Twin Momentum: Fundamental Trends Matter

Transcription

1 Twin Momentum: Fundamental Trends Matter Abstract Using the time-series trends of a collection of firms major fundamentals, we find that fundamental trends matter. A fundamental momentum strategy that goes long stocks in the top quintile in fundamental trends and short stocks in the bottom quintile earns a monthly average return of 88 bps, and is comparable with price momentum. Combining price momentum and fundamental momentum yields a twin momentum, which has an average return more than the sum of price momentum and fundamental momentum. Twin momentum cannot be spanned by extant risk factors, nor can it be explained by short-sale impediments and investor sentiment. Keywords: Price momentum; Fundamental momentum; Twin momentum; Sticky expectation JEL Classification: G12, G14

2 1 Introduction Price momentum that simultaneously buys past winers and sells past losers is one of the most persistent anomalies in finance. Indeed, since the seminal work of Jegadeesh and Titman (1993), its validity continues to hold over time and across asset classes, and has stimulated a large number of theories for understanding its sources. 1 Since both fundamental and price information are used in making investment decisions in the real world (see, e.g., Schwager, 1989), it is not surprising that there are many studies on the relationship between firm fundamentals and asset returns (e.g., Haugen and Baker, 1996, Fama and French, 1992, and Novy-Marx, 2013). However, little attention has been paid to the trends of fundamentals except studies on earnings momentum (e.g., Chan, Jegadeesh, and Lakonishok, 1996; Novy-Marx, 2015). Existing studies typically find that fundamentals are less profitable than price momentum In this paper, we show that there is fundamental momentum in the stock market that can earn an abnormal return as high as price momentum. Particularly, based on seven major fundamental variables (return on equity, return on assets, earnings per share, accrual-based operating profitability, cash-based operating profitability, gross profitability, and net payout ratio) and their time-series trends, we construct a measure of fundamental implied return (FIR hereafter), and show that it has strong forecasting power for future returns. 2 Similar to price momentum, fundamental momentum is constructed by buying stocks in the top FIR quintile and selling stocks in the bottom FIR quintile. The spread earns an average return of 88 basis points (bps) per month over the sample period from April 1976 to September 2015, which is comparable to price momentum (93 bps per month) and cannot be explained by existing factor models. Fundamental momentum and price momentum have different return patterns with a low correlation of While price momentum reverts after four months since portfolio formation, fundamental momentum arrives at its maximum after 12 months and reverts thereafter. With a 5 5 independent double sort on past return (11-month cumulative return from month t 12 to month t 2) and FIR, fundamental momentum is significant within each past return quintile, and so does price momentum. Specifically, fundamental momentum delivers a monthly average return of 1.16%, 0.76%, and 0.61% in the lowest, middle, and highest past return quintiles, respectively, and price momentum earns a monthly average return of 1.55%, 0.91%, and 1% in the lowest, middle, and highest FIR quintiles, respectively. Moreover, the Fama-MacBeth results 1 The Google citation of Jegadeesh and Titman (1993) has exceeded 9,397 as of October Our choice of multiple fundamental variables is in the spirit of Graham and Dodd (1934), who propose to screen stocks based on 10 value- and quality-related characteristics (Lee, 2014). 1

3 of regressing one-month-ahead stock returns on past return and FIR, as well as other firm characteristics, suggest that both past return and FIR are complementary to each other in jointly predicting future stock returns. Fundamental momentum can be similarly explained by theories on price momentum. For example, the model of Hong and Stein (1999) may be applied to interpret that the better returns that follow higher fundamental trends are due to investors underreactions to fundamentals. However, fundamental momentum and price momentum do differ. According to Campbell, Giglio, Polk, and Turley (2017), if the profit is due to cash flow shocks, its effect is likely to be permanent in the sense that rational investors have no reasons to expect the stock price to rebound to previous levels. On the other hand, if the profit is due to discount rate shocks or variance shocks, its effect is likely to be temporary in the sense that rational investors expect the stock price to rebound to previous levels. We find that, although both price momentum and fundamental momentum have significant discount rate and variance betas, price momentum has greater exposures on both discount rate shocks and variance shocks, suggesting that price momentum has a stronger reversal and is more sensitive to investment environment. To exploit price information and fundamental information jointly, we construct a twin momentum strategy. We take a long position in stocks that lie in the top past return and FIR quintiles and a short position in stocks that lie in the bottom past return and FIR quintiles. The resulting twin momentum earns an average return of 2.16% per month, more than doubling that of price momentum (0.93%) or fundamental momentum (0.88%). Moreover, its standard deviation is only 8.34%, compared favorably to that of price momentum (6.78%). Its monthly Sharpe ratio is 0.26, which is much higher than that of both price momentum (0.16) and fundamental momentum (0.14), as well as the market portfolio (0.13) in our sample period. Consistent with fundamental momentum, the superior performance of twin momentum cannot be explained by extant risk-based factor models either. From the investment perspective, an important question is whether twin momentum adds value beyond existing factors. To address this question, we test whether twin momentum can expand the mean-variance frontier. With six mean-variance spanning tests under various distribution assumptions of the data, we find strong evidence that twin momentum cannot be spanned by any of the above factors, suggesting that it can significantly improve investors portfolio performance without taking any additional risk. In the anomaly literature, Stambaugh, Yu, and Yuan (2012, 2014) discover a pervasive pattern that, due to arbitrage asymmetry, many mispricing-driven anomalies are mainly from short-legs. To investigate 2

