Realized Return Dispersion and the Dynamics of

Winner-minus-Loser and Book-to-Market Stock Return Spreads1

Chris Stivers

Terry College of Business

University of Georgia

Athens, GA 30602

Licheng Sun

College of Business

Old Dominion University

Norfolk, VA 23529

This version: June 23, 2008

1We 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 (e-mail: cstivers@terry.uga.edu; phone: (706) 542-3648) or to

Licheng Sun (e-mail: LSun@odu.edu; phone: (757) 683-6552). Stivers acknowledges financial support from

a Terry-Sanford research grant.

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

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 high-

minus-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 or1By 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).2In 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

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

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 broad-

market 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

3Since 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.4Our 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

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-WML-

spreads 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-to-

market/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 corre-

sponding 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’ expla-

nation, 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 sub-

jective 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 ex-

planatory variables. Finally, the RD-spread relations also remain reliably evident when using a5Here, 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

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 medium-

run 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 medium-

run 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 time-

series 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

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.

2.1. 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 cross-

sectional variation in expected returns may have a material role in understanding momentum6However, 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

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 cross-

sectional 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.

2.2. 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 as-

sociated 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

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 two-

state framework to analyze how regime-switching can influence WML return behavior. Our frame-7Consistently, 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.8By ‘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

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 regime-

switching 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 dur-

ing 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 pre-

diction 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. Fundamen-

tal 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.9This 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

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 1962. Our sample extends through December 2005.

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).

3.2. 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 skip-

a-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 80th 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

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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 20th 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, WMLjt , 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 industry-

level 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

are all reliably positive, consistent with the value-versus-growth phenomenon, and (4) all of the

spreads have an appreciable proportion of negative outcomes.

3.3. 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:

RDt =

√√√√[

1n− 1

n∑

i=1

(Ri,t −Rµ,t)2]

(1)

where n is the number of individual stocks (or disaggregate portfolios) that is used for the particular

RD metric, Ri,t is the return of individual stock i (or disaggregate portfolio i) in month t, and Rµ,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-to-

market/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

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, εt, from the following regression:

RDt = λ0 + λ1 |RM,t|+ λ2D−t |RM,t|+ εt (2)

Where RDt is the month’s simple RD from equation (1), |RM,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 λ1 is reliably positive

(λ1=0.328, t-statistic=7.30) and λ2 is reliably negative (λ2=-0.097, t-statistic=-2.37). For the same

estimation with the book-to-market/size RD, we find that λ1 is reliably positive (λ1=0.154, t-

statistic=6.80) and λ2 is essentially zero (λ2=-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.

3.4. 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 RD1−3, 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 to10With 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

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 RD1−3,8−19; the 18-month spreads

use an RD-trend denoted as RD1−3,20−31; and the 36-month spreads use an RD-trend denoted

as RD1−3,38−49. 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.11The 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.

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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.

4.1. 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:

∆WMLjt = β0 + β1 RD1−3,(j+2)−(j+13) + β2 StRt1−12 + εt (3)

∆WMLjt = β3 + β4 RRD1−3,(j+2)−(j+13) + εt (4)

where ∆WMLjt 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; RD1−3,(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)’;

RRD1−3,(j+2)−(j+13) is the same as RD1−3,(j+2)−(j+13) except the ‘market-adjusted relative RD’

replaces the simple RD; StRt1−12 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 forward-

looking spread and the lagged reference spread is reasonable because: (1) the earlier spread just

predates the RD1−3 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

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 autocorrelation-

consistent 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:

∆HMLjt = β0 + β1 RD1−3,(j+2)−(j+13) + β2 StRt1−12 + εt (5)

∆HMLjt = β3 + β4 RRD1−3,(j+2)−(j+13) + εt (6)

where ∆HMLjt 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.

4.2. Main Empirical Results

4.2.1. 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 sub-

sequent change in the 6-month WML spread. For both the firm-level and industry-level spreads, the

estimated β1 coefficients on the RD-trend and the β4 coefficients on the RRD-trend are reliably neg-

ative with a 0.1% p-value. The RRD-trend, by itself, explains around 14% of the variation for both

16

the firm-level and industry-level ‘change in spread’ variables. Subperiod results are qualitatively

consistent in all cases with reliably negative β1 and β4 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 β1 coefficients on the RD-trend and the β4 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 β1 and β4 coefficients, except for the first-half subperiod with the

industry-level WML spreads (where the estimated β1 and β4 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 β1 coefficients on the RD-trend and the β4 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 β1 and β4 coefficients.

Also, for model (3), note that the statistical reliability of the β1 coefficient on the RD-trend

term is greater than that of the β2 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 β2 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.

4.2.2. 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 β1 coefficient on the RD-

17

trend and the β4 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 β1 coefficient on the RD-trend

and the β4 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 β1 coefficient on the RD-trend and

the β4 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.

4.2.3. 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

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 -10.51% (+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 -48.91%

(+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 RRD-

trend 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 6.

4.3. 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 intertem-

poral 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

We estimate the following two models for each spread horizon:

∆WMLjt = θ0 +θ1RRD1−3,(j+2)−(j+13) +θ2StRt1,36 +θ3dyst−1 +θ4divt−1 +θ5termt−1 +θ6yd3t−1 + εt (7)

∆HMLjt = θ0 +θ1RRD1−3,(j+2)−(j+13) +θ2StRt1,36 +θ3dyst−1 +θ4divt−1 +θ5termt−1 +θ6yd3t−1 + εt (8)

where ∆WMLjt is the change in the firm-level WML spread for the j-month horizon; ∆HMLj

t

is the change in the HML spread for the j-month horizon, where j equals 6, 18, or 36 months.

The ∆WML, ∆HML, and RRD-trend terms are as defined in Tables 3 through 6. StRt1,36 is

the lagged 36-month stock market return over months t − 1 to t − 36; dyst−1, divt−1, termt−1,

yd3t−1 are the lagged business cycle variables; and the θ’s are coefficients to estimated. dys is the

default spread equal to the yield difference between Moodys BAA and AAA bonds, div is the stock

market’s aggregate dividend yield, term is the difference between the yield of 10-year T-bonds and

3-month T-bills, and yd3 is the yield of 3-month T-bills. For these two models, we use the lagged

3-year stock market return, as suggested by results in Cooper, Hameed, and Gutierrez (2004).

Table 8 reports the results from estimating equations (7) and (8). For the three WML spread

horizons, we find that the estimated θ1 coefficients on the RRD-trend term remain negative, sizable,

and highly statistically significant (p-values < 0.1%). The four business-cycle variables add very

little explanatory power. Only the coefficient on the default yield spread is statistically significant

and only for the 36-month spread horizon.

For the three HML spread horizons, we find that the estimated θ1 coefficients on the RRD-

trend term remain positive and statistically significant (p-values < 0.1%). For the HML spreads,

the four business-cycle variables add some explanatory power with the coefficients on the default

yield spread being statistically significant for the 18-month and 36-month spread horizon.

To summarize, our primary RD-spread results in Tables 3 through 7 remain reliably evident

when controlling for the four business-cycle variables suggested by Chordia and Shivakumar (2002)

and Avramov and Chordia (2006). Of the six explanatory terms in the two models, our collective

results indicate that the RRD-trend term is the dominant explanatory term. The default yield

spread is the only other explanatory variable that ever comes in statistically significant and in only

three of the six cases.

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4.4. Robustness Evidence

In this subsection, we report a series of additional evidence that bears on robustness and inter-

pretation. For brevity in our main text, we only summarize results here. Prominent results are

provided in tabular form in Appendix B.

First, we are interested in whether our RD-WML findings are substantially driven by small-

cap firms. We examine an alternate WML series that includes only stocks that are in the top

20th percentile of NYSE/AMEX firms by market capitalization (see Section 3.2). The patterns in

average large-firm WML spreads are similar to that for our primary WML spreads (with a mean

6-month WML spread of 4.6%, t-stat=4.4; a mean 18-month WML spread of 0.4%, t-stat=0.2;

and a mean 36-month WML spread of -16.3%, t-stat=-2.5). Consistent with our primary findings

in Tables 3 through 5, we find that the relation between the RD-trend and the ‘change in WML

spread’ is also negative, sizable, and highly reliable for the large-firm WML series (see Table B.1).

Next, we are interested in whether our RD-WML findings are substantially driven by stocks that

have extreme returns in the ranking period. We examine ‘Decile-9 minus Decile-2’ WML spreads,

that are long the decile-9 winners and short the decile-2 losers over the respective ranking period

(thus omitting the extreme 10% of winners and 10% of losers from the spread). The patterns in

average ‘Decile 9 minus Decile 2’ WML spreads are similar to that for our primary WML spreads

(with a mean 6-month WML spread of 2.8%, t-stat=4.5; a mean 18-month WML spread of 3.0%,

t-stat=1.5; and a mean 36-month WML spread of -5.1%, t-stat=-1.0). Again consistent with our

primary findings in Tables 3 through 5, we find that the relation between the RD-trend and the

‘change in WML spread’ is also negative, sizable, and highly reliable for the ‘Decile-9 minus Decile-2’

WML spreads (see Table B.1).

Next, we are interested in whether our RD-spread findings might be because RD is related to

investor sentiment. We form a sentiment-adjusted RD-trend that is orthogonal to the comparable

trend in the investor sentiment index of Baker and Wurgler (2006). When we re-estimate our models

with the sentiment-adjusted RD-trend, the results are nearly identical to our primary results in

Tables 3 through 6. Further, we are unable to find any reliable relation between our ‘change in

spread’ variables and the sentiment index.

Next, we find that all four of our alternate RD metrics (the large-firm RD, the broad firm-level

RD, the industry-level RD, and the portfolio-level RD using 100 book-to-market/size portfolios)

21

yield consistent results for our primary models given by equations (3) through (6) (see Table B.2).

Thus, while the large-firm RD tends to be the best performer for the WML spreads and the book-

to-market/size RD tends to be the best performer for the HML spreads, our results are consistent

with alternate dispersion measures.

Next, we also re-estimate our primary models in Section 4.1, but with the simple 3-month

moving average of the large-firm RD replacing the RD-trend. Our primary RD-spread results are

also evident, but weaker, with this simpler RD measure (see Table B.2).

Next, a month’s RD is a measure of cross-sectional volatility that is substantially positively

skewed (as are volatility measures, by their nature). Accordingly, we form alternate RRD-trend

measures that use the log transformation of a month’s RD to reduce the skewness, in place of the

raw RD. We then re-estimate our RRD-trend models using the log version of the RRD-trend. We

find nearly identical results to those reported in Tables 3 through 6.

We also investigate alternate timing for: (1) the gap between spreads for the ‘change in spread’

term, (2) the length of the RD moving averages in the the RD-trend term, and (3) the gap between

the RD moving averages in the RD-trend term. Our results are robust to other alternate timings

that are similar in concept to our primary measures (see Table B.3).

Finally, recall that the RD is positively related to the absolute market return. We also re-

estimate our primary models with a market-volatility-trend replacing the RD-trend. The market-

volatility-trend is defined identically as the respective RD-trend, but with the month’s absolute

market return replacing the month’s RD. We find that the estimated coefficients for the market-

volatility-trend have the same algebraic sign as the corresponding coefficients on the RD-trend in

all cases, but the explanatory power is much weaker and the relation is only statistically significant

at the 6-month WML spread horizon.

4.5. The Two Components of the ‘Change in Spreads’ and the RD-trend

Of course, our ‘change in spread’ variables may be decomposed into the forward-looking spread

component (spanning months t to either t + 5, t + 17, or t + 35 for the 6, 18, and 36-month spread

horizons) and the lagged, reference-spread component (spanning months t−4 to either t−9, t−21,

or t− 39 for the 6, 18, or 36-month spread horizons). This spread timing is relative to the 3-month

RD moving average from the RD-trend term, which is defined over months t− 1 to t− 3.

22

A natural question is whether the RD-trend is more informative about the forward-looking

spread or the lagged reference spread. In our view, our ‘market-state transition’ hypothesis suggests

that the RD-trend would be informative about both components, but in opposite directions.

In Table 9, we re-estimate our primary models from Tables 3 to 6, but where the dependent

variable is now one of the two components of the respective ‘change in spread’ term. For the WML

spreads, we find that the RD-trend term is both negatively related to the forward-looking WML

spread and positively related to the earlier reference WML spread. The estimated coefficients are

statistically significant for all horizons and for both the firm-level and industry-level WML spreads,

except for the firm-level 18-month WML reference spread (where the estimated coefficients are

consistently positive, but with t-statistics of only about 1.5). These component results seem to fit

well with our ‘market-state transition’ hypothesis.

For the HML spreads, we find that the RD-trend term is negatively related to the earlier

reference HML spread for all three spread horizons and positively related to the forward-looking

HML spread at the 18-month and 36-month horizon. The relation between the RD-trend and the

forward-looking HML spreads at the 6-month horizon is insignificant. The HML component results

for the 18 and 36-month horizon again seem to fit with our ‘market-state transition’ hypothesis.

5. Towards Interpreting the Return-Dispersion Trend

In Section 4, we documented a reliable negative (positive) relation between the market’s RD-trend

and the subsequent change in WML spreads (HML spreads) at the 6, 18, and 36-month spread

horizons. In terms of the competing hypotheses that we offered in Section 2, the evidence strongly

favors the ‘market-state transition’ hypothesis. The robustness and pervasiveness of our findings

supports a broad systematic view for understanding our results, rather than the possibility that

small-cap stocks or fringe stocks (whose valuations are presumably more tied to investor sentiment)

are largely driving our results. A pervasive view of WML spreads is consistent with recent evidence

in Fama and French (2007), who show that momentum is evident in micro-cap, small-cap, and large-

cap stocks; and Dittmar, Kaul, and Lei (2007), who argue that momentum is not an anomaly.

In a nutshell, the evidence suggests that RD is a leading indicator of market-state changes and

that market cyclicality is important in understanding the behavior of WML and HML spreads. If

23

so, we would expect to find other evidence that supports this interpretation of RD. In this section,

we offer additional evidence that bears on the ‘market-state transition’ hypothesis.

5.1. Empirical Estimation of Regime-Switching in the Stock Market

Next, we report on an empirical estimation of two-state regime-switching in the stock market.

Here, we are primarily interested in whether higher realizations of our lagged RD-trend measures

are associated with a higher probability of regime change. Secondarily, we are interested in whether

the regime-specific mean returns and regime durations are consistent with the premise of our two-

state framework in Section 2.2. Our estimation here features the monthly returns of the 48 value-

weighted industry returns from the French data library. We use industry portfolios because: (1)

differences in cyclicality across industries are widely accepted, (2) each industry has monthly return

observations over the entire sample, and (3) this approach limits the complexity.

Our bivariate regime-switching model features a more-cyclical and a less-cyclical portfolio of

industry returns, where each portfolio contains six industries (of the 48 in our sample). The more-

cyclical (less-cyclical) portfolio contains the six industries with the highest (lowest) market beta,

with the market beta estimated over the entire sample. We choose a bivariate, two-state model as

a parsimonious approach that appeals to the intuition of a two-state market with “more cyclical”

and “less cyclical” industry portfolios.

Our model allows for the probability of shifting regimes to vary with the lagged RD-trend. More

specifically, we estimate the following two-state, bivariate regime-switching model on the monthly

returns of two different portfolios of industries, a more-cyclical portfolio (denoted MCP) and a

less-cyclical portfolio (denoted LCP):

rmc,t = µsmc + σs

mc ηmc,t (9)

rlc,t = µslc + σs

lc ηlc,t (10)

Where rmc,t and rlc,t are the monthly returns of our MCP (mc subscript) and our LCP (lc sub-

script), µsmc and µs

lc are regime-specific mean returns for each respective series, σsmc and σs

lc are

regime-specific standard deviations for each respective series, ηmc,t and ηlc,t are bivariate, standard,

normally-distributed random variables, and s is the state variable where s either equals one for the

good regime or two for the bad regime.

24

The s state variable is modeled with time-varying transition probabilities (pjj(t)), as follows:

pjj(t) =ecj+djRD1−3,8−19

1 + ecj+djRD1−3,8−19(11)

where j = 1 (the good regime) or j = 2 (the bad regime); pjj(t) equals the probability that st = j

(the second subscript), given that st−1 = j (the first subscript); and RD1−3,8−19 is our large-

firm RD-trend that is featured for the 6-month WML spread results. The µss, σss, cjs, djs, and

correlations between the ηts are regime-specific parameters to be estimated. We estimate the

model by maximizing the log-likelihood function for the bivariate normal density while allowing

for regime-switching. This specification for the transition probabilities follows from Diebold, Lee,

and Weinbach (1994). The ‘market-state transition’ hypothesis suggests that the dj coefficients

will be negative, which would indicate that the probability of shifting regimes increases with the

RD-trend.

We acknowledge that our simple framework here is clearly not rich enough to capture actual

market return behavior, but we feel it meets the hurdle of usefulness. Our results are as follows.

First, we note that the market-beta of our MCP is 1.33 and the market-beta of our LCP is 0.70.

This contrast in market-betas fits with the intuition of our MCP and LCP distinction.12

Next, we feel the time-series behavior of the estimated regimes seems plausible. The estimation

suggests average durations of about 29 months for the good regime and about 12 months for the bad

regime. For the CRSP value-weighted index returns, the mean and standard deviation of returns

over the good-regime months (bad-regime months) are 1.10% and 3.35% (0.38% and 6.59%) per

month, respectively. Every NBER recessionary period is approximately associated with episodes

of the bad regime. These observations seem to fit with the intuition of a predominant good regime

with lower volatility and a less common bad regime with higher volatility.

Table 10 reports the estimated parameters. Note that the estimated d1 and d2 coefficients are

both negative and that the estimated d1 is reliably negative with a 0.1% p-value (for d2, the p-value

is 0.154). This indicates that the transition probability of shifting regimes increases with the lagged

RD-trend, especially for the good-to-bad transition. For example, when the RD-trend is at its 10th

percentile (90th percentile), the probability of shifting from the good to bad regime is 1.3% (19.4%).12The six industries in the LCP are: utilities, precious metals, tobacco, food products, communications, and

petroleum and natural gas. The six industries in the MCP are: electronic equipment, measuring and control equip-

ment, business services, entertainment, construction, and healthcare.

25

Next, note the substantial contrast in the differences in regime-specific means when comparing

the MCP and the LCP. For the MCP, the estimated good-regime mean is 1.53%/month and the bad-

regime mean is 0.18%/month, so the difference in regime-specific means is about 1.35%/month. For

the LCP, the estimated good-regime mean is 1.14%/month and the bad-state mean is 0.84%/month,

so the difference in regime-specific means is about 0.30%/month. To sum up, the estimated negative

dj coefficients, the regime durations, and the differences in regime-specific means are all consistent

with the premise of our two-state framework in Section 2.2 and the ‘market-state transition’ hy-

pothesis.

5.2. Relating RD to Economic Contractions

We explore the relation between our RD measures and economic recessions per the NBER. Reces-

sions are uncommon (only 12.3% of our sample’s months) and are associated with transition and

stress in the stock market. We are interested in whether recessionary months tend to have a higher

lagged RD, which would suggest that the RD may be informative of market-state changes.

We examine whether the 3-month RD moving average (which is featured in our RD-trend

variables) is different when the subsequent month t is in an economic recession. We find the following

over our 1962 to 2005 sample period. For our large-firm RD, when month t is a recessionary month

(expansionary month), then the mean of the 3-month RD moving average over months t − 1 to

t− 3 is 8.04% (6.67%). This difference of 1.37% is statistically significant with a 0.1% p-value. For

the book-to-market/size RD, when month t is a recessionary month (expansionary month), then

the mean of the 3-month RD moving average over months t − 1 to t − 3 is 3.43% (2.97%). This

difference of 0.46% is statistically significant with a 1.8% p-value. Thus, recessionary months tends

to be preceded by a high RD environment.

5.3. The RD-trend and Variations in Size-based Spreads and the Market Return

Finally, given our RD results for WML and HML spreads, a natural question is whether the RD-

trend is similarly related to size-based spreads or the excess market return (following from the other

two terms in Carhart’s four-factor asset pricing model). We investigate this question and find much

weaker results, as compared to our WML and HML spreads. For size-based spreads and the excess

market return, comparable RD relations are not reliably evident: (1) for the 6-month horizon, in

26

either of our one-half subperiods, (2) for the 18 and 36-month horizon in our first-half subperiod.

For our second-half subperiod only, the RD-trend is associated with a decrease in the market return

at the 18 and 36-month horizon with a related increase in the small-minus-big size spreads.13 In our

view, these additional findings are unsurprising because: (1) there is some evidence that the RD-

trend is associated with transitions to a weaker market-return state, consistent with the negative

d1 coefficient in Section 5.1 (second-half subperiod only), and (2) the excess market return and

size-based spreads do not fit our suggested framework of cross-sectional valuation cycles as well as

WML and HML spreads do (see Section 2.2 and Appendix A).

6. Conclusions

Return spreads based on past relative return strength and book-to-market equity ratios have a

prominent place in current financial practice and theory. By definition, return spreads require

cross-sectional dispersion in realized returns. A natural question is whether the market’s realized

return dispersion is informative about the time-series behavior of these prominent return spreads.

In this paper, we document that the stock market’s RD trend is negatively related to the

subsequent change in winner-minus-loser spreads and positively related to the subsequent change

in high-minus-low spreads based on stocks’ book-to-market equity ratios, where ‘change’ is defined

relative to recent realized spreads. Over our 1962 to 2005 sample, these regularities are reliably

evident and economically sizable at the 6, 18, and 36-month spread horizon. Our findings are robust

in subperiods, to different RD metrics, to variations in RD-trend timing, to variations in timing

for the ‘change in spread’ variables, and for alternate WML strategies implemented on large-firms

only or value-weighted industry-level returns.

We offer a ‘market-state transition’ hypothesis for interpreting our results, which suggests that

RD is a leading indicator of market-state changes and that market cyclicality is important in

understanding WML and HML spreads. Consistent with this interpretation, we also show that the

market’s RD-trend is negatively related to the forward-looking component of the ‘change in WML13We re-estimate our model (4) at all three horizons with a comparable ‘change in size-based spread’ or ‘change

in excess market return’ variable replacing the ‘change in WML spread’ variable. Since a SMB size spread is short

a large-firm portfolio that is highly correlated with the market return, a negative relation with the market return

implies a likely positive relation with the SMB spread. Detailed results are available from the authors by request.

27

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. Further,

we show that RD is informative about the likelihood of market-state changes in a regime-switching

estimation and that a high RD tends to lead recessionary months.

We also show that a two-state return-generating framework suggests that average medium-run

WML spreads (average longer-run WML spreads) should be higher than (lower than) suggested

solely by the cross-sectional variation in unconditional expected returns. This aspect of our ana-

lytical framework is attractive because it is consistent with the stylized facts on how average WML

spreads vary with a spread’s horizon.

The pervasiveness and nature of our findings suggest a systematic, market-wide interpretation

for our results; rather than an interpretation tied to small stocks or fringe stocks that presumably

have more subjective valuations tied to investor sentiment. Thus, in terms of further clarifying

the ‘market-state transition’ hypothesis, we tend to favor a ‘rational cycle in cross-sectional stock

valuation’ interpretation for our results over a behavioral interpretation tied to intuition from

Daniel, Hirshleifer, and Subrahmanyam (1998) and Baker and Wurgler (2006).

To conclude, we document a robust and sizable new regularity in the dynamics of WML and

HML spreads. These regularities bear on theoretically understanding WML and HML spreads.

While we offer a ‘market-state transition’ hypothesis for understanding our results, theorists are

likely to refine this idea or propose new ideas that might explain the empirical regularities and gen-

erate additional empirical implications. In practice, our findings may prove important for investors

who try to vary their loadings on spread-type strategies. Our findings also imply that the stock

market’s RD environment may serve as a state variable in the sense of Cochrane (2005).

28

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31

Table 1: Descriptive Statistics for the WML and HML Return Spreads

This table reports the means, standard deviations, minimum, maximum, and the percentage of negative

observations for the primary WML and HML return spreads used in this study. All return statistics are in

percentage units corresponding to the entire cumulative period. WMLjt is the j-month winner-minus-loser

return spread, where the holding period is over t to t + (j − 1) and the ranking period is over t − (j + 1)

through t − 2. ∆WMLjt 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. j is either 6, 18, or 36

months, denoting the three different spread horizons. Statistics are reported for both the firm-level and

industry-level WML spreads, as defined in Section 3.2. HMLjt is the j-month high-minus-low book-to-

market spread over months t to t + (j − 1), as defined in Section 3.2. ∆HMLjt 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. For the means of the average spread level, a t-statistic is in parentheses

which indicates whether the mean is reliably different than zero, calculated with heteroskedastic- and

autocorrelation-consistent standard errors. The sample period is 1962 through 2005.

Mean Std. Dev. Minim. Maxim. % Negative Obs.

Firm-level WML6t 7.26 (7.48) 11.78 -45.22 49.23 24.4

Firm-level WML18t 4.57 (1.82) 20.29 -63.59 71.23 34.7

Firm-level WML36t -10.76 (-1.38) 40.33 -136.91 112.10 58.2

Firm-level ∆WML6t 0.03 18.07 -59.22 61.91 49.5

Firm-level ∆WML18t -0.32 28.30 -91.33 98.75 52.8

Firm-level ∆WML36t 2.09 64.31 -149.28 220.17 46.5

Industry-level WML6t 3.30 (4.59) 8.74 -34.70 34.83 34.7

Industry-level WML18t 2.48 (1.03) 18.61 -72.33 59.67 39.9

Industry-level WML36t -3.62 (-0.51) 36.40 -88.76 107.30 51.0

Industry-level ∆WML6t -0.02 13.15 -49.82 43.76 49.7

Industry-level ∆WML18t -0.24 27.57 -113.40 91.78 45.8

Industry-level ∆WML36t 2.60 56.83 -173.94 147.54 42.1

Firm-level HML6t 2.83 (3.13) 9.57 -20.69 39.47 42.2

Firm-level HML18t 8.66 (3.09) 18.44 -33.27 67.82 35.9

Firm-level HML36t 16.72 (2.97) 25.81 -35.25 88.20 26.7

Firm-level ∆HML6t -0.04 13.23 -37.942 41.61 53.5

Firm-level ∆HML18t 0.70 28.32 -66.76 89.29 50.5

Firm-level ∆HML36t 0.24 40.55 -87.57 88.11 55.3

32

Table 2: Descriptive Statistics for the Stock Market’s Cross-sectional Return Dispersion

This table reports the means, standard deviations, autocorrelations, and cross-correlations for five

alternate RD measures. The five alternate RD measures are as follows: (1) RDLarge is the RD formed

from the largest decile of NYSE/AMEX stocks, by market capitalization; (2) RDBroad is the RD formed

from all NYSE/AMEX stocks except for those in the smallest quintile, by market capitalization; (3)

RDIndustry is the RD formed from the 48 value-weighted industry returns from the French data library;

(4) RDBM&Sz is the RD formed from 100 book-to-market/size portfolio returns from the French data

library, where the portfolios are formed from a 10x10 double-sort on size and book-to-market; and (5)

RRDLarge is the market-adjusted, relative return dispersion of the largest decile of NYSE/AMEX stocks,

as defined in Section 3.3. Panel A reports univariate statistics for the monthly values of each alternate RD.

Panel B reports the cross-correlations for the 3-month moving RD average, RD1−3, (on the upper diagonal)

and for the RD-trend, RD1−3,8−19, (on the lower diagonal), constructed for each each alternate measure.

RD1−3 is the RD moving average over months t− 1 through t− 3 and RD1−3,8−19 is the RD-trend variable

that is equal to ‘the RD moving average over months t− 1 through t− 3’ minus ‘the RD moving average

over months t− 8 through t− 19’. The sample period is 1962 through 2005.

Panel A: Univariate Monthly RD Statistics, in %

RDLarge RDBroad RDIndustry RDBM&Sz RRDLarge

Mean 6.84 9.94 4.22 3.02 0.00

Std. Deviation 2.09 2.23 1.45 1.29 1.91

Autocorrelation(1) 0.679 0.603 0.559 0.588 0.624

Autocorrelation(2) 0.621 0.491 0.518 0.536 0.579

Autocorrelation(3) 0.665 0.513 0.532 0.520 0.630

Panel B: Cross-Correlations of RD1−3 (upper diagonal) and RD1−3,8−19 (lower diagonal)

RDLarge RDBroad RDIndustry RDBM&Sz RRDLarge

RDLarge 1 0.850 0.854 0.772 0.964

RDBroad 0.773 1 0.812 0.755 0.782

RDIndustry 0.801 0.785 1 0.793 0.809

RDBM&Sz 0.747 0.686 0.626 1 0.756

RRDLarge 0.924 0.635 0.675 0.717 1

33

Table 3: Change in 6-month WML Return Spreads and the Lagged Return Dispersion

This table reports how WML spreads for a symmetric 6-month strategy vary with the lagged

RD-trend:

Model 1 : ∆WML6t = β0 + β1 RD1−3,8−19 + β2 StRt1−12 + εt

Model 2 : ∆WML6t = β3 + β4 RRD1−3,8−19 + εt

where ∆WML6t is the difference between the 6-month WML spread over holding months t to t +5 and

the 6-month WML spread over holding months t − 9 to t − 4; RD1−3,8−19 is the large-firm RD-trend

variable that is equal to ‘the RD moving average over t− 1 through t− 3’ minus ‘the RD moving average

over t− 8 through t− 19’; RRD1−3,8−19 is the same as RD1−3,8−19 except the ‘market-adjusted relative

RD’ replaces the simple RD; StRt1−12 is the 12-month aggregate stock market return over months t−1

to t− 12; and the β’s are coefficients to be estimated. Panel A reports on the firm-level WML spreads

and Panel B reports on the industry-level WML spreads, where both spreads are defined in Section

3.2. The sample period is 1962 to 2005. T-statistics are in parentheses, based on heteroskedastic- and

autocorrelation-consistent standard errors. Columns 3 through 5 report on Model 1, and Columns 6

through 7 report on Model 2.

Regression Model 1 Regression Model 2Sample Dates β1 β2 R2 β4 R2

Panel A: 6-month WML Spreads for Individual Stock StrategyFull 8/1962- -5.17 0.092 17.2% -5.47 14.5%

7/2005 (-5.76) (0.97) (-4.90)

1st half 8/1962- -5.35 -0.081 13.8% -6.55 15.2%1/1984 (-3.83) (-0.60) (-4.02)

2nd Half 2/1984- -6.11 0.310 26.0% -4.89 14.3%7/2005 (-5.98) (3.03) (-3.35)

Panel B: 6-month WML Spreads for Industry-level StrategyFull 8/1962- -3.61 0.064 15.8% -3.83 13.7%

7/2005 (-5.02) (1.07) (-4.66)

1st half 8/1962- -3.77 0.056 14.8% -4.51 12.8%1/1984 (-3.45) (0.77) (-3.62)

2nd Half 2/1984- -3.54 0.068 16.8% -3.47 14.5%7/2005 (-3.40) (0.65) (-3.11)

34

Table 4: Change in 18-month WML Return Spreads and the Lagged Return Dispersion

This table reports how WML spreads for a symmetric 18-month strategy vary with the lagged

RD-trend:

Model 1 : ∆WML18t = β0 + β1 RD1−3,20−31 + β2 StRt1−12 + εt

Model 2 : ∆WML18t = β3 + β4 RRD1−3,20−31 + εt

where ∆WML18t is the difference between the 18-month WML spread over holding months t to t + 17

and the 18-month WML spread over holding months t− 21 to t− 4; RD1−3,20−31 is the large-firm RD-

trend variable that is equal to ‘the RD moving average over t− 1 through t− 3’ minus ‘the RD moving

average over t− 20 through t− 31’; RRD1−3,20−31 is the same as RD1−3,20−31 except the ‘market-

adjusted relative RD’ replaces the simple RD; StRt1−12 is the 12-month aggregate stock market return

over months t − 1 to t − 12; and the β’s are coefficients to be estimated. Panel A reports on the

firm-level WML spreads and Panel B reports on the industry-level WML spreads, where both spreads

are defined in Section 3.2. The sample period is 1962 to 2005. T-statistics are in parentheses, based on

heteroskedastic- and autocorrelation-consistent standard errors. Columns 3 through 5 report on Model

1, and Columns 6 through 7 report on Model 2.

Regression Model 1 Regression Model 2Sample Dates β1 β2 R2 β4 R2

Panel A: 18-month WML Spreads for Individual Stock StrategyFull 8/1962- -8.24 -0.291 40.0% -9.17 34.8%

7/2004 (-8.54) (-2.68) (-7.09)

1st half 8/1962 - -9.17 -0.466 39.5% -10.57 33.3%1/1984 (-8.01) (-3.63) (-6.08)

2nd Half 2/1984 - -8.30 -0.108 42.7% -8.60 37.1%7/2004 (-6.24) (-0.57) (-5.31)

Panel B: 18-month WML Spreads for Industry-level StrategyFull 8/1962- -6.52 -0.200 25.8% -8.15 29.0%

7/2004 (-3.87) (-1.74) (-4.40)

1st half 8/1962 - -3.24 -0.314 9.0% -4.84 8.8%1/1984 (-1.40) (-1.55) (-1.51)

2nd Half 2/1984 - -9.24 0.096 44.8% -9.75 44.4%7/2004 (-4.60) (0.56) (-4.76)

35

Table 5: Change in 36-month WML Return Spreads and the Lagged Return Dispersion

This table reports how WML spreads for a symmetric 36-month strategy vary with the lagged RD

trend:

Model 1 : ∆WML36t = β0 + β1 RD1−3,38−49 + β2 StRt1−12 + εt

Model 2 : ∆WML36t = β3 + β4 RRD1−3,38−49 + εt

where ∆WML36t is the difference between the 36-month WML spread over holding months t to t + 35

and the 36-month WML spread over holding months t− 39 to t− 4; RD1−3,38−49 is the large-firm RD-

trend variable that is equal to ‘the RD moving average over t− 1 through t− 3’ minus ‘the RD moving

average over t− 38 through t− 49’; RRD1−3,38−49 is the same as RD1−3,38−49 except the ‘market-

adjusted relative RD’ replaces the simple RD; StRt1−12 is the 12-month aggregate stock market return

over months t − 1 to t − 12; and the β’s are coefficients to be estimated. Panel A reports on the

firm-level WML spreads and Panel B reports on the industry-level WML spreads, where both spreads

are defined in Section 3.2. The sample period is 1962 to 2005. T-statistics are in parentheses, based on

heteroskedastic- and autocorrelation-consistent standard errors. Columns 3 through 5 report on Model

1, and Columns 6 through 7 report on Model 2.

Regression Model 1 Regression Model 2Sample Dates β1 β2 R2 β4 R2

Panel A: 36-month WML Spreads for Individual Stock StrategyFull 8/1962- -19.13 0.110 42.3% -20.91 39.9%

1/2003 (-4.64) (0.32) (-4.31)

1st half 8/1962 - -28.2 0.389 67.8% -33.12 66.7%1/1984 (-11.70) (1.09) (-10.50)

2nd Half 2/1984 - -12.35 -0.267 26.2% -13.04 23.6%1/2003 (-3.67) (-0.57) (-4.03)

Panel B: 36-month WML Spreads for Industry-level StrategyFull 8/1962- -16.38 -0.055 39.6% -19.57 44.7%

1/2003 (-7.88) (-0.18) (-9.88)

1st half 8/1962 - -15.64 0.250 44.5% -19.93 50.9%1/1984 (-8.51) (0.85) (-7.69)

2nd Half 2/1984 - -16.81 -0.365 38.8% -19.42 41.9%1/2003 (-6.22) (-0.73) (-8.21)

36

Table 6: The Change in HML Return Spreads and the Lagged Return Dispersion

This table reports how the HML spreads vary with the lagged RD-trend for the 6, 18, and 36-month

spread horizon.

