Why Compare Paid and Incurred Link Ratios?
Paid and incurred triangles describe the same claims from different perspectives. A paid triangle records payments made to date. An incurred triangle combines those payments with estimates of amounts still expected to be paid on open claims. Comparing their link ratios at the same development ages helps explain whether observed development is coming mainly from new payments, changing case reserves, or both.
The comparison is meaningful only when the two triangles use the same accident-year grouping, valuation dates, coverage, loss and expense definitions, currency, and other relevant data conventions. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
For example, a paid link ratio from 12 to 24 months compares cumulative paid amounts at those two ages. Its incurred counterpart compares incurred amounts for the same accident year at the same ages. Matching ages and accident years isolates the comparison: a ratio from one period should not be compared as if it were the same observation as a ratio from a different period.
What Each Ratio Measures
A paid link ratio measures the next age’s cumulative payments relative to the payments already made. An incurred link ratio measures the next age’s cumulative paid-plus-case-reserve estimate relative to the earlier incurred amount. Neither ratio is itself a percentage of claims settled, a reserve adequacy measure, or a direct estimate of ultimate losses.
The difference can be seen by separating the changes in payments and reserves. Write \(\Delta P=P_{i,j+1}-P_{i,j}\) and \(\Delta R=R_{i,j+1}-R_{i,j}\). Then the age-to-age ratios can be written as follows:
The paid ratio reflects the payment increment relative to prior cumulative payments. The incurred ratio reflects the payment increment plus the change in case reserves, relative to prior cumulative incurred losses. Consequently, even when payments increase, incurred losses can increase by less, remain unchanged, or decrease if case reserves fall enough.
Worked Example: Payments Rise While Incurred Development Is Smaller
For a hypothetical accident year, all amounts are in thousands of dollars. At 12 months, cumulative paid losses are 1,200 and case reserves are 800. At 24 months, cumulative paid losses are 1,800 and case reserves are 600. Calculate both link ratios and explain the difference. Assume the loss basis is unchanged between valuations.
Method. Calculate the paid and incurred amounts at both ages, then divide each 24-month amount by its corresponding 12-month amount. This comparison avoids confusing a change in case reserves with a payment.
At 12 months, incurred losses are \(1{,}200+800=2{,}000\). At 24 months, incurred losses are \(1{,}800+600=2{,}400\).
The paid increment is \(1{,}800-1{,}200=600\). The paid link ratio is \(1{,}800/1{,}200=1.500\).
The case reserve change is \(600-800=-200\), so the incurred increment is \(600+(-200)=400\). The incurred link ratio is \(2{,}400/2{,}000=1.200\).
Interpretation and check. Payments increased by 600, but case reserves decreased by 200, leaving an incurred increase of 400. The ratios agree with the increment formulas: \(1+600/1{,}200=1.500\) for paid and \(1+400/2{,}000=1.200\) for incurred. The incurred ratio is lower here because of both the reserve release and its larger starting amount.
Common mistake avoided: treating the paid increase of 600 as the incurred increase. Incurred amounts include case reserves, so the reserve decrease must also be included.
Why the Ratios Can Differ
The formulas provide a direct explanation for the comparison. Because the denominators differ, even identical dollar increments need not produce identical ratios. In addition, the incurred increment includes changes in case reserves, whereas the paid increment does not. There is therefore no general rule that the incurred ratio must be above or below the paid ratio.
More formally, when the starting amounts are positive, the incurred ratio is greater than the paid ratio exactly when:
This condition depends on the relative increments and starting amounts, not simply on whether case reserves rose or fell. A reserve increase can coincide with a lower incurred ratio if the incurred starting amount is sufficiently large compared with the paid starting amount; a reserve decrease does not, by itself, determine the ranking either.
Worked Example: The Relative Size Can Reverse Across Accident Years
A hypothetical insurer reviews two accident years at the same transition, from 12 to 24 months. All amounts are in thousands of dollars, and both years use the same coverage and valuation conventions. Calculate the paid and incurred link ratios for each year, then compare their relative sizes.
Accident year A. Paid losses rise from 400 to 700, while case reserves fall from 1,600 to 1,500. Incurred losses are \(400+1{,}600=2{,}000\) at 12 months and \(700+1{,}500=2{,}200\) at 24 months.
The paid link ratio is \(700/400=1.750\). The incurred link ratio is \(2{,}200/2{,}000=1.100\). The payment increment is 300 and the reserve change is \(-100\), so incurred losses rise by \(300-100=200\).
Accident year B. Paid losses rise from 1,000 to 1,300, while case reserves rise from 1,000 to 1,400. Incurred losses are \(1{,}000+1{,}000=2{,}000\) at 12 months and \(1{,}300+1{,}400=2{,}700\) at 24 months.
The paid link ratio is \(1{,}300/1{,}000=1.300\). The incurred link ratio is \(2{,}700/2{,}000=1.350\). The payment increment is 300 and the reserve change is \(400\), so incurred losses rise by \(300+400=700\).
Interpretation and check. For year A, the paid ratio is greater than the incurred ratio; for year B, the incurred ratio is greater than the paid ratio. The calculations reconcile to the stated incurred changes: \(2{,}200-2{,}000=200\) for A and \(2{,}700-2{,}000=700\) for B. The reversal illustrates why the relative ranking must be calculated rather than assumed.
