Lesson 1What the method assumes
The chain ladder assumes that the future development of each origin year will, on average, be proportional to what has been observed, with a common factor for all years at the same age, and that origin years are independent of one another.
\[\hat C_{i,j+1}=\hat f_j\,C_{i,j},\qquad \hat f_j=\frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}}\]
Its limits follow from that: it fails when the speed of reporting or payment changes, when case reserving practice changes, or when the latest year has so little information that any deviation is multiplied by a large cumulative factor.
Lesson 2How to select factors
The volume-weighted average is the method's natural estimator and gives more weight to larger years. The simple average treats every year equally. Averages of the latest three or five years respond to recent trends, and averages that exclude the highest and lowest values reduce the effect of an outlier.
Selecting is not picking the lowest number: it is choosing the average that best describes the future and documenting why. If the diagnostics showed a regime change, a long average mixes two worlds; if they showed a large loss, it is better to exclude it from the factor and reserve it separately.
Lesson 3The tail factor
When the triangle does not reach full maturity, the last observed factor is not enough: development beyond the last age is missing. That development is the tail factor. It is estimated by fitting a curve to the factors minus one (exponential or inverse power, as Sherman proposed), from industry benchmarks, or from the incurred-to-paid relationship at mature ages.
The tail multiplies every origin year, so a small error in it weighs heavily on the total reserve. That is why it is always worth showing the sensitivity: how much IBNR changes if the tail moves by one point.
Lesson 4From factor to reserve, and validation
\[CDF_j=f_{\text{tail}}\prod_{k\ge j}\hat f_k,\qquad \hat U_i=C_{i,n+1-i}\,CDF_{n+1-i},\qquad IBNR_i=\hat U_i-C_{i,n+1-i}\]
The inverse of the CDF is the percentage reported at that age. The result is validated in at least three ways: the implied ultimate loss ratio by year should be consistent with pricing and experience; IBNR for the latest year is checked against another method; and at the next valuation, reported development is compared with what the method expected.