Lesson 3 of 7 · All lessons in this module
  1. Portfolio expected return and varianceVideo lesson
  2. Covariance and correlation
  3. The covariance matrix and many-asset portfolios
  4. Correlation and diversification
  5. Covariance from a joint probability function
  6. Shortfall risk and Roy's safety-first criterion
  7. Shortfall probability, the Sharpe ratio and VaR

Lesson 3 of 7 · 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: σp2=∑i∑jwiwjCov(Ri,Rj)\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 nn assets: n2n^2 entries, nn variances, n(n−1)n(n-1) off-diagonal entries, and n(n−1)/2n(n-1)/2 distinct covariances.
  • As nn grows, covariance terms dominate portfolio variance.
  • A correlation matrix follows from the covariance matrix: σ\sigma = square root of each diagonal entry, then divide.

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