Why Diagnose Adjusted Incurred Losses?
An adjusted incurred-loss triangle is useful only if its entries represent a coherent, consistently adjusted view of claim experience. A calculation can follow the intended reserve-adequacy procedure and still produce data that deserve investigation: an adjusted reserve may be negative, development may be unexpectedly large or small, one accident period may break from the surrounding pattern, or the triangle may not reconcile to its underlying paid and reserve components.
Diagnostics do not automatically prove an adjustment is wrong. They identify results that should be explained before the adjusted data are used in later analysis. A genuine change in claim experience can produce unusual development; a coding, timing, or aggregation error can produce a similar-looking signal. The analyst’s task is to distinguish a defensible feature of the data from a defect in the adjustment or its inputs.
The case reserve in this identity is the adjusted reserve amount represented in the triangle, not a reserve adjustment added a second time to an already adjusted incurred value. The components must refer to the same claims, valuation date, coverage, and treatment of recoveries. If they do not, a difference may reflect inconsistent scope rather than an arithmetic error.
A Structured Diagnostic Review
The checks below are complementary. A reconciliation check tests arithmetic and scope at individual cells. Reserve-sign checks examine whether the implied adjusted reserve is plausible. Development checks compare changes across ages, while accident-period checks look for discontinuities at a common age. A pattern that appears in more than one check is a stronger reason to investigate, but it is still not a conclusion by itself.
Verify accident-period definitions, development ages, valuation dates, coverage, currency, and whether losses are gross or net of recoveries. Confirm that paid and reserve components cover the same claim population.
Calculate paid losses plus adjusted case reserves, then compare the result with adjusted incurred losses. Investigate any difference beyond a stated rounding tolerance.
Calculate \(R^{adj}_{i,j}=I^{adj}_{i,j}-P_{i,j}\), then examine its sign and size. Calculate age-to-age ratios and changes, with attention to both unusually high and unusually low movements.
Look for a break in the pattern across periods, allowing for known changes in exposure, mix, trend, claim handling, and data maturity. Do not treat a raw difference as proof of an adjustment defect.
Recheck claim-level inputs, adjustment factors, status treatment, aggregation, and valuation timing. Document whether the item is corrected, explained, or retained as a genuine feature of experience.
Negative Case Reserves and Reconciliation
For a cell on a consistent gross-loss basis, the implied adjusted case reserve is the adjusted incurred amount less cumulative paid losses. If that difference is negative, the result means adjusted incurred is below paid losses. This is an important warning, but it is not automatically impossible: net recoveries, accounting conventions, or a deliberately net basis may make negative amounts possible. First establish the basis; then determine whether the sign has a valid explanation.
Worked Example: Testing an Implied Adjusted Case Reserve
A hypothetical liability portfolio has, at 24 months, cumulative paid losses of $610,000 and adjusted incurred losses of $575,000. At 12 months for the same accident period, paid losses were $420,000 and adjusted incurred losses were $500,000. Assume both ages are stated on the same gross-loss basis and the figures are rounded to the nearest $1,000. Calculate the implied adjusted case reserve at each age and assess the movement.
Method. Use the cell identity \(R^{adj}=I^{adj}-P\), then compare the two ages.
At 12 months: $$ R^{adj}_{12}=\$500{,}000-\$420{,}000=\$80{,}000. $$
At 24 months: $$ R^{adj}_{24}=\$575{,}000-\$610{,}000=-\$35{,}000. $$
The implied adjusted reserve fell by $$ -\$35{,}000-\$80{,}000=-\$115{,}000. $$ Paid losses increased by \(\$610{,}000-\$420{,}000=\$190{,}000\), while adjusted incurred losses increased by \(\$575{,}000-\$500{,}000=\$75{,}000\). The component changes reconcile: \(\$190{,}000-\$115{,}000=\$75{,}000\).
