Abstract
Despite the widespread use of chain-ladder models, so far no theory was available to test for model specification. The popular over-dispersed Poisson model assumes that the over-dispersion is common across the data. A further assumption is that accident year effects do not vary across development years and vice versa. The log-normal chain-ladder model makes similar assumptions. We show that these assumptions can easily be tested and that similar tests can be used in both models. The tests can be implemented in a spreadsheet. We illustrate the implementation in several empirical applications. While the results for the log-normal model are valid in finite samples, those for the over-dispersed Poisson model are derived for large cell mean asymptotics which hold the number of cells fixed. We show in a simulation study that the finite sample performance is close to the asymptotic performance.
Highlights
Despite the widespread use of chain-ladder models, so far no theory was available to test for model specification
While the results for the log-normal model are valid in finite samples, those for the over-dispersed Poisson model are derived for large cell mean asymptotics which hold the number of cells fixed
We show that the asymptotic distribution of the Bartlett test and the two-sample F-test for common over-dispersion match the finite sample distribution of the test for common log data variance in the log-normal model
Summary
“Can we trust chain-ladder models?” is a central question in non-life insurance claim reserving. The popular over-dispersed Poisson chain-ladder model assumes a fixed variance to mean ratio across the run-off triangle. We show that testing for a violation of central assumptions is straightforward in two popular chain-ladder models: over-dispersed Poisson and log-normal. We note that while this model replicates the classical chain-ladder point forecasts, it differs from the over-dispersed Poisson model and so far lacks a distributional framework that would allow for a rigorous statistical theory. While it is a popular model, we do not consider it further in this paper.
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