Formula, in Risk & Portfolio Construction
R-Squared Formula
R-squared is the proportion of a portfolio's return variance that is explained by movements in its benchmark. In a single-factor regression it equals the square of the correlation between portfolio and benchmark returns. A high R-squared makes beta and alpha meaningful; a low one warns that the benchmark is the wrong yardstick.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
R^2 = 1 - ( SS_res / SS_tot )
Single factor: R^2 = rho(p, b)^2
where SS_res = sum of squared regression residuals, SS_tot = sum of squared deviations of R_p from its mean Variables
SS_res
Residual sum of squares
Sum of squared differences between actual portfolio returns and the returns predicted by the benchmark regression. It is the variation the benchmark fails to explain.
SS_tot
Total sum of squares
Sum of squared deviations of portfolio returns from their own mean, the total variation to be explained.
rho(p, b)
Correlation with benchmark
Correlation coefficient between portfolio and benchmark returns. In the one-factor case R-squared is simply its square, which is why R-squared always lies between 0 and 1.
R^2
Coefficient of determination
Fraction of variance explained, from 0 (benchmark explains nothing) to 1 (benchmark explains everything). Index funds run near 1.0; market-neutral and alternative strategies run low.
Step by step
- 1
Regress portfolio returns on benchmark returns, or compute the correlation between the two series.
The correlation between fund and index monthly returns is 0.92.
- 2
In the single-benchmark case, square the correlation to get R-squared.
0.92^2 = 0.8464.
- 3
Interpret the result as the share of variance explained by the benchmark.
About 85% of the fund's return variation is driven by the index.
- 4
Use R-squared to judge whether beta and alpha are trustworthy: low R-squared means the benchmark is a poor reference and its beta is noisy.
An R-squared of 0.30 means 70% of variation is unexplained, so a reported beta against this benchmark should be treated cautiously.
Worked example
Validating a benchmark for a large-cap fund
Correlation with benchmark
0.92
R-squared = rho^2 = 0.92^2 = 0.8464.
R-squared of about 0.85. Roughly 85% of the fund's return variance comes from the benchmark, so beta and the index are appropriate, and any alpha estimate is meaningful. The remaining 15% is idiosyncratic, the portion where active selection lives. If R-squared had come in near 0.30, the reported alpha and beta would be unreliable because the benchmark explains too little.
Common variations
Run the numbers next
- Calculators Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
- Calculators Returns Distribution Analyzer
Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
Sources
- Applied Linear Statistical Models , Kutner, Nachtsheim, Neter, Li, McGraw-Hill (2005)
- Components of Investment Performance , Eugene F. Fama, Journal of Finance 27(3), 551-567 (1972)
Keep the topic connected
- Formula Correlation Formula
The Pearson correlation formula: covariance of two return series over the product of their standard deviations. The key diversification input.
- Formula Beta Formula
The beta formula: covariance of asset and market returns over market variance. The CAPM regression slope measuring systematic risk, with an example.
- Formula Tracking Error Formula
The tracking error formula: the standard deviation of a portfolio's active return versus its benchmark. How tightly a fund tracks its index.
- Formula CAPM Formula
The CAPM formula: expected return equals the risk-free rate plus beta times the market risk premium. The basis for pricing systematic risk.