To solve this portfolio optimization problem, we'll approach it step by step.

Part (a) - Global Minimum-Variance Portfolio:

The global minimum-variance portfolio is obtained by minimizing portfolio variance subject to the condition that the sum of portfolio weights is 1. This is done without considering the returns of the assets.

The formula for the weights of the global minimum-variance portfolio is:

$\mathbf{w}_{\text{GMV}} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}}$

Where:

• $\Sigma$ is the covariance matrix of asset returns.
• $\mathbf{1}$ is a column vector of ones (i.e., $\mathbf{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$).
• $\Sigma^{-1}$ is the inverse of the covariance matrix.

Step-by-Step:

1. Invert the covariance matrix $\Sigma$.

   Given $\Sigma = \begin{bmatrix} 0.4 & 0.2 & 0 \\ 0.2 & 0.4 & 0.2 \\ 0 & 0.2 & 0.4 \end{bmatrix}$, find $\Sigma^{-1}$.

2. Multiply $\Sigma^{-1}$ by $\mathbf{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$.

3. Normalize the weights by dividing the result by $\mathbf{1}^T \Sigma^{-1} \mathbf{1}$.

Part (b) - Optimal Portfolio for a Required Return:

For this part, we are finding the weights of the optimal risky portfolio that satisfies a required return $z = 0.075$. The formula for the optimal weights is:

$\mathbf{w} = \lambda \Sigma^{-1} \mathbf{1} + \mu \Sigma^{-1} \mathbf{r}$

Where:

• $\lambda$ and $\mu$ are constants to be determined.
• $\mathbf{r} = \begin{bmatrix} 0.04 \\ 0.08 \\ 0.06 \end{bmatrix}$ is the vector of expected returns.
• $z$ is the target return (in this case, 0.075).

The constants $\lambda$ and $\mu$ are found by solving the following system: