Incurred Chain Ladder by Accident Year

Why Incurred Chain Ladder?

A loss triangle shows how cumulative claim amounts develop as claims mature. In the incurred chain ladder, the amounts being developed are cumulative incurred losses: paid losses plus case estimates for amounts expected to be paid on reported claims. Applying selected development factors to an accident year's observed incurred amount projects that year's ultimate loss.

The projected ultimate is not itself the unpaid claim liability. To estimate the liability remaining at the valuation date, subtract paid losses from ultimate losses. Subtracting incurred losses instead gives the difference between ultimate and incurred, not the total unpaid liability. This distinction matters because incurred losses already include case estimates.

Definition: Let $C_{i,j}$ be cumulative incurred losses for accident year $i$ at development age $j$. An incurred age-to-age development factor $f_j$ estimates the ratio of cumulative incurred losses at the next age to those at age $j$. If accident year $i$ is observed at age $k$, and selected factors from age $k$ onward are $f_k,\ldots,f_{m-1}$, its chain ladder ultimate estimate is $$ \widehat{U}_i=C_{i,k}\prod_{j=k}^{m-1}f_j. $$ If $P_i$ is cumulative paid loss for that accident year at the valuation date, the estimated unpaid claim liability is $$ \widehat{R}_i=\widehat{U}_i-P_i. $$

The factors used in a projection are assumed to be appropriate for the relevant development ages and the accident years being projected. They may be selected from historical experience or otherwise supplied for the analysis. This tutorial takes the factors as given; it does not select them. A factor extending from the last observed age to ultimate may incorporate a previously selected tail factor.

The chain ladder is a projection method, not a guarantee that actual future development will equal the estimate. It is most useful when historical development patterns are informative for the accident years under review. Changes in claim handling, case reserving practices, coverage, or the mix of claims can make historical factors less representative. Incurred amounts can also decrease when case estimates are released, so a factor need not always exceed 1.

From a Triangle to Accident-Year Estimates

At a valuation date, accident years are typically observed at different development ages. For each year, start with its latest available cumulative incurred amount and multiply only by the factors needed to reach ultimate. Then calculate unpaid liabilities using the paid amount as of that same valuation date.

Identify the current incurred amount and age.
Use the latest cumulative incurred value available for each accident year, and record its development age.

Build the age-to-ultimate multiplier.
Multiply the selected factors from the current age through ultimate. Do not include factors for ages the accident year has already passed.

Project ultimate losses.
Multiply current cumulative incurred losses by the age-to-ultimate multiplier.

Calculate unpaid liabilities.
Subtract paid losses to date from projected ultimate losses, then sum accident-year estimates if an aggregate is required.

Why the Projection Formula Works

The formula follows directly from the definition of a development factor. At one development age, multiplying $C_{i,j}$ by $f_j$ projects the next cumulative amount. Repeating this operation carries the estimate forward one age at a time:

\widehat{C}_{i,k+1}=C_{i,k}f_k,\qquad \widehat{C}_{i,k+2}=C_{i,k}f_kf_{k+1}.

Continuing in this way through the final selected factor gives $C_{i,k}\prod_{j=k}^{m-1}f_j$. This chaining is appropriate when the selected factors represent the expected proportional development at each age. It does not require every accident year to have the same current incurred amount; it applies the age-specific pattern to each year's own observed amount.

Worked Example: Projecting a Three-Year Incurred Triangle

A hypothetical insurer has three accident years observed at a valuation date. The latest cumulative incurred amounts are: AY 2022 at 36 months, $2,400,000; AY 2023 at 24 months, $1,800,000; and AY 2024 at 12 months, $1,200,000. Selected factors are 12–24 months, 1.50; 24–36 months, 1.20; and 36 months to ultimate, 1.10. Paid losses to date are $2,100,000, $1,500,000, and $800,000, respectively. Estimate ultimate losses and unpaid liabilities by accident year.

