Why Compare Paid and Incurred Development?
A loss triangle can track amounts already paid or amounts estimated to be owed for claims. These views describe different stages of claim emergence. Payments record cash outflow as it occurs, while incurred amounts combine payments with an estimate of the outstanding obligation. Comparing their development patterns helps an actuary understand what information supports a projection and where uncertainty enters.
The distinction matters in chain ladder because paid and incurred triangles can produce different link ratios and different projected ultimates. Neither pattern is automatically correct in every situation. The choice depends on how quickly claims are paid, how reliable case reserves are, and whether the data have been assembled on a consistent basis.
Cumulative paid amounts generally do not decrease: payments made remain part of cumulative paid losses. Incurred amounts can increase or decrease because case reserves can be strengthened or released as claim information changes. Thus, a change in incurred losses from one age to the next is not necessarily a new payment or a new claim; it includes the change in the estimate of outstanding obligations.
Separate Triangles and Separate Development Factors
A paid triangle and an incurred triangle are built from the same accident periods and development ages when the underlying data allow it, but the cells contain different measures. For either measure, an age-to-age link ratio compares the cumulative amount at the later age with the amount at the earlier age. If the selected factor from age \(j\) to age \(j+1\) is \(f_j\), the chain-ladder projection is \(C_{i,j+1}=C_{i,j}f_j\). Paid factors must be calculated from paid cells, and incurred factors from incurred cells.
A volume-weighted factor is the sum of the observed later-age amounts divided by the sum of the corresponding earlier-age amounts. It gives more weight to accident periods with larger earlier-age amounts. Factors are selected separately for paid and incurred development; using a paid factor on an incurred amount, or vice versa, mixes different measures.
Worked Example: Projecting the Latest Period From Both Triangles
A hypothetical insurer has the following cumulative loss triangles, in thousands of dollars. The last row is the latest accident period, observed only at age 12 months.
| Accident period | Paid at 12 | Paid at 24 | Paid at 36 | Incurred at 12 | Incurred at 24 | Incurred at 36 |
|---|---|---|---|---|---|---|
| 1 | 100 | 160 | 180 | 200 | 220 | 240 |
| 2 | 120 | 180 | — | 240 | 260 | — |
| 3 | 150 | — | — | 300 | — | — |
Method. Calculate volume-weighted age-to-age factors separately for the paid and incurred triangles. Use their products to project the age-12 amounts for accident period 3 to age 36. No tail beyond age 36 is included.
Paid factors. For ages 12 to 24, the observed pairs are \(100\) to \(160\) and \(120\) to \(180\). Therefore, $$ f^P_{12,24}=\frac{160+180}{100+120}=\frac{340}{220}=\frac{17}{11}=1.54545\ldots. $$ For ages 24 to 36, only accident period 1 is observed, so $$ f^P_{24,36}=\frac{180}{160}=\frac{9}{8}=1.125. $$ The cumulative factor from age 12 to age 36 is $$ \frac{17}{11}\times\frac{9}{8}=\frac{153}{88}=1.738636\ldots. $$ The projected paid amount is \(150(153/88)=260.79545\ldots\) thousand dollars. The projected unpaid amount through age 36 is \(260.79545\ldots-150=110.79545\ldots\) thousand dollars.
Incurred factors. For ages 12 to 24, $$ f^I_{12,24}=\frac{220+260}{200+240}=\frac{480}{440}=\frac{12}{11}=1.090909\ldots. $$ For ages 24 to 36, $$ f^I_{24,36}=\frac{240}{220}=\frac{12}{11}=1.090909\ldots. $$ The age-12-to-36 cumulative factor is \((12/11)^2=144/121=1.1900826\ldots\). The projected incurred amount is \(300(144/121)=357.02479\ldots\) thousand dollars. The difference from current incurred is \(357.02479\ldots-300=57.02479\ldots\) thousand dollars.
Check and interpretation. The projected paid amount is $260,795.45 and the projected incurred amount is $357,024.79, rounded to the nearest cent after converting from thousands. Their respective projected increases are $110,795.45 and $57,024.79. These are distinct projections from distinct patterns, not two amounts to add together. They differ because the observed paid and incurred factors differ. The common mistake this method avoids is applying the incurred cumulative factor to the paid amount, or treating the two projections as if they were components of one estimate.
Why Incurred Development Can Differ
At any valuation, incurred losses include both paid losses and the estimated case reserve. Comparing two valuations gives a direct explanation for how the incurred pattern changes. Write \(I_1=P_1+R_1\) at the earlier age and \(I_2=P_2+R_2\) at the later age. Subtracting the first identity from the second gives $$ I_2-I_1=(P_2-P_1)+(R_2-R_1). $$ The change in incurred is therefore the change in paid plus the change in case reserves. If reserves fall, the reserve-change term is negative and incurred development is smaller than paid development. This identity explains why incurred development is not simply a record of payments.
Worked Example: Separating Payments From Reserve Changes
For a hypothetical accident period, at 12 months cumulative paid losses are $125,000 and case outstanding reserves are $50,000. At 24 months cumulative paid losses are $200,000 and case outstanding reserves are $20,000. Find the incurred amount at each age, then compare the paid and incurred changes.
