Why Compare Paid and Incurred Triangles?
A paid triangle records the cumulative amounts already paid on claims. An incurred triangle also reflects the current estimates of amounts still owed on reported claims. The two triangles therefore describe the same underlying claim experience from different perspectives: cash payments made to date and the combination of payments made plus outstanding case-reserve estimates.
Comparing the triangles is useful because a payment pattern and a case-reserve pattern can differ substantially. Claims may be paid slowly even when their estimated ultimate costs are relatively stable. Conversely, case reserves may be revised as claim information changes, causing incurred amounts to change even when no payment occurs. A meaningful comparison requires the same origin periods, development ages, valuation dates, loss basis, and currency units.
In the examples below, all figures are hypothetical, gross of reinsurance, in thousands of dollars, and undiscounted. No interest rate, mortality assumption, probability distribution, or stochastic loss model is assumed; the examples concern the accounting relationship among observed paid amounts, case reserves, and incurred amounts.
Match the Cells Before Comparing Patterns
A cell in one triangle should be compared with the cell for the same origin period and development age in the other triangle. Comparing different ages can create a false impression: a more mature paid cell may naturally contain more payments than a less mature incurred cell. Development age and valuation date must therefore be identified before interpreting a difference.
At a matched cell, the difference between incurred and paid loss is the case reserve:
This identity explains why incurred loss is usually at least as large as paid loss when the data contain nonnegative gross case reserves. It is a relationship within a valuation cell, not a promise that incurred loss will rise from one development age to the next. Reserve estimates can be reduced, increased, or, where claims are reopened or estimates are revised, otherwise changed. The observed incurred amount can consequently rise more slowly than paid loss or even decline.
Worked Example: Reconciling One Origin Period Across Ages
A hypothetical liability insurer reports the following cumulative amounts for one origin period. All values are in thousands of dollars. Find the case reserves at each age, the changes in paid and incurred amounts between ages, and the interpretation of the pattern.
| Development age | Cumulative paid \(P\) | Cumulative incurred \(I\) | Case reserve \(C=I-P\) |
|---|---|---|---|
| 12 months | 40 | 100 | ? |
| 24 months | 90 | 140 | ? |
| 36 months | 135 | 150 | ? |
Method. Apply the cell-level identity \(C=I-P\), then compare adjacent ages using changes in each quantity.
At 12 months, \(C=100-40=60\). At 24 months, \(C=140-90=50\). At 36 months, \(C=150-135=15\). Each reserve is nonnegative, and in every cell incurred equals paid plus case reserve: \(40+60=100\), \(90+50=140\), and \(135+15=150\).
From 12 to 24 months, paid loss increases by \(90-40=50\), while the case reserve decreases by \(50-60=-10\). Thus incurred increases by \(50-10=40\), matching \(140-100=40\).
From 24 to 36 months, paid loss increases by \(135-90=45\), while the case reserve decreases by \(15-50=-35\). Incurred therefore increases by \(45-35=10\), matching \(150-140=10\).
The paid amounts rise by 50 and then 45, but incurred rises by 40 and then only 10. The declining case reserve accounts for the difference. The result is not evidence that payments were reversed: paid loss is cumulative and increased at both steps in this example.
How Reserve Changes Translate into Incurred Changes
The relationship between successive valuation ages follows directly from the definition. Subtract the incurred amount at the earlier age from the amount at the later age. Because incurred is paid plus case reserve at each age, the change is the sum of the paid change and the reserve change.
This is an accounting identity, not a statistical development assumption. It holds when the paid and incurred amounts use consistent definitions and scope at both valuations. The first difference is the additional paid amount between valuations; the second is the change in the case-reserve estimate. If reserves are strengthened, the second term is positive. If reserves are released, it is negative. Incurred increases only when the total of these two changes is positive.
Worked Example: Comparing Two Origin Periods at Matched Ages
A hypothetical insurer has two origin periods, North and South. The table gives their cumulative amounts at 12 and 24 months, in thousands of dollars. Calculate the case reserves, the 12-to-24-month paid and incurred increases, and compare the observed patterns.
| Origin period | Age | Paid \(P\) | Incurred \(I\) |
|---|---|---|---|
| North | 12 months | 80 | 120 |
| North | 24 months | 150 | 180 |
| South | 12 months | 35 | 100 |
| South | 24 months | 110 | 160 |
Method. For each matched cell, calculate \(C=I-P\). Then compare amounts within each origin period from 12 to 24 months.
For North at 12 months, \(C=120-80=40\); at 24 months, \(C=180-150=30\). Paid increases by \(150-80=70\), while reserves decrease by \(30-40=-10\). Incurred increases by \(70-10=60\), which agrees with \(180-120=60\).
For South at 12 months, \(C=100-35=65\); at 24 months, \(C=160-110=50\). Paid increases by \(110-35=75\), while reserves decrease by \(50-65=-15\). Incurred increases by \(75-15=60\), which agrees with \(160-100=60\).
Both periods show the same incurred increase of 60, even though South has a larger paid increase. Its reserve release is also larger, offsetting the extra 5 of paid development. At 12 months, South has lower paid loss than North but a higher case reserve; the matched incurred amounts are 100 and 120, respectively. These comparisons describe the displayed amounts; by themselves, they do not establish why the periods differ.
