Why Selection Can Differ by Development Age
A link ratio at one development age need not be selected in the same way as a link ratio at another. Early development may show a recent change in reporting or settlement patterns, while later ages may have fewer observations, different volume patterns, or a more stable process. A defensible selection therefore considers each age on its own evidence, rather than applying one averaging rule or one observation window mechanically to the entire triangle.
The aim is not to find a method that produces the largest or smallest factor. It is to select a factor that reasonably represents expected development on a consistent claims basis, using observations that are both sufficiently mature and relevant to the projected claims.
The observation window is the set of accident years included in estimating a factor at a particular age. The available window usually becomes shorter at older ages because fewer accident years have emerged through those ages. “Use all available data” is not automatically correct: older observations may reflect a different operational process. Conversely, “use only the latest years” can discard useful experience when recent observations are sparse or volatile.
Match the Method to the Evidence
Three common summaries are the simple arithmetic average, the volume-weighted average, and the geometric average. Their differences matter when the observations or their underlying volumes differ.
| Method | Calculation | Potential reason to use it | Key caution |
|---|---|---|---|
| Simple arithmetic average | Average the selected observed ratios | Each included accident year should have comparable influence | Can give a small-volume observation as much weight as a large one |
| Volume-weighted average | Weight each ratio by its cumulative losses at the beginning of the age interval | More weight is appropriate for observations with more underlying loss volume | A large denominator does not cure a change in the underlying process |
| Geometric average | Take the geometric mean of positive observed ratios | Relative, multiplicative variation is more relevant than additive variation | Not defined for nonpositive ratios and sensitive to the assumed scale |
For observations \(R_{i,j}\) with beginning-age volumes \(C_{i,j}\), the volume-weighted selection is \(f_j=\sum_i C_{i,j}R_{i,j}/\sum_i C_{i,j}\). Because \(C_{i,j}R_{i,j}=C_{i,j+1}\), this is also the ratio of the corresponding totals. That identity describes the calculation; it does not establish that weighting is appropriate for every data set.
For a positive set of ratios, a geometric average is \(f_j=\left(\prod_i R_{i,j}\right)^{1/n}\). One justification follows from a multiplicative-error perspective. Suppose a single factor \(f\) is chosen to minimize squared deviations on the log scale, \(Q(f)=\sum_i(\ln R_{i,j}-\ln f)^2\), with \(f>0\). Differentiating with respect to \(\ln f\) and setting the derivative to zero gives \(\sum_i(\ln R_{i,j}-\ln f)=0\). Thus \(\ln f\) is the mean of the log ratios, and exponentiating gives the geometric average. This rationale is useful when proportional differences are meaningful; it is not a reason to use a geometric average when ratios are nonpositive or when a multiplicative model is implausible.
An age-specific selection should also be consistent with the evidence at nearby ages. This does not mean the selected factors must decline smoothly or follow a predetermined pattern. A sharp change may be supportable if it reflects real development behavior, but it should prompt investigation of the data, maturity, and claims process before being accepted.
A Practical Age-by-Age Selection Process
Confirm that cumulative amounts are on a consistent paid or incurred basis, identify the development ages, and calculate each usable link ratio. Investigate unusual or invalid denominators before averaging.
At each age, determine which accident years have emerged and whether older observations represent the current claims and operational environment.
State which observations are included and why. Decide whether equal, volume-based, or log-scale treatment best reflects the data at that age.
Compare the selected factors across ages with observed behavior and known changes. Investigate surprising results; do not impose smoothness merely for appearance.
For a projection through multiple observed development intervals, multiply the selected age-to-age factors. Keep the age labels clear so no factor is applied twice or skipped.
Worked Selections at Several Ages
Worked Example: Recent Volume at Early Development, Broader Evidence Later
A hypothetical paid-loss triangle has the following usable observations. The beginning-age amounts are in thousands of dollars. Select factors for ages 12–24, 24–36, and 36–48 months. For 12–24 months, assume the most recent two years best represent the current early reporting process. At 24–36 months, assume all four observations are mature and comparable. At 36–48 months, assume the four observations have comparable relevance and no meaningful volume differences for selection.
| Age interval | Observed link ratios | Beginning-age amounts (thousands) |
|---|---|---|
| 12–24 months | 1.30, 1.22, 1.18, 1.15, 1.12 (oldest to newest) | 100, 150, 220, 300, 400 |
| 24–36 months | 1.10, 1.08, 1.06, 1.05 | 450, 600, 750, 900 |
| 36–48 months | 1.04, 1.02, 1.08, 1.03 | Comparable influence assumed |
12–24 months: recent volume-weighted average. The stated window is the latest two observations, 1.15 and 1.12, with beginning amounts 300 and 400. Therefore,
24–36 months: volume-weighted average of all four observations. The observations are assumed mature and comparable, so use all four amounts:
36–48 months: simple arithmetic average. Given the assumption that the four observations have comparable influence, each receives equal weight:
The cumulative factor through 48 months is the product of the three selected age-to-age factors:
The choices differ by age because the assumed relevance and weighting evidence differ by age. A common mistake would be to use the latest two observations at every age simply because they are most recent. That would ignore the stated maturity and comparability of the broader later-age experience.
