What Is R-Squared? Definition, Calculation, and Interpretation

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R-squared (R², the coefficient of determination) is a statistic that measures the share of the variation in a dependent variable that a regression model explains through its independent variable or variables. It is usually reported between 0 and 1, or 0% to 100%. An R² of 0.81 means the model accounts for 81% of the variation in the outcome around its average. The other 19% is left unexplained.

In investing, the dependent variable is usually a fund’s or stock’s returns and the independent variable is a benchmark’s returns, so R-squared tells you how much of the asset’s movement lines up with the benchmark’s.

What R-Squared Measures

Every set of observations varies around its average. Monthly fund returns, for example, are above the mean in some months and below it in others. A regression model tries to explain that variation using other variables. R-squared answers one question: of all the variation present, how much does the model’s fitted line (or curve) capture?

• R² = 1 (100%): the fitted values match every observation exactly. All of the dependent variable’s movement is explained by the inputs.
• R² = 0: the model explains nothing beyond what you would get by predicting the average every time.
• Values in between: the model captures part of the variation, and the remainder sits in the residuals (the gaps between actual and predicted values).

The statistic describes fit to the data used. It is a measure of association within that sample, not proof that one variable causes the other or that the relationship will hold in the future.

The R-Squared Formula

The standard formula is:

R² = 1 − (SS_res / SS_tot)

where:

• SS_res (residual sum of squares) is the unexplained variation: the sum of (actual value − predicted value)² across all observations.
• SS_tot (total sum of squares) is the total variation: the sum of (actual value − mean of actual values)².

The ratio SS_res / SS_tot is the fraction of variation the model failed to explain. Subtracting it from 1 gives the fraction it did explain.

For a simple linear regression with one independent variable and an intercept, R² also equals the square of the Pearson correlation coefficient (r) between the two variables. That is where the name comes from. If two series have a correlation of 0.9, a straight-line regression of one on the other has an R² of 0.81.

How to Calculate R-Squared: A Worked Example

Hypothetical example. A fund’s monthly returns are compared with an index over five months. These figures are invented to show the mechanics.

Step 1: find the means. The index averages 3% and the fund averages 4%.

Step 2: fit the line. The deviations of X from its mean are −2, −1, 0, 1, 2. For Y they are −2, −1, 1, 0, 2. Multiplying the paired deviations and summing gives 9. The sum of squared X deviations is 10. The slope is 9 / 10 = 0.9, and the intercept is 4 − (0.9 × 3) = 1.3. The fitted line is: predicted fund return = 1.3 + 0.9 × index return.

Step 3: predicted values. 2.2, 3.1, 4.0, 4.9 and 5.8.

Step 4: residuals. Actual minus predicted gives −0.2, −0.1, 1.0, −0.9 and 0.2. Squared, they are 0.04, 0.01, 1.00, 0.81 and 0.04. SS_res = 1.90.

Step 5: total variation. The squared deviations of Y from its mean are 4, 1, 1, 0 and 4. SS_tot = 10.

Step 6: R-squared. 1 − (1.90 / 10) = 1 − 0.19 = 0.81.

As a cross-check, the correlation is 9 / √(10 × 10) = 0.9, and 0.9² = 0.81. In this example, 81% of the variation in the fund’s monthly returns is associated with the index’s returns, and 19% comes from something else. The slope of 0.9 is the fund’s beta against the index in this sample.

Spreadsheets and statistical software do these steps automatically. A chart trendline labeled “R² =” reports the same calculation for the fitted line or curve, which is why a curved (polynomial) fit can show a higher R² than a straight line on the same data.

How to Interpret R-Squared

Read R² as a percentage of variation explained. An R² of 0.50 means roughly half of the observed variation is accounted for by the model’s inputs. A small value such as 0.0673 means the inputs explain about 6.73% of the variation, leaving 100 − 6.73 = 93.27% unexplained.

Several points keep the interpretation honest:

• R² describes fit, not accuracy of individual predictions. A high-R² model can still miss individual observations by wide margins if the outcome is volatile. Typical error size is shown by the standard error of the regression or the residuals.
• What counts as “good” depends on the field. In some social sciences, an R² around 0.5 may be strong because human behavior is noisy, while physical or engineering measurements can be held to 0.9 or higher.
• Context decides direction. Whether you want a high or a low R² depends on what you are evaluating, as the investing uses below show.

How R-Squared Is Used in Investing

In portfolio analysis, R-squared is computed by regressing a fund’s or security’s returns on a benchmark’s returns over a chosen period. A bond fund might be measured against a bond index; a U.S. large-cap stock fund against the S&P 500. The result tells you what share of the fund’s return movements tracked the benchmark during that window.

