Reserving 101

Every claim has a few key dates. The accident date (or occurrence date) is when the loss actually happened. The report date is when the insurer first learns about it. The gap between the two — which can be days for an auto claim or years for a latent liability claim — is the report lag.

After it's reported, a claim stays open while it's investigated, litigated, and paid, sometimes over many years, before it finally closes. Reserving analyses look at all of this activity as of a fixed snapshot in time called the valuation date (or evaluation date), and describe how "old" a claim is using its development age — the time elapsed since the start of its accident period, usually measured in months (12, 24, 36, ...).

Report lag is the reason reserving exists at all: at any valuation date, there are claims that have already happened but haven't been reported yet, and the insurer has to estimate for them without knowing they exist individually.

Reserving data is usually organized into a loss triangle: accident years down the rows, development age across the columns. Each cell holds the total losses for that accident year, evaluated as of that development age — a cumulative triangle. (An incremental triangle instead shows just the change since the previous evaluation; you can always get one from the other.)

The triangle shape falls out naturally: a more mature accident year has more columns filled in than a recent one, since it's had more time to develop. Here's a small example, cumulative paid losses in $000s:

Accident year 2024 only has a 12-month value because, as of this valuation date, it hasn't had time to reach 24 months yet — that's the "staircase" edge you'll see in every loss triangle.

An age-to-age factor (or link ratio) measures how much cumulative losses grow from one development age to the next: it's simply the later cumulative value divided by the earlier one.

For AY 2022 above, the 12-to-24 month factor is 1,500 / 1,000 = 1.500 — losses grew 50% between the two evaluations.

Using the same triangle, work out the 12-to-24 month age-to-age factor for AY 2023 (12 mo = $1,100k, 24 mo = $1,650k).

With several accident years in a triangle, you get several age-to-age factors for each development period — one per pair of diagonal cells. The actuary has to pick a single selected factor to represent each period, usually from an average: a straight average (equal weight to each year) or a volume-weighted average (equivalent to summing all the numerators and dividing by the sum of all the denominators), sometimes after excluding an obvious outlier year.

Multiplying the selected factors together, from a given age out to the end of the triangle, gives the cumulative development factor (CDF) for that age — how much further a claim at that age is still expected to grow. Beyond the last age the triangle actually observes, losses on long-tail lines like general liability often keep developing — that extra, unobserved development is captured by a tail factor, typically estimated by fitting a curve to the observed pattern or using industry benchmarks.

The chain ladder method puts the pieces together:

1. Build the cumulative triangle.
2. Compute age-to-age factors between each pair of columns, and select one factor per development period (often a volume-weighted average).
3. Multiply the selected factors together to get a cumulative development factor (CDF) to ultimate for each accident year's current age.
4. Multiply each accident year's latest diagonal value by its CDF to ultimate — that gives the projected ultimate loss.
5. Subtract what's been paid (or reported) so far from the ultimate to get the reserve still needed.

Worked example: AY 2024 is currently at $1,200k (12 months). Suppose the selected 12-to-24 factor is 1.50 (matching the pattern from AY 2022/2023), and the CDF from 24 months to ultimate is separately estimated at 1.20. The CDF from 12 months to ultimate is therefore 1.50 × 1.20 = 1.80.

Chain ladder has a weakness for very recent, immature accident years: their CDF to ultimate is large, so a small, noisy change in early reported losses gets multiplied into a big swing in the ultimate estimate. A single early large claim on a brand-new accident year can move the indicated ultimate dramatically.

Bornhuetter-Ferguson (BF) addresses this by blending an independent, a priori expected loss estimate (from pricing or industry benchmarks, expressed as an expected loss ratio) with the losses actually reported so far:

Ultimate_BF = Losses to date + Expected losses × (1 − 1/CDF)

The term (1 − 1/CDF) is the expected share of ultimate losses still unreported. Early on, when CDF is large, most of the BF estimate comes from the stable a priori expectation rather than the thin, volatile actual data. As the accident year matures and its CDF shrinks toward 1, that term shrinks toward zero and BF converges smoothly to the chain ladder answer — it isn't an on/off switch between two different methods.

The total unpaid claim liability breaks down into a few named pieces:

• Case reserve — the claim adjuster's own per-claim estimate of what's still owed on a claim that's already been reported and is open.
• IBNR ("incurred but not reported") — the reserve for claims that have already happened but haven't been reported to the insurer at all yet.
• IBNER ("incurred but not enough reported," i.e. development on known claims) — the extra amount actuaries expect reported claims to develop beyond their current case reserves, because case reserves are often systematically too low (or too high) early in a claim's life.

The actuarial indicated total reserve — ultimate loss minus paid to date, from a method like chain ladder or BF — covers all of it at once. In everyday use, "IBNR" is often used loosely to mean that whole indicated reserve minus the case reserves already booked (i.e. pure IBNR plus IBNER together), which is why the term can be a little slippery depending on who's using it.

Here's the shape of it, end to end:

• Report lag means some claims exist but aren't on the books yet — that's why reserving is necessary at all.
• Loss triangles organize claims data by accident year and development age.
• Age-to-age factors describe how losses at each age tend to develop toward their next evaluation.
• Chain ladder chains those factors together into a cumulative development factor, and applies it to project each accident year's ultimate loss.
• Bornhuetter-Ferguson blends in an a priori expectation to stabilize immature, volatile accident years.
• The final reserve is made up of case reserves, IBNR, and IBNER.

From here, the natural next steps are the CAS's "Estimating Unpaid Claims Using Basic Techniques" study note for a fuller treatment of these and other methods (Cape Cod, expected loss ratio, frequency-severity), and working actual triangles by hand until the mechanics feel automatic.

Accident year

The 12-month period in which claims occurred, used to group a loss triangle's rows.

Development age

Time elapsed since the start of the accident period, usually in months (12, 24, 36, ...).

Valuation date

The snapshot date as of which a reserving analysis is performed (also "evaluation date").

Loss triangle

Claims data arranged by accident year and development age, cumulative or incremental.

Age-to-age (link) factor

The ratio of cumulative losses at one development age to the age before it.

Cumulative development factor (CDF)

The product of selected age-to-age factors from a given age out to ultimate.

Tail factor

Extra development expected beyond the end of the observed triangle.

Case reserve

An adjuster's per-claim estimate of what remains to be paid on a reported, open claim.

IBNR

Reserve for claims that have occurred but not yet been reported to the insurer.

IBNER

Expected development on already-reported claims, beyond their current case reserves.

Ultimate loss

The total loss an accident year is projected to cost once fully developed and settled.

Chain ladder

A method that projects ultimate losses by applying cumulative development factors to current data.

Bornhuetter-Ferguson

A method that blends an a priori expected loss with chain-ladder-implied emergence.

You've completed Reserving 101.