## Abstract

The low-volatility anomaly is often attributed to limits to arbitrage, such as leverage, short-selling, and benchmark constraints. One would therefore expect hedge funds, which are typically not hindered by these constraints, to be the smart money that is able to benefit from the anomaly. This article finds that the return difference between low- and high-volatility stocks is indeed a highly significant explanatory factor for aggregate hedge fund returns, but with the opposite sign, that is, hedge funds tend to bet not against the low-volatility anomaly, rather than on it. This finding suggests that limits to arbitrage are not the key driver of the low-volatility anomaly and that concerns about low-volatility having become an “overcrowded” trade may be exaggerated. Another contribution of this study is that it identifies a new, highly significant explanatory factor for hedge fund returns.

There is a vast amount of evidence that low-volatility and low-beta stocks earn higher returns than predicted by the Capital Asset Pricing Model (CAPM). The relation between market beta and return was already observed to be flatter than predicted by the CAPM in early studies such as Black, Jensen, and Scholes (1972), Fama and MacBeth (1973), and Haugen and Heins (1975). Two decades later, Fama and French (1992) concluded that if one controls for size effects, market beta is unrelated to expected stock returns. Blitz and van Vliet (2007) find that during the subsequent decades the risk-return relation even turned negative, not just in the U.S. but also in international equity markets, and that this anomaly is stronger when stocks are selected on past volatility instead of market beta. In other words, low-volatility stocks have significantly lower risk yet higher returns than the market, while high-volatility stocks have significantly higher risk but lower returns than the market. This phenomenon is typically referred to as the low-volatility (or low-beta) anomaly, although it is just as much a high-volatility (or high-beta) anomaly. Additional confirmation is provided by recent studies such as Baker, Bradley, and Wurgler (2011), Baker and Haugen (2012), and Frazzini and Pedersen (2014).

Blitz, Falkenstein, and van Vliet (2014) provide an extensive overview of the explanations for this phenomenon that have been proposed in various streams of literature. One of the most popular explanations is that the anomaly results from limits to arbitrage, such as leverage, short-selling, and benchmark constraints:

• In the early days of the CAPM, Brennan (1971) and Black (1972) already showed that leverage (or borrowing) constraints may cause low-beta stocks to have higher returns than predicted by the model. Black (1993) argues that leverage restrictions have become stronger over time. Frazzini and Pedersen (2014) also attribute the anomaly to leverage constraints, showing that when funding constraints tighten, betas tend to be compressed toward 1, and the risk–return relation becomes flatter.

• Miller (1977) argued that short-selling constraints can cause high-risk stocks to become overpriced due to a phenomenon known as the “winner’s curse.” In a market with little or no short selling, the demand for a particular security will come from the minority who hold the most optimistic expectations about it. As divergence of opinion tends to increase with risk, high-risk stocks are more likely to be overpriced than low-risk stocks, because their owners will have the greatest bias.

• Blitz and van Vliet (2007) argue that benchmark constraints may explain the low-volatility anomaly, as benchmark-relative performance evaluation gives delegated portfolio managers the incentive to shun low-volatility stocks and focus on high-volatility stocks instead. Baker, Bradley, and Wurgler (2011) link the strengthening of the anomaly over time to the rise in institutional ownership. Falkenstein (2009) formally shows that when everyone is a benchmark-relative investor, the equilibrium relation between risk and return is no longer positive, but entirely flat.

Leverage, short-selling, and benchmark constraints may indeed prevent many investors from exploiting the low-volatility anomaly, but such limits to arbitrage are much less of a concern for hedge funds, as these funds tend to be characterized by an absolute return objective and ample flexibility to apply leverage and shorting. Based on the limits to arbitrage explanation, one would therefore expect hedge funds to be the smart money that actively takes advantage of the opportunity provided by low-volatility stocks. This article empirically tests this hypothesis by regressing aggregate hedge fund returns on the return difference between low- and high-volatility stocks.^{1}

The main finding is that the return difference between low- and high-volatility turns out to be among the strongest explanatory factors for aggregate hedge fund returns, but with a sign that is negative instead of positive. In other words, hedge funds seem to be betting against the low-volatility anomaly instead of trying to exploit it; that is, their return behavior is consistent with going long high-volatility stocks and short low-volatility stocks. This result contributes to the literature on hedge fund performance evaluation, as the return difference between low- and high-volatility stocks turns out to be a stronger explanatory factor for hedge fund returns than many previously documented factors.