Method. Apply incurred chain ladder factors from each year’s current age through ultimate, then subtract paid losses. The paid amounts are needed for the unpaid liability calculation, but they are not the starting values for the incurred projection.

AY 2022. This year is at 36 months, so its age-to-ultimate multiplier is 1.10. Its ultimate estimate is $2{,}400{,}000\times1.10=\$2{,}640{,}000$. Its unpaid liability is $2{,}640{,}000-2{,}100{,}000=\$540{,}000$.

AY 2023. This year is at 24 months, so its multiplier is $1.20\times1.10=1.32$. Its ultimate estimate is $1{,}800{,}000\times1.32=\$2{,}376{,}000$. Its unpaid liability is $2{,}376{,}000-1{,}500{,}000=\$876{,}000$.

AY 2024. This year is at 12 months, so its multiplier is $1.50\times1.20\times1.10=1.98$. Its ultimate estimate is $1{,}200{,}000\times1.98=\$2{,}376{,}000$. Its unpaid liability is $2{,}376{,}000-800{,}000=\$1{,}576{,}000$.

Aggregate check. Total ultimate losses are $2{,}640{,}000+2{,}376{,}000+2{,}376{,}000=\$7{,}392{,}000$. Total paid losses are $2{,}100{,}000+1{,}500{,}000+800{,}000=\$4{,}400{,}000$. Therefore aggregate unpaid liabilities are $7{,}392{,}000-4{,}400{,}000=\$2{,}992{,}000$, which equals $540{,}000+876{,}000+1{,}576{,}000$. The totals reconcile.

Common mistake avoided. Subtracting incurred losses instead of paid losses would give $7{,}392{,}000-(2{,}400{,}000+1{,}800{,}000+1{,}200{,}000)=\$1{,}992{,}000$. That is not the total unpaid liability: incurred includes case estimates as well as paid losses.

Worked Example: Different Ages and a Factor to Ultimate

For a hypothetical specialty insurer, AY A has cumulative incurred losses of $1,800,000 at 24 months and paid losses of $1,550,000. AY B has cumulative incurred losses of $1,200,000 at 12 months and paid losses of $700,000. The selected factors are 12–24 months, 1.25; 24 months to ultimate, 1.08. Estimate ultimate losses and unpaid liabilities for both years and in aggregate.

Method. Chain the factors required by each year’s current age. Then subtract each year’s paid losses—not its incurred losses—from its ultimate estimate.

AY A. At 24 months, only the 24-month-to-ultimate factor applies. Ultimate is $1{,}800{,}000\times1.08=\$1{,}944{,}000$. Unpaid liability is $1{,}944{,}000-1{,}550{,}000=\$394{,}000$.

AY B. At 12 months, both factors apply. The age-to-ultimate multiplier is $1.25\times1.08=1.35$. Ultimate is $1{,}200{,}000\times1.35=\$1{,}620{,}000$. Unpaid liability is $1{,}620{,}000-700{,}000=\$920{,}000$.

Aggregate check. Total ultimate is $1{,}944{,}000+1{,}620{,}000=\$3{,}564{,}000$. Total paid is $1{,}550{,}000+700{,}000=\$2{,}250{,}000$. Aggregate unpaid is $3{,}564{,}000-2{,}250{,}000=\$1{,}314{,}000$, also equal to $394{,}000+920{,}000$.

Common mistake avoided. Using the incurred observations as the amounts to subtract would produce $3{,}564{,}000-1{,}800{,}000-1{,}200{,}000=\$564{,}000$. That is the difference between ultimate and incurred, not the total unpaid liability. The calculation must subtract paid losses of $1,550,000 and $700,000.

Worked Example: A Factor Below One

A hypothetical accident year has cumulative incurred losses of $950,000 at 36 months and paid losses of $720,000. Based on the selected incurred pattern, the factor from 36 months to ultimate is 0.96. Calculate ultimate losses and unpaid liabilities, and explain why the factor is not automatically replaced by a value greater than one.

