Aggregating Asset Contributions to Portfolio Variance

Summary

The document defines portfolio variance as the quadratic form of asset weights and their return covariance matrix. It then proposes using a matrix square root multiplied by the weight vector to obtain signed asset contributions whose squared values sum to total portfolio variance. This decomposition depends on the chosen matrix square root, but the text does not specify which square root to use beyond the defining identity.

The question is how to aggregate contributions for a group of assets, especially when two perfectly correlated positions hedge one another. In that case, the desired group contribution is zero. The document gives no answer or group-level formula, so it identifies a limitation of the proposed asset-level decomposition rather than presenting a complete method. The example highlights that individual signed contributions or their squares may not straightforwardly represent the variance attributable to a group when covariance and hedging are involved.

Key ideas

• Portfolio variance is expressed as asset weights multiplied by the return covariance matrix and weights again.
• The proposed signed contribution vector uses a matrix square root applied to the portfolio weights.
• The squared asset contributions are intended to sum to total portfolio variance.
• Group attribution is unresolved when covariance and offsetting positions affect the combined risk.
• Perfectly correlated, perfectly hedged assets should have zero combined variance contribution in the stated example.

Tags

# Decomposition of Contribution to Variance


# Decomposition of Contribution to Variance












$C$ is a $N\times N$ covariance matrix of stock returns. Assuming $w$ is a vector of positions in each asset, the total variance of the portfolio is $$w^TCw$$ The contribution to total variance of the N stocks is $$\text{contributiontoVariance} = \sqrt Cw$$ where we use the matrix square root such that $\sqrt C\sqrt C = C$.

$\text{contributiontoVariance}$ gives me a signed measure of the contribution to total variance of each asset in the portfolio, such that the sum of the squares of the contributions equals the total portfolio variance $w^TCw$.

How do I calculate the contribution of a group of assets to the total variance? If, say, asset $A$ and asset $B$ are perfectly correlated and are perfectly hedged against each other, I would want their combined contribution to variance to be zero.

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