From Incremental Losses to Link Ratios
A loss triangle may be presented as incremental amounts: each cell records loss added during one development interval. An age-to-age link ratio, however, compares the cumulative loss observed at one development age with the cumulative loss observed at the next age. The conversion from increments to cumulative values is therefore essential: dividing one increment by another does not produce the standard link ratio.
This tutorial uses hypothetical loss triangles on a consistent basis. Within a row, the origin period, coverage, loss definition, currency, and valuation convention are held fixed. Development ages are ordered, and each incremental cell represents the amount added between the preceding age and that age. A missing cell is not treated as a zero increment; a ratio can be calculated only when the required adjacent cumulative observations are available.
The cumulative value at an age includes all increments from the first development interval through that age. Thus, the first cumulative value is the first increment, the second is the sum of the first two increments, and so on. If the next incremental amount is known, the next cumulative amount can also be written \(C_{i,j+1}=C_{i,j}+I_{i,j+1}\).
Why the Ratio Uses Cumulative Amounts
A link ratio describes how the observed total for an origin period changes from one development age to the next. The numerator is the total accumulated by the later age; the denominator is the total accumulated by the earlier age. The incremental amount at the later age is only the new addition, not the later-age total.
The relationship between an increment and a link ratio follows directly from the definition. Substitute \(C_{i,j+1}=C_{i,j}+I_{i,j+1}\) into the ratio:
This identity explains the calculation. The ratio is one plus the next increment divided by the cumulative amount already observed. It is not the next increment divided by the prior increment. When the next increment is positive and the current cumulative amount is positive, the ratio exceeds one. If an increment is negative, the cumulative amount can decline and the ratio can be below one, provided the denominator remains positive.
Confirm the origin period, the ages, the units, and what each increment represents. Do not fill an unobserved cell with zero.
Starting at the first observed age, add each new increment to the cumulative amount at the preceding age.
For a transition from age \(j\) to age \(j+1\), divide the cumulative amount at age \(j+1\) by the cumulative amount at age \(j\).
Multiply the earlier cumulative amount by the calculated ratio. It should reproduce the later cumulative amount, apart from any displayed rounding.
Worked Calculations
Worked Example: A Four-Age Incremental Row
A hypothetical insurer has the following incremental paid losses for one origin period, in thousands of dollars: \(16,\ 24,\ 12,\ 8\), at successive development ages. Convert the row to cumulative losses and calculate every observed age-to-age link ratio. Report ratios to four decimal places.
Method and assumptions. Treat the four entries as successive increments for the same origin period and on the same loss basis. The first cumulative value equals the first increment; each following cumulative value is the preceding cumulative value plus the new increment.
Cumulative losses. At the first age, \(C_1=16\). At the second age, \(C_2=16+24=40\). At the third age, \(C_3=40+12=52\). At the fourth age, \(C_4=52+8=60\). The cumulative row is therefore \(16,\ 40,\ 52,\ 60\), in thousands of dollars.
Link ratios. From the first age to the second, \(f_1=40/16=2.5\). From the second to the third, \(f_2=52/40=1.3\). From the third to the fourth, \(f_3=60/52=1.153846\ldots\), which rounds to \(1.1538\).
Check. The earlier cumulative amounts times their respective ratios give \(16(2.5)=40\), \(40(1.3)=52\), and \(52(1.153846\ldots)=60\). Each reproduces the next cumulative value.
Common mistake avoided. Dividing the second increment by the first gives \(24/16=1.5\), but that is not the age-one-to-age-two link ratio. The correct numerator is the age-two cumulative amount, \(16+24=40\), so the link ratio is \(40/16=2.5\).
Worked Example: Calculating Link Ratios in a Partial Triangle
A hypothetical specialty insurer reports these incremental losses in thousands of dollars. Each row is an origin period, and entries are shown only for observed development ages.
Origin period A: \(30,\ 15,\ 15,\ 10\). Origin period B: \(36,\ 18,\ 9\). Origin period C: \(42,\ 21\). Convert each row to cumulative losses and calculate every link ratio for which both adjacent ages are observed. Give ratios to four decimal places.
Method and assumptions. Accumulate within each origin-period row, not down a column. A link ratio requires two cumulative values from the same row at adjacent ages. The unshown later ages are unavailable, not zero.
Origin period A. The cumulative amounts are \(30\), \(30+15=45\), \(45+15=60\), and \(60+10=70\). The ratios are \(45/30=1.5\), \(60/45=1.3333\ldots\), and \(70/60=1.1666\ldots\). To four decimal places these are \(1.5000,\ 1.3333,\ 1.1667\).
Origin period B. The cumulative amounts are \(36\), \(36+18=54\), and \(54+9=63\). The two observed ratios are \(54/36=1.5000\) and \(63/54=1.1666\ldots=1.1667\).
Origin period C. The cumulative amounts are \(42\) and \(42+21=63\), giving one observed ratio, \(63/42=1.5000\). No later-age ratio is available for this row.