The observed relation can be interpreted in multiple ways. One interpretation is that limits to arbitrage may not be the main explanation for the low-volatility anomaly after all, since reduced leverage, short-selling, and benchmark constraints apparently do not suffice for inducing investors to exploit the anomaly. If hedge funds are not arbitraging, but instead are contributing to the low-volatility anomaly, this also suggests that the popular concern that the anomaly is disappearing or becoming an “overcrowded” trade is exaggerated. The multi-trillion hedge fund industry appears to be on the other side of the low-volatility trade.

Other interpretations of the results are also possible though. For instance, it cannot be ruled out that hedge funds have been making an intentional bet against the low-volatility anomaly, because they saw reasons to expect a reversal of fortunes for high- versus low-volatility stocks. This explanation seems less plausible though, given that low-volatility stocks did not lose their attractiveness during the sample period. Some concerns about possible overcrowding of low-volatility stocks did arise in recent years, but the exposure of hedge funds against low-volatility is found to be structural and pretty much stable over the entire, much longer sample period.

Another alternative interpretation is that the low-minus-high volatility factor could be picking up an exposure to some other, not yet identified systematic factor, which happens to be correlated with the return differential between low- and high-volatility stocks. However, this also does not seem to be a very likely explanation, given the large number of control factors included in the analysis, and because the observed negative relation between hedge fund returns and low- versus high-volatility stock returns is so strong. Resolving this concern would require holdings data for each individual hedge fund.

## DATA

Hedge fund data is obtained from Thomson Reuters Datastream. The sample consists of hedge fund indexes from two leading providers of such data, Hedge Fund Research (HFR) and Credit Suisse (CS). HFR indexes are equally weighted, while CS indexes are asset weighted. The main focus in this article is on the aggregate indexes, which include hedge funds from all categories, but the analysis is also conducted for the various sub-category indexes that are available from these index providers. A priori, it seems more likely that hedge fund categories that mainly invest in individual stocks will exhibit an exposure toward the return difference between low- and high-volatility stocks, than hedge fund categories with a focus on other asset classes or investment instruments will show. All hedge fund returns are taken in excess of the risk-free return provided by Kenneth French.

When analyzing hedge fund returns, it is important to control for known explanatory factors. To this end, the following control factors are included in this study:

• The equity premium (the U.S. equity market excess return provided by Kenneth French), denoted by Mkt;

• The 1-month lagged equity premium, denoted by Mkt-1;

• The return difference between emerging and U.S. equity markets (MSCI Emerging Markets index return minus the U.S. equity market return), denoted by EM-US;

• The term premium (Barclays US Treasury index return minus the risk-free return), denoted by Term;

• The credit risk premium on investment grade corporate bonds (Barclays US Corporate Investment Grade index return minus Barclays US Treasury index return), denoted by IG-Tr;

• The credit risk premium on high yield corporate bonds (Barclays US Corporate High Yield index return minus Barclays US Treasury index return), denoted by HY-Tr;

• The return on time-series momentum strategies applied to commodities, equity indexes, bonds, and currencies, based on Moskowitz, Ooi, and Pedersen (2012),

^{2}denoted by TSmom;• The return on a variance risk premium strategy (the Risklab Variance Premium Trading index minus the risk-free return),

^{3}denoted by VRP;• The standard Fama-French size, value, and momentum factors,

^{4}denoted by SMB, HML, and WML respectively.

As the total number of control factors is quite large, a concern might be that the regressions are over-parameterized. Unreported tests, however, show that none of the conclusions in this study change if statistically insignificant factors are dropped from the regressions.^{5}

The standard Fama-French factors are augmented with a factor that captures the return difference between Low-Volatility and High-Volatility stocks (LV-HV). This LV-HV factor is self-constructed, by applying the standard Fama-French factor construction methodology to past 36-month volatility of total returns. This means that every month all stocks in the CRSP database are first classified as either large or small, using the NYSE median market capitalization as breakpoint, and next value-weighted portfolios consisting of the 30% lowest and 30% highest volatility stocks are created within each of these size groups. The LV-HV factor return is then calculated as the average return of the two low-volatility portfolios minus the average return of the two high-volatility portfolios over the subsequent month.

