R 2 and Momentum Kewei Hou, Lin Peng, and Wei Xiong April 13, 2005 Abstract This paper examines the relationship between price momentum and investors private information, using R 2 -based information measures. By regressing a firm s stock returns onto market and industry returns, we obtain the R 2 statistic that our study will employ as an inverse measure of the amount of private information investors have about the firm. We hypothesize that the amount of private information positively affects the magnitude of price momentum. We find that stocks with lower R 2 exhibit more pronounced price momentum, and this negative relation between R 2 and momentum persists after controlling for several alternative effects. We also find that, after controlling for public information proxies, lower R 2 stocks tend to have more informative prices, confirming R 2 as a useful measure of private information. Our results suggest that investors overreaction to private information is an important factor in determining price momentum. Preliminary please do not quote without permission. Comments are welcome. Fisher School of Business, Ohio State University. Email: hou 28@cob.osu.edu Baruch College, City University of New York. Email: lin peng@baruch.cuny.edu Princeton University and National Bureau of Economic Research (NBER). Email: wxiong@princeton.edu
1 Introduction Price momentum is an empirical phenomenon that has attracted great attention in the finance literature. Following the initial work of Jegadeesh and Titman (1993), many studies have documented that a simple strategy based on buying recent winners and shorting recent losers can generate economically and statistically significant trading profits in the US. This result also appears to be robust across countries: Rouwenhorst (1998) documents similar results in a sample of twelve European markets, and Griffin, Ji, and Martin (2003) show that momentum strategies are profitable in emerging markets. The recent literature has provided several behavioral models to explain price momentum based on investors biased reactions to information, e.g., Barberis, Shleifer, and Vishy (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Hong and Stein (1999). By using R 2 based information measures, we investigate how price momentum is related to investors information. A firm s return R 2 (R-squared) is the R 2 statistic derived from regressing a firm s stock returns to market and industry returns. Roll (1988) proposes R- squared as a useful measure of investors private information about the firm when more firm-specific information is incorporated into the firm s stock price through investors trading, the information causes more firm-specific return variation and therefore a lower return R- squared. 1 Roll further points out that return R-squared appears to be related to investors private information since it does not change significantly on days when public news about the firm appears in the financial press. We hypothesize that if price momentum is driven by investors biased reactions to their private information, then the amount of this information directly affects the degree of price momentum. Our study is directly motivated by the model of Daniel, Hirshleifer, and Subrahmanyam (1998, hereafter DHS), which explicitly shows that investors overreaction to private information, combined with self-attribution bias, generates price momentum. Through a simple extension of the DHS model, we illustrate that when overconfident investors possess more private information about a firm, their overreaction leads to a higher firm-specific re- 1 The recent studies by Durnev, Morck, Yeung, and Zarowin (2003) and Durnev, Morck, and Yeung (2004) confirm that stock prices of firms with lower return R-squared do incorporate more firm-specific information. They find that these firms current returns are more informative of their future earnings growth and that these firms tend to make more efficient investment. 1
turn variation, equivalent to a lower R 2. This result provides a testable hypothesis for our empirical analysis: firms about which investors have more private information, as measured by lower return R-squared, should have stronger price momentum. To examine this hypothesis, we sort all NYSE/AMEX/NASDAQ stocks into different groups based on their return R-squared, and compare the magnitude of price momentum across these groups. We find that there is a negative and significant relation between firms return R-squared and price momentum. A momentum strategy of buying recent winners and selling recent losers generates a significant value-weighted profit of 181 basis points per month in the lowest R-squared quintile. The momentum profit decreases monotonically across R- squared quintiles, and it drops to an insignificant 51 basis points per month in the highest R-squared quintile. This negative relation between R-squared and momentum profits remain robust when alternative R-squared measures, sorting procedures, and return adjustments are used. We also examine the long-run performance of the R 2 -sorted momentum portfolios for various holding periods. We find that the momentum profits for all five R-squared quintiles dissipate after two years, indicating a price reversal in the long-run. The existence of the long-run price reversal is consistent with the overreaction explanation of price momentum. Several other factors may also affect the relation between R-squared and price momentum. First, return R-squared may be correlated with systematic risk factors. However, the empirical evidence available to ascertain whether systematic risk can explain momentum profits remains mixed. 