R-Squared – Definition

Cite this article as:"R-Squared – Definition," in The Business Professor, updated July 31, 2019, last accessed May 26, 2020, https://thebusinessprofessor.com/lesson/r-squared-definition/.

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R-Squared Definition

R-squared, also referred to as the coefficient of determination, is a measure of statistics that gives relationship’s estimate between dependent variable’s movements based on the movement of the independent variable. In finance, it is a measure of statistics between the performance of an investment and an identified benchmark index.

A Little More on What is R-Square

Coefficient of determination is the one that will try to tell you the number of data points that fall within the line’s formed results through the regression equation. Linear regression is expressed in percentage, and when the coefficient is higher, so is the points’ percentage which the line passes through when plotting of data points, and the line is done. Note that 1 or 0 values are an indication that either the regression line represents all the data or none at all. Also, R-squared is negative, when a model being used is not a good fit for the data. In addition, if an intercept is not set, then the coefficient of determination will definitely be negative too.

How R-Squared Works (Steps in Calculating R-squared)

Generally, the calculation of Coefficient of determination (R-squared) goes through several steps. They are as follows:

Step one: Take the observations (data points) of the dependent and independent variables, then use a regression model to find the line of best fit.

Step two: Calculate the values that have been predicted by subtracting the actual values, and then squaring the results. After the calculation, you will notice that it will yield several errors squared. When the errors are summed up, they will equal the explained variance.

Step three: Here the variance is calculated by simply subtracting the average real value from the values that have been predicted. You will take results of this square them and then sum them.

Step four: You will divide the first errors’ sum (Explained variance) by the sum of the second errors (total Variance). You will then minus the derivation from 1 in order to get the R-squared (coefficient of determination.

The formula of R-squared, therefore, looks like this:

R² = Total variance/Explained variation

Note that the coefficient of determination’s range value is 0 to 1, which are commonly expressed as a percentage from 0% to 100%. A coefficient of 100% is an indication that all the security’s movement (dependent variable) is explained by the movements in the independent variable(s) that are of interest to you.

Generally, in investment, a high coefficient of determination between 85 percent and 100 percent is an indication that the stock’s performance is relatively moving in line with the index. On the other hand, a stock with R-squared that is low (either at 70 percent or below) is a sign that the movement performance is not in line with the index.

Note that a higher R-squared value indicates a beta figure that is useful. For instance, a stock with R-squared value that is near 100%, but with a beta below a figure of 1, it is probably producing higher risk-adjusted returns.

The Difference Between R-Squared and Adjusted R-Squared

For R-squared to work as expected, a simple linear regression model with one variable that is explanatory must be applied. R-squared is usually adjusted by the use of various independent variables that has multiple regressions. Note that it is the adjusted R-squared which compares the regression model’s descriptive power, which is inclusive of diverse predictors’ numbers.

Also, it is important to note that a rise in R-squared is as a result of each one of the predictors added to a model. In other words, the addition of predictors to a model does not decrease it but rather increases it. Therefore, more terms on a model have a good fit, whereas the adjusted R-squared compensates the added variables.

Generally, R-squared incorrectly high value which decreases prediction ability happens under overfitting condition. However, this does not happen with the adjusted R-squared. While the comparison of a standard can compare the goodness of two or more models, adjusted R-squared is not an ideal metric when it comes to comparing multiple linear regressions or nonlinear models.

The Difference between R-Squared and Beta

R-squared and Beta are correlation measures which are related and at the same time different. However, beta is a measure of relative riskiness. In other words, it is a mutual fund that has a high R-squared which correlates with a benchmark.

Note that when they are used together, the beta is also usually high and is likely to give higher returns than the benchmark. This more likely to happen in the bull market where the r-square measures the closeness of each change in the asset’s price and how it correlates to the benchmark. In this case, the beta would measure the magnitude of such changes in relation to a benchmark. This way, it is able to give investors a clear picture of the asset’s managers’ performance.

R-Squared Uses

R-square has many uses. Some of its uses are as highlighted below:

  • Investors can use R-square measurement to make a comparison between the performances of portfolios with the broader market, predicting the possible occurrence of future trends.
  • Secondly, R-squared can be a measure that investors use to determine the history movement of funds in mutual fund performance industry and its correlation with a benchmark index.
  • Also, R-square can be used by investors to hedge funds. In other words, they use it to determine their model’s risk level and its association with given factors.
  • Finally, investors can use R-squared to assist them in determining how their stocks are moving and its market correlation. To determine this, investors will need R-squared. Note that when a coefficient of determination is close to one, it is an indication that most stock the movement of the stock can be explained by the movement of the market.

 R-Squared Limitations

  • R-square will not show you the model’s reliability you have chosen. Meaning you are not in a position to tell if it is good or bad.
  • It also does not tell whether or not your predictions and data are biased. In other words, it does not tell the model’s reliability.
  • With R-square, there is a possibility that you will end up with a low r-squared when you use a good model and at the same time, high R-square for a model that is poorly fitted and vice versa. This means that it will not tell you how adequate the regression model is.

References for “R-Squared”

https://www.investopedia.com › Investing › Financial Analysis

https://en.wikipedia.org/wiki/Coefficient_of_determination

https://blog.minitab.com/…/regression-analysis-how-do-i-interpret-r-squared-and-asses…

https://people.duke.edu/~rnau/rsquared.htm

https://www.displayr.com/8-tips-for-interpreting-r-squared/

 

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