The data frequency is monthly, and all returns are total returns in U.S. dollars. The sample period is from January 2000 until December 2016, which is the longest period for which data for all the control factors is available. Although hedge fund index data is available with a longer history, the early data is known to be unreliable as it suffers from severe biases (survivorship bias, backfilling bias, voluntary reporting bias). Moreover, many investors were still unaware about the low-volatility anomaly during the 1990s, so it is unlikely that hedge funds would have been trying to exploit it back then. Results for the first half and second half of the sample period separately are examined in a robustness analysis.

### Main Results

Exhibit 1 shows the results of regressions of HFR hedge fund index returns on the set of explanatory factors over the 2000–2016 period. The first row contains results for the aggregate Fund Weighted Composite Index, which consists of all individual hedge funds, and the second row contains results for the aggregate Fund of Funds Composite Index, which consists of all funds of hedge funds. If hedge funds, on aggregate, were trying to systematically exploit the low-volatility anomaly, one would expect to find statistically significantly positive exposures toward the low-minus-high volatility factor. The exposures of the two indexes toward the LV-HV factor turn out to be statistically significant indeed, but the sign of these exposures is negative instead of positive. In other words, the return behavior of hedge funds is consistent with going long high-volatility stocks and short low-volatility stocks; that is, betting against the low-volatility anomaly. The statistical significance of this result is very strong, with *t*-statistics of −7.75 and −4.75 respectively.^{6} Compared to the other right-hand side variables, the LV-HV factor is clearly among the most significant explanatory factors for hedge fund returns. The other rows in Exhibit 1 show results for the five main sub-categories used by HFR. The estimated exposures toward the LV-HV factor are all negative, and statistically significantly so in four out of five instances. With a *t*-statistic of −7.75, the strongest exposure is found for the Equity Hedge Index, which consists of funds that predominantly invest in individual stocks.

Exhibit 2 shows the results for the CS hedge fund aggregate and sub-category indexes. For the aggregate Hedge Fund Index, the exposure toward the low-minus-high volatility factor is again highly significantly negative, with a *t*-statistic of −5.23. The exposure toward the LV-HV factor is also statistically significantly negative for seven out of the twelve sub-categories in the CS classification, and not significantly different from zero in the remaining five instances. Again, this makes it one of the most powerful explanatory factors for hedge fund index returns. The significance of the LV-HV factor appears to be related to whether the hedge funds in a certain category invest in individual stocks, for example, strong for categories such as Event Driven and Long/Short Equity, but weak for categories such as Fixed Income Arbitrage and Managed Futures. Interestingly, the LV-HV factor is insignificant for the Equity Market Neutral category, which might be because the market neutrality requirement prevents these funds from either going long low-beta and short high-beta stocks, or the other way around.^{7}

The results for the control factors included in the regressions in Exhibits 1 and 2 are mixed. In some cases they are highly significant, and these exposures are generally in line with intuition and the existing literature. For instance, the aggregate indexes and many of the sub-category indexes exhibit a highly significant exposure toward the current and lagged equity market factor; the Emerging Markets indexes exhibit a highly significant exposure toward the EM-US factor; the Fixed Income Arbitrage index exhibits the strongest exposure toward the two credit risk premium factors; and the Managed Futures index exhibits a huge exposure toward the time-series momentum factor. On the other hand, many control factors are typically insignificant for any given hedge fund index. Unreported tests show that the insignificant factors remain insignificant if the LV-HV factor is dropped from the regression, which implies that it is not the addition of this new factor that renders them insignificant. Another interesting observation is that many hedge fund indexes exhibit significant exposures toward the momentum (WML) and value (HML) factors, while their exposures toward the size (SMB) factor are typically negative or insignificant. Several hedge fund indexes also load on the variance risk premium factor. Altogether, the picture that emerges from the regressions is that the main systematic exposures provided by hedge funds are toward classic betas (the equity risk premium, the emerging versus developed equity return, and the default risk premium), various forms of momentum, and a bet against the low-volatility anomaly.

Exhibit 3 shows results for the three aggregate indexes over the first and second half of the sample period separately. The coefficient for the low-minus-high-volatility factor is negative and statistically significantly so in all instances. On balance, the exposure toward the LV-HV factor appears to be about equally strong in both periods. These results indicate that the bet of hedge funds against the low-volatility anomaly has been persistent and robust over time.