4 whether twin momentum displays this pattern, we consider its long- and short-leg portfolios separately, and find that the outstanding performance comes from both legs. They are unlikely attributed to high arbitrage costs alone. Indeed, the average monthly return is 1.72% (t = 5.34) for the long-leg and 0.44% (t = 1.06) for the short-leg. Although the risk-adjusted return from the long-leg is smaller in magnitude than that from the short-leg, it is still statistically significant at the conventional level. Across various asset pricing factor models, the maximum alpha from the long-leg is 0.80% (t = 3.61) and the minimum is 0.37% (t = 2.34). Correspondingly, the maximum alpha in absolute value from the short-leg is 1.75% (t = 6.42) and the minimum is 0.86% (t = 2.40). Therefore, twin momentum is such an anomaly that its log-leg is important too. We conduct several robustness checks on our findings. First, twin momentum is not driven by size effect, which exists in most anomalies and plagues existing risk-factor models (Fama and French, 2015; Hou, Xue, and Zhang, 2015). Following Stambaugh, Yu, and Yuan (2015), we construct twin momentum by eliminating firms with market capitalization falling in the bottom 20%, 40%, 60%, and 80% percentiles of the stock universe, respectively, and find that significant momentum profits still exist in all size groups. For example, in the megacap group (excluding firms below the 80% percentile of all firms), twin momentum earns a monthly average return of 1.42% and its monthly alpha is at least 0.63%, significant at the 1% level. Second, twin momentum is not attributable to transaction costs. Its turnover ratio is about 87.89% per month, comparable to that of price momentum (Martin and Grundy, 2001; Novy-Marx and Velikov, 2016). Following Grundy and Martin (2001) and Barroso and Stanta-Clara (2015), the break-even transaction cost that exactly offsets the twin momentum profit is 2.46%. Third, twin momentum is not driven by investor sentiment. It has an average monthly return of 1.75% in low sentiment periods, and 2.56% in high sentiment periods, which are economically sizeable and statistically significant. Fourth, twin momentum is not driven by research design. For example, it continues to hold when we use different periods of historical fundamental information or different sorting methods. Finally, we show that mutual funds that actively trade on fundamental momentum or twin momentum deliver better performance. Econometrically, there are two major reasons for why our results are much stronger than previous studies. First, instead of using one variable, we use a collection of seven major fundamentals jointly in forming the long and short portfolios. One variable can only measure one aspect of a firm s prospect and it is clearly short in reflecting the firm s overall strength. By using seven major fundamentals, we are able to capture more about the entire economic future of a firm, thereby yielding greater predictive power on its 3

5 returns. Second, existing studies often reply on a single lagged value of a fundamental variable. This does not utilize the full information. For example, good earnings in one quarter does not necessarily suggest that the firm is good, but consistent good earnings does. Therefore, we use in our paper not only the lagged value, but also the trends of the fundamentals, captured by various moving averages of the fundamental variables. Because we use more variables as well as their trends, we are able to obtain more accurate forecasts of future returns. As a result, we find that fundamentals trends matter after all. Our paper is closely related to a study on profit trends. Akbas, Jiang, and Koch (2017) seem the first to explicitly addresses the importance of fundamental trends, but they focus on quarterly gross profit and define the trend as the regression slope of gross profit on a time trend. In contrast, we use moving averages and let the data to weight the relative importantance of short- and long-trend trends. Empirically, over the sample period , the average return of their spread decile portfolio is 0.42% per month (t = 2.55), much smaller than ours. Moreover, the Fama-French (2015) five-factor alpha is 0.07% with an insignificant t-statistic of 0.43, and the Hou, Xue, and Zhang (2015) four-factor alpha has a puzzling sign, 0.36% (t = 2.24). Our paper is also related to existing studies on earnings momentum. Chan, Jegadeesh, and Lakonishok (1996) find that earnings surprises can generate momentum and contain information different from price momentum. However, the magnitude of earnings momentum is small, typically less than half of price momentum. Griffin, Ji and Martin (2005) and many others observe similar weak effects internationally. Because of this, many studies on earnings momentum turn attention to the interaction between price momentum and earning momentum. For example, Chordia and Shivakumar (2006) find that price momentum is captured by the systematic component of earnings momentum for especially small stocks. Recently, Novy-Marx (2015) shows further that earnings momentum explains the performance of various strategies based on price momentum among large cap stocks too. In short, existing studies only find weak predictability of fundamentals. In contrast, we find strong predictability. This paper is related as well to a nascent literature that explores the joint cross section of stock returns by multiple firm characteristics, such as Ou and Penman (1989), Holthausen and Larcker (1992), Haugen and Baker (1996), Piotroski (2000), Hanna and Ready (2005), Piotroski and So (2012), Asness, Frazzini, and Pedersen (2013), Lewellen (2015), Cosemans, Frehen, Schotman, and Bauer (2016), Stambaugh and Yuan (2016), Bartram and Grinblatt (2017), Green, Hand, and Zhang (2017), Light, Maslov, and Rytchkov (2017), and Yan and Zheng (2017), among others. We contribute to the literature by introducing a new 4

