Why Volume Matters in a Link Ratio Selection
A link ratio compares cumulative losses at two successive development ages. When selecting one factor from several origin-year observations, a volume-weighted selection gives more influence to origin years with larger earlier-age cumulative losses. The calculation is useful when the selected factor is intended to describe the development of the combined loss volume represented by the eligible observations.
The key is to use corresponding observations: each later-age cumulative loss must be paired with the earlier-age cumulative loss from the same origin year. The selection is not obtained by averaging the individual link ratios equally. Instead, add the eligible later-age cumulative losses and divide by the sum of their matching earlier-age cumulative losses.
This is a deterministic calculation on the selected triangle observations; it does not require an interest-rate, mortality, or probability-distribution assumption. The central modeling judgment is which observations belong in \(W\). Once that set is chosen, the arithmetic is direct. The earlier material on triangle construction and link ratios emphasizes the importance of consistent definitions and correctly paired development ages; the same data discipline is necessary here.
Calculation and Interpretation
For each selected origin year, the earlier-age cumulative loss is a component of the denominator and its matching later-age cumulative loss is a component of the numerator. If the triangle is reported in thousands of dollars, both sums are in thousands and their ratio is dimensionless. The selected factor is commonly reported to three or four significant figures, while keeping full precision in intermediate calculations.
A useful way to justify the formula is to ask what single factor would map the selected observations' combined earlier-age cumulative loss to their combined later-age cumulative loss. If that factor is \(f\), the aggregate relationship is
Provided the denominator is positive, solving this equation for \(f\) gives the volume-weighted formula. Thus, the factor represents a multiplier for the combined earlier-age volume in the selected set. It does not assert that every origin year individually develops at exactly that rate.
Specify the link being selected, such as 12-to-24 months, and list the origin years eligible for that link.
For each eligible origin year, record the earlier-age and later-age cumulative losses from the same row.
Add the earlier-age values separately from the later-age values, using exactly the same origin years in both sums.
Divide the later-age total by the earlier-age total. Confirm the denominator is positive, the units cancel, and the reported precision is appropriate.
Worked Applications
Worked Example: Selecting a 12-to-24-Month Factor
A hypothetical commercial property portfolio has the following cumulative paid losses, stated in thousands of dollars. Select the volume-weighted 12-to-24-month link ratio using all three origin years. This calculation uses no interest, mortality, or distribution assumption.
| Origin year | 12 months | 24 months |
|---|---|---|
| 2022 | 1,200 | 1,800 |
| 2023 | 1,500 | 2,100 |
| 2024 | 900 | 1,260 |
Step 1: Confirm the eligible pairs. All three origin years have cumulative losses at both 12 and 24 months, so \(W=\{2022,2023,2024\}\).
Step 2: Sum the earlier-age losses. \(1{,}200+1{,}500+900=3{,}600\) thousand dollars.
Step 3: Sum the later-age losses. \(1{,}800+2{,}100+1{,}260=5{,}160\) thousand dollars.
Step 4: Divide.
The selected 12-to-24-month factor is 1.433 to three decimal places. As a check, \(3{,}600\times1.433333\ldots=5{,}160\), so the selected factor maps the combined earlier-age total to the combined later-age total. The common mistake avoided here is averaging the three origin-year ratios equally instead of using the specified volume-weighted calculation.
Worked Example: Applying a Recent-Year Observation Window
A hypothetical insurer is selecting a 24-to-36-month factor. Four origin years have complete observations, but the analyst has decided to use only the three most recent eligible origin years. All amounts are in thousands of dollars. Determine the selection. No interest, mortality, or distribution assumption is needed.
| Origin year | 24 months | 36 months | Use? |
|---|---|---|---|
| 2020 | 1,800 | 2,160 | No |
| 2021 | 2,100 | 2,520 | Yes |
| 2022 | 2,400 | 3,000 | Yes |
| 2023 | 2,700 | 3,240 | Yes |
Step 1: Apply the stated window. The 2020 pair is excluded. The eligible set is \(W=\{2021,2022,2023\}\). Excluding a row from both sums is essential: using different origin years in the numerator and denominator would not produce the intended selection.
Step 2: Add the 24-month values in the window. \(2{,}100+2{,}400+2{,}700=7{,}200\) thousand dollars.
Step 3: Add the corresponding 36-month values. \(2{,}520+3{,}000+3{,}240=8{,}760\) thousand dollars.
Step 4: Calculate the factor.
The volume-weighted 24-to-36-month selection is 1.217 to three decimal places. The check is \(7{,}200\times1.216666\ldots=8{,}760\). Including 2020 despite the stated window would answer a different question, even though its data are complete.
