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Abstract

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

We study the effect of market cycles on both medium-run and long-run relative strength trading strategies. We find that payoffs for both strategies tend to be relatively higher within a market state (rising or falling markets), but substantially lower over transitions between states. Since shorter duration strategies are relatively less likely to include market transitions, our results help reconcile the puzzling fact that medium-run strategies are profitable, but long-run strategies are not. We find that the market's cross-sectional return dispersion: 1) tends to be higher around market transitions, and 2) is negatively related to the subsequent payoffs for both medium-run and long-run strategies.

The profitability suggested by simple relative strength strategies has generated much interest in the recent literature and has important theoretical and practical implications. Jegadeesh and Titman (1993) document that over periods of 3 to 12 months, past winners tend to outperform past losers. Recent empirical evidence suggests that these medium-run momentum profits persist in the 1990s, are present in both large-cap and small-cap stocks, and exist in a number of international markets.1 Curiously, in contrast to medium-run strategies, longer-run strategies tend to exhibit reversals, where past losers tend to outperform past winners.2

In this paper, we study the implications of market cycles on relative strength strategies at both the medium-run and longer-run horizons. Our focus is on a better understanding of: 1) the joint time-series behavior of both medium-run and longer-run strategy payoffs and 2) the horizon-based contrast in average relative strength payoffs where medium-run strategies have higher average profits and longer-run strategies have lower average profits, as compared to those suggested solely by the cross-sectional variation of unconditional mean returns.

A few earlier studies have considered the possibility that market cycles or market states may play a role in understanding relative strength strategies. However, these recent studies have focused solely on medium-term momentum and their market-state empirical approach, evidence, and interpretation are appreciably different than ours.

Cooper, Gutierrez, and Hameed (2004) present evidence that momentum profits depend upon the state of the market. Average momentum profits, using a lagged six-month ranking period, are appreciably positive (marginally negative) following an up (down) market state, defined as a nonnegative (negative) lagged 36-month market return. Cooper et al. (2004) argue that if overconfidence is higher following market increases, then their findings are consistent with the overconfidence behavioral model of Daniel, Hirshleifer, and Subrahmanyam (1998). Cooper et al. (2004) conclude that “models of asset pricing, both rational and behavioral, need to incorporate (or predict) such regime switches.”

Asem and Tian (2010) find that momentum profits are higher when markets stay in the same state (either up or down) and lower for market transitions. Their interpretation also appeals to Daniel et al. (1998), with the assumption of greater overconfidence for both up-state and down-state continuations. Asem and Tian (2010) use: 1) a lagged six-month ranking period, 2) the lagged 12-month market return to categorize the state, and 3) the subsequent one-month market return and one-month momentum profit to assess market continuations and momentum.

Sagi and Seasholes (2007) introduce a rational model where the documented correlation between market states and momentum may be explained by a firm's growth options. In their model, during up markets, firms tend to move closer to exercising their growth options, which tends to increase return autocorrelations (and, hence, momentum). During down markets, firms tend to move closer to financial distress, which tends to decrease return autocorrelations (and, as such, momentum). They provide supportive simulated evidence.3

Grundy and Martin's (2001) focus is not on market cycles, but they recognize that the market trend over the ranking period should result in dynamic factor loadings for momentum strategies. For ranking periods over an up-market, they note that momentum strategies should tend to place a positive “market-beta bet” as the past relative winners should tend to have higher betas. They analyze a momentum strategy with a lagged six-month ranking period and a subsequent one-month holding period, and find that the strategy's risk-adjusted performance improves when they take into account the dynamic factor loadings. The dynamic factor loadings suggest that transitions between market up and down states are likely to be associated with lower momentum payoffs, a notion that we explore further in our study.

Our study begins by providing a simple statistical decomposition of relative strength profits in a two-state market, with the goal of providing intuition and helping to frame our empirical investigation. Our notion of “up-state means” and “down-state means” refers to the realized mean stock return over a sizable but modest market period, such as several months to several years, where the realized means are associated with changing economic conditions. Consider that both Fama and French (1989) and Zhang (2005) argue that the market's price of risk is countercyclical, with weak economic conditions associated with a higher risk premium. If so, then transitions to a weaker economic state should be associated with lower average realized returns as stock prices respond negatively to both negative shocks to expected near-term cash flows and an increasing risk premium.4 Thus, we suggest a rational avenue for the observed contrast between up-state and down-state means, but behavioral influences might also presumably contribute.

With plausible regime durations, our statistical decomposition suggests both higher profits for medium-run strategies and lower profits for longer-run strategies, as compared to the profits implied solely by the cross-sectional variation in unconditional mean returns. The intuition is that cross-sectional differences in up-state and down-state mean returns can affect the relative strength profits in two opposing ways. First, the cross-sectional variance in state-specific means is likely to be greater than the cross-sectional variance in unconditional mean returns. As such, strategy outcomes when within-state should be relatively higher.5 However, across-state outcomes should be associated with lower payoff outcomes since stocks that perform relatively well in one state tend to perform relatively poorly in the other state. Thus, medium-run strategies are likely to benefit from market cyclicality since a relatively high proportion of the strategy's outcomes should be within-state. Conversely, long-run strategies are likely to be harmed by market cyclicality as a relatively higher proportion of the strategy's outcomes will include market transitions.

Empirically, from 1926 to 2010, we study both firm-level and industry-level symmetric relative strength strategies at both medium-run horizons (6 and 12 months) and longer-run horizons (18, 24, and 36 months). Our firm-level strategies are decile-based, that go long (short) the top-decile (bottom-decile) of past relative winners (losers). We evaluate similar industry-level strategies as a complement to our firm analysis because: 1) industry returns might better capture cyclical variation, and 2) Moskowitz and Grinblatt (1999) and Lewellen (2002) demonstrate the importance of industries in understanding medium-run momentum.

We provide new empirical evidence that suggests an important link between market cycles and relative strength strategies. Using two approaches to identify market states, one based on return dispersion as an ex ante indicator and a second based on realized market returns, we document that the payoffs of both medium-run and longer-run strategies tend to be relatively higher (lower) for within-state (across-state) outcomes.

In our first approach, we propose using the market's cross-sectional return dispersion (RD) as a leading indicator of market state transitions as supported by findings in Gomes, Kogan, and Zhang (2003), Stivers and Sun (2010), and related evidence in our study. Since market transitions should be associated with lower relative strength payoffs in our setting, our empirical prediction is a negative relation between RD and the subsequent relative strength payoffs. Strikingly, this is exactly what we find for all 10 of our strategies. For example, when our lagged three-month RD moving average is above its 75th percentile (below its 25th percentile), then the average of the subsequent payoffs is –0.70% and –0.78% per month (+0.91% and +0.44% per month) for our 12- and 24-month firm-level strategies, respectively.

In our second approach, we evaluate a simple ex post up-state and down-state classification method, based on peak-to-trough equity market behavior. We feel that this approach provides a more accurate classification of up and down market cycles, as compared to the methods in Cooper et al. (2004) and Asem and Tian (2010) that rely solely on the lagged cumulative market return. We find that the average payoffs of relative strength strategies are appreciably higher (lower) when the ranking and holding period are from the same (different) market state.

To conclude, we believe our collective findings suggest an intuitive market cycle perspective that contributes toward understanding both the joint time-series behavior of medium-run and long-run strategies, and the contrast in the average payoffs of medium-run versus long-run strategies. This article proceeds as follows. Section I presents our two-state statistical decomposition. Section II describes our data and relative strength strategies. Sections III and IV present our main empirical results, while Section V provides our conclusions.

I. Relative Strength Strategies in a Two-State Cyclical Market

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

In this section, we further discuss related literature and offer a simple statistical decomposition of relative strength profits in a market where the realized means of stock returns vary across two different market states. As previously discussed, our concept of up-state and down-state means refers to realized means tied to an economic outcome or transition rather than ex ante expected returns.

Lo and MacKinlay (1990) propose a relative strength strategy termed the weighted relative strength strategy (WRSS) that has been widely used in the literature (Conrad and Kaul, 1998; Jegadeesh and Titman, 2002; Lewellen, 2002). Under the WRSS, investors buy or short stocks in proportion to how the individual stock returns over the ranking period differ from the average stock return over the ranking period. Specifically, the investment weight assigned to stock i at time t is given by

  • image(1)

where rit-1 equals the return of stock i over period t− 1 and inline image 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

  • image(2)

The WRSS is analytically convenient. Lo and MacKinlay (1990) and Conrad and Kaul (1998) demonstrate that expected profits from the WRSS can be decomposed into three distinct sources:

  • image(3)

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 inline image is the cross-sectional variance in the unconditional expected returns.

