Deriving Covariance Between Two Portfolio Returns

Summary

The document derives the covariance between returns on two portfolios from the asset-level covariance matrix. Each portfolio return is a weighted sum of individual asset returns. Applying covariance’s linearity in each argument expands portfolio covariance into a double sum: each pair of portfolio weights multiplies the covariance between the corresponding assets. Those pairwise covariances are the entries of the covariance matrix, so the sum is the bilinear matrix expression formed by the two weight vectors and that matrix.

The result applies to arbitrary portfolios, not only efficient portfolios or special cases such as minimum-variance and maximum-Sharpe portfolios. The derivation assumes portfolio returns are represented as linear combinations of the same asset return variables and that the covariance matrix uses their pairwise covariances. It gives an algebraic identity, not an empirical estimate or investment strategy; the document does not discuss estimation error, constraints, or changing weights through time.

Key ideas

• A portfolio return is the weighted sum of its underlying asset returns.
• Covariance linearity expands covariance between two portfolio returns into pairwise asset covariances.
• The asset covariance matrix turns that double sum into a bilinear expression in the two weight vectors.
• The identity holds for general portfolios and does not depend on mean-variance efficiency.

Tags

# Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$


# Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$












How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to

$$w_A^\top\Sigma w_B$$

where $w_i$ is a unique portfolio weight vector, and $\Sigma$ is the covariance matrix of asset returns.

Is there a source that goes over this in detail for portfolios in general, not just the minimum-variance and max Sharpe portfolios?

## Answer by steveo'america (score 2, accepted)

https://quant.stackexchange.com/a/59470

When $x_i$ is the return of the $i$th asset, the returns of portfolio $\vec{w}$ are $\sum_i w_i x_i$. The covariance of the returns of two portfolios, $\vec{w}$ and $\vec{v}$ are then $$ \sum_i \sum_j w_i v_j \operatorname{cov}\left(x_i, x_j\right). $$ Now note that $\Sigma_{i,j} = \operatorname{cov}\left(x_i,x_j\right)$. The rest is confirming that this expression is the bilinear form $\vec{v}^{\top}\Sigma \vec{w}$.

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