What is the Correlation Coefficient?

The correlation coefficient is like watching a flock of birds…

Some will stay in tight formation (be highly correlated); some will stray further afield (be only weakly correlated). Some may fly in the complete opposite direction (be negatively correlated).

Correlation describes the relationship between two different variables. That relationship could be causal or not. A causal relationship would be one where X happens because of Y. You most commonly use correlation to show the degree of a linear relationship through the use of a scatter plot (a line plotting the relationship with a bunch of dots scattered around it).

For investors, correlation is useful to show a relationship between variables and to try to understand movements in assets or compare a security to a benchmark index. For example, If the price of gold increases, then the stock of a company that owns gold mines may also increase. It makes sense because gold mines sell the ore they are mining, and if the price of gold increases, then their profits will also potentially increase.

The correlation coefficient measures the degree to which two variables are related. It is a way to understand the strength of the relationship (correlation). For an investor, two variables may show correlation, but it’s useful to know how strong the link is, and if it is positive correlation or negative correlation.

You calculate the correlation coefficient as a range between -1.0 and 1.0. If the two variables have a perfect negative correlation (-1), then they move in exactly opposite directions at the same rate. If they have a perfect positive correlation (1), then they travel in the same direction, at the same magnitude. A value of zero (0) means the two variables are not correlated, and you can’t determine a relationship.

As an investor, you can use the correlation coefficient to help you diversify your portfolio by including assets that have a negative correlation. This can reduce the volatility, or the impact of large price swings, compared to only having a single type of asset or industry.

In other words, it’s useful to understand that the price of gold and gold mines are correlated. However, it’s even more helpful to know the degree to which they are connected and whether that correlation is negative (they move in opposite directions) or positive (they move in the same direction).

The coefficient of variation shows the dispersion of data points (how spread out they are) compared to the mean (average) of the data set. You typically see it visualized as a scatter plot (a line with dots all around it). If the returns are far away from the average return, there is high variation or volatility. If the returns are tightly grouped, then there is low variation, and the performances are relatively close to the average. For investors, this is a useful measure that can help you understand how much risk you are assuming compared to the reward you are expecting.

The coefficient of variation is essentially a risk management tool that can also help you create a portfolio that suits your appetite for volatility (or risk). If you have a lower risk tolerance, you can choose investments with a lesser degree of volatility or risk compared to the return you are aiming for. If your appetite for risk is higher, then you can choose investments with slightly higher volatility and aim for higher returns.

The correlation coefficient helps an investor measure the strength of the relationship between two different variables — such as gold prices and mining stocks. You could use it to help understand a trend in some of your investments. For example, as the price of gold increases, then the price of gold mining stocks will most likely increase as well. Since gold mines sell the gold they unearth, their stock prices tend to be highly correlated with the raw material they are mining. Oil companies act similarly concerning oil prices.

It is also an excellent tool for portfolio diversification to help you choose different assets or funds that are unrelated or negatively correlated. One example is when funds use a mix of stocks and bonds. In many cases, these two different assets show a low correlation and help decrease the overall volatility of the portfolio.

There are several different types of correlation coefficients. The most commonly used one is the Pearson correlation coefficient — also known as the Pearson product-moment correlation coefficient.

The Pearson correlation coefficient measures the strength of the linear correlation (relationship) between two different variables. The calculation yields a range of -1.0 to 1.0. A coefficient of -1 means the two variables have a negative relationship⁠ — They move in opposite directions. A measurement of 0 means they are not correlated at all. A coefficient of 1 means they have a positive correlation and travel in the same direction and at the same rate.

There are some drawbacks to using the Pearson correlation coefficient. It is not able to determine the difference between dependent and independent variables. For example, you could run a test to look for correlation between Alzheimer’s and a poor diet. You might find a high correlation of 0.85, which suggests a poor diet leads to the disease. However, you can switch the two variables around and get the same result, leading you to believe Alzheimer's leads to a high-calorie diet. So while it's useful in understanding the strength of a relationship, it can be misleading when looking for causal relationships.

To calculate the correlation coefficient, you first need to understand the covariance and standard deviation of both variables.

Covariance measures the directional relationship between the two variables. With a positive relationship, both variables move in the same direction. If it is negative, they move in opposite directions. In investing, you typically use covariance to help you diversify your portfolio by understanding the relationship between the returns of different assets.

Standard deviation measures the dispersion (how spread out) data points are from the mean (average) of the data set. With a high standard deviation, the points are further from the mean — more spread out. If the standard deviation is lower, then the data points are closer to the mean and less spread out. For investors, the standard deviation can help you understand market volatility and the risk associated with your portfolio.

To calculate the correlation coefficient, you first find the covariance and then divide it by the product (multiply them together) of the two variables.

The easiest way to calculate the correlation coefficient in Excel is to use the built-in formula for correlation. Check out this PDF to see the formula in action.

The correlation coefficient helps you understand the strength of the relationship between two different variables. Using it can help you understand how a stock is performing relative to its peers or the rest of the industry, as well as create more diversification within your portfolio. The range for the correlation coefficient is -1.0 to 1.0. It means the values returned will not be less than -1 (so not -2) and won’t be more than 1 (so not 2).

If the value is less than zero, then there is a negative relationship. It means the two variables will most likely move in opposite directions. If it is more than zero, there is a positive relationship. It means the variables likely move in tandem.

If the value is zero, then there is no correlation. Sometimes the numbers can be very close to zero, such as -0.1 or 0.1. In these cases, you typically interpret this as not having any correlation.

The strength of the relationship depends on the value of the correlation coefficient. The closer the value is to 1 or -1, the stronger the correlation. As an example, an amount of -0.19 means there is a negative correlation. However, it is probably too weak to form a conclusion ⁠— It is insignificant. Many times, you don’t consider a correlation to be significant until it goes above 0.8 (or -0.8 if it is negative). Anything that is 0.9 (or -0.9 if negative) or higher means the relationship is very strong.

If the two variables have a perfect positive correlation, a value of 1, it means that when one variable moves higher or lower, then the other one travels in the same direction and at the same rate.

If the two variables have a perfect negative correlation, a value of -1, then they tend to move in opposite directions, at the same magnitude. So if one variable increases, the other variable decreases at roughly the same rate.

It’s important to remember that the strength of the relationship will vary over time and may change. Two variables are rarely perfectly correlated or moving together all of the time. For example, the S&P 500 is a snapshot of the 500 largest companies in America. It has a negative correlation with the Vanguard Total Bond Market ETF (BND). But the strength of the relationship changes depending on the period you measure. As of December 23rd, 2019, over five days, the correlation coefficient was -0.9. But if you look at it over five years, the coefficient is -0.5.

Whether a correlation is negative or positive doesn’t necessarily imply that one security is better or worse than another. It merely tells you the relationship and the strength of that relationship. It can be a powerful tool to help an investor diversify by including a mix of securities that are positively and negatively correlated.