What Is Correlation?

Correlation, in the finance and investment industries, is a statistic that measures the degree to which two securities move in relation to each other. Correlations are used in advanced portfolio management, computed as the correlation coefficient, which has a value that must fall between -1.0 and +1.0.

KEY TAKEAWAYS

• Correlation is a statistic that measures the degree to which two variables move in relation to each other.
• In finance, the correlation can measure the movement of a stock with that of a benchmark index, such as the S&P 500.
• Correlation is closely tied to diversification, the concept that certain types of risk can be mitigated by investing in assets that are not correlated.
• Correlation measures association, but doesn't show if x causes y or vice versa—or if the association is caused by a third factor.
• Correlation may be easiest to identify using a scatterplot, especially if the variables have a non-linear yet still strong correlation.

Correlation

What Correlation Can Tell You

Correlation shows the strength of a relationship between two variables and is expressed numerically by the correlation coefficient. The correlation coefficient's values range between -1.0 and 1.0.

A perfect positive correlation means that the correlation coefficient is exactly 1. This implies that as one security moves, either up or down, the other security moves in lockstep, in the same direction. A perfect negative correlation means that two assets move in opposite directions, while a zero correlation implies no linear relationship at all.

For example, large-cap mutual funds generally have a high positive correlation to the Standard and Poor's (S&P) 500 Index or nearly one. Small-cap stocks tend to have a positive correlation to the S&P, but it's not as high or approximately 0.8.

However, put option prices and their underlying stock prices will tend to have a negative correlation. A put option gives the owner the right but not the obligation to sell a specific amount of an underlying security at a pre-determined price within a specified time frame.

Put option contracts become more profitable when the underlying stock price decreases. In other words, as the stock price increases, the put option prices go down, which is a direct and high-magnitude negative correlation.

How to Calculate Correlation

There are several methods of calculating correlation. The most common method, the Pearson product-moment correlation, is discussed further in this article. The Pearson product-moment correlation measures the linear relationship between two variables. It can be used for any data set that has a finite covariance matrix. Here are the steps to calculate correlation.

1. Gather data for your "x-variable" and "y variable.
2. Find the mean for the x-variable and find the mean for the y-variable.
3. Subtract the mean of the x-variable from each value of the x-variable. Repeat this step for the y-variable.
4. Multiply each difference between the x-variable mean and x-variable value by the corresponding difference related to the y-variable.
5. Square each of these differences and add the results.
6. Determine the square root of the value obtained in Step 5.
7. Divide the value in Step 4 by the value obtained in Step 6.

To avoid the complex manual calculation, consider using the CORREL function in Excel.

Formula for Correlation

Using the Pearson product-moment correlation method, the following formula can be used to find the correlation coefficient, r:

\begin{aligned}&r = \frac { n \times ( \sum (X, Y) - ( \sum (X) \times \sum (Y) ) ) }{ \sqrt { ( n \times \sum (X ^ 2) - \sum (X) ^ 2 ) \times ( n \times \sum( Y ^ 2 ) - \sum (Y) ^ 2 ) } } \\&\textbf{where:}\\&r=\text{Correlation coefficient}\\&n=\text{Number of observations}\end{aligned}​r=(n×∑(X2)−∑(X)2)×(n×∑(Y2)−∑(Y)2)​n×(∑(X,Y)−(∑(X)×∑(Y)))​where:r=Correlation coefficientn=Number of observations​

Example of Correlation

Investment managers, traders, and analysts find it very important to calculate correlation because the risk reduction benefits of diversification rely on this statistic. Financial spreadsheets and software can calculate the value of correlation quickly.

As a hypothetical example, assume that an analyst needs to calculate the correlation for the following two data sets:

X: (41, 19, 23, 40, 55, 57, 33)

Y: (94, 60, 74, 71, 82, 76, 61)

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

SUM(X) = (41 + 19 + 23 + 40 + 55 + 57 + 33) = 268

SUM(Y) = (94 + 60 + 74 + 71 + 82 + 76 + 61) = 518

SUM(X,Y) = (41 x 94) + (19 x 60) + (23 x 74) + ... (33 x 61) = 20,391

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) = (41^2) + (19^2) + (23^2) + ... (33^2) = 11,534

SUM(Y^2) = (94^2) + (60^2) + (74^2) + ... (61^2) = 39,174

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

\begin{aligned}&r = \frac { n \times ( \sum (X, Y) - ( \sum (X) \times \sum (Y) ) ) }{ \sqrt { ( n \times \sum (X ^ 2) - \sum (X) ^ 2 ) \times ( n \times \sum( Y ^ 2 ) - \sum (Y) ^ 2 ) } } \\&\textbf{where:}\\&r=\text{Correlation coefficient}\\&n=\text{Number of observations}\end{aligned}​r=(n×∑(X2)−∑(X)2)×(n×∑(Y2)−∑(Y)2)​n×(∑(X,Y)−(∑(X)×∑(Y)))​where:r=Correlation coefficientn=Number of observations​

In this example, the correlation would be:

r = (7 x 20,391 - (268 x 518) / SquareRoot((7 x 11,534 - 268^2) x (7 x 39,174 - 518^2)) = 3,913 / 7,248.4 = 0.54

Correlation and Portfolio Diversification

In investing, correlation is most important in relation to a diversified portfolio. Investors who wish to mitigate risk can do so by investing in non-correlated assets. For example, consider an investor who owns airline stock. If the airline industry is found to have a low correlation to the social media industry, the investor may choose to invest in a social media stock understanding that an negative impact to one industry may not impact the other.

This is often the approach when considering investing across asset classes. Stocks, bonds, precious metals, real estate, cryptocurrency, commodities, and other types of investments each have different relationships to each other. While some may be heavily correlated, others may act as a hedge to diversify risk if they are not correlated.