Common Chain-Ladder Selection and Projection Errors

Why Chain-Ladder Checks Matter

A chain-ladder projection can be arithmetically consistent and still be wrong if a factor is reversed, a triangle mixes cumulative and incremental amounts, or an unavailable cell is treated as observed data. These mistakes can materially change projected ultimates and unpaid reserves. A short, ordered audit helps identify them before the results are used.

The examples use hypothetical cumulative loss triangles. They focus on errors in selecting and applying development factors, not on choosing among reserving methods. For a standard chain-ladder projection, accident periods are assumed to follow a sufficiently stable cumulative development pattern for the selected factors to be applied. A selected factor must be appropriate for the development ages and data being projected; the method does not make an unsuitable factor reliable.

Definition: Let $C_{i,j}$ be the cumulative amount for accident period $i$ at development age $j$, where $j$ increases as development proceeds. The incremental amount for a period at age $j$ is $I_{i,j}=C_{i,j}-C_{i,j-1}$, for $j>1$. A link ratio from age $j$ to the next age is the ratio of cumulative amounts at those two ages. A projected ultimate is the estimated cumulative amount at ultimate; the unpaid reserve for a period is its projected ultimate less its latest observed cumulative amount.

For example, a link ratio from 12 to 24 months must describe development from 12 months to 24 months. A common volume-weighted selection is the sum of the observed cumulative amounts at 24 months divided by the sum at 12 months, using only accident periods with both observations available. Once selected, the factor multiplies the cumulative amount at 12 months to project the cumulative amount at 24 months.

f_{12\to24} = \frac{\sum_i C_{i,24}}{\sum_i C_{i,12}}, \qquad \widehat{C}_{i,24}=C_{i,12}f_{12\to24}

The summation includes only periods with observed values at both development ages. For a selected set of successive factors, projection proceeds in age order: multiply the latest observed cumulative amount by each factor needed to reach ultimate. The reserve is not the ultimate itself; it is the difference between that ultimate and the latest observed cumulative amount.

A Short Justification: Why Factors Are Multiplied

Suppose the selected factor from age $j$ to age $j+1$ is $f_j$. By definition of the factor, the projected cumulative amount at the next age is the current cumulative amount multiplied by $f_j$. Applying the next factor to that projected amount gives:

\widehat{C}_{i,j+2} = \widehat{C}_{i,j+1}f_{j+1} = (C_{i,j}f_j)f_{j+1} = C_{i,j}f_jf_{j+1}.

Repeating the same step shows why a projection across several development ages uses the product of the intervening factors. It also explains why reversing a factor or adding a factor to an incremental amount breaks the intended calculation: neither operation represents the stated sequence of cumulative development.

Worked Examples: Audit the Direction and the Amount Basis

Worked Example: Reversing a Link Ratio and Mixing Cumulative With Incremental Values

A hypothetical paid-loss triangle has the following cumulative amounts, in thousands of dollars. Use volume-weighted factors to project each incomplete period to 36 months. Then calculate the unpaid reserve. Identify the result of reversing the first factor.

Method. Select each link ratio as later-age cumulative amounts divided by earlier-age cumulative amounts, using only rows with both observations. Apply the factors to cumulative amounts, then subtract the latest observed cumulative amount to obtain each reserve.

For 12 to 24 months, the available observations are 100 to 150 and 120 to 180. Thus the selected factor is:

f_{12\to24} = \frac{150+180}{100+120} = \frac{330}{220} = 1.5.

For 24 to 36 months, only the first accident period has both amounts observed:

f_{24\to36} = \frac{180}{150} = 1.2.

The second period has a latest cumulative amount of 180 at 24 months. Its projected 36-month ultimate is $180(1.2)=216$, so its unpaid reserve is $216-180=36$, or $36,000.

The third period has a latest cumulative amount of 140 at 12 months. Its projected 24-month cumulative amount is $140(1.5)=210$; its projected ultimate is $210(1.2)=252$. Its unpaid reserve is $252-140=112$, or $112,000. The total reserve for the two incomplete periods is $36+112=148$, or $148,000.

