R-Squared

R-squared is a statistic that reports what proportion of the variation in one series is accounted for by a model of it, on a scale from 0 to 1. The NIST/SEMATECH e-Handbook of Statistical Methods, published by the National Institute of Standards and Technology, describes it as "the fraction of the total variability in the response that is accounted for by the model." In fund analysis the model is normally a single benchmark index, so the number answers a narrow question: of everything the fund's return did over the measurement period, how much of it moved with the index.

Explaining the naming is worth a sentence, because two different confusions attach to it. Its formal name is the coefficient of determination, and the two are the same statistic; the informal name is the one almost everyone uses, including regulators writing about funds. It is not the correlation coefficient, which is usually written with a lowercase r. In the simple case of one series regressed on one other, R-squared is the square of that correlation, which is why the names get swapped and why the swap matters: squaring turns a correlation of 0.7 into an R-squared of 0.49, and reporting the first as the second overstates how much is explained. Neither name is defined in the Securities and Exchange Commission's investor glossary, which carries no entry for either. The formal name does appear in federal regulation, at 40 CFR 1065.602(k) among other places, but in an emissions test method for engines rather than anywhere in securities law, so the finance reading of the statistic rests on practice rather than on an official definition.

The scale is variance, and that is the single most misread feature of the number. Variation here means squared deviation, not percentage movement. So the complement of R-squared is the unexplained share of the variance, and converting it back into the units a reader thinks in requires a square root. A fund with an R-squared of 0.36 against its benchmark has 64 percent of its return variance unexplained, which is 80 percent of its return volatility. Put the other way, a fund needs an R-squared of 0.25 before even half of its volatility is attributable to the benchmark. Readings that sound moderate are weaker than they look.

A high R-squared says the two series moved together. It does not say the model is right. This is NIST's own emphasis rather than a caveat added here: the handbook's discussion of judging model fit runs under the heading "R² Is Not Enough!" and opens by observing that validation often "seems to consist of nothing more than quoting the R² statistic from the fit," before stating that "unfortunately, a high R² value does not guarantee that the model fits the data well." Its remedy is to look at the residuals, the pattern of what the model missed, because a model can explain most of the variation while being systematically wrong in a way a single summary number cannot show.

In fund analysis the practical consequence is about the benchmark, not the fund. R-squared is always computed against something, so a low reading has two possible causes and they call for opposite conclusions. Either the fund is genuinely doing something the index does not, or the index is the wrong index. Nothing in the statistic distinguishes them. A domestic large-company fund showing a low R-squared against a domestic large-company index is interesting; the same reading against an unrelated index is not information about the fund at all.

Its most useful job is to qualify the other statistics on the same page. Beta and alpha are both outputs of a regression of the fund's returns on the benchmark's, and R-squared reports how much of the fund's movement that regression actually accounts for. When it is low, the fitted relationship describes only a small part of what happened, so the beta is a poorly determined description and the alpha computed from it inherits that weakness. Neither figure is wrong, but neither is carrying much information. Read in that order, R-squared stops being a fourth statistic competing for attention and becomes the one that tells you how seriously to take the first three.

Two limits that follow from the definition rather than from finance. It cannot indicate direction or size: a fund moving twice as far as its benchmark and one moving half as far can post the same R-squared, because that is beta's job. And in a model with several explanatory variables R-squared cannot fall when a variable is added, whether or not the variable is meaningful, so it cannot be used to choose between models of different sizes. Fund reporting is usually a single-variable case where the second point does not bite, but it is the reason statisticians treat the raw figure with suspicion. The statistic has also been studied as a fund-level variable in its own right rather than only as a diagnostic: Amihud and Goyenko's 2013 article in the Review of Financial Studies takes a mutual fund's R-squared against its benchmark as its subject.

The statistic comes from fitting a line: the fund's returns are regressed on the benchmark's over a series of periods, and R-squared reports the fraction of the fund's return variance that the fitted line accounts for. Equivalently, it is one minus the unexplained variance divided by the total variance, which is the form easiest to check.

A hypothetical example. Over the measurement period a fund's monthly returns have an annualized standard deviation of 15.0 percent. After the part explained by its benchmark is stripped out, the leftover, unexplained part of its returns has a standard deviation of 6.0 percent.

Convert both to variances by squaring: total variance corresponds to 15.0² = 225, unexplained to 6.0² = 36. The unexplained share is 36 ÷ 225 = 0.16, so R-squared is 1 − 0.16 = 0.84. The benchmark accounts for 84 percent of this fund's return variance.

Now read the same figures the way they are usually misread. The unexplained piece is 6.0 against 15.0, which is 40 percent as large in the units on the page. It contributes only 16 percent of the variance. Squaring is what collapses it, and it is why a fund can look substantially independent of its benchmark in volatility terms while posting a high R-squared. Reverse the arithmetic to see the same effect from the other side: a fund with an R-squared of 0.36 has 1 − 0.36 = 0.64 of its variance unexplained, and the square root of 0.64 is 0.80, so four-fifths of its volatility has nothing to do with the benchmark.

Pros

• Answers a specific, checkable question about how much of one series' movement another accounts for.
• Bounded between 0 and 1, so readings are comparable across funds in a way unbounded statistics are not.
• Qualifies beta, alpha and any other regression output on the same report, which no other single figure on a fact sheet does.
• Requires no assumption about what a good return is, so it cannot be gamed by choosing a flattering period the way a return figure can.

Cons

• Says nothing about whether returns were good, or even about direction and magnitude of the relationship, which is beta's job.
• A high reading is not evidence the model or the benchmark comparison is correct, which is the point NIST puts in a section heading.
• Depends entirely on the benchmark chosen, and a low reading cannot distinguish a genuinely different portfolio from a badly chosen index.
• Measured in variance, so it is routinely over-read: an unexplained share that looks small as a variance is considerably larger as volatility.
• Backward-looking and period-dependent, so a reading computed over one window need not describe the next one.