Why Compare Development Across Accident Periods?
A link ratio selected from several accident periods summarizes how cumulative losses have developed from one age to the next. That summary is most informative when the periods being combined have similar development patterns. If newer accident periods consistently show larger ratios than older periods, a single factor can blend two different patterns and may not describe either one well.
The diagnostic is direct: calculate the observed age-to-age ratios by accident period, then compare periods at the same development age. This keeps the comparison focused on whether development has changed across accident periods rather than on the fact that ratios often differ between early and late development ages.
Only compare ratios that refer to the same development transition. For example, a ratio from age 1 to age 2 is not directly comparable to a ratio from age 3 to age 4: the two transitions can have different expected levels even when the underlying pattern is stable. Also note how many accident periods support each comparison. A possible shift at a development age observed for only one recent period is weaker evidence than a consistent difference across several periods.
A Practical Comparison Method
For each development age, arrange the observed ratios by accident period. Look for a change in level, a gradual trend, or a single unusual value. Then check the underlying cumulative amounts and the triangle’s calendar-year diagonals. Calendar-year effects can also create unusual patterns; the earlier material on calendar-year distortions explains why diagonal patterns merit attention.
Compare, for example, all observed age 2-to-3 ratios. Do not mix different ages in one comparison.
Divide each period’s cumulative amount at the later age by its cumulative amount at the earlier age.
Check whether higher or lower ratios persist across periods, how many observations support the pattern, and whether one observation dominates.
A pattern may warrant further investigation or a different selection approach. A difference in ratios alone does not prove its cause or automatically determine the selected factor.
Why a Pooled Ratio Can Hide a Shift
A volume-weighted link ratio across selected accident periods is the sum of the later-age cumulative amounts divided by the sum of the earlier-age cumulative amounts. If the selected periods have different ratios, the pooled result lies between the smallest and largest individual ratios when the denominators are positive.
The weights are the proportions of the earlier-age cumulative amounts, so they are nonnegative and sum to 1. This identity explains why the pooled ratio can be dominated by high-volume periods. If older periods have lower ratios and newer periods have higher ratios, the pooled factor is a weighted compromise, not evidence that every period follows that compromise.
The identity requires nonzero total denominator, and interpreting the weights as nonnegative proportions requires nonnegative earlier-age cumulative amounts. When amounts are zero or negative, a ratio comparison and its weighted-average interpretation may fail; do not apply this argument mechanically.
Worked Example: A Shift at the Age 2-to-3 Transition
A hypothetical insurer has the following cumulative losses, in thousands of dollars. Compare the age 2-to-3 ratios across the accident periods and calculate the pooled ratio. State what the comparison suggests.
| Accident period | Age 1 | Age 2 | Age 3 |
|---|---|---|---|
| 1 | 100 | 150 | 180 |
| 2 | 120 | 180 | 216 |
| 3 | 140 | 210 | 252 |
| 4 | 160 | 240 | 336 |
| 5 | 180 | 270 | Not observed |
Method: compare same-age ratios, then calculate the pooled ratio. For accident period 1, the age 2-to-3 ratio is \(180/150=1.20\). For period 2 it is \(216/180=1.20\). For period 3 it is \(252/210=1.20\). For period 4 it is \(336/240=1.40\). Period 5 has no observed age 3 amount, so its age 2-to-3 ratio cannot be calculated.
The first three observed ratios are all 1.20, while the fourth is 1.40. This is a higher ratio in a later accident period, but only one later period supplies evidence at this transition. It is a signal to examine, not yet a demonstrated sustained shift.
For the pooled ratio, add the observed age 2 amounts: \(150+180+210+240=780\). Add their corresponding age 3 amounts: \(180+216+252+336=984\). Therefore,
The pooled factor of about 1.262 is above 1.20 but below 1.40. It blends the three earlier ratios and the one higher ratio; it does not describe the pattern in all four accident periods. Common mistake: treating 1.262 as proof that every period develops at about that rate. The correction is to inspect the individual ratios and note that the apparent change is supported by only one later observation.
Look for Persistence Across Periods
A single ratio can be unusual because of random variation, a large individual claim, an error, or some other feature of the data. Greater confidence in a development change comes from persistence: several successive accident periods show a difference in the same direction at a given development age. A related pattern at another development age can add context, but it should still be assessed using comparisons at that age separately.
Worked Example: Checking Whether a Change Persists
For a hypothetical insurance portfolio, compare early and later accident periods in this cumulative triangle. Amounts are in thousands of dollars.
| Accident period | Age 1 | Age 2 | Age 3 |
|---|---|---|---|
| 1 | 100 | 130 | 156 |
| 2 | 120 | 156 | 187.2 |
| 3 | 150 | 195 | 234 |
| 4 | 180 | 252 | 315 |
| 5 | 200 | 280 | Not observed |
Method: compare periods separately at each transition. For age 1 to age 2, the first three ratios are \(130/100=1.30\), \(156/120=1.30\), and \(195/150=1.30\). The next two are \(252/180=1.40\) and \(280/200=1.40\). The change from 1.30 to 1.40 appears in two successive later periods.
For age 2 to age 3, the first three ratios are \(156/130=1.20\), \(187.2/156=1.20\), and \(234/195=1.20\). Accident period 4 gives \(315/252=1.25\). Period 5 has no age 3 observation, so it cannot confirm whether this higher ratio persists at the later transition.
