Paid Chain Ladder by Accident Year

What Paid Chain Ladder Estimates

A paid loss triangle organizes cumulative claim payments by accident year and development age. An accident year with more elapsed time generally has more paid development observed than a recent accident year. Paid chain ladder uses the latest observed cumulative paid amount for each accident year and selected development factors to project that amount to ultimate.

The result is an estimated ultimate paid loss for each accident year. Subtracting paid losses to date from that estimate gives the estimated unpaid claim liability on the same basis. The method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $C_{a,j}$ denote cumulative paid losses for accident year $a$, evaluated at development age $j$. A paid development factor $f_j$ is the selected multiplicative factor from development age $j$ to the next age. A tail factor $t$ extends the last modeled development age to ultimate. The latest observed cumulative paid amount for an accident year is its current paid amount for this projection.

For example, if $C_{a,24}$ is the latest observed amount for an accident year, the factor $f_{24}$ projects it from 24 to 36 months, not from 24 months directly to ultimate unless the selection explicitly defines it that way. A 36-to-48-month factor and any applicable tail factor must also be included.

Projection Formula and Its Justification

Suppose an accident year is currently evaluated at age $k$, and selected factors carry development through age $m$, followed by a tail factor to ultimate. Under the chain ladder assumption that the selected age-to-age factors represent the future development pattern for this accident year, the projection is:

\widehat{U}_a = C_{a,k} \left(\prod_{j=k}^{m} f_j\right)t

Here, $\widehat{U}_a$ is the estimated ultimate paid loss for accident year $a$. If the selected last age is already ultimate, no additional tail factor is applied; equivalently, the tail factor is 1. The estimated unpaid amount is:

\widehat{R}_a=\widehat{U}_a-C_{a,k}

This subtraction is appropriate because $\widehat{U}_a$ is the estimated total paid loss over the full claim life, while $C_{a,k}$ is the portion already paid. The difference is the amount expected to be paid after the valuation date, on the same loss basis.

The multiplication in the projection formula follows from the definition of the factors. If the factors are represented by successive cumulative paid ratios, then $f_k$ approximates $C_{a,k+1}/C_{a,k}$, $f_{k+1}$ approximates $C_{a,k+2}/C_{a,k+1}$, and so forth. Multiplying successive ratios cancels the intermediate cumulative amounts:

\frac{C_{a,k+1}}{C_{a,k}} \cdot \frac{C_{a,k+2}}{C_{a,k+1}} \cdots \frac{C_{a,m+1}}{C_{a,m}} = \frac{C_{a,m+1}}{C_{a,k}}

Thus the product of age-to-age factors is a cumulative multiplier from the current age to the last modeled age. Multiplying by the tail factor extends that multiplier to ultimate. In practice, the factors are selected estimates, not known ratios for each future accident year. The calculation applies those selections consistently to the current paid amount.

A Practical Accident-Year Procedure

A triangle’s diagonal contains the latest observations at a valuation date: the oldest accident years are usually observed at later development ages, while recent years are observed at earlier ages. Project each accident year from its own latest observed cumulative paid amount. Do not project every cell in its row and add the results; those cells are successive cumulative views of overlapping payments.

Find the latest observed cell.
For each accident year, identify its most recent cumulative paid amount and its development age. Use that amount as the projection starting point.

List the remaining factors.
Starting at the current development age, include each selected factor needed to reach the last modeled age, then include the tail factor if one has been selected.

Calculate ultimate and unpaid.
Multiply current cumulative paid by the applicable cumulative multiplier. Subtract current cumulative paid from the resulting ultimate estimate.

Aggregate and reconcile.
Sum accident-year ultimates and sum accident-year unpaid estimates. Check that aggregate unpaid equals aggregate ultimate minus aggregate current paid.

The simple procedure assumes that the selected paid development pattern is suitable for the accident years being projected. In particular, historical development must be sufficiently informative about future development, and changes in payment speed, coverage, claim handling, or the underlying business should not make the selected factors inappropriate without adjustment. A factor pattern should not be applied mechanically when those conditions are materially violated.

Worked Example: Projecting Three Accident Years at Different Ages

A hypothetical insurer has the following latest cumulative paid amounts: accident year 2023 is at 12 months with $420,000 paid; accident year 2022 is at 24 months with $690,000 paid; and accident year 2021 is at 36 months with $920,000 paid. Selected factors are $f_{12}=1.50$, $f_{24}=1.20$, and $f_{36}=1.08$, followed by a tail factor of 1.03 from 48 months to ultimate. Find each ultimate and unpaid estimate, as well as the aggregate unpaid estimate.

