Simple Arithmetic Averages of Link Ratios

Why Average Link Ratios?

A link ratio summarizes how cumulative losses changed between two development ages for one origin period. Historical link ratios at the same ages often differ. A selected factor is one way to represent that observed development when projecting a cumulative amount forward. This tutorial focuses on the simple arithmetic average: calculate each selected link ratio, then give every selected ratio the same weight in the average.

A simple average is easy to calculate and interpret, but its equal weighting has a specific meaning. It gives the same influence to a ratio from a small origin-period loss amount as to a ratio from a large one. It does not give every dollar of loss equal influence. The selection of which observations to average is therefore important, as is keeping the development ages and loss definitions consistent.

Definition: For selected origin periods $i=1,\ldots,n$, let $C_{12}^{(i)}$ and $C_{24}^{(i)}$ be the cumulative losses at 12 and 24 months. If each $C_{12}^{(i)}$ is positive, the 12-to-24-month link ratio for observation $i$ is $f_{12-24}^{(i)}=C_{24}^{(i)}/C_{12}^{(i)}$. The simple arithmetic average of $n$ selected link ratios is $\bar f_{12-24}=(1/n)\sum_{i=1}^{n} f_{12-24}^{(i)}$.

The ages in this definition are illustrative; the same calculation applies to any adjacent development ages, such as 24 to 36 months. The numerator is the cumulative amount at the later age and the denominator is the cumulative amount at the earlier age. Before averaging, make sure that each pair represents the same origin period, development ages, coverage, and loss basis. A paid link ratio, for example, should not be mixed with an incurred link ratio.

Calculation and Interpretation

The calculation has two distinct steps. First compute each selected observation's ratio. Then average those ratios, rather than averaging the cumulative amounts and taking a ratio afterward. If the selected ratios are 1.04, 1.10, and 1.16, the simple average is $(1.04+1.10+1.16)/3=1.10$. Each of the three observations contributes one-third of the result.

A factor greater than 1 indicates an increase in cumulative losses from the earlier age to the later age. A factor of 1 indicates no change. A factor below 1 indicates a decrease in the observed cumulative amount over that interval. In particular, an incurred link ratio can be below 1 when incurred losses decline. These interpretations describe the observed ratios; they do not by themselves establish that a selected factor is appropriate for future projections.

Worked Example: Average Four Selected Development Ratios

A hypothetical insurer is reviewing its 12-to-24-month paid link ratio. Four origin periods have been selected after confirming that their paid-loss definitions and valuation ages are consistent. Their cumulative paid losses, in thousands of dollars, are 1,000 to 1,120; 2,000 to 2,160; 3,000 to 3,300; and 4,000 to 4,320. Calculate the simple arithmetic average and use it to project a separate 12-month cumulative amount of $7,200,000 to 24 months.

Method. Calculate each 12-to-24-month ratio as later cumulative paid losses divided by earlier cumulative paid losses. Then give each ratio equal weight.

For the first origin period, $1{,}120/1{,}000=1.12$. For the second, $2{,}160/2{,}000=1.08$. For the third, $3{,}300/3{,}000=1.10$. For the fourth, $4{,}320/4{,}000=1.08$.

The sum of the four ratios is $1.12+1.08+1.10+1.08=4.38$. Therefore, the simple average is $4.38/4=1.095$. No rounding is needed for this factor.

Apply the selected factor to the separate 12-month cumulative amount: $ \$7{,}200{,}000 \times 1.095=\$7{,}884{,}000.00$. The projection is $7,884,000.00 at 24 months.

Check. The four observed ratios include two values of 1.08, one of 1.10, and one of 1.12. Their average, 1.095, lies between the smallest and largest observed ratios. The calculation avoids the common mistake of using only the largest or most recent ratio instead of averaging all four selected observations.

Why Each Observation Has Equal Weight

The equal weights follow directly from the definition of an arithmetic mean. If there are $n$ selected observations, the sum of their ratios is divided by $n$. Thus each ratio has coefficient $1/n$, regardless of the size of its earlier cumulative loss. For three observations, each coefficient is $1/3$; for five, each is $1/5$.

