Comparing Paid and Incurred Chain Ladder Reserves

Why Compare Paid and Incurred Chain Ladder Estimates?

Paid and incurred chain ladder methods project ultimate losses from different measures of claim emergence. A paid triangle records amounts paid to date; an incurred triangle records paid amounts plus case reserves. Even for the same accident year, these measures can have different development patterns and selected factors. Their projected ultimates, and the unpaid estimates derived from those ultimates, therefore need not agree.

A useful comparison keeps three quantities distinct: the estimated ultimate loss, the amount already paid, and the amount currently incurred. In particular, a paid-method reserve and an incurred-method reserve are not calculated by subtracting the same observed amount. This distinction is essential when interpreting a difference by accident year. The comparisons in this tutorial use hypothetical data and are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For accident year $i$ and development age $j$, let $C^P_{i,j}$ denote cumulative paid loss and $C^I_{i,j}$ denote cumulative incurred loss. On a consistent basis, $C^I_{i,j}=C^P_{i,j}+K_{i,j}$, where $K_{i,j}$ is the case reserve at that valuation. The paid chain ladder reserve estimate is $R^P_{i,j}=\widehat U^P_i-C^P_{i,j}$; the incurred chain ladder reserve estimate is $R^I_{i,j}=\widehat U^I_i-C^I_{i,j}$. Here $\widehat U^P_i$ and $\widehat U^I_i$ are the respective projected ultimate losses.

Projection and Comparison Framework

A chain ladder projection applies selected development factors to cumulative loss at the accident year's latest observed development age. If $f^P_k$ and $f^I_k$ are the selected paid and incurred age-to-age factors from age $k$ to the next age, their remaining age-to-ultimate factors at age $j$ are:

L^P_j=\prod_{k=j}^{m-1} f^P_k,\qquad L^I_j=\prod_{k=j}^{m-1} f^I_k

The projection at the current age is then:

\widehat U^P_i=C^P_{i,j}L^P_j,\qquad \widehat U^I_i=C^I_{i,j}L^I_j

The reserve estimates follow by subtracting the appropriate cumulative amount:

R^P_{i,j}=\widehat U^P_i-C^P_{i,j},\qquad R^I_{i,j}=\widehat U^I_i-C^I_{i,j}

The product of remaining factors is sometimes called a cumulative development factor or age-to-ultimate factor. A selected tail factor, if used, must also be included in the remaining factor. For a fully developed period with no remaining development, the remaining factor is 1.

The standard chain ladder projection assumes that the selected development pattern is suitable for the accident years being projected: the historical relationship between cumulative amounts at successive ages is informative about future development. Its application also requires usable cumulative values and comparable data and valuation bases. Changes in settlement speed can affect paid patterns; changes in case reserve adequacy can affect incurred patterns. Material calendar-year effects or other changes in the underlying process can also weaken the projection assumption. A factor comparison is not meaningful if, for example, one method is gross of recoveries and the other is net.

What the Difference in Reserves Represents

The two reserve estimates use different current amounts as their starting points. Subtracting the paid reserve from the incurred reserve gives a useful identity:

R^I_{i,j}-R^P_{i,j} = \left(\widehat U^I_i-\widehat U^P_i\right)-K_{i,j}

To see why, substitute the reserve definitions and use $C^I_{i,j}=C^P_{i,j}+K_{i,j}$:

\begin{aligned} R^I_{i,j}-R^P_{i,j} &=(\widehat U^I_i-C^I_{i,j})-(\widehat U^P_i-C^P_{i,j})\\ &=(\widehat U^I_i-\widehat U^P_i)-(C^I_{i,j}-C^P_{i,j})\\ &=(\widehat U^I_i-\widehat U^P_i)-K_{i,j}. \end{aligned}

This identity shows that reserve differences depend on both the difference between projected ultimates and the case reserves currently held. If both methods project the same ultimate, the incurred reserve is lower than the paid reserve by the case reserve amount. If the projected ultimates differ, that difference also matters; neither reserve estimate can be interpreted from the ultimate difference alone.

Worked Example: Comparing Two Ultimate Factors at One Valuation

A hypothetical accident year is valued at 24 months. Cumulative paid loss is $500,000 and cumulative incurred loss is $680,000. The selected paid age-to-ultimate factor is 1.40 and the selected incurred age-to-ultimate factor is 1.20. Calculate both ultimates and reserve estimates, then interpret the comparison. Use the selected-factor chain ladder method.

Step 1: Find the current case reserve. Since incurred equals paid plus case reserves, the case reserve is $680{,}000-500{,}000=\$180{,}000$.

