Covariance and 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:
- \( cov_{AB} = \frac{1}{N-1} \sum_{t=1}^{N}(R_{At} - \bar{R}_A)(R_{Bt} - \bar{R}_B) \)
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:
- \( \rho_{AB} = \frac{cov_{AB}}{\sigma_A \sigma_B} \)
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.
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