Why Inspect a Triangle Before Using It?
A loss development triangle summarizes experience by origin period and development age. Before using it to estimate future development, examine whether individual cells or groups of cells look unusual. A sudden jump may reflect a genuine change in claims, but it may also result from a data-definition change, a processing problem, a large claim, or a shift in operations. A triangle can flag where to investigate; by itself, it usually cannot establish why a value changed.
This inspection is not a substitute for reconciling data to source records or checking coverage, valuation dates, and loss bases. Those checks provide context for interpreting a signal. The objective here is to identify patterns worth explaining, compare like with like, and avoid treating a screening result as an automatic adjustment.
Read Each Cell in Its Triangle Context
A cell has at least three useful coordinates: its origin period, its development age, and the calendar period in which the experience occurred or was valued. These perspectives answer different questions. Comparing origins at the same development age can reveal a change in newer experience. Looking along a calendar-period diagonal can help identify a period-specific influence. Looking within one origin row can identify an unusual change as that experience develops.
First establish whether the triangle is cumulative or incremental. A cumulative value includes amounts from earlier development ages; an incremental value represents amounts assigned to a particular interval. For cumulative values, the increment between consecutive ages is their difference. This distinction is essential when deciding whether an apparent spike is actually a large new amount.
Comparisons also need a sensible reference. Compare cells with the same loss definition, coverage, valuation basis, and development age where possible. Consider whether origin periods are full or partial, whether a row is immature, and whether a difference in exposure could explain different dollar amounts. A high dollar value in a larger origin period is not automatically unusual.
Worked Example: Screening a Large Cumulative Cell
A hypothetical insurer reviews a cumulative incurred triangle, in thousands of dollars. The columns are development ages 12 and 24 months. At age 12, the four origin periods show 350, 370, 360, and 390. At age 24, the first three show 500, 525, and 510; the newest origin has a cumulative value of 830. Screen the newest age-24 cell against the earlier origins, and state what the comparison does and does not establish.
Method: convert cumulative values to the increment for the interval, then compare like development ages. For the newest origin, the 12-to-24-month increment is \(830-390=440\) thousand dollars.
For the earlier origins, the corresponding increments are \(500-350=150\), \(525-370=155\), and \(510-360=150\) thousand dollars. Their median is 150 thousand dollars: the ordered values are 150, 150, and 155.
The newest increment exceeds that median by \(440-150=290\) thousand dollars. Relative to the median, the increase is \(290/150=1.9333\), or 193.3% to one decimal place. The newest increment is \(440/150=2.9333\) times the median.
This is a clear screening flag, but there are only three comparison observations and the calculation does not identify a cause. Follow up by checking source transactions, claim counts, large claims, case-reserving practices, and whether the origins have comparable exposure and definitions.
Common mistake avoided: treating 830 as the new 12-to-24-month amount would compare a cumulative value with incremental values. The correct amount for that interval is \(830-390=440\), because 390 was already included at 12 months.
Look for Patterns Across Origins and Calendar Periods
One unusual cell may be driven by a single claim or a recording issue. A similar movement across multiple origins is more consistent with a broader change, although it still needs investigation. For example, if the same development interval rises across several successive origins, consider whether claim frequency, claim severity, coverage, or operational practice changed. If unusual values cluster in one calendar period, investigate influences that could affect multiple origins during that period.
A calendar-period diagonal contains cells associated with the same calendar period. With annual origin periods and annual development intervals, the calendar year for a cell is the origin year plus the development interval number minus one. This mapping explains why adjacent cells along a diagonal relate to one calendar period, even though they belong to different origins and development ages.
The diagonal can help focus investigation, but amounts on it may represent different development stages. Consequently, a diagonal total or a visually striking diagonal should not be treated as conclusive evidence of a calendar-year change without considering maturity, exposure, and the construction of the triangle.
Worked Example: Interpreting a Calendar-Period Diagonal
A hypothetical incremental paid-loss triangle, in thousands of dollars, has annual origin periods and annual development intervals. For origin 2021, the values are 100, 50, and 30 at intervals 1, 2, and 3. For origin 2022, they are 110 and 75 at intervals 1 and 2. For origin 2023, the interval-1 value is 120. Identify the cells on the 2023 calendar-year diagonal and assess the signal.
Method: map each cell to its calendar year. For origin 2021, interval 3 maps to \(2021+3-1=2023\); its value is 30. For origin 2022, interval 2 maps to \(2022+2-1=2023\); its value is 75. For origin 2023, interval 1 maps to \(2023+1-1=2023\); its value is 120. Thus the diagonal contains 30, 75, and 120 thousand dollars.
The diagonal total is \(30+75+120=225\) thousand dollars. This total is descriptive, but it combines three different development intervals. It should not be compared with a diagonal total from a different calendar year as if that were a like-for-like measure, especially when the number of observed cells may differ.
A development-age comparison gives another view. At interval 1, the values rise from 110 in origin 2022 to 120 in origin 2023: an increase of \(10/110=9.09\%\), or 9.1% to one decimal place. At interval 2, the available values rise from 50 in origin 2021 to 75 in origin 2022: an increase of \(25/50=50.0\%\). The interval-2 movement is worth investigating, but it is based on only two observations and does not alone show that calendar year 2023 caused it.
