Why Plot Link Ratios?
A list of link ratios can hide how the observations relate to one another. The same ratios, arranged by development age or calendar period, can make a systematic pattern easier to see: for example, whether later origin periods have higher ratios at a particular age, or whether ratios associated with a calendar period look unusual across several ages. A plot is a diagnostic display, not a method for automatically selecting a factor or proving why a pattern occurred.
The starting point is a cumulative loss development triangle. Let \(C_{i,j}\) denote cumulative losses for origin period \(i\), evaluated at development age \(j\). In this tutorial, age 1 means 12 months, age 2 means 24 months, and so on. The link ratio from age \(j\) to age \(j+1\) compares successive cumulative values for the same origin period.
For calendar labeling, assume age 1 for origin period \(i\) is valued in calendar year \(i\). The link from age \(j\) to \(j+1\) then ends in calendar year \(i+j\). A link spans two valuations, so state the convention and label the plotted calendar period clearly; otherwise, two analysts can appear to disagree merely because one labels the link by its starting valuation and the other by its ending valuation.
What the Two Views Reveal
A development-age view compares like with like in maturity: every point in a given age group is a link ratio for the same transition, such as 12-to-24 months. This helps reveal whether the ratios at that age have been broadly stable, have moved over successive origin periods, or include an observation that stands apart. It does not, by itself, explain the reason for a movement.
A calendar-period view rearranges those same observations along diagonals of a triangle. It can reveal a pattern shared by ratios associated with a particular calendar period, even though those ratios come from different origin periods and development ages. That arrangement is useful for asking whether the apparent change is concentrated in time rather than at one development age. It does not establish a cause or make unlike development ages directly comparable.
The calendar grouping above uses the ending-valuation convention just defined. In either view, plot the actual observed ratios and retain their labels. A connected line can help show sequence within a group, but should not suggest that unobserved triangle cells have been measured or that the values between points are known.
Worked Example: Reading a Development-Age Pattern
Worked Example: Reading a Development-Age Pattern
A hypothetical specialty insurer has the following cumulative losses, in thousands of dollars, for four origin years. Calculate and organize the observed link ratios, then describe what a development-age plot suggests.
| Origin year | 12 months | 24 months | 36 months | 48 months |
|---|---|---|---|---|
| 2021 | 100 | 150 | 180 | 198 |
| 2022 | 120 | 180 | 216 | 237.6 |
| 2023 | 110 | 176 | 220 | — |
| 2024 | 130 | 208 | — | — |
Step 1: Calculate each observed ratio. For 2021, the three ratios are \(150/100=1.50\), \(180/150=1.20\), and \(198/180=1.10\). For 2022, they are \(180/120=1.50\), \(216/180=1.20\), and \(237.6/216=1.10\). For 2023, the observed ratios are \(176/110=1.60\) and \(220/176=1.25\). For 2024, the observed ratio is \(208/130=1.60\).
Step 2: Group by development age. At 12-to-24 months, the plotted values for origin years 2021–2024 are 1.50, 1.50, 1.60, and 1.60. At 24-to-36 months, the values for 2021–2023 are 1.20, 1.20, and 1.25. At 36-to-48 months, the values for 2021–2022 are both 1.10. Each group has its own horizontal origin-year axis; the later-age groups have fewer observations because the triangle is incomplete.
Step 3: Interpret the display. The two most recent observed 12-to-24-month ratios are each \(1.60/1.50-1=0.0667\), or about 6.67%, above the earlier value of 1.50. The 24-to-36-month ratios are close to 1.20, with the 2023 value slightly higher at 1.25. The two observed 36-to-48-month ratios agree at 1.10, but two points are limited evidence of stability.
Method and check. This is a development-age plot: each comparison holds the age transition fixed. The direct calculations reproduce the listed cumulative amounts—for example, \(176\times1.25=220\)—so the 2023 24-to-36-month ratio is consistent with the triangle. The plot suggests a recent increase at the earliest transition; it does not establish whether that pattern will persist or what produced it.
