Realized Return Dispersion and the Dynamics of. Winner-minus-Loser and Book-to-Market Stock Return Spreads 1

Transcription

1 Realized Return Dispersion and the Dynamics of Winner-minus-Loser and Book-to-Market Stock Return Spreads 1 Chris Stivers Terry College of Business University of Georgia Athens, GA Licheng Sun College of Business Old Dominion University Norfolk, VA This version: June 23, We thank Jonathan Albert, Michael Brandt, Bob Connolly, Jennifer Conrad, Mike Cooper, Ro Gutierrez, Marc Lipson, Cheick Samake, John Scruggs, Lee Stivers, Jeff Wongchoti, Yexiao Xu, Sterling Yan, and seminar participants at the University of Georgia, the University of Missouri, the College of William and Mary, Florida State University, Old Dominion University, the Federal Reserve Bank of Atlanta, the Financial Management Association, and the Southern Economic Association for comments and helpful discussions. Please address comments to Chris Stivers ( cstivers@terry.uga.edu; phone: (706) ) or to Licheng Sun ( LSun@odu.edu; phone: (757) ). Stivers acknowledges financial support from a Terry-Sanford research grant.

2 Realized Return Dispersion and the Dynamics of Winner-minus-Loser and Book-to-Market Stock Return Spreads Abstract We document a striking new regularity in the dynamics of winner-minus-loser (WML) stock return spreads, based on past relative return strength; and high-minus-low (HML) stock return spreads, based on book-to-market equity ratios. Specifically, we find that time-variation in the stock market s cross-sectional return dispersion (RD) is negatively related to the subsequent change in WML spreads and positively related to the subsequent change in HML spreads, where change is defined relative to recent realized spreads. These patterns are reliably evident and economically sizable at the 6, 18, and 36-month spread horizon. We present additional evidence to assist in interpretation, including findings that RD is informative about the likelihood of state changes in a regime-switching estimation on stock returns. Collectively, our results suggest that the stock market s RD is a leading indicator of market-state changes and that market cyclicality is important in understanding WML and HML return spreads. JEL Classification: G12, G14 Keywords: Momentum, Reversals, Book-to-Market Equity Ratio, Return Dispersion

3 1. Introduction Cross-sectional variation in expected stock returns tied to past relative return strength and book-to-market equity ratios has an important role in both current financial practice and theory. The reliability, magnitude, and nature of winner-minus-loser (WML) return spreads and highminus-low (HML) book-to-market return spreads has lead to these spreads being proposed as factor-mimicking portfolios in asset pricing models. However, while it is generally agreed that these spreads are at odds with the classic CAPM, there is an ongoing debate as to whether these prominent spreads represent risk factors or anomalies. 1 Such return spreads, by definition, require cross-sectional dispersion in realized stock returns. Thus, it seems plausible that time-variation in the stock market s realized cross-sectional return dispersion might be informative about the dynamics of WML and/or HML return spreads. 2 This paper documents that the market s trend in return dispersion is negatively related to the subsequent change in WML spreads and positively related to the subsequent change in HML spreads, where change is defined relative to recent realized spreads. Over our 1962 to 2005 sample, these empirical regularities are reliably evident and economically sizable at the 6, 18, and 36-month spread horizon. In our view, the time-series behavior of these spreads is important for several reasons. First, time-series regularities may prove theoretically useful in understanding these prominent spreads. Theories, either rational or behavioral, should explain both a spread s unconditional average and its time-series regularities. Second, time-series behavior may have a practical importance for investors, who might vary their loadings on spread strategies in the sense of Avramov and Chordia (2006). A priori, why might the market s return dispersion (RD) be related to subsequent WML or 1 By WML spreads, we mean the return differential between portfolios of past relative winners and portfolios of past relative losers. By HML spreads, we mean the return differential between portfolios of high book-to-market stocks and low book-to-market stocks. For background on WML spreads and/or HML spreads, see the following (to list just a few): DeBondt and Thaler (1985), Lo and MacKinlay (1990), Jegadeesh and Titman (1993) and (2002), Fama and French (1993), (1996), (1998) and (2007), Carhart (1997), Daniel and Titman (1997), Conrad and Kaul (1998), Moskowitz and Grinblatt (1999), Grundy and Martin (2001), Griffin, Ji, and Martin (2003), Conrad, Cooper, and Kaul (2003), Schwert (2003), Zhang (2005), and Petkova and Zhang (2005). 2 In our paper, the stock market s monthly RD is defined as the cross-sectional standard deviation of monthly individual stock returns or disaggregate portfolio returns, depending upon the particular RD metric. 1