One justification for the arithmetic mean is that it is the single value that minimizes the sum of squared differences from the observed ratios. Let the observed ratios be $f_1,\ldots,f_n$, and consider a candidate factor $a$. The total squared difference is $Q(a)=\sum_{i=1}^n(f_i-a)^2$. Differentiating with respect to $a$, setting the derivative to zero, and solving gives the arithmetic mean:

\begin{aligned} Q(a)&=\sum_{i=1}^{n}(f_i-a)^2,\\ \frac{dQ}{da}&=-2\sum_{i=1}^{n}(f_i-a)=0,\\ na&=\sum_{i=1}^{n}f_i,\\ a&=\frac{1}{n}\sum_{i=1}^{n}f_i=\bar f. \end{aligned}

This derivation explains a mathematical property of the arithmetic mean; it does not prove that the selected observations are representative of future development. It also does not imply that squared error is the only relevant criterion for selecting a development factor. The actuarial judgment lies partly in identifying a suitable set of observations before applying the calculation.

Key takeaway: In a simple arithmetic average, every selected link ratio receives weight $1/n$. The size of the underlying cumulative loss does not change that weight.

Applying an Average Factor

Once an average factor has been selected for a particular development interval, it can be applied to an observed cumulative loss at the earlier age to estimate the amount at the later age. The calculation is multiplication: later-age estimate equals earlier-age cumulative amount multiplied by the selected link ratio. This is an age-to-age projection, not a complete projection to ultimate unless the later age is ultimate or additional development factors are applied.

If the average factor has repeating decimals, keep sufficient precision during the calculation and round only the requested final result. Rounding each individual ratio prematurely can change the average, especially when many observations are selected or the ratios are close together.

Worked Example: Equal Weight Despite Different Loss Volumes

A hypothetical commercial property portfolio has three selected 24-to-36-month cumulative paid-loss pairs, shown in thousands: 600 to 720; 2,400 to 2,640; and 6,000 to 6,300. Calculate the simple average link ratio and project a separate 24-month cumulative paid loss of $3,600,000 to 36 months.

Method. Divide each 36-month amount by its matching 24-month amount, average the resulting three ratios with equal weights, and multiply the selected factor by the separate 24-month amount.

The first ratio is $720/600=1.20$. The second is $2{,}640/2{,}400=1.10$. The third is $6{,}300/6{,}000=1.05$.

The sum of the ratios is $1.20+1.10+1.05=3.35$. The simple average is $3.35/3=1.116666\ldots$. Retain the unrounded value for the projection.

The projected 36-month amount is $\$3{,}600{,}000\times 1.116666\ldots=\$4{,}020{,}000.00$. As a check, using the exact fractional average $3.35/3$ gives $\$3{,}600{,}000\times 3.35/3=\$4{,}020{,}000.00$.

Common mistake avoided. The largest starting amount does not receive the greatest weight in a simple arithmetic average. For example, dividing the sum of the later amounts by the sum of the earlier amounts would give $9{,}660/9{,}000=1.0733\ldots$, not the simple average of 1.1167. That calculation answers a different weighting question. The requested simple average assigns one-third of the weight to each of the three ratios.

Conditions and Selection Cautions

The arithmetic itself is uncomplicated, but the result is meaningful only when the observations can sensibly be averaged. Before calculating, check the following:

Same development interval. Each ratio must refer to the same earlier and later ages. A 12-to-24-month ratio is not an observation of the 24-to-36-month development pattern.
Consistent measure. Use the same type of cumulative loss and expense measure across observations. Differences in coverage, valuation practice, or loss basis may make ratios noncomparable.
Usable denominators. The earlier cumulative amount must be positive for the ordinary ratio calculation. A zero denominator makes the ratio undefined. A very small denominator can produce an extreme ratio, so its inclusion requires review rather than automatic averaging.
Reasoned selection. Equal weighting does not make a data point reliable or appropriate. Investigate known data issues, structural changes, and unusual observations before deciding which historical ratios to include.

