Why Project an Incurred Triangle?
An incurred loss triangle records cumulative incurred amounts at successive development ages for each accident period. Incurred amounts generally combine payments made to date with case estimates for reported claims. As claims develop, those estimates can change, so an incurred amount is not necessarily a final measure of the accident period’s losses.
The chain-ladder method uses observed changes between development ages to project each accident period to an estimated ultimate incurred amount. Comparing that ultimate with the latest incurred amount gives an incurred-based reserve estimate. In this tutorial, all numerical scenarios are hypothetical, and amounts are undiscounted.
An incurred-based reserve is not automatically the same as the total unpaid claim liability. Because incurred amounts include case estimates for reported claims, ultimate incurred less latest incurred is commonly interpreted as the additional development beyond the current incurred estimate, including development on reported claims and amounts for claims not yet reported. A paid-based estimate of total unpaid losses instead compares ultimate losses with cumulative paid losses.
Calculating Factors and Projecting the Triangle
For a transition from development age \(j\) to the next age \(j+1\), the volume-weighted age-to-age factor is the sum of the observed later-age incurred amounts divided by the sum of the corresponding earlier-age amounts. Include only accident periods with observations at both ages.
Here \(C_{i,j}\) denotes the cumulative incurred amount for accident period \(i\) at development age \(j\). A selected factor may be based on this calculation, but the actuary must also judge whether the observed development is suitable for projection. The basic chain-ladder projection assumes that the selected development pattern is relevant to the accident periods being projected, that the data definitions are consistent, and that the observed pattern is not materially distorted by changes such as calendar-year effects or claims-handling changes.
Multiplying successive age-to-age factors gives a cumulative development factor from a given age to the selected ultimate age. If accident period \(i\) is currently observed at age \(j\), its projected ultimate is its latest incurred amount multiplied by that cumulative factor. Its incurred-based reserve is then the difference between the projected ultimate and latest incurred amount.
The selected ultimate age \(m\) is the final age in the factor sequence. If an additional tail factor is selected beyond the observed ages, it is included in the cumulative factor. If no tail factor is used, the projection treats the final modeled age as ultimate.
Why the Factor Projection Works
A development factor is a multiplicative estimate of how much cumulative incurred losses change from one age to the next. For example, if the factor from age 12 to age 24 is \(1.4\), the projection estimates the age-24 amount as \(1.4\) times the age-12 amount. Repeating that reasoning across successive ages gives a product, not a sum.
Thus, the cumulative factor is the product of the age-to-age factors. Subtracting the current incurred amount from the projected ultimate gives the remaining increase in incurred losses implied by the projection. This subtraction also explains why a reserve is not the same as the ultimate estimate itself: the latest incurred amount is already included in the ultimate.
Worked Incurred-Triangle Projection
Worked Example: Projecting a Three-Age Incurred Triangle
A hypothetical insurer has the following cumulative incurred losses, in thousands of dollars. Use volume-weighted factors, project each accident period to age 36, and estimate the total incurred-based reserve. Assume age 36 is ultimate, with no additional tail factor.
| Accident period | Age 12 | Age 24 | Age 36 |
|---|---|---|---|
| A | 100 | 150 | 180 |
| B | 120 | 180 | — |
| C | 140 | — | — |
Method: calculate the observed volume-weighted factors, then project each period from its latest age. For ages 12 to 24, periods A and B have observations at both ages:
For ages 24 to 36, only period A has both observations, so:
Period A is already at age 36. Its ultimate estimate is \(180\), and its reserve is \(180-180=0\). Period B is at age 24, so its cumulative factor to age 36 is \(1.2\):
Period C is at age 12. Its cumulative factor is \(1.5(1.2)=1.8\):
The projected ultimate total is \(180+216+252=648\) thousand dollars. The latest incurred total is \(180+180+140=500\) thousand dollars. Therefore, the total incurred-based reserve is \(648-500=148\) thousand dollars, which agrees with the sum of the accident-period reserves, \(0+36+112=148\) thousand dollars.
Check: Each projected ultimate is at least as large as its latest incurred amount, and the aggregate reserve agrees with the sum by accident period. The calculation uses only positive amounts and factors.