What Each Triangle Shows—and What It Does Not
| Feature | Cumulative paid triangle | Cumulative incurred triangle |
|---|---|---|
| Amount accumulated | Payments made through the valuation date | Payments made plus current case reserves, under the stated definition |
| Effect of a payment | Generally increases cumulative paid loss | Increases paid loss, while the related reserve may also change |
| Effect of reserve revision | No direct effect on paid loss | Changes incurred loss even without a payment |
| Interpretation | Reflects the timing of cash payments | Reflects payments and current estimates for reported claims |
The table is a guide to interpretation, not a claim that either triangle is inherently more accurate. Paid amounts are based on recorded transactions, but payment timing can be affected by settlement practices and administrative delays. Incurred amounts include case estimates, which can respond more quickly to new information but are subject to changes in reserving judgments and procedures.
For this reason, different development patterns do not automatically imply that the underlying claim experience changed. A paid triangle may develop slowly because payments take time. An incurred triangle may appear more developed at an early age because reserves are established before claims are fully paid. Later reserve revisions can also change its pattern. A useful comparison describes these mechanisms without treating an observed pattern as a causal explanation.
Worked Example: Separating a Payment Increase from a Reserve Release
At the first valuation of a hypothetical origin period, cumulative paid loss is 210 and cumulative incurred loss is 280, in thousands of dollars. By the next valuation, paid loss has increased by 55 and the case reserve has decreased by 50. Find the paid amount, case reserve, and incurred amount at the next valuation, and check the results.
Method. First find the initial case reserve, then apply the stated changes to paid loss and case reserve separately.
The initial case reserve is \(280-210=70\). The next paid amount is \(210+55=265\). The next case reserve is \(70-50=20\), which is nonnegative. The next incurred amount is \(265+20=285\).
As a second check, the incurred change is the paid increase plus the reserve change: \(55+(-50)=5\). Therefore the new incurred amount should be \(280+5=285\), agreeing with the direct calculation. All three next-valuation amounts are consistent with the stated relationship.
The reserve release reduces incurred development but does not erase the new payments. The reserve remains positive, and incurred remains greater than paid by exactly 20.
Common Mistakes and Checks
A frequent error is to add a cumulative paid amount at the later age to the earlier incurred amount. This double counts payments already included in the earlier paid amount. For example, suppose paid loss is 80 and incurred loss is 120 at the first valuation, and at the next valuation paid loss is 110 and case reserves are 25. The incorrect calculation \(120+110=230\) treats all 110 of later cumulative paid loss as new payment. The correct calculation is \(110+25=135\). Equivalently, the initial reserve is \(120-80=40\), and incurred changes by the paid increase \(110-80=30\) plus the reserve change \(25-40=-15\), for an increase of 15: \(120+15=135\).
Before accepting a comparison, check that the amounts refer to the same origin period and development age, that cumulative paid and incurred values use the same loss scope, and that incurred less paid equals the stated case reserve. Also distinguish a cumulative amount from an incremental amount: a cumulative payment total already includes payments from earlier ages.
- Do not assume incurred must increase at every age. A sufficiently large reserve release can offset new payments.
- Do not call the paid-incurred difference a payment. Under the definition used here, it is the case reserve.
- Do not compare unmatched cells. A difference caused by different ages or valuation dates is not a like-for-like comparison.
- Do not infer claim quality from the triangle alone. The pattern also reflects claim handling, payment timing, and reserve estimation.
Check Your Understanding
For each hypothetical problem, use the same reported-loss definition: incurred equals cumulative paid plus case reserves. Amounts are undiscounted and in thousands of dollars unless stated otherwise.
- A matched cell has cumulative paid loss of 64 and incurred loss of 92. Find the case reserve and state the relationship that checks your result. (Answer: Case reserve \(=92-64=28\); check \(64+28=92\).)
- At one valuation, paid loss is 70 and the case reserve is 50. By the next valuation, paid loss increases by 35 and the reserve falls by 15. Find the next incurred amount and its change from the first valuation. (Answer: Initial incurred \(=120\); next incurred \(=105+35=140\); increase \(=20\).)
- At 12 months, a period has paid loss 30 and incurred loss 90. At 24 months, paid loss is 75 and incurred loss is 120. Find the case reserves at both ages and reconcile the incurred increase. (Answer: Reserves are 60 and 45; paid increases by 45 and reserves decrease by 15, so incurred increases by \(45-15=30\).)
- At the first valuation, cumulative paid loss is 200 and incurred loss is 260. At the next valuation, paid loss increases by 40 and the case reserve increases by 25. Find the next paid amount, reserve, and incurred amount. (Answer: Next paid \(=240\); initial reserve \(=60\), next reserve \(=85\); next incurred \(=240+85=325\).)
- A later cell has cumulative paid loss of 155 and incurred loss of 190. An analyst claims that the 190 includes an additional 190 of new payments after an earlier valuation. What can be determined from this cell, and why is the analyst’s conclusion unsupported? (Answer: The cell implies a case reserve of \(190-155=35\); it does not identify new payments since an earlier valuation, which require comparing paid amounts at both valuations.)