Parameter and process uncertainty in Mack's method

[the_mighty_onion]

My comment relates to SA3 ASET April 2008 Q1 (vi.). Regarding Mack's method, the ASET says at the end:

"The method estimates both process and parameter uncertainty."

This isn't true. Mack's method by itself only estimates process uncertainty. The parameters in Mack are single values: we just have a single estimated value for each row's mu and sigma; there is no uncertainty in these values.

In conjunction with bootstapping, for instance, Mack's method can definitely estimate parameter uncertainty - but it is the "bootstrapping bit" that is doing that.

As an analogy, saying that Mack's method estimates parameter uncertainty is like saying that fitting a simple normal distribution to some data by estimating mu and sigma estimates parameter uncertainty: it doesn't by itself. It needs coupling with something like bootstrapping or Bayesian/MCMC methods to get a distribution of the fitted parameters.

Sorry for bothering you all again .

 

[Duncan Brydon]

Mack's theory gives a formula for the mean squared error of the reserve (the prediction uncertainty). For each accident year, this can be expressed as the sum of two terms, one relating to the variability in the claims and one relating to the variability around the reserve estimate. The former can be labelled the process uncertainty and the latter the parameter uncertainty.

I hope this helps
Duncan

 

[the_mighty_onion]

Hi Duncan,

Yes, sorry, I see that you are quite right: the parameters fitted in Mack's model do actually have the intent of capturing both process and parameter uncertainty, as stated in Mack's paper.

The thing is though, is that it seems to me that the parameter uncertainty that is caught in Mack's model is with respect to the chain-ladder estimates for the cumulatives, rather than any uncertainty in the mu_j and sigma_j parameters in Mack's model, which are just point estimates for each row. This is a bit different from capturing parameter uncertainty in the mu_j and sigma_j, which bootstrapping would deal with.

In any case, I understand why it is generally said that Mack's model includes parameter uncertainty now - it's not quite what I would think of as parameter uncertainty, since there is no uncertainty in the Mack parameters that are fitted, but I'll know what to say in the exam .

Thanks for this.

 

[Sherwin]

Hi Duncan,

Yes, sorry, I see that you are quite right: the parameters fitted in Mack's model do actually have the intent of capturing both process and parameter uncertainty, as stated in Mack's paper.

The thing is though, is that it seems to me that the parameter uncertainty that is caught in Mack's model is with respect to the chain-ladder estimates for the cumulatives, rather than any uncertainty in the mu_j and sigma_j parameters in Mack's model, which are just point estimates for each row. This is a bit different from capturing parameter uncertainty in the mu_j and sigma_j, which bootstrapping would deal with.

In any case, I understand why it is generally said that Mack's model includes parameter uncertainty now - it's not quite what I would think of as parameter uncertainty, since there is no uncertainty in the Mack parameters that are fitted, but I'll know what to say in the exam .

Thanks for this.

I think that the key point is what the parameter means in Mack method. I think that the parameter in Mack method is the estimator of the ultimate loss, rather than any of distributions. Remember it's distribution-free.

Assuming the estimator is known without any uncertainty, the variance calculated from the data is the process variance, which is the first part of Mack's formula.

The second part of Mack's formula did with the error of the estimator (or the parameter), which implies the parameter variance.

 

[Sherwin]

Error in England's Mack Model?

Recently I found something wrong with England&Verrall's Mack model.

The attached pages are from England and Verrall's paper “Stochastic Claims Reserving in General Insurance”.

I think all the Sigma(k+1) and Lamda(k+1) in calculating Variances should be (k) rather than (k+1).

Anyone noticed that?