What an Observed Link Ratio Measures
A cumulative loss triangle organizes loss amounts by accident period and development age. For a given accident period, the cumulative amount at a later development age includes the amount recorded at the earlier age, together with subsequent development. An observed age-to-age link ratio compares the cumulative amounts for the same accident period at two adjacent development ages.
For example, a ratio from 12 to 24 months compares the 24-month cumulative value with the 12-month cumulative value for the same accident period. It is not a comparison between two accident periods, and it is not a ratio of incremental amounts.
The notation \(j\) can stand for development ages such as 12, 24, and 36 months. The labels \(j\) and \(j+1\) mean successive columns in the chosen development-age sequence. A ratio from 12 to 24 months must use those two columns, not the 12-month and 36-month columns.
To calculate an observed ratio, first locate the two adjacent cumulative columns, then use the same accident-period row in both columns. A complete ratio is available only when both cells are observed and the earlier cumulative amount is nonzero. If a cell is missing, the ratio is unavailable; a blank is not a zero. If the denominator is zero, division does not define a link ratio.
Calculating Ratios From a Triangle
The table below is a hypothetical cumulative paid-loss triangle, with amounts in thousands of dollars. Each row is an accident period; each development column gives the cumulative amount at the stated age.
| Accident period | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2022 | 120 | 180 | 210 |
| 2023 | 150 | 225 | |
| 2024 | 140 |
The observed 12-to-24-month ratios use rows with both a 12-month and a 24-month value. The observed 24-to-36-month ratios use rows with both a 24-month and a 36-month value. Therefore, the 2024 row contributes no ratio, and the 2023 row contributes no 24-to-36-month ratio.
Worked Example: Finding Each Available Ratio
Use the hypothetical triangle above to calculate every available observed age-to-age ratio. State which cells contribute to each ratio. Round final ratios to three decimal places.
Method. For each transition, divide the later cumulative value by the earlier cumulative value in the same accident-period row.
12 to 24 months, accident period 2022. The contributing cells are 180 at 24 months and 120 at 12 months: \(180/120=1.500\).
12 to 24 months, accident period 2023. The contributing cells are 225 at 24 months and 150 at 12 months: \(225/150=1.500\).
24 to 36 months, accident period 2022. The contributing cells are 210 at 36 months and 180 at 24 months: \(210/180=7/6=1.1666\ldots\), which rounds to \(1.167\).
The 2023 row has no observed 36-month cell, so it supplies no ratio for the 24-to-36-month transition. The 2024 row has no observed 24-month cell, so it supplies no 12-to-24-month ratio. The available ratios are therefore 1.500 and 1.500 for 12 to 24 months, and 1.167 for 24 to 36 months.
Check. Each numerator and denominator comes from adjacent development ages in the same row. The 12-to-24-month numerator is larger than its denominator, as expected for these cumulative values.
Common mistake avoided. Dividing 225 by 120 would combine the 2023 value at 24 months with the 2022 value at 12 months. That is not an age-to-age ratio: the accident periods do not match.
Why the Ratio Has This Form
The calculation expresses the multiplicative change between two observed cumulative amounts. If a multiplier \(f_{i,j}\) takes the earlier cumulative amount to the later one for a particular accident period, then by definition it satisfies \(C_{i,j}f_{i,j}=C_{i,j+1}\). When the earlier amount is nonzero, dividing both sides by that amount gives the observed ratio.
This justification is about how to describe the observed pair of cells. It does not establish that the same multiplier will apply to another accident period or to future experience. A calculated ratio records what happened in that row across that transition; deciding how to use observations for a projection is a separate modeling task.
Trace Each Ratio to Its Contributing Cells
A useful way to prevent alignment errors is to name the transition and accident period before doing arithmetic. For a 12-to-24-month ratio from accident period 2021, for example, the contributing data are specifically \(C_{2021,12}\) and \(C_{2021,24}\). The cell at 36 months cannot contribute to that transition, and no cell from another accident period belongs in that ratio.
The data contributing to a single observed ratio are just two cumulative cells: one in the earlier-age column and one in the next-age column, from the same row. Across a triangle, there can be several observed ratios for a transition, because several accident periods may have both required cells. Rows that have not yet reached the later age do not contribute an observation for that transition.
Worked Example: Auditing Which Rows Are Eligible
A hypothetical incurred-loss triangle, in thousands of dollars, has these observed cumulative values. Calculate all available ratios for the two transitions and identify which rows contribute to each transition. Round final ratios to three decimal places.
| Accident period | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2020 | 100 | 160 | 200 |
| 2021 | 120 | 174 | 209 |
| 2022 | 90 | 135 | |
| 2023 | 110 |
Method. Work down each transition separately, and include a row only when it has both adjacent cumulative values.
For 12 to 24 months, the 2020 observation is \(160/100=1.600\). The 2021 observation is \(174/120=1.450\). The 2022 observation is \(135/90=1.500\). The 2023 row has no 24-month value and contributes no ratio. Thus the contributing accident periods are 2020, 2021, and 2022.
