Realised volatility and industry momentum returns

Motivated by the importance of industry volatility and the profitability of industry momentum strategy, this study investigates the relationship between realised volatility and industry momentum returns. The analysis uses daily return data for 48 US industries from July 1969 to June 2021 to calculate realised volatility and to gauge the raw return effect on short- and medium-horizon double-sort momentum-trading strategies. The findings show that past volatility positively relates to industry momentum and that this relationship is stronger after controlling for common risk factors (market, size, value, investment, and profitability). Decomposing the realised total volatility into idiosyncratic and systematic components, this study reveals that both decomposed components are positively related to industry momentum returns. The findings are robust to alternative measures of volatility.


Introduction
M omentum is a trading strategy whereby investors generate abnormal profits by buying (selling) stocks with the best (worst) returns over preceding months. The empirical evidence to date suggests that momentum is pervasive, with studies proving its worth to investment practitioners across markets, time, and asset classes (Li et al., 2010;Asness et al., 2013;Lin et al., 2016;Chen and Lu, 2017;Gao et al., 2018;Grobys et al., 2018;Zaremba, 2018). In doing so, momentum challenges finance theory in the form of the efficient market hypothesis and conventional asset pricing models, and is therefore a financial market phenomenon of acute academic interest. Interestingly, there are subtle variations in momentum-trading strategies, one being that investors could further improve firm-level momentum returns using additionally volatile stocks, and this is also addressed in the literature (Fuertesa et al., 2009;Jacobs et al., 2015;Wang and Xu, 2015). However, few studies link volatility to industry-level momentum, with one exception being  reporting that oil return volatility predicts industry momentum in China.
We find this deficiency of more than passing interest. To start, although many investors prefer to hold individual stocks (Campbell et al., 2001), increasing numbers invest in industry funds, as these may also perform well (O'Neal, 2000;Berk and Van Binsbergen, 2015). Industry momentum strategies are particularly popular and profitable when trading sector funds or individual stocks across industries (Moskowitz and Grinblatt, 1999;Misirli, 2018). Volatility has also a long history in explaining stock returns or firm-level momentum. Barroso and Santa-Clara (2015), for example, report that momentum portfolios exhibit higher volatility than other trading portfolios, such as size and value mimicking portfolios. Interestingly, Campbell et al. (2001) find that individual stock volatility is closely related to industry or market volatility. More importantly, the role of industry volatility is even stronger than that of market volatility (Black et al., 2002;Morana and Sawkins, 2004;Ferreira and Gama, 2005). Noting that industry volatility is closely related to individual stock volatility, Badreddine and Clark (2021) conclude that industry-adjusted volatility is positively related to firm-level momentum returns in the UK.
We also know that stocks within the same industry have similar asset profiles and capital structures, employ comparable production technologies, and operate in like economic, financial, and regulatory environments. These can also translate to similar return distributions and comovements (Parsons et al., 2020;Chen et al., 2021Chen et al., , 2022. Compared with volatility calculated using individual data, industry volatility would then capture risks common to all stocks in the same industry (Wu and Mazouz, 2016). These undiversified industry portfolios typically earn greater gains and exhibit higher volatility risk (particularly idiosyncratic risk) than other well-diversified portfolios (Badreddine and Clark, 2021). Therefore, volatility may serve at the expense of industry momentum gains.
Motivated by the popularity of industry momentum strategy and the significant role of industry volatility, we construct sequential double sorts on 48 US industries following Amaya et al. (2015), Liu et al. (2016), Badreddine and Clark (2021), and Chen et al. (2022). We find that industry momentum returns increase with the level of realised volatility over multiple trading horizons. This relationship is robust to controlling for common risk factors, such as market, size, value, profitability, and investment, and to alternative measurements of volatility. Decomposing the realised total volatility into its idiosyncratic and systematic components, we find that both components are positively related to industry momentum returns. The volatilityadjusted industry momentum strategy following Barroso and Santa-Clara (2015) is also more profitable than the unadjusted strategy, suggesting that momentum gains are compensation for bearing high-volatility risk.
