reported vs paid triangles
When you build a triangle you must choose what 'losses' means in each cell, and there are two main choices. A paid triangle records only cash that has actually gone out the door. A reported (or incurred) triangle records cash paid PLUS the case reserves adjusters have set on still-open claims — in other words, the company's best current guess of total cost so far. The same accident year looks different in the two, and a careful reserving exercise builds both and compares them.
Precisely, paid losses = cumulative payments to date; reported (incurred) losses = paid losses + case reserves on open claims. Because reported losses already include an estimate of the future on known claims, a reported triangle develops more slowly (it has a head start) and has no IBNER reaction once a claim is closed — it can even develop downward if case reserves were set too high and are released. A paid triangle is objective and unmanipulable but matures more slowly and contains no information about open claims. For example, at 12 months a year might show 5 million paid but 8 million reported (the extra 3 million being case reserves); both will march toward roughly the same ultimate, but along different paths, so they imply different development factors.
Comparing the two triangles is one of the most powerful diagnostics in reserving. If the paid and reported methods give very different ultimates, something is going on — perhaps case reserves are being set more conservatively than before, or payments are speeding up. The paid-to-reported ratio over time reveals shifts in claims handling that either triangle alone would hide. The honest caution is that neither is automatically 'right': the reported triangle is contaminated by changing case-reserve practices, while the paid triangle is slow and tells you nothing about the large open claims that may dominate the outcome.
At 12 months accident year 2025 shows 5 million paid and 8 million reported. The paid chain ladder, using a large paid LDF, projects an ultimate of 10.5 million; the reported chain ladder, using a smaller LDF, projects 10.0 million. The half-million gap prompts the actuary to ask whether case reserves have recently been strengthened — a question neither triangle answers alone.
Reported = paid + case reserves; building both and comparing exposes shifts in claims practice.
Neither triangle is inherently more correct. The reported triangle reacts to changing case-reserve adequacy; the paid triangle does not, but it is slower and blind to open claims. Reconciling the two is the actuary's real work.