Loss development: triangles, factors and tail selection

Why losses need developing

For short-tailed lines such as automobile physical damage and homeowners, the ultimate amount is known fairly quickly. For long-tailed lines such as commercial general liability or workers compensation, it can take many years. The ratemaking actuary typically uses the most recent accident year data available, so the history is to some degree immature, and more so if policy year data is used (Werner and Modlin, Chapter 6).

The CAS texts describe the chain ladder method, also called the development technique, as one of the most frequently used approaches. Its key assumption is that claims recorded to date will continue to develop in a similar manner to earlier years. It also assumes that claim processing, the mix of claim types, policy limits and reinsurance retentions stayed reasonably stable across the experience period (Friedland, Chapter 7). The NAIC handbook presents the same method as an illustration and says it is in no way the only or the preferred way to derive ultimate losses.

From triangle to ultimate

Friedland breaks the method into seven steps. The numbers below are invented, to show the mechanics only.

Step 1: the triangle. Cumulative reported losses in thousands, valued at 12, 24, 36 and 48 months:

Step 2: age-to-age factors. The factor for an interval is the later valuation divided by the earlier one. Friedland notes these are also called report-to-report factors or link ratios and are named by interval, so "12-24" is the 12-month to 24-month factor. Here the 12-24 factors are 1.500, 1.527 and 1.550; the 24-36 factors are 1.100 and 1.101; the 36-48 factor is 1.000.

Step 3: averages. Friedland lists the simple average, the medial average (which drops the high and low values), the volume-weighted average (the sum of losses at the later age over the sum at the earlier age, for the same accident years) and the geometric average. For 12-24 the simple average is 1.526 and the volume-weighted average is 1.527. Averages can run over all years or only the latest few. Friedland describes the choice as a trade-off between stability, which favors more periods, and responsiveness, which favors only the most recent ones.

Step 4: selected factors. Suppose the analyst selects 1.52 for 12-24 and 1.10 for 24-36.

Step 5: tail. Suppose the data shows no development after 36 months, so the selected factor from 36 months to ultimate is 1.000. The tail is covered in its own section below.

Step 6: cumulative factors. Multiply from the oldest interval toward the youngest: the factor to ultimate at 36 months is 1.000, at 24 months 1.000 × 1.10 = 1.100, and at 12 months 1.100 × 1.52 = 1.672. Friedland calls these cumulative claim development factors, also known as age-to-ultimate factors.

Step 7: ultimate. Apply each accident year's factor to the latest diagonal.

Choosing factors

Friedland says the selection should look at the history, the age-to-age factors, the averages and, commonly, the prior selection, and lists what the actuary reviews: whether the factors progress smoothly across ages, whether they are stable within an age, the credibility of the experience, changes in patterns, and whether the history applies to the future book. Early intervals are usually the most variable, because claims are least understood when they are youngest. When an insurer's own experience has limited credibility, benchmarks from similar lines or the industry may supplement it, with care that claims practices, coverages and limits are comparable.

Friedland is plain that selections are subjective: different actuaries will reach different factors, and there is more than one reasonable selection. Werner and Modlin say the actuary should select according to what is expected to occur in future periods, which can mean a shorter average when patterns are changing, a weighted average when the line is volatile, or excluding an anomalous high and low value.

Some practical points from the texts:

• Group like claims. Werner and Modlin advise analyzing homogeneous claims, for example property and liability separately, with extraordinary losses removed and material benefit changes adjusted for.
• Pick the basis. For ratemaking in most lines, the interest is usually reported losses including allocated loss adjustment expenses; paid development is also used. Cumulative paid factors are usually larger than reported factors at the same age (Friedland).
• Factors can be below 1.000. Salvage and subrogation recoveries, or case reserves set too high early on, can make development negative.
• Limits matter. The NAIC handbook notes that development factors tend to be higher at higher limits, because large-limit losses take longer to settle.
• Young books. The handbook notes that new or fast-changing books are often triangulated by month or quarter instead of by year.

Tail selection

If the triangle ends before development stops, a tail factor must carry the latest observed age to ultimate. Friedland observes that the point at which development ends varies widely by line, and that some long-tail claims take more than fifteen years to settle. The tail factor multiplies into every accident year's factor to ultimate, so it has disproportionate leverage on the result.

Friedland names three common approaches: industry benchmark development patterns; fitting a curve, such as exponential decay, to the observed factors and extrapolating; and, for paid development when reported development is already at ultimate, using the ratio of reported to paid at the latest age. Werner and Modlin likewise mention fitting curves or running special studies with more years of data.

Development is not trend

The NAIC handbook cautions against confusing the two. Development measures expected future change in a given accident year's losses; trend measures the difference between those developed amounts from one accident year to the next. They answer different questions and do not overlap. The handbook adds that development in a rate filing differs from Schedule P development in the Annual Statement, which is the accounting change in estimated ultimate losses over a year or two, not a projection.

What a development exhibit should show

A reader can only test the selections against the criteria above if the exhibit shows the work:

• The triangle itself, with its basis (paid, reported, counts), valuation dates and whether it includes adjustment expenses.
• The age-to-age factors, the averages calculated and the selected factors beside them.
• The tail and its support.
• The cumulative factors and how they were applied to the latest diagonal.
• The adjustments made to the data and why.

Actuarial Standard of Practice No. 43 covers unpaid claim estimates, not rates, but its general discipline carries over: consider whether the assumptions are reasonable (section 3.6.2) and follow ASOP No. 23 on data quality (section 3.6.3). For where the exhibit sits in a filing, see what goes in a rate filing and the actuarial memorandum.