Glossary

R-Squared (Investing) — Definition, Formula, and Allocator Use

R-squared measures the share of a portfolio's return variance explained by a benchmark. Definition, formula, worked example, and how allocators interpret it.

By NakedPnL Research·May 7, 2026·4 min read

TL;DR

R-squared (R²) is the share of variance in a portfolio's returns explained by a chosen benchmark.
It ranges from 0 (no relationship) to 1 (returns are a linear function of the benchmark).
Allocators read R² alongside beta to judge whether a manager's beta and alpha estimates are statistically meaningful.

Definition

R-squared (R²) is the coefficient of determination from a linear regression. In an investing context, it is the proportion of the variance in a portfolio's returns that is explained by variance in a chosen benchmark's returns. Values close to 1 indicate that benchmark returns explain almost all of the portfolio's return variance; values close to 0 indicate the portfolio's returns are essentially uncorrelated with the benchmark.

Formula

R^2 = 1 - (SS_res / SS_tot)

where, for the regression (R_p - R_f) = alpha + beta * (R_m - R_f) + epsilon:
  SS_res = sum( residual_t^2 )
  SS_tot = sum( (R_p,t - mean(R_p))^2 )

Equivalently for a single-factor regression:
  R^2 = Corr(R_p, R_m)^2

R² is bounded between 0 and 1 for ordinary least-squares regressions with an intercept term.

Worked example

A portfolio is regressed on a broad-market benchmark over 60 monthly observations. The total sum of squared deviations from the portfolio's own mean (SS_tot) is 0.080; the residual sum of squares (SS_res) after the regression is 0.024. R² = 1 − (0.024 / 0.080) = 0.70. Seventy per cent of the portfolio's return variance is explained by the benchmark; thirty per cent is residual or 'idiosyncratic'. A second portfolio that trades a niche strategy might post R² = 0.15 against the same benchmark — the benchmark explains very little of its behaviour.

How allocators use it

Allocators read R² in conjunction with beta and alpha. A high R² (say above 0.80) makes the beta and alpha estimates statistically meaningful — most of what the portfolio does is captured by the benchmark, and any reported alpha is a credible measure of skill against that benchmark. A low R² (below 0.30) means the chosen benchmark is a poor explanation of the portfolio's returns; in that case the regression is the wrong tool and the alpha and beta numbers should be treated with scepticism. R² is also used to identify 'closet indexing' — funds claiming active management whose R² against an index is so high (typically 0.95+) that the active fee is hard to justify.

Beta — slope of the same regression
Alpha — intercept of the same regression
Correlation coefficient — R² is the square of correlation in a single-factor regression
Tracking error

Frequently asked questions

Is high R² good or bad?

Neither inherently. High R² means the benchmark explains the portfolio well; low R² means the portfolio is doing something the benchmark does not capture. Whether either is desirable depends on what the allocator is hiring the manager to do.

Can R² be negative?

Not in an ordinary least-squares regression with an intercept. Some non-OLS goodness-of-fit measures called 'pseudo-R²' can be negative; the standard finance R² cannot.

What R² is required for closet-indexing concerns?

There is no regulatory threshold. In academic and practitioner literature, R² above 0.95 against the stated benchmark, combined with a low active share, is the standard pattern flagged as closet indexing.

References

Coefficient of determination — Wikipedia