Deriving Volume-Weighted Link Ratios

Why Volume-Weight Link Ratios?

At a given development age, accident years can have very different cumulative loss amounts. A link ratio calculated as the ratio of total losses at the next age to total losses at the current age gives more influence to accident years with more current-age loss volume. This is the volume-weighted link ratio, also called the ratio-of-sums selection.

The distinction from a simple average matters: the simple average gives every eligible accident year equal influence, regardless of its loss volume. The volume-weighted method instead asks what single factor would take the aggregate current-age cumulative losses to the aggregate next-age cumulative losses. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $C_{i,j}$ be cumulative losses for accident year $i$ at development age $j$. Let $I_j$ be the set of accident years with observed cumulative losses at both ages $j$ and $j+1$. The volume-weighted link ratio from age $j$ to age $j+1$ is

\hat f_j^{\,\mathrm{VW}} = \frac{\displaystyle\sum_{i\in I_j} C_{i,j+1}} {\displaystyle\sum_{i\in I_j} C_{i,j}}.

The same accident years must appear in both sums. An unobserved next-age value is not a zero.

Here, “volume” refers to the current-age cumulative losses $C_{i,j}$, not the next-age losses. The eligible set $I_j$ can change with development age: a recent accident year may contribute to a 12-to-24-month calculation but not to a 24-to-36-month calculation if its 36-month value is not yet observed.

Derivation: Aggregate Ratio as a Weighted Average

For each eligible accident year, define its individual age-to-age ratio as $r_{i,j}=C_{i,j+1}/C_{i,j}$, provided $C_{i,j}\ne 0$. Suppose, for the usual weighted-average interpretation, that all eligible current-age cumulative losses are positive. Then the aggregate of the next-age values can be rewritten using each accident year's own ratio:

\sum_{i\in I_j} C_{i,j+1} = \sum_{i\in I_j} C_{i,j}r_{i,j}.

Divide both sides by the aggregate current-age losses. The resulting expression is the volume-weighted factor:

\hat f_j^{\,\mathrm{VW}} = \frac{\sum_{i\in I_j} C_{i,j}r_{i,j}} {\sum_{i\in I_j} C_{i,j}} = \sum_{i\in I_j} w_{i,j}r_{i,j}, \qquad w_{i,j} = \frac{C_{i,j}}{\sum_{k\in I_j}C_{k,j}}.

The weights are current-age shares. They are nonnegative and sum to one:

w_{i,j}\ge 0, \qquad \sum_{i\in I_j}w_{i,j} = \frac{\sum_{i\in I_j}C_{i,j}}{\sum_{k\in I_j}C_{k,j}} =1.

This proves the weighted-average result: the factor is an average of the individual ratios, with larger current-age cumulative losses receiving more weight. It also explains the aggregate interpretation. Multiplying the aggregate current-age losses by the selected factor reproduces the observed aggregate next-age losses exactly:

\hat f_j^{\,\mathrm{VW}}\sum_{i\in I_j}C_{i,j} = \sum_{i\in I_j}C_{i,j+1}.

If every accident year followed exactly the same factor, so that $C_{i,j+1}=f_jC_{i,j}$ for each eligible $i$, the ratio of sums would equal that common factor. In actual data, individual ratios may differ; the ratio of sums is a selection convention that preserves the aggregate relationship. It is not, by itself, a guarantee that the selected factor is unbiased or that development is stable.

Condition: The ratio-of-sums formula requires a nonzero aggregate current-age denominator. Its interpretation as a conventional weighted average of individual ratios requires positive current-age cumulative losses for the eligible accident years. If some values are zero, negative, or affected by recoveries, the algebra may still be possible when defined, but the weights may not be positive shares and the usual weighted-average interpretation may fail.

Worked Calculations

Worked Example: Calculating a Volume-Weighted Factor

A hypothetical insurer has the following cumulative losses for three accident years. Calculate the volume-weighted factor from 12 to 24 months, show the individual ratios and weights, and check that the aggregate relationship holds.

Accident year	12-month cumulative losses	24-month cumulative losses
Year A	$100	$150
Year B	$200	$280
Year C	$300	$390

Method: ratio of sums. All three accident years have both observations, so they are all eligible. The current-age total is $100+200+300=\$600$. The next-age total is $150+280+390=\$820$. Therefore,

\hat f_{12\to24}^{\,\mathrm{VW}} = \frac{820}{600} = 1.366666\ldots \approx 1.367.

Check by individual ratios. The ratios are $150/100=1.50$, $280/200=1.40$, and $390/300=1.30$. The current-age weights are $100/600=1/6$, $200/600=1/3$, and $300/600=1/2$. Thus,

\left(\frac16\right)(1.50) + \left(\frac13\right)(1.40) + \left(\frac12\right)(1.30) = 0.250+0.466666\ldots+0.650 = 1.366666\ldots.

Aggregate check. Multiplying the current-age total by the unrounded factor gives $600(820/600)=820$, exactly the observed next-age total. Rounded to three decimal places, the selected factor is 1.367. This method avoids giving the $100-volume accident year the same influence as the $300-volume accident year.

Worked Example: Why the Simple Average Is Different

For another hypothetical portfolio, the matched 12- and 24-month cumulative losses are $80 and $104, $120 and $150, and $200 and $260. Calculate the volume-weighted factor, compare it with the simple average of individual ratios, and identify which result preserves the aggregate relationship.

Step 1: Calculate the ratio of sums. The 12-month total is $80+120+200=\$400$. The 24-month total is $104+150+260=\$514$. Hence,

\hat f_{12\to24}^{\,\mathrm{VW}} = \frac{514}{400} = 1.285.