Check using increments. For the third period, the first projected increment is $210-140=70$, and the next is $252-210=42$. The ultimate reconciles: $140+70+42=252$. The incremental amounts are differences between cumulative amounts; they are not the amounts to which these cumulative link ratios should be applied.

Common mistake avoided. Reversing the first ratio gives $220/330=0.6667$ rather than 1.5. Multiplying 140 by that reversed ratio would project a 24-month cumulative amount below the observed 12-month amount. The direction check is that the numerator must be the later development age and the denominator the earlier one.

Available Cells and Development-Age Alignment

A triangle’s blank cells usually mean that the corresponding development ages have not yet been observed. They are not zero-valued observations. Each link ratio must be selected from accident periods with amounts at both ages in that ratio. Before calculating, confirm that the cells refer to matching development ages and the same cumulative measure, such as paid or incurred amounts.

Worked Example: Excluding Unavailable Cells From Factor Selection

A hypothetical cumulative triangle, in thousands of dollars, is observed at 12, 24, 36, and 48 months:

Accident period 1: 80, 120, 150, 165.
Accident period 2: 100, 145, 174, not yet observed.
Accident period 3: 120, 168, not yet observed, not yet observed.
Accident period 4: 140, not yet observed, not yet observed, not yet observed.

Select volume-weighted factors from 12 to 24, 24 to 36, and 36 to 48 months, then project accident period 4 to ultimate using these factors. Calculate the reserve. For comparison, show what happens if the unavailable 24-month amount for accident period 4 is incorrectly treated as zero when selecting the first factor.

Method. Use only rows with both ages observed for each factor. From 12 to 24 months, accident periods 1 through 3 qualify:

f_{12\to24} = \frac{120+145+168}{80+100+120} = \frac{433}{300} = 1.4433\ldots

From 24 to 36 months, only accident periods 1 and 2 qualify. From 36 to 48 months, only accident period 1 qualifies:

f_{24\to36} = \frac{150+174}{120+145} = \frac{324}{265} = 1.2226\ldots, \qquad f_{36\to48} = \frac{165}{150} = 1.1.

Accident period 4 has an observed 12-month cumulative amount of 140. Projecting sequentially gives:

\widehat{C}_{4,48} = 140\left(\frac{433}{300}\right) \left(\frac{324}{265}\right)(1.1) = 271.7606\ldots

Its projected ultimate is therefore $271.76 thousand, and its reserve is $271.7606\ldots-140=131.7606\ldots$ thousand. Rounded to the nearest cent, the reserve is $131,760.60.

If the unavailable 24-month amount for accident period 4 is incorrectly entered as zero, the first factor becomes $433/(300+140)=433/440$, about 0.9841. That would produce an ultimate of $140(433/440)(324/265)(1.1)=185.2913\ldots$ thousand instead. The incorrect factor treats “not yet observed” as an actual zero and even suggests a decrease from 12 to 24 months. Excluding the unavailable cell gives the correct selection for this calculation.

Check. Accident period 4’s latest observed cumulative amount is 140, so its reserve must equal the projected ultimate less 140. This check also confirms that the answer is not mistakenly reporting the ultimate as the reserve.

Tail Factors: Apply Them After Observed-Age Development

A tail factor represents selected development beyond the last age covered by the ordinary link ratios, through ultimate. It is applied to the projected cumulative amount at that last age, not directly to the latest observed amount if ordinary development factors still need to be applied. Nor is a tail factor itself an incremental percentage of the latest observed amount.

Worked Example: Applying a Tail Factor to the Correct Cumulative Amount

A hypothetical accident period has a latest observed cumulative paid amount of $250,000 at 24 months. The selected factor from 24 to 36 months is 1.20, and the selected tail factor from 36 months to ultimate is 1.05. Calculate the projected ultimate and reserve.

