# The Beta/Correlation Language Barrier

January 18, 2015 2 Comments

*This post originally appeared in my column in Risk Magazine in October, 2014.*

Institutional investors tend to use a different vocabulary when speaking about risk than hedge fund managers. Institutional investors often talk about beta and benchmarks. Hedge funds talk about correlations and absolute returns. Betas and correlations are closely related to each other, but not the way most people think.

Institutional investors as a group first learned about quantifying risk when they were introduced to the concept of beta, probably by a long-only equity manager. The conventional interpretation, they were told a long time ago, is that beta measures the relative risk of their portfolio to their benchmark: if beta is greater than 1, the fund has more volatility than the benchmark. If it’s less than one, the fund has less volatility than the benchmark. This simplification has merit for many long-only funds, but it has led to misinterpretations and a false sense of security when alternatives are included. The issue is compounded by the introduction of a related measure – correlation – from hedge funds’ ubiquitous claims of “uncorrelated returns.” The relationship between beta and correlation is not well understood by many managers and, when misinterpreted, can lead to poor investment decisions.

Both beta and correlation provide a measure of a fund’s relationship to a benchmark or index, but they measure very different aspects of that relationship. Beta measures the strength of the linear relationship between the returns of the fund and the returns of the benchmark/index. Correlation measures to what extend the two are actually related to each other: it can provide the justification for using the beta measure (high correlation) or it can rule out the assumption that makes beta meaningful (low correlation). In other words, correlation tells us whether or not to trust beta.

Beta gives an estimate of what the fund’s performance would be for different returns of the benchmark/index. For example, the usual interpretation of a beta of 0.7 is that when the benchmark/index is up 1%, the fund would be up 70 basis points. A fund with a beta of 1.2 would be up 120 basis points for every 1% rise in the index. Another common interpretation of beta is that it measures relative risk of a fund and its benchmark: the volatility of a fund with a beta of 0.70 is 70% the volatility of the benchmark. Similarly, a fund whose beta is 1.2 has a volatility 120% the volatility of the benchmark. But these interpretations come with a very strong caveat that is all too often ignored: they are only correct when correlation is very close to one.

Mathematically, beta is the slope of the (linear) regression “best fit” line between the fund’s returns and the benchmark’s returns. Figure 1 shows such a linear relationship between a fund and a related benchmark, with a beta of 0.7. Note that the points in this plot line up nicely – in other words, there really is a linear relationship between this fund and this benchmark, and it’s easy to draw a “best fit line” through them. Numerically that is corroborated by the correlation and R^{2} values, both of which are well over 0.9.

But what would line would best fit if the plot looked more like Figure 2A, where the relationship could best be described as “oval-ish”? What beta should be quoted for the fund? Figure 2B shows two possible best-fit lines, **each of which looks as good as the other.** Line A in Figure 2b has a beta (slope) of 1.8 and Line B has a beta of just 0.2. The average of them, 1.0, might be the best guess beta for this situation, but it’s quite clear that there’s a range of acceptable answers, from 0.2 to 1.8. **The fact that such a wide range of betas match the data equally well should reinforce the conclusion that use of a single beta is a bad idea for this fund. **When many answers fit the data, as in this case, the correct conclusion is not that you can pick any of them, but rather, that *all* of them are equally wrong.

Correlation now enters the story and becomes very relevant and important. Ranging from -1 to 1, correlation measures the *extent to which* *there actually is* a relationship between the fund and the benchmark/index. For Figure 1, the correlation is 0.97. In Figure 2, the correlation is only 0.62. Visual, it’s easy to see that Fund 1 has a relationship with its benchmark while Fund 2 is, at most, loosely related to its benchmark. The more circular such a plot looks, the less the fund is related to its benchmark and the farther the correlation moves from one. Said differently, the farther the correlation is from one, the weaker the relationship and therefore the less appropriate it is to speak of “a beta” between the fund and the benchmark. Correlations less than about 0.8 are suspect.

Because beta is a measure of an assumed linear relationship, the use of a beta relies on there actually *being* a linear relationship between the fund and its benchmark. In cases with low correlation, there is no relationship between them, and the concept of a beta becomes meaningless. Correlations less than about 0.8 are typically interpreted as too weak to support the use of beta. But even for such low-correlation funds, it is still possible to calculate a “best” fit line, even if that fit isn’t at all good. **‘Best’ really does not imply ‘good.’**

When measuring beta, most systems will quote an “R-squared” as a measure of the quality of fit. R-squared is nothing more than the square of the correlation, and R-squared of 0.64 corresponds to correlations of 0.8. When a fund has a low correlation to the benchmark, the associated beta (although it can be calculated) is meaningless. But fund managers often ignore the R-squared (who remembers what that really means anyway?!?), and they use beta to make allocations, rebalance and even to make hedging decisions. If a fund has a low correlation to the benchmark (less than about 0.8), its beta is not to be trusted or used. Using beta when correlation is low is simply wrong. It’s like a sailor using an untrue compass that he doesn’t realize is broken, and it can lead to bad portfolio decisions.

The problem hits homes when institution investors insist on using beta even when investing in hedge funds, which tend to have low correlations to the benchmark. For the most hedge funds themselves, beta is quite literally a meaningless concept: it neither describes the hedge fund’s risk nor the hedge fund’s likely response to changes in the benchmark. Indeed, the whole point of hedge funds is to *not* be related to *any* benchmark. “Uncorrelated returns” means that the correlation, and therefore R-squared, are both extremely low.

When correlation is near one, beta is a useful measure: it provides an estimate of how risky the fund is relative to its benchmark and how the fund is likely to respond to changes in the benchmark. When correlation is not near one (less than about 0.8), beta tells you nothing. The danger is that beta can still be calculated and uninformed managers may be tempted to use it because “it’s better than nothing.” But in this situation, it provides no information and should not be used for any purpose.

Conversations between institutional investors and hedge fund managers can be challenging enough, and the historical use of different measures often compounds the confusion, especially about risk. A deeper understanding of how institutional investors use beta, and how correlations affect those measures, can lead to better risk decisions.

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I think you need to read a recent academic article. Its claims have also been empirically tested, but that isn’t out yet. http://swer.wtamu.edu/sites/default/files/Data/swer%2014%20Harris.pdf