Accident year 2023. This year is at 12 months, so it needs all three age-to-age factors and the tail factor. Its cumulative multiplier is:

$$ 1.50(1.20)(1.08)(1.03)=2.00232 $$

Its ultimate estimate is $420{,}000(2.00232)=\$840{,}974.40$. Its unpaid estimate is $\$840{,}974.40-\$420{,}000=\$420{,}974.40$.

Accident year 2022. This year is already at 24 months, so it does not use $f_{12}$. Its multiplier is:

$$ 1.20(1.08)(1.03)=1.33488 $$

Its ultimate estimate is $690{,}000(1.33488)=\$921{,}067.20$. Its unpaid estimate is $\$921{,}067.20-\$690{,}000=\$231{,}067.20$.

Accident year 2021. At 36 months, this year needs $f_{36}$ and the tail factor. Its multiplier is $1.08(1.03)=1.1124$. Its ultimate estimate is $920{,}000(1.1124)=\$1{,}023{,}408.00$, and its unpaid estimate is $\$1{,}023{,}408.00-\$920{,}000=\$103{,}408.00$.

Aggregate check. Current paid is $\$420{,}000+\$690{,}000+\$920{,}000=\$2{,}030{,}000$. Ultimate is $\$840{,}974.40+\$921{,}067.20+\$1{,}023{,}408.00=\$2{,}785{,}449.60$. Therefore aggregate unpaid is $\$2{,}785{,}449.60-\$2{,}030{,}000=\$755{,}449.60$, equal to the sum of the three accident-year unpaid estimates.

Reading a Cumulative Paid Triangle Correctly

The entries in a cumulative paid triangle are not separate payment amounts to be added across a row. For an accident year, the 12-month cumulative amount is included in the 24-month cumulative amount, which is in turn included in later cumulative amounts, subject to the data’s accounting basis. The latest cumulative amount is the starting paid balance for the chain ladder projection.

A useful check is to compare the latest paid amounts with the triangle’s development ages before applying factors. For a year observed at 24 months, the 12-to-24-month factor has already been reflected in its observed 24-month cumulative amount. Applying that factor again would count the same development twice. Conversely, omitting a factor that remains between the current age and ultimate understates projected development.

Worked Example: Using the Latest Diagonal Amounts in a Triangle

A hypothetical paid triangle, in thousands of dollars, has these cumulative observations: accident year A has 250 at 12 months, 430 at 24 months, and 510 at 36 months; accident year B has 320 at 12 months and 540 at 24 months; accident year C has 400 at 12 months. Selected factors are $f_{12}=1.60$, $f_{24}=1.25$, and $f_{36}=1.10$, with a tail factor of 1.04 after 48 months. Estimate the aggregate ultimate and unpaid amounts.

The latest observed amounts are 510 for A at 36 months, 540 for B at 24 months, and 400 for C at 12 months. These latest cells—not the sum of all cells in each row—are the current paid amounts.

Accident year A. From 36 months, the remaining multiplier is $1.10(1.04)=1.144$. Ultimate is $510(1.144)=583.44$ thousand, and unpaid is $583.44-510=73.44$ thousand.

Accident year B. From 24 months, the remaining multiplier is $1.25(1.10)(1.04)=1.43$. Ultimate is $540(1.43)=772.20$ thousand, and unpaid is $772.20-540=232.20$ thousand.

Accident year C. From 12 months, the remaining multiplier is $1.60(1.25)(1.10)(1.04)=2.288$. Ultimate is $400(2.288)=915.20$ thousand, and unpaid is $915.20-400=515.20$ thousand.

Aggregate current paid is $510+540+400=1{,}450$ thousand. Aggregate ultimate is $583.44+772.20+915.20=2{,}270.84$ thousand. Aggregate unpaid is $2{,}270.84-1{,}450=820.84$ thousand, which also equals $73.44+232.20+515.20$ thousand.

Aggregation, Interpretation, and Common Traps

For multiple accident years, aggregate estimates are formed by summing the accident-year estimates. If each year’s current paid amount is $C_a$, then:

\widehat{U}_{\text{total}}=\sum_a \widehat{U}_a, \qquad \widehat{R}_{\text{total}}=\sum_a(\widehat{U}_a-C_a) =\widehat{U}_{\text{total}}-\sum_a C_a

The identity provides a useful arithmetic check. It does not imply that every accident year has the same multiplier: the multiplier depends on the development age of that year. The estimate is an actuarial projection based on selected factors and available paid data; it is not a guarantee that future payments will equal the estimate.

A common error is to use the factor for an age the accident year has already reached, or to omit a factor still needed to reach ultimate. Another is to subtract the wrong paid amount from ultimate, especially when several cumulative cells are shown for a year. Always pair each ultimate with the same latest cumulative paid amount used as its starting point.