Model 1 : ∆HML6t = β0 + β1 RD1−3,8−19 + β2 RRD1−3,8−19 + β3 StRt1−12 + εt

Model 2 : ∆HML18t = β0 + β1 RD1−3,20−31 + β2 RRD1−3,20−31 + β3 StRt1−12 + εt

Model 3 : ∆HML36t = β0 + β1 RD1−3,38−49 + β2 RRD1−3,38−49 + β3 StRt1−12 + εt

where, for Model 1, ∆HML6t is the difference between the 6-month HML spread over months t to t + 5

and months t − 9 to t − 4. For Model 2, ∆HML18t is the difference between the 18-month HML spread

over months t to t + 17 and months t− 21 to t− 4. For Model 3, ∆HML36t is the difference between the

36-month HML spread over months t to t + 35 and months t − 39 to t − 4. The RD-trend, RRD-trend,

and StRt terms are as defined for Tables 3 through 5 where the RD here is the RDBM&Sz, as defined in

Section 3.3, and the β’s are coefficients to be estimated. Panels A, B, and C report on the 6, 18, and

36-month spread horizon, respectively. Each panel reports on two variations of the model. Variation 1

includes the RD-trend term and the 12-month market return (restricts β2 = 0), and Variation 2 includes

only the RRD-trend term (restricts β1 and β3 = 0). The monthly HML spread is defined in Section 3.2.

The sample period is 1962 to 2005.

Variation 1: (β2 = 0) Variation 2: (β1, β3 = 0)Sample Dates β1 β3 R2 β2 R2

Panel A: Model 1, 6-month Spread HorizonFull 8/1962- 4.38 0.114 14.2% 4.89 12.7%

7/2005 (4.71) (1.88) (4.63)1st Half 8/1962- 5.66 0.191 12.1% 6.90 9.6%

1/1984 (2.44) (2.36) (2.48)2nd Half 2/1984- 4.33 0.041 18.4% 4.41 16.9%

7/2005 (4.78) (0.57) (4.31)

Panel B: Model 2, 18-month Spread HorizonFull 8/1962- 9.58 0.252 23.0% 10.84 22.4%

7/2005 (4.09) (1.75) (4.78)1st Half 8/1962- 3.66 0.266 5.2% 4.62 1.8%

1/1984 (0.97) (1.67) (0.77)2nd Half 2/1984- 11.18 0.171 34.0% 11.86 33.4%

7/2005 (4.33) (0.70) (4.86)

Panel C: Model 3, 36-month Spread HorizonFull 8/1962- 16.62 -0.072 25.5% 17.76 25.1%

7/2005 (4.36) (-0.27) (4.19)1st Half 8/1962- 21.10 -0.169 17.7% 25.07 17.9%

1/1984 (2.41) (-0.44) (2.53)2nd Half 2/1984- 16.62 0.108 39.7% 17.20 39.4%

7/2005 (4.81) (0.58) (4.92)

37

Table 7: The ‘Change in Spreads’ when Sorted on the Lagged RRD-trend

This table reports on statistics for subsets of the ‘change in spread’ observations, from sorting on

the lagged RRD-trend. The ‘change in spread’ variables are defined for the different horizons in Tables

3 through 5 for the WML spreads and Table 6 for the HML spreads. The ‘change in spread’ values are

sorted on the lagged RRD-trend term, as defined for models (4) and (6). We report on percentile-based

subset groupings of the spreads when sorted on the RRD-trend. For each sort, row 1 reports the mean

value of the spread over the respective horizon, in percentage units, for the respective subset denoted

by the column heading. For each sort, row 2 reports the percentage of the spread observations that

are negative for the respective subset denoted by the column heading. In the table, WML refers to the

firm-level, decile-based WML spread as used in Tables 3 through 5. HML refers to the book-to-market

spread as used in Table 6. The sample period is 1962 through 2005.

Spread All 0-25 Pctl 25-50 Pctl 50-75 Pctl 75-100 Pctl 90-100 Pctl

∆WML 6-mon Mean, %: 0.03 6.82 2.82 1.01 -10.51 -16.46

(% Neg.) (49.5) (32.3) (42.9) (49.6) (73.2) (78.4)

∆WML 18-mon Mean, %: -0.32 20.02 4.94 -5.22 -20.94 -33.58

(% Neg.) (52.8) (32.2) (40.8) (62.0) (76.0) (89.6)

∆WML 36-mon Mean, %: 2.09 48.74 15.02 -6.37 -48.91 -76.79

(% Neg.) (46.5) (25.9) (31.5) (46.4) (82.1) (100.0)

∆HML 6-mon Mean, %: -0.04 -5.60 -0.52 0.53 5.41 7.77

(% Neg.) (53.4) (72.4) (54.7) (54.3) (32.3) (23.5)

∆HML 18-mon Mean, %: 0.70 -9.20 -7.30 -1.59 20.83 32.34

(% Neg.) (50.5) (62.8) (63.3) (52.9) (23.1) (14.6)

∆HML 36-mon Mean, %: 0.24 -26.61 -14.06 13.52 27.97 40.91

(% Neg.) (55.3) (83.9) (71.1) (42.8) (23.2) (6.7)

38

Table 8: The Changes in Spreads, Return Dispersion, and Business Cycle Variables

This table reports on whether the RD-spread relation remains reliably evident when adding well-

known business-cycle explanatory variables to our primary empirical models. We estimate the following

two models:

∆WMLjt = θ0 +θ1RRD1−3,(j+2)−(j+13) +θ2StRt1,36 +θ3dyst−1 +θ4divt−1 +θ5termt−1 +θ6yd3t−1 + εt

∆HMLjt = θ0 + θ1RRD1−3,(j+2)−(j+13) + θ2StRt1,36 + θ3dyst−1 + θ4divt−1 + θ5termt−1 + θ6yd3t−1 + εt

where ∆WMLjt is the change in the firm-level WML spread for the j-month horizon; ∆HMLj

t is the

change in the HML spread for the j-month horizon, where j equals 6, 18, or 36-months. The ∆WML,

∆HML, and RRD-trend terms are as defined in Tables 3 through 6. StRt1,36 is the lagged 36-month

stock market return over months t−1 to t−36; dyst−1, divt−1, termt−1, yd3t−1 are the lagged business

cycle variables; and the θ’s are coefficients to estimated. dys is the default premium based on the yield

difference between Moodys BAA and AAA rated bonds, div is the stock market’s aggregate dividend

yield, term is the difference between the yield of 10-year T-bonds and 3-month T-bills, and yd3 is the

yield of 3-month T-bills. Panel A reports our primary firm-level WML spreads, as in Panel A for Tables

3 through 5, and Panel B reports on the HML spreads as in Table 6. T-statistics are in parentheses,

based on heteroskedastic- and autocorrelation-consistent standard errors. The sample period is 1962

through 2005.

Spread Horizon θ1 θ2 θ3 θ4 θ5 θ6 R2

Panel A: WML Spreads6-month -6.60 0.068 -0.070 0.095 0.028 0.0075 18.2%

(-5.50) (1.47) (-1.62) (0.04) (0.19) (0.83)

18-month -9.08 -0.014 -0.068 0.137 0.024 0.015 35.7%(-7.15) (-0.16) (-0.78) (0.04) (1.10) (0.93)

36-month -17.55 0.034 -0.49 7.96 0.074 0.013 45.0%(-4.57) (0.12) (-2.16) (0.84) (0.73) (0.20)

Panel B: HML Spreads6-month 4.80 -0.0069 0.051 -1.79 -0.0080 -0.0002 14.7%

(3.88) (-0.18) (1.47) (-1.06) (-0.75) (-0.04)

18-month 12.66 -0.105 0.204 -5.66 -0.0005 -0.0084 31.3%(6.75) (-1.31) (2.94) (-1.52) (-0.02) (-0.56)

36-month 14.93 -0.183 0.415 -9.95 -0.044 -0.0066 37.3%(3.33) (-0.99) (3.43) (-1.32) (-0.79) (-0.28)

39

Table 9: The Two Components of the ‘Change in Spreads’ and the RD Trend

This table reports how the two components of the ‘change in spread’ variables, the forward-looking

spread component and the earlier reference-spread component, vary with the lagged RD trend. We

estimate variations of the following two models:

Model 1 : Spreadjt = λ0 + λ1RD1−3,(j+2)−(j+13) + λ2StRt1,12 + λ3RRD1−3,(j+2)−(j+13) + εt

Model 2 : Spreadjt−(j+3) = λ0 + λ1RD1−3,(j+2)−(j+13) + λ2StRt1,12 + λ3RRD1−3,(j+2)−(j+13) + εt

where Spread refers to either the WML or HML spread. Model 1 refers to the forward-looking spread

covering holding months t to either t+5, t+17, or t+35 for j=6, 18, or 36. Model 2 refers to the earlier

reference spread that is used when calculating the ‘change in spread’ variable for Tables 2 through 8.