6 approach to exploit information in both time series and cross section to construct a spread portfolio. The rest of the paper is organized as follows. Section 2 introduces the data and key variables used in this paper. Section 3 shows that twin momentum exists in the stock market and is different from price momentum. Section 4 proposes a twin momentum trading strategy that can generate significant alpha and add investment value to existing risk factors. Section 5 shows the robustness of twin momentum, which is followed by Section 6 with a brief conclusion. 2 Data Accounting information is collected from the quarterly Compustat database and market information is from the monthly CRSP database. We merge the two datasets through the CCM link table. The Compustat sample period is between January 1973 and August 2015 and the CRSP sample period is from April 1976 to September To construct the sorting variable for fundamental momentum, we use seven fundamental variables common in the literature: return on equity (ROE), return on assets (ROA), earnings per share (EARN), accrual-based operating profitability to equity (APE), cash-based operating profitability to assets (CPA), gross profitability to assets (GPA), and net payout ratio (NPY). These fundamental variables are earnings and profitability related, and are relevant to what investors often use for valuation. Specifically, ROE is defined as income before extraordinary items (IBQ in Compustat) divided by onequarter-lagged book equity as in Baker and Haugen (1996). Book equity is shareholders equity, plus balance sheet deferred taxes and investment tax credit (item TXDITCQ) if available, minus the book value of preferred stocks. Depending on availability, we use stockholders equity (item SEQQ), or common equity (item CEQQ) plus the carrying value of preferred stocks (item PSTKQ), or total assets (item ATQ) minus total liabilities (item LTQ) in that order as shareholders equity. We use redemption value (item PSTKRQ) if available, or carrying value for the book value of preferred stocks. ROA is income before extraordinary items (IBQ) divided by one-quarter-lagged total assets (ATQ) as in Balakrishnan, Bartov and Faurel (2010). EARN is income before extraordinary items (IBQ), plus deferred taxes (TXDIY), minus preferred dividends (DVPY), and divided by one-quarter-lagged common outstanding shares (CSHO) as in Fama and French (1992). APE is revenue (REVTQ) minus cost of goods sold (COGSQ), minus selling, general, and administrative expenses (XSGAQ), minus interest expenses (TIEQ), and divided by onequarter-lagged book equity as in Fama and French (2015). CPA is accrual-based operating profitability on 5

7 assets minus change in accounts receivable (RECTQ) and change in inventory (INVTQ), and plus deferred revenue (DRCQ+DRLTQ), trade accounts payable (APQ) and change in accrued expenses (XACCQ), and scaled by one-quarter lagged total assets as in Ball, Gerakos, Linnaimaa and Nikolaev (2016). GPA is total revenue (REVTQ) minus cost of goods sold (COGSQ) divided by total assets (ATQ) as in Novy-Marx (2013). Finally, NPY is net payout divided by one-quarter-lagged market capitalization of common shares as in Boudoukh, Michaely, Richardson and Roberts (2007). Net payout is measured as dividends per share (DVPSPQ) multiplied by one-quarter-lagged common shares outstanding (CSHOQ) plus total expenditures on the repurchase of common and preferred shares (PRSTKCY) minus the sale of common and preferred stocks (SSTKY). To remove the outlier effect, all of the variables are cross-sectionally winsorized at the 1% and 99% levels in each quarter. Although standard unexpected earnings (SUE) is widely used as well in the literature as the sorting variable for earnings or fundamental momentum (Chan, Jegadeesh, and Lakonishok, 1996; Chordia and Shivakumar, 2006), we do not include it for two reasons. The first reason is that we have already considered earnings per share (EARN) that contains a firm s expected and unexpected earnings, and the second reason is that Hou, Xue, and Zhang (2016) show that ROE subsumes SUE in forecasting future stock returns. Indeed, the major results of our paper continue to hold when choosing different sets of earnings or profitability variables in constructing fundamental momentum, as will be discussed in Section 5. 3 Fundamental Momentum In this section, we show the existence of fundamental momentum and investigate the extent to which it overlaps with and differs from price momentum. 3.1 Fundamental implied return (FIR) As price momentum has been constructed using a variety of metrics in the literature, we follow Daniel and Moskowitz (2016) and form quintile portfolios at the beginning of month t based on past return (i.e., cumulative return over the past 11 months from month t 12 through month t 2) with a requirement of at least eight observations. Stocks in quintile 1 are past losers and stocks in quintile 5 are past winners. We construct a value-weight portfolio with stocks in each quintile and assume a holding period of one month throughout the paper. Price momentum is measured as the difference in returns between the past winner 6

