Compare Link Ratio Averaging Methods

Why Compare Link Ratio Averages?

A set of link ratios can produce different selected factors depending on how the observations are combined. A simple arithmetic average gives every observed ratio equal weight. A volume-weighted average gives more influence to observations with larger starting cumulative losses. A geometric average combines ratios on a multiplicative scale. These distinctions matter because the selected factor is applied to cumulative losses: even a modest difference in factor selection can produce a material difference in a projection.

The methods summarize the same experience in different ways; none automatically makes the underlying observations comparable or appropriate. For a useful comparison, first ensure the link ratios are calculated over the same development-age interval and that their underlying data are eligible for consideration. This tutorial focuses on the effect of the averaging method, not on choosing an observation window.

Definition: Let $C_{i,j}$ be cumulative loss for origin period $i$ at development age $j$. For each eligible origin period, the link ratio from age $j$ to age $j+1$ is $r_{i,j}=C_{i,j+1}/C_{i,j}$. The simple arithmetic selection is the mean of the observed link ratios. The volume-weighted selection is the ratio of the sums of the later-age and earlier-age cumulative losses. The geometric selection is the geometric mean of the observed ratios.

Three Ways to Select a Factor

Suppose there are $n$ eligible observations of the same development-age link ratio, and each ratio $r_{i,j}$ is positive. The simple arithmetic selection, $\bar r_j$, treats each origin period equally. The volume-weighted selection, $r_j^{(w)}$, weights each ratio by its starting cumulative loss. The geometric selection, $r_j^{(g)}$, treats the ratios equally after taking logarithms.

\bar r_j=\frac{1}{n}\sum_{i=1}^{n}r_{i,j}

r_j^{(w)} =\frac{\sum_{i=1}^{n}C_{i,j+1}}{\sum_{i=1}^{n}C_{i,j}} =\frac{\sum_{i=1}^{n}C_{i,j}r_{i,j}}{\sum_{i=1}^{n}C_{i,j}}

r_j^{(g)}=\left(\prod_{i=1}^{n}r_{i,j}\right)^{1/n} =\exp\left(\frac{1}{n}\sum_{i=1}^{n}\ln r_{i,j}\right)

The weighted formula requires a positive sum of starting cumulative losses, with ratios defined for the included observations. The geometric mean requires every included ratio to be strictly positive. If a ratio is zero or negative, its logarithm is not defined as a real number, so the geometric selection cannot be calculated in this form. These formulas also presume the included observations are being used to summarize the same development interval; averaging unlike intervals does not produce a meaningful single link ratio.

Why the Weighted Factor Is a Ratio of Totals

The volume-weighted method is not an arbitrary weighting convention. Since $r_{i,j}=C_{i,j+1}/C_{i,j}$, multiplying each ratio by its starting cumulative loss gives $C_{i,j}r_{i,j}=C_{i,j+1}$. Using the starting cumulative loss as the weight and dividing by the sum of weights therefore gives the ratio of the two cumulative-loss totals.

\frac{\sum_{i=1}^{n}C_{i,j}r_{i,j}}{\sum_{i=1}^{n}C_{i,j}} = \frac{\sum_{i=1}^{n}C_{i,j+1}}{\sum_{i=1}^{n}C_{i,j}}

This identity explains the method’s behavior: each starting cumulative loss contributes in proportion to its size. A link ratio attached to a large denominator has more influence than one attached to a small denominator. By contrast, neither the simple arithmetic mean nor the unweighted geometric mean uses the denominator volumes.

Justification for the geometric mean: If the goal is a single common multiplicative factor whose repeated application matches the product of $n$ observed factors, that factor $g$ must satisfy $g^n=\prod_{i=1}^{n}r_{i,j}$. Taking the positive $n$th root gives $g=(\prod_{i=1}^{n}r_{i,j})^{1/n}$. Equivalently, this is the arithmetic mean of the log ratios, transformed back from the log scale.

Worked Example: Three Selections From One Link Ratio Set

A hypothetical insurer has four eligible observations for the same development interval. Cumulative losses are recorded in thousands of dollars. Calculate the simple, volume-weighted, and geometric selections. State which observation has the greatest influence on the weighted result.