2 Nevertheless, we adjust the momentum profits both by individually using the Fama and French (1993) three-factor model, and by using a characteristic-based matching procedure which accounts for return premia associated with size and book-to-market following Daniel, Grinblatt, Titman, and Wermers (1997). In both cases, we find our results to be robust. Second, price momentum may be driven by investors underreaction to information, as 2 Jegadeesh and Titman (1993) show that momentum is independent of market risk. Fama and French (1996) and Grundy and Martin (2001) demonstrate that the Fama-French three factor model cannot explain momentum profits. Conrad and Kaul (1998) find that the conditional dispersion in stocks expected returns partially explains the momentum profits. Chordia and Shivakumar (2002) provide evidence that momentum profits can be forecasted by lagged macroeconomic variables, but Cooper, Gutierrez, and Hameed (2004) and Griffin, Ji, and Martin (2003) challenge this result. 2
suggested by the slow-information-diffusion theory of Hong and Stein (1999). In addition to capturing investors private information, return R-squared may also depend on the publicinformation environment of the firm. To control for such possibilities, we employ a number of control variables in our analysis. These include firm size and analyst coverage, the two variables used by Hong, Lim, and Stein (2000) to proxy for the speed of information diffusion. We also include institutional ownership, which, together with firm size and analyst coverage, also help to filter out the effects of public information. Third, since return R-squared is partially determined by firms fundamental correlation with the market and their industries, we include firms fundamental R-squared of regressing firms earnings on aggregated market- and industry- earnings as an additional control variable. Fourth, return R-squared may capture the effects of information uncertainty. Investors might have more psychological biases toward firms with greater information uncertainty, thus these firms might exhibit more pronounced price momentum. Jiang, Lee, and Zhang (2004) and Zhang (2004) find supportive evidence using firm size, analyst coverage, return volatility and dispersion in analyst forecasts to proxy for information uncertainty. To control for the effect of information uncertainty, we also incorporate return volatility and dispersion of analyst forecasts as control variables in our analysis. Finally, market illiquidity and other trading frictions might also affect price momentum and R-squared. To control for these potential effects, we use share turnover and the ratio between the average daily absolute return and the average daily dollar trading volume, a illiquidity measure proposed by Amihud (2002), as control variables. To further understand the driving force of the relation between return R-squared and price momentum, we construct residual R-squared measures by regressing firms return R-squared onto the above control variables and analyze momentum profits across quintiles sorted by the residual R-squared measures. We again find negative and monotonic relations between momentum profits and the residual R-squared measures. These results demonstrate that the negative relation between price momentum and return R-squared is not perceptibly driven by the aforementioned effects. We also establish a direct link between return R-squared and a firm s stock price informativeness. We show that the stock returns of low R-squared firms are more informative 3
about the firms future earnings. This negative relation between return R-squared and price informativeness continues to hold true after controlling for the public-information proxies such as firm size, analyst coverage, and institutional ownership. This result directly confirms Roll s proposal that return R-squared is a useful and reasonable measure of investors private information. Our results suggest that investors biased reaction to information can be an important factor for understanding asset price dynamics. Although we focus on investors overreaction to their private information, our results do not exclude other independent mechanisms, such as investor underreaction to other types of information, through which investor behavior can also affect market prices. Our analysis complements earlier empirical studies of price momentum. Hong, Lim, and Stein (2000) show that momentum profits are more pronounced in stocks of a smaller size and which receive less analyst coverage, and suggest that momentum is related to the slow diffusion of information among investors. Cooper, Gutierrez, and Hameed (2004) find that momentum strategies are highly profitable following positive market returns, but become ineffective following market downturns. They suggest that such an asymmetry in momentum profits is consistent with investors self-attribution bias. Jiang, Lee, and Zhang (2004) and Zhang (2004) show that momentum strategies are more profitable in stocks with higher information uncertainty, as proxied by variables such as return volatility, firm size, and analyst coverage. Our study also contributes to the growing literature that analyzes the cross-sectional properties of firms return R-squared, e.g., Morck, Yeung, and Yu (2000), Wurgler (2000), Durnev, Morck, Yeung, and Zarowin (2003), and Durnev, Morck and Yeung (2004). These studies find that return R-squared is negatively associated with more informative stock prices. However, several authors, e.g., West (1988) and Campbell, Lettau, Malkiel, and Xu (2001), also recognize that such a relation is difficult to reconcile within standard models, in which investors react rationally to information. 3 Our model shows that investors overreaction 3 The basic intuition is as follows: When investors process information rationally, more information leads to an earlier resolution of uncertainty and a higher return variation in the current periods, but there will be less uncertainty remaining and therefore less return variation in future periods. As such, the average return variation over time does not increase with the amount of information. 4
to their private information can help explain the negative relationship between R-squared and assets price informativeness. We also demonstrate empirically that such a relationship holds after controlling for public-information proxies, confirming the importance of investors private information in understanding return R-squared. Our results are also consistent with Peng and Xiong (2004), who provide an equilibrium model in which return comovement is determined by investors learning process under an attention constraint. Our paper is organized as follows: Section 2 discusses the overreaction theory and lays out our empirical hypotheses, while section 3 describes the data and variable construction. We study the cross-sectional relation between price momentum and return R-squared in Section 4, and explore the link between return R-squared and stock price informativeness in Section 5. Section 6 concludes. A simple model is also provided in the Appendix to illustrate the theoretical link between return R-squared and investors overreaction to information. 2 Theory and Empirical Hypotheses Our empirical hypothesis is motivated by the theory of Daniel, Hirshleifer, and Subrahmanyam (1998) on investor overconfidence and price momentum. The DHS model incorporates two types of investor biases documented by psychological studies, overconfidence and self-attribution bias. The model shows that overconfident investors overreact to their private information and that this overreaction can become even stronger after a good recent performance, generating price momentum. A natural implication of this overreaction mechanism is that stocks whose prices incorporate more private information from the trading of overconfident investors should display more pronounced price momentum. Investor overconfidence not only leads to price momentum, but also to excessive return variations. In the Appendix, we extend the DHS model by introducing a linear factor structure, comprised of a common market factor and a firm-specific factor, into each asset s fundamentals. Our model, otherwise similar to the DHS model, focuses on analyzing how investors overreaction to their private information affects assets return R-squared. When investors overreact to their private information about a firm, the firm-specific return volatility gets amplified. We show that under reasonable conditions, the magnitude of volatility amplification increases with the amount of the private information. Empirically, firm-specific return 5
variation is inversely related to the R-squared statistic of regressing a firm s stock returns on market and industry returns. Thus, our model establishes a negative association between return R-squared and the amount of overconfident investors private information. Such a relation suggests that we can use return R-squared as an indirect measure of the amount of private information that has been incorporated into stock prices through the trading of overconfident investors. Several researchers, e.g., Roll (1988) and Durnev, Morck, and Yeung (2004), have previously argued that more private information would lead to a lower return R-squared. The empirical study by Durnev et al. (2003) supports this argument by showing that the stock prices of firms with low R-squared tend to be more informative about future fundamentals. However, it would be difficult to derive such a negative relationship between return R-squared and the amount of information in a model where investors react rationally to information. The basic intuition, as shown by our model, is the following. Without investor overreaction, more information only leads to earlier resolution of uncertainty and a more informative stock price, but not to a higher level of return variability. In the case where the discount rate is zero, the return variability is solely