### Robustness

The low-volatility anomaly is closely related to the low-beta anomaly, so one can wonder whether the exposure of hedge funds is predominantly toward high-volatility stocks or high-beta stocks. To this end the regressions in Exhibits 1 and 2 are repeated using a Fama-French style low-minus-high 36-month market beta factor (LB-HB), instead of the Fama-French style low-minus-high 36-month volatility factor. The first column of Exhibit 4 shows the previously estimated *t*-statistics for the LV-HV factor, and the third column of Exhibit 4 shows the t-statistics when the LB-HB factor is used instead.^{8} Even though the LB-HB and LV-HV series exhibit a very high (0.95) correlation, the results for the beta factor turn out to be notably weaker. The t-statistics for the LB-HB factor are still generally negative, but typically smaller, and in several cases not even significant anymore at the 5% level. Also, the R-squared levels (not reported) are consistently lower, with a beta instead of a volatility factor. Thus, even though volatility and beta are closely related metrics, the data clearly shows that the exposure of hedge funds is specifically toward volatility, rather than beta.

By construction, the LV-HV and LB-HB factors have strong negative correlations with the equity market portfolio. Over this sample period, these correlations amount to −0.71 and −0.75, respectively. One might therefore be concerned about multi-collinearity distorting the regression results. A related concern is that the LV-HV and LB-HB factors are dominated by the high-risk legs; that is, the HV and HB components. In order to address these concerns, the analysis is repeated using volatility and beta factor definitions that are (very close to) market neutral, instead of the plain LV-HV or LB-HB factors. In the spirit of Frazzini and Pedersen (2014), these market-neutral factors will be denoted by betting against volatility, or BAV, and betting against beta, or BAB. The BAV and BAB factors are constructed by taking a structural 130% long position in the LV and LB portfolios (borrowing 30% against the risk-free rate) and a structural 70% short position in the HV and HB portfolios (with the remaining 30% invested in the risk-free asset). The second and fourth columns of Exhibit 4 show that regressions results based on these market-neutral factor definitions are fully consistent with the earlier results. The market-neutral BAV factor picks up the same relation as the plain LV-HV factor, with statistical significance levels that are also very similar, or even slightly higher, while the market-neutral BAB factor remains weaker, similar to the plain LB-HB factor. In sum, the results are robust to using market-neutral factor definitions.

Novy-Marx (2014) and Fama and French (2016) find that the low-volatility anomaly is related to the profitability effect. One can therefore also wonder if the exposure of hedge funds is really toward volatility, or more toward profitability. In the last column of Exhibit 5, the regressions are therefore repeated using the Fama and French (2015, 2016) profitability (RMW) factor, obtained from the data library of Kenneth French, instead of the LV-HV factor. The RMW factor has a correlation of 0.80 with the LV-HV factor over this sample period, which indicates that volatility and profitability are indeed related characteristics. The exposure of hedge funds toward the profitability factor also turns out to be negative in almost every instance, and statistically significantly so in many cases. However, the t-statistics are consistently lower compared to those for the LV-HV factor. R-squared levels (not reported) are also consistently lower with the profitability instead of the volatility factor. So although volatility and profitability appear to be related characteristics, the exposure of hedge funds is primarily toward volatility.

Are hedge funds betting against low-volatility stocks and betting on high-volatility stocks at the same time, or is the significant exposure toward the LV-HV factor mainly coming from one leg of the factor? In order to answer this question the regressions are repeated using separate low-volatility minus medium-volatility (LV-MV) and high-volatility minus medium-volatility (HV-MV) factors. These factors are constructed in a similar fashion as the LV-HV factor, using the 40% stocks with medium volatility levels next to the top and bottom 30% volatility stocks.

The first columns in Exhibit 5 show a significant bet against low-volatility stocks as well as a significant bet on high-volatility stocks if the factors are considered separately, with the latter bet appearing to be the stronger one. A similar picture emerges when the two factors are included together, as reported in the last columns of Exhibit 5: hedge funds appear to be betting against low volatility, and, at the same, betting on high volatility, with the latter bet appearing to be the stronger one.

## IMPLICATIONS

The regression results strongly suggest that instead of trying to exploit the low-volatility anomaly, hedge funds seem to be betting against it; that is, going long high-volatility stocks and short low-volatility stocks. Of course, this is a statistical inference, which does not rule out alternative explanations. In particular, the regressions might be picking up an exposure to another, not yet identified systematic factor, which happens to correlate with the return differential between low- and high-volatility stocks. However, the observed exposures toward the LV-HV factor are highly significant, also in the presence of a large number of control factors, so the most obvious explanation is that, on aggregate, hedge funds simply favor high-volatility stocks over low-volatility stocks. This has at least three important implications.