8 portfolio and the past loser portfolio. To construct fundamental momentum, one needs a proxy of the fundamental value as the sorting variable. Unfortunately, a firm s fundamental value is not directly observable, and previous studies often focus on a single observable variable, such as standard unexpected earnings and cumulative three-day abnormal returns (e.g., Chan, Jegadeesh, and Lakonishok, 1996; Chordia and Shivakumar, 2006; Novy-Marx, 2015). This approach is intuitive but unlikely efficient as a firm s fundamental value is from multiple sources and measured by multiple observables in practice. In this paper, with an approach similar to that in Han, Zhou, and Zhu (2016), we extract information from multiple observables to estimate expected future stock returns, denoted by fundamental implied return (FIR). Our approach consists of three steps. In the first step, in each quarter q, we calculate a moving average value (MA) of each fundamental variable over the most recent L quarters of firm i as: MA k i,q,l = Fk i,q + Fk i,q 1 +,+Fk L i,q L+1, k = 1,2,,7; L = 1,2,4,8, (1) where Fi,q k j denotes the realized value of firm i s kth fundamental variable in the jth lagged quarter. The denominator is adjusted accordingly if there are missing observations in the past L quarters. We require at least one observation for each fundamental variable over the past two quarters to be included, two observations over the past four quarters, and four observations over the past eight quarters. In sum, we use 28 fundamental variables, seven fundamental variables times four moving averages of each variable with different lags, to capture a firm s fundamental value. The use of moving averages is another key feature of our paper. The purpose is to capture not only what happens most recently with a firm, but also its fundamental trends. This is clearly important in practice as investors care both earnings and their sustainability for which the trends often provide good indications. The importance of trends has also been documented in the academic literature. For example, Lakonishok, Shleifer, and Vishny (1994) find a negative relationship between five-year sales (log) growth and future stock returns, where the five-year sales growth is indeed the average of the past five years annual sales growths. Recently, Hirshleifer, Hou, Teoh, and Zhang (2004) show that the cumulative net operating income (i.e., average net operating income times the number of observations) contains incremental information about future stock returns. Theoretically, the fundamental moving averages can be justified by sticky expectation models of Coibion and Gorodnichenko (2012, 2015). Following Bouchard, Krüger, Landier, and Thesmar 7

9 (2016), we show in Section 3.3 that, with sticky expectation, moving averages have incremental forecasting power relative to current fundamentals. In the second step, in month t, we merge firm i s excess return, R i,t, with the most recent MA k i,q,l and run a cross-sectional regression across all firms as: R i,t = α t + 7 L=1,2,4,8 k=1 β k L,tMA k i,t 1,L + ε i,t, (2) where α t is the intercept in month t, β k L,t is the regression coefficient of MAk i,t 1,L, and ε i,t is the unexplained return of stock i in month t. Following Hou, Xue, and Zhang (2016), MA k i,t 1,L is used in the months immediately after the most recent public quarterly earnings announcement dates (item RDQ). For a firm to enter regression (2), we require the end of the fiscal quarter that corresponds to its announcement earnings date to be within six months prior to the regression month. In the third step, as in Haugen and Baker (1996), Hanna and Ready (2005), and Han, Zhou, and Zhu (2016), we construct firm i s fundamental implied return, FIR, in month t using the forecasted return for month t + 1 as: FIR i,t = 7 L=1,2,4,8 k=1 E t [β k L,t+1]MA k i,t,l, (3) where E t [βl,t+1 k ] is the expected coefficient on the kth fundamental variable with L lags and is defined as E t [βl,t+1 k ] = β L,t k. In (3), we do not include an intercept because it is the same for all stocks in the crosssectional regression and does not play any role in ranking stocks. It is worth pointing out that, as only information in month t or prior is used to forecast the expected returns in month t + 1, FIR is a pure realtime predictor of stock returns and does not suffer from looking-forward biases. 3.2 Fundamental momentum return Now we are ready to construct fundamental momentum. At the beginning of each month, we sort all stocks into five value-weight portfolios by their FIRs, with the bottom quintile (quintile 1) containing stocks with the lowest FIR and the top quintile (quintile 5) containing stocks with the highest FIR. Fundamental momentum is the zero-investment strategy that buys the top FIR quintile portfolio and sells the bottom FIR quintile portfolio. Table 1 reports the monthly average and risk-adjusted returns of spread portfolios based on different 8

10 sorting variables, where the benchmark asset pricing models include the CAPM, Fama and French (FF3, 1993) three-factor model, FF3 plus a price momentum factor model (FF3M), Hou, Xue and Zhang (HXZ, 2015) four-factor model, and the Fama and French (FF5, 2015) five-factor model. To highlight the importance of multiple variables and trends in capturing the expected stock returns, we consider four cases in Table 1. Panel A forms quintile portfolios based on a single variable, such as the most recent quarterly ROE, Panel B based on a single variable and its trends, which are captured by the moving averages with lags 2, 4, and 8, Panel C based on the seven fundamental variables but not using trend information, and Panel D based on the seven variables and their trends. Empirically, Panel A of Table 1 shows that a single fundamental variable can generate significant average returns, but these returns can be described by the most recent developed asset pricing models, HXZ and FF3, which advocate investment and profitability as new risk factors. Panel B shows that trends can improve the spread portfolio returns, however, this improvement is not large enough to generate abnormal returns. Panel C shows that simultaneously considering multiple variables significantly improves the performance, which can be improved further if the trend information is incorporated. Table A1 shows the monthly average and risk-adjusted returns of FIR quintile portfolios, and suggests that FIR is indeed aligned with expected returns in the cross section: the average portfolio return monotonically increases from quintile 1 to quintile 5. Table A2 shows that the forecasting power of FIR is significant in industries such as consumer, manufacturing, high tech, and others, but is insignificant in health. Before ending this section, we address that unlike most studies that attempt to explain average stock returns by sorting on a few (usually less than three) firm characteristics or fundamental variables, we construct fundament momentum from cross-sectional regressions that allow us to incorporate multiple firm fundamental proxies. Including or excluding one or more variables is easily implemented in our framework. In the literature, there is no consensus on the proxy of a firm s fundamental value, and we do not take a stance as to which is the most important variable in constructing fundamental momentum. No matter how one chooses the variables, a collection of common fundamentals collectively have a strong and consistent impact on stock returns, but the impact is weak with only one observable. 9