A Larger Triangle and an Aggregate Check
Worked Example: Calculating Factors From Several Observations
A hypothetical specialty insurer records the following cumulative incurred amounts, in thousands of dollars. Calculate the volume-weighted factors, project all periods to age 48, and find the total incurred-based reserve. Assume age 48 is ultimate and no tail factor applies.
| Accident period | Age 12 | Age 24 | Age 36 | Age 48 |
|---|---|---|---|---|
| A | 100 | 140 | 168 | 189 |
| B | 200 | 280 | 336 | — |
| C | 300 | 420 | — | — |
| D | 400 | — | — | — |
Method: calculate each factor from all eligible observed pairs. At ages 12 to 24, periods A, B, and C are eligible:
At ages 24 to 36, periods A and B are eligible:
At ages 36 to 48, only period A is eligible:
Project each accident period from its latest observed age. Period A is already at age 48, so its ultimate is \(189\) and its reserve is zero. Period B is at age 36:
Period C is at age 24, so its cumulative factor is \(1.2(1.125)=1.35\):
Period D is at age 12, so its cumulative factor is \(1.4(1.2)(1.125)=1.89\):
The ultimate total is \(189+378+567+756=1{,}890\) thousand dollars. The latest incurred total is \(189+336+420+400=1{,}345\) thousand dollars. Their difference is \(1{,}890-1{,}345=545\) thousand dollars. The period reserves also sum to \(0+42+147+356=545\) thousand dollars.
Check: The factor for ages 12 to 24 uses three paired observations, while the factor for ages 36 to 48 uses only one. Each factor uses the observations available at its own transition; missing later cells are not treated as zero.
Applying Selected Factors at Different Maturities
In practice, the selected factors may be given or chosen after reviewing development experience. The key calculation is to use the cumulative factor appropriate to each accident period’s latest development age. Periods at different ages do not share one common multiplier to ultimate.
Worked Example: Projecting Mixed Maturities With a Tail Factor
A hypothetical liability portfolio has three accident periods. Amounts are in thousands of dollars. The latest incurred amounts are 240 at age 36 for period A, 180 at age 24 for period B, and 100 at age 12 for period C. Selected factors are \(f_{12,24}=1.4\), \(f_{24,36}=1.25\), and a tail factor of \(1.05\) from age 36 to ultimate. Estimate each ultimate and reserve, and reconcile the total.
Method: multiply all selected factors between the latest age and ultimate. Period A is at age 36, so only the tail factor remains:
Period B is at age 24, so its cumulative factor is \(1.25(1.05)=1.3125\):
Period C is at age 12. Its cumulative factor is \(1.4(1.25)(1.05)=1.8375\):
The ultimate total is \(252+236.25+183.75=672\) thousand dollars. The latest incurred total is \(240+180+100=520\) thousand dollars. Thus the aggregate incurred-based reserve is \(672-520=152\) thousand dollars, equal to \(12+56.25+83.75=152\) thousand dollars.
Check: The least mature period has the largest cumulative factor, \(1.8375\), because it requires all three development steps. The calculation retains the unrounded values throughout; the stated amounts are exact for these inputs.
Common Mistake: Using the Wrong Factor or Reserve Base
A frequent error is to apply the age-12-to-ultimate factor to every period, even when some periods have already developed beyond age 12. In the first worked example, period B’s latest incurred amount is 180 at age 24. Applying the age-12-to-ultimate factor \(1.5(1.2)=1.8\) gives the incorrect projection \(180(1.8)=324\). That repeats the age-12-to-24 development even though the amount 180 is already at age 24.
The correction is to begin at the period’s actual latest age. For period B, only the age-24-to-36 factor applies, giving \(180(1.2)=216\). Its incurred-based reserve is \(216-180=36\), not \(324-180=144\).
A second related error is to report projected ultimate losses as the reserve. Ultimate includes the losses already reflected in the latest incurred amount. The incurred-based reserve is the difference between those quantities, calculated accident period by accident period or as the difference between their aggregate totals.
Check Your Understanding
For each hypothetical scenario, calculate the requested factor, ultimate, or incurred-based reserve. Amounts are in thousands of dollars unless stated otherwise.
- A period has latest incurred losses of 260 at age 12. Selected factors are \(f_{12,24}=1.3\) and \(f_{24,36}=1.1\), with age 36 treated as ultimate. Find its ultimate and reserve. (Answer: ultimate \(371.8\); reserve \(111.8\).)
- A period has latest incurred losses of 500 at age 24. The selected factor from age 24 to age 36 is \(1.2\), and age 36 is ultimate. Find the incurred-based reserve. (Answer: \(100\).)
- For a factor from age 12 to age 24, two eligible accident periods have incurred amounts \(90\) and \(110\) at age 12, and \(126\) and \(154\) at age 24. Calculate the volume-weighted factor. (Answer: \(1.4\).)
- A period has latest incurred losses of 300 at age 24. The selected factor from age 24 to age 36 is \(1.15\), and the tail factor from age 36 to ultimate is \(1.04\). Find the ultimate and reserve. (Answer: ultimate \(358.8\); reserve \(58.8\).)
- Period A has latest incurred losses of 240 at age 36; period B has 100 at age 12. The selected factors are \(f_{12,24}=1.4\), \(f_{24,36}=1.2\), and a tail factor of \(1.05\). Find the combined incurred-based reserve. (Answer: \(88.4\), comprising \(12\) for A and \(76.4\) for B.)
Careful attention to maturity, factor sequence, and reserve definition makes an incurred-triangle projection auditable from the observed data through the aggregate estimate. This worked method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.