Momentum Life Cycle Hypothesis Revisited

Transcription

1 Momentum Life Cycle Hypothesis Revisited Tsung-Yu Chen, Pin-Huang Chou, Chia-Hsun Hsieh January, 2016 Abstract In their seminal paper, Lee and Swaminathan (2000) propose a momentum life cycle (MLC) hypothesis, which states that turnover plays a crucial role in explaining the coexistence of short-term momentum and long-term reversal, and that high-turnover losers and low-turnover winners experience stronger persistence in performance. We show that the hypothesis holds only partially because only high-turnover losers experience persistent performance. Further analysis indicate, however, that the turnover effect and price momentum are two separate effects driven by different combinations of fundamental and behavioral forces. In particular, the explanatory power of the MLC hypothesis disappears after returns are controlled for the turnover effect and the price momentum. JEL Classification: G12; G14. Key Words: trading volume; price momentum; momentum life cycle. All authors are from Department of Finance, National Central University, Jhongli, Taiwan 320. Chou acknowledges financial support from the National Science Council of Taiwan (grant number H MY3). Corresponding author: Chou; choup@cc.ncu.edu.tw. Tel: ext Fax: Address: 300, Jhong-Da Rd., Jhongli City, Taiwan

2 1 Introduction The co-existence of short-term (or intermediate-term) momentum and long-term reversals in stock returns has been one of the most robust puzzles in financial research. 1 Lee and Swaminathan (2000), however, document that the puzzle exhibits a pattern related to turnover (i.e., trading volume) in that low-turnover winners (losers) exhibit stronger (weaker) persistence in performance than high-turnover winners (losers). They therefore propose a momentum life cycle (hereafter MLC) hypothesis, which argues that stocks experience periods of investor favoritism and neglect, and it turns out that low volume (high volume) firms would exhibit many characteristics associated with value (glamor) stocks (see Figure 4 of Lee and Swaminathan (2000)). As a result, early-stage momentum strategies constructed by take a long position on low-turnover winners and an equal short position on high-turnover losers are more likely to persist, whereas late-stage momentum strategies composed of high-turnover winners and low-turnover losers are more likely to reverse quickly. The novelty of the MLC hypothesis is that it unifies information on turnover and past returns, and helps to explain the coexistence of short-term momentum and long-term reversals. Apparently, turnover plays a crucial role in reconciling the short-term underreaction and long-term overreaction effects. While Lee and Swaminathan (2000) show that trading volume provides an important link between short-term price momentum and long-term reversals, they also acknowledge that there are some limitations with the MLC hypothesis. First, the MLC hypothesis holds only at the portfolio levels, but not with individual stocks. Second, the MLC hypothesis does not provide satisfactory answers concerning why 1 Famous models like Daniel, Hirshleifer, and Subrahmanyam (hereafter DHS, 1998), Barberis, Shleifer, and Vishny (hereafter BSV, 1998), and Hong and Stein (1999)) all try to explain the coexistence of short-term momentum and long-term reversals documented in the U.S. stock markets, a puzzling phenomenon that cannot be explained by neoclassical risk-based theories. 1

3 price momentum is more pronounced among high turnover stocks and why turnover might decline as a stock falls out of favor. In our view, although their empirical evidence indicates that there are some interactions between turnover and past returns, it is not clear if the two variables are really correlated in accordance with the prediction of the MLC hypothesis. As it is well documented that there exists a low-turnover effect whereby low turnover stocks earn higher average returns than high turnover stocks (Datar et al., 1998; Amihud et al., 2005; Chou et al., 2013), the fact that the return persistence in low-turnover winners and high-turnover losers may actually reflect the multiplication of the low turnover and price momentum effects. In other words, it is possible that the turnover effect and the price momentum may in fact be two separate effects. Hence, the purpose of this paper is to examine how the two effects interact with each other and especially how they are related to the MLC hypothesis. Specifically, we conduct various tests to examine how the three effects, namely the low-turnover effect, the price momentum effect, and the MLC hypothesis, compete with each other in explaining returns over various holding horizons ranging from 1 year to 5 years. With an extensive sample of monthly stock returns for , we find that the profitability of the early momentum strategies persists for up to 3 years, but the profits mainly come from the persist underperformance of the high-turnover price losers; the low-turnover winners earn significant profits only for one year following formation. Perhaps the most striking result is that the profitability of the early momentum strategies completely disappears after controlling for price momentum and the turnover effects in the regression model, thus supporting our conjecture that the MLC hypothesis simply reflects the multiplication of two separate effects. Further analysis indicates that price momentum and the turnover premium are indeed two separate 2

