Outlier Link Ratios and Robust Selection

Why Outlier Link Ratios Require Judgment

A chain ladder factor summarizes observed development from one age to the next. An unusually high or low link ratio can materially influence that summary, especially when only a few accident years are available. Yet an extreme observation is not automatically an error: it may reflect a data problem, a one-time event, or genuine development that could recur. The analyst’s task is to investigate the observation and select a factor that is appropriate for the intended projection, rather than to remove values simply because they look unusual.

This tutorial focuses on the effect of outliers on factor selection. It uses the simple-average, median, and volume-weighted methods as comparison tools, then considers retaining, excluding, or down-weighting a suspect observation. These are alternative judgment choices, not a mechanical rule for deciding what the “correct” factor must be.

Definition: For accident year $i$, let $C_{i,j}$ denote cumulative loss at development age $j$. When $C_{i,j}$ is positive, the observed link ratio from age $j$ to age $j+1$ is $r_{i,j}=C_{i,j+1}/C_{i,j}$. An outlier link ratio is an observation that is unusually high or low relative to the other ratios being considered. Being unusual is a diagnostic signal, not proof that the observation is invalid.

A useful assessment starts with the data behind the ratio. Verify that the two cumulative values have the intended valuation dates, definitions, and units, and that the ratio was calculated correctly. Then investigate whether a known event or change explains the development. The observation’s size, the number of other years available, and whether the cause is expected to recur all matter when selecting a factor.

How an Outlier Affects Common Selections

A simple-average factor gives each observed link ratio equal weight. A median factor is the middle ordered ratio (or the average of the two middle ratios when there is an even number). A volume-weighted factor gives more influence to observations with larger cumulative losses at the beginning of the development period.

\hat f_{\text{simple}}=\frac{1}{n}\sum_{i=1}^{n} r_{i,j}, \qquad \hat f_{\text{median}}=\operatorname{median}(r_{1,j},\ldots,r_{n,j}), \qquad \hat f_{\text{volume}}= \frac{\sum_{i=1}^{n} C_{i,j+1}}{\sum_{i=1}^{n} C_{i,j}}.

The volume-weighted factor can also be written as a weighted average of the link ratios, with starting cumulative losses as weights:

\hat f_{\text{volume}} = \frac{\sum_{i=1}^{n} C_{i,j}r_{i,j}}{\sum_{i=1}^{n} C_{i,j}}.

This identity follows directly from the link ratio definition. Since $C_{i,j+1}=C_{i,j}r_{i,j}$, summing the next-age values gives $\sum_i C_{i,j+1}=\sum_i C_{i,j}r_{i,j}$. Dividing by $\sum_i C_{i,j}$ yields the weighted-average form. Thus, an extreme ratio from a large starting cumulative value may have more effect on the volume-weighted selection than the same ratio from a small value.

Each method answers a different summarization question. None automatically identifies whether an unusual observation is erroneous or predictive. In particular, the median may resist the numerical influence of an extreme ratio, but that alone does not justify choosing it: it can also disregard meaningful information about the volume or circumstances behind the observation.

Worked Example: Measuring the Impact of One High Ratio

A hypothetical liability portfolio has five observed link ratios at one development age: 1.20, 1.22, 1.18, 1.90, and 1.21. Investigation confirms that the ratio of 1.90 is calculated from the stated cumulative values, but its underlying event is not expected to recur. Compare the simple average, median, and average after excluding that observation. For an accident year with cumulative loss of $600,000 at this age, project ultimate loss using each selected factor and a subsequent factor of 1.08.

Method. Calculate each candidate factor, then multiply the current cumulative loss by the selected factor and the subsequent factor. The comparison quantifies the outlier’s influence; the stipulated nonrecurring cause provides a rationale to consider an adjusted selection.

Simple average retaining all observations:

\hat f_{\text{simple}} =\frac{1.20+1.22+1.18+1.90+1.21}{5} =\frac{6.71}{5} =1.342.

Median retaining all observations: In order, the ratios are 1.18, 1.20, 1.21, 1.22, and 1.90. The middle value is 1.21.

Simple average excluding 1.90:

\hat f_{\text{exclude}} =\frac{1.20+1.22+1.18+1.21}{4} =\frac{4.81}{4} =1.2025.

