What is R-Squared
R-squared is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable. In investing, R-squared is generally considered the percentage of a fund or security's movements that can be explained by movements in a benchmark index.
For example, an R-squared for a fixed-income security versus 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 versus the S&P 500 index, or any other relevant index.
BREAKING DOWN R-Squared
R-squared values range from 0 to 1 and are commonly stated as percentages from 0% to 100%. An R-squared of 100% means all movements of a security (dependent variable) are completely explained by movements in the index (independent variable). A high R-squared, between 85% and 100%, indicates the stock or fund's performance moves relatively in line with the index. A fund with a low R-squared, at 70% or less, indicates the security 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 Calculation Example
The actual R-squared equation is calculated as:
R-Squared = 1 - (Explained Variation / Total Variation)
However, the actual calculation requires several steps. This includes taking the data points of dependent and independent variables and finding the line of best fit. From there you would calculate predicted values, subtract actual values and square the results. This yields a list of errors squared, which is then summed and equals the explained variance.
To calculate the total variance, you would subtract the average actual value from the predicted values, square the results and sum them. From there, divide the first sum of errors (explained variance) by the second sum (total variance), subtract the result from one, and you have the R-squared.
R-Squared Limitations and Interpretations
R-squared will give you an estimate of the relationship between movements of a dependent variable based on an independent variable's movements. 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-square isn't necessarily good or bad, as it doesn't convey the reliability of the model, nor whether you've chosen the right regression. You can get a low R-squared for a good model, or a high R-square for a poorly fitted model, and vice versa.