The correlation coefficient (ρ) is a measure that determines the degree to which two variables' movements are associated. The most common correlation coefficient, generated by the Pearson product-moment correlation, may be used to measure the linear relationship between two variables. However, in a non-linear relationship, this correlation coefficient may not always be a suitable measure of dependence.

### Key Takeaways

• Correlation coefficients are used to measure the strength of the relationship between two variables.
• Positive correlation is a relationship between two variables in which both variables move in tandem—that is, in the same direction.
• Negative correlation or inverse correlation is a relationship between two variables whereby they move in opposite directions.
• Negative correlation is a key concept in portfolio construction, as it enables the creation of diversified portfolios that can better withstand portfolio volatility and smooth out returns.

## Understanding Correlation

The range of values for the correlation coefficient is -1.0 to 1.0. In other words, the values cannot exceed 1.0 or be less than -1.0 whereby a correlation of -1.0 indicates a perfect negative correlation, and a correlation of 1.0 indicates a perfect positive correlation. Anytime the correlation coefficient, denoted as r, is greater than zero, it's a positive relationship. Conversely, anytime the value is less than zero, it's a negative relationship. A value of zero indicates that there is no relationship between the two variables.

Correlation among variables does not (necessarily) imply causation.

In the financial markets, correlation coefficient is used to measure the correlation between two securities. When two stocks, for example, move in the same direction, the correlation coefficient is positive. Conversely, when two stocks move in opposite directions, the correlation coefficient is negative.

• If the correlation coefficient of two variables is zero, it signifies that there is no linear relationship between the variables. However, this is only for a linear relationship; it is possible that the variables have a strong curvilinear relationship.
• When the value of ρ is close to zero, generally between -0.1 and +0.1, the variables are said to have no linear relationship or a very weak linear relationship. For example, suppose the prices of coffee and of computers are observed and found to have a correlation of +.0008; this means that there is no correlation, or relationship, between the two variables.

## Calculating ρ

To calculate correlation, one must first determine the covariance of the two variables in question. Next, one must calculate each variable's standard deviation. The correlation coefficient is determined by dividing the covariance by the product of the two variables' standard deviations.

Standard deviation is a measure of the dispersion of data from its average. Covariance is a measure of how two variables change together, but its magnitude is unbounded, so it is difficult to interpret. By dividing covariance by the product of the two standard deviations, one can calculate the normalized version of the statistic. This is the correlation coefficient.

﻿$\text{Correlation}=\rho=\frac{\text{cov}(X,Y)}{\sigma_X\sigma_Y}$﻿

## Positive Correlation

A positive correlation, when the correlation coefficient is greater than 0, signifies that both variables move in the same direction or are correlated. When ρ is +1, it signifies that the two variables being compared have a perfect positive relationship; when one variable moves higher or lower, the other variable moves in the same direction with the same magnitude.

The closer the value of ρ is to +1, the stronger the linear relationship. For example, suppose the value of oil prices are directly related to the prices of airplane tickets, with a correlation coefficient of +0.8. The relationship between oil prices and airfares has a very strong positive correlation since the value is close to +1. So if the price of oil decreases, airfares follow in tandem. If the price of oil increases, so does the prices of airplane tickets.

In the chart below, we compare one of the largest U.S. banks JPMorgan Chase & Co. (JPM) with the Financial Select SPDR ETF (XLF). As you can imagine J.P. Morgan should have a positive correlation to the banking industry as a whole.

We can see the correlation coefficient (bottom of the chart) is currently at .7919, which is close to signaling a strong positive correlation. A reading above .50 typically signals a strong positive correlation.

Understanding the correlation between two stocks or a stock and its industry can help investors gauge how the stock is trading relative to its peers. All types of securities, including bonds, sectors and ETFs can be compared with the correlation coefficient.

## Negative Correlation

A negative (inverse) correlation occurs when the correlation coefficient is less than 0 and indicates that both variables move in the opposite direction. In short, any reading between 0 and -1 means that the two securities move in opposite directions. When ρ is -1, the relationship is said to be perfectly negative correlated; in short, if one variable increases, the other variable decreases with the same magnitude, and vice versa. However, the degree to which two securities are negatively correlated might vary over time and are almost never exactly correlated, all the time.

For example, suppose a study is conducted to assess the relationship between outside temperature and heating bills. The study concludes that there is a negative correlation between the prices of heating bills and the outdoor temperature. The correlation coefficient is calculated to be -0.96. This strong negative correlation signifies that as the temperature decreases outside, the prices of heating bills increase and vice versa.

When it comes to investing, negative correlation doesn't necessarily mean that the securities should be avoided. The correlation coefficient can help investors diversify their portfolio by including a mix of investments that have a negative or low correlation to the stock market. In short, when reducing volatility risk in a portfolio, sometimes opposites do attract.

As an example, assume you have a \$100,000 balanced portfolio that is invested 60% in stocks and 40% in bonds. In a year of strong economic performance, the stock component of your portfolio might generate a return of 12%, while the bond component may return -2% because interest rates are on a rising trend. Thus, the overall return on your portfolio would be 6.4% ((12% x 0.6) + (-2% x 0.4). The following year, as the economy slows markedly and interest rates are lowered, your stock portfolio might generate -5% while your bond portfolio may return 8%, giving you an overall portfolio return of 0.2%.

What if, instead of a balanced portfolio, your portfolio was 100% equities? Using the same return assumptions, your all-equity portfolio would have a return of 12% in the first year and -5% in the second year, which are more volatile than the balanced portfolio's returns of 6.4% and 0.2%.

## The Bottom Line

The correlation coefficient can be helpful in determining the relationship between your investment and the overall market or other securities.

This type of statistic is useful in many ways in finance. For example, it can be helpful in determining how well a mutual fund is behaving compared to its benchmark index, or it can be used to determine how a mutual behaves in relation to another fund or asset class. By adding a low or negatively correlated mutual fund to an existing portfolio, diversification benefits are gained.