4 HML spreads? We suggest competing possibilities. First, a high RD might correspond to market conditions where there is a high dispersion in expected returns. Then, since WML and HML have been proposed as factors to explain cross-sectional variation in expected returns, one might observe a positive intertemporal relation between RD and subsequent WML or HML spreads. A different possibility is that a high RD might be associated with market-state transitions. As economic and financial conditions change, the relative performance of different sectors is likely to change due to changes in expected future cash flows (in the sense of Veronesi (1999)) or changes in risk premia (in the sense of Fama and French (1989)). This suggests substantial cross-sectional changes in valuation during state transitions, which may translate to a high realized RD. If a high RD is associated with market-state transitions, then rational cross-sectional valuation cycles might result in an RD-spread relation. Consider a shift toward a weak economic state (or a market crisis). With the shift to the weak state, the more cyclical stocks (that were the relative winners in the prior good state) may transition to underperformance (relative to less cyclical stocks) and the winners over the past ranking period could suddenly become the current relative losers. If so, then WML spreads should be lower around the market-state transition. Similarly, if the changes in market state are also associated with changes in the relative performance of growth versus value stocks, then RD may also be related to changes in HML spreads. Alternately, if a high RD is associated with market-state transitions, then the intuition from Daniel, Hirshleifer, and Subrahmanyam (DHS) (1998) and Baker and Wurgler (2006) suggests a second potential reason that a high RD might be associated with changing spreads. In the framework of DHS, medium-run momentum profits are generated by price overreaction, attributed to overconfidence with biased self-attribution. At some point, the overreaction in prices will be corrected as investors revise their valuations back toward fundamental values. Thus, if economic transitions or financial crises are associated with changes in the relative valuations of past winners and losers due to shifts in investor sentiment (in the sense of Baker and Wurgler (2006)), then a higher RD might be associated with lower subsequent WML spreads. Similarly, if changes in the relative valuations of value versus growth stocks tends to occur around these transitional times due to changing sentiment, then the RD may be informative about subsequent HML spreads. In this paper, we study how the time-series of both WML and HML spreads are related to time-variation in the market s realized RD. We examine spreads at the following sizable economic 2

5 horizons: (1) the 6-month horizon, because of its prominence in the momentum literature with reliable WML spreads that survive standard risk adjustments (Fama and French (1996)), (2) the 36- month horizon, because this longer-run horizon is in the spirit of sizable economic cycles and because of the long-run reversals in relative strength strategies (DeBondt and Thaler (1985)), and (3) the 18-month horizon, as an in-between comparison. With three horizons, we also hope to further our understanding of how medium-run momentum and long-run-reversal behavior interrelate. We are interested in an RD metric that captures whether the recent RD is relatively high or low, as compared to the market s longer-term past RD environment. We focus on an RD-trend variable that is defined as the difference between the most recent 3-month RD moving average and an older 12-month RD moving average. We examine four alternative RD metrics: a broadmarket RD in individual stocks, a large-firm RD in individual stocks, an industry-level RD using 48 industries, and the RD in 100 book-to-market and size double-sorted portfolios. We investigate both the simple RD-trend and a market-adjusted relative return dispersion (or RRD) trend. 3 For the return spreads, our empirical work differs from prior time-series work by focusing on changes in the realized spreads (rather than the spread level). In our view, the spread changes are attractive because they: (1) are likely to be sharper in picking up changes in market conditions, and (2) should be relatively insensitive to any very long-term trends in the spread levels. For the change in spread variables, we focus on the difference in the realized spread over months t to t + (j 1) and an earlier realized spread over months t 4 to t (j + 3) (relative to the RD-trend that features RD over months t 3 to t 1). The j indicates either 6, 18, or 36 months for the three spread horizons. We stress that our results are robust to alternate timing variations that are similar in concept, for both the RD-trend and the change in spread variables. Over our 1962 to 2005 sample, we document a new striking empirical regularity that describes the time-series of WML and HML spreads. First, we find that the market s RD trend is negatively and substantially related to subsequent changes in WML spreads at the 6, 18, and 36-month return horizons, and for strategies implemented on both individual stocks and industry-level portfolios. 4 3 Since a month s RD should vary with the magnitude of the month s market s return, due to dispersion in firm s market-betas, we construct a monthly RRD that is orthogonal to the month s absolute market return. 4 Our symmetric WML spreads go long (short) stocks whose returns were above (below) a percentile threshold over the ranking period (deciles for the firm WML spreads and quartiles for the industry WML spreads); with the ranking period and holding period having the same length. 3

6 For example, by itself, the variation in the market-adjusted RRD-trend of large-firm stocks explains over 14%, 34%, and 39% of the variability in the subsequent changes in firm-level WML spreads at the 6, 18, and 36-month horizons, respectively. Further, for observations corresponding to the RRD-trend s top quartile (bottom quartile) of values, the mean of the subsequent changes-in-wmlspreads is -10.5%, -20.9%, and -48.9% (6.8%, 20.0%, and 48.7%) for the 6, 18, and 36-month spread horizon, respectively. Subperiod results are consistent. Next, we find that the market s RD trend is positively related to subsequent changes in HML spreads. For example, by itself, the variation in the market-adjusted RRD-trend of the book-tomarket/size portfolios explains over 12%, 22%, and 25% of the variability in the subsequent changes in HML spreads at the 6, 18, and 36-month horizons, respectively. 5 Further, for observations corresponding to the RRD-trend s top quartile (bottom quartile) of values, the mean of the subsequent changes-in-hml-spreads is 5.4%, 20.8%, and 27.9% (-5.6%, -9.2%, and -26.6%) for the 6, 18, and 36-month spread horizon, respectively. Subperiod results are again consistent. Collectively, the intertemporal RD-spread relations suggests that a high RD is associated with market-state transitions. Consistent with this view, we also document that the RD-trend is both negatively related to the forward-looking component of the change in WML spread variables and positively related to the lagged component of the change in WML spread variables; and vice versa for the two components of the change in HML spread variables. Next, recall that we proposed two possible mechanisms for the market-state transition explanation, with the second one tied to investor sentiment. Accordingly, we present auxiliary evidence to explore whether our primary findings appear to be: (1) pervasive with a market-wide economic interpretation, or (2) concentrated in small stocks or fringe stocks that presumably have more subjective sentiment-related valuations. We first document that the RD-WML relations are evident in WML spreads that include only large firms and for WML spreads that exclude the extreme 10% of past winner and losers. Second, the RD-spread relations remain virtually unchanged after controlling for other state variables suggested by related time-series work in Chordia and Sarkar (2002) and Cooper et al (2004). Further, in our setting, the RD-trend dominates these other explanatory variables. Finally, the RD-spread relations also remain reliably evident when using a 5 Here, we refer to the difference between the average return for the two highest decile-portfolios and the two lowest decile-portfolios from sorting stocks on their book-to-market equity ratio, from the Kenneth French data library. 4