The age 1-to-2 comparison shows a repeated later-period increase, while the age 2-to-3 comparison shows one higher observed later ratio. The evidence therefore appears stronger for a change at the first transition than for a sustained change at the second. Common mistake: calling both transitions equally changed because one newer ratio is higher at each. The correction is to report the number of later observations supporting each comparison and avoid overstating what the sparse age 2-to-3 evidence establishes.
Distinguish a Shift from an Isolated Ratio
Ratios can be sensitive to the individual amounts that make them up. A comparatively large increase at one development age may be driven by one claim or by a small earlier-age cumulative amount. Check the cumulative cells and the corresponding changes in amount before concluding that accident-period development has shifted. Also confirm that the ratio was calculated from the intended cells and that the triangle is internally consistent.
Worked Example: An Isolated High Ratio
A hypothetical specialty insurer records these cumulative amounts, in thousands of dollars. Assess the age 1-to-2 ratios and state what the newest observation supports.
| Accident period | Age 1 | Age 2 | Age 3 |
|---|---|---|---|
| 1 | 80 | 100 | 120 |
| 2 | 100 | 125 | 150 |
| 3 | 120 | 150 | 180 |
| 4 | 140 | 175 | 210 |
| 5 | 160 | 200 | Not observed |
| 6 | 180 | 252 | Not observed |
Method: calculate the same transition for every observed period. The first five age 1-to-2 ratios are \(100/80=1.25\), \(125/100=1.25\), \(150/120=1.25\), \(175/140=1.25\), and \(200/160=1.25\). The newest observed ratio is \(252/180=1.40\).
The newest increase is \(252-180=72\), which is \(72/180=0.40\), or 40% of the age 1 amount. The earlier ratios all indicate a 25% increase. Thus the newest period is distinctly higher, but it is one observation, not a repeated shift. The age 2-to-3 ratios for the first four periods are each \(1.20\); the newer periods have no age 3 data, so this transition cannot yet provide a later-period comparison.
This result supports flagging the newest age 1-to-2 ratio for review. It does not establish why the ratio is higher or whether future accident periods will follow it. Common mistake: either immediately accepting 1.40 as the new pattern or discarding it because it differs from the older observations. The correction is to recognize it as an unusual observation, verify the underlying data, and distinguish a possible change from a confirmed repeated pattern.
Interpreting the Diagnostic
Comparing accident periods is a diagnostic, not a mechanical rule for selecting a factor. Its value is that it makes the evidence visible: which ages show a difference, which periods support it, and whether the pattern is consistent or isolated. A selected link ratio should be considered in light of that evidence and the reasons for any observed differences.
- Consistent ratios across periods: there is little evidence of a change at that particular development age.
- Repeatedly higher or lower ratios in later periods: the pattern may indicate a development shift that a single pooled factor could conceal.
- One unusual ratio: investigate the observation, but do not treat it alone as proof of a lasting change.
- Few comparable periods: state the limitation; lack of observed ratios is not evidence that the pattern is stable.
A careful conclusion names the development transition, identifies the accident periods being compared, and describes how many ratios support the apparent difference. For example, “The age 1-to-2 ratios are 1.30 for the first three periods and 1.40 for the next two, suggesting a repeated increase at this transition” is more informative than “development has changed.” The former states what the triangle shows without claiming a cause that the ratios alone cannot establish.
Check Your Understanding
For each problem, compare like development ages and distinguish a repeated pattern from an isolated observation.
- A hypothetical triangle has age 1 and age 2 cumulative amounts of 90 and 117, 110 and 143, 130 and 169, and 150 and 210 for four successive accident periods. Calculate the age 1-to-2 ratios and state whether the last period alone demonstrates a repeated shift. (Answer: The ratios are 1.30, 1.30, 1.30, and 1.40; the last period is higher, but one observation alone does not demonstrate a repeated shift.)
- At age 2 and age 3, three accident periods have cumulative amounts \(160,192\), \(200,240\), and \(250,312.5\), respectively. Calculate the three ratios and describe the pattern. (Answer: The ratios are 1.20, 1.20, and 1.25; the third is higher, but it is one differing observation.)
- At a particular development age, three periods have earlier-age amounts 100, 200, and 300 and link ratios 1.10, 1.20, and 1.30. Calculate the pooled ratio and explain why it differs from the unweighted arithmetic mean. (Answer: The pooled ratio is \((110+240+390)/(100+200+300)=740/600=1.2333\); the unweighted mean is 1.20, and the pooled result gives more weight to periods with larger earlier-age amounts.)
- Four successive periods have observed age 1-to-2 ratios of 1.50, 1.50, 1.80, and 1.50. What is the strongest conclusion supported by these ratios alone? (Answer: The third period is isolated and higher; the ratios alone do not establish a sustained development shift.)
- At age 1 to age 2, three earlier accident periods each have ratio 1.25, and the next two each have ratio 1.40. At age 2 to age 3, the first three ratios are 1.20 and only the first of the two newer periods is observed, at 1.25. Which transition has stronger evidence of a repeated later-period increase? (Answer: Age 1 to age 2; both newer observed periods are higher, while only one newer period supports an increase at age 2 to age 3.)