If one assumes that stock prices follow a random walk process, as considered in Conrad and Kaul (1998), then all of the covariance terms will vanish in Equation (3). Thus, under the simple Conrad and Kaul (1998) framework, the only source of relative strength profits is the cross-sectional variance in unconditional mean returns. Using the cross-sectional variance of sample means as a measure of inline image, Conrad and Kaul (1998) argue that inline image appears capable of explaining a substantial portion of medium-run momentum profits.

However, Jegadeesh and Titman (2002) argue that it is improper to use the cross-sectional variance of sample means as a measure of inline image because the measurement error imparts an upward bias in the estimated inline image. They argue that inline image seems capable of explaining only a very modest percentage of total medium-run momentum profits. Further, Jegadeesh and Titman (2002) argue that reversals in longer-run relative strength strategies are inconsistent with a simple inline image explanation.

Regime-switching models have been widely used to describe random structural breaks in time series data and have been particularly successful in modeling financial data (Hamilton, 1989; Turner, Startz, and Nelson, 1989; Gray, 1996; Ang and Bekaert, 2002). Our focus in this section is on the analytical relationship between relative strength profits and the parameters of a two-state regime-switching process. In addition to the intuition of business cycles, our motivation for a regime-switching perspective also includes recent results in Chordia and Shivakumar (2002), Grundy and Martin (2001), Cooper et al. (2004), Stivers and Sun (2010), and Asem and Tian (2010), which collectively suggest the time-series behavior of medium-run momentum profits may be related to economic cycles or the market state.6

More formally, we assume the return-generating process for a stock i can be written as

  • image(4)

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:

  • image(5)

where p11= prob (st= 1|st− 1= 1) and p22= prob (st= 2|st− 1= 2). In this simple specification, the transition probabilities are constant and they dictate the persistence of the regimes, where the expected duration of regime i in periods Di is defined as inline image. To contribute to relative strength profits, the process only requires regime shifts in the mean return.

Denote inline image as the unconditional probability that the process is in regime 1 inline image. Timmermann (2000) finds that the first order autocovariance function for this two-state regime-switching process can be written as

  • image(6)

From Equation (6), note that the first-order autocovariance will be positive if (p11+p22) > 1.

Proposition 1 below demonstrates how a regime-switching process can contribute to relative strength profits beyond the profits suggested solely by inline image. This proposition gives the average WRSS profit per period, where a period equals the length of the strategy's ranking and holding horizon.

Proposition 1: Under the regime-switching process for stock returns as described in Equations (4) and (5), the expected WRSS profit, inline image, is given by

  • image(7)

where inline imageinline image is the cross-sectional variance of regime-mean differences, and inline image is the cross-sectional variance of unconditional expected returns. A stock's regime-mean difference is the difference between the stock's up-state mean and its down-state mean (for the proof, see the Appendix).

Thus, in this framework, expected WRSS profits can be attributed to three sources: 1) the cross-sectional variance of unconditional expected returns inline image, 2) the cross-sectional variance of regime-mean differences inline image, and 3) the expected regime durations implied by the transition probabilities. The first source is well known (Conrad and Kaul, 1998; Bulkley and Nawosah, 2009). However, the other two factors have not been considered in the literature.

It is important to note that the transition probabilities and durations are measured in terms of multiples of “base periods,” where a base period is the horizon of a particular strategy. Thus, Equation (7) must be applied separately for strategies with different horizons, and Equation (7) is derived under the assumption that the expected duration of the market's up- and down-state will be some multiple of the strategy's horizon. For example, consider a market with an expected duration of 54 months for the up-state and 18 months for the down-state. For a six-month relative strength strategy, the up-state (down-state) has nine expected periods (three expected periods). Then, for this six-month strategy, the value of “A” would be 0.104 with p11 equal to 0.89 and p22 equal to 0.67 and average six-month strategy profits would be greater than suggested solely by the cross-sectional variation in unconditional expected returns. However, for an 18-month strategy, this same cyclical market would imply an up-state (down-state) with three expected periods (one expected period). Then, for the 18-month strategy, the value of A would be –0.0625, with p11 equal to 0.67 and p22 equal to 0.0 and average 18-month relative strength profits would be less than suggested solely by the cross-sectional variation in unconditional mean returns.

We stress that the primary purpose of our two-state statistical decomposition is to: 1) formalize the intuition that market cycles can generate medium-run (longer-run) relative strength profits that are greater than (lower than) that suggested solely by the cross-sectional variation in unconditional mean returns and 2) suggest implications for the time-series behavior of both medium-run and longer-run strategies. In Section IV.A, we briefly discuss how the observed cyclical behavior of industry returns supports the empirical relevance of our two-state framework here. However, we do not argue that this simple decomposition can adequately model or fully explain all aspects of the performance of relative strength strategies. Accordingly, we next move to the data to evaluate some empirical implications suggested by this market cycle perspective.

II. Description of Our Data and the Relative Strength Strategies

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

A. Raw Return Data and Sample Periods

Our empirical work features return data from two sources. For US individual stock return data, we examine NYSE and AMEX stock returns from the Center for Research in Security Prices (CRSP) monthly return file. For US industry-level return data, we examine monthly returns for the 30 value-weighted industry portfolios from the K. French data library.

To estimate the cross-sectional return dispersion across US disaggregate stock portfolio returns, we use the 100 size and book-to-market equity ratio portfolios from the K. French data library. We choose value-weighted portfolio returns from the French library, rather than equally-weighted portfolios because: 1) Moskowitz and Grinblatt (1999) examine value-weighted industry portfolios and 2) value-weighted returns should mitigate microstructure concerns, such as autocorrelation in portfolio returns, due to high nonsynchronous trading in small-cap stocks.

In our empirical work, we examine relative strength strategies implemented on monthly stock returns from July 1926 to December 2010. We choose July 1926 as the start date, rather than the CRSP start date of January 1926, because the industry data in the French library begin in July 1926 and we want the firm-level and industry-level return data to coincide. In all our empirical work, we report on separate estimations for the following five sample periods: 1) the full period from July 1926 to December 2010, 2) the modern period from January 1962 to December 2010, where the start date roughly corresponds to early momentum studies such as Jegadeesh and Titman (1993), 3) the first half of the modern period from January 1962 to June 1986, 4) the second half of the modern period from July 1986 to December 2010, and 5) the older July 1926 to December 1961 period. With these choices, our second-half modern period is substantially out-of-sample (relative to early momentum studies) and the pre-1962 sample predates early momentum studies.

B. Methodology for Relative Strength Strategies

Our empirical work examines percentile-based relative strength strategies (rather than the WRSS as in the Section I development) due to their prevalence in the literature and the more straightforward intuition of interpreting the resulting profits. Jegadeesh and Titman (1993) find that returns from the WRSS and the decile-based strategy have a correlation of 0.95 in their sample. The percentile-based strategies form a zero cost portfolio by starting with an equally sized long and short position. All of our strategies follow the common practice of skipping a month between the ranking period and the investment holding period to avoid microstructure distortions.

We strictly examine symmetric strategies, where the ranking period and subsequent holding period are of the same length. We make this choice: 1) because of its prevalence in the literature (Conrad and Kaul, 1998; Griffin et al., 2003), 2) because symmetric strategies fit with our analytical framework in Section I, and 3) for brevity to limit the strategy variations. We examine symmetric 6, 12, 18, 24, and 36-month relative strength strategies. We begin with the six-month strategy due to its prevalence in the literature as a medium-run strategy with reliably positive average profits that survive risk adjustments. We extend out to a 36-month horizon due to the reversals of long-run relative strength strategies first noted in DeBondt and Thaler (1985).

For our relative strength strategies on firm-level returns, we rank NYSE and AMEX individual stocks into deciles based on their j-month ranking period return. The decile-based strategies take a long position in the top decile (the past relative winners) and a short position in the bottom decile (the past relative losers) with equally weighted portfolios. After skipping a month, the positions are held for the subsequent j-month investment holding period. We use a price screen of one dollar at the beginning of each holding period to minimize microstructure issues related to illiquid stocks. For our strategies implemented on the 30 industries, we perform a similar procedure except we use a percentile strategy where the past winner and past loser portfolios each contain five industries.

In our time-series work, we use the following convention. The relative strength payoff for month t, inline image, refers to the return difference between the past winner and past loser portfolios for the j-month investment holding period that begins in month t and concludes in month t+(j-1) on a per month basis. To standardize the RS payoffs to a per month basis, we take the cumulative return difference over the holding period and divide by the number of months in the holding period. The corresponding ranking period is over months t− (j+1) to t− 2. In regard to our different subsamples, the date for inline image determines the payoff's subsample classification. Since our analysis features overlapping payoffs, the standard errors for the estimated regression coefficients must be corrected to take into account the correlated lags. In our regression analysis, we calculate heteroskedastic and autocorrelation consistent standard errors using the Newey-West approach where the number of correlated lags is set equal to the number of months in the respective relative strength strategy.