7 sentiment-adjusted RD that is orthogonal to the sentiment index of Baker and Wurgler (2006). We also directly consider spread behavior in a two-state regime-switching market, where certain stocks perform relatively better in good states and other stocks perform relatively better in poor states. Our regime-switching results show that: (1) market-state transitions should be associated with lower subsequent WML returns, and (2) a higher RD-trend is associated with market-state transitions, especially good-to-bad state transitions. Relatedly, we also document that NBER recessionary months tend to be preceded by relatively high RD values. Collectively, our auxiliary results suggest a pervasive, market-wide economic interpretation for the RD-spread relations. To sum up, we document a sizable and pervasive relation between the market s RD and both WML and HML spreads. We offer an interpretation which suggests that RD is a leading indicator of market-state changes and that market cyclicality is important in understanding WML and HML spreads. However, this interpretation does not flow from a formal theoretical model but rather from a two-state, return-generating analytical framework, intuition, and related empirical evidence. Thus, our results pose a challenge to theorists working on the behavior of WML and HML spreads. The horizon consistency in our findings indicates there is a common temporal influence in mediumrun momentum and longer-run reversals. Further, our results take a step towards understanding temporal commonalities in WML and HML spreads. Regardless of the theoretical underpinnings, our findings seem likely to have a practical importance for investors, such as hedge funds, who might vary their loadings on spread strategies. Section 2 discusses related literature and hypothesis development. Section 3 presents our data and variable construction. Section 4 presents our main empirical findings and Section 5 presents additional auxiliary evidence to assist in interpretation. Section 6 concludes. 2. Related Literature and Hypothesis Development The performance of WML spreads remains an ongoing puzzle in financial economics, with mediumrun WML spreads (in the 3 to 12 month range) exhibiting reliable profits but with longer-run WML spreads tending to be negative. However, there has been relatively little work on the timeseries of WML spreads. Our WML work is novel in that we examine how the change in WML spreads is related to the market s realized RD and we jointly examine the dynamics of medium-run 5

8 momentum and long-run reversals. The literature on WML spreads is vast, especially for the medium-run momentum phenomenon. Here, we only discuss recent studies that suggest momentum profits are related to the market state or the business cycle. Chordia and Shivakumar (2002) find evidence that momentum profits can be linked to business cycles and predicted by lagged macroeconomic variables. 6 Cooper, Gutierrez, and Hameed (2004) present evidence that momentum profits are only reliably positive following positive 3-year market returns. They conclude that models of asset pricing, both rational and behavioral, need to incorporate (or predict) such regime switches. Avramov and Chordia (2006) show that an optimizing investor who conditions on business cycle variables can successfully vary their momentum exposure during different economic times. Return spreads based on book-to-market equity ratios have also been widely documented and debated in the finance literature. The so-called value-versus-growth spread refers to the observed phenomenon where stocks with a high book-to-market ratio tend to have higher average returns than stocks with a low book-to-market ratio. See, e.g., Fama and French (1993), (1996) and (1998), Daniel and Titman (1997), Conrad, Cooper, and Kaul (2003), Zhang (2005), and Petkova and Zhang (2005) for perspective and recent evidence on HML spreads. Our HML work is novel in that we examine how the change in HML spreads at various horizons is related to the market s realized RD The Time-variation in the Dispersion of Expected Returns Hypothesis Cross-sectional variation in expected returns is one source behind generating dispersion in realized returns. Thus, if: (1) a high dispersion in realized returns is associated with economic times that have a higher cross-sectional variation in expected returns, and (2) a higher cross-sectional variation in expected returns is associated with higher subsequent WML spreads and/or HML spreads, then a relatively high RD may be associated with higher subsequent WML and/or HML spreads. Evidence in Conrad and Kaul (1998) and Bulkley and Nawosah (2007) suggest that crosssectional variation in expected returns may have a material role in understanding momentum 6 However, Cooper, Gutierrez, and Hameed (2004) find that the results in Chordia and Shivakumar are not robust to methodological adjustments that guard against market frictions and penny stocks driving the results. Griffin, Ji, and Martin (2003) find that the results in Chordia and Shivakumar tend to not hold in other countries. 6