11 3.3 Why do fundamental trends have forecasting power? In this subsection we provide a simple way to understand why the trends of fundamentals can increase the forecasting power for future stock returns. Our analysis draws on insights about investor sticky expectation on fundamental information (Coibion and Gorodnichenko, 2012, 2015). Following Bouchard, Krüger, Landier, and Thesmar (2016), risk-neutral investors forecast future profits using a signal about the fundamentals with sticky belief dynamics. Each period, they update their beliefs using all available information with a probability 1 λ. With probability λ, they stick to their previous beliefs. When solving the model, we show that future stock returns can be forecasted by the trends of fundamentals. Specifically, suppose firm profits π t+1 can be predicted with a signal s t as π t+1 = s t + ε t+1, (4) s t+1 = ρs t + u t+1, (5) where ρ < 1 and ε t+1 N(0,σ 2 ε ) and u t+1 N(0,σ 2 u ) are white noises. Let Ẽ t (π t+h ) and E t (π t+h ) be the subjective and objective expectations formed at time t about profits π t+h at time t + h, which follows a process as Ẽ t (π t+h ) = (1 λ)e t (π t+h ) + λẽ t 1 (π t+h ). (6) The coefficient λ indicates the extent of expectation stickiness. When λ = 0, expectations are perfectly rational. When 0 < λ < 1, expectations are sticky and investors will insufficiently incorporate new information into their forecasts. When λ < 0, investor will overreact to new information. In the finance literature, Daniel and Titman (2006) find no evidence that stock prices overreact to any fundamental growth measures (tangible information), such as those variables explored in this paper. Because the subjective expectations are sticky, i.e., 0 < λ < 1, Bouchard, Krüger, Landier, and Thesmar (2016) show that investors do not immediately and fully incorporate the information in profits, generating a positive relation between current profits and future stock returns: Cov(R t+1,π t ) = (1 + mρ) λ 2 ρσu 2 > 0, (7) 1 λρ2 10

12 where m = (1 λ)/(1 + r ρ) and r is the risk-free rate. In terms of this paper, we have empirically shown in Section 3.2 that the trends of firm profits have incremental forecasting power relative to current profits, which suggests that investors do not fully incorporate the information in past profits either. Moreover, trends may serve as a consistency measure of firm profits. Good earnings in one quarter does not necessarily suggest the firm is good, but consistent good earnings does suggest a good future of the firm. Mathematically, define the profit trend as MA t,l = π t + π t 1 + π t L+1. With some algebra, we can show that this trend predicts the return R t+1 as: Cov(R t+1,ma t,l ) = (1 + mρ) λ 2 ρσ 2 u 1 λρ 2 1 (λρ) L 1 λρ, (8) which is increasing with respect to L if λ > 0. Apparently, the lagged profits π t L for L > 0 have incremental forecasting power relative to π t on future stock returns. Hence, the trend MA t,l has greater forecasting power compared with π t. 3.4 Fundamental momentum versus price momentum We conduct three exercises, including portfolio sort, Fama-MacBeth regression, and Campbell, Giglio, Polk, and Turley (2017) decomposition, to show that fundamental momentum and price momentum exploit different sources of information and complement each other Bivariate portfolio analysis In this section, we investigate the performance of fundamental momentum with controlling for price momentum, as there is no concensus in the literature as to whether price momentum and fundamental momentum are a tale of two anomalies, where the latter is constructed usually based on standard unexpected earnings or cumulative three-day abnormal returns (e.g., Chan, Jegadeesh, and Lakonishok, 1996; Chordia and Shivakumar, 2006; Hou, Peng, and Xiong, 2009; Novy-Marx, 2015). At the beginning of each month t, we independently sort stocks into five groups based on their past returns (11-month cumulative return from month t 12 to month t 2) and five groups based on their fundamental implied returns (FIRs), and construct price momentum by buying the highest past return quintile portfolio and selling the lowest past return quintile portfolio, and fundamental momentum by buying 11

13 the highest FIR quintile portfolio and selling the lowest FIR quintile portfolio. The intersections of the two sorts produce 25 value-weight portfolios. Table 2 reports the average return of each portfolio in excess of the risk-free rate (the one-month U.S. Treasury-bill rate) and the associated Newey-West t-statistic with 12 lags that tests whether the average monthly excess return is different from zero. For consistency, all t-statistics are calculated in this way throughout the paper. The results suggests that there is fundamental momentum, which is different from price momentum (e.g. Hou, Peng, and Xiong, 2009). Within each past return quintile, the average return of the fundamental momentum portfolio monotonically increases in FIR, and within each FIR quintile, the average return of the price momentum portfolio increases in past return. The higher FIR is, the higher the average excess return fundamental momentum earns. For example, in the lowest past return quintile, the average excess return of the FIR portfolios increases from 0.83% (t = 2.07) per month for quintile 1 to 0.33% (t = 0.71) per month for quintile 5, suggesting that the fundamental momentum strategy, measured by the spread return, earns a significant average return of 1.16% (t = 4.02) per month. Similarly, in the highest past return quintile, the average excess return of the FIR portfolios increases from 0.72% (t = 2.11) per month for quintile 1 to 1.33% (t = 4.13) per month for quintile 5, suggesting that fundamental momentum earns a significant average return of 0.61% (t = 2.26) per month. One interesting finding in the last column of Table 2 is that the fundamental momentum profit generally decreases in the past return quintile rank, which suggests that fundamental momentum exists not only in the past winner stocks but also in the past loser stocks with even stronger performance. The last row of Table 2 shows that price momentum exists in each FIR quintile, and its performance is even stronger than that of fundamental momentum with the same sorting rank. The average monthly return of price momentum is 1.55% (t = 4.43) in the lowest FIR quintile and is 1.00% (t = 2.58) in the highest FIR quintile. To summarize Table 2, fundamental momentum exists in stock returns and is not a manifestation of price momentum Fama-MacBeth regression A complementary approach to portfolio sort is to run a Fama-MacBeth regression of stock return in month t on past return (i.e., 11-month cumulative return from month t 12 to month t 2) and FIR in month t 1 directly. The advantage of this cross-sectional regression is that one can control for other firm characteristics, which may contain information in the variables of interest. As such, we choose five firm characteristics that have been commonly used in the literature, including short-term reversal (stock return in month t 1), long- 12

