Question

Explain Key Statistical Relationships between Covariance and Correlation of Returns

Answer 1

Covariance and Correlation are two terms which are exactly opposite to each other, they both are used in statistics and regression analysis, covariance shows us how the two variables vary from each other whereas correlation shows us the relationship between the two variables and how are they related.

Covariance is an indicator of the degree to which two random variables change with respect to each other. Correlation, on the other hand, measures the strength of this relationship. The value of correlation is bound on the upper by +1 and on the lower side by -1. Thus, it is a definite range. However, the range of covariance is indefinite. It can take any positive value or any negative value (theoretically the range is -∞ to +∞). You can rest assured that a correlation of .5 is greater than .3 and the first set of numbers (with correlation as .5) are more dependent on each other than the second set (with correlation as .3) Interpreting such a result would be very difficult from covariance calculations.

Change of scale affects covariance. For example, if the value of two variables is multiplied by similar or different constants, then this affects the calculated covariance of these two numbers. However, applying the same mechanism for correlation, multiplication by constants does not change the previous result. This is because a change of scale does not affect correlation.

Unlike covariance, correlation is a unit-free measure of the inter-dependency of two variables. This makes it easy for calculated correlation values to be compared across any 2 variables irrespective of their units and dimensions.

Covariance can be calculated for only 2 variables. Correlation, on the other hand, can be calculated for multiple sets of numbers. Another factor that makes the correlation desirable to analysts compared to covariance.

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Covariance vs Correlation

Home » Excel » Statistical Functions in Excel » Covariance vs Correlation

correlation vs covariance

Difference Between Covariance and Correlation

Covariance and Correlation are two terms which are exactly opposite to each other, they both are used in statistics and regression analysis, covariance shows us how the two variables vary from each other whereas correlation shows us the relationship between the two variables and how are they related.

Correlation and covariance are two statistical concepts that are used to determine the relationship between two random variables. Correlation defines how a change in one variable will impact the other while covariance defines how two items vary together. Confusing? Let’s dive in further to understand the difference between these closely related terms.

What is Covariance?

Covariance measures how two variables move with respect to each other and is an extension of the concept of variance (which tells about how a single variable varies). It can take any value from -∞ to +∞.

Higher this value, more dependent is the relationship. A positive number signifies positive covariance and denotes that there is a direct relationship. Effectively this means that an increase in one variable would also lead to a corresponding increase in the other variable provided other conditions remain constant.

On the other hand, a negative number signifies negative covariance which denotes an inverse relationship between the two variables. Though covariance is perfect for defining the type of relationship, it is bad for interpreting its magnitude.

What is Correlation?

Correlation is a step ahead of covariance as it quantifies the relationship between two random variables. In simple terms, it is a unit measure of how these variables change with respect to each other (normalized covariance value).

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Unlike covariance, the correlation has an upper and lower cap on a range. It can only take values between +1 and -1. A correlation of +1 indicates that random variables have a direct and strong relationship.

On the other hand, correlation of -1 indicates that there is a strong inverse relationship and an increase in one variable will lead to an equal and opposite decrease in the other variable. 0 indicates that the two numbers are independent.

Formula for Covariance and Correlation

Let’s express these two concepts mathematically. For two random variables A and B with mean values as Ua and Ub and standard deviation as Sa and Sb respectively:

Effectively the relationship between the 2 can be defined as:

Formula

Both correlations and covariance find application in fields of statistical and financial analysis. Since correlation standardizes the relationship, it is helpful in comparison of any two variables. This help analyst in coming up with strategies like pair trade and hedging for not only efficient returns on the portfolio but also safeguarding these returns in terms of adverse movements in the stock market.

Correlation vs Covariance Infographics

Let’s see the top difference between Correlation vs Covariance.

Infographics correlation and covariance

Key Differences

Covariance is an indicator of the degree to which two random variables change with respect to each other. Correlation, on the other hand, measures the strength of this relationship. The value of correlation is bound on the upper by +1 and on the lower side by -1. Thus, it is a definite range. However, the range of covariance is indefinite. It can take any positive value or any negative value (theoretically the range is -∞ to +∞). You can rest assured that a correlation of .5 is greater than .3 and the first set of numbers (with correlation as .5) are more dependent on each other than the second set (with correlation as .3) Interpreting such a result would be very difficult from covariance calculations.

Change of scale affects covariance. For example, if the value of two variables is multiplied by similar or different constants, then this affects the calculated covariance of these two numbers. However, applying the same mechanism for correlation, multiplication by constants does not change the previous result. This is because a change of scale does not affect correlation.

Unlike covariance, correlation is a unit-free measure of the inter-dependency of two variables. This makes it easy for calculated correlation values to be compared across any 2 variables irrespective of their units and dimensions.

Covariance can be calculated for only 2 variables. Correlation, on the other hand, can be calculated for multiple sets of numbers. Another factor that makes the correlation desirable to analysts compared to covariance.

Covariance vs Correlation Comparative Table

Basis Covariance Correlation

Meaning Covariance is an indicator of the extent to which 2 random variables are dependent on each other. A higher number denotes higher dependency. Correlation is an indicator of how strongly these 2 variables are related provided other conditions are constant. The maximum value is +1 denoting a perfect dependent relationship.

Relationship Correlation can be deduced from covariance Correlation provides a measure of covariance on a standard scale. It is deduced by dividing the calculated covariance with standard deviation.

Values The value of covariance lies in the range of -∞ and +∞. Correlation is limited to values between the range -1 and +1.

Scalability Affects covariance Correlation is not affected by a change in scales or multiplication by a constant.

Units Covariance has a definite unit as it is deduced by the multiplication of two numbers and their units. Correlation is a unitless absolute number between -1 and +1 including decimal values.

Conclusion

Correlation and covariance are very closely related to each other and yet they differ a lot. Covariance defines the type of interaction, but correlation defines not only the type but also the strength of this relationship. Due to this reason correlation is often termed as the special case of covariance. However, if one must choose between the two, most analysts prefer correlation as it remains unaffected by the changes in dimensions, locations, and scale. Also, since it is limited to a range of -1 to +1, it is useful to draw comparisons between variables across domains. However, an important limitation is that both these concepts measure the only linear relationship.