Understand, interpret, and apply techniques for estimating outstanding claims, using Expected Loss Ratio, Chain-Ladder, Bornhuetter-Ferguson, Bayesian, and Frequency and Severity methods.

Reserving asks one question: how much will you ultimately pay for claims already incurred? Five standard methods give five different answers. Knowing which to trust, and when, is the whole game.

The reserving setup. Losses develop over time. By accident year $i$, the cumulative paid or reported amount at development age $j$ is $C_{i,j}$. The ultimate $U_i$ is what $C_{i,j}$ converges to as $j\to \infty$. Outstanding reserve equals $U_i-C_{i,j}$, where $j$ is the most recent observation.

Earned premium is the slice of written premium covering coverage already provided. A one-year policy written July 1 for $1,200 has earned only $600 by December 31. The other $600 sits as unearned premium until next calendar year.

The unearned premium reserve (UPR) is a liability that recognizes the carrier still owes coverage time. Regulators force its accrual so that profit is not booked before it is earned. Calendar-year reserving questions almost always start with a UPR carryover.

For a policy with annual premium $P$, written on date $d$, with term $T$ and evaluation date $e$:

Common mistakes

• Confusing reported and paid triangles. ELR and BF can sit on either, but the CDF must match the triangle used for $C_{i,j}$.
• Applying inflation to severity twice. Once it is inside the severity LDF, do not also trend with a separate factor.
• Using $1+1/\text{CDF}$ instead of $1-1/\text{CDF}$ for BF's unreported percent. The correct unreported share is $1-1/\text{CDF}$.

Bottom line

• Expected Loss Ratio (ELR): Ultimate = ELR × Earned Premium. Ignores actual emergence, so it is best only for green, immature accident years and breaks down as data matures.
• Chain-Ladder: Ultimate = latest cumulative × product of future age-to-age factors. Trusts the data and ignores priors, so a noisy latest diagonal distorts it.
• Bornhuetter-Ferguson: Ultimate = Reported + (1 − 1/CDF) × Expected Ultimate. A weighted blend of ELR and Chain-Ladder; its weakness is the subjective a priori.
• Bayesian: prior on the loss ratio, posterior weights data by credibility $Z$. BF is a special case with fixed $Z=1-1/\text{CDF}$.

Exam shortcut

Memorize BF as "reported plus unreported share of a priori". The percent unreported is $1-1/\text{CDF}$, not $1/\text{CDF}$. When the triangle includes counts, expect a frequency-severity sub-question. Project counts and average severity in parallel, then multiply. For the latest accident year, expect BF to land between ELR and Chain-Ladder. If your number is outside that range, recheck the CDF calculation.

The full lesson (about 4,786 words, 32 min read) adds 4 worked examples, all 12 common mistakes, a self-check, free in the app.

Learning objectives

• 6a

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