Worked Paid-Triangle Chain-Ladder Projection

Introduction

A paid loss triangle records how claim payments accumulate as accident periods mature. Chain ladder uses the observed development pattern to project the unpaid amounts for less mature periods. The calculations are compact, but they depend on reading each row correctly, selecting factors consistently, and subtracting the amount already paid from the projected ultimate.

This tutorial works through a cumulative paid triangle, shows how volume-weighted link ratios are calculated, and then converts projected ultimates into unpaid reserves. All amounts are hypothetical and stated in thousands of dollars. The projections are undiscounted; no interest adjustment or tail factor is applied.

Definition: Let $C_{i,j}$ be cumulative paid losses for accident period $i$, measured through development age $j$. A link ratio from age $j$ to the next age is $C_{i,j+1}/C_{i,j}$, when both amounts are observed and $C_{i,j}>0$. A selected link ratio $f_j$ is the factor used to project cumulative paid losses from age $j$ to age $j+1$. The chain-ladder ultimate estimate for accident period $i$ is the latest observed cumulative paid amount multiplied by all selected link ratios needed to reach the assumed ultimate age. Its unpaid reserve is that ultimate estimate less cumulative paid losses to date.

Reading the Triangle and Selecting Factors

Each row represents an accident period, and each column represents a development age. A blank cell means the amount is not yet observed; it is not a zero. In a paid triangle, the observed entries are usually cumulative payments, so a later entry in a row includes the payments in earlier development periods.

A common selection is the volume-weighted link ratio. For the transition from age $j$ to age $j+1$, add the observed cumulative amounts at age $j+1$, then divide by the corresponding total at age $j$:

f_j=\frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}},

where the sums use only accident periods with both ages observed. This selection gives more weight to periods with larger cumulative paid amounts. It is not the same as taking an unweighted arithmetic average of the individual accident-period ratios. A different method may be appropriate if the problem specifies one or if the data require a considered selection; the method used should be stated.

Worked Example: Projecting a Four-Age Paid Triangle

A hypothetical insurer has the following cumulative paid losses, in thousands of dollars. The 48-month age is assumed to be ultimate, and no tail factor is needed. Calculate volume-weighted link ratios, project ultimate losses for each accident period, and find the total unpaid reserve.

Accident period	12 months	24 months	36 months	48 months
Year 1	120	180	216	237.6
Year 2	100	150	180
Year 3	130	195
Year 4	110

Step 1: Calculate the 12-to-24-month factor. The periods with both ages observed are Years 1, 2, and 3. Their 24-month total is $180+150+195=525$, and their 12-month total is $120+100+130=350$. Therefore:

f_{12\text{--}24}=\frac{525}{350}=1.5.

Step 2: Calculate the 24-to-36-month factor. Years 1 and 2 have both ages observed. The 36-month total is $216+180=396$, and the 24-month total is $180+150=330$. Thus:

f_{24\text{--}36}=\frac{396}{330}=1.2.

Step 3: Calculate the 36-to-48-month factor. Only Year 1 has both ages observed, so:

f_{36\text{--}48}=\frac{237.6}{216}=1.1.

Step 4: Project ultimates and reserves. Year 1 is already observed at the assumed ultimate age, so its ultimate is $237.6$, and its unpaid reserve is $237.6-237.6=0$. Year 2 is observed through 36 months:

\text{Ultimate}_2=180(1.1)=198,\qquad \text{Reserve}_2=198-180=18.

Year 3 is observed through 24 months and needs both remaining factors:

\text{Ultimate}_3=195(1.2)(1.1)=257.4,\qquad \text{Reserve}_3=257.4-195=62.4.

Year 4 is observed through 12 months and needs all three factors:

\text{Ultimate}_4=110(1.5)(1.2)(1.1)=217.8,\qquad \text{Reserve}_4=217.8-110=107.8.

The total unpaid reserve is $0+18+62.4+107.8=188.2$ thousand dollars, or $\$188{,}200$. As a check, total projected ultimate losses are $237.6+198+257.4+217.8=910.8$; total latest observed cumulative paid losses are $237.6+180+195+110=722.6$. Their difference is $910.8-722.6=188.2$, which agrees with the sum of accident-period reserves. The method is volume-weighted factor selection followed by multiplication through the remaining development ages. It avoids treating each individual link ratio as equally weighted when the requested selection is volume-weighted.