One important difference between our approach and previous time-series work in Chordia and Shivakumar (2002), Griffin et al. (2003), and others, is that their relative strength profits for a given month use an average across the last n investment portfolios reflecting n different ranking periods, where n is the number of months for the ranking and holding period (typically six). Our alternate timing convention is necessary as the relative strength profit for month t corresponds directly to the single most recent ranking period. For our purposes, it would be inappropriate to mix portfolios formed from different ranking periods due to the timing importance of relating market transitions to specific profit outcomes. Figure 1 illustrates the variability in the six-month firm-level and industry-level strategy payoffs.

image

Figure 1. The Time Series of Six-Month Relative Strength Strategies and the Market State

This figure displays the time series of the payoffs for six-month relative strength strategies, implemented on returns from July 1926 to December 2010. The observation for month t is for the holding-period payoff over months t to t+ 5. The units are in percentage returns per month, equal to the cumulative return payoff divided by six. Panel A (Panel B) displays the firm-level (industry-level) strategy's payoffs. In March and April 2009, the values for the firm-level strategy are off the chart at −30.8 and −33.8, respectively. The additional binary series in each panel indicates our primary ex post 15% trough-to-peak indicator, where a high value indicates a market down-state and a value of zero indicates a market up-state.

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C. Average Relative Strength Payoffs and a Strategy's Horizon

Our two-state framework suggests a decline in the average profits of relative strength strategies as we increase the strategy's horizon from a medium six-month horizon to our longer horizons (on a standardized profit per month basis). While other studies have noted a declining performance in relative strength strategies as the horizon increases (see Conrad and Kaul (1998), who have a sample end date of 1989), we re-examine this behavior using return data from 1926 to 2010 and we examine both firm-level and industry-level strategies.

Table I reports the average payoff for our symmetric relative strength strategies implemented on both firm and industry returns at the 6, 12, 18, 24, and 36-month horizons. Table I, Panel A reports the average profits for our firm strategies. On a profit per month basis, over our full sample period (modern sample period), we note that the average profits are +100.1 (+112.4) basis points per month for the six-month strategy with a t-statistic of 6.22 (4.69) to reject a null hypothesis that the average profits are zero. Over our full sample period (modern sample period), the average profits decline monotonically as the horizon lengthens to +21.9 (+34.6), −11.1 (−5.0), −18.6 (−17.0), and −83.2 (−79.8) basis points per month for the 12, 18, 24, and 36-month strategies, respectively. For the 36-month strategy, the average profits are negative and reliably different than zero over both our full and modern sample periods. The other subperiods also tend to reflect these same patterns, with only the six-month strategy yielding reliably positive profits.

Table I.  Variation in Average Relative Strength Payoffs with a Strategy's Horizon
  1. Full 1926–2010 2. Modern 1962–2010 3. 1st Half Modern 1962–1986.06 4. 2nd Half Modern 1986.07–2010 5. Older 1926–1961
  1. a

    This table reports on average relative strength payoffs for the following symmetric strategies: 6, 12, 18, 24, and 36-month strategies. We report on relative strength strategies implemented on both firm-level and industry-level stock returns. For the firm-level strategies in Panel A, we implement a decile-based strategy, where the holding period goes long the top 10% of past relative winners and goes short the bottom 10% of past relative losers over the ranking period. For the industry-level strategy in Panel B, we evaluate 30 value-weighted industry portfolios, where the holding period goes long the top five past relative winner industries and goes short the bottom five past relative loser industries over the ranking period. For all of the strategies, we skip a month between the ranking and holding periods. The table reports the average payoff, where the holding period profit is the difference between the cumulative return of the past winner portfolio and the cumulative return of the past loser portfolio. The average payoffs are reported in basis points per month by dividing the average holding period profit by the number of months in the strategy. We report on all five of our separate periods of interest with dates denoted in the column headings. i-statistics are in parentheses that indicate whether the average payoffs are statistically different than zero, calculated with heteroskedastic and autocorrelation consistent standard errors.

Panel A. Firm-Level Strategies
6-month100.1(6.22)112.4(4.69)139.7(6.40)84.7(1.99)82.9(4.37)
12-month21.9(1.36)34.6(1.50)77.3(4.53)−9.7(−0.23)4.2(0.20)
18-month−11.1(−0.71)−5.0(−0.22)17.6(0.76)−29.1(−0.75)−19.7(−0.95)
24-month−18.6(−1.10)−17.0(−0.72)−18.9(−0.54)−15.0(−0.48)−20.9(−0.88)
36-month−83.2(−2.72)−79.8(−2.83)−86.9(−2.13)−71.6(−1.94)−88.5(−1.42)
Panel B. Industry-Level Strategies
6-month54.0(5.30)55.4(3.92)71.7(4.09)38.8(1.77)52.1(3.62)
12-month27.4(2.54)28.5(2.00)46.5(2.91)9.9(0.42)25.9(1.57)
18-month7.6(0.70)11.0(0.77)10.4(0.62)11.7(0.50)2.8(0.17)
24-month2.9(0.23)6.5(0.40)−0.8(−0.04)13.5(0.53)−2.2(−0.12)
36-month−5.5(−0.31)6.6(0.31)−24.7(−1.01)42.0(1.41)−22.6(−0.78)

In Table I, Panel B, we implement a similar examination of our industry strategies. Again, the profits decline monotonically as we lengthen the horizon with average monthly profits of +54.0, +27.4, +7.6, +2.9, and −5.5 basis points at the 6, 12, 18, 24, and 36 month strategies, respectively, over our full sample period. Subperiod results are also largely consistent, with only the six-month strategy yielding reliably positive profits.7

Under plausible parameter values, our statistical decomposition also suggests that the proportion of negative payoffs for the relative strength payoffs should increase appreciably as we increase the strategy's horizon from six to 36 months. Consistent with this implication, we find that 22.9% (33.6%), 32.4% (38.2%), 44.2% (47.6%), 50.6% (47.7%), and 61.6% (53.4%) of the payoffs are negative for the 6, 12, 18, 24, and 36-month firm-level strategies (industry-level strategies), respectively, over our full sample period.

III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

Both our two-state statistical decomposition and earlier related literature suggest that relative strength payoffs are likely to be relatively higher for within-state outcomes and relatively lower for across-state outcomes (Grundy and Martin, 2001; Asem and Tian, 2010). If so, empirical measures that lead market-state transitions should be associated with lower subsequent relative strength payoffs. In Section III.A, we first propose and justify using the stock market's cross-sectional RD as a leading indicator of market transitions. Our justification includes both past literature and new evidence. Then, in Section III.B, we investigate how the market's lagged RD is related to the subsequent relative strength payoffs.

Before proceeding, we note that this time-series implication suggests a positive co-movement between firm-level and industry-level relative strength payoffs. Consistently, over our 1926–2010 sample, we find that the simple correlations between the firm-level and industry-level payoffs for the different strategy horizons are all appreciably positive at 0.632, 0.637, 0.573, 0.574, and 0.598 for the 6, 12, 18, 24, and 36-month strategies, respectively.

A. Return Dispersion as a Leading Indicator of Market-State Changes

Stivers and Sun (2010) determine that the stock market's cross-sectional RD is informative about time variation in both medium-run momentum and value versus growth strategies, with a higher RD associated with lower subsequent momentum payoffs and higher subsequent value versus growth strategy payoffs. Their evidence suggests that the market's RD may serve as a leading state variable that tends to be high around weaker market states and market transitions. Gomes et al. (2003) provide a theoretical perspective that links RD variation to different market states. Their conditional capital asset pricing model (CAPM) model's framework suggests that RD should be higher around weak market states due to the countercyclical nature of both aggregate return volatility and the dispersion in conditional market betas. Stivers and Sun (2010) evaluate only six-month momentum strategies over the 44 year period from 1962 to 2005. In contrast, in this section, we investigate how the market's lagged RD is related to the subsequent relative strength payoffs for: 1) both medium-run and long-run strategies (6, 12, 18, 24, and 36-month strategies), 2) both firm-level and industry-level strategies, and 3) the 85 year period from 1926 to 2010.

Following from Stivers and Sun (2010), we use a three-month RD moving average, where the RD is the cross-sectional dispersion in monthly returns for 100 value-weighted portfolios formed from double sorts on size and book-to-market equity ratios.8 An RD from these 100 portfolios, rather than an RD calculated from firm-level returns, should better capture dispersion attributed to common factor exposure for factors that have proven useful in explaining the cross-sectional variation in expected stock returns. Additionally, a RD from disaggregate portfolio returns should mitigate idiosyncratic noise that does not reflect market-wide conditions. A three-month moving average is chosen as it should be responsive to changes in market conditions, but should also remove some of the noise in month-to-month variations. We denote this moving average as RDt− 1,t− 3 and it is formed from the monthly RD over months t-3 through t-1, relative to strategy payoffs with a holding period that commence in month t. To work with an RD variable that is closer to normally distributed, we use the natural log of RDt− 1,t− 3, denoted as log   (RDt− 1,t− 3) in our subsequent empirical work with the RD units in percentage returns per month.