4 effects driven by different combinations of fundamental and behavioral forces. For price momentum, we find that more than half of its short-term profitability can be explained by a macroeconomic factor proposed by Chen, Roll, and Ross (hereafter CRR, 1986). The rest of its profitability exhibits patterns related to investor sentiment as claimed by Antoniou, Doukas, and Subrahmanyam (ADS, 2013). For the turnover premium, our empirical results indicate that it persists all over the 5-year holding period, and mainly comes from the persistent underperformance of high-turnover stocks. Although risk-based models such as CRR s macroeconomic factor model explains a great proportion of its premium, high-turnover stocks persistently earn significant negative returns during the 5-year holding period. Further analysis indicates that high-turnover stocks exhibit strong return patterns related to idiosyncratic volatility, a proxy commonly used to measure arbitrage risk. Thus, just as the short-term price momentum profit that is composed of both fundamental and behavioral components, the turnover premium is also affected by both rational and behavioral forces, though the underlying behavioral component is not entirely the same. Overall, we find strong evidence against the MLC hypothesis for two reasons. First, its predictability disappears after the two effects on price momentum and turnover are controlled for. Second, price momentum and the turnover premium are in fact two separate effects driven by different market forces, rather than an interrelated effect as claimed by Lee and Swaminathan s MLC hypothesis. The rest of the paper is organized as follows. In section 2, we describe our data and some preliminary results. Section 3 presents the major empirical results, whereas Section 4 provides further analysis from both rational and behavioral perspectives. The last section concludes. 3

5 2 Data and preliminary result 2.1 Data Our sample is composed of the ordinary common equities of all firms (share codes 10 and 11) listed on the NYSE and AMEX return files from January 1963 to December We exclude Nasdaq firms from our analysis because trading volume for NAS- DAQ stocks is inflated relative to NYSE and AMEX stocks. We obtain daily data on stock returns, share prices, trading volume and the number of shares outstanding, as well as monthly data on stock return and market equity from the Center for Research in Security Prices (CRSP) database. We define trading volume as the average daily turnover in percentage during the formation period, where the daily turnover is calculated as the number of shares traded, divided by the number of shares outstanding at the end of each trading day. Share price is the closing price per share at the end of the month. We exclude stocks whose share price as of formation date was below $1 to minimize the impact of microstructure effects on returns. 2.2 Preliminary results We begin with some preliminary examinations on investment strategies formed on the basis of past returns and turnover. Specifically, the price winner (loser) portfolio is the portfolio of 10% of the stocks with the highest (lowest) past 6-month returns, while the high (low) turnover portfolio is the portfolio of 10% of the stocks with the highest (lowest) past 6-month average turnover. The price momentum strategy is formed by buying the price winner portfolio and selling an equal position of the price loser portfolio. Similarly, the turnover-based strategy is formed by buying the low 4

6 turnover portfolio and selling the high turnover portfolio. We shall also refer to the returns of the turnover-based strategy as the turnover premium. All portfolios are equally weighted Momentum profits and turnover premium Table 1 reports the average monthly returns of price momentum and turnover strategies over various holding horizons ranging from 1 year to 5 years following formation. 2 [Insert Table 1 about here] We first focus on the results for price momentum strategies, which are reported in the top of each panel. In Panel A, price winners earn an average return of 1.354% per month for 1-12 months, while price losers gain an average return of 0.842% per month over the same holding period, thus resulting in an average momentum profit of % per month (t-statistic = 3.24). Outside of January, Panel B indicates that the average momentum profit becomes even larger for 1-12 months; the price momentum profit earns 0.807% (t-statistic = 4.87) per month. When we extend the holding period to longer horizons, a long-lasting return reversal pattern begins to emerge. Starting from the second year, the momentum profits become significantly negative. However, the reversal pattern is weaker with the exclusion of January observations, suggesting that a proportion of return reversals is related to January seasonality. Overall, the results confirm the coexistence of shortterm momentum and long-term return reversals well documented in the literature. 2 As a common practice to alleviate potential problems associated with bid-ask bounce and nonsynchronous trading, we skip one month between ranking and holding periods. The empirical results, however, are not sensitive to whether we skip one month or not. 5

7 Next, we turn our attention to the performance of the turnover-based strategies that long the low-turnover decile portfolio, while shorting an equal position of the high-turnover decile portfolio. The result is reported in the bottom of each panel. Table 1 shows that, regardless of the inclusion of January observations, the profits of the turnover strategies are positive all throughout the 5-year holding period, but are statistically significant only for the first 2 years; Panel A indicates that the average profit is 0.475% (t-statistic=2.52) for the first year, and is still 0.327% (tstatistic=1.80) for the second year. The profits remain about the same when January observations are excluded, suggesting that the turnover premium is not related to January seasonality. Interestingly, the turnover premium becomes insignificant for holding period beyond the second year following formation; there is, however, no evidence of return reversals for turnover-based strategies. Overall, the preliminary analysis identifies different return patterns for price momentum and turnover effects, especially in the longer holding horizons. It is unclear, however, how the two effects interact with each other, which is examined in the following subsection Pairwise comparisons of price momentum and turnover strategies To examine how price momentum and turnover effects interact with each other, we conduct pairwise comparisons of returns on strategies constructed on the interaction of past returns and turnover. Specifically, for each month all stocks are classified into 100 equal-weighted portfolios based on independent sorts on average daily turnover and returns over the past 6 months. With the two-way sorting portfolios, the interaction of price momentum and turnover premium can be compared. To save space, we only focus on the results of the highest and lowest groups. Also, the performance of the early-stage price momentum that longs the low-turnover price winner portfolio 6

8 and shorts the high-turnover price loser portfolio can be examined. The results are reported in Table 2. [Insert Table 2 about here] We first examine how price momentum strategies perform after controlling for the turnover effect. Panel A indicates that the return pattern of price momentum strategies remain about the same as that reported in Panel A of Table 1, except the profitability is smaller. For both within low-turnover and high-turnover groups, the average profit of price momentum is positive for the first year, and becomes negative thereafter. For example, within the low (high) turnover group, the price momentum earns an average profit of 0.179% (0.168%) for the first year following formation, but the raw profit reported in Table 1 is 0.512%. When January observations are excluded, the profitability of price momentum becomes weaker, also consistent with the result reported in Panel B of Table 1. The results indicate that the pattern on short-term price momentum and long-term reversals remain qualitatively intact even after controlling for the turnover effect. We next examine the turnover premium by controlling for the price momentum effect. The results in Table 2 indicate that turnover premium is consistently positive throughout the 5-year holding period, no matter when it is evaluated within the price winner or price loser groups. Similar to the results of price momentum reported above, however, the turnover premium is also weaker after controlling for the price momentum effect. The turnover premium is significantly positive only in the first year following formation, and lasts for two years when January observations are excluded. Specifically, Panel A indicates that the first-year turnover premium is 0.215% (tstatistic=2.39) within the price winner group, and is 0.203% (t-statistic=2.27) within the price loser group. Outside of January, Panel B indicates that the second-year 7

9 turnover premium is still 0.152% (t-statistic=1.65) within the price winner group, and is 0.168% (t-statistic=1.85) within the price loser group. Overall, the patterns of the turnover premium and the price momentum profits remain mostly the same as those reported in Table 1, suggesting that the two effects, while somehow correlated, are two separate ones. As a final examination, Table 2 also reports the performance of the early-stage momentum strategy by longing low-turnover winners and shorting high-turnover losers. The results are reported in the bottom of each panel. Panel A indicates that the early-stage momentum strategy earns an average return of 0.382% (t-statistic=2.91) in 1-12 months, but becomes insignificant thereafter. Outside of January, the profitability of the early-stage momentum strategy appears to be more persistent in that it is more profitable in 1-12 months, and is also profitable in the third year following formation; the average profit is 0.550% (t-statistic=4.04) for the first year, and is 0.171% (t-statistic=1.65) for the third year. As there is no significant return reversals, the preliminary result, after excluding the January seasonality, appears to support the momentum life cycle hypothesis. However, as the performance persistence of the early-stage momentum is weak, it is not clear whether it is driven by the multiplication of the low-turnover and price momentum effects. We shall see in later sections that the persistence disappears once we control for the two effects and other confounding factors. 8

10 3 Price momentum, turnover premium, and the momentum life cycle hypothesis 3.1 Fama-MacBeth regression This section examines the performance of the price momentum and turnover-based strategies by using a Fama-MacBeth (1973) style cross-sectional regression advocated by George and Hwang (2004) and Grinblatt and Moskowitz (2004) to measure and compare the return patterns related to past returns and turnover. We also include the ratioofpriceto52-weekhighintheregressionmodelasacontrolvariabletocontrolfor the 52-week high effect documented in George and Hwang(2004). Specifically, George and Hwang (2004) use the ratio of the current price level to the past 52-week high (hereafter the 52-week high or 52wh for short) to measure the performance of stocks, and demonstrate that the 52-week high momentum strategy subsumes Jegadeesh and Titman s price momentum strategy and Moskowitz and Grinblatt s (1999) industry momentum strategy. As the returns on the 52-week high strategy do not exhibit longterm reversals, they claim that short-term momentum and long-term return reversals are separate phenomena. 3 The cross-sectional monthly regression model is as follows: R i,t = b 0j,t +b 1j,t R i,t 1 +b 2j,t Size i,t 1 +b 3j,t 52whW i,t j +b 4j,t 52whL i,t j + b 5j,t PriceW i,t j +b 6j,t PriceL i,t j +b 7j,t TurnoverL i,t j + b 8j,t TurnoverH i,t j +ε i,t, (1) where R i,t 1 and Size i,t 1 are the return and the market capitalization of stock i in month t 1, and the variables are included to control for size effect and the microstruc- 3 George and Hwang (2007) further show that long-term return reversals occur only for winner stocks, and argue that the long-term return reversals can be explained by a rationality-based story based on capital-gain tax, thus supporting their argument that short-term momentum and long-term return reversals are two separate phenomena generated by different market forces. 9

11 tureeffect duetobid-askbounce. Thedummy variable52whw i,t j (52whL i,t j )takes the value of 1 if stock i s 52-week high measure as ranked in the top (bottom) 10% in month t j. PriceW i,t j (PriceL i,t j ) is dummy variable which takes the value of 1 if stock i s past 6-month return is ranked in the top (bottom) 10% in month t j. TurnoverL i,t j (TurnoverH i,t j ) is the low (high) turnover dummy which takes the value of 1 if stock i s past 6-month turnover is ranked in the bottom (top) 10% in month t j. We perform 12-month (j=1,...,12 to j=49,...,60) cross-sectional monthly regressions. Thus, the return of the pure price winner (loser) portfolio with the 12-month holding period in the month t is calculated as b 5j,t = j=1 b 5j,t (b 6j,t = j=1 b 6j,t), and the difference between b 5j,t and b 6j,t is the profit of the price momentum strategy. The t-statistics of the coefficient estimates are adjusted by the Newey-West (1987) robust standard errors. Note that unlike common Fama-MacBeth regression applications where variables of interest that normally take the form of continuous variables are used as explanatory variables, here dummy variables are used. As a result, the coefficient of a dummy variable has a quantitative meaning in that it captures the net return of the portfolio related to that dummy variable. For example, b 0j,t captures the average return of the rest of the sample stocks (i.e., the medium portfolio) that are neither winners nor losers, which are formed in month t j and held in month t. Thus, b 5j,t and b 6j,t, respectively, capture the incremental returns of winner and loser portfolios over the medium portfolio. A major advantage of the regression approach over the traditional portfolio formation method as used in Jegadeesh and Titman (1993) is that it can isolate out the confounding effects such as the size effect and the bid-ask bounce. A second advantage of this method is that it allows for comparing the competing performances of various trading strategies simultaneously. [Insert Table 3 about here] 10

12 The results are reported in Table 3. We begin with the performance of price momentum strategies. Table 3 indicates that the profit of price momentum strategy is positive in the first year, but reversed to become negative thereafter. Specifically, the average monthly return is 0.588% (t-statistic=5.11) in the first year following formation, but is % (t-statistic=-2.38) in the second year, and % (tstatistic=-2.87) in the fifth year. The results are similar when January observations are excluded. A closer look at the performance of winner and loser portfolios indicates that, when excluding the January seasonality, the short-term momentum profitability mainly comes from the underperformance of the price losers, whereas the long-term return reversals, which are statistically significant only in the fifth year, can be attributed to reversals of price winners. Specifically, outside of January, the average return of price loser portfolio for the first year is % (t-statistic=-8.22), but that of price winner portfolio is 0.117% (t-statistic=1.27). By contrast, for the return reversals in the fifth year, the average return of price winner portfolio for the fifth year is % (t-statistic=-2.75) per month, but that of price loser portfolio is 0.024% (t-statistic=0.27). By comparing the performance of winner and loser portfolios, it can be seen that winners experience reversals after the first year following formation, whereas losers experience stronger persistence in performance, especially outside of January. Outside of January, losers earn negative average returns for four consecutive years following formation, and the returns are significant in the first and the third years. We next examine the performance of turnover-based strategies. Two interesting features emerge in Table 3. First, the profits of turnover strategies are almost all significantly positive over the entire 5-year holding period; the only exception is that in the third year, the profit is not statistically significant. The turnover premium is only slightly weaker when January observations are excluded. In comparison with the 11

13 results reported in Table 1, Table 3 indicates that the turnover premium is stronger after controlling for the confounding effect and other strategies. Second, perhaps the most striking result about the turnover premium is that it can be entirely attributed to the persistent underperformance of the high-turnover stocks. Specifically, high-turnover stocks earn persistent and negative future returns all over the 5-year holding period, regardless of the inclusion of January observations. The results are consistent with the findings of the low-turnover effect whereby lowturnover stocks earn higher average return than high-turnover stocks (e.g., Datar et al., 1998; Amihud et al., 2005; Chou et al., 2013). To briefly summarize, the empirical results based on the regression analysis are mostly consistent with the findings reported earlier with one exception: the turnover effect is stronger when we control for other confounding factors. 3.2 The momentum life cycle hypothesis Our empirical results so far show that price momentum and the turnover effect appear to be two separate effects, thus casting doubt on the momentum life cycle hypothesis. In this subsection, we formally examine whether the momentum life cycle hypothesis holds in a regression context that controls for other confounding factors. As a first test, we examine the performance of early-stage momentum strategies by incorporating two interaction terms on low-turnover winners and high-turnover losers into the regression model. The Fama-MacBeth regression results are reported in Table 4. [Insert Table 4 about here] Table 4 indicates that the early-state momentum strategy is significantly profitable in the first two years following formation, but becomes insignificant thereafter. The 12

14 early-stage momentum strategy earns an average profit of 1.428% (t-statistic = 4.78) in 1-12 months, and 0.415% (t-statistic=1.73) in months. Outside of January, the profitability of the early-stage momentum strategy persists for up to 3 years following formation; the average profit is still 0.529% (t-statistic=1.88) per month in months. The results in Table 4 indicate that the early-stage momentum profitability is more persistent after controlling for confounding effects due to the size effect and the bid-ask bounce. Note that in Table 3 for the first year the price momentum profit is 0.588%, and the turnover premium is 0.457%; the sum of the two is 1.045%, which is much smaller than that of the early-stage price momentum, which is 1.428% per month. The results for the second year are also similar. This suggests that price momentum and the turnover effect are correlated in a way that gives rise to a higher profit of the early-stage momentum strategy, thus supporting the MLC hypothesis. However, a closer look at the results in Table 4 reveals that the persistence of the early-state momentum profitability mainly comes from the high-turnover price losers. The average estimates of the interaction term on high turnover and price losers are negative all over the 5-year holding period, and are mostly statistically significant, especially outside of January. In contrast, the average return on the lowturnover price winner portfolio is significant only for the first year; the average return is 0.403% (t-statistic = 2.13). There is no persistence of performance for low-turnover price winners, especially outside of January. Thus, our empirical results indicate that, after controlling for some confounding factors, the evidence only partially supports the MLC hypothesis. In other words, only the early-stage price losers perform persistently in accordance with the prediction of the MLC hypothesis. There is no evidence supporting the winner side of the MLC story. 13

