# Correlation Coefficient

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 σ_{x} and σ_{y} 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.

Written by Obaidullah Jan