A covariance matrix summarizes how asset returns vary individually and together. It is a square, symmetric matrix: diagonal entries are variances, while off-diagonal entries are covariances. Covariance combines volatility and correlation: Cov(rᵢ,rⱼ) = σᵢσⱼCorr(rᵢ,rⱼ). section 2.2
For portfolio weights w and covariance matrix Σ, portfolio volatility is √(wᵀΣw). This makes Σ a central risk input: it captures both each asset’s standalone variability and the interaction among assets rather than treating holdings independently. section 2.2 section 12.1.1
Build a matrix from correlations and volatilities
Given a correlation matrix Corr and volatility vector σ, form Σ as diag(σ) @ Corr @ diag(σ). For three assets, the resulting matrix has each squared volatility on its diagonal and a matching covariance in positions (i,j) and (j,i). Portfolio standard deviation then follows from (w @ Cov @ w.T) ** 0.5. section 2.2
Listing 2.3 Computing the covariance matrix from volatilities and correlations
Corr = [[ 1. , 0.1 , 0.17],
[ 0.1 , 1. , 0.26],
[ 0.17, 0.26, 1. ]]
Cov = np.diag(sigma) @ Corr @ np.diag(sigma) #1
print('Covariance matrix: \n', Cov)
This listing shows the direct NumPy construction that scales correlations into covariances. section 2.2
Historical return series provide a direct estimate: in pandas, annualize the daily covariance matrix with stock_returns.cov() * 252. This is straightforward for a few assets, but estimating all pairwise relationships becomes impractical for thousands of securities; multifactor models reduce the number of correlations that must be estimated. section 3.2.1
A usable correlation or covariance matrix must be positive definite, checked by confirming that all eigenvalues are positive. Historical sample estimates also miss volatility clustering, so estimated risk can change as market volatility changes. section 3.3 section 3.2.1
Covariance matrices also enter allocation models. A diagonal Σ is a simplifying assumption that removes cross-asset covariances; under that assumption, a mean-variance solution can yield weights inversely proportional to asset volatilities. In Black–Litterman, a separate covariance matrix Ω represents uncertainty and correlation among investor views. section 11.1.3 section 12.3.2