15 But how does the MLC hypothesis perform in conjunction with price momentum and the turnover effect? To examine this, we incorporate 6 dummy variables into the regression model: two dummies on past returns, two dummies on turnover, and two interaction terms capturing the early-stage momentum. The cross-sectional monthly regression model is as follows: R i,t = b 0j,t +b 1j,t R i,t 1 +b 2j,t Size i,t 1 +b 3j,t PriceW i,t j +b 4j,t PriceL i,t j + b 5j,t TurnoverL i,t j +b 6j,t TurnoverH i,t j +b 7j,t TurnoverL i,t j PriceW i,t j + b 8j,t TurnoverH i,t j PriceL i,t j +ε i,t, (2) While the interaction terms are intended to capture the returns on low-turnover price winner portfolio and high-turnover price loser portfolio, it should be noted that the coefficients of the two interaction terms, b 7 and b 8, do not capture the total or net returns on the two portfolios, but the incremental returns beyond b 3 through b 6. Essentially, if the two effects are uncorrelated, then b 3 through b 6 would suffice to reflect return patterns related to them. However, if the two effects are correlated as suggested by the MLC hypothesis, one would expect b 7 to be positive and b 8 to be negative, and the signs persist over a longer period of time. The results are reported in Table 5. [Insert Table 5 about here] The most striking result in Table 5 is that the profitability of early-stage momentum strategies totally disappears. The results presented at the bottom of Table 5 indicate that the returns are generally very small, suggesting that the MLC hypothesis loses its explanatory power after controlling for the price momentum and turnover effects in the regression model. Specifically, in comparison with the results in Table 4, Table 5 indicates that both low-turnover price winners and high-turnover price losers 14

16 no longer exhibit persistence in performance. Outside of January, the returns of the low-turnover price winners, which are predicted to be positive according to the MLC hypothesis, are even negative for the first two years following formation. Although the high-turnover price losers still earn negative returns, the returns are statistically insignificant all over the 5-year holding period. In contrast, the turnover premium remains persistently positive all over the 5-year holding period, but it is statistically significant only for the first two years. Again, the turnover premium can be mostly attributed to the negative performance of highturnover stocks. For price momentum strategies, the profitability only lasts for the first year following formation, and is reversed to become negative thereafter. The first-year momentum profit is stronger outside of January, which is 1.007% (t-statistic=5.99) per month, and mainly comes from losers, whose average monthly return is % (tstatistic=-5.43). The long-term return reversals are significant in the second and the fifth years. Specifically, the profit is 0.708% (t-statistic=4.34) per month for the first year, but becomes % (t-statistic=-2.62) in the second year, and % (t-statistic=-3.19) in the fifth year. Overall, the return patterns of price momentum and turnover strategies are the similar to those reported in Table 3. Thus, the empirical evidence does not support the MLC hypothesis, suggesting that the persistence of early-stage momentum strategies is driven by the multiplication of the price momentum and turnover effects. It also suggests that price momentum and the turnover effect might be two separate effects. However, as the evidence is somehow indirect in that it is purely based on statistical tests, it would be of interest to examine whether the two effects are indeed driven by different forces or investor behavior, which is the goal of the next section. 15

17 4 What drives price momentum and the turnover premium? This section explores the driving forces behind price momentum and the turnover premium from both rational and perspectives. As both effects are widely examined in the literature, our purpose is not to solve the two puzzles, but to investigate the extent to which the two effects are affected by fundamental and behavioral forces. Essentially, our purpose in this section is to provide more direct evidence from an economic perspective, rather than from a statistical viewpoint. We first explore whether the two effects are rationally driven by examining if their returns can be explained by well-known asset pricing models. We then examine whether the two effects exhibit behavioral patterns. 4.1 Risk-based tests This subsection examines whether price momentum profits and the turnover premium, can be explained by risk-based theories. To this end, we consider two well-known asset-pricing models that have been used to evaluate the performance of the price momentum strategy in the literature: Fama and French s (1993) three-factor model and the macroeconomic factor model proposed by Chen, Roll, and Ross (1986); the macroeconomic model is also referred to as the CRR model. The CRR model is used because Liu and Zhang (2008) show that the growthrelated macroeconomic factor on industrial production, denoted MP, explains more than half of the price momentum profit. Their empirical results echo the findings of Chordia and Shivakumar (2002) and Cooper, Gutierrez, and Hameed (2004), who show that price momentum profits are strong in economic expansions, but not in 16