We begin by describing the RD series over our 1926–2010 sample. The unconditional average of log   (RDt− 1,t− 3) over our sample is 1.280 with a standard deviation of 0.455. If month t is categorized as being in a National Bureau of Economic Research (NBER) recession (19.7% of the months), then the average of log   (RDt− 1,t− 3) is 1.526 (vs. an average of 1.219 for the expansion months), with a t-statistic of 3.98 that rejects a null of no difference between these recession and expansion averages.

Next, we perform a simple empirical exercise to further probe the notion that the market's RD may serve as a leading indicator of changes in the market state. If the market's RD is predictive about changes in the market state, then a high three-month RD moving average should suggest a large absolute difference between the 12-month stock market return that follows the high RD moving average and the 12-month stock market return that precedes the high RD moving average. We estimate the following two regressions to evaluate this notion:

  • image(8)
  • image(9)

where inline image is the absolute difference between the two 12-month stock market returns over months t to t− 11 and t− 15 to t− 26; log   (RDt− 12,t− 14) is the natural log of the rolling three-month RD moving average (MA) over months t-12 to t-14 that occurs between the two 12-month stock-market returns of interest; inline image is a dummy variable that is equal to one when the RD MA over months t-12 to t-14 is in its inner 50th percentile; inline image is a dummy variable that is equal to one when the RD MA over months t-12 to t-14 is above its 75th percentile; inline image is the residual; and the α's and γ's are coefficients to be estimated. For the stock market return, we use the CRSP value-weighted stock index. We report on separate estimations for our full, modern, and older sample periods.

Table II reports the results from estimating Equations (8) and (9). For Equation (8) in Table II, Panel A, we find that the estimated inline image's are positive and statistically significant with a 1%p-value for all three estimation periods. For Equation (9) in Table II, Panel B, we determine that the estimated inline image's are positive and statistically significant with a 5%p-value or better for all three estimation periods. The results in both panels indicate that a higher RD is associated with a larger absolute difference between the subsequent and preceding 12-month stock market returns. For example, for our full sample, the average absolute difference between the subsequent and preceding 12-month stock market returns is over 37% when our RD MA is high (above its 75th percentile) versus only about 16% when our RD MA is low (below its 25th percentile). In our view, the collective evidence in this subsection solidly supports the view that the lagged RD may serve as a leading change-of-state variable over our 1926–2010 sample.

Table II.  Market Transitions and Cross-Sectional Return Dispersion
Panel A. Absolute Market Return Changes and the Lagged RDMA
1. Sample Period 2. Intercept α0 (t-stat) 3. RD term α1 (t-stat) 4. R2
Full−0.28(−0.05)19.37(4.15)16.3%
(1926–2010)     
Modern2.70(0.46)16.99(2.96)8.3%
(1962–2010)     
Older−4.96(−0.56)21.93(3.32)15.9%
(1926–1961)     
Panel B. Absolute Market Return Changes with Lagged RDMA Quartile Breakpoints
1. Sample Period 2. Average Low RD γ0 3. Average Mid RD γ01 4. Average High RD γ02 5. Difference High-Lo γ2 (t-stat) for Diff.
  1. a

    This table demonstrates how the absolute change in 12-month stock-market returns is related to the market's cross-sectional return dispersion (RD). We report on the following two specification, with Panel A reporting on the first regression and Panel B reporting on the second regression:

    • image
    • image

    where inline image is the absolute difference between the two 12-month stock market returns over months t to t −11 and t −15 to t −26; log   (RDt− 12,t− 14) is the rolling three-month RD moving average (MA) over months t −12 to t −14 that occurs between the two 12-month stock market returns of interest; inline image is a dummy variable that is equal to one when the RD MA over months t −12 to t− 14 is in its inner 50th percentile; inline image is a dummy variable that is equal to one when the RD MA over months t −12 to t −14 is above its 75th percentile; inline image is the residual; and the α's and γ's are coefficients to be estimated. For month t, RDt is the cross-sectional standard deviation of monthly returns for the 100 value-weighted portfolios formed from size and book-to-market equity ratio sorts. The reported t-statistics indicate whether the coefficient is statistically significant. We report on separate estimations for our full, modern, and older sample periods.

Full15.62%22.19%37.04%21.42%(4.33)
(1926–2010)
Modern16.00%20.76%25.84%9.84%(2.27)
(1962–2010)
Older20.80%26.33%44.30%23.50%(3.81)
(1926–1961)

B. RD and Subsequent Relative Strength Payoffs

Next, to investigate whether the relative strength payoffs tend to be lower following a relatively high market RD, we estimate the following model for each of our 10 relative strength strategies:

  • image(10)

where inline image is the cumulative holding period payoff for the j month relative strength strategy, divided by the number of months in the strategy; month t is the holding period's first month, month t+ (j −1) is the holding period's last month, month t− (j +1) is the ranking period's first month, and month t −2 is the ranking period's last month; j is either 6, 12, 18, 24, or 36 months; log   (RDt− 1,t− 3) is the natural log of the rolling three-month average of the market's RD over months t −1 to t −3; inline image is the residual; and the λ's are coefficients to be estimated. We report on separate estimations for each of our five sample periods.

We begin with this simple linear specification in Equation (10) as it relies solely on the lagged three-month moving average of the RD without a look-ahead requirement to know the full distribution of the market's RD. The disadvantage is that the estimated coefficients do not readily convey the economic magnitude of the variation tied to RD and a linear relation is assumed. Later, we present additional evidence to better convey the economic magnitude of the RD-related variations.

Table III, Panel A (Panel B) reports the results from estimating Equation (10) on our five firm (industry) strategies. For the 10 strategies across the five estimation periods, we find that all 50 of the estimated inline image coefficients are negative and 43 of the 50 estimates are negative and statistically significant at a 10%p-value or better. The estimated inline image is statistically significant for all 10 strategies over our modern 1962–2010 sample period with a 5%p-value. These estimates indicate that the subsequent relative strength payoffs tend to decline with the lagged RD.

Table III.  Relative Strength Payoffs and the Lagged Cross-Sectional Return Dispersion
1. Strategy 2. Full 1926–2010 λ1 (t-stat) 3. Modern 1962–2010 λ1 (t-stat) 4. 1st Half Modern 1962–1986.06 λ1 (t-stat) 5. 2nd Half Modern 1986.07–2010 λ1 (t-stat) 6. Older 1926–1961 λ1 (t-stat)
  1. a

    This table investigates whether the market's lagged cross-sectional return dispersion (RD) has forward-looking information for the subsequent relative strength payoff with a simple linear specification. We report on the following model for each of our 10 relative strength strategies.

    • image

    here inline image is the cumulative holding-period payoff for the j month relative strength strategy, in percentage returns, divided by the number of months in the strategy; the first month of the holding period is month t, the last month of the holding period is month t + (j −1), the first month of the ranking period is month t − (j +1), and the last month of the ranking period is month t −2 (with the skip-a-month); j is either 6, 12, 18, 24, or 36 months; log   (RDt− 1,t− 3) is the natural log of the rolling three-month average of the market's RD over months t −1 to t −3, with RD expressed as the cross-sectional standard deviation of the monthly percentage returns; inline image is the residual; and the λ's are coefficients to be estimated. For the market's RD, we use the cross-sectional standard deviation in monthly returns for 100 value-weighted portfolios formed from double sorts on size and book-to-market equity ratios. We estimate the model separately over our 1926–2010 period and four subperiods of interest. t-statistics are in parentheses, calculated with heteroskedastic and autocorrelation consistent standard errors.

Panel A. Firm-Level Strategies
6-month−0.91 (−2.67)−2.75 (−2.22)−1.89 (−2.11)−3.01 (−1.84)−0.22 (−0.55)
12-month−1.17 (−3.42)−3.18 (−3.06)−1.40 (−1.91)−3.74 (−2.82)−0.55 (−1.34)
18-month−1.15 (−3.68)−3.41 (−4.49)−1.92 (−2.36)−3.92 (−3.91)−0.57 (−1.98)
24-month−0.82 (−2.94)−2.55 (−7.30)−1.82 (−2.08)−2.85 (−8.04)−0.46 (−1.89)
36-month−1.37 (−2.63)−2.25 (−6.07)−1.74 (−2.06)−2.46 (−6.36)−1.70 (−3.08)
Panel B. Industry-Level Strategies
6-month−0.53 (−2.25)−1.47 (−2.61)−1.80 (−2.61)−1.32 (−1.84)−0.35 (−1.26)
12-month−0.50 (−1.90)−1.75 (−2.65)−0.96 (−1.39)−2.01 (−2.61)−0.13 (−0.59)
18-month−0.65 (−3.13)−2.01 (−5.53)−0.83 (−1.43)−2.45 (−6.71)−0.30 (−1.31)
24-month−0.70 (−3.56)−1.91 (−5.43)−1.18 (−2.40)−2.20 (−5.64)−0.46 (−2.25)
36-month−0.98 (−5.02)−1.32 (−3.49)−1.27 (−1.91)−1.32 (−2.60)−1.08 (−4.71)

Next, we contrast the average of relative strength payoffs that follow a relatively low RD realization (below the 25th percentile) versus the average of relative strength payoffs that follow a relatively high RD realization (above the 75th percentile). This simple empirical exercise allows us to report differences in average payoffs as tied to the lagged RD value, which readily indicates the economic magnitude. For our quartile breakpoints for RD for each period and subperiod, we use the actual RD distribution from that respective period or subperiod, so that each evaluation has about 25% of the observations for the low RD case and 25% of the observations for the high RD case. Again, we stress that this exercise has a forward-looking focus since we are interested in the relative strength payoffs that follow after the RD realization. However, in Table IV, there is a look-ahead issue in that our method requires knowledge of the RD distribution over the respective period or subperiod.

Table IV.  Average Relative Strength Payoffs and Cross-Sectional Return Dispersion
1. Strategy 2. Average Low-RD 3. Average Mid-RD 4. Average High-RD 5. Difference High-Lo (t-stat) for Diff.
  1. a

    This table demonstrates how the average relative strength payoffs vary depending upon the market's lagged cross-sectional return dispersion (RD). For each of our 10 strategies, we report the average strategy payoffs when the lagged three-month rolling average RD is: 1) below its 25th percentile or Low RD, 2) between its 25th and 75th percentile or Mid RD, and 3) above its 75th percentile or High RD. For each strategy, the timing is as in Table III where the first month of the holding period is month t, the last month of the holding period is month t + (j-1), j is either 6, 12, 18, 24, or 36 months, and the RD moving average is over months t −1 to t −3. The average payoffs are reported in basis points per month by taking the average holding period payoff and dividing by the number of months in the strategy. The final column reports the difference between the average payoffs following a High RD and a Low RD, with t-statistics in parentheses that indicate whether the difference is statistically significant, calculated with heteroskedastic and autocorrelation consistent standard errors. For the market's RD, we use the dispersion in monthly returns for 100 value-weighted portfolios formed from double sorts on size and book-to-market equity ratios. Panel A reports on firm-level strategies, while Panel B provides industry-level strategies.

Panel A. Firm-Level Strategies
A.1. Full Sample Period: 1926–2010
6-month158.8107.626.3−132.5(−2.48)
12-month91.132.8−69.8−160.9(−3.18)
18-month66.2−8.7−95.2−161.4(−3.81)
24-month43.6−21.4−77.8−121.4(−3.46)
36-month−19.6−58.5−198.5−178.9(−2.38)
A.2. Modern Sample Period: 1962–2010
6-month170.8147.1−15.4−186.2(−2.37)
12-month100.667.7−98.2−198.8(−2.88)
18-month76.824.1−146.2−223.0(−4.07)
24-month59.1−8.7−112.8−171.9(−4.96)
36-month−18.5−68.3−166.4−147.9(−4.16)
A.3. Modern Sample Period, First-Half: 1962–1986.06
6-month189.4138.093.2−96.2(−1.79)
12-month121.773.440.5−81.2(−1.97)
18-month66.422.8−41.4−107.8(−2.27)
24-month27.3−15.6−71.5−98.8(−1.91)
36-month−54.6−78.2−135.4−80.8(−1.56)
A.4. Modern Sample Period, Second-Half: 1986.07–2010
6-month173.5131.7−100.4−273.9(−1.90)
12-month96.545.1−226.1−322.6(−2.74)
18-month89.620.3−252.4−342.0(−3.72)
24-month79.46.8−158.2−237.6(−6.67)
36-month11.4−55.0−195.1−206.5(−5.49)
A.5. Older Sample Period: 1926–1961
6-month108.473.875.1−33.3(−0.63)
12-month36.89.6−39.9−76.7(−1.21)
18-month−3.8−2.5−69.7−65.9(−1.51)
24-month7.5−19.7−53.6−61.1(−1.79)
36-month32.4−110.4−178.5−210.9(−2.41)
Panel B. Industry-Level Strategies
B.1. Full Sample Period: 1926–2010
6-month67.862.523.4−44.4(−1.51)
12-month53.773.7−11.5−65.2(−2.01)
18-month56.04.2−35.0−91.0(−3.40)
24-month47.32.2−41.9−89.2(−3.81)
36-month33.613.5−83.5−117.1(−3.71)
B.2. Modern Sample Period: 1962–2010
6-month82.371.5−3.7−86.0(−2.18)
12-month64.739.3−30.0−94.7(−2.31)
18-month64.618.4−58.4−123.0(−3.96)
24-month64.210.4−61.5−125.7(−4.30)
36-month42.410.6−39.5−81.9(−2.42)
B.3. Modern Sample Period, First-Half: 1962–1986.06
6-month117.278.712.4−104.8(−2.40)
12-month82.142.319.1−63.0(−1.71)
18-month39.67.5−13.7−53.3(−1.48)
24-month41.4−4.3−36.2−77.6(−2.70)
36-month0.3−16.9−64.6−64.9(−1.60)
B.4. Modern Sample Period, Second-half: 1986.07–2010
6-month72.146.1−9.8−81.9(−1.29)
12-month53.733.5−81.4−135.1(−1.96)
18-month84.129.3−100.2−184.3(−3.99)
24-month78.730.0−84.6−163.3(−3.55)
36-month67.954.5−10.8−78.7(−1.45)
B.5. Older Sample Period: 1926–1961
6-month43.479.75.8−37.6(−1.00)
12-month15.644.5−0.4−16.0(−0.43)
18-month−10.326.6−30.3−20.0(−0.68)
24-month2.117.2−44.5−46.6(−2.01)
36-month26.0−10.3−99.4−125.4(−3.21)

Table IV provides the conditional averages for our five firm and five industry strategies, respectively, for each of our five sample periods of interest. The final column in the table reports the key result, the difference in average payoffs for the high-RD case versus the low-RD case. For the 10 strategies across five alternate evaluation periods, this difference in average payoff is negative for all 50 evaluations and negative and statistically significant for 38 of the 50 evaluations at a 10%p-value or better. The differences in averages are sizable. For example, for all five of our firm strategies over our modern period from 1962 to 2010 (Table IV, Panel A.2), the average payoff following a high RD is at least 148 basis points per month lower than the average payoff following a low RD.

In our view, the collective evidence in this section solidly supports the conclusion that the lagged RD contains reliable forward-looking information regarding the subsequent payoffs of both medium-run and longer-run relative strength strategies.

IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

In this section, our purpose is to evaluate how the relative strength payoffs are different for within-state versus across-state outcomes using a simple ex post classification of market up and down states based on the market's peak-to-trough behavior. Our empirical prediction is that the average relative strength payoffs will be appreciably higher (lower) for payoffs when the ranking period and holding period are from the same state (different states).

We proceed as follows in this section. We first explain and justify our primary method for classifying the market states, ex post, in Section IV.A. Section IV.B then presents our primary findings regarding the variation in average relative strength payoffs, while Section IV.C evaluates robustness.

A. An Ex Post Categorization of Up States and Down States from Stock Return Behavior

We propose a simple ex post method for categorizing the market state, based on the peak-to-trough movement in aggregate stock market returns. Our procedure is as follows. We calculate the cumulative gross return of the US stock market from July 1926 to December 2010 using the monthly total return of the CRSP value-weighted stock index. Figure 2 depicts this growth, with June 1926 arbitrarily assigned a value of one. Then, we track the stock market's growth in value and use a minimum 15% cumulative growth requirement (or decline) from the previous market trough (or peak) to indicate a market UP state (DOWN state). The UP (DOWN) state classification covers the entire realized trough-to-peak (peak-to-trough) period, so the method is ex post in nature. In our view, this approach is a simple objective method that lets the actual stock market growth (or decline) indicate the market state. For our framework in Section I, it is the stock-return state that matters, not the economic state, especially since the stock market is generally considered a leading economic indicator.9

image

Figure 2. The Cumulative Stock Market Return, Market Down-States, and NBER Recessions

This figure displays the cumulative stock market return, our market down-states based on our primary 15% peak-to-trough classification, and NBER recessions from July 1926 to December 2010. We report the stock market's cumulative growth based on the CRSP value-weighted stock index with June 1926 being arbitrarily set to a value of one. The vertical axis is on a logarithmic scale. Our market down-states are denoted with a marker on the value “ten.” From September 1929 to July 1934, the volatility around the Great Depression translates to six down-market states, which blur together in the figure. NBER recessions are denoted with a marker on the value “100.”