Why the Chain-Ladder Projection Uses Products of Factors

The projection follows directly from the definition of a link ratio. If a cumulative amount at age $j$ is projected to age $j+1$, the projection is $C_{i,j}f_j$. If further development remains, apply the next factor to that projected amount, then the next one, and so on. For two remaining ages, for example:

C_{i,j}f_jf_{j+1}.

This also explains why cumulative development factors are products, rather than sums, of link ratios. A factor at one age acts on the amount produced at the preceding age.

A simple development model helps describe the interpretation of a link ratio. Suppose $E[C_{i,j}]=U_i g_j$, where $U_i$ is an accident period's expected ultimate amount and $g_j$ is the expected proportion of that ultimate paid by age $j$. Then:

\frac{E[C_{i,j+1}]}{E[C_{i,j}]}=\frac{U_i g_{j+1}}{U_i g_j}=\frac{g_{j+1}}{g_j}, \qquad g_j>0.

This is a ratio of expected cumulative amounts under the model. It is not generally the expected value of the ratio $C_{i,j+1}/C_{i,j}$. The expectation of a ratio need not equal the ratio of expectations; randomness in the denominator matters. In practice, a selected factor is an estimate of the development ratio, not a claim that the average of random individual ratios must equal the ratio of their averages.

The chain-ladder projection is most credible when the paid amounts use consistent definitions, the observed development pattern is informative about future development, and there are no unaddressed changes that materially distort the pattern. The factor multiplication itself does not correct for changes in claim handling, coverage, inflation, or calendar-year conditions. The selected endpoint also matters: treating the last development age as ultimate is an assumption, not a consequence of the calculation.

Worked Example: Selecting a Volume-Weighted Factor

A hypothetical paid triangle has two observed 12-to-24-month transitions: one accident period grows from $80$ to $120$, and another grows from $120$ to $168$, all in thousands of dollars. The selected 24-to-36-month factor is $1.2$. A third accident period has $100$ paid at 12 months and nothing later observed. Use the volume-weighted factor for 12-to-24 months to project the second and third accident periods. Assume 36 months is ultimate.

Step 1: Calculate the two observed ratios. They are $120/80=1.5$ and $168/120=1.4$. Their simple arithmetic average would be $(1.5+1.4)/2=1.45$, but the requested selection is volume-weighted.

Step 2: Calculate the volume-weighted factor. The total at 24 months is $120+168=288$, and the corresponding total at 12 months is $80+120=200$. Therefore:

f_{12\text{--}24}=\frac{288}{200}=1.44.

Step 3: Project the second accident period. It is already observed at 24 months with $168$ paid. Its projected ultimate is $168(1.2)=201.6$, so its reserve is $201.6-168=33.6$ thousand dollars.

Step 4: Project the third accident period. It is observed at 12 months with $100$ paid. Its projected ultimate is $100(1.44)(1.2)=172.8$, so its reserve is $172.8-100=72.8$ thousand dollars. The two projected ultimates total $201.6+172.8=374.4$, and their latest observed amounts total $168+100=268$. The difference, $374.4-268=106.4$, equals the summed reserves $33.6+72.8=106.4$. This check confirms that each reserve is measured against the latest observed cumulative amount.

Working from Incremental Paid Data

Some problems provide incremental payments instead of cumulative amounts. Incremental payments are the amounts paid during individual development periods. Before calculating cumulative link ratios, add each period's incremental payments across the row in order. Do not apply a cumulative development factor directly to an incremental amount.

Worked Example: Converting Incremental Payments Before Projection

A hypothetical triangle reports incremental paid losses in thousands. The selected cumulative link ratios are $1.5$ from 12 to 24 months and $1.2$ from 24 to 36 months. Assume 36 months is ultimate. Convert the data to cumulative amounts, then project ultimates and reserves.

Accident period	Paid at 12 months	Paid during months 12–24	Paid during months 24–36
Year 1	48	24	12
Year 2	60	30
Year 3	75

Step 1: Convert to cumulative paid amounts. Year 1 has cumulative amounts $48$, $48+24=72$, and $72+12=84$. Year 2 has $60$ at 12 months and $60+30=90$ at 24 months. Year 3 has $75$ at 12 months.

