R-Squared: Definition, Calculation Formula, Uses, and Limitations

What Is R-Squared?

R-squared (R²) is defined as a number that tells you how well the independent variable(s) in a statistical model explain the variation in the dependent variable. It goes from 0 to 1, where 1 indicates a perfect fit of the model to the data.

Whereas correlation explains the strength of the relationship between an independent and a dependent variable, R-squared explains the extent to which 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.

Key Takeaways

R-squared is a statistical measure that indicates how much of the variation of a dependent variable is explained by an independent variable in a regression model.
In investing, R-squared is generally interpreted as the percentage of a fund’s or security’s price movements that can be explained by movements in a benchmark index.
An R-squared of 100% means that all movements of a security (or other dependent variable) are completely explained by movements in the index (or whatever independent variable you are interested in).

Formula for R-Squared

$\begin{aligned} &\text{R}^2 = 1 - \frac{ \text{Unexplained Variation} }{ \text{Total Variation} } \\ \end{aligned}$

The calculation of R-squared requires several steps. steps. This includes taking the data points (observations) of dependent and independent variables and conducting regression analysis to find the line of best fit, often from a regression model. This regression line helps to visualize the relationship between the variables. From there, you would calculate predicted values, subtract actual values, and square the results. These coefficient estimates and predictions are crucial for understanding the relationship between the variables. This yields a list of errors squared, which is then summed and equals the unexplained variance.

To calculate the total variance, you would subtract the average actual value from each of the actual values, square the results, and sum them. This process helps in determining the total sum of squares, which is an important component in calculating R-squared. From there, divide the first sum of errors (unexplained variance) by the second sum (total variance), subtract the result from one, and you have the R-squared.

What R-Squared Can Tell You

In investing, R-squared is generally interpreted as the percentage of a fund’s or security’s movements that can be explained by movements in a benchmark index. For example, an R-squared for a fixed-income security vs. a bond index identifies the security’s proportion of price movement that is predictable based on a price movement of the index.

The same can be applied to a stock vs. the S&P 500 Index or any other relevant index. It may also be known as the co-efficient of determination.

R-squared values range from 0 to 1 and are commonly stated as percentages from 0% to 100%. An R-squared of 100% means that all of the movements of a security (or another dependent variable) are completely explained by movements in the index (or whatever independent variable you are interested in).

In investing, a high R-squared, from 85% to 100%, indicates that the stock’s or fund’s performance moves relatively in line with the index. A fund with a low R-squared, at 70% or less, indicates that the fund does not generally follow the movements of the index. A higher R-squared value will indicate a more useful beta figure. For example, if a stock or fund has an R-squared value of close to 100%, but has a beta below 1, it is most likely offering higher risk-adjusted returns.

R-Squared vs. Adjusted R-Squared

R-squared only works as intended in a simple linear regression model with one explanatory variable. With a multiple regression made up of several independent variables, the R-squared must be adjusted.

The adjusted R-squared compares the descriptive power of regression models that include diverse numbers of predictors. This is often assessed using measures like R-squared to evaluate the goodness of fit. Every predictor added to a model increases R-squared and never decreases it. Thus, a model with more terms may seem to have a better fit just for the fact that it has more terms, while the adjusted R-squared compensates for the addition of variables; it only increases if the new term enhances the model above what would be obtained by probability and decreases when a predictor enhances the model less than what is predicted by chance.

In an overfitting condition, an incorrectly high value of R-squared is obtained, even when the model actually has a decreased ability to predict. This is not the case with the adjusted R-squared.

R-Squared vs. Beta

Beta and R-squared are two related, but different, measures of correlation. Beta is a measure of relative riskiness. A mutual fund with a high R-squared correlates highly with a benchmark. If the beta is also high, it may produce higher returns than the benchmark, particularly in bull markets.

R-squared measures how closely each change in the price of an asset is correlated to a benchmark. Beta measures how large those price changes are relative to a benchmark. Used together, R-squared and beta can give investors a thorough picture of the performance of asset managers. A beta of exactly 1.0 means that the risk (volatility) of the asset is identical to that of its benchmark.

Essentially, R-squared is a statistical analysis technique for the practical use and trustworthiness of betas of securities.

Limitations of R-Squared

R-squared will give you an estimate of the relationship between movements of a dependent variable based on an independent variable’s movements. However, it doesn’t tell you whether your chosen model is good or bad, nor will it tell you whether the data and predictions are biased.

A high or low R-squared isn’t necessarily good or bad—it doesn’t convey the reliability of the model or whether you’ve chosen the right regression. You can get a low R-squared for a good model, or a high R-squared for a poorly fitted model, and vice versa.

What Does R-Squared Tell You?

R-squared tells you the proportion of the variance in the dependent variable that is explained by the independent variable(s) in a regression model. It measures the goodness of fit of the model to the observed data, indicating how well the model's predictions match the actual data points.

Can R-Squared Be Negative?

No, R-squared cannot be negative. It always falls within the range of 0 to 1, where 0 indicates that the independent variable(s) do not explain any of the variability in the dependent variable, and 1 indicates a perfect fit of the model to the data.

What Is a Good P-Value?

In statistical hypothesis testing, a good p-value is typically one that is less than or equal to 0.05. This indicates that there is strong evidence against the null hypothesis, suggesting that the observed results are statistically significant and not due to random chance. However, the interpretation of p-values should also consider the context of the study and the specific research question.

What Does R-Squared Not Tell You?

While R-squared provides valuable insight into the goodness of fit of a regression model, it does not indicate whether the chosen model is appropriate or whether the predictions are biased. Additionally, R-squared does not provide information about the causal relationship between the variables or the reliability of the model's coefficients.

Why Is R-Squared Value So Low?

A low R-squared value suggests that the independent variable(s) in the regression model are not effectively explaining the variation in the dependent variable. This could be due to factors such as missing relevant variables, non-linear relationships, or inherent variability in the data that cannot be captured by the model.

What Is a "Good" R-Squared Value?

What qualifies as a “good” R-squared value will depend on the context. In some fields, such as the social sciences, even a relatively low R-squared value, such as 0.5, could be considered relatively strong. In other fields, the standards for a good R-squared reading can be much higher, such as 0.9 or above. In finance, an R-squared above 0.7 would generally be seen as showing a high level of correlation, whereas a measure below 0.4 would show a low correlation. This is not a hard rule, however, and will depend on the specific analysis.

What Does An R-Squared Value of 0.9 Mean?

Essentially, an R-squared value of 0.9 would indicate that 90% of the variance of the dependent variable being studied is explained by the variance of the independent variable. For instance, if a mutual fund has an R-squared value of 0.9 relative to its benchmark, this would indicate that 90% of the variance of the fund is explained by the variance of its benchmark index.

Is a Higher R-Squared Better?

Here again, it depends on the context. Suppose you are searching for an index fund that will track a specific index as closely as possible. In that scenario, you would want the fund’s R-squared value to be as high as possible since its goal is to match—rather than trail—the index. On the other hand, if you are looking for actively managed funds, then a high R-squared value might be seen as a bad sign, indicating that the funds’ managers are not adding sufficient value relative to their benchmarks.

The Bottom Line

R-squared can be useful in investing and other contexts, where you are trying to determine the extent to which one or more independent variables affect a dependent variable. However, it has limitations that make it less than perfectly predictive.