Covariance, Correlation Coefficient

Interactive Risk through Covariance

When two securities are held in a portfolio, the risk involved is measured by the Covariance, also known as Interactive risk.

Covariance between securities

Covariance measures how two securities move in relation to each other:

• Positive Covariance: Indicates that the rates of return of two securities move in the same direction.
• Negative Covariance: Implies that the rates of return of two securities move in opposite directions.
• Zero Covariance: Suggests that the rates of return of two securities are independent of each other.

The formula for Covariance ((cov_{AB})) between two securities A and B is given by:

• ( cov_{AB} = \frac{1}{N-1} \sum_{t=1}^{N}(R_{At} - \bar{R}A)(R{Bt} - \bar{R}_B) ){: style="font-size: 150%;"}

where:

• ( N ) is the number of observations.
• ( R_{At} ) and ( R_{Bt} ) are the rates of return for securities A and B at time ( t ), respectively.
• ( \bar{R}_A ) and ( \bar{R}_B ) are the average rates of return for securities A and B, respectively.

Coefficient of Correlation

The Coefficient of Correlation (( \rho )) indicates the similarity or dissimilarity in the behavior of two variables. It is standardized covariance and gives a value between -1 and 1.

• -1.0 indicates a perfect negative correlation.
• 1.0 indicates a perfect positive correlation.
• 0 indicates that the variables are uncorrelated.

The formula for the Correlation Coefficient between two securities A and B is given by:

• ( \rho_{AB} = \frac{cov_{AB}}{\sigma_A \sigma_B} ){: style="font-size: 150%;"}

where:

• ( cov_{AB} ) is the covariance between securities A and B.
• ( \sigma_A ) and ( \sigma_B ) are the standard deviations of securities A and B, respectively.

The Correlation Coefficient normalizes the Covariance by the product of the standard deviations of the variables, which allows for a comparison that is independent of the units of measure.