9 profits. The intuition is that the realized returns of stocks with high expected returns should tend to be relative winners and stocks with lower expected returns should tend to be the relative losers. Next, differences in firms book-to-market equity ratios have been proposed to proxy for crosssectional differences in firms exposure to a distress risk factor. If economic times with a particularly high value-minus-growth risk premium are associated with a higher realized RD, then RD may be associated with higher subsequent HML spreads. See Cohen, Polk, and Vuolteenaho (2003) for recent evidence on time-varying book-to-market ratios and HML return spreads. The possibilities discussed in this subsection imply a time-varying dispersion in expected returns hypothesis, where RD may be positively related to both subsequent WML spreads and HML spreads The Market-state Transition Hypothesis It is well documented that the mean of HML spreads and 6-month WML spreads are reliably positive. However, the realized spreads exhibit substantial time-series variability. For example, in our sample, the realized HML spreads (WML spreads) are negative for 42.2%, 35.9%, and 26.7% (24.4%, 34.7%, and 58.2%) of the time for the 6, 18, and 36-month spread horizons, respectively. The market-state transition hypothesis follows from the possibility that: (1) a high RD is associated with market-state transitions, and (2) the time-varying behavior of the WML and HML spreads is related to changes in the market state. Here, we first offer an example to illustrate the intuition behind the market-state transition hypothesis. Consider a two-state stock market, where the good-regime is the predominant regime with an expected duration of 48 months. The bad-regime has an expected duration of 18 months and is presumably associated with recessions or other financial crises. The true market state is unknown, in real time, but investors can learn about the state in the sense of Lewellen and Shanken (2002). Next, assume three different stock types; Stock A (representing highly cyclical stocks), Stock B (representing stocks of average cyclicality), and Stock C (representing less cyclical stocks), which have unconditional one-month expected returns of 1.2%, 1%, and 0.8%, respectively. Further, assume that Stocks A, B, and C have one-month mean returns of 1.70%, 1.30%, and 0.60% in the good-regime and -0.13%, 0.20%, and 1.33% in the bad-regime, respectively. Such differences in regime-specific mean returns could be attributed to at least two factors. First, times that were classified as a bad-regime (good-regime), ex post, are likely to have experienced 7

10 negative (positive) earnings surprises in real time, especially for highly cyclical stocks. Second, Fama and French (1989) argue that market-wide risk-premia are higher when economic conditions are weak. If so, then stock prices should tend to fall during transitions to the bad-regime as the market-wide risk-premium increases (and vice versa). These two effects could translate to variation in realized mean returns across regimes. Thus, the regime-specific means are interpreted as realized subset means associated with an economic outcome, rather than conditional risk premia. 7 The differences in regime-specific means affect the WML spreads in two opposing ways. First, the cross-sectional variance in regime-specific means is greater than the cross-sectional variance in unconditional expected returns. Thus, WML spreads for within regime outcomes will be greater than the spreads implied solely by the unconditional expected returns. 8 However, across-regime outcomes should be associated with negative WML spreads, because stocks that perform relatively well in one regime tend to perform relatively poorly in the other regime. The net impact of regime switching to WML spreads is unclear and will vary with the spread horizon. We calculate the average WML spreads in this market for both a symmetric 6-month and 24- month strategy that buys the relative winner and shorts the relative loser over the ranking period. The average 6-month WML spread is nearly twice that suggested by the cross-sectional variation in unconditional expected returns (0.76% versus 0.40% per month), with 18% of the realized WML spreads being negative. Conversely, the average 24-month WML spread is negative and appreciably lower than that suggested by the cross-sectional variation in unconditional expected returns (-0.24% versus 0.40% per month), with about 63% of the realized 24-month WML returns being negative. Thus, in this simple two-state framework, changes in the market state should be associated with subsequent negative WML spreads. Further, consistent with the WML stylized facts, average medium-run (long-run) WML spreads are appreciably greater than (lower than) that suggested solely by the cross-sectional variation in unconditional expected returns. To formalize the intuition from this illustrative example, we also offer a formal analytical twostate framework to analyze how regime-switching can influence WML return behavior. Our frame- 7 Consistently, over our 1962 to 2005 sample, the mean return of the CRSP value-weighted stock index is 0.40% per month during recessionary months and 0.98% per month during expansionary months. 8 By within regime ( across regime ), we mean outcomes for the WML strategy where the ranking period and the holding period are within the same uninterrupted regime (across different regimes). By outcome, we mean the profit from a single ranking-period/holding-period event. 8

11 work starts from the decomposition of the weighted related strength strategy in Lo and MacKinlay (1990) and Conrad and Kaul (1998). We then incorporate the autocovariance function with regimeswitching from Timmermann (2000). For brevity, details are in Appendix A. During market-state transitions, RD may be high because of sizable cross-sectional variation in equity re-valuations due to: (1) changes in expected cash flows with economic sector reallocations as the relative performance of more cyclical versus less cyclical stocks shifts with the market state (or the relative performance of value versus growth stocks), and/or (2) shifts in risk premia with the changing market state. 9 Prior studies report some evidence that a higher RD may be associated with market-state changes. For example, Stivers (2003) notes that RD is higher during economic recessions and finds that RD has incremental information about subsequent market volatility. Loungani, Rush, and Tave (1990) find that RD tends to lead unemployment, which suggests a link between RD and economic reallocation across firms. The two-state framework in this subsection clearly suggests a negative relation between the realized RD and subsequent WML spreads. However, under this subsection s framework, the prediction between RD and subsequent HML spreads is unclear. If the RD is equally associated with transitions to a value-over-growth state and a growth-over-value state, then we would not expect to see a relation between RD and subsequent HML spreads (since the HML is a unidirectional value-over-growth state). However, it is possible that RD may tend to be relatively larger for the transition to a value-over-growth state than for a growth-over-value state (or, vice versa). If so, then one may observe a relation between RD and subsequent HML spreads (although the direction of the relation is unpredictable, a priori). Thus, the two-state framework in this subsection implies a market-state transition hypothesis, where: (1) the RD may serve as a leading indicator of market-state transitions, and (2) market cyclicality is important in understanding the dynamics of WML and HML spreads. Fundamental cross-sectional valuation cycles might generate these time-series patterns. However, another possibility is that stock valuation cycles might follow from time-varying investor sentiment. 9 This possible interpretation of RD also seems consistent with the prior use of RD-type metrics as a measure of aggregate firm-level information flows (see, e.g., Bessembinder, Chan, and Seguin (1996) and Lowry, Officer, and Schwert (2006)). Since the cross-sectional dispersion in market-betas should generate some realized RD, a month s RD should vary positively with the month s absolute market return. Thus, by a high RD, we mean high beyond the variation tied to the market s return. 9

