Somewhere in the exhibits of your actuarial report is a table of decimal numbers: 2.150, 1.380, 1.120, 1.045, 1.010. These are the loss development factors, and they are doing more work than anything else in the analysis. The chain ladder multiplies by them directly. Bornhuetter-Ferguson uses them to decide how much weight your actual claims get. Even the reserve on your oldest accident year moves when one of them changes.
Most buyers skip this table. That is a mistake. The development factors are where the actuary’s judgment enters the analysis, and judgment is exactly what you are paying to evaluate. This article explains what the factors are, how they are calculated, how they become the cumulative factors that project your ultimate losses, and the specific selection decisions worth questioning.
What a loss development factor is
Claims grow after they are reported. An injured worker’s medical treatment extends. An adjuster raises a case estimate as facts emerge. A claim that looked closed reopens. A loss development factor measures that growth: it is the ratio of a cohort’s losses at one age to the same cohort’s losses at an earlier age.
The raw material is the loss development triangle, which organizes losses by accident year and evaluation age. Dividing each accident year’s losses at 24 months by the same year’s losses at 12 months gives an age-to-age factor (also called a link ratio) for the 12-to-24 interval. Do that for every accident year and every interval and you have a triangle of factors, one observation per year per interval.
An age-to-age factor of 1.380 for the 12-to-24 interval says that, for that accident year, losses at 24 months were 38% higher than at 12 months. A factor of 1.010 near the end of the triangle says the cohort was almost done developing. A factor below 1.000 says losses went down, which happens routinely on reported triangles when case reserves close below their estimates, and rarely on paid triangles (mostly through recoveries such as subrogation or salvage).
From observations to selections
The factor triangle gives the actuary several observations for each interval: one per accident year. The next step is choosing a single representative factor per interval, and this is where arithmetic ends and judgment begins.
The standard toolkit is a set of averages, each with a different bet embedded in it:
- All-year average. Uses every observation. Stable, but slow to recognize a genuine change in the development pattern.
- Latest three- or five-year average. Uses only recent diagonals. Responds faster to change, at the cost of more noise, which matters for the thin triangles most self-insureds have.
- Volume-weighted average. Weights each year by its loss volume, so a large year counts more than a small one. This dampens the distortion from one small, erratic year.
- Medial or trimmed average. Drops the highest and lowest observation before averaging, a mechanical way to exclude outliers.
A well-documented report shows the candidate averages side by side and a “selected” row indicating which value the actuary chose for each interval. The selection does not have to equal any of the averages. An actuary who believes the pattern is lengthening may select above every average. That can be defensible, but it should be visible and explained, because each selection compounds into the reserve.
Chaining selections into CDFs
Selected age-to-age factors become useful when they are chained together. Multiplying the selected 36-to-48 factor by the 48-to-60 factor, and so on through the end of the pattern, produces a cumulative development factor (CDF) from 36 months to ultimate. The CDF is the number that actually touches your data: current losses x CDF = projected ultimate losses.
The reciprocal of the CDF is just as useful. A CDF of 1.333 means current losses are expected to be 75% of ultimate (1 divided by 1.333). Actuaries call this the percent reported (or percent paid). It is the maturity gauge for the accident year, and it is the weight that Bornhuetter-Ferguson uses to decide how much your actual experience counts versus the prior expectation.
Chaining is also why small selection differences compound. Nudge each of six selected factors up by 1% and the CDF moves up by about 6%. On an immature accident year with a CDF of 4.00, that is the difference between an ultimate of $4.0 million and $4.25 million on the same $1.0 million of reported losses, from selection changes too small to notice individually.
The tail factor
The triangle only shows development as old as your data. If your oldest accident year is ten years old but workers compensation claims in your state keep developing for thirty, the pattern beyond year ten has no observations at all. The actuary covers that gap with a tail factor, a single multiplier for all development beyond the edge of the triangle.
Tail factors are estimated from industry benchmarks, curve fits to your own factors, or judgment, and on long-tailed lines they can be the largest single source of uncertainty in the reserve. A tail of 1.05 versus 1.10 on a workers compensation program changes every accident year’s ultimate, because the tail multiplies into every CDF. The considerations are the same ones covered in Tail Factor Selection for Captives, and they apply to any self-insured program with a long-tailed line, captive or not.
Why the factors are judgment, not arithmetic
Two competent actuaries given the same triangle will produce similar age-to-age observations, because those are arithmetic. They can still select materially different factors, because selection answers questions the arithmetic cannot:
Which history is representative? If the claims operation changed (new TPA, new case reserving philosophy, a litigation initiative), the older diagonals describe a process that no longer exists. Excluding them is defensible. So is keeping them. The choice moves the reserve.
Are the outliers signal or noise? A 12-to-24 factor of 3.50 in a triangle of 2.00s might be one late-reported large claim (exclude it) or the first evidence of a lengthening pattern (do not). For thin self-insured triangles, one observation can be a third of the data for an interval.
Is there a calendar-year effect? Factors elevated along a recent diagonal, across all accident years at once, point to something that happened in a calendar period: a case reserve adequacy push, an inflation shock, a fee schedule change. Averaging across diagonals smears that effect into every projection. The diagnostic signs are covered in Case Reserve Strengthening.
Friedland’s text, the CAS’s standard reference on these techniques, treats factor selection as an explicit, documented judgment rather than a mechanical output, and lists the averages above as candidates the actuary weighs, not rules the actuary follows (Friedland, ch. 7).
Where the leverage is
Development factors do not carry equal weight. The factors applied to your most recent accident years sit on top of the most IBNR, so errors there are the expensive ones. A 10% error in a 1.020 factor on a nine-year-old accident year is rounding. A 10% error in a 4.00 CDF on the current year is 10% of a mostly-unreported ultimate.
This is the same leverage problem described in the chain ladder explainer, viewed from the factor side: the chain ladder is only as good as the factors, and the factors are weakest exactly where the method leans on them hardest. It is also why actuaries shift to expectation-anchored methods for the greenest years, where even a carefully selected factor multiplies a base too thin to trust.
What to ask about the factor selections
You do not need to re-derive the triangle to supervise this part of the analysis. Five questions cover the ground:
1. Can I see indicated versus selected? The report should show the candidate averages and the selected factor for each interval. If only the selections appear, ask for the comparison. Selections that consistently sit above or below every average are a finding, not a formality.
2. Which averaging basis did you rely on, and why? The honest answer names the trade-off: responsiveness versus stability, and what in this program’s history justified the choice.
3. What did you exclude, and where is that documented? Excluded observations and the reason for each should be in the report. An exclusion without a reason is an invisible assumption.
4. What supports the tail factor? Benchmark source, curve fit, or judgment, plus a sensitivity: what does the reserve look like at a reasonably higher and lower tail?
5. Did the factors change from last year’s study, and what did that do to the reserve? Factor selections should evolve slowly. A quiet reselection can move the indicated reserve as much as a genuine change in claim experience, and it deserves the same visibility. This is one of the checks in How to Evaluate an Actuary’s Report.
Further reading
For the triangle the factors come from, see How to Read a Loss Development Triangle. For the method that multiplies by the CDF directly, see Chain Ladder. For how the percent-reported form of the CDF drives the credibility weighting in BF, see Bornhuetter-Ferguson. For the method landscape in one place, see How Actuaries Estimate Your Unpaid Claims.