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Exam ASTAM · Development Data and Chain Ladder · Tutorial 691 of 979

Why Chain Ladder Uses Volume-Weighted Ratios

See how the size of cumulative losses determines each accident period’s influence on a volume-weighted development factor.

Beginner 9 min read

What You'll Learn

  • Calculate a volume-weighted development factor from eligible cumulative loss pairs
  • Rewrite the factor as a weighted average of individual link ratios
  • Compare the influence of large and small cumulative loss amounts
  • Distinguish the aggregate development rate from the typical accident-period ratio
  • Identify when an observation should not be included in the calculation

Why the Average Matters

At a given development age, different accident periods can have different observed link ratios. One period may have a relatively small cumulative loss amount and a high link ratio; another may have a much larger amount and a lower ratio. The way these observations are averaged determines how much each period influences the selected development factor.

The conventional volume-weighted factor gives more influence to periods with larger cumulative losses at the starting development age. This is useful when the intended summary is the development of the combined observed volume, rather than the link ratio for a “typical” accident period. The distinction matters: the two summaries answer different questions and need not be close.

Definition: Let \(C_{i,j}\) denote the cumulative loss for accident period \(i\) at development age \(j\). For each accident period with both \(C_{i,j}\) and \(C_{i,j+1}\) observed, its individual link ratio is \(r_i=C_{i,j+1}/C_{i,j}\). The volume-weighted development factor from age \(j\) to age \(j+1\) is the sum of the next-age cumulative losses divided by the sum of the corresponding current-age cumulative losses.
$$ f_j=\frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}} $$

The sums use the same set of accident periods: only periods with both values observed are included. For the usual interpretation, each included starting cumulative loss is positive. Amounts must also be on a consistent basis, such as all paid losses or all incurred losses, and the comparison must be at the same pair of development ages.

How Volume Weighting Works

A ratio of sums can also be expressed as a weighted average of individual link ratios. This identity explains why large starting cumulative losses have greater influence.

$$ f_j =\frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}} =\frac{\sum_i C_{i,j}r_i}{\sum_i C_{i,j}} =\sum_i w_i r_i, \qquad w_i=\frac{C_{i,j}}{\sum_k C_{k,j}}. $$

To justify the identity, start from the definition \(r_i=C_{i,j+1}/C_{i,j}\). Multiplying both sides by \(C_{i,j}\) gives \(C_{i,j}r_i=C_{i,j+1}\). Summing over the eligible accident periods gives \(\sum_i C_{i,j}r_i=\sum_i C_{i,j+1}\). Dividing by \(\sum_i C_{i,j}\) yields the stated factor. The weights are nonnegative and sum to one because

$$ \sum_i w_i =\sum_i \frac{C_{i,j}}{\sum_k C_{k,j}} =\frac{\sum_i C_{i,j}}{\sum_k C_{k,j}} =1. $$

Thus a period with 600 in starting cumulative losses has six times the weight of a period with 100, all else equal. That does not mean the larger period’s ratio is necessarily more accurate or more representative in every situation. It means that this particular calculation summarizes the change in the combined observed cumulative amounts.

Key distinction: The simple average gives every eligible accident period equal weight: \(\bar r=\sum_i r_i/n\). The volume-weighted factor gives each period weight \(C_{i,j}/\sum_k C_{k,j}\). The simple average summarizes the average of the observed ratios; the volume-weighted factor summarizes the ratio of the combined next-age amount to the combined starting amount.

Worked Examples

Worked Example: A Large Period Has More Influence

A hypothetical insurer reviews cumulative paid losses, in thousands of dollars, from development age 12 to age 24. Three accident periods have the following observations: Period A, 100 at age 12 and 120 at age 24; Period B, 300 and 390; Period C, 600 and 720. Calculate the simple average link ratio and the volume-weighted factor. Explain which periods have the greatest influence on the latter.

Method. First calculate each individual ratio by dividing the age-24 amount by the corresponding age-12 amount:

$$ r_A=\frac{120}{100}=1.20,\qquad r_B=\frac{390}{300}=1.30,\qquad r_C=\frac{720}{600}=1.20. $$

The simple average is

$$ \bar r=\frac{1.20+1.30+1.20}{3} =\frac{3.70}{3} =1.233333\ldots $$

For the volume-weighted factor, sum the age-24 amounts and divide by the sum of the age-12 amounts:

$$ f_{12,24}=\frac{120+390+720}{100+300+600} =\frac{1{,}230}{1{,}000} =1.23. $$

The weights are \(100/1{,}000=0.10\), \(300/1{,}000=0.30\), and \(600/1{,}000=0.60\). Checking by the weighted-average form gives \(0.10(1.20)+0.30(1.30)+0.60(1.20)=0.12+0.39+0.72=1.23\). Period C has the greatest influence because it has the largest starting cumulative loss; Period B’s higher ratio is moderated by its 30% weight. The calculations agree with the ratio of sums. The common mistake to avoid is treating each period as equally influential when calculating the volume-weighted factor.

Worked Example: A High Ratio on a Small Amount

For a hypothetical portfolio, cumulative losses at two successive development ages are 20 and 30 for Period A, 800 and 840 for Period B, and 150 and 165 for Period C, all in thousands of dollars. A fourth period has 400 at the earlier age, but its later-age amount is not yet observed. Calculate both averages from the usable observations and explain why the answers differ.

