Cape Cod Method

Study guide: Exam 5

Cape Cod Method (Stanard-Bühlmann) is the Bornhuetter-Ferguson technique with the a priori loss ratio derived from the data itself rather than supplied externally. The ELR is estimated as total reported losses over total used-up (on-level, developed) premium across all years.

$\text{ELR} = \frac{\sum_i C_i}{\sum_i \text{EP}_i \times \tfrac{1}{\text{CDF}_i}}$

$\text{IBNR}_i = \text{ELR} \times \text{EP}_i \times \left(1 - \tfrac{1}{\text{CDF}_i}\right)$

Used-up premium $\text{EP}_i / \text{CDF}_i$ is the share of each year's premium "exposed" to the losses reported so far. Weighting by it puts each accident year on a common maturity footing, so an immature year contributes little to the ELR estimate — as it should, having reported little.
Once the ELR is in hand, the IBNR formula is identical to BF. The whole difference between the two methods is where the a priori comes from: BF imports it, Cape Cod estimates it.
That makes Cape Cod the right choice when no credible external ELR exists — a new programme, a book whose pricing history is unreliable — and it removes the single largest source of BF error. The cost is that the estimate is no longer independent of the data: a systematically distorted triangle now corrupts the a priori as well as the development.
Premium must be on level and losses must be trended to a common cost level before the ELR is computed, or the derived ratio blends years written at different rate levels and different cost levels into one number. This is Cape Cod's most-missed requirement.
Cape Cod converges to the chain ladder as cohorts mature and to the expected claims technique for very immature ones — the same limiting behaviour as BF.
A decay-weighted variant (generalized Cape Cod) weights recent years more heavily in the ELR estimate, recognizing that the loss ratio level genuinely drifts.

Deriving the ELR and the Reserve Three accident years, with on-level earned premium and reported losses ($000s):

AY Maturity EP Reported CDF $\%$ reported

$2022$ $36$ mo $7{,}000$ $1{,}750$ $1.084$ $92.3\%$

$2023$ $24$ mo $6{,}500$ $2{,}600$ $1.265$ $79.1\%$

$2024$ $12$ mo $6{,}000$ $2{,}850$ $1.898$ $52.7\%$

Compute the Cape Cod ELR and total IBNR.
Answer Used-up premium:
 $\begin{align*} 2022: \; 7{,}000 \times 0.923 &= 6{,}461 \\ 2023: \; 6{,}500 \times 0.791 &= 5{,}142 \\ 2024: \; 6{,}000 \times 0.527 &= 3{,}162 \\[4pt] \text{Total} &= 14{,}764 \end{align*}$ 
ELR:

$\text{ELR} = \frac{1{,}750 + 2{,}600 + 2{,}850}{14{,}764} = \frac{7{,}200}{14{,}764} = 48.8\%$

IBNR by year ( $\text{ELR} \times \text{EP} \times \%$ unreported):

AY $\%$ unreported IBNR

$2022$ $7.7\%$ $0.488 \times 7{,}000 \times 0.077 = 263$

$2023$ $20.9\%$ $0.488 \times 6{,}500 \times 0.209 = 662$

$2024$ $47.3\%$ $0.488 \times 6{,}000 \times 0.473 = 1{,}384$

Total $\$2{,}309$ K

Note how the used-up premium weights the years. AY 2022 is mature, so its used-up premium is nearly its full premium and it dominates the ELR estimate. The immature AY 2024 contributes only $3{,}162$ of the $14{,}764$ denominator despite having $6{,}000$ of premium.

AY	Maturity	EP	Reported	CDF	$\%$ reported
$2022$	$36$ mo	$7{,}000$	$1{,}750$	$1.084$	$92.3\%$
$2023$	$24$ mo	$6{,}500$	$2{,}600$	$1.265$	$79.1\%$
$2024$	$12$ mo	$6{,}000$	$2{,}850$	$1.898$	$52.7\%$

AY	$\%$ unreported	IBNR
$2022$	$7.7\%$	$0.488 \times 7{,}000 \times 0.077 = 263$
$2023$	$20.9\%$	$0.488 \times 6{,}500 \times 0.209 = 662$
$2024$	$47.3\%$	$0.488 \times 6{,}000 \times 0.473 = 1{,}384$
Total		$\$2{,}309$ K

When Cape Cod Beats BF, and When It Does Not Two situations: (a) a five-year-old commercial programme whose pricing indications have never been reconciled to results; (b) a book that changed its case reserving philosophy two years ago, strengthening reserves across all open years.

Which method suits each?

Answer (a) Cape Cod. With no reliable external ELR, a BF estimate would rest on an a priori nobody can defend. Cape Cod derives the loss ratio from the programme's own emergence, weighted properly by maturity, and removes the guess entirely.

(b) BF — or neither, until the triangle is fixed. The case strengthening has inflated reported losses on every year at once. Cape Cod computes its ELR from those inflated losses, so the strengthening enters the estimate twice: once through the reported losses in the numerator of each year's ultimate, and again through an ELR that is now too high.

BF with an externally derived ELR — from pricing, adjusted for rate level and trend — is insulated from that particular contamination, because its a priori never touches the distorted triangle.

The general principle: Cape Cod's strength is independence from external judgment; its weakness is dependence on the data. When the data is the thing that is wrong, an external a priori is worth more than a self-consistent one. The proper fix in (b) is a Berquist-Sherman restatement before either method is run.