At 12 months. Incurred losses are \(125{,}000+50{,}000=\$175{,}000\).
At 24 months. Incurred losses are \(200{,}000+20{,}000=\$220{,}000\).
Changes from 12 to 24 months. Cumulative paid increases by \(200{,}000-125{,}000=\$75{,}000\). Incurred increases by \(220{,}000-175{,}000=\$45{,}000\). Case reserves change by \(20{,}000-50{,}000=-\$30{,}000\).
Check. The identity gives incurred change \(=\$75{,}000+(-\$30{,}000)=\$45{,}000\), matching the direct calculation. The common mistake is to assume incurred increased by the full $75,000 of payments. The reserve release offsets $30,000 of that increase.
Incurred factors are often lower than paid factors at early ages because case reserves recognize some expected claim cost before it has been paid. This is a tendency, not a rule. Changes in case estimates can cause incurred factors to be higher, lower, or occasionally below 1. A factor below 1 indicates that the later cumulative incurred amount is lower than the earlier amount for the observations used; it can reflect reserve releases rather than negative payments.
Worked Example: Comparing Weighted Development Information
Two hypothetical accident periods have the following cumulative amounts, in thousands of dollars, from age 12 to age 24. For paid losses, period A develops from 100 to 140 and period B from 300 to 360. For incurred losses, period A develops from 180 to 216 and period B from 420 to 462. Calculate the volume-weighted factor for each measure. Then use each factor to project a third period currently at age 12, with paid losses of 200 and incurred losses of 250.
Paid factor. The sum at age 12 is \(100+300=400\), and the sum at age 24 is \(140+360=500\). Thus the volume-weighted paid factor is \(500/400=1.25\).
Incurred factor. The sum at age 12 is \(180+420=600\), and the sum at age 24 is \(216+462=678\). Thus the volume-weighted incurred factor is \(678/600=1.13\).
Project the third period one age step. The paid projection is \(200(1.25)=250\) thousand dollars, an increase of \(250-200=50\) thousand dollars. The incurred projection is \(250(1.13)=282.5\) thousand dollars, an increase of \(282.5-250=32.5\) thousand dollars.
Check and interpretation. The paid and incurred projected amounts are $250,000 and $282,500 at age 24, respectively. The incurred amount is higher in dollars because its starting amount was higher; its percentage factor is lower. A simple unweighted average would give a paid factor of \((1.40+1.20)/2=1.30\) and an incurred factor of \((1.20+1.10)/2=1.15\), not the requested volume-weighted factors. The common mistake is to average accident-period ratios when the method specified is volume weighting.
What Each Pattern Tells You
The two measures emphasize different information. Paid development is grounded in actual payments and is not directly altered by case-reserve revisions. It can, however, emerge slowly when claims take a long time to settle. Incurred development can provide an earlier view of expected claim costs because it includes estimates of unpaid claims, but it is sensitive to the quality and consistency of those estimates.
A sound comparison checks that both triangles cover the same accident periods and valuation dates and use the same treatment of expenses, recoveries, and other loss adjustments. It also asks whether their factors tell a coherent story. A large difference is a reason to investigate the underlying payment and reserve information, not a reason to average the two factors automatically.
For a chain-ladder projection, use the latest observed cumulative amount from the chosen triangle and multiply by the cumulative development factor selected for that same triangle. If the goal is a paid projection, the result is projected cumulative paid losses; the difference from current paid is the unpaid amount implied by that projection. If the goal is an incurred projection, the difference from current incurred is the implied increase in incurred losses. These interpretations are not interchangeable.
Check Your Understanding
For each question, calculate the requested development measure and distinguish paid amounts from incurred estimates.
- A hypothetical period has paid losses of $80,000 at 12 months and $120,000 at 24 months. Incurred losses are $160,000 and $176,000 at those ages. Find the paid and incurred age-to-age factors. (Answer: paid factor \(1.5000\); incurred factor \(1.1000\))
- Paid losses rise from $90,000 to $150,000 between valuations. Case reserves fall from $60,000 to $35,000. Find incurred losses at both valuations and the incurred change. (Answer: incurred amounts $150,000 and $185,000; incurred increase $35,000, consisting of $60,000 paid development offset by a $25,000 reserve release)
- Two periods have paid amounts of $80,000 to $100,000 and $225,000 to $270,000. Their incurred amounts are $150,000 to $180,000 and $400,000 to $440,000. Find the volume-weighted factor for each measure. (Answer: paid factor \(370/305=1.2131\); incurred factor \(620/550=1.1273\))
- A period has current paid losses of $240,000 and current incurred losses of $310,000. Selected one-step factors are 1.15 for paid and 1.08 for incurred. Find both projected next-age amounts and their increases from current amounts. (Answer: paid projection $276,000 with increase $36,000; incurred projection $334,800 with increase $24,800)
- Incurred cumulative amounts at ages 12, 24, and 36 months are $210,000, $240,000, and $255,000. Find the incurred changes from 12 to 24 months and from 24 to 36 months. (Answer: increases of $30,000 and $15,000; the three cumulative amounts are not added together)