For 24 to 36 months, the 2020 observation is \(200/160=1.250\). The 2021 observation is \(209/174=1.201149\ldots\), which rounds to \(1.201\). The 2022 row has no 36-month value and contributes no ratio. Thus the contributing accident periods are 2020 and 2021.
Check. The 2021 24-to-36-month ratio uses 209 and 174 from the same row; \(174\times1.201149\ldots=209\). No blank cell was treated as a numerical value.
Common mistake avoided. The 2023 row cannot contribute to the 12-to-24-month transition merely because its 12-month amount is known. Both cells of the pair are required.
The eligible rows can differ from one transition to another. In a typical upper-triangular display, fewer accident periods have reached each successively later development age. As a result, the 24-to-36-month transition often has fewer observed ratios than the 12-to-24-month transition. The available cells, rather than an assumed fixed number of observations, determine which ratios can be calculated.
Common Alignment and Calculation Traps
- Mixing accident periods. A ratio is within one row. Do not divide a later-age value from one accident period by an earlier-age value from another.
- Skipping an age column. A ratio from 12 to 36 months is not the adjacent 12-to-24-month ratio. Use the two columns named by the transition.
- Using incremental values. The defined link ratio compares cumulative amounts. A ratio of two incremental amounts is a different calculation and does not give this observed age-to-age ratio.
- Reversing numerator and denominator. The later-age cumulative amount is the numerator. Reversing the division gives the reciprocal, not the link ratio as defined here.
- Treating blanks as zero. A missing later-age cell means the pair is unavailable. It does not imply that no loss developed.
- Over-interpreting one observation. A ratio is a descriptive comparison of two cells. It is not proof that a future period will develop by the same amount or percentage.
Worked Example: Correcting a Misaligned Calculation
In a hypothetical cumulative triangle, accident period A has values 80 at 12 months and 120 at 24 months. Accident period B has values 100 at 12 months and 150 at 24 months. A candidate reports \(150/120=1.250\) as a 12-to-24-month ratio. Identify the error, calculate both valid observed ratios, and check them. Round to three decimal places.
Method. Match the accident period first, then divide each row's 24-month value by its own 12-month value.
For accident period A, the valid ratio is \(120/80=1.500\). For accident period B, the valid ratio is \(150/100=1.500\).
The candidate's calculation \(150/120=1.250\) uses the 24-month value from B and the 12-month value from A. It compares different accident periods and is not an observed ratio for either row.
Check. For A, \(80\times1.500=120\); for B, \(100\times1.500=150\). Each valid ratio reproduces the later cumulative value from its corresponding earlier value.
Common mistake avoided. Selecting cells because they appear near each other in a table is not enough. Confirm both the development-age columns and the accident-period row.
Practical Calculation Checklist
Before recording an observed ratio, check the cells and the arithmetic in a fixed order. This short process is especially helpful when a triangle has missing entries or several development columns.
Write the earlier and later development ages, such as 12 to 24 months.
Confirm that the numerator and denominator both belong to that same row.
If either cell is missing, do not calculate a ratio. If the earlier cumulative amount is zero, the ratio is undefined.
Use the later-age cumulative amount as the numerator and the earlier-age cumulative amount as the denominator.
Verify that multiplying the denominator by the ratio reproduces the numerator, then record the transition and accident period.
This careful labeling matters because a bare number such as 1.25 is ambiguous. Recording “2021, 24 to 36 months: 1.201” makes the contributing cells and the meaning of the observation clear. It also makes later review easier if the triangle is corrected or updated.
Check Your Understanding
For each problem, identify the correct cells and calculate only ratios supported by the stated data.
- A hypothetical accident period has cumulative losses of 64 at 12 months and 96 at 24 months. Calculate its observed 12-to-24-month ratio. (Answer: 1.500)
- A triangle has these 12- and 24-month values: accident period A, 80 and 104; accident period B, 120 and 150. Calculate each row's observed 12-to-24-month ratio. (Answer: A: 1.300; B: 1.250)
- A triangle records accident period P at 90, 117, and 140 for 12, 24, and 36 months; accident period Q at 100 and 130 for 12 and 24 months only. Calculate all available ratios and name the row contributing to the 24-to-36-month transition. (Answer: P, 12-to-24: 1.300; P, 24-to-36: 1.197; Q, 12-to-24: 1.300; only P contributes to 24-to-36 months.)
- Accident period M has cumulative values 75 at 12 months, 120 at 24 months, and 150 at 36 months. Calculate both adjacent-age ratios and state which amount is the numerator for the 24-to-36-month ratio. (Answer: 12-to-24: 1.600; 24-to-36: 1.250; the numerator is 150.)
- For accident period R, the observed cumulative amount is 0 at 12 months and 18 at 24 months. For accident period S, the amounts are 50 and 65 at those ages. Which 12-to-24-month ratio is defined, and what is its value? (Answer: Only S has a defined ratio; \(65/50=1.300\).)