Definition of Covariance

Covariance is defined as the measure of the extent to which two random variables vary together. It tells us whether an increase in one variable would depict an increase or decrease in another variable.

• Positive Covariance: This means that as one variable increases, then so does the other, generally.

• Negative Covariance: It indicates that when the value of one variable increases, the other variable tends to decrease.

• Zero Covariance: Shows that the movements between two variables are not correlated.

Formula For Covariance

Population Covariance Formula: 

                     Cov(X,Y)=     

Sample Covariance Formula:

                          Cov(X,Y)=  

 

Notations in Covariance Formula:

• xi = observed value of x

• yi = observed value of y

• x̄ = mean of x

• ȳ = mean of y

• N = no. of data values.

Relationship between Correlation Coefficient and Covariance Formula

Here, Cov (x,y) stands for covariance between x and y. σx and σy correspondingly represent the standard deviation of x and y. From the formula above, using the covariance or the standard deviations can get us the correlation coefficient formula and vice versa.

Example Question Using Covariance Formula

The following table reports the growth rate in the economy, xi and returns on the S&P 500, yi. Use the covariance formula to compute the covariance between the growth rates in the economy and the returns on the S&P 500. Before you compute the covariance, compute the mean of x, mean of y,

 

x = 2.1, 2.5, 4.0, and 3.6 (economic growth)

y = 8, 12, 14, and 10 (S&P 500 returns)

Find x̄ and ȳ.

Solution:

x̄ =

x̄ = 

x̄ =      

x̄ = 3.1

ȳ = 

ȳ =

ȳ =

ȳ = 11

Now, put these values into the formula for covariance to see if there is any linear relationship between economic growth and returns from S&P 500.

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