Our study contributes to the literature on three counts. First, while many studies identify a close relationship between volatility and momentum returns at the firm level (Fuertesa et al., 2009;Jacobs et al., 2015;Wang and Xu, 2015), there is less consideration of industry momentum. However, sector funds are increasingly common in the stock market (Kacperczyk et al., 2005) and industry momentum strategies exhibit strong profitability when trading industry portfolios or sector funds (Moskowitz and Grinblatt, 1999;Misirli, 2018). Although many firm-level studies have used volatility to explain momentum for individual stocks, few have focused on industry momentum. Our study therefore extends the existing firm-level results to investigate whether industry momentum returns could improve by using additionally volatile industry portfolios. We also use the method of Barroso and Santa-Clara (2015) to construct volatility-adjusted industry momentum, providing a more profitable momentum strategy with a lower crash risk for investors to secure higher gains. Such volatilityadjusted strategy deepens our understanding about the profitability of industry momentum.
Second, industry volatility plays a significant role in explaining firm and market level volatility (Black et al., 2002;Morana and Sawkins, 2004;Badreddine and Clark, 2021). Although the recent work by Badreddine and Clark (2021) uses industry-adjusted volatility to explain individual momentum returns in the UK, no study to our best knowledge has investigated this for industry momentum. As discussed, stock returns in the same industry display comovements (Parsons et al., 2020;Chen et al., 2021Chen et al., , 2022. Industry portfolios with higher gains also exhibit higher risk than other well-diversified trading portfolios (Badreddine and Clark, 2021). Incorporating industry volatility into industry momentum particularly fits the features of our sample data, thereby contributing majorly to the extant literature on industry volatility. Finally, many firm-level studies focus only on systematic volatility, being the diversified component of volatility (e.g., Stivers and Sun, 2010;Wang and Xu, 2015), as they believe that unsystematic risks should, at least theoretically, be diversified away in an efficient market. However, markets are sometimes inefficient. For instance, Arena et al. (2008) and Liu et al. (2016) find that idiosyncratic volatility is closely related to momentum returns. Unlike most firm-level studies, the undiversified industry portfolios in our analysis are better test assets than individual stocks or other diversified portfolios in investigating idiosyncratic risk. Applying sorting strategies to the decomposed volatility, we find that both systematic and idiosyncratic volatility are closely related to industry momentum returns. This significant role of idiosyncratic component also well fits the features of our sample data. Our study thus contributes to the literature by providing a more suitable measurement for volatility, further deepening our understanding of the role of volatility in industry momentum through complementing other firm-level studies.
The rest of this study is organised as follows. Section 'Data' presents the data and Section 'Methods' specifies our methods, including the sequential double-sorting strategies, the measurement of volatility and of returns, common asset pricing models used to adjust returns, and the construction of volatility-adjusted industry momentum. Section 'Empirical results' discusses the empirical results. Section 'Robustness' provides robustness checks using an alternative measurement for volatility and Section 'Conclusion' concludes.

Data
Daily data on value-weighted industry returns of the 48 industry portfolios in the US stock market is sourced from the Kenneth French Data Library. This data has several advantages, including a long sample period to allow for better stability and unbiasedness of the estimates and widespread use to facilitate comparisons with existing work (Stivers and Sun, 2010;Chen et al., 2021Chen et al., , 2022. The sample period starts in July 1969 and ends in June 2021. Table 1 reports the summary statistics for the selected 48 industries, including the average monthly mean and excess returns (in percentages), and realised volatility (not the dispersion of returns from the mean, rather the distance of returns from zero), skewness, as well as kurtosis calculated on monthly returns over the sample period. Other than Coal, Real Estate, and Others, all industries exhibit positive excess returns.
To ascertain whether common asset pricing models [including the Capital Asset Pricing Model (CAPM) and the Fama-French three-and five-factor models] explain the relationship between realised volatility and momentum returns, we source common risk factors, including market, size, value, profitability, and investment factors, from the Kenneth French data library. If the relationship remains significant after controlling for these conventional factors, realised volatility may also be of use for predicting and enhancing industry momentum returns. This table provides summary statistics for the 48 industries in the US, reporting the monthly average mean and excess return (%) and the realised volatility, skewness, and kurtosis calculated on monthly industry returns over the sample period from July 1969 to June 2021.

Methods
Trading strategy. We follow previous studies (e.g., Amaya et al., 2015;Liu et al., 2016;Badreddine and Clark, 2021;Chen et al., 2022) and employ a double-sorting strategy to rank industries based on past volatility and then past returns. For each month t, we rank industries into three groups based on their past realised volatility over the E-month (E = 1, 3, 6, 9, 12, 24) estimation period. Within each volatility-based group, we rank industries into four groups based on their past estimationperiod mean returns. We then form a momentum portfolio based on the second ranking of the past mean returns and hold these positions over the following H-months (H = 1, 3, 6, 9, 12, 24). We calculate equal-weighted raw and adjusted returns over these different holding periods. If momentum returns increase (decrease) as volatility increases, investors may enhance (reduce) the volatility level of their holding portfolios to generate higher industry momentum returns.
