R-Squared (R²)

R-squared (R²), also called the coefficient of determination, is a statistic that shows what share of the movements in one set of numbers is explained by movements in another. In investing, it measures how much of a fund's or security's return pattern can be explained by the returns of a benchmark index. It runs from 0 to 1, or from 0 to 100 when shown as a percentage, as fund data services usually do. The statistic comes from regression analysis and is one of the standard measures of fit used in econometrics, the application of statistical methods to economic data.

An R² of 95 means that 95% of a fund's month-to-month return movements line up with its benchmark; an R² of 40 means most of the fund's behaviour comes from something else. R² is read together with beta and alpha, because it tells the investor whether those two figures can be trusted. It does not say whether a fund performed well or badly.

Imagine walking a dog on a lead. If the lead is short, the dog's path follows yours almost exactly; knowing where you walked tells you nearly everything about where the dog went. If the dog runs free in a park, your path explains very little of its route.

R² measures how short the lead is between a fund and its benchmark. A high number means the fund stays close to the index and its ups and downs are mostly the index's ups and downs. A low number means the fund roams freely, so comparing it with that index says little. R² does not say whether the dog walked faster or slower than its owner; it only describes how closely the two paths match.

In a regression of a fund's returns on a benchmark's returns, R² compares the leftover, unexplained variation with the total variation:

• $SS_{\text{res}}$ is the residual sum of squares: the squared gaps between the fund's actual returns and the returns the regression line predicts from the benchmark.
• $SS_{\text{tot}}$ is the total sum of squares: the squared gaps between the fund's returns and their own average.

Suppose the total sum of squares for a fund's monthly returns is 100 and the residual sum of squares is 20. Then $R^2 = 1 - \frac{20}{100} = 0.80$, or 80: four-fifths of the fund's return variation is explained by the benchmark.

With a single benchmark, R² is simply the square of the correlation coefficient between the two return series:

A fund whose correlation with its index is 0.9 therefore has an R² of 0.9 × 0.9 = 0.81, or 81. A correlation of −0.9 also gives 81, so R² hides the direction of the relationship; beta shows it.

Fund data providers such as Morningstar publish R² against a standard index for each fund category and, in some cases, against a "best-fit" index that matches the fund most closely. Rough ranges used by analysts are:

• Very high, 95 to 100. Typical of index funds and exchange-traded funds that track the benchmark, such as funds following the S&P 500.
• Moderate, about 70 to 95. Common for diversified active equity funds that hold shares from the same market as the index but in different weights.
• Low, below about 70. Seen in concentrated, sector, global or alternative funds, and in many hedge funds, whose returns do not follow a broad index.

These bands are conventions, not rules set by any regulator. Their main use is practical. When R² is low, the beta and alpha calculated against that index are unreliable, because the index explains little of what the fund does. In that case the investor should look for a better-fitting index before judging the manager.

A very high R² combined with high fees can signal a "closet index fund": a fund that charges for active management but holds a portfolio close to the index. Such a fund is unlikely to beat its benchmark after its expense ratio. In a statement of 2 February 2016, the European Securities and Markets Authority (ESMA) used R² alongside active share and tracking error to estimate how many European equity funds sold as active might in fact be closet trackers, and national supervisors such as Luxembourg's CSSF followed up with their own reviews.

A lower R² is not bad in itself. A manager who aims to differ from the index will naturally show a lower R² and a higher tracking error. What matters is whether the different returns are rewarded, which is where the Sharpe ratio and alpha come in. In asset allocation, low R² between two holdings is useful, because it signals that they add diversification.

• R² measures fit, not cause. Two series can move together by coincidence over a short period.
• It depends on the data window. R² over three years of monthly returns can change sharply when the window moves, especially across a market shock.
• Outliers distort it. One extreme month, such as March 2020, can raise or lower R² considerably.
• It assumes a straight-line relationship. Funds holding options or using leverage may respond to the market in non-linear ways that R² understates.
• Adding factors always raises it. In multi-factor models, adjusted R² is used to penalise extra variables that add little explanatory power.

R² is also used well outside fund analysis, for example to judge how well economic models explain movements in inflation or interest rates. The interpretation is the same: the share of variation the model accounts for.

• CSSF conclusions on closet index tracking work