Reserving Methods: Chain Ladder, Bornhuetter–Ferguson, Benktander, and Cape Cod

Once loss development factors are selected (see Loss Development Triangles), several distinct methods can project ultimate losses for a given origin year. They differ in how much weight they place on the origin year's own reported experience to date versus an a priori expectation, and choosing among them is a central reserving judgment.

Chain ladder (CL)

The chain ladder simply multiplies reported (or paid) losses to date by the cumulative development factor:

$\hat U_i = C_i \times CDF_i,$

where $C_i$ is losses at the latest evaluation and $CDF_i$ is the cumulative development factor from that age to ultimate. It is the simplest method and uses only the origin year's own experience — it implicitly assumes the CDF (derived from the full triangle) applies equally well to this specific year. Chain ladder is best suited to mature origin years, where $CDF_i$ is close to 1 so the multiplication is not highly leveraged, and to lines with stable, credible development patterns. It performs poorly for immature years or after a large individual claim distorts the year's reported total — the leverage of a large $CDF_i$ amplifies that distortion directly into the ultimate.

Expected claims method

At the opposite extreme, the expected claims method ignores the year's own reported experience entirely and instead applies an a priori expected loss ratio (ELR), established from pricing, industry benchmarks, or prior accident years, to the year's exposure base $q_i$ (e.g., earned premium on-leveled):

$\hat U_i = q_i \times ELR.$

This is appropriate when the origin year is so immature (e.g., first evaluation of a brand-new line) that reported losses carry essentially no information, or when a known catastrophic distortion makes the reported total misleading.

Bornhuetter–Ferguson (BF)

BF blends the two: it uses reported losses to date for the already-emerged portion, and the a priori expected losses for the not-yet-emerged portion, weighted by the percentage unreported:

\hat U_i = C_i + q_i\times ELR\times\Big(1-\frac{1}{CDF_i}\right).

The quantity $1-1/CDF_i$ is the percentage of ultimate losses expected to still emerge (e.g., if $CDF_i=1.25$, then $1/1.25=0.80$ or 80% has emerged, so 20% remains). BF's key advantage is that it does not multiply reported losses by a large leveraged factor — instead it adds an IBNR provision computed off the stable exposure base — so it is far less sensitive to random noise in early-maturity reported losses than the chain ladder. Its weakness is dependence on a well-chosen a priori ELR; a stale or wrong ELR biases every immature year's reserve.

Benktander (iterated BF)

Benktander's method iterates BF once, using the BF ultimate (rather than the a priori ELR alone) as the expected-claims basis for a second pass:

$\hat U_i^{Bktr} = C_i + \big(\hat U_i^{BF}\big)\times\Big(1-\frac1{CDF_i}\Big).$

Benktander is a credibility-weighted average of the chain ladder and BF estimates and is known (under certain model assumptions) to have lower mean squared error than either pure CL or pure BF — it responds more to the year's own emerging experience than BF while remaining more stable than chain ladder for immature years.

Cape Cod method

Cape Cod is BF's cousin, but instead of relying on an externally selected ELR, it derives the ELR directly from the triangle's own data, weighting each origin year's reported losses by its "used-up" exposure ($q_i/CDF_i$, i.e., exposure weighted down to reflect only the portion that has had a chance to emerge):

$ELR_{CC} = \frac{\sum_i C_i}{\sum_i q_i/CDF_i}.$

This blended, credibility-adjusted ELR is then applied in the BF formula in place of an externally selected ELR. Cape Cod is attractive when there is no reliable external a priori estimate (e.g., new lines, or after a significant rate or mix change makes historical pricing ELRs unreliable), because it lets the data itself set the expected loss ratio while still damping chain-ladder's leverage problem for immature years.

Worked comparison: one origin year

Origin year 2024, evaluated at 12 months: reported losses $C = 4{,}000{,}000$; selected $CDF = 2.50$ (so 40% of losses expected reported, 60% unreported); on-leveled earned premium $q = 20{,}000{,}000$; a priori expected loss ratio $ELR = 65\%$.

Chain ladder: $\hat U = 4{,}000{,}000\times2.50 = 10{,}000{,}000$.

Expected claims: $\hat U = 20{,}000{,}000\times0.65 = 13{,}000{,}000$.

Bornhuetter–Ferguson: percentage unreported $=1-1/2.50=1-0.40=0.60$. $\hat U = 4{,}000{,}000 + 20{,}000{,}000\times0.65\times0.60 = 4{,}000{,}000+7{,}800{,}000=11{,}800{,}000$.

Benktander: use the BF ultimate (11,800,000) in place of the a priori expected claims (13,000,000): $\hat U^{Bktr}=4{,}000{,}000+11{,}800{,}000\times0.60=4{,}000{,}000+7{,}080{,}000=11{,}080{,}000$. Note this sits between CL (10.0M) and BF (11.8M), as expected.

Cape Cod (illustratively, if the triangle-wide used-up-exposure-weighted ELR comes out to 58% rather than the a priori 65%): $\hat U = 4{,}000{,}000+20{,}000{,}000\times0.58\times0.60=4{,}000{,}000+6{,}960{,}000=10{,}960{,}000$ — close to Benktander here because the derived ELR (58%) sits below the a priori 65%.

Pitfalls

• Using chain ladder on very immature, volatile origin years, where a single large early claim swings the leveraged ultimate wildly.
• Using a stale a priori ELR in BF/expected claims without checking it against recent rate change and mix shift.
• Applying Cape Cod without recognizing it still depends on the same CDFs, so a distorted development pattern still corrupts the derived ELR.
• Presenting only one method's point estimate rather than a reasonable range across methods as a check on model risk.

Exam relevance

CL, BF, Benktander, and Cape Cod are core computational content on CAS Exam 6 and Exam 8.

Further reading

• Friedland, Estimating Unpaid Claims Using Basic Techniques, CAS Study Note.
• Benktander, An Approach to Credibility in Calculating IBNR, ASTIN.
• Clark, LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach, CAS Forum.