Step 2: Calculate the individual ratios. They are $104/80=1.30$, $150/120=1.25$, and $260/200=1.30$. Their simple average is

\frac{1.30+1.25+1.30}{3} = \frac{3.85}{3} = 1.283333\ldots \approx 1.283.

Step 3: Verify the volume weighting. The current-age weights are $80/400=0.20$, $120/400=0.30$, and $200/400=0.50$. Applying them gives

$$ (0.20)(1.30)+(0.30)(1.25)+(0.50)(1.30) = 0.260+0.375+0.650 = 1.285. $$

The ratio of sums, not the simple average, takes the aggregate from $400 to $514: $400(1.285)=514$. The simple average would imply $400(1.283333\ldots)=\$513.33$ when rounded to cents, which does not match the observed aggregate. The difference is small here, but the methods answer different questions.

Worked Example: Using the Correct Eligible Accident Years at Each Age

A hypothetical cumulative paid-loss triangle has the following observed values, in thousands of dollars. Calculate the volume-weighted factors from 12 to 24 months and from 24 to 36 months. Use only matched observations at each age.

Accident year	12 months	24 months	36 months
Year 1	120	180	198
Year 2	150	210	231
Year 3	180	252	277
Year 4	200	280	Not yet observed

12-to-24 months. All four accident years have both ages. The 12-month total is $120+150+180+200=650$. The 24-month total is $180+210+252+280=922$. Therefore,

\hat f_{12\to24}^{\,\mathrm{VW}} = \frac{922}{650} = 1.4184615\ldots \approx 1.418.

24-to-36 months. Only Years 1, 2, and 3 have observed values at both ages. The matched 24-month total is $180+210+252=642$, and the matched 36-month total is $198+231+277=706$. Thus,

\hat f_{24\to36}^{\,\mathrm{VW}} = \frac{706}{642} = 1.0996885\ldots \approx 1.100.

The Year 4 24-month value of 280 is not included in the second denominator because its 36-month value is unobserved. Treating that missing value as zero would incorrectly change both the data set and the selected factor. Factors are rounded here to three decimal places; the aggregate calculations use the unrounded ratios.

Common Mistakes and Practical Conditions

The volume-weighted link ratio is straightforward to calculate, but its meaning depends on careful matching and interpretation. The following traps are especially important when reading a triangle or comparing factor-selection methods.

Using a simple average by mistake. The simple average is $\sum r_{i,j}/n$; the volume-weighted factor is $\sum C_{i,j+1}/\sum C_{i,j}$. They generally differ because the first assigns equal weight to each accident year.
Weighting by next-age losses. The weights in the weighted-average representation are $C_{i,j}/\sum C_{i,j}$, not $C_{i,j+1}/\sum C_{i,j+1}$. The weighting comes from rewriting each next-age value as current-age value times its individual ratio.
Including unmatched observations. Both sums must use the same eligible accident years. A missing next-age cell is excluded, not entered as zero.
Assuming the weights are always valid shares. With positive current-age cumulative losses, the weights are nonnegative and sum to one. If values are zero or negative, individual ratios may be undefined or weights may not have the standard averaging interpretation.
Calling the result a universal best estimate. The ratio of sums exactly matches the aggregate totals in the selected data. That algebra does not establish that the development pattern is stable or that this method is appropriate when the data have structural changes.

A quick reasonableness check is to calculate the factor both ways: directly from the two aggregate totals and as the current-volume-weighted average of individual ratios. With positive current-age values and the same eligible accident years, the two calculations must agree before rounding. If they do not, check the matched rows, the weights, and whether intermediate values were rounded too early.

Key takeaway: A volume-weighted link ratio is the ratio of aggregate next-age losses to aggregate current-age losses over the same eligible accident years. When current-age values are positive, it is exactly a weighted average of individual link ratios, using each accident year's share of current-age cumulative loss volume.

Check Your Understanding

For each problem, identify the eligible observations, calculate as requested, and use unrounded values for checks. Final factors are rounded to four decimal places where needed.

A hypothetical triangle has matched current- and next-age cumulative losses of $50 and $70, $150 and $180, and $300 and $360. Calculate the volume-weighted factor and verify it using current-age weights. (Answer: $610/500=1.2200$; weighted calculation $0.10(1.40)+0.30(1.20)+0.60(1.20)=1.2200$.)
Three accident years have matched values of $90 and $117, $110 and $154, and $300 and $360. Find the ratio of sums and the simple average of individual ratios. Which one matches the aggregate relationship? (Answer: Volume-weighted $631/500=1.2620$; simple average $(1.30+1.40+1.20)/3=1.3000$; the ratio of sums matches the aggregate relationship.)
At one development age, current cumulative losses are $125, $250, and $500, and next-age values are $150, $300, and $575. Find the individual ratios, their current-age weights, and the volume-weighted factor. (Answer: Ratios $1.20,1.20,1.15$; weights $1/7,2/7,4/7$; factor $1025/875=1.1714$.)
For a 12-to-24-month link, values are $100 to $130, $200 to $250, and $300 to $390. For a 24-to-36-month link, only the first two accident years are observed: $130 to $156 and $250 to $275. Calculate both factors using the correct matched years. (Answer: 12-to-24 factor $770/600=1.2833$; 24-to-36 factor $431/380=1.1342$.)
A triangle has 12-month values of $80, $320, $600, and $400. The first three have 24-month values of $104, $352, and $690; the fourth 24-month value is unobserved. Calculate the volume-weighted factor and state the effect of incorrectly treating the missing value as zero. (Answer: Correct factor $1146/1000=1.1460$; incorrectly entering zero gives $1146/1400=0.8186$, because the unmatched $400 is wrongly added to the denominator.)

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