12 3. Data and Variable Construction 3.1. Data Sources Our empirical work features stock return data from two sources. For U.S. individual stocks, we examine monthly NYSE and AMEX stock returns from CRSP. We also use the following monthly, value-weighted portfolio returns from the Kenneth French data library: (1) 48 industry portfolios, (2) decile portfolios based on stocks book-to-market equity ratios, and (3) 100 book-to-market and size-based portfolios, formed using a double-sort (10 x 10) of a stock s book-to-market equity ratio and market capitalization. Following Jegadeesh and Titman (1993) and Conrad and Kaul (1998), we focus on the period from January Our sample extends through December Our study also uses the following: (1) business cycle data from the National Bureau of Economic Research (NBER), (2) the yield of Moody s BAA bonds, Moody s AAA rated bonds, 10-year T- notes, and 3-month T-bills from the Federal Reserve Statistical Release H.15, (3) the aggregate dividend payout data from CRSP, and (2) the sentiment index from Baker and Wurgler (2006) Measuring WML and HML Return Spreads Our work features the percentile-based WML strategy. This strategy forms a zero-cost portfolio by starting with an equally-sized long and short position, based on the relative performance of stock returns over the lagged ranking period. For the ranking periods, we use the standard skip-a-month case (where the ranking period is gapped by one month from the holding period). More specifically, for our firm-level decile strategy, we rank NYSE and AMEX stocks into deciles based on their j-month ranking-period return (months t (j + 1) through t 2 with the the skipa-month). Equally-weighted, decile-portfolios are formed based on this ranking-period sort. Our firm-level WML spread is the return of the top decile portfolio (the winners) less the return of the bottom decile portfolio (the losers). The positions are held for the subsequent j-month period (months t to t + (j 1)). We exclude stocks priced less than five dollars at the beginning of each holding period to minimize microstructure issues related to illiquid and low-priced stocks. For our primary firm-level WML series, we also require a stock to be in the top 80 th percentile by market capitalization in the last month of the ranking period. This choice ensures the smallest micro-cap stocks are not driving our results. As explained in our introduction, we examine spreads from 10

13 symmetric 6-month, 18-month, and 36-month strategies, where the ranking and holding periods are the same length (so j equals either 6, 18, or 36). We also briefly examine two alternate firm-level WML strategies. First, we examine a large-firm only WML series, where a stock s market capitalization must be in the top 20 th percentile in the last month of the ranking period in order for it to be selected for the winner or loser portfolio. Second, we examine a WML series that excludes the extreme 10% of winners and losers. This less-extreme strategy goes long the decile-9 winners and shorts the decile-2 losers. For our industry WML spreads, we perform a similar procedure on the 48 industry returns, except with a quartile threshold so the winner and loser groupings contain a sizable number of 12 industries. Quartiles are close to the 30-percentile threshold in Moskowitz and Grinblatt (1999). In our time-series empirical work, we use the following timing convention. The WML spread for month t, W ML j t, refers to the aggregate WML return for the j-month holding period over month t to month t + (j 1). The corresponding ranking period is over months t (j + 1) to t 2 to allow for the one-month gap between the ranking and holding periods. Thus, one important difference between our approach and previous time-series work in Jegadeesh and Titman (1993), Chordia and Shivakumar (2002), and Griffin, Ji, and Martin (2003), is that their momentum profits for a given month use an averaging across the last n investment portfolios and thus reflect n different ranking periods, where n is the number of months for the ranking and holding period (typically 6). In contrast, in our work, each month s WML spread corresponds to the WML outcome from a single ranking period/holding period event. Our timing convention is more appropriate for our time-series analysis because the WML outcome for month t corresponds directly to the explanatory variables up through month t 1. For our HML spreads, we use the same timing convention as for the WML spreads (but, of course, the ranking period is not applicable for the HML spreads). Our monthly HML spread is the difference between the average return of the two highest decile portfolios and the two lowest decile portfolios, using the book-to-market decile-portfolios from the French data library. Table 1 reports descriptive statistics for the WML and HML spreads featured in this paper. Note that: (1) the 6-month WML series are reliably positive for both the firm-level and industrylevel spreads, consistent with the momentum literature; (2) the 36-month WML series are negative, on average, consistent with the long-run reversals in DeBondt and Thaler; and (3) the HML series 11

