Why Weighting by Earlier-Age Volume Matters
An age-to-age link ratio describes how cumulative losses change between two development ages. When several origin periods are available, each period supplies its own ratio. A volume-weighted selection gives greater influence to periods with more cumulative loss at the earlier age. This is not an arbitrary preference: it makes the selected factor reproduce the ratio of the combined later-age losses to the combined earlier-age losses for the observations included.
The key is to match each later-age amount with its own earlier-age amount. The weight attached to an origin period is based on its earlier-age volume, because that is the denominator used to calculate that period’s link ratio.
For the standard interpretation, the included earlier-age cumulative amounts are positive and all loss values are on the same basis, such as paid losses or incurred losses, with consistent currency and valuation conventions. The formula requires matched observations: each later-age amount in the numerator must correspond to an earlier-age amount in the denominator. If an origin period has no usable later-age observation, it is not part of that matched set.
Deriving the Ratio-of-Totals Formula
Write the unnormalized weight for origin period \(i\) as \(C_{i,j}\). Dividing each weight by the sum of all included earlier-age amounts makes the weights add to one. Substituting the individual link ratio into this weighted average gives the result directly.
The cancellation in the second line is the essential step: for each origin period, multiplying its link ratio by its earlier-age volume recovers its later-age volume. Summing those products therefore gives the total later-age volume. Dividing by the total earlier-age volume produces the ratio of totals. This derivation assumes the earlier-age amounts are positive so the individual ratios and weights are defined; with zero or negative amounts, the usual positive-volume weighting interpretation may not apply.
Worked Examples
Worked Example: Calculating a Weighted Factor From a Small Triangle
A hypothetical liability portfolio has the following cumulative paid losses, in thousands of dollars, at development ages 12 and 24 months. Calculate the volume-weighted 12-to-24-month link ratio and show each origin period’s weight.
| Origin period | 12 months | 24 months |
|---|---|---|
| A | 1,200 | 1,800 |
| B | 3,000 | 3,600 |
| C | 5,800 | 6,380 |
Method. First calculate each individual link ratio. Then weight each ratio by its 12-month cumulative loss divided by the total 12-month cumulative loss.
For A, the link ratio is \(1{,}800/1{,}200=1.5\), and the weight is \(1{,}200/(1{,}200+3{,}000+5{,}800)=0.12\).
For B, the link ratio is \(3{,}600/3{,}000=1.2\), and the weight is \(3{,}000/10{,}000=0.30\). For C, the link ratio is \(6{,}380/5{,}800=1.1\), and the weight is \(5{,}800/10{,}000=0.58\). The weights sum to \(0.12+0.30+0.58=1.00\).
Check using totals: the later-age total is \(1{,}800+3{,}600+6{,}380=11{,}780\), and the earlier-age total is \(10{,}000\). Their ratio is \(11{,}780/10{,}000=1.178\), agreeing with the weighted calculation.
Common mistake avoided: Giving each origin period one-third of the weight would not be volume weighting. The weights are the earlier-age shares, not equal shares merely because there are three observations.
Worked Example: Using Only Matched Observations
A hypothetical commercial property triangle reports cumulative paid losses, in millions of dollars. Two origin periods have complete observations from 18 to 30 months; a third has no 30-month value yet. Calculate the volume-weighted factor using the eligible pairs.
| Origin period | 18 months | 30 months |
|---|---|---|
| 2019 | 2.4 | 3.0 |
| 2020 | 1.0 | 1.5 |
| 2021 | 6.6 | 7.26 |
| 2022 | 4.0 | Not yet observed |
Method. Form the common eligible set first. The 2022 observation cannot supply a link ratio because its later-age value is missing, so include 2019–2021 only.
The eligible earlier-age total is \(2.4+1.0+6.6=10.0\) million. The eligible later-age total is \(3.0+1.5+7.26=11.76\) million. Therefore,
As a check from individual ratios, the ratios are \(3.0/2.4=1.25\), \(1.5/1.0=1.5\), and \(7.26/6.6=1.1\). Their weights are \(2.4/10.0=0.24\), \(1.0/10.0=0.10\), and \(6.6/10.0=0.66\). The weighted sum is \((0.24)(1.25)+(0.10)(1.5)+(0.66)(1.1)=0.30+0.15+0.726=1.176\).