Projection using the simple average: $600{,}000\times1.342=805{,}200$, then $805{,}200\times1.08=\$869{,}616.00$.

Projection using the median: $600{,}000\times1.21=726{,}000$, then $726{,}000\times1.08=\$784{,}080.00$.

Projection using the average excluding 1.90: $600{,}000\times1.2025=721{,}500$, then $721{,}500\times1.08=\$779{,}220.00$.

The calculations show that retaining the outlier in the simple average increases the projected ultimate by $90,396.00 compared with the average excluding it. The median is close to the exclusion result here, but that is a feature of these data, not a general guarantee. The result is checked by applying each factor in sequence to the same starting cumulative loss and subsequent factor. Common mistake avoided: removing 1.90 solely because it is the largest ratio, without investigating its cause or recording why its development should not inform the selection.

Retain, Exclude, or Down-Weight?

These options should be considered in light of the evidence, not applied as automatic outlier rules. Retaining an observation is appropriate when it is valid and represents development that is relevant to the projection. Exclusion may be defensible when a verified one-time event makes the observation unrepresentative of the future pattern; the excluded value and rationale should remain documented. Down-weighting is a middle course when the observation contains some relevant information but should have less influence than other observations.

Down-weighting requires an explicit method. For example, an analyst may compute a weighted arithmetic average in which each observation has a chosen relative weight $w_i$:

\hat f_w=\frac{\sum_i w_i r_{i,j}}{\sum_i w_i}.

A weight is a judgment about influence, not an objectively determined probability that the ratio is correct. It should be supported by a clear explanation, applied consistently where appropriate, and compared with results under other reasonable choices. Do not confuse this discretionary weighted average with the volume-weighted link ratio: volume weighting uses $C_{i,j}$ as the weights, while discretionary weights reflect the analyst’s documented selection judgment.

Worked Example: Assessing a Low Ratio in Incurred Development

A hypothetical incurred-loss triangle produces link ratios of 1.10, 1.11, 0.72, 1.12, and 1.10. The corresponding cumulative incurred losses at the start of development are $100,000, $200,000, $300,000, $400,000, and $500,000. The 0.72 ratio is traced to a large case-reserve release that is confirmed as a one-time event. Compare the volume-weighted factor with and without that observation, and calculate the projected ultimate for a current cumulative incurred amount of $1,000,000 using a subsequent factor of 1.05.

Method. Use the beginning and ending cumulative values to calculate the volume-weighted factor. Since ending values equal beginning values multiplied by their ratios, they are $110,000, $222,000, $216,000, $448,000, and $550,000.

Retaining the observation: The starting total is $100{,}000+200{,}000+300{,}000+400{,}000+500{,}000=\$1{,}500{,}000$. The ending total is $110{,}000+222{,}000+216{,}000+448{,}000+550{,}000=\$1{,}546{,}000$. Therefore:

\hat f_{\text{volume, retain}} =\frac{1{,}546{,}000}{1{,}500{,}000} =1.030666\ldots

The projection is $1{,}000{,}000\times1.030666\ldots=\$1{,}030{,}666.67$, followed by $\$1{,}030{,}666.67\times1.05=\$1{,}082{,}200.00$, rounding the final dollar amount to cents.

Excluding the observation: Remove the third row’s $300,000 starting amount and $216,000 ending amount. The remaining totals are $1,200,000 and $1,330,000, respectively.

\hat f_{\text{volume, exclude}} =\frac{1{,}330{,}000}{1{,}200{,}000} =1.108333\ldots

The projection is $1{,}000{,}000\times1.108333\ldots=\$1{,}108{,}333.33$, followed by $\$1{,}108{,}333.33\times1.05=\$1{,}163{,}750.00$, rounded to cents.

The exclusion case projects $81,550.00 more than the retention case. The confirmed, nonrecurring reserve release provides a reason to test an exclusion, but the selected factor still requires judgment about whether the remaining observations represent future development. Common mistake avoided: averaging the five ratios equally and calling the result volume-weighted. Volume weighting requires the cumulative-loss weights, or equivalently the ratio of the summed next-age to summed starting values.