14 are all reliably positive, consistent with the value-versus-growth phenomenon, and (4) all of the spreads have an appreciable proportion of negative outcomes The Stock Market s Realized Cross-sectional Return Dispersion Our work features the stock market s cross-sectional RD over a calendar month. We evaluate four alternate measures of the dispersion in disaggregate returns. A month s RD is simply the cross-sectional standard deviation of the monthly disaggregate returns, as follows: [ ] RD t = 1 n (R i,t R µ,t ) n 1 2 i=1 (1) where n is the number of individual stocks (or disaggregate portfolios) that is used for the particular RD metric, R i,t is the return of individual stock i (or disaggregate portfolio i) in month t, and R µ,t is the equally-weighted portfolio return of the individual stocks (or disaggregate portfolios) included in the RD metric for month t. First, we construct and evaluate a large-firm RD that is comprised of the largest 10% of NYSE/AMEX stocks by market capitalization, excluding stocks priced less than one dollar, with the size ranking repeated each month. We examine a large-firm RD because large firms may be more indicative of the economic environment, since small firms may add noise through non-synchronous trading or high idiosyncratic volatility. Evidence in Connolly and Stivers (2003) supports this notion. The large-firm RD tends to be the best performer in our setting for the WML spreads. Second, we construct and evaluate an RD from the monthly returns of the 100 disaggregate book-to-market/ size portfolios that are described in Section 3.1. The RD of the 100 book-tomarket/size portfolios tends to be the best performer in our setting for the HML spreads. Third, we construct and evaluate a broad-market RD that uses all individual NYSE/AMEX stocks, except those in the smallest size quintile and those stocks priced less than one dollar. Finally, we construct and evaluate an industry-based RD using the 48 industry returns, as described in Section 3.1. While our large-firm RD and the book-to-market/size RD are the best performers in our setting, we stress that all four of the RD metrics contain similar information. Our work features both the simple realized RD from equation (1) and a market-adjusted relative return dispersion (or RRD). As Stivers (2003) shows, a month s RD should vary with the month s absolute market return, due to dispersion in market betas. Since we are interested in whether the 12

15 RD is relatively high or low beyond the variation tied to the realized market return, we construct a monthly RRD that is orthogonal to the month s simple market return and absolute market return. The RRD is defined as the estimated residual, ɛ t, from the following regression: RD t = λ 0 + λ 1 R M,t + λ 2 D t R M,t + ɛ t (2) Where RD t is the month s simple RD from equation (1), R M,t is the absolute market-level stock return, D t is a dummy variable that equals one when the market return is negative, and the λs are coefficients to be estimated. The CRSP value-weighted market index is used as the market return. When estimating (2) with our large-firm RD over 1962 to 2005, we find that λ 1 is reliably positive (λ 1 =0.328, t-statistic=7.30) and λ 2 is reliably negative (λ 2 =-0.097, t-statistic=-2.37). For the same estimation with the book-to-market/size RD, we find that λ 1 is reliably positive (λ 1 =0.154, t- statistic=6.80) and λ 2 is essentially zero (λ 2 =-0.01, t-statistic=-0.00). The estimations indicate that RD varies positively with the absolute market return, as expected, but firm returns are less disperse for negative market returns (consistent with the asymmetric correlations in Ang and Chen (2002)). The R-squared values are 16.3% for the large-firm RD and 11.5% for the book-to-market/size RD, which indicates that much of the RD variability is not directly tied to the market return Construction of the RD-Trend Variables Our intent is to construct an RD measure that captures times when the RD is relatively high in an economic sense, rather than variation more attributed to very long-term trends or short-term statistical noise. We propose using an RD-trend, which is defined as the difference between a recent, short-term RD moving average and an older, longer-term RD moving average. In our tables, we focus on the following definition of the RD-trend. Relative to the spread observation for month t, the RD-trend is equal to the difference between the recent 3-month RD moving average over months t 1 to t 3 and an older, 12-month RD moving average. 10 We focus on the recent 3-month moving average, denoted as RD 1 3, because: (1) we feel that 3 months is a reasonable compromise that is responsive to market conditions but also removes some of the noise in month to month variations, and (2) the t 1 to t 3 timing seems like a good fit to 10 With longer term trends in volatility (see, e.g., French, Schwert, and Stambaugh (1987) and Campbell, Lettau, Malkiel, and Xu (2001)), a 3-month moving average may not adequately measure whether the RD is economically high, relative to the recent RD environment. 13

16 be informative about differing market conditions between the WML holding periods (which covers months t to t + 5, t + 17, or t + 35) and the WML ranking periods (which covers months t 2 to t 7, t 19 or t 37). For the older RD moving average, we use a 12-month moving average that just predates the ranking period for month t s WML spread, because: (1) we feel that a 12-month RD moving average is long enough to be informative about whether the most recent 3-month RD moving average is relatively high or relatively low, as compared to the recent RD environment, and (2) with this timing, the older RD moving average is not coincident with any return used in the ranking-period or holding-period for month t s WML spread. Thus, the 6-month spreads use an RD-trend denoted as RD 1 3,8 19 ; the 18-month spreads use an RD-trend denoted as RD 1 3,20 31 ; and the 36-month spreads use an RD-trend denoted as RD 1 3, This notation indicates that the RD-trend is equal to the 3-month RD moving average over months t 1 to t 3 minus the 12-month RD moving average over months t (j + 2) through t (j + 13), where j either equals 6, 18, or 36 for the different spread horizons. As discussed in Section 3.3, we also desire to examine an RD-trend that features a monthly RD that is orthogonal to the month s realized market return. Thus, we also construct and evaluate a comparable RRD-trend with the same timing as above, but with the market-adjusted RRD replacing the simple RD. 11 Our results are not unique to this timing for the RD-trend and RRD-trend (see Section 4.4). We elect to adopt this single timing convention for consistency, rather than to experiment and choose the strongest performer for each horizon and for each spread. Table 2 reports descriptive statistics for the four alternate RD measures featured in this paper. Note that each RD series is substantially autocorrelated (Panel A) and that the various RD-trend series are all sizably correlated (Panel B). Figures 1 through 3 exhibit the time series of the WML spreads, the HML spreads, and our primary large-firm RD-trend variable for the 6, 18, and 36- month spread horizon, respectively. 11 The simple RD-trend is attractive because it can be constructed in real time with no estimated parameters. The RRD-trend is attractive because it uses a monthly market-adjusted RD that is orthogonal to the month s absolute and simple market return. 14