Download figure to PowerPoint

While our market-state method in this section is data driven, we acknowledge that the choice of the movement threshold is somewhat subjective. We offer the following justification for our initial choice of the 15% level. First, in our view, this method yields a reasonable number and duration of market states with a strong association to actual NBER recessions. Second, the standard deviation of monthly market-level stock returns over our sample is 5.45%. Thus, 15% is roughly three times this standard deviation, so a 15% movement is appreciable and will generally require multiple months of movement in one direction to reach the cumulative growth or decline requirement. Indeed, all of the down states and up states by this method include multiple months. Third, the 15% value seems intuitively reasonable as halfway between the traditional values for a market correction (10%) and a bear market (20%).10

Figure 2 depicts: 1) the cumulative growth of the stock market return, 2) our primary market state classification, based on a minimum 15% movement trough-to-peak (or peak-to-trough), and 3) NBER recession months. Our method captures the sizable down movements in the stock market and, temporally, our market DOWN states lead six of the seven NBER recessions since 1962. Other DOWN states are associated with well known crises or political stress, such as the Russian default on their foreign debt in the summer of 1998 or growing civil unrest with the Vietnam War in 1966. The volatility around the Great Depression in the early 1930s translates to six DOWN market states from September 1929 to July 1934 (note that our DOWN states over this period blur together in Figure 2).

Over our full 1926–2010 sample, there are 22 UP states and 21 DOWN states with an average duration of 10.2 months for the DOWN states and 36.4 months for the UP states, and with 21.1% of the months existing in a DOWN state. For our modern sample (1962–2010), our method results in 10 UP states and 10 DOWN states, with an average duration of 12.5 months for the DOWN states and 46.3 months for the UP states, with 21.3% of the months occurring in a DOWN state.

Finally, for market cycles to be important to the performance of relative strength strategies in our two-state statistical decomposition, some stocks would need to have relatively high UP state means and relatively low DOWN state means, where relative refers to a cross-sectional comparison. Other stocks would need to have relatively low UP state means and relatively high DOWN state means. To investigate this issue, we estimate the UP state means and DOWN state means for the monthly returns of each industry, using the 15% DOWN states per Figure 2.

We find sizable differences between the UP state means and the DOWN state means across the 30 industries, with a mean return across the 30 industries of 2.21% in the UP state and –3.41% in the DOWN state. There is substantial cross-sectional variation in the industry differences between their UP state and DOWN state realized means, with a cross-sectional standard deviation of this difference (the σ(d) from Section I) equal to 1.155% for the industry returns.11 By comparison, the cross-sectional standard deviation of the unconditional mean returns (the σ(μ) from Section I) across the 30 industries is only 0.128% per month. Thus, in terms of our analytical framework in Section I, the cross-sectional variance of the regime-mean differences is over 80 times as large as the cross-sectional variance of unconditional mean returns. This observation, along with the plausible regime durations, supports the notion that market cycles in state-specific mean returns may be important in understanding relative strength payoffs.

B. Main Results with a 15% Market Cycle Threshold

Figure 1 displays the time-series of both our 15% DOWN states and the payoffs for the six-month firm-level and industry-level strategies. Note that sizably negative payoffs tend to occur around transitions into and out of the DOWN states.

Next, Table V reports on the average relative strength payoffs for both within-state payoffs and across-state payoffs using our primary 15% market-state classification. Specifically, we estimate the following equation for each of our 10 relative strength strategies:

  • image(11)

where inline image is the cumulative holding period payoff for the j month relative strength strategy, as defined in Equation (10); inline image is a within-state dummy variable that takes a value of one when a strategy's ranking period and holding period are considered to be within-state and zero otherwise; inline image is the residual, and the γ's are coefficients to be estimated. j is either 6, 12, 18, 24, or 36 months.

Table V.  Relative Strength Payoffs and Ex Post Market States
1. Strategy 2. Average Within-State (=γ01) 3. Average Across-State (=γ0) 4. Difference-in-Aver. (=γ1) (t-stat) for γ1 5. Percentage Within-State
  1. a

    This table reports whether the average payoffs for a given strategy depend upon whether the payoff's ranking and holding period are from the same market state (within-state) or across different states (across-state). We report on the following model for each of our 10 relative strength strategies.

    • image

    where inline image is the payoff for the j month relative strength strategy in basis points per month units; month t is the holding period's first month, as defined in Table III; inline image is a within-state dummy variable equal to one when a strategy's ranking period and holding period are considered to be within-state and zero otherwise; inline image is the residual; and the γ's are coefficients to be estimated. j is either 6, 12, 18, 24, or 36 months. Panels A and B report on firm-level and industry-level strategies, respectively. The ranking period and holding period are considered to be within-state if the inner two-thirds of the months from the ranking and holding period are categorized as being within the same market state. For this table's results, up states (down states) require at least a 15% market movement trough-to-peak (peak-to-trough). Column Five reports the percentage of observations that are considered to be within-state for each strategy. t-statistics are in parentheses for the inline image coefficient, calculated with heteroskedastic and autocorrelation consistent standard errors.

Panel A. Firm-Level Strategies
A.1. Full Sample Period: 1926–2010
6-month129.124.8104.3(2.24)72.1%
12-month62.8−24.487.2(2.81)53.1%
18-month43.5−47.090.5(3.63)39.7%
24-month59.1−52.6111.8(4.30)30.4%
36-month44.0−111.4155.4(3.57)18.2%
A.2. Modern Sample Period: 1962–2010
6-month149.6−4.2153.8(1.90)75.8%
12-month83.0−25.6108.6(2.34)55.5%
18-month56.5−45.4101.9(2.87)39.6%
24-month60.9−50.8111.7(3.07)30.3%
36-month5.1−97.6102.6(2.58)17.4%
A.3. Modern Sample Period, First-Half: 1962–1986.06
6-month164.567.197.5(1.69)74.4%
12-month99.351.348.0(1.54)54.1%
18-month65.1−11.276.2(1.87)37.8%
24-month84.0−58.6142.6(2.79)27.9%
36-month54.5−108.3162.8(4.10)13.2%
A.4. Modern Sample Period, Second-Half: 1986.07-2010
6-month134.9−85.1220.0(1.39)77.2%
12-month66.8−110.6177.4(2.11)56.9%
18-month48.2−83.9132.1(2.29)41.5%
24-month39.4−41.681.0(1.66)32.8%
36-month-28.8−83.554.7(1.09)22.0%
A.5. Older Sample Period: 1926–1961
6-month96.854.442.4(0.96)67.1%
12-month31.4−22.954.4(1.41)49.9%
18-month25.3−49.474.7(2.28)39.8%
24-month56.8−55.3112.2(3.12)30.7%
36-month93.8−132.0225.9(3.13)19.2%
Panel B. Industry-Level Strategies
B.1. Full Sample Period: 1926–2010
6-month71.64.768.4(2.91)72.1%
12-month56.1−5.161.2(3.25)53.1%
18-month45.1−17.062.1(3.27)39.7%
24-month59.4−21.881.2(4.06)30.4%
36-month64.3−21.085.3(3.57)18.2%
B.2. Modern Sample Period: 1962–2010
6-month71.64.667.0(1.87)75.8%
12-month58.3−8.566.8(2.52)55.5%
18-month44.5−10.955.3(2.24)39.6%
24-month55.6−14.870.4(2.76)30.3%
36-month48.2−2.250.4(1.84)17.4%
B.3. Modern Sample Period, First-Half: 1962–1986.06
6-month83.138.344.8(1.17)74.4%
12-month55.735.620.1(0.76)54.1%
18-month22.82.820.0(0.69)37.8%
24-month40.3−16.757.1(1.83)27.9%
36-month20.5−31.552.0(2.14)13.2%
B.4. Modern Sample Period, Second-half: 1986.07–2010
6-month60.2−33.693.9(1.52)77.2%
12-month60.9−57.4118.3(2.77)56.9%
18-month65.3−26.391.6(2.42)41.5%
24-month69.4−12.581.9(2.07)32.8%
36-month66.934.832.1(0.79)22.0%
B.5. Older Sample Period: 1926–1961
6-month75.44.870.6(2.29)67.1%
12-month52.7−0.853.5(1.95)49.9%
18-month46.0−25.771.6(2.41)39.8%
24-month64.8−31.896.6(3.04)30.7%
36-month85.2−48.3133.5(3.90)19.2%

Initially, for this empirical exercise, inline image is equal to one if the last two-thirds of the months from the ranking period are in the same market state as the first two-thirds of the months from the subsequent holding period. For example, for the 12-month strategy, a ranking-period/holding-period event would be within-state if the eight months over t-9 to t-2 of the ranking period are classified as being in the same market state as the eight months over t to t+7 of the holding period. Month t-1, the skipped month, would also need to be in the same market state. We later evaluate robustness to alternate within-state classifications.

