Covariance measures how two asset returns vary together by combining their volatilities with their correlation. It turns pairwise relationships into inputs for portfolio-risk calculations. section 2.2
For assets i and j, Cov(rᵢ,rⱼ) = σᵢσⱼ Corr(rᵢ,rⱼ). When i = j, covariance becomes the asset’s variance, so variances occupy the diagonal of a covariance matrix and pairwise covariances occupy its off-diagonal elements. section 2.2
Use covariance to calculate portfolio volatility:
- Construct a covariance matrix from a correlation matrix and individual volatilities: Cov = diag(σ) @ Corr @ diag(σ). section 2.2
- Combine asset weights with the matrix: σₚ = (wᵀΣw)¹ᐟ², equivalently the square root of the double sum of weighted covariances. section 2.2
- Estimate covariances from historical returns with pandas; annualize daily covariance values by multiplying by 252. section 3.2.1
Three-asset covariance matrix
Given volatilities and correlations for three assets, the resulting matrix is [[0.0225, 0.003, 0.008925], [0.003, 0.04, 0.0182], [0.008925, 0.0182, 0.1225]]. Applying portfolio weights with w @ Cov @ w.T produces a portfolio standard deviation of 20.7%. 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)
Inspect the NumPy listing that converts volatilities and correlations into a covariance matrix. section 2.2
stock_returns.cov() * 252
Inspect the pandas expression that estimates an annualized covariance matrix from historical returns. section 3.2.1