17 4. Main Empirical Models and Results This section provides our primary empirical results regarding whether there is an intertemporal relation between the market s RD trend and changes in the subsequent WML or HML spread. We investigate spreads over 1962 through 2005 and at the 6, 18, and 36-month spread horizon. Our empirical work differs from prior time-series work by focusing on changes in the realized spreads, rather than the simple spread level Primary Empirical Models We focus on two models initially. For the change in WML spreads, we estimate variations of the following two models where j equals either 6, 18, or 36 for the different spread horizons: W ML j t = β 0 + β 1 RD 1 3,(j+2) (j+13) + β 2 StRt ɛ t (3) W ML j t = β 3 + β 4 RRD 1 3,(j+2) (j+13) + ɛ t (4) where W ML j t is the difference between the j-month WML spread over holding months t to t+(j 1) and the j-month WML spread over holding months t (j +3) to t 4; RD 1 3,(j+2) (j+13) is the RD-trend variable that is equal to the 3-month RD moving average over months t 1 through t 3 minus the 12-month RD moving average over months t (j + 2) through t (j + 13) ; RRD 1 3,(j+2) (j+13) is the same as RD 1 3,(j+2) (j+13) except the market-adjusted relative RD replaces the simple RD; StRt 1 12 is the 12-month aggregate stock market return over months t 1 to t 12; and the β s are coefficients to be estimated. We estimate the models for both the firm-level and industry-level WML spreads, as defined in Section 3.2. For our primary change in spread variables, we feel that this 3-month gap between the forwardlooking spread and the lagged reference spread is reasonable because: (1) the earlier spread just predates the RD 1 3 moving average that is featured in the RD-trend variables, and (2) three months seems a reasonable horizon to consider changes in market conditions. For example, with the 6-month spread, the change in spread is the difference between the WML outcome over months t to t + 5 and the WML outcome over months t 9 to t 4, relative to the 3-month RD moving average over months t 3 to t 1. We stress that our results are robust to alternate variation in the timing that are similar in concept (see Appendix B). 15

18 For model (3), we include the lagged 12-month market return as a control for the market-return state, as suggested by results in Cooper, Gutierrez, and Hameed (2004). We estimate the coefficients by ordinary least squares, but we report t-statistics with heteroskedastic- and autocorrelationconsistent standard errors. The number of correlated residual lags are set to equal the number of months in the strategy s horizon, since our estimation has rolling monthly observations. For the change in HML spreads, we estimate variations of the following two models where j equals either 6, 18, or 36 for the different spread horizons: HML j t = β 0 + β 1 RD 1 3,(j+2) (j+13) + β 2 StRt ɛ t (5) HML j t = β 3 + β 4 RRD 1 3,(j+2) (j+13) + ɛ t (6) where HML j t is the difference between the j-month HML spread over holding months t to t+(j 1) and the j-month HML spread over holding months t (j + 3) to t 4, as defined in Section 3.2; and the other terms are as described for equations (3) and (4). For the monthly RD measure in models (3) through (6), we estimate variations of the model for all four of our alternate RD measures. See Section 3.3 and Table 2 for descriptions of the four different RD measures Main Empirical Results The RD-trend and the Subsequent Change in the WML Spreads Tables 3, 4, and 5 report on whether the RD-trend is related to the subsequent change in WML spreads at the 6, 18, and 36-month spread horizons, respectively. The models are given by equations (3) and (4). Each table reports on spreads using both individual stocks strategies (Panel A) and value-weighted industry portfolio returns (Panel B). For these three tables, we report results using the large-firm RD, but the results are qualitatively consistent with our other three alternate RD measures (as shown in Appendix B). To begin with, Table 3 indicates that the RD-trend contains reliable information about the subsequent change in the 6-month WML spread. For both the firm-level and industry-level spreads, the estimated β 1 coefficients on the RD-trend and the β 4 coefficients on the RRD-trend are reliably negative with a 0.1% p-value. The RRD-trend, by itself, explains around 14% of the variation for both 16