For Model 2, the spread refers to the earlier spread covering holding months t-4 to either t− 9, t− 21,

or t−39 for j=6, 18, or 36. The RD-trend and RRD-trend are as defined in Tables 3 through 6, and the

λs are coefficients to be estimated. We report on our primary firm-level WML spreads, industry-level

WML spreads, and firm-level HML spreads, as defined in Section 3.2. The sample period is 1962 to

2005. T-statistics are in parentheses, based on heteroskedastic- and autocorrelation-consistent standard

errors.

Variation 1 (λ3 = 0) Variation 2 (λ1, λ2 = 0)Spread λ1 λ2 R2 λ3 R2

Firm-level WML6t -2.37 (-3.22) 0.088 (2.69) 9.2% -1.83 (-2.61) 3.8%

Firm-level WML18t -6.35 (-5.67) 0.143 (2.84) 34.8% -6.08 (-4.13) 29.2%

Firm-level WML36t -12.74 (-9.00) 0.406 (3.76) 53.3% -13.22 (-5.82) 43.0%

Firm-level WML6t−9 3.39 (5.55) 0.005 (0.16) 17.0% 3.64 (5.06) 14.9%

Firm-level WML18t−21 2.06 (1.47) 0.139 (2.51) 12.5% 3.09 (1.59) 7.6%

Firm-level WML36t−39 6.65 (2.28) 0.306 (1.71) 21.6% 7.77 (2.37) 13.7%

Industry-level WML6t -1.83 (-3.62) 0.071 (2.92) 10.2% -1.44 (-2.68) 4.2%

Industry-level WML18t -3.28 (-2.26) 0.108 (1.75) 11.2% -3.51 (-2.01) 11.4%

Industry-level WML36t -9.14 (-7.31) 0.183 (1.32) 32.8% -10.71 (-8.84) 36.1%

Industry-level WML6t−9 2.32 (4.43) -0.004 (-0.20) 13.8% 2.40 (3.70) 11.8%

Industry-level WML18t−21 3.10 (2.82) 0.149 (2.95) 24.6% 4.65 (3.28) 20.3%

Industry-level WML36t−39 7.07 (3.80) 0.247 (2.04) 27.0% 8.86 (4.48) 23.5%

Firm-level HML6t -0.04 (-0.06) 0.028 (0.63) 0.3% 0.25 (0.39) 0.1%

Firm-level HML18t 2.52 (1.31) 0.052 (0.44) 3.4% 3.18 (1.73) 4.4%

Firm-level HML36t 3.73 (2.57) -0.140 (-0.60) 3.9% 4.49 (2.36) 3.7%

Firm-level HML6t−9 -4.42 (-4.96) -0.084 (-1.93) 25.1% -4.64 (-4.61) 21.6%

Firm-level HML18t−21 -7.06 (-5.22) -0.199 (-2.15) 29.9% -7.66 (-5.55) 26.4%

Firm-level HML36t−39 -4.19 (-4.19) -0.067 (-0.53) 35.2% -13.27 (-4.03) 32.6%

40

Table 10: A Bivariate Regime-Switching Model for Stock Returns and the RD-trend

This table reports on estimating the following two-state, bivariate regime-switching model for the

monthly returns of a more-cyclical and less-cyclical industry-based portfolio:

rmc,t = µsmc + σs

mcηmc,t

rlc,t = µslc + σs

lcηlc,t

Where rmc,t and rlc,t are the monthly returns of our more-cyclical portfolio of industries and our less-

cyclical portfolio of industries, respectively, where the more-cyclical (less-cyclical) portfolio contains the 6

industries with the highest (lowest) market beta of the 48 industries in our sample; µsmc and µs

lc are regime-

specific mean returns for the respective series; σsmc and σs

lc are regime-specific standard deviations for each

respective series; and ηmc,t and ηlc,t are bivariate, standard, normally-distributed, random variables. The

superscript s refers to the regime, either regime-one (the good regime) or regime-two (the bad regime).

The s state variable is modeled with time-varying transition probabilities (pjj(t)):

pjj(t) =ecj+djRD1−3,8−19

1 + ecj+djRD1−3,8−19,

where j = 1 (regime-one) or j = 2 (regime-two); pjj(t) equals the probability that st = j(the second

subscript), given that st−1 = j (the first subscript); RD1−3,8−19 is our large-firm RD-trend variable used

in Table 2. The µss, σss, cjs, djs, and correlations between the ηts are parameters to be estimated.

We estimate the model by maximizing the log-likelihood function for the bivariate normal density with

regime-switching between the two states. The sample period is 1962 to 2005. The coefficient estimates are

reported below, with standard errors in parentheses. The return parameters are given in percentage units.1, 2, 3, and 4, indicate 0.1%, 1%, 5%, and 10% p-values for whether the regime-specific means, and the

estimated cj and dj coefficients are statistically significantly different than zero.

Good Regime (s = 1, j = 1) Bad Regime (s = 2, j = 2)Coeff. Estimated Parameters Estimated Parametersµs

mc 1.531 (0.271) 0.180 (0.745)

µslc 1.141 (0.164) 0.8444 (0.447)

σsmc 4.85 (0.227) 9.16 (0.545)

σslc 2.94 (0.170) 5.58 (0.324)

ρ 0.672 (0.033) 0.646 (0.044)

cj 2.911 (0.343) 1.981 (0.353)

dj -0.8761 (0.253) -0.283 (0.198)

41

Figure 1: Time-series of 6-month WML and HML Return Spreads and the Lagged RD Trend

This figure plots the time-series of 6-month Winner-minus-Loser return spreads, 6-month High-minus-

Low book-to-market return spreads, and the corresponding lagged large-firm return dispersion (RD)

trend. Panel A plots the 6-month WML return spreads for the decile strategy on individual stocks, as

defined in Section 3.2. Panel B plots the 6-month HML return spreads, as defined in Section 3.2. For

Panels A and B, the month t value refers to the cumulative spread for the holding period over t to t+5

in percentage units. For the WML spread, the ranking period is over t− 7 to t− 2. Panel C plots the

large-firm RD-trend, where the month t value refers to the large-firm RD1−3,8−19, as defined in Section

3.4.

Panel A: Time-series of 6-month WML Return Spreads, Individual Stocks

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-40

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Panel B: Time-series of 6-month HML Return Spreads

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Panel C: Time-series of Large-firm RD-Trend

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`

42

Figure 2: Time-series of 18-month WML and HML Return Spreads and the Lagged RD Trend

This figure plots the time-series of 18-month Winner-minus-Loser return spreads, 18-month High-minus-

Low book-to-market return spreads, and the corresponding lagged large-firm return dispersion (RD)

trend. Panel A plots the 18-month WML return spreads for the decile strategy on individual stocks,

as defined in Section 3.2. Panel B plots the 18-month HML return spreads, as defined in Section 3.2.

For Panels A and B, the month t value refers to the cumulative spread for the holding period over t to

t + 17 in percentage units. For the WML spread, the ranking period is over t − 19 to t − 2. Panel C

plots the large-firm RD-trend, where the month t value refers to the large-firm RD1−3,20−31, as defined

in Section 3.4.

Panel A: Time-series of 18-month WML Return Spreads, Individual Stocks

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2Panel B: Time-series of 18-month HML Return Spreads

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Panel C: Time-series of Large-firm RD-Trend

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43

Figure 3: Time-series of 36-month WML and HML Return Spreads and the Lagged RD Trend

This figure plots the time-series of 36-month Winner-minus-Loser return spreads, 36-month High-minus-

Low book-to-market return spreads, and the corresponding lagged large-firm return dispersion (RD)

trend. Panel A plots the 36-month WML return spreads for the decile strategy on individual stocks,

as defined in Section 3.2. Panel B plots the 36-month HML return spreads, as defined in Section 3.2.

For Panels A and B, the month t value refers to the cumulative spread for the holding period over t to

t + 35 in percentage units. For the WML spread, the ranking period is over t − 37 to t − 2. Panel C

plots the large-firm RD-trend, where the month t value refers to the large-firm RD1−3,38−49, as defined

in Section 3.4.

Panel A: Time-series of 36-month WML Return Spreads, Individual Stocks

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150

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Panel B: Time-series of 36-month HML Return Spreads

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Panel C: Time-series of Large-firm RD-Trend

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44

Appendix A: An Analytical Framework for WML Profits with Regime Switching

Here, we offer a formal analytical framework to analyze how two-state regime-switching can influence WML

spreads. The intuition follows from our simple example in Section 2.2. Our framework starts from the

decomposition of the weighted relative strength strategy (WRSS) in Lo and MacKinlay (1990) and Conrad

and Kaul (1998).

