R-⁠Squared

What is R-squared?

R-squared, or the coefficient of determination, is the share of an outcome’s variation that a fitted regression explains. It compares residual variation with total variation in the outcome. In investing, the outcome is a fund or stock return and the predictor is a benchmark return, so R-squared shows how tightly the returns follow a linear relationship with the benchmark.

A standard benchmark regression puts R-squared between zero and one, and multiplying by 100 gives the percentage: 0.85 is 85%.

R-squared vs beta and correlation

In a one-predictor ordinary least-squares model with an intercept, R-squared equals squared sample correlation. Correlations of +0.90 and −0.90 both give R-squared 0.81.

Beta preserves the direction and scale of the fitted relationship. A series with returns exactly twice benchmark returns has beta 2 and R-squared 1. A series with returns exactly opposite the benchmark has beta −1 and R-squared 1. High R-squared therefore means lockstep movement at the beta’s scale: a fund with beta 2 and R-squared 1 doubles every benchmark move.

Alpha is the return left after the benchmark exposure is paid for, and two funds with the same R-squared can earn very different alpha.

R-squared example: two funds, one benchmark

The chart contains 25 monthly levels, producing 24 monthly returns. Regressing those returns on the benchmark with an intercept gives Fund A beta 0.98 and R-squared 0.98. Fund B has beta 0.76 and R-squared 0.31.

Fund A ends at 97.6, Fund B at 101.3 and the benchmark at 98.2. Fund A hugs the benchmark and finishes below it; Fund B, at R-squared 0.31, is the only one to finish above it. Judge the payoff from return and alpha, and use R-squared to see how much of each fund’s risk belongs to the benchmark.

Adjusted R-squared, added predictors and negative values

Adding predictors to a nested ordinary least-squares fit on the same observations cannot lower ordinary R-squared. It can leave it unchanged. Adjusted R-squared incorporates the number of predictors, while held-out error measures predictions on observations excluded from fitting.

Centered R-squared can be negative for held-out predictions or other model specifications. A constant outcome makes SST zero, so the displayed ratio is undefined.

How investors use R-squared

Choose a benchmark that matches the portfolio question. Read the fit alongside beta, coefficient precision, residual variation, expenses and actual return.

Use R-squared to decide how much weight beta deserves. At R-squared 0.98, a hedge at Fund A’s 0.98 beta removes nearly all of the fund’s return variation in the sample; at 0.31, Fund B’s 0.76 beta leaves most of its variation unexplained.

Limitations of R-squared

• Read direction from beta and payoff from return. A fund that moves exactly opposite the benchmark has R-squared 1, and in the two-fund chart Fund B finishes highest at R-squared 0.31.
• A low reading points two ways. A differentiated strategy and a poorly chosen benchmark both produce it, so test a benchmark that matches the mandate and a second window before you act on the number.
• Compare models with adjusted R-squared. Adding predictors to the same sample never lowers ordinary R-squared, so a higher value from more variables needs adjusted R-squared or held-out error to show the model improved.
• R-squared reads straight-line fit. A symmetric U-shaped relationship scores zero even when one series drives the other, so plot the returns before you call two series unrelated.
• Replace a model whose held-out reading falls below zero, because it predicts worse than the average return. The zero-to-one range applies to in-sample fits with an intercept.

Bottom line

R-squared measures how much of a return series’ variation the benchmark explains, so it tells you how far to trust beta as the fund’s hedge ratio. Read the payoff from return, alpha and costs, and test the fit against a benchmark that matches the portfolio’s real exposures.