Covariance and Correlation — Section 1: Foundations of Probability

Variance measures how a single variable moves around its mean. Covariance measures how two variables move together.

\mathrm{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]\, E[Y]

Positive covariance means $X$ and $Y$ tend to deviate from their means in the same direction. Negative means opposite. Zero means no linear relationship.

The trouble with covariance is that its magnitude depends on the scales of $X$ and $Y$ , so the number itself is hard to interpret. Correlation rescales covariance to $[-1, 1]$ :

\rho(X, Y) = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \sigma_Y}

Now $\rho = +1$ means perfectly linear with positive slope, $\rho = -1$ means perfectly linear with negative slope, and $\rho = 0$ means no linear relationship. Watch out: $\rho = 0$ does not imply independence — it just rules out a linear relationship. Two independent variables always have $\rho = 0$ , but the converse fails (e.g. $Y = X^2$ on a symmetric $X$ ).

Covariance is the engine of portfolio finance. Portfolio variance, factor exposures, and risk decomposition all run through the covariance matrix of asset returns.

Variance measures how a single variable moves around its mean. Covariance measures how two variables move together.

\mathrm{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]\, E[Y]

Positive covariance means $X$ and $Y$ tend to deviate from their means in the same direction. Negative means opposite. Zero means no linear relationship.

The trouble with covariance is that its magnitude depends on the scales of $X$ and $Y$ , so the number itself is hard to interpret. Correlation rescales covariance to $[-1, 1]$ :

\rho(X, Y) = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \sigma_Y}

Covariance is the engine of portfolio finance. Portfolio variance, factor exposures, and risk decomposition all run through the covariance matrix of asset returns.