Simple-Average Link Ratio Selections

Why Average Link Ratios?

In chain ladder estimation, an age-to-age link ratio measures how cumulative losses at one development age compare with cumulative losses at the next. Selecting a factor for each development age turns the observed pattern into a projection rule. One possible selection is the simple arithmetic average of the observed link ratios at that age. Its defining feature is that each eligible accident year contributes one ratio, whether that year has a small or large cumulative loss amount.

That equal-year weighting is easy to apply, but it has an important consequence: a ratio from a low-volume year can affect the selected factor as much as a ratio from a high-volume year. The selection method therefore should be stated explicitly, and the selected factor should be interpreted in light of the pattern and the data behind it.

Definition: Let $C_{i,j}$ be cumulative losses for accident year $i$ evaluated at development age $j$. When both $C_{i,j}$ and $C_{i,j+1}$ are observed and $C_{i,j}\ne 0$, the age-to-age link ratio for accident year $i$ at age $j$ is $$ r_{i,j}=\frac{C_{i,j+1}}{C_{i,j}}. $$ If there are $n_j$ eligible observed ratios at that development age, the simple-average selected link ratio is $$ \bar{f}_j=\frac{1}{n_j}\sum_{i\in E_j}r_{i,j}, $$ where $E_j$ is the set of accident years with usable observations at both ages $j$ and $j+1$. Each eligible accident year has the same weight in this arithmetic average.

Eligibility and Calculation

A ratio is eligible for an age-to-age average only when the same accident year has a cumulative loss observation at both the starting and ending ages of the link. For example, calculating a 12-to-24-month ratio requires both the 12-month and 24-month cumulative values. An accident year with a 12-month value but no 24-month value cannot contribute a ratio to that selection.

The comparison should also be meaningful: the two values should use a consistent loss basis, and the development ages should represent comparable intervals across accident years. A zero starting value makes the ratio undefined. A very small denominator can produce an unusually large ratio, so even a mathematically calculable ratio may warrant investigation before it is included.

The simple average is calculated separately for each development age. It is not an average of every link ratio in the triangle combined. Thus, a 12-to-24-month factor is selected from observed 12-to-24-month ratios, while a 24-to-36-month factor is selected from observed 24-to-36-month ratios. The number of eligible years can differ by age as the triangle becomes less complete at later ages.

Worked Example: Selecting a 12-to-24-Month Factor

A hypothetical insurer has the following cumulative paid losses, in thousands of dollars, for three accident years. Calculate the simple-average 12-to-24-month link ratio. Then use it to project an additional accident year with 300 thousand dollars at 24 months.

Accident year	12-month cumulative losses	24-month cumulative losses
2021	100	150
2022	400	440
2023	200	260

Step 1: Calculate each observed link ratio.

For 2021, $150/100=1.5000$. For 2022, $440/400=1.1000$. For 2023, $260/200=1.3000$. All three years have both required observations, so $n_{12}=3$.

Step 2: Take the arithmetic average.

\bar{f}_{12}=\frac{1.5000+1.1000+1.3000}{3} =\frac{3.9000}{3}=1.3000.

Step 3: Project the additional year. Applying the selected 12-to-24-month factor to 300 thousand dollars gives $300(1.3000)=390$ thousand dollars at 36 months only if the starting observation is at 24 months and the selected factor is for 24-to-36 months. Here the selected factor is specifically 12-to-24 months. Therefore, it cannot be applied to a 24-month value. The requested projection is not supported by this selection alone; a 24-to-36-month factor is needed.

This is an intentional development-age check: the simple average is 1.3000 for the 12-to-24-month interval, but the question’s 300-thousand-dollar value is at 24 months. Applying 1.3000 to it would use a factor at the wrong age. Method: calculate the factor only from matching development-age pairs. This avoids the common error of treating a selected factor as if it applied at every age.

Why the Method Gives Equal Weight to Each Year

The arithmetic average can be justified as the factor that minimizes the sum of the squared differences between the observed ratios and one common selected factor, with each accident year contributing one squared difference. This is a description of the calculation’s equal-observation weighting, not a claim that the factor is always the best choice for every triangle.

Q(f)=\sum_{i\in E_j}(r_{i,j}-f)^2.

Differentiate with respect to $f$, set the derivative equal to zero, and solve:

\frac{dQ}{df}=-2\sum_{i\in E_j}(r_{i,j}-f)=0 \quad\Longrightarrow\quad \sum_{i\in E_j}r_{i,j}-n_jf=0 \quad\Longrightarrow\quad f=\frac{1}{n_j}\sum_{i\in E_j}r_{i,j}=\bar{f}_j.

The second derivative is $d^2Q/df^2=2n_j>0$, so this solution is the minimum. Because each term appears once in the sum, the calculation assigns each eligible accident year equal weight in squared-ratio space. It does not assign equal weight to each dollar of cumulative losses.

Worked Example: Selecting Factors at Two Development Ages

A hypothetical property insurer has the following cumulative losses, in thousands of dollars. Select simple-average factors for 12-to-24 and 24-to-36 months. Then project Accident Year C from its observed 24-month value to 36 months. Accident Year D has only a 12-month observation.

Accident year	12 months	24 months	36 months
A	120	180	216
B	250	275	330
C	160	208	Not observed
D	90	Not observed	Not observed

Step 1: Select the 12-to-24-month factor. A, B, and C each have both required observations. Their ratios are $180/120=1.5000$, $275/250=1.1000$, and $208/160=1.3000$. Therefore,

\bar{f}_{12}=\frac{1.5000+1.1000+1.3000}{3}=1.3000.