Assessment. The 24-month adjusted incurred amount is $35,000 below paid losses, producing a negative adjusted case reserve. On the stated gross basis, investigate whether the adjusted reserve calculation, claim population, and valuation dates are correct. If the data are actually net of recoveries, confirm that the treatment is consistent and documented.
Common mistake avoided. Reversing the subtraction gives \(\$610{,}000-\$575{,}000=\$35{,}000\) and incorrectly labels the reserve positive. The reserve identity is adjusted incurred minus paid, not paid minus adjusted incurred.
The reconciliation identity also gives a short justification for why the component check is decisive. By definition, incurred losses on this basis consist of amounts already paid plus amounts still held as case reserves. Therefore, if the adjusted incurred total and its two components use the same scope and valuation, $$ I^{adj}_{i,j}-(P_{i,j}+R^{adj}_{i,j})=0. $$ A nonzero difference means that at least one component, definition, or calculation does not match the stated basis, apart from any explicitly accepted rounding.
Age-to-Age Movements
An age-to-age ratio summarizes the change in a cumulative adjusted incurred amount between successive development ages. For positive values, define $$ f^{adj}_{i,j}=\frac{I^{adj}_{i,j+1}}{I^{adj}_{i,j}}. $$ The ratio is a diagnostic description, not a rule that adjusted incurred losses must increase. Because case reserves can be strengthened or released, adjusted incurred losses can decline; an age-to-age ratio below 1 is possible. Ratios can also be unstable when the denominator is small or zero, so inspect dollar changes and underlying components as well.
A useful review asks whether a movement is unusual relative to comparable accident periods and whether its cause is visible in paid losses or adjusted reserves. A large ratio can reflect a large emergence in paid losses, a reserve increase, or both. A small ratio can reflect modest paid emergence, reserve release, or a change in the adjusted reserve basis. The ratio alone does not identify which explanation applies.
Worked Example: Locating an Unusual Age-to-Age Movement
A hypothetical portfolio has the following adjusted incurred amounts, in thousands of dollars. All cells use the same adjustment basis. Assess whether the youngest period’s 12-to-24-month movement merits review.
| Accident period | 12 months | 24 months | 36 months |
|---|---|---|---|
| A | 1,000 | 1,280 | 1,400 |
| B | 1,100 | 1,400 | 1,510 |
| C | 1,250 | 1,310 | Not yet observed |
Method. Calculate the relevant age-to-age ratios and compare like development intervals.
For period A, the 12-to-24-month ratio is $$ \frac{1{,}280}{1{,}000}=1.280. $$ For period B, it is $$ \frac{1{,}400}{1{,}100}=1.2727\ldots\approx1.273. $$ For period C, it is $$ \frac{1{,}310}{1{,}250}=1.048. $$
The first two periods show increases of 28.0% and approximately 27.3%; period C increases by 4.8%. The difference is substantial enough to investigate. It does not establish that period C is wrong: a genuine difference in claim emergence, mix, or reserve changes could explain it.
As a separate comparison at the 12-month age, the values rise from 1,000 to 1,100 to 1,250 (in thousands). That progression is not itself a like-for-like development comparison; it is a comparison across accident periods and may reflect changing exposure or loss levels. Keep the two diagnostic questions separate.
Common mistake avoided. Treating every increasing triangle diagonal as a development pattern confuses accident-period differences with changes across development ages. Compare periods at the same age for discontinuities, and compare successive ages within a period for age-to-age movement.
Discontinuities Across Accident Periods
A discontinuity is a break in the pattern across accident periods when values are compared at the same development age or when comparable age-to-age movements are considered. It may arise from a change in case-reserve practice that the adjustment did not adequately standardize, a data or mapping problem, or a real change in risk. It may also arise from different exposure volumes or claim mix. Consequently, raw adjusted incurred amounts across periods should not be expected to match.