19 the firm-level and industry-level change in spread variables. Subperiod results are qualitatively consistent in all cases with reliably negative β 1 and β 4 coefficients. Next, Table 4 indicates that the RD-trend contains reliable information about the subsequent change in the 18-month WML spread. For both the firm-level and industry-level spreads, the estimated β 1 coefficients on the RD-trend and the β 4 coefficients on the RRD-trend are again reliably negative with a 0.1% p-value. The RRD-trend, by itself, explains about 35% of the variation for the firm-level change in spread variable. Subperiod results are qualitatively consistent in all cases with reliably negative β 1 and β 4 coefficients, except for the first-half subperiod with the industry-level WML spreads (where the estimated β 1 and β 4 coefficients remain negative but are not statistically significant). Finally, Table 5 indicates that the RD-trend contains reliable information about the subsequent change in the 36-month WML spread. For both the firm-level and industry-level spreads, the estimated β 1 coefficients on the RD-trend and the β 4 coefficients on the RRD-trend are again reliably negative with a 0.1% p-value. The RRD-trend, by itself, explains around 40% of the variation for both the firm-level and industry-level change in spread variables. Subperiod results are consistent in all cases with reliably negative β 1 and β 4 coefficients. Also, for model (3), note that the statistical reliability of the β 1 coefficient on the RD-trend term is greater than that of the β 2 coefficient on the lagged market return in all cases except for the 18-month industry WML spreads in the first-half period only. For the one-half subperiods, the estimated β 2 coefficient on the lagged market return is only statistically significant for the 18-month firm-level WML spreads in the first half only. For our change in spread variables, we conclude the lagged market-return term is not important The RD-trend and the Subsequent Change in the HML Spreads Table 6 reports on whether the RD-trend is related to the subsequent change in HML spreads at the 6, 18, and 36-month spread horizons. The models are given by equations (5) and (6). Here, we report results using the RD from the book-to-market/size portfolios, but the results are qualitatively consistent for our three alternate RD measures (as shown in Appendix B). To begin with, we find that the RD-trend contains reliable information about the subsequent change in the 6-month HML spread. For the overall sample, the estimated β 1 coefficient on the RD- 17

20 trend and the β 4 coefficient on the RRD-trend are reliably positive with a 0.1% p-value. Subperiod results are consistent. The R-squared values seem sizable at 12.7%, 9.6%, and 16.9% with the RRD-trend only, for the overall period, first-half, and second-half, respectively. Next, we find that the RD-trend contains reliable information about the subsequent change in the 18-month HML spread. For the overall sample, the estimated β 1 coefficient on the RD-trend and the β 4 coefficient on the RRD-trend are reliably positive with a 0.1% p-value. Subperiod results are qualitatively consistent, but the RD-HML relation is statistically insignificant in the first-half subperiod. For the overall period, the R-squared value is sizable at 22.4% for the RRD-trend model. Finally, we find that the RD-trend contains reliable information about the subsequent change in the 36-month HML spread. For the overall sample, the estimated β 1 coefficient on the RD-trend and the β 4 coefficient on the RRD-trend are reliably positive with a 0.1% p-value. Subperiod results are consistent with highly reliable RD-trend coefficients. The R-squared values are sizable at 25.1%, 17.9%, and 39.4% with the RRD-trend only, for the overall period, first-half, and second-half, respectively. To summarize, Tables 3 through 6 indicate a strong, reliable negative relation (positive relation) between the RD-trend and the subsequent change in the WML spread (HML spread) at all three spread horizons. The RD-WML spread tends to be stronger than the RD-HML relation, in terms of the R-squared values and coefficient reliability. When interpreted using our hypothesis development in Section 2, our findings indicate a clear rejection of the time-variation in the dispersion of expected returns hypothesis in favor of the market-state transition hypothesis. In our framework from Section 2.2, it is not surprising that the RD-WML relation tends to be stronger than the RD-HML relation because the change in WML spreads should be negative for both transitions (from the good to the bad state and from the bad to the good state). Our RD-HML results suggest that the RD-trend is more informative about the transition to where value performs relatively better than growth (and not vice versa). Since the HML effect is presumably related to only one of the two market-state transitions, the RD-HML relation should be weaker than the RD-WML relation Sorting the Change in Spreads by the RRD-Trend We next examine the RD-spread relation by sorting the change in spread observations on the RRD-trend. The intent is to re-evaluate the RD-spread relations using a simple intuitive method 18

21 that clearly depicts the spread variation tied to the return dispersion. We examine each of the change in spread series that were featured in Tables 3 through 6. We sort each change in spread series on the respective lagged RRD-trend series, as defined in the corresponding table. We then report statistics on change-in-spread percentile subsets, based on the RRD-trend sort. Table 7 reports the results from this sorting exercise. First, consider the WML sorts. We find a striking contrast across groupings. For the 6-month WML spreads, the mean change in WML spread for the top RRD-trend quartile (bottom RRD-trend quartile) of values is % (+6.82%) per 6 months and the observations are negative for 73.2% (32.3%) of the time. The contrast is also strong for the longer-run horizons. For the 36-month WML spreads, the mean change in WML spread for the top RRD-trend quartile (bottom RRD-trend quartile) of values is % (+48.74%) per 36 months and the observations are negative for 82.1% (25.9%) of the time. Next, the HML sorts also indicate a striking variation with the RRD-trend. For the 6-month HML spreads, the mean change in HML spread for the top RRD-trend quartile (bottom RRDtrend quartile) of values is 5.41% (-5.60%) per 6 months and the observations are negative for 32.3% (72.4%) of the time. The contrast is also strong for the longer-run horizons. For the 36-month HML spreads, the mean change in HML spread for the top RRD-trend quartile (bottom RRD-trend quartile) of values is 27.97% (-26.61%) per 36 months and the observations are negative for 23.2% (83.9%) of the time. When comparing the high RRD-trend quartile of observations to the low RRD-trend quartile of observations, the differences-in-means for the change in spread variables are reliably different at a 0.4% p-value, or better, for all six cases in Table 7. Thus, this simple sorting exercise reinforces our primary findings in Table 3 through The RD-Spread Relation with Business-Cycle Explanatory Variables Chordia and Shivakumar (2002) and Avramov and Chordia (2006) find that business-cycle variables are informative about variation in momentum profits. We next investigate whether the intertemporal RD-spread relations, as depicted in Tables 3 through 7, remain evident when including the four business-cycle explanatory variables suggested in these papers. 19