Covariance Estimation and Correlation

Covariance measures joint variability in units; correlation rescales it to a dimensionless sensitivity between −1 and 1.

Explanation

Covariance captures how two variables move together but depends on their units and scales.

Correlation normalises covariance by the product of standard deviations, so it is unit-free and lies between −1 and 1.

Sample covariances and correlations are noisy estimates; their stability depends on sample size and the underlying structure.

covariancecorrelationestimation

Interactive visualisation

This scatter shows simulated returns for two assets X and Y. The blue cloud is the sample; the amber ellipse represents the one-standard-deviation contour implied by the target covariance.

Target ρ: 0.70 · Sample ρ̂: 0.66

Target correlation ρ: 0.70Volatility σ_X: 2.0%Volatility σ_Y: 3.0%Number of observations n: 150

Numbers

Sample var(X) ≈ 3.08e-4 · var(Y) ≈ 7.52e-4

Sample cov(X, Y) ≈ 3.16e-4

Target ρ = 0.700 · sample ρ̂ ≈ 0.656

Implied population cov ≈ 4.20e-4

Interpretation

Scaling σ_X or σ_Y stretches the cloud and the covariance, but correlation stays the same: it measures shape (tilt) rather than absolute size. That is why covariance depends on units and correlation does not.

With small n the sample covariance and ρ̂ jump around their target values; as n grows, they stabilise and the cloud matches the amber ellipse more closely. The mental model: covariance is “joint variance in units”, correlation is the same object viewed through a scale-free lens, and estimation error comes from limited data.