Step 2: Select the 24-to-36-month factor. Only A and B have both a 24-month and a 36-month value. Their ratios are $216/180=1.2000$ and $330/275=1.2000$. C is not eligible for this factor because its 36-month value is missing. Thus,

\bar{f}_{24}=\frac{1.2000+1.2000}{2}=1.2000.

Step 3: Project Accident Year C. C has 208 thousand dollars at 24 months. The factor selected for 24-to-36 months applies at this age, so the projection is $208(1.2000)=249.6$ thousand dollars.

The result is 249.6 thousand dollars at 36 months. A does not enter the 24-to-36-month average more than once because it has a larger loss amount; B likewise contributes one ratio. D contributes to neither calculation because it lacks the later observation required to form a link ratio. Method: make a separate eligibility set and average for each link age. This avoids including incomplete years or carrying a factor from one development age to another.

Equal Accident-Year Weight Is Not Equal Loss-Volume Weight

A simple-average selection treats the observed ratios as the values being averaged. Consequently, an accident year with a small denominator has the same influence on the selected factor as one with a large denominator. This can be useful when the actuary wants each year’s observed development experience to have equal standing, but it can also make the result sensitive to a volatile ratio.

A frequent exam trap is to add the ending cumulative losses and divide by the sum of the starting cumulative losses, then call the result the simple average of the ratios. Those are different calculations. The first calculation combines dollars before dividing; the simple average first forms each accident year’s ratio, then averages those ratios. The following example makes the distinction visible.

Worked Example: A High Ratio From a Low-Volume Year

A hypothetical specialty insurer observes the following 12-to-24-month cumulative paid losses, in thousands of dollars. Calculate the simple-average factor and project a separate accident year with 300 thousand dollars at 12 months.

Accident year	12-month losses	24-month losses	12-to-24 ratio
2020	1,000	1,100	1.1000
2021	50	100	2.0000
2022	500	600	1.2000

Step 1: Average the three ratios.

\bar{f}_{12}=\frac{1.1000+2.0000+1.2000}{3} =\frac{4.3000}{3}=1.4333\overline{3}.

Step 2: Apply the factor to the target year. The projected 24-month amount is $300(1.4333\overline{3})=430$ thousand dollars.

Common mistake and correction. A mistaken calculation combines the totals first: $ (1{,}100+100+600)/(1{,}000+50+500)=1{,}800/1{,}550$, approximately 1.1613. That is not the arithmetic average of the three observed ratios. The required simple average is 1.4333 to four decimal places, so the projection is 430 thousand dollars. The 2.0000 ratio from the smallest accident year receives one-third of the simple-average weight, just like each other ratio. The large difference between the two calculations demonstrates why the method must be identified rather than inferred from the word “average.”

When to Use the Selection Carefully

A simple arithmetic average is a transparent way to summarize observed development at a particular age. It is most interpretable when the eligible accident years are reasonably comparable and the ratios are credible representations of the development pattern intended for projection. The calculation itself does not establish that those conditions hold. Judgment is still needed about whether the years are comparable and whether unusual observations reflect data issues or genuine development.

The method can be sensitive to extreme ratios because the arithmetic average includes each ratio’s full numerical value. Before selecting it, check that cumulative values are on a consistent basis, that denominators are nonzero and not anomalously small, and that each ratio belongs to the intended development interval. If an observed ratio is unusual, investigate the underlying experience and selection rationale rather than silently omitting it.

Do not confuse the arithmetic averaging rule with the broader chain ladder projection process. The selected factor applies to its own development age. To project an immature accident year through several ages, apply the relevant factors in sequence, with each factor matched to the corresponding age interval. If an age-specific factor is unavailable or not considered appropriate, the simple average alone does not supply a justified replacement.

Key takeaway: For a simple-average link ratio selection, calculate one ratio for every eligible accident year at the same development age, add those ratios, and divide by the number of eligible years. The result weights accident years equally, not loss dollars equally.

Check Your Understanding

For each problem, identify the applicable development age before averaging. Answers are rounded to four decimal places for factors unless stated otherwise.

At one development age, three hypothetical accident years have cumulative values of 100 and 130, 200 and 240, and 50 and 75, respectively. What is the simple-average link ratio? (Answer: 1.3333)
For 24-to-36 months, two eligible years have cumulative losses of 180 and 216, and 250 and 300. A third year has only a 24-month value of 400. Select the factor and project a different year with 315 at 24 months. (Answer: factor 1.2000; projected 36-month losses 378)
Three eligible ratios at a particular age are 0.9500, 1.0500, and 1.2000. Calculate the simple-average factor and project 600 at the starting age. (Answer: factor 1.0667; projected amount 640)
Two years have starting and ending cumulative values of 10 and 20, and 90 and 99. Calculate the simple-average factor. Explain why the quotient of summed ending values divided by summed starting values is not the requested factor. (Answer: simple-average factor 1.5500; the quotient of sums is 1.1900 and is not the arithmetic average of the two ratios)
A year has 200 at 12 months. The selected simple-average factors are 1.2500 for 12-to-24 months and 1.1200 for 24-to-36 months. Project its 36-month amount by applying the factors in sequence. (Answer: 280)

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