Why Calculate Age-to-Age Link Ratios?
A loss development triangle records cumulative amounts for origin periods at successive development ages. Comparing two adjacent ages for the same origin period gives an observed age-to-age link ratio. The ratio expresses the later cumulative amount relative to the earlier one, so it is a compact description of the change between those ages.
The calculation is simple division, but the setup matters: the two amounts must belong to the same origin period and comparable loss definitions, and the earlier-age amount must not be zero. Interpretation also requires care. A ratio above one indicates numerical growth only when the starting amount is positive; with a negative starting amount, the signs and actual values must be examined.
This is an arithmetic definition, not a probability model: it does not require a particular distribution for claim amounts. To make the resulting ratio meaningful as a comparison, however, the two cells should use consistent coverage, loss and expense bases, valuation rules, currency, and cumulative measurement. A ratio cannot repair incompatible data.
A triangle may have blank cells because a later development age has not yet been observed for an origin period. A missing amount is not a zero amount. If either cell needed for the comparison is missing, the link ratio cannot be calculated from that pair. If the earlier amount is recorded as zero, division is undefined, so the ratio is also unavailable. These distinctions are important when reading a triangle built from claim records.
Calculate a Ratio From the Correct Pair of Cells
For each calculation, first identify the origin period, then the two adjacent development ages, then the earlier-age denominator and later-age numerator. Do not divide amounts from different origin periods or skip an age unless the requested comparison explicitly defines a different interval.
Worked Example: Calculating Links Across a Small Triangle
A hypothetical insurer has assembled cumulative paid losses, in thousands of dollars, for three accident years. Calculate every available adjacent-age link ratio. State which accident year has no ratio yet.
| Accident year | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2022 | 120 | 180 | 216 |
| 2023 | 150 | 210 | — |
| 2024 | 180 | — | — |
Method. Divide each later cumulative amount by the amount at the immediately preceding development age for the same accident year.
For 2022, the 12-to-24-month ratio is \(180/120=1.50\). The 24-to-36-month ratio is \(216/180=1.20\). Both pairs are present and their earlier-age denominators are nonzero.
For 2023, the 12-to-24-month ratio is \(210/150=1.40\). There is no 36-month amount, so no 24-to-36-month ratio can be calculated.
For 2024, only the 12-month amount is present. There is no later-age cell, so no adjacent-age ratio is available.
Check. Multiplying each earlier amount by its ratio recovers the later amount: \(120(1.50)=180\), \(180(1.20)=216\), and \(150(1.40)=210\). The common mistake avoided here is using a value from another accident year to fill a missing cell.
What the Ratio Says—and What It Does Not
The ratio has a precise multiplicative meaning:
When the earlier amount is positive, the ratio can also be translated into a percentage change. The change in amount is \(C_{i,j+1}-C_{i,j}\); dividing that change by the earlier amount gives
Thus, for a positive starting amount, \(f_{i,j}=1.20\) means the later amount is 20% greater than the earlier amount. A ratio of \(0.80\) means it is 20% lower. This percentage interpretation follows from the equation and assumes the denominator is positive.
If the starting amount is negative, the ratio still satisfies the multiplication identity, but comparison with one alone does not tell whether the amount increased or decreased numerically. For example, moving from \(-10\) to \(-5\) gives a ratio of \(0.50\), even though the amount increased numerically. Moving from \(-10\) to \(-20\) gives a ratio of \(2.00\), even though the amount decreased numerically. If the earlier amount is zero, the ratio is undefined rather than zero or infinite.
These qualifications matter particularly for incurred amounts, which can decline when case reserves are reduced or claims close. Paid cumulative amounts can also be affected by recoveries, corrections, or other negative transactions. A ratio describes the two observed cell values; by itself it does not explain why they changed or establish that the same change will occur for another origin period.
Worked Example: Interpreting Ratios With Negative Incurred Amounts
In a hypothetical runoff review, two origin periods have signed cumulative incurred values, in thousands of dollars. For the first period, the amount changes from \(-40\) at 12 months to \(-20\) at 24 months. For the second, it changes from \(-30\) to \(-45\). Calculate each link ratio and describe the numerical change without relying on the ratio’s position relative to one.
