# Covariance

Covariance measures the extent to which two variables, say x and y, move together. A positive covariance means that the variables move in tandem and a negative value indicates that the variables have an inverse relationship. While covariance can indicate the direction of relation, the correlation coefficient is a better measure of the strength of relationship.

Covariance is an important input in estimation of diversification benefits and portfolio optimization, calculation of beta coefficient, etc.

In manual calculation of covariance, the following steps are involved:

• Calculate mean of each variable i.e. µx and µx,
• Find deviation of each value of x and y from their respective means i.e. (xi - µx) and (yi - µy)
• Multiply deviation of x corresponding deviation of y i.e. (xi - µx) × (yi - µy)
• Sum up all the products of deviations
• Divide by total number of observations N.

## Formula

The following equation describes the relationship:

$$\text{Cov}(\text{x} \text{,} \text{y})=\sum _ {\text{i}}^{\text{N}}\frac{(\text{x}\ -\ \mu _ \text{x})(\text{y}\ -\ \mu _ \text{y})}{\text{N}}$$

Covariance can also be calculated using Excel COVAR, COVARIANCE.P and COVARIANCE.S functions.

If we know the correlation coefficient, we can work out covariance indirectly as follows:

$$\text{Cov}(\text{x} \text{,} \text{y})=\rho\times\sigma _ \text{x}\times\sigma _ \text{y}$$

Where ρ is the correlation coefficient, \sigmax is the standard deviation of x and \sigmay is the standard deviation of y.

## Example

Let’s calculate covariance using the same data set as in correlation coefficient.

The data set bellows shows monthly closing prices of SPDR Oil & Gas Exploration and Production ETF (designed as y) and Brent Crude (designed as x):

Date x y
1/1/2014 109.95 65.75
2/1/2014 108.16 69.69
3/1/2014 108.98 71.83
4/1/2014 105.7 77.61
5/1/2014 108.63 77.04
6/1/2014 109.21 82.28
7/1/2014 110.84 75.29
8/1/2014 103.45 79.05
9/1/2014 101.12 68.83
10/1/2014 94.57 60.87
11/1/2014 84.17 51.08
12/1/2014 70.87 47.86
1/1/2015 55.27 46.18
2/1/2015 47.52 50.81
3/1/2015 61.89 51.66
4/1/2015 55.73 55.09
5/1/2015 64.13 49.53
6/1/2015 64.88 46.66
7/1/2015 62.01 38.35
8/1/2015 52.21 36.00
9/1/2015 49.56 32.84
10/1/2015 47.69 36.61
11/1/2015 49.56 37.13
12/1/2015 44.44 30.22
1/1/2016 37.28 28.49
2/1/2016 34.24 24.60
3/1/2016 36.81 30.35
4/1/2016 38.67 35.74
5/1/2016 48.13 35.52
6/1/2016 49.72 34.81
7/1/2016 50.35 34.25
8/1/2016 42.14 36.79
9/1/2016 45.45 38.46
10/1/2016 49.06 35.35
11/1/2016 48.14 41.93
12/1/2016 53.94 43.18
1/1/2017 56.82 40.08
2/1/2017 56.8 37.86
3/1/2017 56.36 37.44
4/1/2017 52.83 34.95
5/1/2017 51.52 32.57
6/1/2017 50.63 31.92
7/1/2017 47.92 32.52
8/1/2017 51.78 30.16
9/1/2017 52.75 34.09
10/1/2017 57.54 34.28
11/1/2017 60.49 35.72
12/1/2017 63.73 37.18

$$\text{Mean} \text{of} \text{x} = \ \mu _ \text{x} = \text{63.83}$$

$$\text{Mean} \text{of} \text{y} = \ \mu _ \text{y} = \text{45.34}$$

$$\text{Sum of products of deviations} = (\text{x}\ -\ \mu _ \text{x})(\text{y}\ -\ \mu _ \text{y}) = \text{16,467}$$

$$\text{Cov}(\text{x} \text{,} \text{y})=\sum _ {\text{i}}^{\text{N}}{\frac{\text{16,467}}{\text{48}}=\text{343.06}}$$

Covariance of SPDR XOP ETF with Brent Crude is positive which indicates that they both move together. However, we can’t say conclusively how strong the relationship because the covariance value depends on the units used. Correlation coefficient is a better measure which works out to 0.93 for the given data set. We can arrive at covariance value if we have the value for correlation coefficient and individual standard deviations of x and y, which are 23.63 and 15.87 respectively.

$$\text{Cov}(\text{x} \text{,} \text{y})=\text{0.93}\times\text{23.63}\times\text{15.87}=\text{343}$$