14 term reversal (cumulative stock return between month t 60 and month t 13), log market capitalization (log size), book-to-market (B/M), and idiosyncratic volatility (IVOL) estimated from the FF3 model using the past month daily returns, with a requirement of at least 16 observations. The regression results are reported in Table 3. The first two columns are the results of regressing stocks returna on past return or FIR alone, and the regression slopes are positive and statistically significant, suggesting that past return and FIR have forecasting power for future stock returns. Column 3 is the result of regressing stock returns on past return and FIR simultaneously, and the regression slopes are positive and statistically significant too, consistent with Table 2 that fundamental momentum exists in stock returns. When we include the other five firm characteristics as controls in Columns 4 to 6, the regression slopes on past return and FIR are quantitatively unchanged and remain statistically significant. Overall, Tables 2 and 3 provide congruent evidence that fundamental momentum and price momentum coexist in the stock market and neither one subsumes the other Cash flow, discount rate, and variance betas The previous results with double sort and regression analysis suggest that fundamental momentum and price momentum are different and complementary. We investigate the difference in more depth and attempt to link it to possible sources in this section. Campbell and Vuolteenaho (2004) show that the unexpected returns on the market portfolio can be decomposed into two components: shocks about future cash flows and shocks about discount rates that investors apply to these cash flows. Campbell, Giglio, Polk, and Turley (2017) extend this model by assuming that the variance of stock returns is stochastic and the unexpected returns also respond to variance shocks. As such, investment opportunities may deteriorate because the expected return on the stock market declines (discount rate shocks) or because the variance of the stock market increases (variance shocks), and a conservative long-term investor has to hedge against two types of shocks to the investment opportunity set. This means that the single CAPM beta can be decomposed into three betas: one reflecting the covariance with news about future cash flows, one with news about discount rates, and one with news about variance. Following Campbell, Giglio, Polk, and Turley (2017), we run time-series regressions of price momentum and fundamental momentum returns on shocks about future cash flows (N c f ), discount rate ( N dr ), and 13

15 variance (N v ) of the aggregate market, 3 r i = β 0 + β c f N c f + β dr ( N dr ) + β v N v + ε i, where r i is the quarterly price momentum or fundamental momentum returns over the sample period 1976Q2 2011Q4. Table 4 reports the regression results, which reveals two interesting facts. First, price momentum and fundamental momentum are related. Specifically, both price momentum and fundamental momentum have negative cash flow beta and discount rate beta, but positive variance beta, which are consistent with the signs of the UMD and RMW factors in Campbell, Giglio, Polk, and Turley (2017). Moreover, their cash flow betas are insignificant, whereas discount rate betas and variance betas are significant, suggesting that the profits of price momentum and fundamental momentum do not come from cash flow shocks. According to Campbell and Shiller (1988) and Campbell and Vuolteenaho (2004), the significant discount rate beta suggests that the effect of discount rate shocks is temporary in the sense that rational investors expect the stock price to rebound to previous levels. As a result, the price momentum and fundamental momentum profits should be reverting some time after portfolio formation. Second, price momentum and fundamental momentum are different. Their discount rate betas are 0.35 (t = 2.00) and 0.16 (t = 1.97), and their variance betas are 1.06 (t = 3.09) and 0.37 (t = 2.00), respectively. In magnitude, the effect of discount rate shocks on price momentum is about two times and the effect of variance shocks is about three times as large as that on fundamental momentum. The relative effect of variance shocks vs. discount rate shocks is stronger for price momentum too. Hence, the three shocks in Campbell, Giglio, Polk, and Turley (2017) describe more variation of price momentum than fundamental momentum in terms of regression R 2 (8.61% vs. 2.71%). To have an intuitive understanding of the difference between price momentum and fundamental momentum, Figure 1 plots their cumulative returns over time after portfolio formation, where the cumulative returns are calculated by adding monthly returns from formation month t to month t + i. As expected, the price momentum profit arrives its maximum after four months since portfolio formation and reverts to be negative after eight months. In contrast, the fundamental momentum profit reaches its maximum after 12 months and reverts to be negative after 18 months. 3 We thank Christopher Polk for providing the data on his web page. 14