An arithmetic mean can be sensitive to an unusually high or low ratio because that ratio still receives the same one-observation weight. Do not delete an observation merely because it differs from the others; first determine whether there is a data error or a relevant change in the underlying process. Conversely, do not assume that every available historical ratio belongs in the average simply because it can be calculated.

Worked Example: A Selected Window With an Unrounded Average

A hypothetical specialty insurer has selected four 12-to-24-month incurred link ratios for an analysis: 1.08, 1.12, 1.00, and 1.20. The selection review has already determined that all four are usable and comparable. Find the simple average and apply it to a separate 12-month incurred amount of $7,250,000. Report the factor to four decimal places and the projected amount to the nearest cent.

Method. Add the four selected ratios and divide by four. Then apply the unrounded average to the separate cumulative amount.

The sum is $1.08+1.12+1.00+1.20=4.40$. Dividing by four gives $4.40/4=1.10$. To four decimal places, the selected factor is 1.1000.

The projected amount is $\$7{,}250{,}000\times1.10=\$7{,}975{,}000.00$. The factor is applied to the 12-month amount because the factor covers the 12-to-24-month interval.

Check. The selected factor, 1.10, is within the observed range from 1.00 to 1.20. The projection is 10% above the 12-month amount, consistent with multiplying by 1.10. The common mistake avoided is applying this factor to a cumulative amount already at 24 months, which would project from the wrong development age.

Common Exam Traps

The most frequent errors are usually not difficult arithmetic; they are mistakes about what is being averaged or what the resulting factor applies to.

Reversing the ratio. For a 12-to-24-month link ratio, divide the 24-month cumulative amount by the 12-month cumulative amount, not the other way around.
Using the wrong number of observations. Divide the sum by the number of selected ratios, not by the number of cumulative amounts in the data table or by the number of development ages.
Changing to a ratio of totals. Averaging individual ratios gives each ratio equal weight. Combining all later-age amounts and dividing by all earlier-age amounts is not the same calculation.
Projecting from the wrong age. A factor applies to an amount at its stated starting age. Check that the cumulative amount and the factor interval match.
Rounding too early. Retain unrounded ratios and averages through the projection when possible, then round the final answer as requested.

A short calculation check can catch several of these errors. Confirm that the selected factor is the arithmetic mean of the stated ratios, that it lies between the smallest and largest ratio, and that the projected amount is the starting cumulative amount multiplied by the factor for the correct interval.

Check Your Understanding

For each problem, calculate the requested simple average using equal weight for each selected observation. Keep the development ages and units in view.

A hypothetical insurer selects three 12-to-24-month ratios: 1.04, 1.10, and 1.16. Find the simple arithmetic average. (Answer: 1.1000)
Three selected 24-to-36-month cumulative paid-loss pairs, in thousands, are 1,500 to 1,650; 2,500 to 2,750; and 4,000 to 4,200. Find the simple average ratio and project a separate 24-month amount of $6,000,000 to 36 months. (Answer: average factor 1.0833; projected amount $6,500,000.00)
Four selected 12-to-24-month incurred ratios are 0.98, 1.02, 1.10, and 1.06. Find their simple arithmetic average and state whether it represents an increase or decrease from 12 to 24 months. (Answer: 1.0400, an average increase of 4%)
Three selected 36-to-48-month ratios are calculated from cumulative amounts, in thousands: 2,400 to 2,760; 4,000 to 4,200; and 9,000 to 9,900. Find the simple average and project a separate 36-month amount of $5,250,000 to 48 months. (Answer: average factor 1.1000; projected amount $5,775,000.00)
A selected 12-to-24-month set contains three ratios: 1.05, 1.15, and 1.10. A candidate instead calculates a factor by dividing the sum of the three 24-month amounts by the sum of the three 12-month amounts. Is that necessarily the simple arithmetic average? State the correct simple average of the three given ratios. (Answer: No; the simple average is 1.1000, and a ratio of aggregate amounts is not necessarily the simple arithmetic average.)

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