Decomposing Portfolio Variance into Variance and Covariance Terms

Summary

The document explains how a portfolio variance expression separates into contributions from individual asset variances and from covariances between different assets. In the covariance matrix, the variance terms lie on the diagonal, where an asset is paired with itself; off-diagonal entries represent relationships between distinct assets. The corresponding portfolio contributions use squared asset weights for variances and cross-products of weights for covariances.

The question concerns why a factor of 1/n appears outside a summation in a textbook formula and whether the count of observations should instead appear once. The answer points back to the general portfolio variance formula: with equal weights of 1/n, applying that formula yields the same result. The explanation is conceptual rather than a worked derivation, and it does not spell out the full indexing or distinguish sample covariance estimation conventions. Its main lesson is to interpret the terms by their positions in the covariance matrix, rather than treat the two parts as separate observation counts.

Key ideas

• Diagonal covariance-matrix entries are the individual asset variances.
• Off-diagonal entries represent covariances between distinct assets.
• Portfolio variance weights variance contributions by squared asset weights.
• Applying the general portfolio formula with equal weights reproduces the decomposed expression.

Tags

# Portfolio Variance - Explanation for equation : Investments by Zvi Bodie


# Portfolio Variance - Explanation for equation : Investments by Zvi Bodie












Source: Investments 10th Edition by Bodie, Zvi. Page 227 Chapter 7

In Equation 7.17, the book breaks the variance into two parts. I can't seem to understand why the 1/n is represented outside the summation part in the first part of the equation. Also, shouldn't there be a single n given that there are n observations representing variance in the covariance table?

## Answer by AlRacoon (score 3)

https://quant.stackexchange.com/a/43443

The first part of equation 7.17 is just the contribution of the portfolio variance from the variance of the individual assets (j=i, the asset equals itself). It is the squared weights times just the diagonal of the variance-covariance matrix. The second part of the equation 7.17 is the contribution of the covariances of all the assets. It is the contribution of all the other elements of the variance-covariance matrix other than the diagonal. As you can see it is where the asset does not equal itself (j<>i).

If you just use equation 7.16, using 1/n as the weights, you will get the same answer.

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