Worked Example: Geometric Selection and a Changed Late-Age Process
A hypothetical incurred-loss triangle has positive 12–24 month ratios of 1.00, 1.10, and 1.21. Assume the relative, multiplicative variation is the appropriate scale and that the observations are relevant, but their volumes should not determine influence. For 24–36 months, three mature observations have ratios 1.10, 1.08, and 1.06 and beginning amounts of 100, 200, and 300. For 36–48 months, a documented case-reserving process change makes only the two most recent observations relevant; their ratios are 1.03 and 1.02.
12–24 months: geometric average. The product of the three positive ratios is \(1.00(1.10)(1.21)=1.331\), and \(1.10^3=1.331\). Hence the geometric selection is exactly 1.10.
24–36 months: volume-weighted average. Use all three mature observations and their stated beginning amounts:
36–48 months: simple average of the relevant recent pair. The process change is the reason to exclude older experience; it is not a general rule to discard older observations. The selected factor is
Combining the age-to-age selections gives \(1.10(1.073333\ldots)(1.025)=1.2102\), rounded to four decimal places. The common mistake is to treat the geometric selection as universally preferable because the first set has multiplicative variation. That justification applies only to the stated early-age model; the later-age choices use different evidence and assumptions.
Worked Example: Checking a Mixed Selection Against Its Assumptions
A hypothetical specialty-liability triangle has three usable 12–24 month observations. The ratios are 1.12, 1.10, and 1.08, with beginning-age amounts of 100, 300, and 600 thousand dollars. Assume the volumes are meaningful and the observations are comparable. At 24–36 months, assume three mature observations have ratios 1.06, 1.04, and 1.02, but no reliable volume weighting is available and each accident year should have equal influence. At 36–48 months, only two comparable mature observations are available: 1.03 and 1.01.
12–24 months: volume-weighted average. Calculate the weighted numerator as \(100(1.12)=112\), \(300(1.10)=330\), and \(600(1.08)=648\). The total numerator is \(112+330+648=1090\), and the total beginning amount is \(100+300+600=1000\). Therefore \(f_{12-24}=1090/1000=1.09\).
24–36 months: simple average. With equal influence assumed, \(f_{24-36}=(1.06+1.04+1.02)/3=3.12/3=1.04\).
36–48 months: simple average of available mature experience. The average is \(f_{36-48}=(1.03+1.01)/2=1.02\). The smaller number of observations is a limitation to disclose, not a reason to substitute an unsupported method.
The factor through 48 months is \(1.09(1.04)(1.02)=1.156272\), or 1.1563 to four decimal places. The check is that each component is an age-to-age factor and all three adjacent intervals are included exactly once. A common mistake is to use the 12–24 weighted factor of 1.09 as though it were cumulative to 48 months; the required projection across all three intervals multiplies all three selections.
Judgment, Documentation, and Common Traps
A selected factor is an actuarial judgment supported by data and assumptions, not simply the output of a formula. Document, for each age, the included accident years, the averaging method, the reason the window is appropriate, and any known operational or claims-process changes. If an unusual observation is omitted or given reduced influence, state the evidence for that treatment rather than hiding it inside a calculation.
- Using one method everywhere. Different ages can have different volume, maturity, and stability characteristics. Reconsider the method separately at each age.
- Confusing recent with representative. Recent data may reflect current conditions, but a very short window can be volatile or incomplete.
- Weighting without a reason. Volume weighting is not automatically superior; it gives high-volume observations greater influence, which is appropriate only if that influence matches the selection objective.
- Ignoring maturity. An accident year that has not yet emerged through the selected development age cannot supply a link ratio for that interval.
- Forcing a smooth pattern. A smooth sequence can be a useful diagnostic expectation, but changing a factor merely to make a curve look smooth can override credible evidence.
- Applying a factor to the wrong interval. Label selected factors by their start and end ages, then multiply only the consecutive intervals needed for the projection.
Check Your Understanding
For each problem, identify the method or window implied by the facts, show the calculation, and compare your result with the answer.
- At one development age, three comparable accident years have link ratios 1.12, 1.08, and 1.10. Equal influence is appropriate. What is the simple average factor? (Answer: 1.1000)
- Two relevant observations have ratios 1.10 and 1.05, with beginning-age amounts of 200 and 400. Calculate the volume-weighted factor. (Answer: 1.0667)
- Positive ratios at an early age are 1.00, 1.20, and 1.44. Under a multiplicative-error rationale, calculate their geometric average. (Answer: 1.2000)
- At a later age, only the latest three observations are considered representative. Their ratios are 1.12, 1.08, and 1.05, with beginning-age amounts 100, 200, and 400. Calculate the selected volume-weighted factor for that window. (Answer: 1.0686)
- A hypothetical projection uses selected factors 1.08 from 12–24 months, 1.05 from 24–36 months, and 1.02 from 36–48 months. Calculate the cumulative factor through 48 months and state the interval check. (Answer: 1.1567; all three consecutive intervals are included once.)