First, it argues against limits to arbitrage, such as leverage, short-selling, and benchmark constraints, being the main driver of the low-volatility anomaly. Hedge funds have the flexibility to use leverage and short-sell stocks, and they can focus on generating absolute returns, but apparently these conditions are not sufficient for exploiting the low-volatility anomaly. The strong preference of hedge funds for high-volatility stocks suggests that other explanations that have been proposed for the low-volatility anomaly may be more important. For instance, Baker and Haugen (2012) argue that portfolio managers are willing to overpay for high-volatility stocks in order to maximize the expected value of their option-like payoff contracts. This optionality is clearly present in hedge fund fee structures, which are characterized by a base fee plus a performance fee, which kicks in when the realized return exceeds a certain threshold, so that might explain why hedge funds seem to prefer high-volatility stocks even when not hampered by limits to arbitrage. However, it also cannot be ruled out that hedge funds have been betting against the low-volatility anomaly for the simple reason that they expected a reversal of fortunes in the performance of high-volatility versus low-volatility stocks, based on their analysis of prevailing market conditions. This does not seem very plausible though, given that throughout all or most of the sample period the low-volatility anomaly remained a largely unexploited phenomenon, with a similar performance potential (and realization) as in earlier decades.

The finding that hedge funds are betting against the low-volatility anomaly suggests that the concern of some investors that the anomaly has been largely arbitraged away already, or that it may have turned into an ‘overcrowded’ trade, is exaggerated. These concerns are driven by rising valuations of low-volatility strategies and rising assets under management in both active and passive low-volatility strategies. However, the finding that the multi-trillion hedge fund industry is, on balance, strongly betting on high-volatility stocks suggests that this large group of investors is on the other side of the low-volatility trade. Such a vast amount of smart money betting against low-volatility makes it seem unlikely that the anomaly is close to having been arbitraged away or has turned into an overcrowded trade. Again, however, the opposite interpretation cannot be ruled out either: it might also be that hedge funds have been betting against the low-volatility anomaly intentionally, for the very reason that they considered it to be an overcrowded trade already. This does not seem very likely though, since concerns about overcrowding only started to arise in recent years, while the bet of hedge funds against the low-volatility anomaly is a structural phenomenon throughout the entire sample period.

The final contribution of this study is that it identifies a new factor with strong explanatory power for hedge fund returns, which adds to the existing literature on hedge fund performance evaluation. In fact, the return difference between low- and high-volatility stocks turns out to be a stronger explanatory factor for hedge fund returns than many previously documented factors.

## ACKNOWLEDGMENTS

The author thanks Pim van Vliet, Thijs Markwat, Matthias Hanauer, and Laurens Swinkels for various useful comments and suggestions.

## ENDNOTES

↵

^{1}In other words, these tests provide insight into the collective behavior of hedge funds. Individual hedge funds may, of course, exhibit very different behavior, given the high degree of idiosyncrasy in hedge fund returns.↵

^{2}Data from https://www.aqr.com/library/data-sets/time-series-momentum-factors-monthly; the website of AQR.↵

^{3}The Risklab Variance Premium Trading Index systematically sells variance swaps on EuroStoxx50 and S&P500 to provide exposure to the differential between (forward looking) implied equity index volatility and subsequently realized volatility. It is designed to generate positive performance during periods where implied volatility levels are greater than their subsequently realized levels, and is motivated by the historical observation that implied volatility tends to overestimate future realized volatility, resulting in a negative risk premium of volatility or variance.↵

^{4}Data from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html; the data library of Kenneth French.↵

^{5}Unreported tests also show that the Fung and Hsieh (2001) factors, which are based on look-back straddle options, provide little or no additional explanatory power. In most cases the estimated exposures toward these factors are statistically insignificant, have the wrong sign, or are heavily influenced by one or two outliers.↵

^{6}This result is consistent with Asness et al. (2015), who also examine the exposure of hedge fund indexes toward various factor premiums, and briefly note that hedge funds exhibit a significant exposure toward high-risk stocks.↵

^{7}The CS Equity Market Neutral index returns contain a huge outlier of −39% in November 2008, due to the Madoff bankruptcy, which is replaced by zero in order to prevent spurious regression results.↵

^{8}Because the exposures toward the other right-hand side factors are not materially affected by replacing the LV-HV factor with related metrics, these other exposures are not reported in this section for the sake of brevity.

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