Method. The fourth period has no observed pair of successive-age amounts, so it cannot contribute a link ratio or either sum. For the three complete pairs:

$$ r_A=\frac{30}{20}=1.50,\qquad r_B=\frac{840}{800}=1.05,\qquad r_C=\frac{165}{150}=1.10. $$

The simple average is

$$ \bar r=\frac{1.50+1.05+1.10}{3} =\frac{3.65}{3} =1.216666\ldots $$

The volume-weighted factor is

$$ f=\frac{30+840+165}{20+800+150} =\frac{1{,}035}{970} =1.0670103\ldots $$

For a check, the weights are \(20/970\), \(800/970\), and \(150/970\); multiplying those weights by \(1.50\), \(1.05\), and \(1.10\), respectively, gives the same factor \(1.0670103\ldots\). Period A’s ratio of 1.50 is striking, but its starting amount is only \(20/970\), about 2.06% of the included total. Period B contributes about 82.47% of that total, and its ratio is 1.05. The weighted factor is therefore much closer to 1.05 than the simple average is. The common mistake is to include the fourth period’s 400 in the denominator even though its next-age observation is missing; the numerator and denominator must cover the same complete set of periods.

Worked Example: Checking the Weights Before Averaging

A hypothetical insurer has three complete cumulative-loss pairs, in thousands: 500 followed by 540, 250 followed by 280, and 50 followed by 60. Calculate the two factors, show each starting-volume weight, and state what the difference reveals.

Method. The individual ratios are

$$ r_1=\frac{540}{500}=1.08,\qquad r_2=\frac{280}{250}=1.12,\qquad r_3=\frac{60}{50}=1.20. $$

The simple average is

$$ \bar r=\frac{1.08+1.12+1.20}{3} =\frac{3.40}{3} =1.133333\ldots $$

The starting total is \(500+250+50=800\), and the next-age total is \(540+280+60=880\). Thus the volume-weighted factor is \(880/800=1.10\). The individual weights are \(500/800=0.625\), \(250/800=0.3125\), and \(50/800=0.0625\). The weighted-average check is

$$ 0.625(1.08)+0.3125(1.12)+0.0625(1.20) =0.675+0.350+0.075 =1.10. $$

The largest ratio, 1.20, belongs to the smallest starting amount and therefore receives only 6.25% weight. The simple average answers what the arithmetic mean of the three ratios is; the volume-weighted factor answers how the combined observed amount changed. The common mistake is to calculate the correct individual ratios and then assume their unweighted average must be the conventional volume-weighted factor.

What the Factor Does—and Does Not—Say

The ratio-of-sums calculation is directly tied to the combined observations: its numerator is the total cumulative loss at the later age for the eligible periods, and its denominator is the total at the earlier age for those same periods. Equivalently, it is an average of the individual ratios with weights proportional to the starting cumulative amounts. This is the reason the volume-weighted calculation gives larger periods greater influence.

A selected factor is a summary of the observations used to calculate it; the arithmetic alone does not establish that future development will follow that factor. Nor does volume weighting automatically make the result superior for every purpose. If the question asks for the typical accident period’s link ratio, the simple average may be the relevant descriptive quantity. If it asks for the conventional volume-weighted ratio, use the ratio of sums. The intended summary should be clear before doing the arithmetic.

Keep the data pairing consistent. Use the same development-age transition for each ratio, include only observations available at both ages, and do not combine incompatible loss measures. When values are rounded, retain sufficient precision during the calculation and round the reported factor at the end. If the factor is reported to four decimal places, for instance, the factor \(1.0670103\ldots\) is reported as 1.0670.

Exam trap: Suppose two periods have ratios 1.30 and 1.05, with starting cumulative losses of 100 and 400. The wrong calculation for a volume-weighted factor is the simple average \((1.30+1.05)/2=1.175\). The correct calculation is \((100(1.30)+400(1.05))/(100+400)=(130+420)/500=1.10\). The larger period supplies 80% of the starting volume, so its ratio supplies 80% of the factor’s weight.

Check Your Understanding

For each question, distinguish the simple average from the ratio of sums when requested. Treat all amounts as hypothetical cumulative losses in the same units.

  1. Two complete pairs are 100 followed by 110 and 200 followed by 250. Calculate the simple average link ratio and the volume-weighted factor. (Answer: simple average \(1.175\); volume-weighted factor \(1.200\).)
  2. Three pairs are 40 followed by 52, 360 followed by 378, and 500 followed by 525. Calculate the volume-weighted factor and the simple average. (Answer: volume-weighted factor \(955/900=1.0611\); simple average \((1.30+1.05+1.05)/3=1.1333\).)
  3. Three pairs are 80 followed by 88, 120 followed by 138, and 300 followed by 345. Calculate the volume-weighted factor and identify the weight of the third period. (Answer: factor \(571/500=1.1420\); third-period weight \(300/500=0.6000\).)
  4. Three complete pairs are 25 followed by 35, 175 followed by 192.5, and 800 followed by 840. A fourth period has an earlier-age amount of 500 but no later-age amount. Calculate the factor using the complete pairs only. (Answer: \(1{,}067.5/1{,}000=1.0675\).)
  5. A small period has starting cumulative losses of 100 and a link ratio of 1.30. A larger period has a link ratio of 1.05. What starting amount must the larger period have for the volume-weighted factor to be 1.10? What share of the total starting amount then belongs to the small period? (Answer: the larger starting amount is 400; the small period’s share is \(100/500=0.2000\).)

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