Common mistake avoided: Do not place all nine ratios into one unlabelled sequence and interpret adjacent points as comparable. A 12-to-24-month ratio and a 36-to-48-month ratio describe different transitions.
Why the Calendar Diagonal Is Meaningful
A calendar-period plot is a reordering, not a new calculation. To see why the diagonal assignment follows, let origin period \(i\) be valued at age 1 in calendar period \(i\). The age-\(j\) valuation is then in period \(i+j-1\), and the age-\(j+1\) valuation is in period \(i+j\). Thus the link \(L_{i,j}\), which compares those two valuations, ends in calendar period \(i+j\). Holding \(i+j\) fixed traces a diagonal through the triangle.
This assignment gives a useful diagnostic: ratios at different ages and from different origins can be displayed together when their later valuations fall in the same calendar period. If several ratios on one diagonal move together, that alignment warrants investigation. The link ratios may still reflect different development transitions, so the visual alignment is a clue rather than a direct, controlled comparison.
Worked Example: Assigning Ratios to Calendar Periods
Worked Example: Assigning Ratios to Calendar Periods
A hypothetical insurer reports the following cumulative amounts, in thousands, for a new line of coverage. Calculate the ratios and place each into the calendar period of its later valuation.
| Origin year | 12 months | 24 months | 36 months | 48 months |
|---|---|---|---|---|
| 2021 | 80 | 120 | 144 | 158.4 |
| 2022 | 90 | 135 | 162 | 210.6 |
| 2023 | 100 | 160 | 208 | — |
| 2024 | 110 | 143 | — | — |
Step 1: Calculate the ratios by origin year. For 2021, the values are \(120/80=1.50\), \(144/120=1.20\), and \(158.4/144=1.10\). For 2022, they are \(135/90=1.50\), \(162/135=1.20\), and \(210.6/162=1.30\). For 2023, they are \(160/100=1.60\) and \(208/160=1.30\). For 2024, the one observed ratio is \(143/110=1.30\).
Step 2: Identify the later valuation calendar period. Under the stated convention, the 2021 links end in 2022, 2023, and 2024, respectively. The 2022 links end in 2023, 2024, and 2025. The 2023 links end in 2024 and 2025. The 2024 link ends in 2025.
| Later valuation calendar period | Link ratios plotted in that period |
|---|---|
| 2022 | 1.50 |
| 2023 | 1.20, 1.50 |
| 2024 | 1.10, 1.20, 1.60 |
| 2025 | 1.30, 1.30, 1.30 |
Step 3: Read the diagonal pattern cautiously. The three observations assigned to 2025 are all 1.30, while the three assigned to 2024 vary from 1.10 to 1.60. The 2025 alignment is a pattern worth investigating, but the ratios come from three different transitions: 36-to-48 months, 24-to-36 months, and 12-to-24 months. Their agreement is informative as a diagnostic display, not proof of a single cause.
Method and check. This is calendar grouping by the later valuation. The assigned values total \(1.30+1.30+1.30=3.90\) for 2025, matching the three calculated ratios from 2022, 2023, and 2024. A common error is to assign the ratios by origin year alone; that would place the three 2025 observations in three different groups and erase the diagonal alignment.
Recognizing Instability Without Overreacting
Plots should make variation visible, not make every difference look like a structural change. A single high or low point can arise for several reasons, including limited volume, data issues, or a genuine change in experience. The display identifies where to ask questions; it cannot decide which explanation applies. In particular, do not discard a point solely because it looks unusual. Preserve the observation, verify its underlying data and context, and document the investigation.
A useful first pass is to examine the scale and the pattern together. Ask whether values at one age move gradually across origins, whether a change appears across several ages on a calendar diagonal, and whether one observation is isolated from its peers. Note the number of observations: a flat line based on two points is not strong evidence that a pattern is stable. A plot also cannot repair a ratio whose denominator is zero or negative; such ratios require separate care before ordinary ratio interpretation is appropriate.
