Some important properties of chain ladder models
Contents
- Section I. Mack method - the regression formulation of volume-weighted average (chain ladder) link ratios
- Section II. The Chain ladder (Mack) method does not distinguish between accident years and development years
A paper on the Mack Method and the corresponding Bootstrap predictions is available here.
A paper on the importance of diagnostic assessment of models and associated bootstrap technique is available here.
Section II. Chain ladder does not distinguish between accident years and development years
It is a property of the chain ladder that, when applied to an incremental triangle, the incremental forecasts are identical to those for the equivalent procedure where accident years and development years are interchanged.
Equivalently, you get the same incremental forecasts, total reserves and reserves by calendar year whether cumulation runs to the right, across development years, or down, across accident years.
This applies to any model whose forecasts reproduce those of the standard chain ladder, so Mack's model and the two-way cross-classification quasi-Poisson GLM both have this property in respect of the mean forecasts.
Let α denote the sum of the values in the rectangle labelled α.
Let β be the sum of the values in the column labelled β.
Let γ be the sum of the values in the column labelled γ.
The value α represents the sum of the row totals which is also the sum of the column totals in rectangle labelled α, as arithmetic is commutative.
Similarly for the rectangle labelled α + β (extended by one period to the right) and the rectangle labelled α + γ extended by one period down.
Cumulating across development years (traditional approach), the Mack link ratio is
Therefore the projection of the incremental p is
which is symmetric in γ and β.
If instead you cumulate across the accident years, the Mack link ratio is
Therefore,
So it does not matter which way you do it.
An algebraic proof that the incremental forecasts are of the form (β γ) / α is given in Barnett, Zehnwirth and Dubossarsky (2005).
This formula is symmetric (the formula is the same if we interchange accident and development periods - i.e. transpose the incremental array, since β * γ / α = γ * β / α.
Consequently, it is always the case that the chain ladder works the same if we treat the accident years as if they were the development years and vice-versa.
This is a worrisome property, because we know that the accident and development year directions are different. It appears to make no sense to cumulate downward and take ratios running down, but in fact the usual across version makes just as much sense. Some of the consequences are discussed in detail in Barnett et al (2005).
Example: ABC
The data in the following example are from the triangle ABC in Barnett and Zehnwirth (2000). Note that we do not use the exposures in this example.
Firstly, let's see what the residuals from a Mack model tell us about the suitability of a ratio model.
We clearly see below strong changes in trend in the calendar year direction; neither Mack nor the quasi-Poisson GLM version of the chain ladder can deal with this. This is not an unusual circumstance.
Let's now examine standard chain ladder forecasts for this data. These forecasts are the same for Mack and the quasi-Poisson GLM.
Incremental chain ladder forecasts for the ABC data (not exposure adjusted):
Incremental chain ladder forecasts for the transpose of the ABC data:
It should be noted, however, that the Mack standard deviations of forecasts (or outstandings or ultimates) and their coefficients of variation do not have the transpose property; because of the way the conditioning is set up; when transposed, the conditioning in the variance is not symmetric.
Mack method forecast from to PL(C) and PL(C)transpose. Note that Calendar year means are the same but aggregate (Total) SDs are different. Mack SDs are conditional on the first development period data when we cumulate across development periods, but are conditional on the first accident period data when we cumulate across accident years.
The symmetry in the forecasts (specifically, that the transpose of the forecast and the forecast of the transpose of the triangle are equal) is not a desirable property. It is not an indication of "robustness" - forecasts from ratio models are highly sensitive to particular observations, and completely insensitive to other observations. It is not an indication of suitability of ratio models in general or of the chain ladder in particular, nor of either the Mack or quasi-Poisson GLM versions of it.
These plots show the ABC data by development year and by accident year. Each plot has a clear and distinct pattern. A model for the data should be able to account for the structures visible here. The symmetry of the chain-ladder means it lacks the resolution to do so.
If we run a structurally similar symmetric model from PTF (a two-way cross-classification model with log-link), of course the model won't be suitable either:
- however, a more suitable model can be obtained by allowing for some CY trend changes (though it would still be overparameterized and suffer from some of the other deficiences mentioned in Barnett et al, 2005)
This model does have symmetry in forecast standard errors and CVs. For example, the following table is the same for both the original and transposed array.
The same would be true for the quasi-Poisson GLM version of the chain ladder.
As already mentioned, symmetry is not of itself a desirable property, but if you want to impose that symmetry it would be interesting to ask why we should expect it to hold for the mean but not the standard deviation.
References
Barnett, G. and B. Zehnwirth (2000) Best Estimates for Reserves, PCAS No 87, p245-303.
Barnett G., B. Zehnwirth and E. Dubossarsky (2005) When Can Accident Years Be Regarded As Development Years?, PCAS No 92, p249-256.
Mack, Th. (1994), "Which stochastic model is underlying the chain ladder method?" Insurance Mathematics and Economics, Vol 15 No. 2/3, 1994, pp. 133-138.
