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Lesson 3 of 13 · 13 min

The covariance matrix and many-asset portfolios

With more than two assets, portfolio variance is the weighted sum of every entry in the covariance matrix, and the covariance entries quickly outnumber the variances.

In short

General formula: $\sigma_p^2 = \sum_i\sum_j w_iw_j\text{Cov}(R_i,R_j)$ , summing over every pair including each asset with itself.
The covariance matrix has variances on the diagonal and covariances off it; it is symmetric.
For $n$ assets: $n^2$ entries, $n$ variances, $n(n-1)$ off-diagonal entries, and $n(n-1)/2$ distinct covariances.
As $n$ grows, covariance terms dominate portfolio variance.
A correlation matrix follows from the covariance matrix: $\sigma$ = square root of each diagonal entry, then divide.

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