Recall that:

Portfolio VaR = VaRₚ = ασₚW = α√(x′Σx)

The first step is to calculate the variance of the portfolio dollar returns. Let x be the dollar amounts allocated to each risk factor, in millions. We first calculate the product:

Σx = \begin{bmatrix} 0.065^2 & 0 \\ 0 & 0.1^2 \end{bmatrix} \begin{bmatrix} \`4 \\ \3` \end{bmatrix} = \begin{bmatrix} 0.065^2 \times \`4 + 0 \times \3` \\ 0 \times \`4 + 0.1^2 \times \3` \end{bmatrix} = \begin{bmatrix} \`0.0169 \\ \0.03` \end{bmatrix}

Therefore, the portfolio value in dollar returns is:

σₚ²W² = x′Σx = \begin{bmatrix} \`4 & \3` \end{bmatrix} \begin{bmatrix} \`0.0169 \\ \0.03` \end{bmatrix} = \`4 \times 0.0169 + \$3` \times 0.03 = 0.1576$

The dollar volatility is:

√0.1576 = $0.396989 million

With α = 1.65, the portfolio VaR is given as:

VaRₚ = 1.65 × $396,989 = $655,032