# Inverse Correlation

## What does 'Inverse Correlation' mean

An inverse correlation, also known as negative correlation, is a contrary relationship between two variables such that they move in opposite directions. For example, with variables A and B, as A increases, B decreases and as A decreases, B increases. In statistical terminology, an inverse correlation is denoted by the correlation coefficient "r" having a value between -1 and 0, with r = -1 indicating perfect inverse correlation.

## BREAKING DOWN 'Inverse Correlation'

In financial markets, the best example of an inverse correlation is probably the one between the U.S. dollar and gold. As the U.S. dollar depreciates against major currencies, gold is generally perceived to rise, and as the U.S. dollar appreciates, gold declines in price. Two points need to be kept in mind with regard to negative correlation. First, the existence of negative correlation, or positive correlation for that matter, does not necessarily imply a causal relationship. Second, the relationship between two variables is not static and fluctuates over time, which means the variables may display an inverse correlation during some periods and a positive correlation during others.

## Inverse Correlation Calculation Example

Calculating correlation is important because the risk reduction benefits of portfolio diversification rely on this statistic. The below example shows how to calculate the statistic.

Assume an analyst needs to calculate the correlation for the following two data sets:

X: 55, 37, 100, 40, 23, 66, 88

Y: 91, 60, 70, 83, 75, 76, 30

There are three steps involved in finding the correlation. First, add up all the X values to find SUM(X), add up all the Y values to find SUM(Y) and multiply each X value with its corresponding Y value and sum them to find SUM(X,Y):

SUM(X) = (55 + 37 + 100 + 40 + 23 + 66 + 88) = 409

SUM(Y) = (91 + 60 + 70 + 83 + 75 + 76 + 30) = 485

SUM(X,Y) = (55 x 91) + (37 x 60) + (100 x 70) + ... (88 x 30) = 26,926

The next step is to take each X value, square it and sum up all these values to find SUM(x^2). The same must be done for the Y values:

SUM(X^2) = (55^2) + (37^2) + (100^2) + ... (88^2) = 28,623

SUM(Y^2) = (91^2) + (60^2) + (70^2) + ... (30^2) = 35,971

Noting there are seven observations, n, the following formula can be used to find the correlation coefficient, r:

r = (n x (SUM(X,Y) - (SUM(X) x (SUM(Y))) / Square Root((n x SUM(X^2) - SUM(X)^2) x (n x SUM(Y^2) - SUM(Y)^2))

In this example, the correlation is:

r = (7 x 26,926 - (409 x 485) / Square Root((7 x 28,623 - 409^2) x (7 x 35,971 - 485^2)) = 9,883 / 23,414 = -0.42

The two data sets have an inverse correlation of -0.42.