What Is Inverse Correlation? How It Works and Example Calculation (2024)

What Is an Inverse Correlation?

An inverse correlation, also known as negative correlation, is a contrary relationship between two variables such that when the value of one variable is high then the value of the other variable is probably low.

For example, with variables A and B, as A has a high value, B has a low value, and as A has a low value, B has a high value. In statistical terminology, an inverse correlation is often denoted by the correlation coefficient "r" having a value between -1 and 0, with r = -1 indicating perfect inverse correlation.

Graphing Inverse Correlation

Two sets of data points can be plotted on a graph on an x and y-axis to check for correlation. This is called a scatter diagram, and it represents a visual way to check for a positive or negative correlation. The graph below illustrates a strong inverse correlation between two sets of data points plotted on the graph.

Example of Calculating Inverse Correlation

Correlation can be calculated between variables within a set of data to arrive at a numerical result, the most common of which is known as Pearson's r. When r is less than 0, this indicates an inverse correlation. Here is an arithmetic example calculation of Pearson's r, with a result that shows an inverse correlation between two variables.

Assume an analyst needs to calculate the degree of correlation between the X and Y in the following data set with seven observations on the two variables:

• 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):

$\begin{aligned} \text{SUM}(X) &= 55 + 37 + 100 + 40 + 23 + 66 + 88 \\ &= 409 \\ \end{aligned}$SUM(X)​=55+37+100+40+23+66+88=409​

$\begin{aligned} \text{SUM}(Y) &= 91 + 60 + 70 + 83 + 75 + 76 + 30 \\ &= 485 \\ \end{aligned}$SUM(Y)​=91+60+70+83+75+76+30=485​

$\begin{aligned} \\\text{SUM}(X,Y) &= (55 \times 91) + (37 \times 60) + \dotso + (88 \times 30) \\&= 26,926 \\\end{aligned}$SUM(X,Y)​=(55×91)+(37×60)+…+(88×30)=26,926​

The next step is to take each X value, square it and sum up all these values to find SUM(x²). The same must be done for the Y values:

$\text{SUM}(X^2) = (55^2) + (37^2) + (100^2) + \dotso + (88^2) = 28,623$SUM(X2)=(552)+(372)+(1002)+…+(882)=28,623

$\text{SUM}(Y^2) = (91^2) + (60^2) + (70^2) + \dotso + (30^2) = 35,971$SUM(Y2)=(912)+(602)+(702)+…+(302)=35,971

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

$r = \frac{[n \times (\text{SUM}(X,Y) - (\text{SUM}(X) \times ( \text{SUM}(Y) ) ]} {\sqrt{[(n \times \text{SUM}(X^2) - \text{SUM}(X)^2 ] \times [n \times \text{SUM}(Y^2) - \text{SUM}(Y)^2)]}}$r=[(n×SUM(X2)−SUM(X)2]×[n×SUM(Y2)−SUM(Y)2)]​[n×(SUM(X,Y)−(SUM(X)×(SUM(Y))]​

In this example, the correlation is:

• $r = \frac{(7 \times 26,926 - (409 \times 485))} {\sqrt{((7 \times 28,623 - 409^2) \times (7 \times 35,971 - 485^2))}}$r=((7×28,623−4092)×(7×35,971−4852))​(7×26,926−(409×485))​
• $r = 9,883 \div 23,414$r=9,883÷23,414
• $r = -0.42$r=−0.42

The two data sets have a correlation of -0.42, which is called an inverse correlation because it is a negative number.

What Does Inverse Correlation Tell You?

Inverse correlation tells you that when one variable is high, the other tends to be low. Correlation analysis can reveal useful information about the relationship between two variables, such as how the stock and bond markets often move in opposite directions.

The correlation coefficient is often used in a predictive manner to estimate metrics like the risk reduction benefits of portfolio diversification and other important data. If the returns on two different assets are negatively correlated, then they can balance each other out if included in the same portfolio.

In financial markets, a well-known 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, the dollar price of gold is generally observed to rise, and as the U.S. dollar appreciates, gold declines in price.

Limitations of Using Inverse Correlation

Two points need to be kept in mind with regard to a negative correlation. First, the existence of a negative correlation, or positive correlation for that matter, does not necessarily imply a causal relationship. Even though two variables have a very strong inverse correlation, this result by itself does not demonstrate a cause-and-effect relationship between the two.

Second, when dealing with time series data, such as most financial data, the relationship between two variables is not static and can change over time. This means the variables may display an inverse correlation during some periods and a positive correlation during others. Because of this, using the results of correlation analysis to extrapolate the same conclusion to future data carries a high degree of risk.