Lo and MacKinlay (1990) propose a momentum strategy termed the weighted relative strength strategy

(WRSS), which has also been widely used in the literature; see, e.g. Conrad and Kaul (1998), Jegadeesh

and Titman (2002), and Lewellen (2002). Jegadeesh and Titman (1993) find that returns from the WRSS

and the decile-based strategy have a correlation of 0.95 in their sample. Here, we use the WRSS because it

is analytically convenient to decompose the WML profits.

Under the WRSS, investors buy or short stocks in proportion to how the individual stock return over

the ranking period differs from the average stock return over the ranking period. Specifically, the investment

weight assigned to stock i at time t is given by:

wit =1N

(rit−1 − r̄t−1) (12)

where rit−1 equals the return of stock i over period t − 1 and r̄t−1 is the return on an equally-weighted

portfolio of all N stocks in the sample. The weights sum to zero. The profits over period t from this strategy

can be expressed as

πt =1N

N∑

i=1

rit(rit−1 − r̄t−1) (13)

Lo and MacKinlay (1990) and Conrad and Kaul (1998) show that expected profits from the WRSS can

be decomposed into three distinct sources.

E(πt) = −Cov(r̄t, r̄t−1) +1N

N∑

i=1

Cov(rit, rit−1) + σ2(µ) (14)

where the first term is the negative of the autocovariance of the market, the second term is the cross-

sectional average of the autocovariances of individual stocks, and σ2(µ) is the cross-sectional variance in the

unconditional expected returns.

Next, we consider the influence of regime-switching on WML return behavior.14 Our focus is on the an-

alytical relation between WML spreads and the parameters of the regime-switching process. The framework

only requires that certain stocks have relatively higher realized means in good market states and other stocks

have relatively higher realized means in poor market states. More formally, we assume the return-generating

process for a stock i can be written as:

rit = µsi + σs

i ηit (15)

14See Turner, Startz, and Nelson (1989), Veronesi (1999), Maheu and McCurdy (2000), and Ang and Bekaert

(2002) for examples of regime-switching applications in finance.

45

where s denotes the unobserved regime indicator (1 or 2) and η is a zero-mean random variable that is

identically and independently distributed. Following Hamilton (1989), we assume that s follows a two-state,

first-order Markov process with the following transition probability matrix

P =

p11 1− p11

1− p22 p22

(16)

where p11 = prob(st = 1|st−1 = 1) and p22 = prob(st = 2|st−1 = 2). In this specification, the transition

probabilities dictate the persistence of the regimes, where the expected duration of regime i in periods, Di,

is defined as Di = 11−pii

. To contribute to WML spreads, the process only requires regime shifts in the

mean return. However, volatility is also likely to change with the regime, so equation (15) also allows for

the volatility to switch.

Denote λ1 as the unconditional probability that the process is in regime 1 (λ1 = 1−p222−p11−p22

). Timmermann

(2000) shows that the autocovariance function for this two-state regime-switching model can be written as

follows:

Cov(rt, rt−n) = λ1(1− λ1)(µ1 − µ2)2vec(Pn)′v1 (17)

where v1 = ((1− λ1),−(1− λ1),−λ1, λ1)′. Then, the first-order autocovariance is given by

Cov(rt, rt−1) = λ1(1− λ1)(µ1 − µ2)2(p11 + p22 − 1). (18)

From equation (18), first note that the larger the magnitude of the regime-mean difference (µ1 − µ2), the

higher the autocorrelation, other things being equal. Second, note that the first-order autocovariance will be

positive if (p11 +p22) > 1. Numerical results in Timmermann (2000) suggest that this basic regime-switching

model can easily generate monthly autocorrelations comparable to that in the data.

Proposition 1 shows how a regime-switching process can contribute to determining the average WML

profit, beyond the profit suggested by σ2(µ). This proposition gives the expected WML profit per period,

where a period is the length of the strategy’s symmetric ranking and holding horizon.

Proposition 1 Under the regime-switching process for stock returns as described in equations (15) and (16),

the expected WRSS profit, E[πt], is given by:

E[πt] = Aσ2(d) + σ2(µ), (19)

where A = λ1(1 − λ1)(p11 + p22 − 1), σ2(d) is the cross-sectional variance of regime-mean differences, and

σ2(µ) is the cross-sectional variance of unconditional expected returns. The regime-mean difference for a

stock is the difference between its mean return in one state and the other state.

To prove Proposition 1, we substitute equation (18) into equation (14). Then, we denote the regime

mean for the equally-weighted market index as µ̄s for regime s, where:

µ̄s =1N

N∑

i=1

µsi (20)

46

Next, define di ≡ µ1i − µ2

i as the regime-mean difference for stock i. The expected WRSS profit becomes:

E[πt] = λ1(1− λ1)(p11 + p22 − 1)

[1N

N∑

i=1

(µ1i − µ2

i )2 − (µ̄1 − µ̄2)2

]+ σ2(µ) (21)

= λ1(1− λ1)(p11 + p22 − 1)

[1N

N∑

i=1

(di − d̄)2]

+ σ2(µ) (22)

= λ1(1− λ1)(p11 + p22 − 1)σ2(d) + σ2(µ) (23)

Hence, in this framework, expected average WML profits can be attributed to three sources: the cross-

sectional variance of regime-mean differences (σ2(d)), the cross-sectional variance of unconditional expected

returns (σ2(µ)), and the transition probabilities. As long as (p11 + p22 − 1) > 0, A will be positive, which

is strongly suggested by the data for the 6-month horizon. Therefore, average medium-run WML profits

will be positive and will also be greater than suggested by the cross-sectional variation in unconditional

expected returns. It is important to note that this proposition is defined in terms of one-base period and

the framework does not allow for regime-switching at a smaller horizon than this defined base period.

Conceptually, as the strategy’s horizon increases to near the expected duration of the less persistent

regime, the transition probability p22 will approach zero. As p22 shrinks, (p11 + p22) will become less than

one and the first term in equation (19) will become negative and act to reduce the expected WRSS profits (as

compared to the profits implied solely by σ2(µ)). Therefore, for longer horizon spreads where the strategy’s

horizon approaches the duration of the shorter regime, average WML spreads will be less than suggested by

the cross-sectional variation in unconditional expected returns, and the expected spread could be negative.

For example, assume regime-one has an expected duration of 48 months and regime-two has an expected

duration of 24 months. When 6 months is the spread period in (19), the implied p11 and p22 are 0.875 and

0.75, respectively (with durations of eight and four 6-month periods for the two regimes, respectively). Thus,

for a 6-month WML spread, the ‘A’ value is positive at 0.14 and WML profits would be greater than implied

solely by the cross-sectional variation in expected returns. Next, assume that 24-months is the spread period

in (19). Now, the implied p11 and p22 are 0.5 and 0, respectively (with durations of two and one 24-month

period for the two regimes, respectively). Thus, for a 24-month WML spread, the ‘A’ value is negative at

-0.11 and WML profits would be less than implied solely by the cross-sectional variation in expected returns.

Appendix B: Robustness Results

In this appendix, we provide some details for the robustness and additional evidence that is summarized in

Section 4.4. In each table, we re-estimate variations of our primary models, given by equations (3) through

(6), but with alternate spreads or alternate explanatory RD terms. Details of the models and results are

provided in Tables B1 through B3.

47

Appendix B:

Table B1: Robustness - Alternate Firm-level WML Spreads and the Lagged RD

This table reports on two alternate firm-level WML return spreads. Panel A reports on large-

firm WML return spreads, where the decile-based WML strategies are implemented only on the largest

quintile of NYSE/AMEX stocks. The market capitalization screen is from the last month of the ranking

period (month t − 2, relative to the holding period that starts in month t). Panel B reports on less

extreme WML return spreads, where the WML strategies go long the decile-9 stocks (the second-best

decile of winners) and short the decile-2 stocks (the second-worst decile of losers). The ‘9-2’ strategies

are implemented on all NYSE/AMEX stocks after our screens explained in Section 3.2. We report on the

the 6-month, 18-month, and 36-month horizon. Columns 2 through 4 report on Model 1 and Columns

5 through 6 report on Model 2, where the models correspond to Tables 3 through 5 (depending upon

the return horizon). The coefficients of interest are β1 on the RD-trend and β4 on the RRD-trend. T-

statistics are in parentheses, based on heteroskedastic- and autocorrelation-consistent standard errors.

The sample period is 1962 through 2005.