Step 2: Project Year 2. The latest cumulative amount is $90$ at 24 months. Its ultimate is $90(1.2)=108$, and its unpaid reserve is $108-90=18$ thousand dollars.

Step 3: Project Year 3. Its latest cumulative amount is $75$ at 12 months. Its 24-month projection is $75(1.5)=112.5$, and its ultimate projection is $112.5(1.2)=135$. Thus its reserve is $135-75=60$ thousand dollars.

Year 1 is observed at the assumed ultimate age, so its reserve is zero. Total projected ultimate losses are $84+108+135=327$, while total latest cumulative paid losses are $84+90+75=249$. The total reserve is $327-249=78$ thousand dollars, matching $0+18+60=78$. The calculation uses cumulative amounts for projection and uses each accident period's latest observed cumulative amount when subtracting paid losses.

Common Mistakes and Practical Checks

A correct chain-ladder calculation is more than a sequence of multiplications. Before accepting the result, confirm the data type, the applicable development ages, the factor-selection rule, and the reserve basis.

Averaging ratios when a volume-weighted factor is required. In the second worked example, the arithmetic average is $1.45$, while the volume-weighted factor is $1.44$. Using $1.45$ would project the third period's ultimate as $100(1.45)(1.2)=174$, rather than the correct $172.8$ under the stated selection. The correction is to divide the sum of next-age cumulative amounts by the sum of matching current-age amounts.
Confusing incremental and cumulative data. An incremental amount is not the cumulative amount at that age. First add the incremental payments within each accident-period row.
Subtracting an earlier development value. The reserve is ultimate less the latest cumulative paid amount, not ultimate less the amount at the first development age.
Applying too few or too many factors. Start with the latest observed age in each row. Apply only the factors from that age through the assumed ultimate age.
Assuming a last age is ultimate without being told. If development beyond the observed ages is expected, an appropriate tail factor or other treatment is needed. In these examples, the endpoint is explicitly assumed to be ultimate.
Interpreting a ratio of expectations as an expected ratio. The development model gives a ratio of expected cumulative amounts. It does not, by itself, establish the expectation of a random ratio.

For a final arithmetic check, add the accident-period reserves and compare that sum with total projected ultimates less total latest observed cumulative paid. These totals must agree when the same set of accident periods is included in both calculations.

Check Your Understanding

For each problem, state the factor or factors used and show how the projected ultimate is converted to an unpaid reserve. Amounts are in thousands of dollars unless stated otherwise.

A period has cumulative paid losses of $40$ at 12 months and $52$ at 24 months. The selected 24-to-36-month factor is $1.1$, and 36 months is ultimate. Find its ultimate and reserve. (Answer: Ultimate $57.2$; reserve $5.2$.)
A triangle has cumulative rows $50,75,90$; $70,105$; and $80$, observed through 36, 24, and 12 months respectively. Use the observed factors and assume 36 months is ultimate. Find the total unpaid reserve. (Answer: Factors $1.5$ and $1.2$; reserves $0$, $21$, and $64$; total $85$.)
Two observed transitions from 12 to 24 months are $100$ to $140$ and $200$ to $260$. Use a volume-weighted 12-to-24-month factor of $400/300$, a selected 24-to-36-month factor of $1.1$, and a 12-month cumulative amount of $90$ for a newer period. If 36 months is ultimate, find that period's reserve. (Answer: Ultimate $132$; reserve $42$.)
Assume selected factors are $1.25$ from 12 to 24 months and $1.2$ from 24 to 36 months. One period has $150$ paid at 36 months, another has $160$ paid at 24 months, and a third has $100$ paid at 12 months. Treat 36 months as ultimate. Find the total reserve. (Answer: Reserves $0$, $32$, and $50$; total $82$.)
Incremental paid amounts are $50$ at 12 months and $25$ during months 12–24 for one period; $40$ and $20$ for a second period; and $30$ at 12 months for a third. Use selected cumulative factors $1.5$ and $1.2$, with 36 months assumed ultimate. Convert to cumulative data and find the total reserve. (Answer: Cumulative amounts are $50,75$; $40,60$; and $30$. The reserves are $0$, $12$, and $24$, for a total of $36$.)

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