Table V reports on the within-state and across-state average payoffs for our firm and industry strategies, respectively, per Equation (11). The difference in average payoffs for the two categories is quite large and reliable (see Column Four in the table). For all 10 strategies over our full sample period (Panel A.1 and B.1), the differences in payoffs are sizable with a minimum t-statistic of 2.24 for tests of whether the differences are statistically significantly different than zero. For the firm strategies, the average payoffs are 104.3, 87.2, 90.5, 111.8, and 155.4 basis points per month less for the across-state outcomes versus the within-state outcomes for the 6, 12, 18, 24, and 36-month strategies, respectively. The comparable differences for the industry-level strategies are all greater than 60 basis points per month.

Additionally, Table V reports on separate estimations of Equation (11) for our four subperiods of interest for both our firm strategies (Panels A.2-A.5) and industry strategies (Panel B.2-B.5). The estimated inline image's are positive and sizable for all 40 estimations, and positive and statistically significant at a 10%p-value or better for 30 of the 40 estimations. We feel this subperiod consistency is striking and reinforces the notion of a strong tie between relative strength payoffs and market transitions.

Finally, for the 6 and 12-month strategies, we also find that the average payoffs are higher than normal for both the UP within-state and the DOWN within-state payoffs. For example, for the six-month firm (industry) strategies: 1) the UP within-state payoffs encompass 64.3% of the observations, with a mean that is 100 (60) basis points per month larger than the mean for the across-state payoffs; and 2) the DOWN within-state payoffs encompass 8.0% of the observations, with a mean that is 141 (133) basis points per month larger than the mean for the across-state payoffs. We do not investigate this issue for the longer-run horizons as there are too few DOWN within-state observations.

C. Robustness with Alternate Classifications for the Ex Post Market State

Next, we turn to the issue of robustness. Table VI reports the results from estimating Equation (11) with five alternate classification methods for the within-state and across-state categorization of relative strength payoffs. The estimations are over our full 1926–2010 period, with Panel A (Panel B) reporting on our firm-level (industry-level) strategies.

Table VI.  Relative Strength Payoffs and Ex Post Market States: Robustness
1. Strategy 2. Average Within-State (=γ01) 3. Average Across-State (=γ0) 4. Difference-in-Aver. (=γ1) (t-stat) for γ1 5. Percentage Within-State
  1. a

    This table reports robustness evidence for whether the average payoffs for a given strategy depend upon whether the payoff's ranking and holding period are within-state or across-state. We report on five variations of the following model for each of our 10 relative strength strategies.

    • image

    where the terms are as defined for the model in Table V. Panels A and B report on firm-level and industry-level strategies, respectively, with each estimation over our full sample period from 1926 to 2010. We consider the following five variations for the within-state categorization: 1) 20% peak to trough, with two-thirds of the ranking and holding period's months within-state; 2) 17.5% peak to trough, with two-thirds of the ranking and holding period's months within-state; 3) 15% peak to trough, with all of the ranking and holding period's months within-state; 4) 20% peak to trough, with all of the ranking and holding period's months within-state; and 5) 15% peak to trough, with two-thirds of the ranking and holding period's months within-state, but only for down states associated with a NBER recession. Column Five reports the percentage of observations that are considered to be within-state for each strategy. t-statistics are in parentheses for the inline image coefficient, calculated with heteroskedastic and autocorrelation consistent standard errors.

Panel A. Firm-Level Strategies
A.1. 20% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month128.9−34.5163.5(2.51)82.3%
12-month55.6−51.0106.6(2.58)68.4%
18-month17.5−52.069.6(2.27)58.8%
24-month6.6−45.251.8(1.66)51.3%
36-month−24.8−123.198.3(1.85)40.6%
A.2. 17.5% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month121.630.491.2(1.73)76.3%
12-month60.8−35.996.7(2.82)59.8%
18-month29.2−49.779.0(2.86)48.9%
24-month20.4−46.266.6(2.09)41.4%
36-month−30.9−106.275.4(1.37)30.6%
A.3. 15% Threshold, Within-State Requires All Ranking and Holding Period Months
6-month129.850.779.1(2.08)62.4%
12-month61.5−4.365.8(2.38)39.8%
18-month57.6−36.594.1(3.67)27.0%
24-month48.6−33.181.7(3.69)17.7%
36-month65.7−95.5161.2(2.73)7.6%
A.4. 20% Threshold, Within-State Requires All Ranking and Holding Periods Months
6-month131.55.5126.0(2.50)75.0%
12-month50.9−19.270.1(2.01)58.7%
18-month18.6−38.457.0(2.07)47.9%
24-month9.1−36.745.9(1.59)39.5%
36-month−11.5−112.3100.8(2.08)28.9%
A.5. Recessions with 15% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month123.7−21.9145.6(1.98)83.7%
12-month49.1−43.993.0(2.11)70.8%
18-month11.1−47.358.4(1.90)61.9%
24-month−2.6−38.535.9(1.25)55.3%
36-month−77.9−87.69.7(0.16)44.8%
Panel B. Industry-Level Strategies
B.1. 20% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month67.5−8.876.3(2.41)82.3%
12-month50.0−21.371.3(3.18)68.4%
18-month35.8−32.568.3(3.70)58.8%
24-month34.8−30.865.7(3.39)51.3%
36-month33.4−32.065.4(2.19)40.6%
B.2. 17.5% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month69.54.265.3(2.49)76.3%
12-month57.1−16.573.4(3.78)59.8%
18-month46.1−29.275.3(4.19)48.9%
24-month49.4−30.049.3(4.05)41.4%
36-month29.1−20.749.8(1.78)30.6%
B.3. 15% Threshold, Within-State Requires All Ranking and Holding Period Months
6-month72.922.850.1(2.36)62.4%
12-month56.28.447.8(2.45)39.8%
18-month46.4−6.753.1(2.62)27.0%
24-month43.9−5.949.8(2.90)17.7%
36-month74.2−12.181.7(2.59)7.6%
B.4. 20% Threshold, Within-State Requires All Ranking and Holding Periods Months
6-month67.413.953.5(2.02)75.0%
12-month48.9−3.152.1(2.44)58.7%
18-month31.7−14.546.1(2.58)47.9%
24-month31.1−15.646.7(2.09)39.5%
36-month40.8−24.365.1(2.01)28.9%
B.5. Recessions with 15% Threshold, Within-State Requires 2/3 Ranking and Holding Period Months
6-month65.5−5.070.5(2.08)83.7%
12-month48.8−24.373.1(3.18)70.8%
18-month31.1−30.461.4(3.22)61.9%
24-month28.5−28.857.2(2.90)55.3%
36-month13.8−21.235.1(1.13)44.8%

Table VI, Panels A.1 and B.1 (A.2 and B.2), report on a 20% market threshold (17.5% market threshold) in regard to the minimum peak-to-trough movement, used in place of our primary 15% market threshold from Table V. We choose to evaluate a 20% threshold and 17.5% threshold due to the common practice of referring to a 20% drop as a bear market, with 17.5% existing half way between our primary 15% and the larger 20% level. The estimation results are very similar to the comparable results in Table V. The estimated inline image's are all positive, with a minimum value of about 50 basis points. For the 20 separate estimations, all but one of the estimated inline image's are positive and statistically significant at a 10%p-value, or better.

Next, Panels A.3 and B.3 (A.4 and B.4) report on a 15% market threshold (20% market threshold), where all of the months in a ranking-period/holding-period realization must be in the same market state, rather than two-thirds of the months as in our Table V estimations. We evaluate both the 15% threshold and 20% threshold to represent a smaller and larger threshold. With this more stringent requirement for a within-state classification, note that relatively fewer payoffs are classified as within-state. Even so, the results are very similar to the comparable results in Table V. The estimated inline image's are all positive, with a minimum value of 46 basis points. For the 20 estimations, all but one of the estimated inline image's are positive and statistically significant at a 5%p-value or better.

Finally, Panels A.5 and B.5 report on a classification method that is tied to both NBER recessions and market return cycles. We only categorize the market state as being in a DOWN state if the market movement is at least 15% peak-to-trough and at least some of the months in the downturn coincide with an NBER recession. Since there are less NBER recessions than there are market downturns, this method produces relatively more within-state classifications. Given this observation, we would expect that this recession-related method would perform less well than our prior methods. Our findings indicate the following: 1) the estimated inline image's are positive for all 10 estimations and positive and statistically significant for seven of the 10 estimations and 2) the estimated inline image is appreciably weaker for the longest 36-month horizon strategies, as compared to our prior results. In our view, the relatively high proportion of within-state 36-month strategy payoffs is intuitively too high with this method at nearly 45% (vs. 18% for our primary method in Table V), so a low value for the estimated inline image is not surprising.