16 4 Twin Momentum After establishing the fact that fundamental momentum and price momentum capture different information sources, it is interesting to combine these two trading strategies into one to improve investment performance. We name this new strategy as twin momentum, which buys stocks in the intersection of the top past return and FIR quintiles and sells stocks in the intersection of the bottom past return and FIR quintiles. This approach is intuitive and commonly used in the literature (see, e.g., Da and Warachka, 2011). Our approach has two potential advantages. The first is that the twin momentum can better utilize the unfavorable information that is conveyed by stock-level characteristics. The second is that price momentum is a relatively fast-moving characteristics, since returns from year to year are relatively independent, while FIR is a relatively slow-moving characteristic, since it is based on levels and trends rather than changes in market values. 4.1 Average returns Panel A of Table 5 reports summary statistics of the portfolio returns of price momentum, fundamental momentum, and twin momentum, which include the monthly average return, t-statistic (from the test of the average return is zero), volatility (standard deviation), skewness, kurtosis, and the probability of the strategy producing a positive excess return. Over the whole sample period from April 1976 to September 2015 (henceforth ), the average return of price momentum is large, 0.93% (t = 2.97) per month with a standard deviation (volatility) of 6.78%. However, the skewness of 1.21 suggests that price momentum has a higher probability of experiencing big crashes than what a normal distribution implies. Fundamental momentum earns an average return of 0.88% (t = 3.57) per month, which is comparable with price momentum. The skewness is 0.73, suggesting that fundamental momentum has a higher chance of generating positive returns. The average return of twin momentum is impressive, 2.16% (t = 5.64) per month, which is higher than the simple sum of the average returns of price momentum and fundamental momentum. The skewness of twin momentum is only 0.04 and is between that of price momentum and fundamental momentum. One reason may be that twin momentum chooses stocks with positive skewness from fundamental momentum and negative skewness from price momentum, and as a result, their aggregate skewness is neutralized. Another reason may be that twin momentum does not choose stocks with positive or negative skewness at 15

17 all, and therefore, its skewness is close to zero. As such, twin momentum is more likely to follow a normal distribution. Table 5 also shows that price momentum, fundamental momentum, and twin momentum have higher probabilities to produce positive rather than negative returns, which are 0.62, 0.58, and 0.66 over the whole sample period, respectively. The last column of Table 5 shows that the correlation between price momentum and fundamental momentum is as low as This value lends direct support to the success of twin momentum. Indeed, if price momentum and fundamental momentum are strongly positively correlated, combining them will not generate significant improvement. In addition, twin momentum has a correlation of 0.69 with price momentum and 0.61 with fundamental momentum, implying that price momentum and fundamental momentum are equally important to twin momentum and neither one dominates the other. To explore whether our results are driven by specific time periods, we split the whole sample period into four subperiods. As our sample covers almost four decades, each subperiod is about 10 years. The summary statistics for each subperiod are reported in Panel B of Table 5. Four observations follow this panel immediately. First, twin momentum exists in each subperiod. In contrast to the recent literature, such as Chordia, Subrahmanyam, and Tong (2014) and McLean and Pontiff (2016), that anomaly returns are decaying over time, twin momentum does not display a declining pattern. Its monthly average return is 1.64% (t = 2.66) over , whereas it is 2.10% (t = 4.44) over , 2.82% (t = 2.62) over , and is 2.06% (t = 2.70) over , respectively. Second, although fundamental momentum does not decay over time, price momentum does, with average returns significant in the first two subperiods but insignificant in the last two subperiods. However, the declining performance in price momentum does not reduce the twin momentum profit as price momentum and fundamental momentum intensify each other. That is, the twin momentum profit is much larger than the sum of price momentum and fundamental momentum in latter periods. For example, the average return of twin momentum is 2.06% per month over , 44 bps higher than the sum of price momentum (0.73%) and fundamental momentum (0.79%). Third, while the skewness of price momentum is always negative, it is mainly from the last subperiod, , over which, the value is This pattern is consistent with Daniel and Moskowitz (2016) who show that three of ten top momentum crash months occur in 2009 (March, April, and August). Instead, the skewness of fundamental momentum is always positive with one exception in , over which the value is negative but close to zero, Lastly, twin momentum always has a higher probability than 16

18 price momentum and fundamental momentum to generate positive returns. To visualize the outperformance, Figure 2 plots the time series returns (upper panel) and the log cumulative wealth (lower panel) for investing in price momentum, fundamental momentum, and twin momentum, respectively. To iron out idiosyncratic returns, we smooth portfolio returns in the upper panel with 12-month moving average values. With an eyeball test, one can easily conclude that twin momentum generates much better performance than price momentum and fundamental momentum. Table 5 also shows that the volatility of twin momentum is relatively high, and Figure 2 tells the reason: twin momentum has a higher chance of generating big positive profits and therefore, the high volatility is more likely from the upside movements. 4.2 Risk-adjusted returns We examine whether the outperformance of twin momentum can be explained by existing risk factor models. If an asset pricing model completely captures the average return of twin momentum, the intercept is indistinguishable from zero in a regression of the twin momentum portfolio return on the model s factor returns. Table 6 reports the regression results of the twin momentum portfolio return on each model s factor returns. The first row is alpha (abnormal return) in percentage from each risk factor model, the last row is the associated R 2, and other rows are the corresponding loadings that measure how much risk twin momentum exposes to the factors. The Newey-West t-statistics with 12 lags are reported in parentheses. Table 6 makes several statements about twin momentum. First, the abnormal returns with the five factor models are economically and statistically significant, varying from 1.37% (t = 2.45) per month with the HXZ model to 2.50% (t = 6.69) per month with the FF3 model, suggesting that the high return delivered by twin momentum cannot be fully explained by these extant risk factors. Second, the FF3M, HXZ, and FF5 models have some power to explain the twin momentum profit (i.e., the alphas are less than the average return). The reason is that twin momentum has a positive exposure to the price momentum factor in the FF3M model, to the ROE factor in the HXZ model, and to the CMA factor in the FF5 model. Instead, the risk-adjusted returns become higher than the excess return with the CAPM and FF3 models, as they push twin momentum from 2.16% per month to 2.31% (t = 6.34) and 2.50% (t = 6.69) per month, respectively. The reason is that twin momentum has a negative exposure of 0.26 (t = 1.94) to MKT in the CAPM 17