Volatility. Following Chen et al. (2021), we transform the raw industry returns into log returns to reflect the dollar value of the returns generated by investors over multiple investment horizons: where x i,t is the daily raw return in percentage as provided in the French Data Library and r i,t is the log return of industry i at day t.
Following Amaya et al. (2015), we use realised volatility to measure the variance of past returns over the estimation period as follows: where r i,t is the daily log return for each industry i on each trading day t and N is the number of observations over an estimation period T. The estimation-period realised volatility is the standard deviation of the sum-squared daily returns. The main justification for the choice of realised volatility is that it outperforms any alternatives. For example, model-dependent methods, such as autoregressive conditional heteroskedasticity and stochastic volatility models, require many parameter assumptions, whereas realised volatility is a model-free method that reduces estimation errors (Yin et al., 2016), save when the number of observations within the estimation period is very small. In our analysis, this is for the one-month estimation period only, as the other estimation periods exceed three months and contain more than 60 observations. Although unsystematic risks theoretically should be diversified away in an efficient market, markets are sometimes inefficient. More importantly, stock returns in the same industry show comovements, so industry portfolios are normally undiversified (Parsons et al., 2020;Chen et al., 2021Chen et al., , 2022. We therefore decompose the realised total volatility into idiosyncratic and systematic volatility to further examine which component better drives the relationship between volatility and industry momentum returns. Following Bali and Cakici (2008), we apply the CAPM over each estimation period as follows: where R i,t − R f,t is the excess return of industry i, R m,t − R f,t is the market premium, and β i is the sensitivity to the market risk. ε i,t is an error term that measures idiosyncratic return. For each estimation period, realised idiosyncratic volatility is the realised volatility of idiosyncratic return (ε i,t ). We then measure another component, systematic volatility, as total realised volatility minus realised idiosyncratic volatility.
Raw returns. We obtain the estimation or holding-period log returns by simply adding daily excess returns (daily returns exceeding the risk-free rate of return) together within that period: where r i;T is the average monthly excess return of industry i over the estimation (or holding) period, N is the number of trading days in that trading period, and T is the number of months over the estimation or holding period.
Adjusted returns. Following Conrad et al. (2013), Amaya et al. (2015), and Chen et al. (2021), we adjust the raw momentum returns for common risk factors. We commence with the CAPM: where R i,T − R f,T here is the holding-period monthly excess return of the industry momentum portfolio. Given the CAPM's restrictive assumptions and traditional failure to explain returns, French (1992, 1993) construct their three-factor model by adding size and value factors, as both empirically closely relate to returns: where SMB T is the size mimicking portfolio return, and HML T is the value mimicking portfolio return.
To even better price expected returns, Fama and French (2015) extend the three-factor model to a five-factor model by adding profitability and investment factors. This accords well with the dividend discount model in that both help determine dividends: where RMW T is the profitability mimicking portfolio return, and CMA T is the investment mimicking portfolio return.
In brief, if we obtain significant abnormal returns (α i ) with these models, common risk factors do not fully capture the relationship between realised volatility and momentum returns at the industry level.
Volatility-adjusted industry momentum. As momentum gains are related to volatility, Barroso and Santa-Clara (2015) adjust firm-level momentum returns using realised volatility and find higher gains for their measure of risk-managed momentum. We follow their method to construct volatility-adjusted industry momentum, scaling the industry momentum portfolio by its realised volatility to obtain constant volatility over time.
We rank the industries based on their past returns into six groups to construct an industry momentum portfolio of buying the highest return group and selling the lowest return group. Following Barroso and Santa-Clara (2015), the ranking is based on the returns from month t -12 to t -2. We then scale the industry momentum return using realised volatility as follows: where r rml,t is the unscaled excess return of industry momentum; d RVOL t is a realised volatility of momentum calculated at each month t by using daily returns over the past 6 months. RVOL target is a target constant volatility around 13%, measured as the annualised volatility, and this target volatility is close to that used for individual momentum (12%) in Barroso and Santa-Clara (2015). If volatility-adjusted industry momentum excess return (r * wml; t ) is higher than the unscaled return, managing realised volatility can enhance industry momentum gains.