Method. Use later amount divided by earlier amount, retaining both signs.
For the first period,
The multiplication check is \((-40)(0.50)=-20\). Numerically, the value increased from \(-40\) to \(-20\), because \(-20\) is greater than \(-40\). Therefore, saying “the factor is below one, so the amount decreased” would be wrong.
For the second period,
The check is \((-30)(1.50)=-45\). Numerically, the value decreased from \(-30\) to \(-45\), because \(-45\) is less than \(-30\). Here a factor above one accompanies a numerical decrease.
Check and interpretation. Both ratios are positive because each pair consists of two negative values. In each case the ratio reproduces the later value exactly, but the direction of numerical movement must be read from the signed amounts. The common mistake avoided is treating “above one” or “below one” as a universal direction rule.
Calculation Checks and Common Traps
A reliable calculation is more than a correctly executed division. Check that the comparison is defined, the ages are adjacent as requested, and the two cells have the same origin period and measurement basis. Then verify the result by multiplying the denominator by the computed ratio.
Identify one origin period and the earlier and later development ages being compared. Confirm that both cells are observed, not blank.
If the earlier-age value is zero, the ratio is undefined. If it is negative, retain its sign and plan to interpret the actual amounts.
Use later-age cumulative losses as the numerator and earlier-age cumulative losses as the denominator.
Check that the denominator multiplied by the ratio gives the numerator. Use “increase” or “decrease” from comparison with one only when the starting amount is positive.
Worked Example: A Zero Denominator and an Adjacent-Age Check
A hypothetical cumulative paid-loss record for one origin period is \(0\) at 6 months, \(24\) at 12 months, and \(30\) at 18 months, in thousands of dollars. Determine which adjacent-age ratios can be calculated, and calculate and interpret the available ratio.
Method. Check each earlier-age denominator before dividing. The 6-to-12-month ratio would be \(24/0\), which is undefined; it is not a zero ratio. The 12-to-18-month ratio is defined because its denominator is \(24\), not zero.
The multiplication check is \(24(1.25)=30\). Because the starting amount is positive, the later amount is 25% greater: \((30-24)/24=6/24=0.25\). The common mistake avoided is either dividing by the zero 6-month value or treating a missing or undefined ratio as a numeric factor.
The ratio is useful because it standardizes a pair of cumulative values into a multiplier that can be compared across observed origin periods. At this stage, each ratio remains an individual observation. Calculating a ratio does not itself decide which observations should be combined, nor does it determine a future amount. Those tasks require additional methods and assumptions.
Check Your Understanding
For each problem, show the division order and check the result by multiplication.
- A hypothetical origin period has cumulative losses of $64,000 at 12 months and $80,000 at 24 months. Calculate the link ratio and the percentage change. (Answer: \(80{,}000/64{,}000=1.25\); increase of 25%.)
- For one origin period, cumulative incurred values are \(-$18,000\) at 12 months and \(-$9,000\) at 24 months. Calculate the ratio and state the numerical direction of change. (Answer: ratio \(0.50\); numerical increase from \(-$18,000\) to \(-$9,000\).)
- A triangle has values of $0 at 6 months, $15,000 at 12 months, and $18,000 at 18 months. Which adjacent-age ratio is undefined, and what is the other ratio? (Answer: 6-to-12 months is undefined; 12-to-18 months is \(18{,}000/15{,}000=1.20\).)
- For a single origin period, cumulative amounts are $42,000 at 12 months, $50,400 at 24 months, and $55,440 at 36 months. Calculate both adjacent-age ratios and verify each. (Answer: 12-to-24 months \(=1.20\), checked by \(42{,}000(1.20)=50{,}400\); 24-to-36 months \(=1.10\), checked by \(50{,}400(1.10)=55{,}440\).)
- For an origin period, a signed cumulative amount changes from \(-$25,000\) to \(-$35,000\) at the next age. Calculate the ratio and explain why comparing it with one alone is misleading. (Answer: ratio \(1.40\); the amount decreased numerically from \(-$25,000\) to \(-$35,000\), despite the ratio being above one.)