Common mistake avoided: Including the 2022 amount of 4.0 million in the denominator while omitting its unavailable later-age amount would compare different sets of origin periods. The ratio-of-totals identity requires matched pairs.
Worked Example: Translating Selected Ratios Into Totals
A hypothetical specialty insurer has three eligible origin periods with earlier-age cumulative incurred losses of 400, 1,600, and 2,000 thousand dollars. Their observed link ratios are 1.40, 1.15, and 1.05. Calculate the volume-weighted factor, the implied later-age total, and each origin period’s contribution to that total.
Method. Use the earlier-age amounts as the unnormalized weights. Multiplying each amount by its link ratio gives the corresponding later-age amount.
The total earlier-age volume is \(400+1{,}600+2{,}000=4{,}000\) thousand dollars. The normalized weights are \(400/4{,}000=0.10\), \(1{,}600/4{,}000=0.40\), and \(2{,}000/4{,}000=0.50\).
The later-age amounts implied by the individual ratios are \(400(1.40)=560\), \(1{,}600(1.15)=1{,}840\), and \(2{,}000(1.05)=2{,}100\), all in thousands. Their total is \(560+1{,}840+2{,}100=4{,}500\).
The ratio-of-totals check is \(4{,}500/4{,}000=1.125\). Applying the selected factor to the earlier-age total also gives \(4{,}000(1.125)=4{,}500\) thousand dollars. Thus, the aggregate later-age amount is preserved exactly for these observations.
Common mistake avoided: Using the later-age amounts as weights would answer a different weighting question. The derivation shows that the weights that reproduce the ratio of totals are proportional to the earlier-age denominators.
Interpretation and Practical Checks
The weight is not a separate judgment about the credibility or quality of an origin period. It follows mechanically from the volume-weighted formula. If one period has 60% of the total earlier-age cumulative losses, it receives 60% of the weight, even if its individual link ratio is not typical of the other periods. A selected factor can therefore be strongly influenced by a large-volume period.
The ratio-of-totals calculation is useful as both a calculation method and a check. If the weighted average of individual ratios does not equal the ratio of the corresponding totals, check the arithmetic, the weights, and whether the same observations appear in both totals. Also verify that the data are comparable: mixing paid and incurred losses, changing the valuation cutoff inconsistently, or using different currency units can make the result unsuitable even if the arithmetic identity holds.
A frequent exam trap is to confuse “volume weighted” with weighting by the amount that emerges at the later age. The correct weight for an age-to-age ratio is its denominator, \(C_{i,j}\). This is why multiplying the factor by the combined earlier-age total reproduces the combined later-age total. The identity is exact for the included data; it does not by itself establish that the selected factor is appropriate for projecting future development.
Check Your Understanding
For each problem, use the stated matched observations and show how the earlier-age volumes determine the result.
- A hypothetical accident-year triangle has paid losses of 900 and 1,080, and 2,100 and 2,310, at ages 12 and 24 months, respectively. Find the volume-weighted factor. (Answer: 1.130)
- At ages 24 and 36 months, three eligible hypothetical origin periods have cumulative losses, in thousands, of 500 to 650, 1,500 to 1,650, and 3,000 to 3,300. Calculate the volume-weighted factor and state the largest weight. (Answer: Factor 1.120; largest weight 0.600, assigned to the third period.)
- Earlier-age volumes are 200, 800, and 1,000, with individual link ratios 1.50, 1.25, and 1.10. Calculate the normalized weights and the weighted factor. (Answer: Weights 0.100, 0.400, and 0.500; factor 1.200)
- A hypothetical triangle has matched cumulative paid losses, in millions, of 4 to 5 for 2018, 2 to 2.5 for 2019, and 4 to 4.4 for 2020. The 2021 earlier-age value is 3, but its later-age value is unavailable. Calculate the factor using eligible pairs only. (Answer: 1.190)
- Two hypothetical origin periods have earlier-age losses of 500 and 2,000. The first has a link ratio of 1.40. The ratio of the combined later-age total to the combined earlier-age total is 1.16. Find the second link ratio. (Answer: 1.100)