What is R-Squared?

R-squared is like trying to understand how a class full of teenagers is going to act based on what the teacher is doing…

Some students are going to listen to the teacher and do what they say (a high r-squared), while some are going to do what they want (a low r-squared). It’s the same with trying to understand the movements of stocks based on what an index is doing.

In statistics, r-squared looks at the scattering of data points around a regression line. It’s usually visualized in a scatter plot — a diagonal line with dots scattered around it. The tighter the grouping of the dots to the regression line, the higher the r-squared value. The more spread out the dots are, the lower the r-squared.

In finance, r-squared tells you how much of your investment’s movement can be explained by another variable — like the S&P 500 index. R-squared is typically shown as a percentage between 0 – 100%.

There are broad ranges for interpreting r-squared:

• Low r-squared: An r-squared between 1-40% generally means there is a weak correlation between your investment and the index’s returns. In other words, when your investment goes up or down, very little of the movement is due to the change in the index.
• Medium r-squared: An r-squared between 40-70% generally means there is an average correlation. It means changes in the index can explain some of the movement, but not the majority.
• High r-squared: An r-squared between 70-100% generally means there is a high correlation and that the benchmark index can explain the majority of the returns on your investment.

Adjusted r-squared helps you understand how much of your investment’s movement can be explained by several other variables — like the S&P 500 and Nasdaq indexes. Adjusted r-squared is typically shown as a percentage between 0 – 100%.

A high adjusted r-squared means that the changes in the other variables can explain most of the variance of your investment. A low adjusted r-squared tells you that very little of those changes are due to the movement in the other variables.

R-squared and adjusted r-squared are similar to one another with one key difference. Both models tell you how much of your investment can be explained by another variable(s), and they are both expressed as a percentage between 0 – 100%. The critical difference is that adjusted r-squared can use several variables within its model without distorting the r-squared value.

For example, if you use multiple variables within a standard r-squared model, it may give you a higher r-squared value simply because you included more variables. Adjusted r-squared takes this into account and only increases if the new variable fits the model beyond what normal probability would suggest.

R-squared tells you how much of the movement of your investment can be explained by the change of another variable, such as a benchmark index. It essentially tells you how closely the fluctuations of your investment and the other variable correlate to one another. It’s a great tool to help you allocate your assets based on how closely you want them to correlate to a benchmark index.

Beta is a different type of tool. It indicates how volatile (risky) your investment is compared to the overall stock market. The entire market has a beta of 1. A beta that is < 1 (less than 1) suggests the stock will fluctuate less dramatically than the rest of the market. A beta > 1 (greater than 1) means the stock will be more volatile than the market as a whole. While a lower beta implies a lower level of risk, it can also point to a lower chance of reward.

R, often referred to as the correlation coefficient, tells you how strong a relationship is between two different variables. It is usually measured between -1 to +1. When r = 1, it means that an increase in one variable will lead to a proportional increase in the second variable. For example, when your foot grows, your shoe size also increases a similar amount.

When r = 0, it means the two variables are not correlated and share no relationship. When r = -1, then the two variables are inversely related. So when one of them increases, the other will decrease proportionally. For example, when you step on the gas pedal, your car’s speed increases, but the amount of gas in the tank decreases.

R-squared tells you how well the variance of one variable explains the variance of another variable — Variance is how dispersed random numbers are from their average value. In investing, r-squared helps you understand how much the movement of your investment can be explained by the change of another variable — such as a benchmark index like the S&P 500. It is usually expressed as a percentage between 0 – 100%. A high r^2 (r-squared) means that most of the changes can be explained by the benchmark index. A low r^2 means that very little of the movement is due to changes in the index.

The simplified formula for r-squared is:

If you enjoy Greek, then you could also calculate R (correlation coefficient) and square the result.

For our purposes, let’s use the simplified formula for r-squared.

In the explanation below, you can see why the formula containing the sum of errors makes sense even if you don’t immediately understand the terminology.

Calculating r-squared is a multi-step process. First, you take all the data points of both the dependent (your investment) and independent variables (benchmark index) and find the line of best fit (a straight line that best represents the data points on a scatter plot) using a regression model.

Next, you work on the explained variation. You calculate the predicted values, subtract the actual values, and then square the results. This gives you a list of errors squared. You then add these together to get the explained variance or the sum of first errors.

After this, you start working on total variance. You subtract the average actual value from the predicted values. Then you square the results to get a list of errors squared. Then you add them together to get the total variance (second sum of errors).

Next, you divide the explained variance (first sum of errors) by the total variance (second sum of errors. You take this number, subtract it from 1, and this gives you r-squared.

One way to think about r-squared is that it is simply the correlation squared. If you look at it this way, then there are two different ways to calculate r-squared in Excel.

In this example, we are looking at the weekly returns of Apple (AAPL) and the S&P 500 between September 29 and December 9, 2019. Check out this PDF to see R Squared in action.

R-squared is a useful tool if you want to understand how movements in a single variable (like the S&P 500) affect the changes in your investment. However, introducing multiple variables may give you a higher r-squared value simply because there are more data points.

It also doesn’t tell you whether the most appropriate set of variables was chosen. Perhaps the Nasdaq would be better to use than the S&P 500 to explain the movements in your investment.

To compensate for multiple variables, you can use adjusted r-squared. It is similar to r-squared, but adjusted r-squared can use numerous variables within its model without disrupting the r-squared value. Adjusted r-squared only increases if the additional variable fits the model beyond what probability or chance would suggest. Like r-squared, you get a percentage between 0 – 100%.

First, let’s take a look at correlation to understand the relationship with r^2 (r-squared).

Correlation tells you how strong the relationship between the two variables is. It is measured between -1 to + 1 with one meaning that the variables are perfectly correlated — They move in the same direction and the same proportion. A correlation of -1 means they are perfectly un-correlated — They move in opposite directions. You typically express correlation as “R” (correlation coefficient).

R-squared is simply the correlation (or R) squared. R-squared tells you how much of the variance in your investment can be explained by another variable — such as the S&P 500. There is no right or wrong r-squared value for correlation. It depends on what you are looking for. If you want to make an investment that moves in close step with a benchmark index, then you will look for a higher correlation and higher r-squared. If you are after more diversification, then you may want an investment with a low correlation and low r-squared.

Linear regression is a way to model the relationship between a dependent and an independent variable. You usually see the linear model visualized in a scatter plot — a diagonal line with dots scattered around it.

In statistics, r-squared looks at the dispersion (scattering) of different data points around a regression line. The more tightly the dots are grouped around the line, the higher the r-squared value is. The more scattered the dots are, the lower the r-squared.

You use linear regression to help model r-squared. What is a good value? It depends on what you are looking to achieve. If you want your investment to track your independent variable (benchmark index) tightly, then you will want a high r-squared and the dots to be tightly grouped around the regression line.

If you want your investment to move more independently from the benchmark, you will aim for a lower r-squared with the dots more scattered on the regression line.