Introduction
An age-to-age factor is calculated from observed cumulative losses at two successive development ages. Before doing the arithmetic, an analyst must decide which historical observations to use. That decision matters: changing the included accident periods can change the factor, even when every individual link ratio is calculated correctly.
The aim is to make the selection explicit and reproducible. State the development ages, the intended observation window, and which observations in that window have both required cumulative amounts. Then calculate the weighted ratio from those selected observations. This tutorial concerns selecting and calculating one age-to-age factor; it does not project ultimate losses.
Define the Selection Before Calculating
A historical observation is eligible for a factor from age \(j\) to age \(j+1\) only when both cumulative amounts, \(C_{i,j}\) and \(C_{i,j+1}\), are available for the same accident period and are measured on a consistent basis. The denominator must also be positive for the individual link ratio to be defined. A missing next-age cell is not an observed amount of zero: it means the pair is not available for that factor.
There are two related decisions. First, specify the intended historical window—for example, a named set of accident periods or the most recent specified number of periods. Second, determine which periods within that window actually have both required ages. The selected set \(I_j\) consists of the periods meeting both conditions. Being in the intended window does not make an incomplete pair usable.
The selection should be based on the question's stated rule, not on whether a period produces a high or low factor. If the problem specifies the periods to include, use them when the required cells are available. If it specifies a window but one period lacks the next-age amount, identify that limitation and calculate from the available eligible pairs. Do not silently replace a missing amount with zero or switch to a different window without saying so.
Write the starting and ending development ages, such as 12 to 24 months. This fixes which two columns are needed.
Use the periods specified in the problem or state the selection rule being applied.
Include a period only if both cumulative values are observed, comparable, and its starting value is positive.
Add the selected next-age cumulative losses and divide by the sum of the selected starting-age cumulative losses.
Report the selected periods, the two totals, and the resulting factor. Check that it lies between the smallest and largest selected individual ratios.
Why the Weighted Formula Has This Form
The weighted factor is the ratio of the aggregate next-age amount to the aggregate starting-age amount for the selected periods. To see its relationship to the individual ratios, write each next-age amount as \(C_{i,j+1}=C_{i,j}L_{i,j}\). Substitution gives
Each individual ratio is therefore weighted by its starting-age cumulative amount as a proportion of the selected total. The weights are nonnegative and sum to one when all selected starting amounts are positive. This also justifies the range check: the weighted factor must be between the smallest and largest individual ratios. If a calculated factor falls outside that range, recheck the selected cells, totals, and arithmetic.
Worked Example: Selecting Three 12-to-24-Month Observations
A hypothetical insurer is reviewing cumulative paid losses in thousands of dollars. For the 12-to-24-month factor, the stated selection window is accident periods A, B, and C. Calculate the volume-weighted factor and check it against the individual ratios.
| Accident period | \(C_{i,12}\) (thousands) | \(C_{i,24}\) (thousands) |
|---|---|---|
| A | 100 | 140 |
| B | 900 | 990 |
| C | 1,000 | 1,300 |
Method and eligible observations. Use the specified A-to-C window. Each period has both the 12-month and 24-month cumulative amount, and each starting amount is positive, so all three periods are eligible.
Individual ratios. For A, \(140/100=1.40\). For B, \(990/900=1.10\). For C, \(1{,}300/1{,}000=1.30\).
Selected totals. The 12-month total is \(100+900+1{,}000=2{,}000\) thousand dollars. The 24-month total is \(140+990+1{,}300=2{,}430\) thousand dollars.
Weighted factor.
The result is between the smallest individual ratio, 1.10, and the largest, 1.40, so it passes the range check. The factor is 1.215, or 1.2150 to four decimal places.
Common mistake avoided. The simple average of the three ratios is \((1.40+1.10+1.30)/3=1.2667\) to four decimal places. That is not the requested volume-weighted ratio: it gives each period equal weight despite the different starting amounts. The selected aggregate ratio is \(1.215\).
Incomplete Windows and Missing Cells
Triangles are unevenly mature: recent accident periods may have a starting-age amount but not yet have a next-age amount. Such a period can fall inside the intended window while remaining ineligible for the particular age-to-age calculation. Record the distinction rather than pretending that the triangle is rectangular.
For example, if a problem requests the 12-to-24-month factor for accident periods 2019 through 2022, but 2022 has no observed 24-month amount at the valuation date, the usable pairs are 2019, 2020, and 2021, assuming their cells are available and otherwise meet the stated rule. The 2022 observation is not a zero and cannot contribute a link ratio. Report both the requested window and the eligible subset used.
Worked Example: Applying a Window When the Latest Pair Is Incomplete
A hypothetical property portfolio has the following cumulative paid losses, in thousands of dollars. The requested factor is from 12 to 24 months, and the intended window is accident periods 2019 through 2022. The 2022 period has not yet reached 24 months.
| Accident period | \(C_{i,12}\) (thousands) | \(C_{i,24}\) (thousands) |
|---|---|---|
| 2019 | 800 | 1,040 |
| 2020 | 1,200 | 1,500 |
| 2021 | 2,000 | 2,400 |
| 2022 | 2,500 | Not observed |
Eligibility. The intended window contains four periods. The 2019, 2020, and 2021 periods each have both required amounts and positive starting values. The 2022 period does not have a 24-month value, so it is not an eligible pair.
Sum eligible starting amounts. \(800+1{,}200+2{,}000=4{,}000\) thousand dollars.
Sum eligible next-age amounts. \(1{,}040+1{,}500+2{,}400=4{,}940\) thousand dollars.
