Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?

Question

What I want to do is the following:

Let's say I have two assets 1 and 2, and have a 2x2 covariance matrix.

Then I have two portfolios A and B made of weights from assets 1 and 2.

What I would like to do is create a 4x4 covariance matrix of assets 1 and 2 and portfolios A and B.

I know how to calculate the covariance of the portfolios to the assets, I'm interested if there's a 'shortcut' to creating the 4x4 matrix using matrix algebra vs. building it from parts.

Answer 1

If your two assets are denoted by random variables $X_1$, $X_2$, with 2x2 covariance matrix $\mathbf{Q}$ and the portfolios:

$$ Z_1 = w_{11} X_1 + w_{12} X_2 $$ $$ Z_2 = w_{21} X_1 + w_{22} X_2 $$

Then,

$Cov(Z_1, X_1) = w_{11}Cov(X_1,X_1) + w_{12} Cov(X_2, X_1)$ , etc.

In matrix algebra:

$$ \mathbf{Z} = \mathbf{W} \mathbf{X}$$

The 4x4 covariance matrix, is:

$$ \begin{bmatrix} \mathbf{Q} & \mathbf{QW^T} \\ \mathbf{WQ} & \mathbf{WQW^T} \\ \end{bmatrix} $$

Where W is the identity matrix you can verify this reduces to your intuition.