Step 2: Project paid ultimate. Apply the paid factor to cumulative paid loss: $500{,}000\times1.40=\$700{,}000$. The paid reserve estimate is $700{,}000-500{,}000=\$200{,}000$.

Step 3: Project incurred ultimate. Apply the incurred factor to cumulative incurred loss: $680{,}000\times1.20=\$816{,}000$. The incurred reserve estimate is $816{,}000-680{,}000=\$136{,}000$.

Step 4: Check and interpret. The incurred ultimate exceeds the paid ultimate by $\$816{,}000-\$700{,}000=\$116{,}000$. The incurred reserve is lower by $\$136{,}000-\$200{,}000=-\$64{,}000$. The identity gives $116{,}000-180{,}000=-\$64{,}000$, as required. The different selected factors and current amounts together explain why a higher incurred ultimate can coexist with a lower incurred reserve. The common mistake avoided here is comparing the ultimates alone and assuming the reserve comparison must have the same direction.

Reading Differences by Accident Year

A comparison should be made at each accident year's actual valuation age. One year may have substantial paid emergence and relatively little remaining paid development, while a younger year may still have a large paid reserve estimate. Incurred losses may appear more mature because case reserves recognize estimated unpaid amounts, but that does not guarantee that an incurred projection is more reliable or that its remaining factor is smaller. The factor selections and the adequacy of the case reserves matter.

For each accident year, a disciplined comparison records the same basic items: age, cumulative paid, cumulative incurred, case reserves, remaining paid and incurred factors, projected ultimates, and reserves on each basis. This prevents an age mismatch or a basis mismatch from being mistaken for a genuine method difference.

Align the data.
Use the same accident year, valuation date, currency, loss definition, and gross-or-net basis for paid and incurred amounts.

Identify the observed starting amounts.
Record cumulative paid and incurred losses at the latest available age, and calculate case reserves as incurred less paid when the data use the stated convention.

Apply each method's selected remaining factor.
Use paid factors for paid cumulative losses and incurred factors for incurred cumulative losses. Do not substitute one method's factors for the other's.

Calculate both reserves and inspect the components.
Subtract paid from paid ultimate and incurred from incurred ultimate. Compare the ultimate difference and case reserve amount as well as the final reserve estimates.

Worked Example: Comparing Remaining Development Paths

For a hypothetical accident year at 12 months, cumulative paid loss is $260,000 and cumulative incurred loss is $390,000. Selected paid factors are 1.50 from 12 to 24 months and 1.20 from 24 months to ultimate. Selected incurred factors for those periods are 1.20 and 1.10. Calculate the projected ultimates and reserves. The method is sequential chain ladder projection; the common mistake to avoid is applying only the first age-to-age factor.

Step 1: Calculate the current case reserve. It is $390{,}000-260{,}000=\$130{,}000$.

Step 2: Project paid loss through each stage. At 24 months, paid loss projects to $260{,}000\times1.50=\$390{,}000$. Continuing to ultimate gives $390{,}000\times1.20=\$468{,}000$. Equivalently, the remaining paid factor is $1.50\times1.20=1.80$. The paid reserve estimate is $468{,}000-260{,}000=\$208{,}000$.

Step 3: Project incurred loss through each stage. At 24 months, incurred loss projects to $390{,}000\times1.20=\$468{,}000$. Continuing to ultimate gives $468{,}000\times1.10=\$514{,}800$. The remaining incurred factor is $1.20\times1.10=1.32$. The incurred reserve estimate is $514{,}800-390{,}000=\$124{,}800$.

Step 4: Check the reserve difference. The incurred ultimate exceeds the paid ultimate by $514{,}800-468{,}000=\$46{,}800$. The reserve difference is $124{,}800-208{,}000=-\$83{,}200$, which equals $46{,}800-130{,}000=-\$83{,}200$. Applying just the first factor would stop at a projected 24-month amount and omit the remaining development to ultimate.

Comparing Multiple Accident Years

When accident years are at different ages, the remaining factors must reflect those ages. The table below illustrates a hypothetical selection of age-to-ultimate factors. For example, a paid factor of 2.00 at 12 months represents all remaining paid development from 12 months to ultimate; it is not a single 12-to-24-month factor.

Current age	Paid age-to-ultimate factor	Incurred age-to-ultimate factor
12 months	2.00	1.50
24 months	1.50	1.20
36 months	1.20	1.05
Ultimate	1.00	1.00

Worked Example: Comparing Estimates at Different Ages

A hypothetical portfolio has two accident years. Accident Year A is at 24 months, with cumulative paid loss of $360,000 and cumulative incurred loss of $450,000. Accident Year B is at 12 months, with cumulative paid loss of $200,000 and cumulative incurred loss of $310,000. Use the age-to-ultimate factors in the table above. Calculate each accident year's two ultimates and reserves, and state what the comparison shows. The method is age-specific chain ladder projection; the common mistake avoided is applying the 12-month factor to Accident Year A.