We conclude that plausible market states, derived from simple peak-to-trough market behavior, are reliably tied to payoff variations for relative strength strategies. Both medium-run and longer-run strategies exhibit strong within-state and weak across-state performance.

V. Conclusions

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References

Relative strength strategies tend to exhibit reliable momentum for medium-run strategies, but modest reversals for longer-run strategies. This puzzling horizon contrast has presented a challenge to financial economists. In this paper, we study the implications of market cycles on relative strength strategies, with an eye toward this horizon contrast issue and the joint time-series behavior of medium-run and longer-run strategies. Empirically, we study both firm-level and industry-level symmetric relative strength strategies from 1926 to 2010, at both medium-run horizons (6 and 12 months) and longer-run horizons (18, 24, and 36 months).

We contribute with both new analytical and empirical findings. We begin by offering a simple statistical decomposition of relative strength profits in a two-state cyclical market. This decomposition provides insight for our main empirical investigation and demonstrates that market cycles can naturally lead to the puzzling horizon contrast in average relative strength payoffs.

In our main investigation, we offer empirical evidence that supports the practical importance of market cycles in understanding relative strength payoffs. Using two alternate empirical approaches, we document a common cyclical variation in the payoffs of both medium-run and longer-run relative strength strategies. The evidence from both approaches suggests that relative strength payoffs are higher for within-state outcomes and lower for across-state outcomes.

In our first empirical approach, we use the market's cross-sectional return dispersion as a proposed leading indicator of market state transitions and find that relative strength payoffs are negatively related to the market's prior three-month RD moving average. This predictive relation is strikingly evident and robust for all 10 of our relative strength strategies.

In our second empirical approach, we evaluate an ex post market-state classification method, based on peak-to-trough market behavior. We find that all 10 strategies exhibit average payoffs that are sizably and reliably higher when the ranking and holding periods are from the same ex post market state, rather than from different market states.

Our results suggest a systematic cyclical perspective that appears to have an appreciable role in understanding relative strength strategies. If such market cycles are rational, then our study suggests a rational perspective that contributes to jointly understanding medium-run and long-run relative strength strategies. The link between the stock market state and the underlying economic state would seem to support a rational interpretation. Regardless of the underlying cause of the two-state behavior (rational, behavioral, or some combined effect), our study indicates that market-wide cycles have important implications for relative strength strategies.

Further, our study suggests that relative strength payoffs are more likely to be negative during times of economic transition and weak economic times. Since the marginal utility of wealth and consumption seems likely to be relatively high during such times, this suggests a partial risk-based explanation for the positive average profits for medium-run momentum, consistent with Ahn et al. (2003).

Our findings and observations reinforce earlier studies that suggest market cycles may help to understand medium-run momentum (Grundy and Martin, 2001; Cooper et al. 2004; Asem and Tian, 2010). Consistent with our evidence, Asem and Tian (2010) also find that relative strength payoffs are unusually high for both “up market, within-state” and “down market, within-state” payoffs. However, they rely on a behavioral explanation tied to overconfidence during both up-state and down-state continuations. Further, their empirical scope is much narrower since they only evaluate an asymmetric momentum strategy with a six-month ranking period and one-month holding period, whereas our symmetric strategies extend out to 36 months.

The expansive literature on relative strength strategies suggests that the underlying economics behind momentum and reversals are multifaceted and complex. As such, it seems unlikely that any single perspective could explain all aspects of such strategies. Undoubtedly, our cyclical market perspective is also only a partial contributor toward understanding the behavior of relative strength strategies. For example, our study does not address the short-run reversals that tend to be evident in weekly stock returns. Rather, the framework and evidence in our paper seems to be more useful for jointly understanding medium and longer horizon strategies.

Footnotes
  • 1

    See Jegadeesh and Titman (2001), Fama and French (2008), Rouwenhorst (1998), and Griffin, Ji, and Martin (2003). The profitability of medium-run momentum has survived risk-adjustments by the capital asset pricing model (CAPM) and Fama-French three-factor asset pricing models (Fama and French, 1996; Grundy and Martin, 2001). This apparent shortcoming of risk-based asset pricing has led to several interesting behavioral models that may generate momentum; see, e.g., Daniel, Hirshleifer, and Subrahmanyam (1998) and Hong and Stein(1999), where overconfidence and momentum traders are featured, respectively. More recent papers, with alternate approaches, present evidence that risk may be importantly tied to momentum profits (Ahn, Conrad, and Dittmar, 2003; Liu and Zhang, 2008; Agarwal and Taffler, 2008).

  • 2

    See DeBondt and Thaler (1985) and Conrad and Kaul (1998) for evidence regarding longer-run strategies. We use the term “relative strength” strategies, rather than momentum, as a more general term that encompasses both the medium-run momentum horizon and the longer-run reversal horizon.

  • 3

    Chordia and Shivakumar (2002) find that momentum profits may be linked to business cycles and predicted by lagged macroeconomic variables. However, Cooper et al. (2004) find that the results of Chordia and Shivakumar (2002) are not robust to widely used methodological adjustments that guard against market frictions and penny stocks driving the results. Further, Griffin et al. (2003) determine that the Chordia and Shivakumar (2002) results do not generally hold in other countries. For other perspectives, see Grinblatt and Han (2005) and Hur, Pritamani, and Sharma (2010), who link momentum to the disposition effect, and Loh (2010), who suggests investor inattention may induce momentum.

  • 4

    Conversely, transitions to a stronger economic state are likely to exhibit higher realized mean returns. Over our 1926–2010 sample, the average stock-market return over National Bureau of Economic Research (NBER) recessions (expansions) is –0.19% per month (1.19% per month). Recessions tend to have higher volatility and higher volatility is associated with different cross-sectional variation in subsequent returns (Gulen, Xing, and Zhang, 2011).

  • 5

    By within-state (across-state), we refer to the outcomes for a relative strength strategy where the ranking period and the holding period are within the same uninterrupted state (across different states). By outcome, we mean the payoff from a single ranking-period/holding-period event.

  • 6

    Relatedly, on the theoretical side, both Berk, Green, and Naik (1999) and Johnson (2002) present theoretical models where time-varying expected returns can generate momentum in a nonlinear fashion.

  • 7

    One interesting difference between the firm-level and industry-level strategies is that the 36-month firm-level strategy generates sizably negative average profits of –0.83% per month versus near zero average profits of –0.055% per month for the 36-month industry-level profits. A contributing factor in this contrast may be that the industry strategy is implemented on value-weighted industry returns and, as such, focus more on larger cap stocks as compared to the firm-level strategies. Consistent with this notion, we compute the comparable relative strength profits for strategies implemented on equally weighted industry portfolio returns and find that the 36-month strategy's profits are appreciably lower at –0.32% per month. Additionally, we note that the component stocks of a given holding period portfolio do not change for our firm-level strategy. In contrast, stocks within the industry-level portfolios may change annually with the French industry portfolios.

  • 8

    RDt is the cross-sectional standard deviation of the 100 disaggregate portfolio returns for month t.

  • 9

    Another approach would be to use the formal economic recession and expansion categorization by NBER to categorize the market state. However, such an approach would assume that the stock market is a coincident indicator of the economic state, rather than a leading indicator as generally acknowledged and as suggested by our data.

  • 10

    Section IV.C considers robustness, including alternate movement thresholds. Cooper et al. (2004) and Asem and Tian (2010) rely on lagged rolling stock market returns to classify the market state. For our purposes in this section, we feel such a backward-looking method would not be a good fit. First, for our empirical work in this section, it is critical to classify market states accurately. With the lagged long-term stock return as the indicator as in Cooper et al. (2004) and Asem and Tian (2010), the classification method can substantially delay the change of state. For example, in the recent bear market and recovery from 2007 to 2009, the “lagged three-year market return” method of Cooper et al. (2004) does not classify a month as a down state until November 2008, yet the bear market decline began in November 2007. Further, their three-year method has not shifted back to an up state through December 2010, even though the market went up about 58% from March to December 2009.

  • 11

    The largest difference between the UP state mean and DOWN state mean is 7.83 for the games industry and the smallest is 2.89 for the smoke industry. The recreation, electrical equipment, fabricated products, and automobile industries have the highest UP state means and among the lowest DOWN state means. The tobacco, food, healthcare, and telecommunication industries have the highest DOWN state means and among the lowest UP state means.

References

  1. Top of page
  2. Abstract
  3. I. Relative Strength Strategies in a Two-State Cyclical Market
  4. II. Description of Our Data and the Relative Strength Strategies
  5. III. Relative Strength Payoffs Following a High Cross-Sectional Dispersion in Stock Returns
  6. IV. Average Relative Strength Payoffs for Within-State and Across-State Outcomes Using an Ex Post State Classification
  7. V. Conclusions
  8. Appendix
  9. References