Empirical results
Realised total volatility and industry momentum returns. To test the relationship between realised volatility and industry momentum, we sort the industries into three groups using the terciles of realised volatility over the estimation periods (first or highest, second, and third or lowest terciles), and then construct momentum strategies within each volatility group. If there is a significant relationship between volatility and raw industry momentum returns and after controlling for common risk factors, volatility may explain these returns. This suggests that investors may enhance momentum returns by observing the pattern of past volatility. Table 2 reports the relationship between the realised total volatility and industry momentum returns. Overall, industry momentum returns increase with the level of volatility over most tested trading periods, except for the 1/1 (1-month estimation and 1-month holding) trading strategy. Most raw returns of the zero-cost portfolios with high volatility are significant, except those of the 1/3, 24/1, and 24/24 trading horizons. For instance, the high-volatility portfolio of the 12/1 trading strategy generates a 1.004% momentum return per month, which is significant at the 5% level. Sharpe ratios adjusting for standard deviation (Sharpe, 1998) also show similar patterns: Sharpe ratios of the momentum portfolios under the high-volatility tercile are higher than those under the low-volatility tercile over most trading horizons, except for the 1/1 and 24/1 trading strategies. Those results indicate a positive relationship between realised total volatility and industry momentum returns. Table 3 presents the momentum returns adjusted for the common risk factors. Consistent with the raw results shown in Table 2, adjusted momentum returns of the high-volatility groups are higher than those of the low-volatility groups over most trading horizons. Most adjusted returns of momentum portfolios with high volatility are significant at the 5% level, except for the returns adjusted by the CAPM of the 12/24 and 24/24 trading strategies. Some even display very large t-statistics, such as those of the 12/12, 24/12, and 24/24 trading strategies. Those riskadjusted results indicate that common risk factors cannot explain the positive relationship between realised total volatility and industry momentum returns, and therefore, additionally volatile industries could enhance industry momentum returns.
Table 3 also shows that adjusted momentum returns are even stronger than raw results reported in Table 2. The greater sensitivity to the common risks of high-volatility loser industries may explain the stronger positive relationship between volatility and momentum returns after controlling for common risk factors shown in Table 3 (Rouwenhorst, 1998;Jegadeesh and Titman 1993;Arena et al., 2008). The returns of high-volatility losers may be lower after controlling for common risk factors if they are sensitive to risk factors, resulting in higher momentum returns of high-volatility groups. These results indicate that momentum returns compensate for bearing high-volatility risk. Investors may then profitably select high-volatility industries when constructing momentum-trading strategies to secure larger abnormal returns.
Realised decomposed volatility and industry momentum returns. Industry portfolios are normally undiversified, so we then decompose the realised total volatility into idiosyncratic and systematic volatility to examine which component best drives the relationship between volatility and industry momentum returns. Table 4 provides the Sharpe ratios and raw/adjusted returns for momentum portfolios under each idiosyncratic volatility group over selected trading horizons. The results for additional trading horizons are unreported but available upon request. Consistent with the results for total volatility, Table 4 shows that both the raw and adjusted momentum returns of the high idiosyncratic volatility group are higher and more significant than those of the low-volatility group over most trading horizons, except for the 1/1 trading window. The Sharpe ratios also yield similar patterns, except for the 1/1 and 1/3 trading horizons. These results suggest that idiosyncratic volatility matters for industry momentum returns, even after controlling for common risk factors. Table 5 reports the results for systematic volatility over selected trading horizons. Once again, both the raw and adjusted momentum returns of the high systematic volatility group are higher and more significant than those of the low-volatility group over most trading horizons, except for those of the 24/24 trading strategy, which are insignificant. The Sharpe ratios also increase with the level of systematic volatility over most tested trading periods, except for the 1/1 and 1/3 trading strategy where the middle-ranked volatility groups earn the highest Sharpe ratios for momentum portfolios. These results indicate that systematic volatility is positively related to industry momentum returns, even after controlling for common risk factors.