Accident Year A at 24 months. Its current case reserve is $450{,}000-360{,}000=\$90{,}000$. The paid ultimate is $360{,}000\times1.50=\$540{,}000$, so the paid reserve is $540{,}000-360{,}000=\$180{,}000$. The incurred ultimate is $450{,}000\times1.20=\$540{,}000$, so the incurred reserve is $540{,}000-450{,}000=\$90{,}000$. The ultimate estimates agree, while the paid reserve exceeds the incurred reserve by exactly the $90,000 case reserve.

Accident Year B at 12 months. Its current case reserve is $310{,}000-200{,}000=\$110{,}000$. The paid ultimate is $200{,}000\times2.00=\$400{,}000$, and the paid reserve is $400{,}000-200{,}000=\$200{,}000$. The incurred ultimate is $310{,}000\times1.50=\$465{,}000$, and the incurred reserve is $465{,}000-310{,}000=\$155{,}000$. The incurred ultimate exceeds the paid ultimate by $65,000; the incurred reserve is lower by $65{,}000-110{,}000=-\$45{,}000$.

Check. For Accident Year A, $540{,}000-540{,}000-90{,}000=-\$90{,}000$ is the reserve difference. For Accident Year B, $65{,}000-110{,}000=-\$45{,}000$. Both results agree with the reserve identity. The comparison is accident-year specific: the two methods produce equal ultimates for A but different ultimates for B, while both years have lower incurred reserve estimates after accounting for their case reserves.

Common Comparison Errors

Subtracting paid from an incurred ultimate. The incurred reserve is incurred ultimate minus cumulative incurred, not minus cumulative paid.
Comparing ultimate differences as if they were reserve differences. The reserve comparison also reflects case reserves already included in incurred loss.
Using the same factor for both methods without justification. Paid and incurred development patterns are selected from different observed quantities and can differ.
Ignoring valuation age. An age-to-ultimate factor for 12 months generally includes more remaining development than the corresponding factor for 24 months.
Over-interpreting a numerical difference. A difference is a signal to inspect data, selected patterns, and assumptions; it does not by itself establish which estimate is preferable.

Key takeaway: Compare paid and incurred chain ladder results in two stages: first compare their projected ultimate losses, then calculate each unpaid estimate using its own observed cumulative amount. The case reserve is the bridge between the two reserve bases, while the selected development patterns determine how each method projects future emergence.

Check Your Understanding

For each hypothetical situation, calculate the requested paid and incurred results using the stated selected factors.

At the current valuation, cumulative paid loss is $240,000 and cumulative incurred loss is $330,000. The paid age-to-ultimate factor is 1.75 and the incurred factor is 1.25. Find both ultimates and reserves. (Answer: Case reserves $90,000; paid ultimate $420,000 and reserve $180,000; incurred ultimate $412,500 and reserve $82,500.)
At 12 months, paid loss is $400,000 and incurred loss is $520,000. The paid factors are 1.40 from 12 to 24 months and 1.15 from 24 months to ultimate; the incurred factors are 1.20 and 1.05. Find both ultimates and reserves. (Answer: Paid ultimate $644,000 and reserve $244,000; incurred ultimate $655,200 and reserve $135,200.)
A fully developed chain ladder projection gives the same ultimate of $600,000 under both methods. Current cumulative paid is $420,000 and cumulative incurred is $510,000. Calculate both reserves and verify the difference. (Answer: Case reserves $90,000; paid reserve $180,000; incurred reserve $90,000; incurred minus paid reserve is $-\$90,000$.)
At 12 months, paid loss is $150,000 and incurred loss is $240,000. The remaining paid and incurred factors are 2.40 and 1.60. Calculate both ultimates and reserves, then compare the reserve difference with case reserves. (Answer: Case reserves $90,000; paid ultimate $360,000 and reserve $210,000; incurred ultimate $384,000 and reserve $144,000; incurred minus paid reserve is $-\$66,000$.)
At 24 months, paid loss is $500,000 and incurred loss is $650,000. Selected paid factors to ultimate are 1.30 from 24 to 36 months and 1.10 thereafter; selected incurred factors are 1.15 and 1.05. Find the remaining factors, ultimate losses, and reserves. (Answer: Remaining factors 1.43 paid and 1.2075 incurred; paid ultimate $715,000 and reserve $215,000; incurred ultimate $784,875 and reserve $134,875.)

What You'll Learn