Worked Example: A Large Origin Year Has More Influence
A hypothetical specialty insurer is selecting a 12-to-24-month factor from two origin years. The first year has a smaller cumulative loss volume but a larger individual link ratio. Select the volume-weighted factor, then use it to project a separate current origin year's 12-month cumulative loss of $3,200 thousand to 24 months. The projection uses the selected factor only; no interest, mortality, or distribution assumption applies.
| Origin year | 12 months ($000) | 24 months ($000) |
|---|---|---|
| A | 500 | 750 |
| B | 2,500 | 3,000 |
Step 1: Add the 12-month amounts. \(500+2{,}500=3{,}000\) thousand dollars.
Step 2: Add the paired 24-month amounts. \(750+3{,}000=3{,}750\) thousand dollars.
Step 3: Calculate the volume-weighted factor.
The selected factor is 1.250. For comparison, the individual ratios are \(750/500=1.50\) and \(3{,}000/2{,}500=1.20\). Their simple arithmetic average is \((1.50+1.20)/2=1.35\), which is not the volume-weighted selection. The larger-volume origin year has more influence on the selected factor.
Step 4: Apply the selected factor to the separate current origin year. \(3{,}200\times1.25=4{,}000\) thousand dollars, or $4,000,000. The check is that the projected cumulative amount is 25% greater than $3,200,000, consistent with a factor of 1.25. The key trap is to use the arithmetic mean of the individual ratios when the requested method is volume-weighted.
Conditions, Checks, and Common Traps
A volume-weighted selection is only meaningful when its input pairs are comparable. Before calculating, verify that the cumulative values use the same loss definition, coverage, treatment of expenses, valuation basis, currency, and development-age convention. The earlier material on consistent triangle definitions and reconciliation to source data is relevant: a neat ratio cannot correct a mismatch in what the cells measure.
The formula also requires a usable denominator. In the ordinary application, the sum of the selected earlier-age cumulative losses is positive. If it is zero, the ratio is undefined; if values are negative or otherwise atypical, the ordinary interpretation as a multiplier for positive cumulative volume may fail. Those cases require investigation rather than mechanical use of the standard selection.
Keep the observation window explicit. A triangle may contain many origin years, but only rows with observed values at both ages can contribute to that link. If the selection calls for a particular subset, include only that subset in both sums. Do not add a later-age value whose matching earlier-age value is missing, and do not treat a blank cell as zero.
- Wrong calculation: average all individual link ratios equally when asked for a volume-weighted selection. Correction: sum the later-age cumulative losses and divide by the sum of their paired earlier-age cumulative losses.
- Wrong calculation: use one set of origin years in the numerator and another in the denominator. Correction: use exactly the same eligible set \(W\) for both sums.
- Wrong interpretation: assume the selected factor is the exact development rate for every origin year. Correction: it is the factor relating combined later-age and earlier-age totals for the selected observations.
- Wrong unit treatment: report a factor in dollars or thousands of dollars. Correction: the units cancel, so the factor is dimensionless.
Check Your Understanding
For each problem, use the stated window and keep intermediate sums in the units given.
- A hypothetical fleet portfolio has 12-month cumulative losses of $800 thousand and $1,200 thousand, paired with 24-month values of $1,040 thousand and $1,560 thousand. Calculate the volume-weighted 12-to-24-month factor. (Answer: 1.300)
- A hypothetical insurer uses all three listed origin years for a 12-to-24-month selection: 12-month values are $600 thousand, $1,400 thousand, and $2,000 thousand; corresponding 24-month values are $900 thousand, $1,820 thousand, and $2,500 thousand. Calculate the factor to three decimals. (Answer: 1.305)
- For a 24-to-36-month selection, use only the two most recent eligible origin years. The older eligible pair is $2,000 thousand to $2,400 thousand, and the two selected pairs are $3,000 thousand to $3,600 thousand and $1,000 thousand to $1,400 thousand. Calculate the selected factor. (Answer: 1.250)
- A triangle has 12-month values of $1,000 thousand and $3,000 thousand with corresponding 24-month values of $1,300 thousand and $3,600 thousand. A third origin year has a 24-month value of $900 thousand but no observed 12-month value in the selected data extract. Using only complete pairs, calculate the factor and explain whether the unpaired value belongs in either sum. (Answer: 1.225; exclude the unpaired value from both sums.)
- For a hypothetical book, the selected 24-to-36-month pairs, in thousands of dollars, are $500 to $650, $1,500 to $1,800, and $4,000 to $4,400. Calculate the factor to four decimals, then project a separate $2,400 thousand 24-month cumulative loss to 36 months. (Answer: factor 1.1417; projected cumulative loss $2,740,000.)