Why Observation Windows Differ by Development Age
A link ratio at one development age need not be based on the same accident years as a link ratio at another age. The latest accident years may provide useful information at early development ages, while later-age links can only be calculated for older, more mature years. In addition, older observations may become less relevant when a known change has altered the claims process or the data being measured.
Selecting an observation window means deciding which eligible accident years to use for a particular age-to-age link ratio. This decision comes before applying an averaging method. A carefully calculated average cannot correct for using observations that are unavailable or no longer relevant to the expected future development.
A row contributes to the link from age \(j\) to \(j+1\) only when both development ages are observed. Thus, in a triangle valued through age 48 months, the 36-to-48-month link may be available for mature accident years, but the 48-to-60-month link is not yet observable. The number of eligible observations therefore usually decreases at later ages.
A Framework for Selecting a Window
There is no universal rule that the same number of accident years must be used at every age. The selection should reflect both availability and relevance. A wider window may provide more observations and more volume, but it can also include experience produced under conditions that no longer represent the business being projected. A narrow window may be more current, but can leave a factor dependent on only a few observations.
For each development age, identify which accident years have values at both ages in the link. Do not treat a missing later diagonal as a zero or as an observed ratio.
Consider whether the observations represent the process and data expected to apply to the losses being projected. Use documented changes and prior investigation; age alone does not make an observation irrelevant.
Count the eligible observations and consider the underlying cumulative-loss volume. A few observations may be unstable; many observations can still be misleading if they span an important change.
Record the accident years included for this particular age-to-age link. Apply the chosen averaging method to that set and document why the window is appropriate.
For a volume-weighted selection, the selected factor is the ratio of the sum of next-age cumulative losses to the sum of current-age cumulative losses over the selected years. This follows directly from weighting each accident-year ratio by its current-age cumulative loss:
The sums here must use the same selected observation window. The identity explains the calculation for that method; it does not determine which years belong in the window. Window selection and averaging-method selection are related but distinct judgments.
Worked Examples
Worked Example: Use Different Windows at Early and Later Ages
A hypothetical insurer has the following cumulative paid losses, in thousands of dollars. The insurer documents that a payment-handling change beginning with accident year 2022 affects early development. It finds no comparable change affecting the 24-to-36-month link. Select a defensible window for each link and calculate a volume-weighted factor.
| Accident year | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2021 | 1,000 | 1,400 | 1,540 |
| 2022 | 1,200 | 1,620 | 1,782 |
| 2023 | 1,500 | 1,950 | 2,106 |
| 2024 | 1,800 | 2,250 | Not yet available |
Method: select the window separately at each age, then use a volume-weighted ratio of totals. For 12-to-24 months, all four ratios are observable, but 2021 predates the documented change. Use 2022–2024 as the relevant window. Its current-age total is \(1{,}200+1{,}500+1{,}800=4{,}500\), and its next-age total is \(1{,}620+1{,}950+2{,}250=5{,}820\). Therefore:
For 24-to-36 months, 2024 is not eligible because its 36-month value is unavailable. The three eligible observations are 2021–2023, and the stated investigation found no comparable process change for this link. Their current-age total is \(1{,}400+1{,}620+1{,}950=4{,}970\); their next-age total is \(1{,}540+1{,}782+2{,}106=5{,}428\). Thus:
The results are checked against the underlying data: the selected 12-to-24-month ratios are \(1.35\), \(1.30\), and \(1.25\), so a weighted result of \(1.2933\) is plausible. The 24-to-36-month ratios are \(1.10\), \(1.10\), and \(1.08\), consistent with \(1.0922\). Common mistake avoided: using the same accident years for both links would either include an irrelevant early observation or wrongly include the immature 2024 observation at 24-to-36 months.
Worked Example: Retain a Broader Window at a Sparse Later Age
A hypothetical specialty insurer has no identified change in its claims process for the 36-to-48-month link. The following five eligible accident years have cumulative paid losses, in thousands of dollars. The later-age data are limited, so assess the full available window rather than automatically restricting it to the most recent years.
| Accident year | 36 months | 48 months |
|---|---|---|
| 2019 | 2,000 | 2,140 |
| 2020 | 2,200 | 2,332 |
| 2021 | 2,400 | 2,520 |
| 2022 | 2,600 | 2,704 |
| 2023 | 2,800 | 2,912 |
Method: use the full relevant set of eligible observations and calculate the volume-weighted factor. The 36-month total is \(2{,}000+2{,}200+2{,}400+2{,}600+2{,}800=12{,}000\). The 48-month total is \(2{,}140+2{,}332+2{,}520+2{,}704+2{,}912=12{,}608\). Therefore:
The row-level ratios are \(1.07\), \(1.06\), \(1.05\), \(1.04\), and \(1.04\), which support the calculated result. Including all five observations retains more available experience at a later age where few years mature far enough to contribute. Common mistake avoided: “use only the latest three years” is not a universal selection rule; a narrower window would discard relevant observations without a stated reason.
