \(1{,}102.94\) thousand; reserve \(352.94\) thousand.)
- A period has premium \(1{,}000\) thousand, reported losses \(250\) thousand, \(p=0.30\), and BF selected ratio \(0.70\). Calculate BF ultimate and reserve, then calculate the one-period Cape Cod ratio and ultimate. (Answer: BF ultimate \(740\) thousand and reserve \(490\) thousand; Cape Cod ratio \(250/300=0.8333\) and ultimate \(833.33\) thousand.)
- A portfolio's newer business has a materially different underwriting mix from its older experience. Which method's expected loss level is directly estimated from the pooled included experience, and what must be assessed before applying it to the newer business? (Answer: Cape Cod; assess whether the pooled experience supports a representative expected loss ratio for the newer risk mix.)
- A hypothetical period has premium \(1{,}200\) thousand, reported losses \(240\) thousand, \(p=0.20\), and selected BF ratio \(0.50\). Calculate BF ultimate and reserve, and the Cape Cod one-period ratio and ultimate. (Answer: BF ultimate \(720\) thousand and reserve \(480\) thousand; Cape Cod ratio \(1.00\) and ultimate \(1{,}200\) thousand.)