As unsystematic risks theoretically should be diversified away in an efficient market, many firm-level studies focus only on systematic volatility (e.g., Stivers and Sun, 2010;Wang and Xu, 2015). However, markets are sometimes inefficient. For instance, Arena et al. (2008) and Liu et al. (2016) find that idiosyncratic volatility matters, and it is closely related to momentum returns. More importantly, stocks in the same industry exhibit comovements and similar characteristics, so industry portfolios are normally undiversified (Parsons et al., 2020;Chen et al., 2021Chen et al., , 2022. Therefore, the findings that both systematic and unsystematic volatility can enhance industry momentum returns fits the features of our sample data. The finding of this positive relationship between realised industry volatility and industry momentum returns is consistent with the results explaining firm-level momentum gains using volatility. For example, both total and idiosyncratic volatility are positively related to firm-level momentum formed on Islamic stocks (Narayan and Phan, 2017). Arena et al. (2008) find comparable results by focusing on idiosyncratic volatility for individual stock momentum. Measuring industry volatility, recent work by Badreddine and Clark (2021) suggests that momentum returns increase with the levels of industry-adjusted volatility, even after controlling for the varying levels of liquidity in the UK.
These studies suggest some potential explanations for the positive relationship. For instance, Zhang (2006) finds that volatile stocks underreact to news, and therefore, return spreads between past winners and losers will be wider. In addition, high volatility is associated with the limits to arbitrage (Cao and Han, 2016;Guidolin and Ricci, 2020). For instance, risk-averse investors are unwilling to engage in arbitrage with highvolatility stocks, leading to stronger price persistence and momentum effects (Arena et al., 2008;Liu et al., 2016).
Other explanations relate to the relationship between idiosyncratic volatility and momentum returns. From an investment perspective, for example, Arena et al. (2008) find that momentum is related to underreaction to firm-specific information. Similar arguments by Rachwalski and Wen (2016) also indicate that investors may underreact to unsystematic volatility risk, and therefore, the slow price correction mechanism will result in stronger momentum returns for securities with high unsystematic volatility. Chan (2003) also argues that stronger momentum Table 2 Realised total volatility and industry momentum returns.
Volatility-adjusted industry momentum. As industry momentum is related to volatility, we follow Barroso and Santa-Clara (2015) to adjust industry-level momentum returns using realised volatility. Table 6 compares the unscaled industry momentum with the volatility-adjusted results. Consistent with firm-level results in Barroso and Santa-Clara (2015), Table 6 shows that the volatility-adjusted industry momentum return is much higher with a larger t-statistic than the unscaled return. Volatility adjustment also reduces left skewness from −0.254 to -0.015 and kurtosis from 3.036 to 0.546. These results indicate that adjusting realised volatility enhances industry momentum gains and reduces the crash risk. The results are also in line with our portfolio sorting results discussed previously where momentum return represents a compensation for bearing highvolatility risk.

Robustness
Given realised volatility assumes a zero mean, drift-adjusted realised volatility is an alternative that adjusts daily returns by the mean return (Amaya et al., 2015): where r i,t is the daily log excess return for each industry i in each trading day t and N is the number of observations over an estimation period T. Table 7 provides the results of the relationship between driftadjusted volatility and industry momentum returns over   This table reports Sharpe ratios (S.R.), log excess returns (H-L) in percentile (%), and adjusted returns (α c , α 3 , α 5 ) of the momentum portfolio under drift-adjusted volatility terciles (Low, Middle, and High) calculated on industry returns over selected estimation and holding periods (E/H). The t-statistics are in parentheses under returns. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively.
selected trading horizons. The results of drift-adjusted volatility are broadly like those obtained with realised volatility in Tables  2 and 3. Most momentum returns for the high-volatility group exceed those for the low-volatility groups, except for the 1/1 trading strategy. The strongest momentum effect is over the 9/9 trading strategy with a 0.797% momentum return for the high drift-adjusted volatility group.

Conclusion
Given the demonstrated interest in momentum-trading strategies, it is of importance to gauge whether realised volatility can enhance the performance of these strategies. Extant studies consider only firm-level relationships, despite strong investor interest in industry-level findings. Using a comparable approach, we find that selecting industries based on their realised volatility (and its decomposed idiosyncratic and systematic volatility) have positive and economically significant impacts on raw and riskadjusted abnormal returns when employing momentum strategies over 1-24 months estimation and holding periods. The findings are robust to the use of the drift-adjusted realised volatility. The volatility-adjusted industry momentum strategy is also more profitable than the unadjusted results, suggesting that industry momentum gains are compensations for bearing highvolatility risk.

Data availability
The datasets analysed during the current study are available in the Kenneth French Data Library: https://mba.tuck.dartmouth.edu/ pages/faculty/ken.french/data_library.html.