A rule of thumb in fund analysis treats an R² of 85% to 100% as meaning the fund moves largely in line with its index, and 70% or below as meaning it does not generally follow it. Other conventions put “high” at about 0.7 and “low” below 0.4. These are rough guides, and results depend on the benchmark and period used.

R² is used in three main ways:

Judging an index fund. An index fund or ETF that aims to replicate a benchmark should have an R² very close to 100% against that benchmark. A lower figure suggests the fund’s holdings or methods are causing it to drift from the index.

Judging an active manager. For an actively managed fund, a very high R² can be a warning sign: the investor may be paying active management fees for something close to index exposure. A lower R² shows the fund behaves differently from the benchmark, which is the point of active management, though it says nothing about whether the difference helped or hurt returns.

Checking whether beta and alpha are meaningful. Beta and alpha are both estimated from a regression against a benchmark. When R² is low, the benchmark explains little of the fund’s behavior, so beta and alpha computed against it describe the fund weakly, and a different index may fit better. When R² is high, they rest on a relationship that actually held in the data.

R² is one piece of a fund analysis. Return measures and risk-adjusted measures such as the Sharpe ratio address performance, which R² does not.

R-Squared vs. Adjusted R-Squared

In ordinary least squares regression, adding another independent variable cannot lower R² on the same data, and it usually raises it slightly, even if the new variable is irrelevant. A model with many predictors can therefore look better simply because it has more terms. Pushed far enough, this produces overfitting: the model fits the noise in the sample and predicts new data poorly, while R² stays misleadingly high.

Adjusted R-squared penalizes extra predictors:

Adjusted R² = 1 − (1 − R²) × (n − 1) / (n − k − 1)

where n is the number of observations and k is the number of independent variables.

Adjusted R² rises only when a new variable improves the model by more than would be expected by chance, and falls when it does not. It is always at or below R², and the gap widens when the sample is small relative to the number of predictors.

Applied to the five-month example (n = 5, k = 1): 1 − (0.19 × 4 / 3) = 1 − 0.253 ≈ 0.75. With so few observations, the penalty takes 0.81 down to about 0.75; with a single predictor and a large sample, the two are nearly identical. For comparing models with different numbers of predictors, adjusted R² is the more appropriate figure.

R-Squared vs. Beta

Beta and R² come from the same regression of an asset’s returns on a benchmark’s returns, but they measure different things:

• Beta is the slope. It measures how large the asset’s moves have been relative to the benchmark. A beta of 1.0 means the asset has moved one for one with the benchmark on average; 1.5 means 1.5% for each 1% benchmark move; 0.9, as in the example, means 0.9%.
• R-squared measures how consistently the asset’s moves lined up with the benchmark’s, regardless of their size.

Two funds can share a beta of 1.2 yet have very different R² values. One tracks the index closely with amplified moves (high R²), while the other has the same average sensitivity but large swings unrelated to the index (low R²). For the second fund, the beta is a much less dependable description of its behavior. Used together, the two statistics show both how much a fund tends to move with its benchmark and how reliable that estimate is.

R-Squared vs. Correlation

Correlation (r) measures the strength and direction of a linear relationship between two variables and ranges from −1 to +1. R-squared measures how much of one variable’s variance is explained by the model and, in the simple linear case, equals r².

Squaring removes the sign. A correlation of −0.9 and one of +0.9 both give an R² of 0.81, so R² alone cannot tell you whether two series moved together or in opposite directions. It also turns correlation into a share of variance, which shrinks moderate values considerably: a correlation of 0.5 corresponds to an R² of only 0.25, meaning a quarter of the variation is explained.

Limitations of R-Squared

• It does not validate the model. A poorly specified model can produce a high R², and a correct model can produce a low one when the outcome is inherently noisy. R² does not reveal bias in the data or predictions.
• It assumes the form of the relationship. A linear R² measures fit to a straight line, so a strong nonlinear relationship can show a low linear R².
• It is sensitive to the sample. In finance, R² against a benchmark changes with the time period, the return frequency (daily, monthly) and the benchmark chosen. Outliers can move it substantially.
• It says nothing about performance. A fund can have a 95% R² and still lag its index after fees, or a 40% R² and outperform.

Why R-Squared Can Be Low

A low R² means the independent variables capture little of the variation in the dependent variable. Common reasons are missing relevant variables, a relationship that is not linear, or an outcome with a lot of inherent randomness. Raising R² is not the goal in itself: changes to a model should make it more accurate on new data, which is why adjusted R² and out-of-sample testing matter more than the in-sample figure.

FAQs

The Bottom Line

R-squared matters to investors checking how closely a fund tracks its benchmark and whether its beta and alpha mean much. A high figure confirms the benchmark explains the fund's moves, and a low one suggests another index may fit better. Because R² says nothing about returns or causation, pair it with performance and risk-adjusted measures before judging a fund.