Ask whether the affected cells share a change in reporting, settlement, coverage, or claim handling; also compare claim counts and exposure. Different ages, origin sizes, and the small number of observations prevent a firm conclusion from this triangle alone.
Common mistake avoided: calling the diagonal total a calendar-year trend without checking that the cells are comparable. The sum of 225 is correctly calculated, but the three components are at different development ages.
Use Counts and Exposure to Investigate a Shift
When loss amounts change by origin, separate the possible sources of the change. Claim counts, exposure, and average claim size can help distinguish a frequency movement from a severity movement. This decomposition does not prove causation, and a triangle may not contain all the information needed to make it. It provides targeted questions for a source-data or operational review.
For a simple origin-period summary, loss per exposure equals claim frequency per exposure multiplied by average loss per claim, provided the count and loss amounts refer to the same claims and basis. The identity follows directly: claim count divided by exposure, multiplied by total loss divided by claim count, equals total loss divided by exposure. If the claim count is zero, average loss per claim is undefined, so this decomposition cannot be used in that form.
Worked Example: Separating Frequency and Severity Changes
A hypothetical portfolio has the following origin-period results. In 2019 there are 1,200 exposures, 200 claims, and total loss of $2,400,000. In 2020 there are 1,350 exposures, 225 claims, and total loss of $2,700,000. In 2021 there are 1,300 exposures, 260 claims, and total loss of $3,900,000. Calculate loss per exposure, claim frequency per exposure, and average loss per claim for each period; then describe what the pattern suggests investigating.
Method: calculate each component on a consistent basis. For 2019, frequency is \(200/1{,}200=0.1667\) claims per exposure, average loss is \(\$2{,}400{,}000/200=\$12{,}000\), and loss per exposure is \(\$2{,}400{,}000/1{,}200=\$2{,}000\).
For 2020, frequency is \(225/1{,}350=0.1667\), average loss is \(\$2{,}700{,}000/225=\$12{,}000\), and loss per exposure is \(\$2{,}700{,}000/1{,}350=\$2{,}000\).
For 2021, frequency is \(260/1{,}300=0.2000\), average loss is \(\$3{,}900{,}000/260=\$15{,}000\), and loss per exposure is \(\$3{,}900{,}000/1{,}300=\$3{,}000\).
From 2020 to 2021, frequency changes by \(0.2000/0.1667-1=20.0\%\) using the exact fractions \(260/1{,}300\) and \(225/1{,}350\). Average loss changes by \(\$15{,}000/\$12{,}000-1=25.0\%\). Their combined effect is \(1.20\times1.25=1.50\), a 50.0% increase in loss per exposure, matching \(\$3{,}000/\$2{,}000-1=50.0\%\).
The pattern suggests investigating both claim frequency and average claim size in 2021, rather than attributing the loss-per-exposure increase to only one component. Check whether claim counts use the same claim definition and whether large claims, exposure mix, or changes in claim handling contributed. These origin-period summaries are diagnostic; they do not by themselves explain a particular triangle cell.
Common mistake avoided: comparing total losses alone would make exposure growth or contraction difficult to distinguish from a change in claim experience. Here, 2021 has fewer exposures than 2020, yet its loss per exposure is higher.
From a Flag to an Investigation
A useful review records the observation, the comparison used, and a specific follow-up question. Do not remove, cap, or alter a value just because it is unusual. First determine whether it is valid and whether the comparison is appropriate. If an error is confirmed, correct it from the underlying records and preserve a reconciliation. If the value is valid, document its cause or leave it as experience with an appropriate qualification.
Name the cell or pattern, its origin and development age, and whether the triangle is cumulative or incremental.
Check similar development ages and origins, while considering exposure, maturity, coverage, and loss basis.
Map calendar-period patterns and, where available, compare counts, exposure, and average claim size.
Review source records and operational context; distinguish a confirmed data issue from a valid but unusual observation.
Common causes to investigate include a large individual claim, a change in claim reporting or settlement speed, a reserving practice change, a coverage or limit change, a transaction-processing change, or an inconsistent data extract. These are hypotheses, not diagnoses. The evidence should determine the conclusion.
Check Your Understanding
For each problem, identify the relevant comparison and interpret the result cautiously.
- A cumulative incurred cell is 720 at 24 months and 480 at 12 months. What is the 12-to-24-month increment? (Answer: 240, in the same monetary units as the cell values.)
- Three comparable incremental cells at the same development age are 80, 90, and 90. A newer origin has 180. Using the median of the three as a screen, calculate the percentage above the median. (Answer: 100.0% above the median.)
- With annual origins, which calendar year does origin 2022 at development interval 3 map to? (Answer: 2024.)
- A portfolio's claim count increases from 150 to 180 while exposure increases from 1,000 to 1,200. Calculate frequency per exposure in each period and state whether frequency changed. (Answer: 0.1500 in both periods; frequency did not change.)
- Average loss per claim rises from $8,000 to $10,000, while frequency per exposure is unchanged. By what percentage does loss per exposure change, assuming consistent claim and loss bases? (Answer: 25.0% increase.)