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R-squared definition and tutorial

A scale from 0 to 1 makes this difficult-to-calculate measure easy to interpret.
  1. Define - Define R-squared.
  2. Context - Use R-squared in a sentence.
  3. Video - See the video.
  4. Script - Read the transcript.
  5. Quiz - Test your knowledge.
by Paul Alan Davis, CFA, June 29, 2016
Updated: December 22, 2018
Many in Finance race to interpret R-squared without thinking about its context. Let's look at it more deeply.

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R-squared (r2) for Investment Modeling

Intermediate

R-squared is a measure of the strength of the relationship between two data sets. For investments it is used for statistical interpretations. The R refers to Pearson's R, or r, from correlation and is calculated by squaring the correlation between a pair of securities. This transforms the correlation scale, of -1 to +1, to an easily interpreted scale of 0 to 1. On the one end, 0 indicates no relationship and 1 indicates a perfect linear relationship.

Synonym: coefficient of determination, r2

In a Sentence

Doc:  Who remembers why it is important to review R-squared  with beta.
Mia:  Even a random shotgun pattern has a line of best fit.

Video

This video can be accessed in a new window or App , at the YouTube Channel or from below.

R-Squared definition for investment modeling (4:31)

Video Script

The script includes two sections where we visualize and demonstrate the concept of R-squared.

Visualize

We're sitting right here in Excel and this is a snippet from our boot camp course.

This is one depiction of R-squared, from a scatterplot of returns for one stock Merck versus a basket of stocks, the Market, for 60 periods. Think about each dot here as returns for the Market on the x-axis and Merck on the y-axis, for each month.

So if stocks exhibit co-movements, as they appear to here, then a pattern will look linear. A random shotgun pattern would have low R-squared.

Once we square correlation, the direction of the line goes away, and we can only interpret the goodness of fit of the line versus the data points.

The best way to understand R-squared is with an example, so let's head there now.

Demonstrate

We'll walk all the way through a calculation for two stocks, Microsoft and eBay.

We have six monthly returns for each stock from April to September 2003. Column F is the return on Microsoft, eBay is in column G.

Next we compute the average of each, here 2.38% and 3.98%. Then we move those over to columns H and I.

In column J take the return minus this average which gives us 3.24%. That's 5.62% minus 2.38%. For eBay it is 8.91% minus 3.98% or 4.93%. Carry that formula down, and let's call these demeaned returns.

Next, in column L, multiply these together. As you notice, when the stocks move together, like in April, the product is negative. And when they move in opposite directions, the product is negative.

Next, using the =SUM function, add up the products to get -0.0037 for the pair of stocks. Next, divide by 6 observations for a covariance of -0.0006 for the pair of stocks. This isn't interpretable as the units are returns-squared, so we translate to correlation by dividing by the product of the two standard deviations.

So to interpret, the R-squared explains what percentage of the total errors is explained by the x-variable.

Quiz

Click box for answer.

To calculate R-squared divide the covariance between a pair of assets by the product of their standard deviations and then square the result. | True or False?

True. The first part is correlation.

Questions or Comments?

Still unclear on R-squared? Leave a question in the comments section on YouTube or check out the Quant 101 Series, specifically How to interpret Correlation and R-squared.

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