Panel A: WML Spreads for Large-firm Stocks Only

Regression Model 1 Regression Model 2Return Horizon β1 β2 R2 β4 R2

6-month -5.87 0.059 17.8% -6.26 15.7%(-5.43) (0.69) (-4.93)

18-month -9.42 -0.134 32.4% -10.94 33.2%(-5.45) (-0.83) (-5.89)

36-month -18.55 0.247 37.0% -20.96 36.9%(-5.13) (0.66) (-4.78)

Panel B: WML Spreads Using a ‘Decile-9 minus Decile-2’ Strategy

Regression Model 1 Regression Model 2Return Horizon β1 β2 R2 β4 R2

6-month -3.36 0.096 17.4% -3.24 11.7%(-6.05) (2.02) (-4.92)

18-month -5.56 -0.190 27.5% -5.94 21.6%(-5.54) (-1.89) (-4.54)

36-month -11.74 0.013 34.6% -12.66 31.8%(-4.12) (0.07) (-3.74)

48

Appendix B:

Table B2: Robustness - WML and HML Spreads with Alternate RD Measures

This table reports how both WML and HML return spreads vary with alternate RD measures. We

estimate variations of the following two models:

Model 1 : ∆WMLjt = β0 + β1 RD1−3 + β2 RRD1−3,(j+2)−(j+13) + εt

Model 2 : ∆HMLjt = β0 + β1 RD1−3 + β2 RRD1−3,(j+2)−(j+13) + εt

where ∆WMLjt is the change in WML return spread for the j-month horizon, as for Panel A in Tables

3, 4, and 5 for the 6, 18, and 36-month strategies, respectively; ∆HMLjt is the change in HML return

spread for the j-month horizon, as in Table 6 for the 6, 18, and 36-month strategies; RD1−3 is the lagged

3-month moving average of RD for months t − 1 to t − 3; RRD1−3,(j+2)−(j+13) is the lagged trend in

the market-adjusted relative return dispersion. Each panel reports on a different RD measure. Panel

A reports on our large-firm RD, with the RD-trend timing as in Tables 3 through 5 for the different

horizons. Panel B reports on our large-firm RD, but with a simple 3-month moving average rather

than an RD-trend. Panel C reports on a firm-level, broad market RD, with the RD-trend timing as

in Tables 3 through 5 for the different horizons, and with a firm-level RD that is calculated using the

largest 80th percentile of NYSE/AMEX stocks. Panel D reports on an industry-level RD, using the 48

value-weighted industry returns, and the RD-trend timing as in Tables 3 through 5 for the different

horizons. Panel E reports on the RD of 100 double-sorted book-to-market/size portfolio returns (10 x

10), using value-weighted returns from the K. French data library, and with the RD-trend timing from

Tables 3 through 5 for the different horizons. T-statistics are in parentheses, based on heteroskedastic-

and autocorrelation-consistent standard errors. The sample period is 1962 through 2005.

Panel A: Large-firm RRD-trend, (restricts β1 = 0)

Firm-level Industry-level

Model, Horizon β2 R2 β2 R2

Model 1, 6-month ∆WML -5.47 (-5.81) 14.5% -3.83 (-4.66) 13.7%

Model 1, 18-month ∆WML -9.17 (-7.09) 34.8% -8.15 (-4.40) 29.0%

Model 1, 36-month ∆WML -20.91 (-4.31) 39.9% -19.57 (-9.88) 44.7%

Model 2, 6-month ∆HML 2.22 (2.09) 4.4% n/a

Model 2, 18-month ∆HML 6.17 (2.74) 15.7% n/a

Model 2, 36-month ∆HML 11.99 (6.49) 33.0% n/a

49

Table B2: (continued)

Panel B: Large-firm RD 3-month Moving Average, (restricts β2 = 0)

Firm-level Industry-level

Model, Horizon β1 R2 β1 R2

Model 1, 6-month ∆WML -2.66 (-2.67) 7.3% -1.78 (-2.10) 6.1%

Model 1, 18-month ∆WML -4.80 (-3.06) 9.6% -5.71 (-2.50) 14.4%

Model 1, 36-month ∆WML -17.67 (-3.80) 25.0% -18.39 (-8.93) 34.7%

Model 2, 6-month ∆HML 1.48 (2.14) 4.2% n/a

Model 2, 18-month ∆HML 5.42 (2.79) 12.3% n/a

Model 2, 36-month ∆HML 11.61 (5.58) 27.2% n/a

Panel C: Firm-level Broad Market RRD-trend, (restricts β1 = 0)

Firm-level Industry-level

Model, Horizon β2 R2 β2 R2

Model 1, 6-month ∆WML -4.31 (-4.91) 10.9% -2.20 (-2.86) 5.4%

Model 1, 18-month ∆WML -6.93 (-4.78) 20.5% -5.48 (-2.71) 13.5%

Model 1, 36-month ∆WML -16.42 (-3.88) 25.9% -17.49 (-6.10) 37.6%

Model 2, 6-month ∆HML 2.24 (2.60) 5.5% n/a

Model 2, 18-month ∆HML 5.34 (2.29) 12.1% n/a

Model 2, 36-month ∆HML 12.42 (5.58) 37.3% n/a

Panel D: Industry-level RRD-trend, (restricts β1 = 0)

Firm-level Industry-level

Model, Horizon β2 R2 β2 R2

Model 1, 6-month ∆WML -5.88 (-4.23) 7.2% -3.62 (-3.24) 5.1%

Model 1, 18-month ∆WML -12.47 (-5.53) 24.2% -9.99 (-2.98) 16.4%

Model 1, 36-month ∆WML -24.38 (-4.09) 23.7% -26.76 (-8.18) 36.6%

Model 2, 6-month ∆HML 3.20 (2.16) 3.9% n/a

Model 2, 18-month ∆HML 8.02 (2.05) 10.0% n/a

Model 2, 36-month ∆HML 16.94 (5.77) 28.8% n/a

Panel E: RRD-trend of 100 Size and Book-to-Market Portfolios, (restricts β1 = 0)

Firm-level Industry-level

Model, Horizon β2 R2 β2 R2

Model 1, 6-month ∆WML -6.55 (-6.15) 12.3% -4.59 (-5.76) 11.5%

Model 1, 18-month ∆WML -10.1 (-4.25) 19.1% -11.37 (-5.50) 26.4%

Model 1, 36-month ∆WML -20.70 (-3.45) 13.5% -24.79 (-8.61) 25.2%

Model 2, 6-month ∆HML 4.89 (4.63) 12.8% n/a

Model 2, 18-month ∆HML 10.84 (4.78) 22.4% n/a

Model 2, 36-month ∆HML 17.76 (4.19) 25.0% n/a

50

Appendix B:

Table B3: Robustness - Alternate Timing for the ‘Change in Spreads’ and RD Trend

This table reports on our primary empirical results using alternate timing for the ‘change in spread’

variables and the RD-trend. We estimate variations of the following two models for the 6, 18, and

36-month spread horizon:

Model 1 : ∆WMLjt = λ0 + λ1RD1−4,(j+5)−(j+22) + λ2StRt1,12 + λ3RRD1−4,(j+5)−(j+22) + εt

Model 2 : ∆HMLjt = λ0 + λ1RD1−4,(j+5)−(j+22) + λ2StRt1,12 + λ3RRD1−4,(j+5)−(j+22) + εt

where the gap between spread values for the ‘change in spread’ term is now 4 months instead of 3

months. For example, here, the ‘change in spread’ for the 6-month horizon is equal to the spread over

months t to t+5 less the spread over months t-10 to t-5. Here, the RD-trend variables also uses a longer

nearby moving average of 4 months, months t-1 to t-4, instead of the 3 months used in Tables 3 through

6. Also, in the RD-trend, the longer reference RD moving average is now 18 months instead of the 12

months used in Tables 3 through 6. Finally, in the RD-trend, the older RD moving average is earlier

than before. For example, for the 6-month horizon, the older RD moving average in the RD-trend

term covers the month t-28 to t-11, which predates the older spread value used in the change-in-spread

term. The RD is our large-firm RD for the WML spreads and the book-to-market/size RD for the HML

spreads, other terms are as defined in Tables 3 through 6, and the λs are coefficients to be estimated.

The sample period is 1962 to 2005. T-statistics are in parentheses, based on heteroskedastic- and

autocorrelation-consistent standard errors.

Variation 1 (λ3 = 0) Variation 2 (λ1, λ2 = 0)

Spread λ1 λ2 R2 λ3 R2

Firm-level ∆WML6t -3.37 (-4.01) 0.112 (1.10) 10.4% -3.39 (-3.09) 7.5%

Firm-level ∆WML18t -6.78 (-5.65) -0.315 (-2.67) 31.7% -7.31 (-4.89) 25.0%

Firm-level ∆WML36t -18.58 (-4.30) 0.021 (0.05) 37.3% -20.36 (-4.12) 35.4%

Industry-level ∆WML6t -2.29 (-3.06) 0.054 (0.90) 8.2% -2.28 (-2.42) 6.2%

Industry-level ∆WML18t -5.84 (-3.17) -0.144 (-1.38) 22.4% -7.14 (-3.44) 24.7%

Industry-level ∆WML36t -16.66 (-7.27) -0.123 (-0.42) 38.4% -19.56 (-8.49) 42.0%

Firm-level ∆HML6t 4.25 (4.49) 0.118 (1.79) 15.5% 4.87 (4.58) 14.3%

Firm-level ∆HML18t 7.80 (2.19) 0.256 (1.68) 15.7% 8.66 (2.40) 14.9%

Firm-level ∆HML36t 17.53 (4.02) -0.047 (-0.17) 26.8% 18.80 (3.94) 26.4%

51