R-Squared

What Is R-Squared?

R-squared, also written R², measures how well a regression analysis fits the data it was built from. It is also called the coefficient of determination. In most standard cases its value runs from 0 to 1. A value of 0 means the model explains none of the variation in the outcome being studied, while a value of 1 means it explains all of it.

In investing, R-squared is most often used to compare a fund or stock with a benchmark index, such as a broad market index. It answers a simple question: how much of this security's return movement can be accounted for by movements in the benchmark?

How It Is Calculated

R-squared compares the error left over by the model with the total variation in the data. The standard formula is:

R² = 1 − (Residual Sum of Squares ÷ Total Sum of Squares)

The residual sum of squares measures the gap between the actual values and the values the model predicts. The total sum of squares measures how far the actual values spread around their average. The smaller the leftover error relative to total variation, the closer R-squared gets to 1. In a simple regression with one benchmark, R-squared equals the square of the correlation coefficient between the two series.

Why It Matters to Investors

R-squared helps judge how much weight to give other statistics. For example, beta describes how sensitive a stock is to its benchmark, but beta is a more reliable guide when R-squared is high. Consider a hypothetical fund with an R-squared of 0.90 against a large U.S. stock index. About 90% of its return variation would track that index, so the index would be a fairly good explanation of its performance. A fund with an R-squared of 0.20 would be driven mostly by other factors, and its beta against the same index would say little.

Limitations to keep in mind:

• A high R-squared does not mean the model is correct or that one variable causes the other.
• Adding more variables to a regression almost always raises R-squared, so adjusted R-squared is often used to penalize extra variables.
• R-squared does not show whether a fund beat its benchmark, which is what alpha measures.