R-Squared vs. Adjusted R-Squared: What’s the Difference?

R-Squared vs. Adjusted R-Squared: An Overview

R-squared and adjusted R-squared enable investors to measure the performance of a mutual fund against that of a benchmark. Investors may also use them to calculate the performance of their portfolio against a given benchmark.

In the world of investing, R-squared is expressed as a percentage between 0 and 100, with 100 signaling perfect correlation and zero no correlation at all. The figure does not indicate how well a particular group of securities is performing. It only measures how closely the returns align with those of the measured benchmark. It is also backwards-looking—it is not a predictor of future results.

Adjusted R-squared can provide a more precise view of that correlation by also taking into account how many independent variables are added to a particular model against which the stock index is measured. This is done because such additions of independent variables usually increase the reliability of that model. For investors, that means the correlation with the index.

R-Squared

R-squared (R²) is a statistical measure that represents the proportion of the variance for a dependent variable that’s explained by an independent variable or variables in a regression model. R-squared explains to what extent the variance of one variable explains the variance of the second variable. So, if the R² of a model is 0.50, then approximately half of the observed variation can be explained by the model’s inputs.

An R-squared result of 70 to 100 indicates that a given portfolio closely tracks the stock index in question, while a score between 0 and 40 indicates a very low correlation with the index. Higher R-squared values also indicate the reliability of beta readings. Beta measures the volatility of a security or a portfolio.

While R-squared can return a figure that indicates a level of correlation with an index, it has certain limitations when it comes to measuring the impact of independent variables on the correlation. This is where adjusted R-squared is useful in measuring correlation.

Adjusted R-Squared

Adjusted R-squared is a modified version of R-squared that has been adjusted for the number of predictors in the model. The adjusted R-squared increases when the new term improves the model more than would be expected by chance. It decreases when a predictor improves the model by less than expected. The adjusted R-squared is positive, not negative. It is always lower than the R-squared.

Adding more independent variables or predictors to a regression model tends to increase the R-squared value, which tempts makers of the model to add even more variables. This is called overfitting and can return an unwarranted high R-squared value. Adjusted R-squared is used to determine how reliable the model is and how much of its correlation is improved with additional independent variables.

In a portfolio model that has more independent variables, adjusted R-squared will help determine how much of the correlation with the index is due to the addition of those variables. The adjusted R-squared compensates for the addition of variables and only increases if the new predictor enhances the model above what would be obtained by chance. Conversely, it will decrease when a predictor improves the model less than what is predicted by chance.

Key Differences

The most obvious difference between adjusted R-squared and R-squared is simply that adjusted R-squared considers and tests different independent variables against the stock index and R-squared does not. Because of this, many investment professionals prefer using adjusted R-squared because it has the potential to be more accurate. Furthermore, investors can gain additional information about what is affecting a stock by testing various independent variables using the adjusted R-squared model.

R-squared, on the other hand, does have its limitations. One of the most essential limits to using this model is that R-squared cannot be used to determine whether or not the coefficient estimates and predictions are biased. Furthermore,  in multiple linear regression, the R-squared cannot tell us which regression variable is more important than the other.

Adjusted R-Squared vs. Predicted R-Squared

The predicted R-squared, unlike the adjusted R-squared, is used to indicate how well a regression model predicts responses for new observations. So where the adjusted R-squared can provide an accurate model that fits the current data, the predicted R-squared determines how likely it is that this model will be accurate for future data.

Example of R-Squared vs. Adjusted R-Squared

Imagine a hedge fund that wants to predict the performance of different stock prices. The fund starts by constructing a model based on known values like the stock price, revenue, earnings per share, and so on. Then, the fund tests the model against actual data and calculates the R-squared to determine how well it explains the variation of stock prices. They might also tweak the variables to try to improve the R-squared.

The problem with this approach is that it encourages the researchers to overfit the model by including irrelevant variables. Every time an additional term is introduced, the R-squared of the model either increases or stays the same.

To counteract this tendency, the fund can also calculate adjusted R-squared, a measure that decreases if new variables are added that do not improve the model. Using adjusted R-squared effectively penalizes the researchers for adding irrelevant variables, allowing them to zero in on the ones with the most predictive power.

Special Considerations

R-Squared and Goodness-of-Fit

The basic idea of regression analysis is that if the deviations between the observed values and the predicted values of the linear model are small, the model has well-fit data. Goodness-of-fit is a mathematical model that helps to explain and account for the difference between this observed data and the predicted data. In other words, goodness-of-fit is a statistical hypothesis test to see how well sample data fit a distribution from a population with a normal distribution.

Low R-Squared vs. High R-Squared Value

One misconception about regression analysis is that a low R-squared value is always a bad thing. This is not so. For example, some data sets or fields of study have an inherently greater amount of unexplained variation. In this case, R-squared values are naturally going to be lower. Investigators can make useful conclusions about the data even with a low R-squared value.

In a different case, such as in investing, a high R-squared value—typically between 85% and 100%—indicates the stock or fund’s performance moves relatively in line with the index. This is very useful information to investors, thus a higher R-squared value is necessary for a successful project.

Explain Like I’m Five

Statisticians and economists often construct mathematical models to explain the relationships between different variables. R-squared measures how well a model explains the variation in actual observed data.

The problem with R-squared is that it doesn’t tell you if a variable fails to improve the model. Some researchers might be tempted to add extra variables, even if they don’t add much explanatory power. The adjusted R-squared addresses this problem by penalizing variables that do not help explain the data.

The Bottom Line

R-squared and adjusted R-squared enable investors to measure the performance of a mutual fund against that of a benchmark. Many investors have found success using adjusted R-squared over R-squared because of its ability to make a more accurate view of the correlation between one variable and another.