The covariances for the portfolio will be:
Covariance$_{\text{AA}} = \rho_{\text{AA}} \sigma_{\text{A}} \sigma_{\text{A}} = 1 * 0.03 * 0.03 = 0.0009$
Covariance$_{\text{AB}} = \rho_{\text{AB}} \sigma_{\text{A}} \sigma_{\text{B}} = 0.4 * 0.03 * 0.05 = 0.0006$
Covariance$_{\text{BA}} = \rho_{\text{BA}} \sigma_{\text{B}} \sigma_{\text{A}} = 0.4 * 0.05 * 0.03 = 0.0006$
Covariance$_{\text{BB}} = \rho_{\text{BB}} \sigma_{\text{B}} \sigma_{\text{B}} = 1 * 0.05 * 0.05 = 0.0025$

$$\begin{bmatrix} \text{Cov}(A, A) & \text{Cov}(A, B) \\ \text{Cov}(B, A) & \text{Cov}(B, B) \end{bmatrix}$$

Thus, our covariance matrix is:

$$\begin{bmatrix} 0.0009 & 0.0006 \\ 0.0006 & 0.0025 \end{bmatrix}$$

The standard deviation for the portfolio can be calculated by multiplying the covariance matrix with the amount of investment: We do this by first solving $\beta_{\text{h}} C$, followed by $\beta_{\text{h}} C \beta_{\text{v}}$, and then finding the square root.

$$\beta_{\text{h}} C = \begin{bmatrix} 3 & 7 \end{bmatrix} \begin{bmatrix} 0.0009 & 0.0006 \\ 0.0006 & 0.0025 \end{bmatrix} = (3 * 0.0009 + 7 * 0.0006,\; 3 * 0.0006 + 7 * 0.0025) = [0.00690\; 0.0193]$$ $$\beta_{\text{h}} C \beta_{\text{v}} = [0.00690\; 0.0193] \begin{bmatrix} 3 \\ 7 \end{bmatrix} = 0.009252 \times 20 + 0.010384 \times 10 \\ = 3 \times 0.0069 + 7 \times 0.0193 \\ = 0.1558$$

Thus, $\sigma_p = \sqrt{0.1558} = 0.3947$

\text{VaR}_p = \sigma_p \alpha \sqrt{x} = \`1`\ \text{million} \times 0.3947 \times 2.33 \sqrt{1} = 0.92\ \text{million}$

Where: $\beta_{\text{h}}$ is the horizontal $\beta$ vector of invested amounts
$\beta_{\text{Y}}$ is the vertical $\beta$ vector of invested amounts
$C$ is the covariance matrix of the returns of the assets.