A separate arithmetic trap is to calculate the product correctly but then miscompute the subtraction or aggregate. For instance, with $700,000 paid at 36 months, $f_{36}=1.10$, and tail factor 1.02, the multiplier is $1.10(1.02)=1.122$. Ultimate is $\$700{,}000(1.122)=\$785{,}400$, so unpaid is $\$85{,}400$, not $84,140. If another year has $500,000 paid at 24 months with $f_{24}=1.25$, $f_{36}=1.10$, and tail 1.02, its multiplier is 1.4025, ultimate is $701,250, and unpaid is $201,250. Aggregate unpaid is therefore $\$201{,}250+\$85{,}400=\$286{,}650$.

The same discipline matters in larger aggregations. For example, $800,000 at 12 months with factors 1.40, 1.25, and a 1.04 tail projects to $1,456,000. Separately, $1,050,000 at 24 months with the 1.25 factor and 1.04 tail projects to $1,365,000. Aggregate current paid is $1,850,000; aggregate ultimate is $2,821,000; and aggregate unpaid is $971,000. Each year must be projected with only the factors still applicable at its current age before the totals are combined.

Key takeaway: For each accident year, start with its latest cumulative paid amount, multiply only the age-to-age factors remaining from its current age plus any applicable tail factor, and subtract that same current paid amount from ultimate. Then verify the aggregate unpaid amount by subtracting aggregate current paid from aggregate ultimate.

Worked Example: Aggregating Two Accident-Year Projections

A hypothetical specialty insurer has paid $800,000 for one accident year at 12 months and $1,050,000 for another at 24 months. The selected factors are $f_{12}=1.40$, $f_{24}=1.25$, and a tail factor of 1.04 after the next modeled age. Calculate aggregate current paid, aggregate ultimate, and aggregate unpaid.

12-month accident year. This year requires $f_{12}$, $f_{24}$, and the tail. The multiplier is $1.40(1.25)(1.04)=1.82$. Its ultimate is $800{,}000(1.82)=\$1{,}456{,}000$, and its unpaid estimate is $\$1{,}456{,}000-\$800{,}000=\$656{,}000$.

24-month accident year. This year has already reached the age for $f_{12}$, so it uses only $f_{24}$ and the tail. The multiplier is $1.25(1.04)=1.30$. Its ultimate is $1{,}050{,}000(1.30)=\$1{,}365{,}000$, and its unpaid estimate is $\$1{,}365{,}000-\$1{,}050{,}000=\$315{,}000$.

Aggregate current paid is $\$800{,}000+\$1{,}050{,}000=\$1{,}850{,}000$. Aggregate ultimate is $\$1{,}456{,}000+\$1{,}365{,}000=\$2{,}821{,}000$. Aggregate unpaid is $\$2{,}821{,}000-\$1{,}850{,}000=\$971{,}000$, which agrees with $\$656{,}000+\$315{,}000$. The method avoids the mistake of either applying the 12-month factor to the 24-month year or summing mismatched accident-year totals.

Check Your Understanding

For each problem, identify the applicable remaining factors before calculating. Answers are stated in dollars unless the problem specifies otherwise.

An accident year has $480,000 cumulative paid at 24 months. The selected factors are $f_{24}=1.22$, $f_{36}=1.08$, and a tail factor of 1.03 after 48 months. Find ultimate and unpaid. (Answer: ultimate $651,421.44; unpaid $171,421.44)
An accident year has $600,000 cumulative paid at 36 months. Use $f_{36}=1.07$ and a tail factor of 1.02. Find its ultimate and unpaid estimates. (Answer: ultimate $654,840; unpaid $54,840)
One year has $300,000 paid at 12 months and another has $500,000 paid at 24 months. Use $f_{12}=1.40$, $f_{24}=1.20$, $f_{36}=1.10$, and a tail factor of 1.05 after 48 months. Find aggregate current paid, ultimate, and unpaid. (Answer: current paid $800,000; ultimate $1,275,120; unpaid $475,120)
A triangle’s latest cells are $360,000 at 12 months for one year and $520,000 at 24 months for another. Apply $f_{12}=1.50$, $f_{24}=1.20$, $f_{36}=1.10$, and a 1.04 tail after 48 months. Find the two ultimate estimates and aggregate unpaid. (Answer: ultimates $741,312 and $713,856; aggregate unpaid $575,168)
A year has cumulative paid of $250,000 at 12 months and $430,000 at 24 months. If the year is currently valued at 24 months, explain which paid amount is the projection starting point and why adding both cells would be incorrect. (Answer: use $430,000; the 12-month cumulative amount is already included in the 24-month cumulative amount, so adding them double counts prior paid loss.)

What You'll Learn