Calculate.
The individual eligible ratios are \(1{,}040/800=1.30\), \(1{,}500/1{,}200=1.25\), and \(2{,}400/2{,}000=1.20\). The selected factor, 1.235, is within their range. It is 1.2350 to four decimal places.
Common mistake avoided. Treating the unobserved 2022 value as zero would give a fabricated ratio of \(0/2{,}500=0\) and would change the denominator as well. Missing means unavailable, not zero. The correct calculation uses the three eligible pairs and reports that 2022 was excluded because its next-age cell is not observed.
Selection Rules and Reporting
A factor is not fully described by a number alone. A reproducible calculation states what it connects and which periods contributed. A concise report might read: “For the 12-to-24-month factor, the intended window was 2019–2022; 2019–2021 had both required cumulative amounts, so the selected volume-weighted factor was \(4{,}940/4{,}000=1.235\).”
If a problem gives an explicit selected set, use that set rather than substituting a different one. If it gives only a selection criterion, apply the criterion consistently to every period. A selection window is not the same as an eligibility test: a period may satisfy the window but fail the cell-availability condition. Conversely, a period outside the specified window should not be added merely because its data are available.
When the data are stated in thousands, keep both sums in thousands; the units cancel in the ratio. Keep enough precision during the calculation to avoid premature rounding. Report the factor to the requested precision, and use the individual-ratio range as an arithmetic check. This check does not decide whether a period should be selected; the selection rule does.
Worked Example: Selecting a Stated Recent Window for 24-to-36 Months
A hypothetical liability portfolio has complete cumulative losses for accident periods 2020, 2021, and 2022 at both 24 and 36 months. The instruction is to use exactly these three periods for the 24-to-36-month volume-weighted factor. Losses are in thousands of dollars.
| Accident period | \(C_{i,24}\) (thousands) | \(C_{i,36}\) (thousands) |
|---|---|---|
| 2020 | 1,600 | 2,000 |
| 2021 | 2,400 | 3,000 |
| 2022 | 4,000 | 5,400 |
Confirm the selected set. The instruction names 2020, 2021, and 2022, and all three have both required amounts and positive 24-month values. Therefore, use all three, without adding any other accident period.
Calculate the individual ratios. For 2020, \(2{,}000/1{,}600=1.25\). For 2021, \(3{,}000/2{,}400=1.25\). For 2022, \(5{,}400/4{,}000=1.35\).
Calculate the totals. The 24-month total is \(1{,}600+2{,}400+4{,}000=8{,}000\) thousand dollars. The 36-month total is \(2{,}000+3{,}000+5{,}400=10{,}400\) thousand dollars.
Calculate and check.
The result is between 1.25 and 1.35, the smallest and largest selected ratios. The factor is 1.30, or 1.3000 to four decimal places.
Common mistake avoided. Adding a fourth period merely because it has available data would contradict the stated selection rule. First honor the specified set, then check eligibility within it; do not let the calculation itself determine the selection.
Common Exam Traps
- Using a period with only one required age. A link ratio needs both cumulative amounts for the same accident period. Do not fill a missing cell with zero.
- Changing the selection window without explanation. Use the stated periods or stated rule. If a period in the window is incomplete, identify it and report the eligible subset.
- Confusing the simple average with the weighted ratio. The simple average gives equal weight to individual ratios. The volume-weighted factor is the ratio of the selected column totals.
- Combining inconsistent cells. The starting and ending amounts must refer to successive development ages and a consistent loss basis, such as the same paid-loss triangle.
- Rounding each individual ratio too soon. For the weighted calculation, sum the selected cumulative amounts directly and divide once; retain precision until the final reported factor.
- Reporting only a factor. State the ages and selected periods so another reader can reproduce the calculation.
Check Your Understanding
For each question, state which periods are eligible before calculating the requested factor.
- A hypothetical triangle gives 12-month cumulative losses of 400, 600, and 1,000 and corresponding 24-month losses of 520, 720, and 1,500, in thousands. All three periods are specified for selection. Calculate the volume-weighted 12-to-24-month factor. (Answer: \(2{,}740/2{,}000=1.3700\))
- For a specified 2019–2022 window, a hypothetical portfolio has 12-month amounts 800, 1,200, 2,000, and 2,500; the observed 24-month amounts are 960, 1,560, and 2,600 for 2019–2021, while 2022 is not yet observed. Calculate the eligible volume-weighted factor. (Answer: \(5{,}120/4{,}000=1.2800\))
- A hypothetical triangle has 24-month amounts 500, 1,500, and 3,000, and corresponding 36-month amounts 650, 1,800, and 3,600. Use exactly these three periods. Calculate the volume-weighted factor and verify that it is within the individual-ratio range. (Answer: \(6{,}050/5{,}000=1.2100\), between 1.20 and 1.30)
- A 12-to-24-month factor is requested for three specified periods. Their 12-month values are 1,000, 2,000, and 3,000; the 24-month values are 1,200, not observed, and 3,900, respectively. Calculate the factor from eligible pairs and state why one period is omitted. (Answer: \((1{,}200+3{,}900)/(1{,}000+3{,}000)=1.2750\); the second period lacks its 24-month value)
- For a stated 24-to-36-month selection, two eligible periods have starting amounts 2,000 and 8,000 and next-age amounts 2,600 and 9,600, in thousands. Calculate the weighted factor and compare it with the simple average of the individual ratios. (Answer: weighted factor \(12{,}200/10{,}000=1.2200\); simple average \((1.30+1.20)/2=1.2500\))