R-Squared

R squared, also called the coefficient of determination, measures how much of the variation in one series is explained by another. In investing, it usually shows how much of a fund's or strategy's return movements are explained by its benchmark. A US index fund might have an R squared near 1.00 against the S&P 500; a market neutral hedge fund might have one near 0. R squared does not say whether performance was good. It says how much the benchmark comparison, and therefore beta and alpha, can be trusted.

What it means#

R squared = 1 - (Unexplained variation / Total variation)

For a simple regression on one benchmark, R squared equals the correlation between the two series, squared. See Covariance and Correlation and Regression Analysis.

Uses of R squared#

R squared in model building#

In forecasting, R squared measures how much of the target's variation a model explains. In trading, out of sample R squared values for return forecasts are usually tiny, often below 1%, yet can still be economically valuable when applied across many bets. An in sample R squared that looks high is often a sign of overfitting. See Overfitting and Curve Fitting and Information Ratio and Tracking Error.

Traps#

1. High R squared is not good performance: an index fund has R squared near 1 and zero alpha before fees.
2. Trending series: regressing one trending price series on another can give a high R squared with no real relationship, a spurious regression. Use returns, not prices. See Stationarity, Differencing and Unit Roots.
3. Adding variables always raises in sample R squared: use adjusted R squared or out of sample tests.
4. Outliers: a few extreme points can create or destroy R squared. See Outliers and Robust Statistics.
5. Non linear relationships: option strategies may relate strongly to the market in a non linear way that a linear R squared understates.

Adjusted R squared#

Adjusted R squared penalises extra explanatory variables:

Adjusted R squared = 1 - (1 - R squared) × (n - 1) / (n - k - 1)

where n is the number of observations and k the number of explanatory variables. It rises only when a new variable improves the model more than expected by chance.

Frequently asked questions#

What does R squared mean in investing?#

The share of a portfolio's return variation explained by its benchmark or a model, from 0 (none) to 1 (all).

What is a good R squared for a mutual fund?#

It depends on the goal. Index funds should be near 1; active funds with R squared above about 0.95 may be closet indexers, and low values suggest the benchmark is a poor fit.

Does a high R squared mean a good investment?#

No. It only measures how closely returns follow the benchmark, not whether they were high or low.

Next, move from measuring performance to building portfolios in Portfolio Construction.