Common mistake avoided: concluding that an incurred ratio must exceed a paid ratio whenever case reserves increase. The reserve change is only one part of the comparison; both denominators and the size of the payment increment matter.
Interpreting a Decline in Incurred Losses
Cumulative paid losses ordinarily do not decrease when they are measured consistently as payments to date and there are no recoveries, reversals, or data corrections. Incurred losses can decrease because case reserves are estimates that may be revised. An incurred link ratio below 1.000 means the ending incurred amount is below its starting amount; it does not mean that cumulative payments decreased.
Worked Example: A Reserve Release Produces an Incurred Ratio Below One
For a hypothetical accident year, all amounts are in thousands of dollars. At 24 months, cumulative paid losses are 900 and case reserves are 500. At 36 months, cumulative paid losses are 1,000 and case reserves are 300. Calculate the 24-to-36-month paid and incurred link ratios, and interpret the result.
Method. Form each cumulative amount at both ages before calculating the ratio. This prevents a reserve change from being mistaken for a change in cumulative payments.
At 24 months, incurred losses are \(900+500=1{,}400\). At 36 months, they are \(1{,}000+300=1{,}300\).
The paid link ratio is \(1{,}000/900=1.1111\) to four decimal places. The paid increment is \(1{,}000-900=100\).
The case reserve change is \(300-500=-200\). Thus, the incurred increment is \(100-200=-100\), and the incurred link ratio is \(1{,}300/1{,}400=0.9286\) to four decimal places.
Interpretation and check. Paid losses increased by 100, but case reserves decreased by 200, so incurred losses decreased by 100. The direct amounts confirm the ratios: \(1{,}000/900=1.1111\) and \(1{,}300/1{,}400=0.9286\), rounded to four decimals.
Common mistake avoided: treating a ratio below one as an arithmetic error because payments rose. The incurred calculation includes the reserve release, which is larger than the payment increase.
A Practical Comparison Process
A disciplined comparison keeps the arithmetic separate from the interpretation. First verify that the matched observations are genuinely comparable; then calculate each ratio from its own cumulative amounts; finally explain the difference using payment and reserve changes.
Use the same accident year and the same beginning and ending development ages for paid and incurred amounts. Confirm that the coverage, loss basis, and valuation conventions are consistent.
At each age, add cumulative paid losses and case reserves: \(I_{i,j}=P_{i,j}+R_{i,j}\).
Divide the ending paid amount by the beginning paid amount; separately divide the ending incurred amount by the beginning incurred amount.
State the paid increment and case reserve change. Their sum is the incurred increment, which explains whether incurred losses rose or fell.
What the Comparison Can and Cannot Say
A comparison can describe how payment emergence and case reserve changes are reflected in the observed link ratios. For instance, a relatively high paid ratio may reflect substantial new payments compared with prior cumulative payments. An incurred ratio that differs from it may reflect reserve strengthening or release, as well as the different starting denominator.
The comparison alone cannot establish whether case reserves are adequate, whether claims will ultimately develop in a particular direction, or which ratio is appropriate for a future projection. A ratio is an observed relationship between two amounts at specified valuations. Its interpretation should remain tied to those amounts and their definitions.
A frequent exam trap is to compare unmatched ages or inconsistent bases—for example, a paid ratio from 12 to 24 months with an incurred ratio from 24 to 36 months, or paid indemnity with incurred losses that include an additional expense component. Such a comparison mixes differences in development timing or measurement with differences between paid and incurred emergence. Check the data basis first, then compare like with like.
Check Your Understanding
For each hypothetical scenario, calculate the requested ratios from the stated amounts and interpret the relationship. Amounts are in thousands of dollars unless otherwise stated.
- A liability portfolio has paid losses of 240 at 12 months and 360 at 24 months. Case reserves are 760 and 640, respectively. Calculate both 12-to-24-month link ratios. (Answer: Paid \(=1.500\); incurred \(=1.000\).)
- For a fleet accident year, paid losses rise from 800 to 920 between 12 and 24 months, while case reserves rise from 1,200 to 1,280. Calculate both link ratios and state which is larger. (Answer: Paid \(=1.150\); incurred \(=1.100\); the paid ratio is larger.)
- For a hypothetical equipment coverage, paid losses rise from 500 to 650 between 12 and 24 months, while case reserves fall from 1,500 to 1,450. Calculate incurred losses at both ages and both link ratios. (Answer: Incurred amounts \(=2{,}000\) and \(2{,}100\); paid ratio \(=1.300\); incurred ratio \(=1.050\).)
- At 24 months, a hypothetical accident year has paid losses of 1,200 and case reserves of 800. At 36 months, paid losses are 1,500 and case reserves are 500. Calculate both link ratios and explain why they differ. (Answer: Paid \(=1.250\); incurred \(=1.000\); the 300 payment increase is offset by a 300 reserve decrease.)
- For the same 12-to-24-month transition, accident year A has paid losses of 600 then 900 and case reserves of 1,400 then 1,300. Accident year B has paid losses of 1,000 then 1,200 and case reserves of 1,000 then 1,300. Calculate both ratios for each year and identify which year has the larger incurred ratio. (Answer: A: paid \(=1.500\), incurred \(=1.100\); B: paid \(=1.200\), incurred \(=1.250\); year B has the larger incurred ratio.)