9.6 GLM Bootstrap Model

Limitations of ODP Bootstrap carry over from chainladder

1. Does not adjust for CY trend

2. May over fit the data from using too many parameters

We can solve the above by going back to the GLM framework instead of using the Simplified GLM when we’re at Step 2 of the simulation

GLM benefits

1. Not forced to use a specific number of parameters (e.g. GLM Variation 1 and 2)

2. Allows for CY trend

3. Can work with shapes that are non triangles (e.g. data with only the last \(x\) diagonals)

   • We can forecast past the end of the triangle (e.g. have the \(\beta\)’s continue the decay)

   • Also see section on practical issues

GLM drawbacks

1. Solving GLM at each iteration can slow down the process

2. Model not explainable using development factors

Below subsections are just examples of the variations discussed above

9.6.1 GLM Variation 1: Reduce Row Parameters

Use only 1 AY parameter \(\alpha_1\)

• Similar to Venter Cap Cod method 5.3 with \(h(w) = h\)

• Moves away from the Chainladder assumption that each AY has its own level

• Also note that the residual in cell (3,1) will no longer be zero since it shares the \(\alpha_1\) with all the other rows

• Below is our \(3 \times 3\) triangle example analogous to our example in (9.5)

\[\begin{equation*} \begin{array}{ccccc} \mathbf{Y} & = & \mathbf{X} &\times & \mathbf{A} \\ & & \alpha_1 \:\:\: \beta_2 \:\:\: \beta_3 & &\\ \begin{bmatrix} ln[q(1,1)] \\ ln[q(2,1)] \\ ln[q(3,1)] \\ ln[q(1,2)] \\ ln[q(2,2)] \\ ln[q(1,3)] \\ \end{bmatrix} & = & \begin{bmatrix} 1 & - & - \\ 1 & - & - \\ 1 & - & - \\ 1 & 1 & - \\ 1 & 1 & - \\ 1 & 1 & 1 \\ \end{bmatrix} & \times & \begin{bmatrix} \alpha_1 \\ \beta_2 \\ \beta_3 \\ \end{bmatrix} \end{array} \end{equation*}\]

9.6.2 GLM Variation 2: Reduce Column Parameters

Use only 1 development year parameter \(\beta_2\)

• This just assumes the losses decay by \(e^{\beta_2}\) for all ages

• Also note that the residual in cell (1,3) will no longer be zero since it shares the \(\beta_2\) with all the other columns

• Below is our \(3 \times 3\) triangle example analogous to our example in (9.5)

\[\begin{equation*} \begin{array}{ccccc} \mathbf{Y} & = & \mathbf{X} &\times & \mathbf{A} \\ & & \alpha_1 \:\:\: \alpha_2 \:\:\: \alpha_3 \:\:\: \beta_2 & &\\ \begin{bmatrix} ln[q(1,1)] \\ ln[q(2,1)] \\ ln[q(3,1)] \\ ln[q(1,2)] \\ ln[q(2,2)] \\ ln[q(1,3)] \\ \end{bmatrix} & = & \begin{bmatrix} 1 & - & - & - \\ - & 1 & - & - \\ - & - & 1 & - \\ 1 & - & - & 1 \\ - & 1 & - & 1 \\ 1 & - & - & 2 \\ \end{bmatrix} & \times & \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \beta_2 \\ \end{bmatrix} \end{array} \end{equation*}\]

9.6.3 GLM Variation 3: Reduce Row and Column Parameters

Further reduce the parameters to just 1 row and 1 column parameter \(\alpha_1\) and \(\beta_2\)

• Flexibility of the GLM Bootstrap so that we’re not always stuck with \(p = 2n-1\) as stated earlier

• This will gives us 6 residuals to sample from (the corners will not longer be 0’s)

• Below is our \(3 \times 3\) triangle example analogous to our example in (9.5)

• See diagnostics section on how to determine which parameters are statistically significant

\[\begin{equation*} \begin{array}{ccccc} \mathbf{Y} & = & \mathbf{X} &\times & \mathbf{A} \\ & & \alpha_1 \:\:\: \beta_2 & &\\ \begin{bmatrix} ln[q(1,1)] \\ ln[q(2,1)] \\ ln[q(3,1)] \\ ln[q(1,2)] \\ ln[q(2,2)] \\ ln[q(1,3)] \\ \end{bmatrix} & = & \begin{bmatrix} 1 & - \\ 1 & - \\ 1 & - \\ 1 & 1 \\ 1 & 1 \\ 1 & 2 \\ \end{bmatrix} & \times & \begin{bmatrix} \alpha_1 \\ \beta_2 \\ \end{bmatrix} \end{array} \end{equation*}\]

9.6.4 GLM Variation 4: Calendar Year Parameter

We can also add calendar year trend \(\gamma_k\) where \(k\) is the CY

• \(\gamma_2\) is the 2^nd diagonal and etc

• \(\gamma_k\)’s are incremental decay similar to the \(\beta_d\)’s

  \(\therefore\) The total impact on the 3^rd diagonal is \(e^{\gamma_2 + \gamma_3}\)

• Note that the model here have 7 parameters and 6 values

  \(\therefore\) It has no unique solution

• Below is our \(3 \times 3\) triangle example analogous to our example in (9.5)

\[\begin{equation*} \begin{array}{ccccc} \mathbf{Y} & = & \mathbf{X} &\times & \mathbf{A} \\ & & \alpha_1 \:\:\: \alpha_2 \:\:\: \alpha_3 \:\:\: \beta_2 \:\:\: \beta_3 \:\:\: \gamma_2 \:\:\: \gamma_3 & &\\ \begin{bmatrix} ln[q(1,1)] \\ ln[q(2,1)] \\ ln[q(3,1)] \\ ln[q(1,2)] \\ ln[q(2,2)] \\ ln[q(1,3)] \\ \end{bmatrix} & = & \begin{bmatrix} 1 & - & - & - & - & - & -\\ - & 1 & - & - & - & 1 & -\\ - & - & 1 & - & - & 1 & 1\\ 1 & - & - & 1 & - & 1 & -\\ - & 1 & - & 1 & - & 1 & 1\\ 1 & - & - & 1 & 1 & 1 & 1\\ \end{bmatrix} & \times & \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \beta_2 \\ \beta_3 \\ \gamma_2 \\ \gamma_3 \\ \end{bmatrix} \end{array} \end{equation*}\]

9.6.5 GLM Variation 5: One Parameter for Each Dimension

Again we can simplify things by having only 1 parameters for each dimension: \(\alpha_1\), \(\beta_2\), and \(\gamma_2\)

• Use this as a starting point then add or remove parameters as needed

• Below is our \(3 \times 3\) triangle example analogous to our example in (9.5)

\[\begin{equation*} \begin{array}{ccccc} \mathbf{Y} & = & \mathbf{X} &\times & \mathbf{A} \\ & & \alpha_1 \:\:\: \beta_2 \:\:\: \gamma_2 & &\\ \begin{bmatrix} ln[q(1,1)] \\ ln[q(2,1)] \\ ln[q(3,1)] \\ ln[q(1,2)] \\ ln[q(2,2)] \\ ln[q(1,3)] \\ \end{bmatrix} & = & \begin{bmatrix} 1 & - & - \\ 1 & - & 1 \\ 1 & - & 2 \\ 1 & 1 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 2 \\ \end{bmatrix} & \times & \begin{bmatrix} \alpha_1 \\ \beta_2 \\ \gamma_2 \\ \end{bmatrix} \end{array} \end{equation*}\]