determined by the underlying fundamental uncertainty and is independent of investors information. As a result, the return R-squared, which is inversely related to the firm-specific return variation, is also independent of investors information. Using a more general setup, West (1988) has provided a similar result. In particular, he shows that, if the discount rate is positive, more information actually reduces the return variability when investors are rational. The capacity of investor overconfidence to generate both price momentum and the negative association between return R-squared and the amount of private information serves as the foundation of our empirical analysis. While the amount of private information possessed by overconfident investors is not directly observable, we can use return R-squared as a proxy and analyze momentum profits across different groups of stocks sorted by return R-squared. More specifically, the theory described above motivates the following empirical hypothesis: Hypothesis 1. momentum. All else equal, stocks with lower return R-squared exhibit stronger price 6
Hypothesis 1 is the focal point of our empirical analysis. We also examine two additional questions. First, if price momentum is driven by investor overreaction, stock prices should also exhibit long-run reversals as they eventually converge to the fundamental values. Second, the relation between momentum and R-squared might also be related to other effects, such as those generated by slow information diffusion, information uncertainty, and market illiquidity. To control for these effects, we employ several control variables including firm size, degree of analyst coverage, percentage of institutional ownership, fundamental R-squared, return volatility, dispersion of analyst forecasts, share turnover and a market illiquidity measure. The overreaction theory also predicts a negative relation between stocks price informativeness and return R-squared because, all else equal, lower R-squared is associated with stocks for which more private information is incorporated through overconfident investors trading. This prediction is summarized in the following hypothesis: Hypothesis 2. Even after controlling for the amount of public information, stocks with lower return R-squared should have more informative prices. Testing Hypothesis 2 allows us to directly verify that return R-squared is related to investors overreaction to their private information. To control for the amount of public information, we use firm size, degree of analyst coverage, and fraction of institutional ownership as the control variables. 3 Data Description 3.1 Data and Sample Selection Our sample includes all NYSE/AMEX/NASDAQ listed securities on the Center for Research in Security Prices (CRSP) data files with sharecodes 10 or 11 (we exclude ADRs, closedend funds, REITs) from July 1963 to December 2002. For our analysis, we require firms to have information on a number of balance sheet and income statement items from the COMPUSTAT database. To ensure that the accounting variables are known before the period that stock returns are measured, we match CRSP stock returns from July of year t to June of year t+1 with accounting variables for fiscal year ending in year t-1. We obtain the following variables from COMPUSTAT with the data item numbers in 7
parenthesis. Book equity is defined as stockholder s equity (216) (or common equity (60) plus preferred stock par value (130) or asset (6) minus liabilities (181)), minus preferred stock (liquidating value (10), or redemption value (56), or par value (130)), plus balance sheet deferred taxes and investment tax credit (35) if available, minus post retirement asset (330) if available. Earnings is earnings before interest, which is income before extraordinary items (18) plus interest expense (15) plus income statement deferred taxes (50) when available. Asset is total asset (6). Firm size (Size) is measured by multiplying CRSP number of shares outstanding with share price at the end of June of year t. BE/ME is calculated by dividing book equity by market capitalization measured at the end of year t-1. We also obtain analysts coverage and institutional ownership data from the Institutional Brokers Estimate System (I/B/E/S) and the Standard & Poors, respectively. Analyst coverage (Num Analyst) is defined as the monthly number of analysts providing current fiscal year earnings estimates, averaged over the previous year. It is available from 1976 onwards. Institutional ownership (Inst) is measured in December of the year t-1 and is available from 1980 onwards. To measure information uncertainty surrounding a firm, we employ two additional proxies: total return volatility (Tvol) and analyst dispersion (Disp). Total return volatility is the standard deviation of a firm s weekly returns over the past one year or the entire sample period. Analyst dispersion is the monthly standard deviation of analysts annual earnings forecast divided by the absolute value of the mean forecast, averaged over the previous year, as in Diether, Malloy, and Scherbina (2002). It is available after 1976. Finally, we employ monthly share turnover (Turnover), defined as the monthly number of shares traded divided by shares outstanding, averaged over the previous year, and Amihud s (2002) illiquidity measure (Illiq), which is the average daily absolute return divided by daily dollar trading volume over the previous year, as measures of stock liquidity. Both liquidity proxies are available over the entire sample period from 1963 to 2002. 3.2 Construction of Return R 2 Weekly returns are employed to measure each firm s return R-squared. The weekly frequency balances between lower precision at monthly frequencies and more confounding microstruc- 8