Worked Example: Distinguishing a Broad Pattern from One Unusual Point
Worked Example: Distinguishing a Broad Pattern from One Unusual Point
For a hypothetical portfolio, the cumulative amounts at 24 and 36 months are as follows, in thousands. Calculate the 24-to-36-month ratios and assess what a development-age plot communicates.
| Origin year | 24 months | 36 months |
|---|---|---|
| 2020 | 200 | 216 |
| 2021 | 250 | 275 |
| 2022 | 300 | 336 |
| 2023 | 400 | 580 |
Step 1: Calculate the four ratios. They are \(216/200=1.08\), \(275/250=1.10\), \(336/300=1.12\), and \(580/400=1.45\). The points for 2020–2022 form a narrow sequence from 1.08 to 1.12, while the 2023 point is higher.
Step 2: Quantify the contrast as a descriptive check. The four-point arithmetic mean is \((1.08+1.10+1.12+1.45)/4=4.75/4=1.1875\). The median is \((1.10+1.12)/2=1.11\), and the range is \(1.45-1.08=0.37\). These summaries do not replace the plot: the plot shows that the highest ratio is the latest point, rather than a gradual increase across all four origin years.
Step 3: State the diagnostic conclusion. The 2023 observation merits investigation because it is separated from the other three. The evidence is not enough to declare it erroneous, remove it, or conclude that the underlying process changed. Confirm the cumulative values and seek relevant information before deciding how the observation should influence a factor selection.
Method and check. This is an age-specific plot with simple descriptive summaries. The ratios reproduce their numerators when multiplied by their denominators; for example, \(400\times1.45=580\). The common trap is to calculate the mean and immediately call 1.1875 the representative link ratio. The mean is pulled upward by the separated observation and, by itself, conceals the sequence visible in the plot.
A Practical Plotting Sequence
Check that the displayed values are cumulative losses on a consistent basis and that each ratio compares successive ages for the same origin period.
Record the origin period, development-age transition, and calendar-period convention for every observed link. Do not fill unobserved triangle cells with assumed values.
Plot by development age to compare origin periods at a common transition; rearrange by calendar period to inspect diagonal alignment.
State the visible pattern and its limits: how many points support it, which ages or periods are involved, and whether the signal is broad or isolated. Seek supporting information before treating the pattern as an explanation or changing a selection.
Careful plotting is useful because factor selection depends on understanding the observations being considered, not merely applying a calculation to them. The development-age view helps expose movement at a common maturity; the calendar view helps expose diagonal alignment; and the plotted points make sparse data and isolated instability difficult to overlook. Neither view substitutes for judgment, verification, or a reasoned selection.
Check Your Understanding
For each problem, identify the plot or calculation that supports a sound diagnostic conclusion.
- A hypothetical origin year has cumulative losses of $240,000 at 12 months and $300,000 at 24 months. Calculate the 12-to-24-month link ratio. If age 1 is valued in the origin calendar year 2022, state the calendar year assigned to this link using the later-valuation convention. (Answer: 1.25; calendar year 2023)
- At 24-to-36 months, four successive origin-year ratios are 1.14, 1.15, 1.16, and 1.28. What is the mean of the first three, and which observation should the development-age plot prompt the analyst to review? (Answer: mean 1.15; review the 1.28 observation)
- Three ratios are observed in calendar year 2026: one from 12-to-24 months, one from 24-to-36 months, and one from 36-to-48 months. What does their placement in the same calendar group mean, and what does it not prove? (Answer: their later valuations fall in 2026, so the alignment warrants review; it does not prove a common cause or make the development ages identical)
- A triangle has origin years 2021–2024. Under the convention that age 1 is valued in the origin year, in which calendar year does the 2022 origin year's 24-to-36-month link end? (Answer: 2024)
- At one development age, ratios are 1.06, 1.07, 1.08, and 1.31. Calculate the range and state the appropriate initial conclusion from the plot. (Answer: range 0.25; the 1.31 point is separated and warrants verification, but the plot alone does not justify removing it or assigning a cause)