18 recessions. Note, however, that the CRR model in its original form is not a pricing model, but a return generating process in the spirit of Ross s (1976) arbitrage pricing theory. To come up with a pricing formula, we need to estimate the factor risk premium associated with each of the macroeconomic factors. To do so, we choose 49 industry, 10 BM, and 10 Size portfolios, a total of 69 portfolios, as the test assets. 4 For each month from January 1963 to November 2011, factor loadings are estimated for each test asset over the prior 60 months. Fama-MacBeth cross-sectional regression of portfolio returns on the factor loadings is then estimated, which gives the estimates of factor risk premiums. The factor risk premiums are plugged back into the factors, resulting the estimates of the returns on the factor portfolios. For each month from January 1963 to November 2011, factor loadings are estimated for each test asset over the prior 60 months. Fama-MacBeth cross-sectional regression of portfolio returns on the factor loadings is then estimated, which gives the estimates of factor risk premiums. The factor risk premiums are plugged back into the factors, resulting the estimates of the returns on the factor portfolios. As in George and Hwang (2004), we estimate the risk-adjusted return on a certain portfolio as the intercept from a time series regression of monthly returns of the portfolio (the average coefficient of the corresponding dummy variable) on the con- 4 Liu and Zhang (2008) use 10 size, 10 book-to-market, and 10 momentum one-way sorted portfolios as the test assets. We exclude momentum portfolios to avoid potential spurious look-ahead bias suggested by Lo and MacKinlay (1990). Industry portfolios are used because Lewellen, Nagel, and Shanken (2010) show that inferences based on the size-bm portfolios alone are misleading. More specifically, they show that the size-bm portfolios exhibit a strong factor structure is that more than 90% of the time-series variation of the portfolio returns and more than 75% of the cross-sectional variation in the average return are explained by popular empirical asset-pricing models such as the Fama-French 3-factor model. The performance of most existing models is disappointing when they implement the tests on samples expanded beyond the size-bm portfolios. They therefore suggest that researchers use larger samples, beyond the size-bm portfolios, such as expanding the size-bm portfolios with industry portfolios, to allow for more powerful inferences. 17

19 temporaneous factors. The empirical results are reported in Table 6. To save space, the results for intercept and the control variables are not reported. Panel A reports the results based on the Fama-French 3-factor risk adjustment, while Panel B reports the results based on the CRR risk adjustment. 5 [Insert Table 6 about here] Panel A of Table 6 indicates that the return patterns on price momentum strategies remain the same under the Fama-French 3-factor risk adjustment. However, the turnover premium becomes strongly significant all throughout the 5-year holding period. The profits of early-stage momentum strategies remain insignificant. Clearly, the Fama-French 3-factor model fails to explain either of the two effects. Panel B shows that both price momentum and the turnover premium are weaker under the CRR risk-adjustment. For price momentum strategies, the first-year profit is 0.358% per month, which is only about 35.6% of the profit reported in Table 5, which is 1.007%. In other words, the CRR model explains 64.4% of the first-year momentum profit. In particular, the returns on price winner portfolio become mostly insignificant all over the 5-year holding period, consistent with the finding of Liu and Zhang (2008). Liu and Zhang (2008) argue that winner stocks are more sensitive to changes in expected growth than loser stocks because of their higher expected growth. Perhaps a bit puzzling is that the third-year momentum profit becomes significant; the average momentum profit is 0.300% (t-statistic=2.83) per month, and is 0.282% (t-statistic=2.75) after excluding the January seasonality. For the turnover premium, Panel B indicates that the returns on low-turnover portfolio remain mostly significant all over the 5-year holding period, but the turnover 5 The data for the Fama-French three factors are obtained from Kenneth French s data library: The data for CRR factors are downloaded from Xiaolei Liu s website at: fnliu/research.html 18

20 premium is significant only for the first year. In comparison with the results in Table 5, however, the low-turnover returns are generally smaller after returns are risk adjusted using the CRR macroeconomic factor model. As a result, the turnover premium is significant only for the first year following formation, suggesting that the CRR model also provides very good account of the turnover premium, except for the first year premium. Specifically, the first-year turnover premium is 0.406% (tstatistic=2.94) per month. In comparison with the raw turnover premium, 0.556%, the CRR model explains about 27% of the first-year premium. Outside of January, the first-year risk-adjusted turnover premium drops to 0.294% (t-statistic=2.06) per month, which is about 56.21% of the non-january premium reported in Table 5, 0.523% (i.e., 0.294/0.523= %). Overall, the evidence suggests that the CRR macroeconomic factor model provides some explanatory power on both price momentum and the turnover premium. 4.2 Behavioral tests Thus far we have documented that price momentum profits and the turnover premium cannot be fully explained by well-known asset-pricing models. In this subsection, we explore whether behavioral theories help to explain the two effects. Recall from Table 3 that high-turnover stocks exhibit strong persistence in negative returns for up to five years, and that winner stocks also exhibit persistent return reversals starting from the second year following formation. Table 6 indicates that even after risk adjustment, the underperformance of high-turnover stocks remains persistent, though weaker. Regardless of risk adjustment, loser stocks experience persistent negative performance for up to three years. The results seem to suggest that such long-lasting underperformance, at least in part, is due to mispricing. 19