Worked Example: Down-Weighting a Partly Informative High Ratio

A hypothetical portfolio has ratios 1.10, 1.12, 1.11, 1.13, and 1.60. The high ratio is valid, but review indicates that its unusual event may be partly relevant to future development. The analyst documents a decision to give that observation weight 0.25 and each other observation weight 1. Compare the weighted factor with the equal-weight average and with the result from excluding 1.60. For a current cumulative loss of $800,000, apply each factor followed by a subsequent factor of 1.05.

Method. Apply the stated weighted-average formula; the weights are an explicit selection judgment.

Equal-weight average: The sum is $1.10+1.12+1.11+1.13+1.60=6.06$, so the factor is $6.06/5=1.212$.

Excluding 1.60: The remaining sum is $1.10+1.12+1.11+1.13=4.46$, so the factor is $4.46/4=1.115$.

Down-weighting 1.60: The weighted numerator is $4.46+(0.25\times1.60)=4.46+0.40=4.86$. The total weight is $4+(0.25)=4.25$. Thus:

\hat f_w=\frac{4.86}{4.25}=1.143529\ldots

With the equal-weight factor, the projection is $800{,}000\times1.212=\$969{,}600$, then $\$969{,}600\times1.05=\$1{,}018{,}080.00$. With the exclusion factor, it is $800{,}000\times1.115=\$892{,}000$, then $\$892{,}000\times1.05=\$936{,}600.00$. With the down-weighted factor, it is $800{,}000\times1.143529\ldots=\$914{,}823.53$, then $\$914{,}823.53\times1.05=\$960{,}564.71$, rounding projected dollar amounts to cents.

The down-weighted result falls between the retain and exclude results, as expected for this choice of positive weights. This is a numerical check, not proof that 0.25 is the uniquely correct weight. Common mistake avoided: presenting a discretionary weight as if it were mandated by the link ratio data. The file should state why this observation receives less influence and show the sensitivity to plausible alternatives.

Selection Discipline and Common Traps

A factor selection should be supported by both calculation and context. A concise record can identify the unusual observation, its cause or unresolved uncertainty, the method and weights used, and the effect on projected ultimates under alternatives. If the cause is unknown, it is preferable to show sensitivity and qualify the judgment rather than assert that the ratio is an error.

The chain ladder projection applies selected development factors to cumulative losses at the applicable age. As a result, an outlier can affect not only its own age-to-age factor but also every projection that uses that factor. The size of the ultimate impact depends on the current cumulative amount and any subsequent factors. Comparing only the link ratios can understate the practical importance of the selection.

“The most extreme ratio must be removed.” An unusual ratio may be valid and predictive. Investigate before changing its influence.
“The median is always the robust answer.” The median limits the direct influence of a magnitude-extreme ratio, but it does not account for cumulative volume or establish that an observation should be ignored.
“Exclusion means the data are deleted.” Preserve the observation in the analysis record even if a selected calculation excludes it.
“Any weighted average is volume-weighted.” Volume weighting uses starting cumulative losses. Other weights are separate judgment choices and should be labelled accordingly.
“A small change in the factor is always immaterial.” Translate the factor difference into projected dollars, especially when the cumulative losses or remaining development are large.

Key takeaway: An outlier is a reason to investigate and test sensitivity, not an automatic instruction to remove data. Retain, exclude, or down-weight only with a supportable rationale, and show how the choice changes the projected ultimate.

Check Your Understanding

For each problem, calculate the requested factor or projection and check the influence of the unusual observation.

At one age, five link ratios are 1.05, 1.07, 1.08, 1.09, and 1.41. Calculate the equal-weight simple average and the median. (Answer: simple average 1.140; median 1.080)
Ratios are 1.08, 1.10, 0.65, 1.11, and 1.09. If the 0.65 ratio is excluded for a documented one-time event, calculate the simple average of the remaining ratios. (Answer: 1.095)
Four rows have starting cumulative losses of $100,000, $200,000, $300,000, and $400,000, with respective link ratios 1.05, 1.10, 1.20, and 1.15. Calculate the volume-weighted factor. (Answer: 1.145)
Ratios are 1.20, 1.22, 1.18, and 1.80. Give the first three observations weight 1 and the last weight 0.5. Calculate the weighted average factor. (Answer: 1.2857, to four decimal places)
A selected factor is 1.08 and the subsequent factor is 1.04. Project ultimate loss from cumulative loss of $2,500,000 at the current age. (Answer: $2,808,000.00)

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