Variance, Covariance, and Correlation

Variance measures the spread or dispersion of a distribution, indicating how much a random variable deviates from its mean. For a random variable \( X \), the variance is defined as:

\[\text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]\]

A higher variance implies greater uncertainty or volatility in outcomes.

Covariance measures how two random variables, such as asset returns \( X \) and \( Y \), move together:

\[\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]\]

A positive covariance suggests that the variables tend to increase or decrease together, while a negative covariance indicates an inverse relationship.

Correlation standardizes covariance to a scale from \( -1 \) to \( 1 \), providing a unit-free measure of linear dependence:

\[\rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}\]

Correlation is essential for portfolio construction, guiding diversification and hedging decisions by quantifying how assets co-move.