Correlation Coefficient

by Obaidullah Jan, ACA, CFA

Correlation coefficient measures the degree to which two variables move together. Its value ranges between -1 and 1. -1 indicates perfectly negative relationship, 1 shows a perfectly positive relationship and zero means there is no linear relationship between the variables. Correlation doesn’t necessarily mean causation.

Correlation coefficient is a very important number in finance because it helps tell whether there is a relationship between say population growth and GDP growth, crude oil price and stock price of oil and gas companies, a mutual fund and the broad market index, etc. However, correlation coefficient must be used with a caveat: it doesn’t infer causation. Two variables might have a very high correlation, but it might not necessarily mean that one causes the other.

Formula

The most common measure of correlation is called the Pearson correlation which can be calculated using the following formula:

$$ r=\frac{n \sum{xy} − \sum{x} \sum{y}}{\sqrt{\left[ n \sum{x^2} − \left( \sum{x} \right)^2 \right] \times \left[ n \sum{y^2} − \left( \sum{y} \right)^2 \right]}} $$

If you already know the covariance between two investments, you can find correlation coefficient using the following formula:

$$ Correlation\ Coefficient\ (r)=\frac{Covariance(x,y)}{\sigma_x\sigma_y} $$

Where σx and σy represent the standard deviation of variable x and y respectively.

Correlation coefficient can also be calculated using Excel CORREL function. It’s syntax is CORREL(array1, array2). array1 and array2 represent the series of x and y values.

Example

The following table shows the monthly close data for SPDR S&P Oil & Gas Explore & Prod. (ETF) (XOP) (designated as y variable) and Brent Crude Oil Price (designated as x variable) from 1 January 2014 to 31 December 2017:

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

After working out the necessary intermediate numbers, our Pearson correlation equation looks like this:

$$ Correlation\ Coefficient\ (r)\\=\frac{48 \times 155,384 − 3,064 \times 2,177}{\sqrt{\left[ 48 \times 221,774 − 3,065^2 \right] \times \left[ 48 \times 110,526 − 2,177^2 \right]}}\\=0.93 $$

We get the same result using Excel CORREL function as illustrated in the attached Excel Worksheet.