Worked Example: Exclude Experience Made Irrelevant by a Known Change
A hypothetical insurer documents that a claims-handling change beginning in 2023 altered the 12-to-24-month paid-loss emergence. The change does not affect the comparison requested here because the task is to select an observation window for that same early link under the current process. Cumulative paid losses, in thousands of dollars, are:
| Accident year | 12 months | 24 months |
|---|---|---|
| 2020 | 1,000 | 1,500 |
| 2021 | 1,200 | 1,740 |
| 2022 | 1,400 | 1,960 |
| 2023 | 1,600 | 2,000 |
| 2024 | 1,800 | 2,160 |
| 2025 | 2,000 | 2,300 |
Method: restrict the window to years under the current process, then calculate the selected volume-weighted factor. Use 2023–2025. The 12-month total is \(1{,}600+1{,}800+2{,}000=5{,}400\); the 24-month total is \(2{,}000+2{,}160+2{,}300=6{,}460\). Hence:
For comparison, including all six years would give a 12-month total of \(9{,}000\) and a 24-month total of \(11{,}660\), or \(11{,}660/9{,}000=1.2956\). This calculation is not the selected factor: it mixes the documented earlier process with the current one. The post-change row ratios are \(1.25\), \(1.20\), and \(1.15\), consistent with the selected weighted result. Common mistake avoided: more observations do not automatically mean a better window when the older observations are known to represent a different process.
Common Traps and Practical Checks
Observation-window choices matter because the selected link ratios are inputs to development factors and projected losses. A window that is too narrow may put too much weight on a small amount of experience; a window that is too broad may combine conditions that should not be treated as comparable. The objective is not to maximize the number of ratios or to prefer recency in every case. It is to select the available experience that best informs the intended estimate.
- Confusing available with relevant. An accident year can have both values needed for a ratio and still be excluded for a documented relevance reason. Conversely, an older year is not automatically unusable.
- Applying one window at every age. Later-age links have fewer eligible accident years, and relevance can vary by development age. State the selected years separately for each link.
- Ignoring the denominators and volume. Counting observations alone does not describe how much cumulative-loss experience supports a volume-weighted factor. Check the selected current-age totals as well.
- Treating a method as a window decision. Arithmetic, geometric, and volume-weighted methods summarize a chosen set in different ways. Choosing an averaging method does not resolve which accident years belong in that set.
For example, a common incorrect calculation is to take all available accident years for every link, even when an early process change makes some years irrelevant. The correction is to first identify the eligible years for that link, then remove observations that are not representative for a stated reason, and only then calculate the factor from the selected set. The converse trap is to discard older years merely because they are older, without evidence that they are less relevant.
Check Your Understanding
For each problem, identify the appropriate eligible or relevant years before using the supplied data to calculate or explain a factor selection.
- A triangle is valued through 48 months. Which age-to-age link is the latest one for which a factor can be calculated, and can a 48-to-60-month factor be calculated from this triangle? (Answer: The latest observable link is 36-to-48 months; no 48-to-60-month factor is observable.)
- A documented change makes accident years 2019 and 2020 irrelevant for a particular link. The selected years have current-age cumulative losses of 900, 1,100, and 1,300, and next-age amounts of 1,170, 1,375, and 1,560. Calculate the volume-weighted factor. (Answer: \(4{,}105/3{,}300=1.2439\).)
- At a later development age, five years are eligible, no process change is known, and the data are not considered anomalous. State one reason to consider retaining all five rather than automatically using only the newest three. (Answer: Retaining all five can provide more relevant observations and volume at a sparse later age; recency alone is not a sufficient reason to discard eligible experience.)
- For an early link, the three post-change years have totals of 4,800 at the current age and 5,760 at the next age. The three pre-change years have totals of 3,600 and 5,040, respectively. If the change makes the earlier years irrelevant, calculate the selected factor and explain why the all-year ratio is not the selection. (Answer: The selected factor is \(5{,}760/4{,}800=1.2000\); the all-year ratio \(10{,}800/8{,}400=1.2857\) mixes pre-change and current-process experience.)
- A candidate selects the same most recent four accident years at 12-to-24 and 24-to-36 months, even though the newest year has no 36-month value. Identify the error and state the correction. (Answer: The newest year is ineligible for the 24-to-36-month link; select only years with both 24- and 36-month values, then assess relevance for that link separately.)