Origin period	Starting cumulative loss $C_{i,j}$ ($000)	Later cumulative loss $C_{i,j+1}$ ($000)	Link ratio $r_{i,j}$
A	1,000	1,100	1.10
B	800	896	1.12
C	100	114	1.14
D	100	160	1.60

Method and conditions. All four ratios refer to the same development interval and are positive, so all three calculations are defined. The ratio of 1.60 is much larger than the other three, but its starting cumulative loss is only $100,000.

Simple arithmetic selection. Add the four ratios and divide by four:

\bar r_j=\frac{1.10+1.12+1.14+1.60}{4} =\frac{4.96}{4}=1.24

Volume-weighted selection. The sum of starting cumulative losses is $1{,}000+800+100+100=2{,}000$ thousand dollars. The sum of later cumulative losses is $1{,}100+896+114+160=2{,}270$ thousand dollars. Thus:

r_j^{(w)}=\frac{2{,}270}{2{,}000}=1.135

The same result follows by weighting each ratio by its starting amount: $1{,}000(1.10)+800(1.12)+100(1.14)+100(1.60)=2{,}270$, divided by $2{,}000$.

Geometric selection. Multiply the ratios, then take the fourth root:

r_j^{(g)} =(1.10\times1.12\times1.14\times1.60)^{1/4} =(2.247168)^{1/4} \approx1.2244

The weighted factor is 1.135, the geometric factor is approximately 1.2244, and the simple factor is 1.24. Origin D has the largest ratio but a small starting volume, so it has only a 5% weight in the volume-weighted calculation. The methods differ because they give different meanings to an “average,” not because the inputs differ.

Common mistake avoided. Weighting by the later cumulative losses would use the wrong base. For example, using the later amounts as weights gives $[1{,}100(1.10)+896(1.12)+114(1.14)+160(1.60)]/2{,}270\approx1.1451$, not the volume-weighted link ratio. The correct weights are the starting cumulative losses, as shown by the ratio-of-totals identity.

How Unequal Volumes and an Extreme Ratio Affect the Result

An extreme ratio does not have the same influence under all three methods. In the simple arithmetic mean, every ratio contributes one observation’s share, regardless of its denominator. In the geometric mean, each ratio contributes equally on the log scale; a large ratio increases the result, but the averaging takes place after logarithms. This can make the geometric result less elevated than the arithmetic result when ratios are uneven, but it does not make the geometric mean a robust outlier-removal method. In the volume-weighted mean, the effect depends on both the ratio and its starting cumulative loss.

The next comparison holds the four ratios from the worked example fixed but changes the denominators. This isolates the role of volume. It is a mathematical comparison, not a claim that actual loss data can be rearranged independently of their observed history.

Worked Example: Changing Volume Weights While Ratios Stay Fixed

Use the ratios 1.10, 1.12, 1.14, and 1.60. In a second hypothetical portfolio mix, suppose their starting cumulative losses are, respectively, $100,000, $100,000, $100,000, and $1,000,000. Calculate all three selections and compare them with the results in the preceding example.

Simple selection. The ratios have not changed, so:

\bar r_j=\frac{1.10+1.12+1.14+1.60}{4}=1.24

Volume-weighted selection. The later cumulative amounts implied by the ratios are $100(1.10)=110$, $100(1.12)=112$, $100(1.14)=114$, and $1{,}000(1.60)=1{,}600$, all in thousands of dollars. Their sum is $110+112+114+1{,}600=1{,}936$, while the starting amounts total $100+100+100+1{,}000=1{,}300$:

r_j^{(w)}=\frac{1{,}936}{1{,}300}\approx1.4892

Geometric selection. Because the ratios are unchanged, the geometric selection also remains unchanged:

r_j^{(g)}=(1.10\times1.12\times1.14\times1.60)^{1/4} \approx1.2244

The high ratio now has the largest denominator, so it receives most of the volume weight and the weighted selection rises from 1.135 to about 1.4892. The simple and geometric selections do not change because neither uses the denominator amounts. This comparison shows why a volume-weighted factor cannot be understood from the ratios alone; the associated volumes are essential.

