Estimate unpaid losses with the development (chain ladder), case outstanding development, expected claims, Bornhuetter-Ferguson, Cape Cod, frequency-severity, Berquist-Sherman, and Benktander techniques, and judge when each fits.

An immature accident year has almost no paid data, so trusting its own development is reckless. A mature year has almost nothing left to emerge, so an a priori guess wastes hard evidence. Every method below trades responsiveness against stability differently.

Define $p=1/\text{CDF}$, the percentage of ultimate already reported (or paid). Then $1-p$ is the unreported percentage. Almost every method is a statement about how much to trust the emerged fraction $p$.

Development (chain ladder). Average the age-to-age link ratios, chain them into a CDF, and multiply the latest diagonal.

$$U_{CL}=\text{Reported}\times \text{CDF}=\frac{\text{Reported}}{p}$$

It assumes future development is proportional to what has emerged. That breaks for the youngest year, where a small early number gets multiplied by a large, leveraged CDF.

The source runs the method in seven steps: (1) compile the claims triangle, (2) calculate age-to-age factors, (3) select development factors, (4) select a tail factor, (5) chain the selected factors into cumulative development factors, (6) project ultimate claims, and (7) subtract paid or...

Common mistakes

• Multiplying a tiny green-year diagonal by a huge CDF. A 12-month figure times a CDF of 5.0 gives a wildly unstable chain ladder ultimate; use BF or expected claims instead.
• Reading age-to-age factors as CDFs. The 1.50, 1.17, 1.03 are link ratios; the 12-month age-to-ultimate factor is 1.50 × 1.17 × 1.03 = 1.81. Multiply the youngest diagonal by 1.81, not 1.50.
• Running Berquist-Sherman in the wrong direction. The latest diagonal is the anchor and never moves. You restate the earlier diagonals up to that adequacy, not the newest diagonal down.

Bottom line

• Chain ladder ultimate equals latest cumulative losses times the cumulative development factor (CDF); fully responsive, unstable for green years.
• Expected claims ultimate equals premium times an a priori loss ratio; ignores actual emergence, best for immature or thin years.
• Bornhuetter-Ferguson ultimate equals reported losses plus expected losses times the unreported percentage, one minus one over the CDF.
• Cape Cod is BF with the loss ratio estimated from the triangle: sum of reported over sum of used-up (on-level) premium.

Exam shortcut

Convert every CDF to $p=1/\text{CDF}$ immediately. Then chain ladder is Reported ÷ $p$, BF is Reported + Expected × $1-p$, and both blends fall out with no extra algebra. Chain link ratios before you touch a diagonal. The age-to-age factors are not CDFs; the 12-month CDF here is 1.50 × 1.17 × 1.03 = 1.81, and a 1.000 tail past the oldest maturity is the default unless told otherwise.

The full lesson (about 4,316 words, 29 min read) adds 5 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• B9

Browse all free Exam 5 lessons or jump into free Exam 5 practice questions.