A more informative review compares measures on a consistent basis. Depending on the available data, the analyst may examine age-to-age ratios, adjusted incurred per exposure, or claim-level components at a common age. The comparison should account for known changes in coverage, limits, inflation, and exposure. These checks are diagnostic: they do not replace the separate selection of development assumptions.
Reconciliation at the Triangle Cell
Reconcile adjusted incurred losses to paid losses and adjusted case reserves at each accident-period and development-age cell, not only at the total-triangle level. Offsetting errors can make a grand total appear correct even when individual cells are inconsistent. Establish a practical tolerance before reviewing results; for data reported in thousands, small residuals may be caused by rounding, but a material residual requires explanation.
Worked Example: Finding a Cell-Level Reconciliation Difference
For a hypothetical accident period at 24 months, cumulative paid losses are $620,000 and the sum of adjusted case reserves is $210,000. The adjusted incurred triangle shows $850,000. At 12 months, the corresponding amounts are paid losses of $360,000, adjusted case reserves of $140,000, and adjusted incurred losses of $500,000. Determine whether the cells reconcile and quantify any difference.
Method. For each age, add paid losses and adjusted reserves, then subtract that sum from the triangle value.
At 12 months: $$ \$360{,}000+\$140{,}000=\$500{,}000, $$ so the difference is \(\$500{,}000-\$500{,}000=\$0\).
At 24 months: $$ \$620{,}000+\$210{,}000=\$830{,}000, $$ so the difference is $$ \$850{,}000-\$830{,}000=\$20{,}000. $$
The 24-month triangle amount exceeds the sum of its stated components by $20,000. The analyst should check whether a claim or reserve component was omitted, whether the triangle value includes a different adjustment, and whether the valuation dates and claim populations match. The available information does not identify which source is responsible.
Common mistake avoided. Checking only that the total triangle and total components agree can hide cell-level differences that cancel elsewhere. Reconcile each cell before drawing conclusions from the aggregate.
Common Diagnostic Traps
- Assuming adjusted incurred must be nondecreasing. Reserve releases can reduce incurred amounts. Investigate an unexpected decline, but do not label every decline an error.
- Calling every negative implied reserve invalid. First confirm whether amounts are gross or net and how recoveries are treated. Then test whether the result is consistent with that basis.
- Comparing unlike cells. Different development ages, valuation dates, or claim populations can create apparent discontinuities. Align the basis before interpreting differences.
- Using a ratio without its components. The same age-to-age ratio can result from different paid emergence and reserve movements. Inspect both components.
- Ignoring small denominators and rounding. A ratio may be extreme when its denominator is small. Review the underlying dollars and the precision of reported values.
Check Your Understanding
Work each problem using the stated basis, then compare your result with the answer.
- At 18 months, a hypothetical cell has paid losses of $275,000 and adjusted incurred losses of $340,000. What is the implied adjusted case reserve? (Answer: $65,000)
- An adjusted incurred amount rises from $800,000 at 12 months to $960,000 at 24 months. Calculate the 12-to-24-month age-to-age ratio and percentage increase. (Answer: ratio \(1.20\); increase \(20.0\%\))
- A cell has paid losses of $480,000 and adjusted case reserves of $170,000, but its adjusted incurred triangle value is $635,000. By how much does the triangle value differ from the reconciled amount, and in which direction? (Answer: reconciled amount $650,000; the triangle value is $15,000 lower)
- Two accident periods have adjusted incurred losses of $1.02 million and $1.31 million at 12 months. Can the difference alone establish an accident-period discontinuity? State one additional comparison that would help. (Answer: No; compare a consistent measure such as adjusted incurred per exposure at the same age, while checking changes in exposure and mix.)
- For one period, adjusted incurred losses decline from $740,000 at 24 months to $725,000 at 36 months. Is the decline by itself proof of an error, and what component check is central? (Answer: No; reserve releases can reduce incurred losses. Reconcile adjusted incurred to paid losses plus adjusted case reserves at both ages.)