Lesson 1Anatomy of a triangle
Each row is an origin period: the accident year (the most common choice in property and casualty), the policy year or the report year. Each column is a development age, measured in months from the start of the origin period: 12, 24, 36 and so on.
Cell \(C_{i,j}\) is the cumulative amount for origin year \(i\) at age \(j\). The last cell in each row forms the latest diagonal, which corresponds to the valuation date: everything below it lies in the future and is what we must estimate.
\[X_{i,j}=C_{i,j}-C_{i,j-1},\qquad C_{i,j}=\sum_{k\le j}X_{i,k}\]
The incremental triangle shows what happened in each period; the cumulative one shows what has accumulated up to each age. Factor methods work on the cumulative triangle; stochastic models such as the bootstrap work on the increments.
Lesson 2What to measure: paid, incurred and counts
Paid losses do not depend on adjuster judgment, but they mature slowly: in long-tail lines the last payment may come years later. Incurred losses (paid plus case reserves) mature faster, but they inherit any change in how case reserves are set.
That is why an incurred triangle estimates both pure IBNR, for unreported claims, and IBNER, the deficiency or redundancy in case reserves on claims already reported. Under Mexican regulation both belong to the reserve for claims incurred but not reported or not fully reported.
Triangles of reported and closed claim counts, and of average severity, do not produce the reserve on their own, but they explain why the amounts move: are there more claims, more expensive claims, or a change in the speed of reporting or settlement?
Lesson 3Individual factors: column, row and diagonal
\[f_{i,j}=\frac{C_{i,j+1}}{C_{i,j}}\]
Down a column you measure how stable development is at a given age: if the 12 to 24 month factors range from 1.80 to 1.88, the uncertainty in the latest year is of that order. Across a row you spot an unusual year, for example one with a large loss or a change in business mix.
Along a diagonal you look for calendar effects: cost inflation, a legal change, a new claims system or a drive to close files all hit every origin year in the same valuation period. A factor method assumes they do not exist, so look for them before you rely on it.
Lesson 4Diagnostics before projecting
A minimum checklist: paid-to-incurred ratios by age (they reveal changes in case reserving); reported loss ratio at each age against earned premium; average severity and closure rates; and a reconciliation of the triangle to the accounting records, because a triangle that does not tie to the balance sheet is not usable.
Large and catastrophe losses are analyzed separately, and segments that behave differently (for example, personal auto and fleets) are not mixed. When there are known changes in case reserving or payment speed, there are formal adjustments such as Berquist and Sherman, covered in the IBNR methods guide.