Check and common mistake avoided. For each calculated transition, multiplying the earlier cumulative amount by its ratio reproduces the later cumulative amount; for example, \(54(1.1666\ldots)=63\) for period B. Do not use the ratio of a later increment to the prior increment: period A's first such increment ratio is \(15/30=0.5\), whereas its age-one-to-age-two cumulative link ratio is \(45/30=1.5\). Also, do not manufacture a final ratio for period C by treating its missing third-age cell as zero.
Worked Example: A Negative Increment and a Declining Cumulative Amount
A hypothetical paid-loss row, in thousands of dollars, contains increments of \(80,\ 20,\ -8,\ 18\) at successive ages. Calculate the cumulative amounts and link ratios, with ratios rounded to four decimal places. Explain whether the ratio across the negative increment can still be calculated.
Method and assumptions. The negative entry is an observed adjustment in the same hypothetical paid-loss series, not a missing value. Accumulate it with its sign. A conventional link ratio is interpretable here when the earlier cumulative denominator is positive and both adjacent ages are observed.
Cumulative losses. The first cumulative amount is \(80\). The second is \(80+20=100\). The third is \(100+(-8)=92\). The fourth is \(92+18=110\). Thus the cumulative row is \(80,\ 100,\ 92,\ 110\).
Link ratios. The first ratio is \(100/80=1.25\). The second is \(92/100=0.92\). The third is \(110/92=1.195652\ldots\), or \(1.1957\) to four decimal places.
Check. The transitions reproduce their later cumulative values: \(80(1.25)=100\), \(100(0.92)=92\), and \(92(1.195652\ldots)=110\). The ratio of \(0.92\) is calculable because its denominator is positive and both cumulative observations exist; it reflects a decline from \(100\) to \(92\).
Common mistake avoided. Replacing the negative increment with zero would give a third-age cumulative amount of \(100\) and an incorrect second link ratio of \(1.0000\). The observed increment is \(-8\), so it must be included as \(-8\), producing a cumulative amount of \(92\) and a link ratio of \(0.92\).
Conditions, Checks, and Common Traps
The arithmetic is simple, but the calculation is meaningful only when the amounts being compared are compatible. The two cumulative values must refer to the same origin period and consistent definitions of loss, coverage, currency, and valuation. The age intervals must also be read in the correct order. If one cell is missing, the adjacent-age transition cannot be calculated from that row unless the missing value is supplied from a valid source.
A zero cumulative denominator makes the ratio undefined. A negative denominator permits division as arithmetic, but the resulting quotient is not ordinarily interpreted as a conventional development factor without further investigation of the data and its basis. A positive denominator is therefore the standard condition for the link-ratio interpretation used here. Likewise, a link ratio below one is not automatically a calculation error: it can arise when an observed negative increment reduces the cumulative amount.
| Task | Correct quantity | Frequent error |
|---|---|---|
| Convert increments at ages 1 through 3 | \(C_1=I_1,\ C_2=I_1+I_2,\ C_3=I_1+I_2+I_3\) | Use only the increment at each age as if it were cumulative |
| Calculate the age-1-to-age-2 ratio | \(C_2/C_1\) | Calculate \(I_2/I_1\) |
| Use an incomplete row | Calculate only transitions with both adjacent cumulative values observed | Treat a missing age as a zero increment |
| Check a calculated ratio | Verify \(C_{i,j}f_{i,j}=C_{i,j+1}\) | Check against an increment rather than the next cumulative amount |
These checks matter because an incorrect ratio changes the description of how losses have developed between ages. In particular, a ratio constructed from increments can look plausible while representing a different quantity. Keeping the accumulated total in both numerator and denominator makes the transition comparable across development ages.
Check Your Understanding
For each problem, convert the supplied increments where needed and use cumulative amounts for the link ratio.
- A hypothetical row has incremental losses of \(12,\ 18,\ 6\), in thousands of dollars. What is the age-one-to-age-two link ratio? (Answer: \(30/12=2.5000\).)
- A row has increments of \(25,\ 10,\ 15\), in thousands. Give the cumulative amount at the third age and the age-two-to-age-three ratio. (Answer: Cumulative amount \(=50\) thousand; ratio \(=50/35=1.4286\).)
- For a hypothetical origin period, observed increments are \(48,\ 12,\ 0\). Calculate the age-one-to-age-two and age-two-to-age-three link ratios. (Answer: \(60/48=1.2500\) and \(60/60=1.0000\).)
- A partial row contains increments \(70,\ -14,\ 21\), in thousands. Calculate the cumulative values and the age-one-to-age-two ratio. (Answer: Cumulative values \(70,\ 56,\ 77\); ratio \(56/70=0.8000\).)
- A triangle row has cumulative losses of \(90\) at age two and \(117\) at age three. If the age-three incremental amount is \(27\), verify the age-two-to-age-three link ratio and explain why \(27/90\) alone is not the ratio. (Answer: \(117/90=1.3000\); \(27/90=0.3000\) is the added increment divided by the earlier cumulative amount, so the link ratio is \(1+0.3000=1.3000\).)