Some important properties of chain ladder models

Contents

• Section I. Mack (chain ladder) is volume-weighted average ratios
• Section II. Chain ladder does not distinguish between accident and development years

A paper on the Mack Method and Bootstrap predictions is available here.

A paper on the importance of diagnostic assessment of bootstrap models is available here.

Section II. Chain ladder does not distinguish between accident and development years

It is a property of the chain ladder that, when applied to an incremental triangle. the incremental forecasts identical to those for the equivalent procedure where accident years and development years are interchanged.

Equivalently, you get the same incremental forecasts whether cumulation, calculation of ratios and projection 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.

Consider the following (toy) triangle, which represents incremental paid losses.

20	10
30	?

It turns out that the value to be forecast (?) = 30 x 10 / 20 = 15

Usual calculation (working across to the right)

Equivalent cumulatives:

20	30
30	?

Ratio: 30/20 = 1.5

Cumulative forecast = 30 x 1.5 = 45

Incremental forecast = 45 - 30 = 15.

Working down rather than across

Cumulating down:

20	10
50	?

Ratio running down: 50/20 = 2.5

"Cumulative" forecast = 10 x 2.5 = 25

Incremental forecast = 25 - 10 = 15.

In the first case, writing everything in terms of the incrementals, the ratio is (20+10)/20 = 1 + 10/20.

The cumulative forecast is 30 x (1 + 10/20) = 30 + 30 x 10/20.

The incremental forecast is 30 + 30 x 10/20 - 30 = 30 x 10 / 20.

In the second case, again writing in terms of incrementals, the ratio is

(20+30)/20 = 1 + 30/20.

The cumulative forecast is 10 x (1 + 30/20) = 10 + 10 x 30/20.

The incremental forecast is 10 + 10 x 30/20 - 10 = 10 x 30 / 20.

In the general case, the incremental forecast (calculated in either direction) if b denotes the sum of incrementals in the same development year, and c denotes the sum of incrementals in the same accident year, and a denotes the sum of all values that are both above and to the left (in earlier accident and development years), then the incremental predicted (forecast) value is (b x c)/a .

Note that if the forecast value is further into the future than the next diagonal (calendar year), that (unknown) future incremental values required for the formula are replaced with their own forecasts.

An algebraic proof that the incremental forecasts are of the form (b c)/a 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 b x c/a = c x b/a.

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.

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.

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.