Method. First project to the last ordinary development age. Then apply the tail factor to that projected cumulative amount:

\widehat{C}_{36} = 250{,}000(1.20) = 300{,}000, \qquad \widehat{C}_{\text{ultimate}} = 300{,}000(1.05) = 315{,}000.

The unpaid reserve is $315{,}000-250{,}000=$$65,000. As a check, the projected increase from 24 to 36 months is $50,000, and the tail adds $315{,}000-300{,}000=$$15,000. Together, the projected development is $65,000.

Common mistake avoided. Adding 5% of the 24-month amount to the 36-month projection would give $300{,}000+0.05(250{,}000)=312{,}500$, understating the ultimate by $2,500. The stated tail factor multiplies the 36-month cumulative amount: the tail increment is 5% of $300,000, not 5% of the earlier $250,000.

A Practical Projection Audit

Use the following checks before accepting a selected-factor projection. They do not establish that the selected pattern is appropriate; they help reveal whether the factors have been selected and applied consistently.

Label the triangle.
Confirm the loss basis, accident periods, development ages, and whether the entries are cumulative or incremental.

Check each factor’s direction.
For a link from an earlier age to a later age, place the later-age cumulative amounts in the numerator and the earlier-age amounts in the denominator.

Check cell availability.
Include only accident periods with observed amounts at both ages used in the factor. Do not enter blanks as zeros.

Project in age order.
Apply each factor to the cumulative amount at its starting age. Use the resulting cumulative amount as the starting point for the next factor.

Reconcile ultimate and reserve.
For each incomplete period, subtract the latest observed cumulative amount from the projected ultimate. Check projected increments against the cumulative change.

Place the tail last.
If a tail factor is selected, apply it after projecting to the age from which that tail begins.

The main exam trap is losing track of what a number represents. A cumulative amount includes development through an age; an incremental amount is the change between ages. A factor links cumulative amounts at two ages. A projected ultimate includes future development; a reserve is only the amount still unpaid relative to the latest observed cumulative value. Keeping these quantities labelled prevents several common errors at once.

Key takeaway: Audit the development-age direction, eligible cells, amount basis, projection sequence, tail starting age, and ultimate-minus-observed reserve calculation. A result should reconcile both as a cumulative projection and, where useful, as the sum of its projected increments.

Check Your Understanding

For each hypothetical situation, identify the correct projection or error check.

A period has a latest cumulative amount of $90,000 at 12 months. The selected factors are 1.40 from 12 to 24 months, 1.10 from 24 to 36 months, and a tail factor of 1.03 from 36 months to ultimate. What are the ultimate and reserve? (Answer: Ultimate $142,758.00; reserve $52,758.00.)
At 12 months two eligible periods have cumulative amounts 80 and 120; at 24 months their cumulative amounts are 112 and 168. What is the volume-weighted factor from 12 to 24 months, and what error results from reversing it? (Answer: The factor is $280/200=1.40$; reversing it gives $200/280=0.7143$, which has the wrong development direction.)
A period has a cumulative amount of $160,000 at 24 months, a factor of 1.25 from 24 to 36 months, and a tail factor of 1.04 from 36 months to ultimate. Calculate the ultimate and reserve. (Answer: Ultimate $208,000; reserve $48,000.)
For a 12-to-24-month factor, eligible rows total $360,000 at 12 months and $450,000 at 24 months. A fourth row has $100,000 at 12 months but no 24-month observation. What factor should be selected, and why should the fourth row not be added to the denominator? (Answer: $450/360=1.25$; the fourth row is not eligible because its 24-month amount is unavailable.)
A period has cumulative amounts of $75,000 at 12 months and $105,000 at 24 months. What is the incremental amount from 12 to 24 months, and what would be wrong with applying a cumulative 12-to-24-month factor directly to that increment? (Answer: The increment is $30,000; the factor links cumulative amounts, so applying it to the increment would not project the 24-month cumulative amount.)

What You'll Learn