Zhe Li

Forecasting Correlation of Returns: Covariance Given a Joint Probability Function

Dec202025

The joint probability function of two random variables X and Y, denoted P(X,Y), gives the probability of joint occurrences of values of X and Y.

A formula for computing the covariance between random variables $R_A$ and $R_B$ is

Cov(R_A, R_B)=\sum_i\sum_jP(R_{A,i},R_{B,j})(R_{A,i-ER_A})(R_{B,j}-ER_B)

The formula tells us to sum all possible deviation cross-products weighted by the appropriate joint probability.

Independence is a stronger property than uncorrelatedness because correlation addresses only linear relationship. The following condition holds for independent random variables and, therefore, also holds for uncorrelated random variables, since for two variables E(XY)=E(X)E(Y)+Cov(X,Y), and when the variables are uncorrelated, Cov(X,Y) = 0.

The expected value of the product of uncorrelated random variables is the product of their expected values.

E(XY)=E(X)E(Y) if X and Y are uncorrelated.