Why Diagnose Link Ratios?
A selected chain ladder factor is applied to observed cumulative losses to project losses at later development ages. Before relying on that factor, an actuary should examine the link ratios behind it. A factor can summarize several observations while hiding a pattern: ratios may change over accident years, vary sharply from year to year, or be heavily influenced by a particular observation.
Diagnostics do not determine a selection automatically. They help identify which observations support a proposed factor and which deserve further investigation. In particular, a visible unusual value is a prompt to check the underlying data and context, not a reason by itself to delete the value or replace it.
The diagnostic is meaningful only in the context of the data and calculation that produced it. Check that the cumulative values use consistent definitions, that the development ages are comparable, and that the denominator is positive. The basic ratio is not defined when \(C_{i,j}=0\); a negative or otherwise problematic cumulative value also requires care rather than routine ratio interpretation.
Reading a Link Ratio Table and Plot
A useful first view is a table with accident years as rows and development ages as columns. Each available cell is one observed link ratio. A plot can then show the ratios at one development age across accident years, or show the development-age profile for each accident year. Keep the accident-year order chronological: rearranging the years can conceal a gradual change or make a run of similar values harder to see.
| Diagnostic view | What it can reveal | What it cannot establish alone |
|---|---|---|
| Ratios at one development age, plotted by accident year | Whether ratios fluctuate, cluster, or appear to rise or fall across years | The cause of a trend or whether it will continue |
| Development-age profiles by accident year | Whether years follow similar shapes across successive ages | That every individual year should use the same factor |
| Table of ratios with cumulative-loss denominators | Which observations have greater denominator volume and may influence a volume-weighted factor more | That large-volume observations are automatically correct or representative |
A straight-line impression on a plot is only descriptive evidence. With a small number of accident years, apparent patterns can arise from ordinary variation. Consider the size and direction of differences, whether they persist across adjacent years, and whether the pattern occurs at one age or across several ages. Also examine the data underlying any surprising cell before interpreting its meaning.
The volume-weighted link ratio at development age \(j\), when all included denominators are positive, is the sum of next-age cumulative losses divided by the sum of current-age cumulative losses. Since \(C_{i,j+1}=C_{i,j}f_{i,j}\), it is a weighted average of the individual ratios, with weights proportional to \(C_{i,j}\):
This identity explains why a plot and a selected factor answer different questions. The plot shows the observations and their pattern; the volume-weighted factor compresses them into one summary that gives more weight to larger denominators. Neither view replaces the other.
Worked Diagnostics
Worked Example: Checking Stability at One Development Age
A hypothetical insurer has the following cumulative losses at 12 and 24 months for five accident years. Calculate the link ratios, describe their pattern, and compare the simple-average and volume-weighted summaries.
| Accident year | 12-month cumulative | 24-month cumulative |
|---|---|---|
| 2019 | $480 | $624 |
| 2020 | $520 | $676 |
| 2021 | $600 | $750 |
| 2022 | $640 | $832 |
| 2023 | $700 | $910 |
Method. Calculate each 12-to-24-month ratio and inspect it in accident-year order. The purpose is to assess consistency, not to infer a causal explanation for variation.
For 2019, \(624/480=1.3000\). For 2020, \(676/520=1.3000\). For 2021, \(750/600=1.2500\). For 2022, \(832/640=1.3000\). For 2023, \(910/700=1.3000\). Thus the chronological series is \(1.3000, 1.3000, 1.2500, 1.3000, 1.3000\). Four observations are 1.3000 and one is 1.2500; the values do not show a progressive rise or fall, although the 2021 observation is lower.
The simple average is \( (1.3000+1.3000+1.2500+1.3000+1.3000)/5=6.4500/5=1.2900\). For the volume-weighted ratio, total 24-month cumulative losses are \(624+676+750+832+910=3{,}792\), and total 12-month cumulative losses are \(480+520+600+640+700=2{,}940\). Therefore the weighted ratio is \(3{,}792/2{,}940=1.2897959\), or 1.2898 to four decimal places.
Check. The weighted ratio, 1.2898, lies between the lowest and highest observations, 1.2500 and 1.3000, as a positive weighted average should. It is close to the simple average because the denominators are of comparable size. The common mistake is to treat the single lower ratio as proof of a sustained trend; the ordered values show no such trend in this small set. The observation may still merit review, but this diagnostic alone does not justify excluding it.
Worked Example: Detecting a Pattern Across Accident Years
For a hypothetical liability portfolio, an analyst tabulates the observed ratios below. Describe the pattern at each development age and decide what the table warrants investigating before a factor is selected.
| Accident year | 12–24 months | 24–36 months |
|---|---|---|
| 2019 | 1.50 | 1.20 |
| 2020 | 1.46 | 1.19 |
| 2021 | 1.40 | 1.20 |
| 2022 | 1.34 | 1.18 |
| 2023 | 1.30 | — |
Method. Compare each development-age column across accident years, and do not treat the missing 24-to-36-month observation for 2023 as a zero. It is unavailable, not an observed ratio of zero.
At 12–24 months, the values decrease in chronological order: 1.50, 1.46, 1.40, 1.34, and 1.30. The total decrease is \(1.50-1.30=0.20\). The four successive changes are \(-0.04\), \(-0.06\), \(-0.06\), and \(-0.04\). This is a consistent downward pattern in the observed series.
At 24–36 months, the available values are 1.20, 1.19, 1.20, and 1.18. Their sum is \(1.20+1.19+1.20+1.18=4.77\), so their simple average is \(4.77/4=1.1925\). The range is \(1.20-1.18=0.02\), much narrower than the range of 0.20 at 12–24 months.
