Building an Expanded Covariance Matrix for Asset Portfolios

Summary

The document shows how to extend the covariance matrix of underlying assets to include portfolios formed from those assets. It represents the asset returns or random variables as a vector X, the portfolio weights as a matrix W, and the portfolio variables as Z = WX. Given the asset covariance matrix Q, the expanded matrix places Q in the asset block, QW-transpose and WQ in the cross-covariance blocks, and WQW-transpose in the portfolio block.

This construction computes covariances among the assets, between assets and portfolios, and among portfolios in one matrix expression. It is a compact alternative to calculating each covariance separately. The result assumes portfolios are linear combinations of the included assets and that the weights are represented consistently across rows of W. It does not address estimation error, changing weights, or portfolios containing additional assets outside the original covariance matrix.

Key ideas

• Portfolio variables formed from assets can be written as Z = WX.
• The asset covariance matrix and weight matrix determine the expanded joint covariance matrix.
• Cross-covariances between assets and portfolios occupy the off-diagonal blocks.
• Portfolio-to-portfolio covariances are given by WQW-transpose.
• The construction assumes linear portfolios made from the assets represented in Q.

Tags

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


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












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 by Attack68 (score 9, accepted)

https://quant.stackexchange.com/a/60897

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.