Statistical Models

Mack-Method

The Mack chain ladder method is a statistical method to estimate developmental factors in the chain ladder method. The method assumes the following:

Distinct rows of the array/matrix [math]C_{ij}[/math] are independent.
[math]\operatorname{E}[C_{i,k+1} | C_{i1},\ldots,C_{ik}] = f_k C_{i,k}[/math] with [math]f_k[/math] a constant.
[math]\operatorname{Var}[C_{i,k+1} | C_{i1},\ldots,C_{ik}] = \sigma_{k}^2 C_{i,k}[/math] with [math]\sigma_k[/math] a constant.

The goal of the Mack-method is to estimate the factors [math]f_k[/math] using the observable [math]C_{ik}[/math]. The estimators, denoted [math]\hat{f}_k[/math], then become selected age-to-age factors. The estimators are defined as follows:

[[math]] \begin{equation} \hat{f}_k = \frac{\sum_{j=1}^{I-k}C_{j,k+1}}{\sum_{j=1}^{I-k}C_{j,k}}. \end{equation} [[/math]]

They have the following desirable properties:

[math]\hat{f_k}[/math] is an unbiased estimator for [math]f_k[/math]: [math]\operatorname{E}[\hat{f_k}] = f_k[/math].
The estimator [math]\hat{f_k}[/math] is a minimum variance estimator in the following sense:

[[math]] \hat{f_k} = \underset{X \in S_k}{\operatorname{argmin}} \operatorname{Var}[ X | A_k ],\, S_k = \{\sum_{i=1}^{I-k}w_i C_{i,k+1}/C_{ik} | \sum_{i=1}^{I-k}w_i = 1\} [[/math]]

with [math]A_k = \cup_{i=1}^{I-k}\{C_{i1},\ldots,C_{ik}\} [/math] the claims information contained in the first [math]k[/math] periods.

Applying the method to the (reported claims) data in Standard Estimation Techniques , we obtain the following selected factors:

12-24	24-36	36-48	48-60
1.186	1.059	1.027	1.012