21 Miller (1977), for example, shows that in the presence of differences of opinion and short-sale constraints, asset prices will be dominated by optimistic investors, thereby causing overpricing, followed by lower subsequent returns as the mispricing is virtually corrected. In a frictionless market, the overpricing would disappear quickly because professional arbitrageurs would have exploited the arbitrage opportunity. Practically though, arbitrage is costly, and any systematic mispricing might not be corrected in situations in which the arbitrage costs exceed the arbitrage benefits (Shleifer and Vishny (1997)). Thus, a limit-of-arbitrage argument provides a reasonable basis for predicting why a mispricing-driven anomaly can persist over time (Shleifer and Vishny, 1997; Ali, Hwang, and Trombley, 2003). If the unerderpermance of low-turnover and winner and loser stocks is a result of mispricing, according to predictions by Shleifer and Vishny (1997) and Ali, Hwang, and Trombley (2003), the magnitude of mispricing should be greater for stocks that are subject to higher degrees of arbitrage risk. Also, the mispricing will exhibit patterns related to investor sentiment, because investors overreaction is deepened during periods of extreme sentiment. We hence explore two behavioral hypotheses in this subsection. The first hypothesis examines whether the two effects is the result of idiosyncratic risk limiting arbitrage. The second hypothesis explores whether the two effects exhibit different patterns during periods of different investor sentiment Idiosyncratic risk Idiosyncratic risk has been widely identified as the primary arbitrage holding cost that prevents arbitrageurs from correcting the mispricing (e.g., Shleifer and Vishny, 1997; Pontiff, 2006). It is also viewed as a proxy for noise trader risk, which is present when the market of the asset is populated with irrational traders, whose trading drives an 20

22 asset s price away from its fundamental value (Barberis and Thaler, 2003). The effect of idiosyncratic risk on price momentum and the turnover premium has been explored in the literature. McLean (2010) tests whether the persistence of the momentum and reversal effects is the result of idiosyncratic risk limiting arbitrage, and finds that reversal is prevalent only in high idiosyncratic risk stocks. However, he finds that price momentum is not related to idiosyncratic risk. As price momentum generates a smaller aggregate return than reversal, McLean (2010) suggests that idiosyncratic risk limits arbitrage in reversal mispricing, and that transaction costs are sufficient to prevent arbitrageurs from eliminating momentum mispricing. For the turnover premium, Chou, Huang, and Yang (2013) show that the premium is greater for stocks with higher idiosyncratic volatility, thus also consistent with the mispricing explanation based on arbitrage risk. By incorporating interaction terms into the Fama-MacBeth regression model, we examine whether idiosyncratic risk plays a role in preventing arbitrageurs from fully correcting the mispricing embedded in the two effects. Specifically, we introduce four interaction terms into the Fama-MacBeth regressions: R i,t = b 0j,t +b 1j,t R i,t 1 +b 2j,t Size i,t 1 +b 3j,t PriceW i,t j +b 4j,t PriceL i,t j +b 5j,t TurnoverL i,t j + b 6j,t TurnoverH i,t j +b 7j,t IV i,t j PriceW i,t j +b 8j,t IV i,t j PriceL i,t j + b 9j,t IV i,t j TurnoverL i,t j +b 10j,t IV i,t j TurnoverW i,t j +ε i,t, (3) whereiv i,t denotesstocki idiosyncraticvolatilityattimet, estimatedasthestandard deviation of the residual from a regression of daily returns on market returns over the past 12 months. The results are reported in Table 7. [Insert Table 7 about here] Let us first focus on the interaction between price momentum and idiosyncratic 21

23 volatility. The results in Table 7 indicate that the interaction term on price winner and idiosyncratic volatility is negative throughout the 5-year holding period, and is mostly statistically significant when January observations are excluded. By contrast, the price winner portfolios earns a significant non-january profit of 0.603% (t-statistic=3.90) per month in the first year, and 0.244% (t-statistic=2.00) in the second year, and no significant return reversals thereafter. The results indicate that, after excluding the January seasonality, winners with larger idiosyncratic volatility exhibit weaker short-term performance persistence and stronger long-term return reversals. For loser stocks, the interaction term is positive in 1-12 months, seemingly suggesting that losers with larger idiosyncratic volatility also exhibit weaker short-term performance persistence. However, the interaction term becomes negative when January months are excluded, thus indicating that losers with larger idiosyncratic volatility experience return reversals mainly in January. However, after excluding January observations, price losers experience significant return reversals in the 4th and 5th years; the average return is 0.326% (t-statistic=1.73) per month in the 4th year, and is 0.330% (t-statistic=1.83) in the 5th year. The result is somewhat surprising because our empirical results so far do not indicate long-term reversals for losers. Interestingly, the interaction term is also significantly negative in the last two years; the estimate of the interaction term is (t-statistic=-2.29) in the 4th year, and is (t-statistic=-1.72) in the 5th year. Thus, the results indicate that outside of January losers with larger idiosyncratic volatility experience weaker long-term return reversals, which is not consistent with the mispricing argument based on arbitrage risk. Overall, the results indicate that winners with larger idiosyncratic volatility exhibit weaker short-term return persistence and stronger long-term return reversals, 22

24 consistent with the findings of McLean (2010). The fact that long-term return reversals are stronger for stocks with larger idiosyncratic volatility is consistent with the limits of arbitrage argument whereby greater arbitrage risk deters price correction from mispricing. However, as losers return pattern associated with the idiosyncratic risk does not conform to the prediction based on limits of arbitrage, thus suggesting that just as risk-based models do not fully explain short-term price momentum and long-term reversals, arbitrage risk alone does not fully explain the two phenomena. We now move on to the relation between idiosyncratic risk and the turnover premium. Consistent with our conjecture, returns on high-turnover stocks exhibit strong relation associated with idiosyncratic risk. The coefficient of the interaction term on high turnover dummy and idiosyncratic volatility IV is significantly negative all over the 5-year holding period, suggesting that the negative returns on highturnover stocks are stronger for stocks with larger arbitrage risk. As the coefficient of the high-turnover dummy becomes insignificantly different from zero throughout the 5-year holding period, the empirical evidence suggests that idiosyncratic risk in the major determinant of the turnover premium Investor sentiment Our final investigation explores the relation between investor sentiment and the two effects. Suppose the market is populated with two kinds of investors: noise traders and rational arbitrageurs. Noise traders suffer from behavioral biases such as cognitive dissonance, and as a result their sentiment causes stock prices to deviate from their fundamental values. By contrast, rational arbitrageurs engage in eliminating mispricing resulting from noise traders misreaction, but are subject to short-sale restrictions in the presence of overpricing. 23