Common mistake avoided. Do not say that a weighted selection is always less affected by an extreme ratio. In the first example the extreme ratio had little volume and its influence was limited. Here it has the largest volume, so it dominates the weighted result.

Translate the Selection Difference Into a Projection

For a single development interval, a selected link ratio $r_j$ projects a starting cumulative loss $C_j$ to $C_jr_j$. This is a one-interval projection only. It does not, by itself, establish an ultimate loss or account for any other development ages.

Worked Example: Comparing One-Interval Projections

A hypothetical portfolio has $5,000,000 of cumulative loss at a given development age. Apply each of the three selections from the first worked example to project the cumulative loss one development interval later. Report projected amounts to the nearest thousand dollars.

Method. Multiply the same starting cumulative loss by each selected factor. The starting amount and development interval are held fixed, so differences in the projections come only from the selected factor.

Simple arithmetic selection: $5{,}000{,}000\times1.24=\$6{,}200{,}000$.

Volume-weighted selection: $5{,}000{,}000\times1.135=\$5{,}675{,}000$.

Geometric selection: using the unrounded factor $1.22436$, $5{,}000{,}000\times1.22436\approx\$6{,}121{,}800$, which rounds to $\$6{,}122{,}000$ to the nearest thousand.

The highest projection is the simple-average result, followed by the geometric result and then the volume-weighted result. The simple and weighted projections differ by $\$525{,}000$. This is the numerical consequence of the first example’s high ratio receiving equal weight in the simple mean but little weight in the volume-weighted mean. The calculation is checked by dividing each projected amount by $5,000,000: the resulting factors are 1.24, 1.135, and approximately 1.2244, respectively.

Common mistake avoided. Do not apply a selected link ratio to the later-age cumulative loss when projecting forward from the starting age. The link ratio is defined as later cumulative loss divided by starting cumulative loss; therefore, the forward projection multiplies the starting amount by the factor.

Comparison and Practical Interpretation

The three calculations answer related but distinct questions. The simple mean asks for the average observed ratio when each origin period counts equally. The volume-weighted mean asks for the factor that takes the total starting cumulative loss across the included observations to the total later cumulative loss. The geometric mean gives the common multiplicative factor implied by the product of the observed ratios.

Method	What receives equal treatment?	How volume matters	Important condition
Simple arithmetic	Each link ratio	It does not enter the formula	Ratios must be calculated for the same development interval
Volume-weighted	Each unit of starting cumulative loss	Larger starting cumulative losses receive greater weight	The sum of starting cumulative losses must be positive
Geometric	Each log link ratio	It does not enter the formula	Every included ratio must be strictly positive

A frequent exam trap is to interpret “weighted” without checking the weight. For these link ratio selections, the volume weight is the starting cumulative loss, not the later cumulative loss and not the number of origin periods. Another trap is to interpret a geometric mean as a guarantee against outliers. A geometric mean changes the scale on which the ratios are averaged; it does not determine whether an observation should be included. The choice of method should be understood in light of the observed data and the intended interpretation.

Key takeaway: Calculate all candidate selections from the same eligible ratios before comparing them. The simple mean gives equal influence to each ratio, the volume-weighted mean gives influence according to starting cumulative loss, and the geometric mean averages multiplicative changes on a log scale. Explain the weighting and the resulting factor in context.

Check Your Understanding

For each original problem, calculate the requested selection or projection. Answers are rounded only as stated.

A hypothetical treaty portfolio has three link ratios for one development interval: 1.08, 1.12, and 1.20. Calculate the simple arithmetic selection. (Answer: 1.1333)
For three eligible observations, starting cumulative losses are $500,000, $300,000, and $200,000, and the corresponding link ratios are 1.10, 1.20, and 1.50. Calculate the volume-weighted selection. (Answer: 1.2100)
Calculate the geometric selection for positive link ratios 1.05, 1.10, and 1.21. Give the answer to four decimal places. (Answer: 1.1180)
A simple selected factor is 1.075 for a development interval. A hypothetical portfolio has $2,400,000 of cumulative loss at the starting age. What is the one-interval projected cumulative loss? (Answer: $2,580,000)

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