Interpretation. The first development age merits investigation because the observed ratios move down steadily across accident years. The second age appears comparatively stable in these four observations, but it has fewer available accident years. The table does not identify why the first pattern occurred, and the missing cell cannot be filled in by assuming it equals the column average.
Check. The calculated 24–36-month mean, 1.1925, is within that column’s observed range of 1.18 to 1.20. The common mistake is to read the dash as zero; doing so would incorrectly change both the average and the apparent pattern. A diagnostic should preserve the distinction between unavailable and observed values.
Worked Example: Measuring the Influence of a Small-Denominator Observation
A hypothetical insurer is reviewing five 12-to-24-month ratios. Four accident years have large 12-month cumulative losses, while the fifth has a much smaller denominator. Calculate the simple and volume-weighted ratios, then quantify how much the fifth observation changes the weighted result.
| Accident year | 12-month cumulative | Link ratio | 24-month cumulative |
|---|---|---|---|
| 2019 | $800 | 1.25 | $1,000 |
| 2020 | $850 | 1.27 | $1,079.50 |
| 2021 | $900 | 1.26 | $1,134 |
| 2022 | $950 | 1.25 | $1,187.50 |
| 2023 | $100 | 1.55 | $155 |
Method. Compare the individual ratios with their denominators, then calculate the weighted result both with and without the final observation. This measures its numerical influence; it does not decide whether it should be included.
The simple average of all five ratios is \((1.25+1.27+1.26+1.25+1.55)/5=6.58/5=1.3160\). The sum of the denominators is \(800+850+900+950+100=3{,}600\). The sum of the next-age amounts is \(1{,}000+1{,}079.50+1{,}134+1{,}187.50+155=4{,}556\). Thus the volume-weighted ratio is \(4{,}556/3{,}600=1.265555\ldots\), or 1.2656 to four decimal places.
Without the fifth observation, the denominator total is \(800+850+900+950=3{,}500\), and the next-age total is \(1{,}000+1{,}079.50+1{,}134+1{,}187.50=4{,}401\). The resulting weighted ratio is \(4{,}401/3{,}500=1.257428\ldots\), or 1.2574. Including the fifth observation therefore raises the weighted ratio by approximately \(1.2656-1.2574=0.0082\), using the rounded displayed results. The fifth ratio raises the simple average much more: \(1.3160-1.2575=0.0585\), where 1.2575 is the simple average of the first four ratios.
Check. The weighted result with all observations, 1.2656, is between the minimum 1.25 and maximum 1.55. The fifth observation is conspicuous in the plot but has only \(100/3{,}600=0.02778\), or about 2.78%, of the total denominator weight. The common mistake is to assume that the largest ratio must dominate a volume-weighted selection. Here it moves the result, but its small denominator limits its influence. The observation should be checked on its merits, not automatically discarded because it looks unusual.
A Practical Diagnostic Sequence
A disciplined review separates calculation, pattern recognition, and actuarial judgment. The following sequence keeps those steps visible.
Confirm that each ratio is the next-age cumulative amount divided by the current-age cumulative amount, with consistent units and definitions. Check denominators and whether missing values are genuinely unavailable.
Tabulate each development age separately and retain chronological accident-year order. Plot the individual ratios rather than only a single selected factor.
Note ranges, clusters, runs, and apparent direction of change. Compare the amount of variation among ages and across years, while recognizing that a small number of observations limits conclusions.
Use the denominators to understand the influence of observations on a volume-weighted summary. Check unusual cells against source data and relevant portfolio information before deciding how they affect selection.
Common Traps and Why the Diagnostics Matter
One common error is to infer stability from a plausible overall average. A factor near 1.30 could summarize both a stable series around 1.30 and a declining series from 1.50 to 1.10. Those patterns have different implications for whether one common factor is a reasonable summary. The ratio table and plot expose information that the average hides.
Another error is to confuse the number of observations with the amount of data they represent. A simple average gives each ratio equal weight; a volume-weighted average gives greater weight to observations with larger current-age cumulative losses. Neither is universally correct in every setting. The point of diagnosis is to understand the data and the consequences of the chosen method, rather than to select a method mechanically.
Finally, a trend across accident years is a signal to investigate, not an explanation. Diagnostics by accident year and development age identify where a pattern appears, but do not establish its cause. Avoid turning a visual pattern into a causal claim without supporting information. The later analysis of outlier selection and calendar-year effects addresses distinct questions; the basic task here is to describe the observed pattern and its potential impact on a selected factor.
Check Your Understanding
For each hypothetical problem, calculate or interpret the requested diagnostic before consulting the answer.
- A portfolio has cumulative losses of $360 at 12 months and $450 at 24 months for one accident year, and $500 and $625 for another. Calculate both link ratios and state whether they match. (Answer: Both ratios are 1.2500; the two observations match.)
- At one development age, chronological accident-year ratios are 1.42, 1.38, 1.34, and 1.30. Calculate the total change and describe the pattern without claiming a cause. (Answer: The total change is \(-0.12\); the observed sequence declines at each step.)
- Four link ratios are 1.10, 1.12, 1.11, and 1.13, with equal denominators. Find the simple average and volume-weighted average. (Answer: Both are 1.1150.)
- Two observations have ratios 1.20 and 1.50 with current-age cumulative losses of $900 and $100, respectively. Calculate the volume-weighted ratio and explain why it differs from the simple average. (Answer: \((900(1.20)+100(1.50))/1{,}000=1.2300\); it is below the simple average of 1.3500 because the lower ratio has the larger denominator weight.)
- A table shows ratios of 1.25, 1.29, a dash, and 1.31 across four successive accident years at one age. What should the analyst do with the dash when describing the observed pattern? (Answer: Treat it as unavailable, not as zero; describe the three observed ratios and investigate why the cell is missing before drawing broader conclusions.)