25 If the two effects are driven by the interaction of noise traders and arbitrageurs, we conjecture that stocks in the long leg (i.e., winner stocks and low-turnover stocks) and short leg (i.e., loser stocks and high-turnover stocks) will react differently under different states of sentiment. 6 Specifically, during periods of optimistic sentiment, due to disposition effect, which is a form of cognitive dissonance, noise traders are more reluctant to sell short-leg stocks, causing such stocks to continue to underperform in the future. Such persistent underperformance is strengthened because the presence of short-sale restrictions refrains rational arbitrageurs from exploiting the mispricing (see, e.g., Antoniou et al, 2013). In contrast, during periods of pessimistic sentiment when the fear of potential loss is more widespread, noise traders reluctance to sell short-leg stocks is less severe; thus their underperformance is less prominent following periods of pessimism. According to the cognitive dissonance argument, they suffer less from realizing the loss of selling short-leg stocks because they can attribute the loss to the overall market condition. For long-leg stocks, as noise traders engage in their trading more actively during periods of optimistic sentiment, the underpricing of such stocks during such periods is weak. In contrast, since during periods of pessimism noise traders are more reluctant to buy long-leg stocks, the underpricing will be more prominent than during periods of optimism. Hence, we establish three testable hypotheses. Hypothesis 1: Stocks in the short leg will continue to underperform following periods of optimism. 6 Our story builds upon the idea of Antoniou et al. (2013), who show that price momentum profits arise only under optimism, during which losers are more underpriced because, due to cognitive dissonance, investors are more reluctant to realize their loss. 24

26 Hypothesis 2: Stocks in the long leg will continue to outperform following periods of pessimism. Hypothesis 3: Due to short-sale constraints, the magnitude of underperformance of stocks in the short leg following of optimism (i.e., Hypothesis 1) is stronger than that of outperformance of stocks in the long leg following period of pessimism (i.e., Hypothesis 2). We use the sentiment index compiled by Baker and Wurgler (2006), and classify a month as being in the optimistic or pessimistic regime, depending on whether its previous month s sentiment level is above or below the median of the whole sample period. 7 We then perform the same Fama-MacBeth regression as in Tables 5 and 6 for optimistic and pessimistic subsamples. To save space, however, we only report the results for the first year and for the entire 5-year holding period. Panel A of Table 8 reports the results based on raw returns, and Panel B reports the results based on returns that are risk adjusted using the CRR model. [Insert Table 8 about here] Table 8 indicates that return patterns on price momentum fully conform to the prediction of the three hypotheses, but those of turnover-sorted portfolios seem to be unrelated to the sentiment-based story. Let us first focus on price momentum. Consistent with Hypothesis 1, Panel A indicates that losers earn a significant average monthly return of % (t-statistic=- 3.19) for the one year following optimism, and the effect is even stronger after excluding January seasonality; the return becomes % (t-statistic=-6.54). Even for 7 The data on the BW sentiment index are downloaded from Jeffrey Wurgler s website at http: //pages.stern.nyu.edu/$\sim$jwurgler/. However, the BW sentiment index is available only up to December

27 the entire 5-year holding period, the returns remain statistically significant. Panel B indicates that losers underperformance is much weaker under CRR risk adjustment, but remains significant for the first year; losers average return (non-january return) becomes % (-0.159%) per month, and are statistically significant. For winners, Panel A indicates that they earn a significant average monthly return of 0.495% (t-statistic=2.84) for the one year following pessimism, and the return is still 0.344%(t-statistic=2.56) after excluding January seasonality, consistent with the prediction of Hypothesis 2. Winners profits, however, become insignificant after CRR risk adjustment, and are also insignificantly different from zero for the 5-year holding period. Thus, the results are also consistent with the prediction of Hypothesis 3. For the turnover premium, in contrast, Table 8 indicates that the return patterns are about same as those reported earlier; low-turnover stocks do not earn significant incremental returns, whereas high-turnover stocks earn persistent negative returns. The results indicate that the turnover-based return predictability is not significantly related to investor sentiment. 5 Conclusion Lee and Swaminathan (2000) propose a famous momentum life cycle hypothesis in which turnover plays a crucial role is explaining the coexistence of short-term momentum and long-term reversals. The MLC hypothesis claims that low-turnover winners and high-turnover losers experience stronger performance persistence. Based on various tests, however, we show that the MLC hypothesis holds only partially in that only high-turnover losers experience persistence in performance. Further analysis indicates that the performance persistence of high-turnover losers is spurious, and in fact reflects the multiplication of price momentum and the high-turnover effects, which we 26