19 model, and negative exposures of 0.38 (t = 2.63) and 0.51 (t = 1.87) to MKT and HML in the FF3 model, which move the adjusted return away from zero. Third, as twin momentum has a negative exposure to MKT in all the five factor models and has a correlation of 0.14 with MKT, twin momentum can serve as a perfect hedge for the market portfolio. Fourth, the size factor in the FF3, FF3M, HXZ, and FF5 models has no power to explain the variation of twin momentum and none of the loadings is significant, which previews the result in Section 5.1 that the size effect cannot explain twin momentum. Finally, the risk factor models explain a small fraction of the twin momentum variation and their regression R 2 s are less than 15%. The only exception is the FF3M model, which augments FF3 with the price momentum factor (obtained from Ken French s website) and is supposed to explain the price momentum variation. As a result, the regression R 2 is 45.8%, suggesting that at least half variation in twin momentum is from fundamental momentum, which is different from price momentum. 4.3 Mean-variance spanning tests From the perspective of asset pricing, Table 6 shows that twin momentum cannot be explained by extant risk factor models. This section explores whether twin momentum adds any investment value from the perspective of an investor who holds a well diversified portfolio, such as the market portfolio or a portfolio spanned by the FF5 factors. The mean-variance spanning test originally proposed by Huberman and Kandel (1987) provides exact the answer we need. The key idea is to show that whether twin momentum lies outside the mean-variance frontier spanned by an asset pricing model s factor returns. As such, we run a time-series regression of the twin momentum portfolio return on the factor returns in each asset pricing model over the whole sample period as: R t = α + M β j f j,t + ε t, (9) j=1 where f j,t is the return of factor j in month t and M is the number of risk factors in the asset pricing model, such as M = 5 in the FF5. Huberman and Kandel (1987) show that the spanning test is equivalent to the test of the following restrictions: H 0 : α = 0, M β j = 1. (10) j=1 18

20 We follow Kan and Zhou (2012) and carry out six tests, which include Wald test under conditional homoskedasticity, Wald test under independent and identically distributed (IID) elliptical distribution, Wald test under conditional heteroskedasticity, Bekerart-Urias spanning test with errors-in-variables (EIV) adjustment, Bekerart-Urias spanning test without the EIV adjustment and DeSantis spanning test. All these tests have an asymptotic chi-squared distribution with 2 degrees of freedom. The results in Table 7 suggest a strong rejection of the hypothesis that twin momentum is inside the mean-variance frontier of any of the five factor models considered in this paper. Therefore, twin momentum is clearly a unique trading strategy that describes the cross section of stock returns unexplained by extant risk factor models, and provides incremental investing value. 4.4 Long- and short-leg portfolios Stambaugh, Yu, and Yuan (2012, 2014, 2015) show that investors face both arbitrage risk and arbitrage asymmetry, where the former refers to the risk that deters arbitrage and the latter refers to the greater ability or willingness of an investor to take a long position as opposed to a short position when perceiving mispricing in a stock. Combining these two concepts, they find that most of the anomaly returns in the finance literature is from the short-leg portfolios as overpricing is more prevalent than underpricing, due to short-sale impediments. 4 Table 8 report the average and risk-adjusted returns of the long- and short-leg portfolios of price momentum, fundamental momentum, and twin momentum, respectively. Price momentum has a high average return, 1.35% (t = 5.01) per month, in the long-leg and a low average return, 0.42% (t = 1.13) per month, in the short-leg. However, its risk-adjusted return (in absolute value) is much smaller in the long-leg than that in the short-leg with the CAPM, FF3, and FF5 models. For example, the FF3 alpha is 0.35% (t = 3.10) per month in the long-leg and 0.78% (t = 3.52) per month in the short-leg. As our price momentum is only a finer sort relative to the Fama-French price momentum factor, our price momentum portfolio is fully explained by the FF3F. It is also explained by the HXZ model, consistent with Hou, Xue, and Zhang (2016) that the HXZ four factors can explain price momentum. Generally, our result on price momentum is consistent with Daniel and Moskowitz (2016) who show that the price momentum winner decile portfolio has a much larger average return (15.3% vs. 2.5% per year) but a much smaller CAPM 4 Nagel (2005) also finds that the overpricing is more pronounced in the short-leg portfolios, especially for those with low institutional ownership. 19