Covariance and Correlation

Explained:

Covariance and correlation are related parameters that indicate the extent to which two random variables co-vary. Suppose there are two technology stocks. If they are affected by the same industry trends, their prices will tend to rise or fall together. They co-vary. Covariance and correlation measure such a tendency.

Let's formalize this. Consider a random vector X whose components are random variables :

[1]

Given any pair of components, and , we denote their covariance as either or . The covariance is defined by the expectation

[2]

where and are the means of and . By definition, covariance is symmetric, with . Also, the covariance of any component with itself is that component’s variance:

[3]

We summarize all the covariance's of a random vector X with a covariance matrix:

[4]

Due to the symmetry property of covariances, this is necessarily a symmetric matrix. It can be shown that covariance matrices are positive definite or positive semidefinite.

The magnitude of a covariance depends upon the standard deviations of the two components and . To obtain a more direct indication of how two components co-vary, we scale covariance to obtain correlation.

Given any pair of components, and , we denote their correlation as either or . The correlation is defined as

[5]

where and are the standard deviations of and .

By construction, a correlation is always a number between –1 and 1. Correlation inherits the symmetry property of covariance: . From [3] and [5], , which indicates that a random variable co-varies perfectly with itself. If and are independent, their correlation is 0. The converse is not true. As with covariances, we can summarize all the correlations of a random vector X with a symmetric correlation matrix:

[6]

Like covariance matrices, correlation matrices must be positive definite or positive semidefinite.