Method. Apply the selected factor as given, then subtract paid losses. A factor below one can arise when incurred estimates are expected to reduce, for example through the release of case reserves.

The ultimate estimate is $950{,}000\times0.96=\$912{,}000$. The unpaid liability is $912{,}000-720{,}000=\$192{,}000$. As a check, current incurred exceeds projected ultimate by $950{,}000-912{,}000=\$38{,}000$, while case reserves currently exceed future unpaid by $950{,}000-720{,}000=\$230{,}000$. Their difference is $230{,}000-38{,}000=\$192{,}000$, matching ultimate less paid.

Common mistake avoided. Forcing every factor above one would ignore the selected development pattern and overstate ultimate losses. A factor below one is not inherently invalid, although it should prompt a review of the data, valuation consistency, and factor selection.

Checks and Common Traps

Before relying on an accident-year projection, verify the starting values, development ages, factor chain, and unpaid calculation. In particular, make sure all amounts use the same valuation date and monetary basis. A paid amount from a different valuation date cannot be subtracted consistently from the projected ultimate.

Check	What to verify
Starting value	Use the latest cumulative incurred amount for the accident year, not an incremental amount or a paid amount.
Factor chain	Include each selected factor from the current age onward, and do not repeat an age already reached.
Unpaid liability	Subtract paid losses to date from ultimate losses; do not substitute incurred losses.
Aggregate reconciliation	Total accident-year unpaid estimates should equal total ultimate losses less total paid losses.

The aggregate reconciliation is an especially useful arithmetic check:

\sum_i \widehat{R}_i =\sum_i(\widehat{U}_i-P_i) =\sum_i\widehat{U}_i-\sum_iP_i.

This equality follows from subtracting each accident year's paid amount from its ultimate estimate and then adding across accident years. If the two aggregate calculations do not match, check for arithmetic errors, omitted accident years, or inconsistent paid values.

Key takeaway: In incurred chain ladder, selected incurred development factors project cumulative incurred losses to ultimate by accident year. The estimated unpaid claim liability is ultimate less paid, because incurred losses already include both paid losses and case estimates.

Check Your Understanding

For each hypothetical situation, project ultimate losses where factors are supplied and distinguish the ultimate estimate from the unpaid liability. Amounts are in dollars.

An accident year has cumulative incurred losses of $640,000 at 24 months, paid losses of $500,000, and a selected 24-month-to-ultimate factor of 1.15. Calculate ultimate losses and unpaid liabilities. (Answer: ultimate $736,000; unpaid $236,000.)
AY A has incurred losses of $1,100,000 at 36 months and paid losses of $900,000. AY B has incurred losses of $800,000 at 24 months and paid losses of $550,000. The selected factors are 24–36 months, 1.20, and 36 months to ultimate, 1.04. Calculate each ultimate and the aggregate unpaid liability. (Answer: AY A ultimate $1,144,000; AY B ultimate $998,400; aggregate unpaid $692,400.)
Two accident years have projected ultimate estimates of $1,944,000 and $1,620,000. Their paid losses to date are $1,550,000 and $700,000. Calculate aggregate unpaid liability and show the aggregate subtraction. (Answer: $1,314,000, calculated as $3,564,000 ultimate less $2,250,000 paid.)
An accident year has $2,000,000 incurred at 12 months and $1,400,000 paid. Selected factors are 12–24 months, 1.30, 24 months to ultimate, 1.10. Calculate the age-to-ultimate multiplier, ultimate, and unpaid liability. (Answer: multiplier 1.43; ultimate $2,860,000; unpaid $1,460,000.)
An accident year has $900,000 incurred at 18 months. The selected factors from 18 to 36 months and from 36 months to ultimate are 1.20 and 1.05. Paid losses are not provided. What is the ultimate estimate, and can the total unpaid liability be determined? (Answer: ultimate $1,134,000; total unpaid liability cannot be determined without paid losses to date.)

What You'll Learn