ture effects such as nonsynchronous trading and bid-ask bounce at daily frequencies. We define weekly returns as compounded daily returns from Wednesday close to the following Wednesday close (e.g., Hou, 2004 and Hou & Moskowitz, 2004). R-squared measures constructed from these weekly returns are then matched with CRSP monthly return series to form portfolios. More specifically, we follow Roll (1988), Durnev et al. (2003) and Durnev et al. (2004) and run a regression of each stock s weekly returns on contemporaneous returns of the market portfolio as well as the industry portfolio to which the stock belongs: r i,t = α i,t + β i r m,t + γ i r I,t + ɛ i,t (1) where r i,t is the return of stock i, and r m,t and r I,t are returns of the value-weighted CRSP market portfolio and industry portfolio in week t. We exclude firm i when calculating both the market return and the industry return. For example, the industry return is computed by j I,j i r I,t w j,t r j,t j I,j i w j,t where w j,t is the market capitalization of firm j in industry I. Excluding firm i when calculating r I,t prevents potential spurious correlations between r i,t and r I,t. The regression R 2 from equation 1 is R 2 1 t ɛ2 i,t t (r i,t r i,t ) 2. We include both the market return and the industry return in regression (1) so that the R-squared from the regression is inversely related to the firm-specific return variation. This way, the R-squared measure is directly related to the amount of private information that is incorporated into stock prices through the trading of overconfident investors, as we discussed in Section 2. Table 1 presents the summary statistics of six R-squared measures that differ in estimation sample period or whether they adjust for the degree of freedom in estimating (1). RP 2 1 is the R-squared estimated using weekly returns over the past one year. The mean of this variable is 0.16 and the median is 0.11, with 25% of the firms having an R-squared value of less than 0.04 and 25% of the firms having a R-squared value of greater than 0.23. R 2 P S is estimated using weekly returns over the entire past sample. Its mean and median are 0.12 and 0.09 9
respectively. RF 2 S is estimated using the full sample of weekly return data. Its mean and median are 0.09 and 0.06. adj.r 2 F S, adj.r2 P 1, and adj.r2 P S are the corresponding adjusted R-squared measures. Panel B of Table 1 presents the correlation matrix of these R-squared measures. As expected, the R-squared and the corresponding adjusted R-squared are highly correlated, with correlations ranging from 0.97 to 1. The correlations among R 2 P 1, R2 P S and R 2 F S are also large and statistically significant, ranging from 0.65 to 0.83. Due to noise in individual stocks weekly returns, the R 2 measures from regression (1) can be quite noisy, especially with limited time series data. RF 2 S employs the largest number of observations, and therefore should be more precise in measuring the average amount of private information that investors overreact to. Since we are interested in understanding the source of momentum phenomenon rather than constructing feasible trading strategies, we will focus our discussions of the momentum effect primarily using RF 2 S, the full sample R-squared measure. We will also report results based on the one-year and the past-sample R-squared measures as robustness checks. As a control variable in our analysis, we also employ a firm s fundamental R-squared the variation in a firm s fundamentals that can be explained by market- and industrywide fundamental movement. We estimate the fundamental R-squared by regressing a firm s fundamental variable on that of the aggregate market and the industry that the firm belongs: E i,t = α E i,t + β E i E m,t + γ E i E I,t + ɛ E i,t (2) where E i,t is earnings scaled by total asset, and E m,t and E I,t are the value-weighted scaled earnings of the market portfolio and the industry portfolio (both excluding the firm itself). To improve precision, equation (2) is estimated over the entire sample period for each firm. The variable FRSQ is defined as the R-squared statistic of this regression. 4 Return R 2 and Price Momentum 4.1 R 2 and Momentum To explore the relation between price momentum and firm-specific return variation, we report in Table 2 returns on momentum portfolios for firms with different levels of R-squared using a doubled-sorted five-by-five grid. At the beginning of each month, all firms in our sample 10
are first ranked by an R-squared measure using NYSE breakpoints and placed into quintile portfolios. Within each R-squared group, stocks are further sorted into quintiles based on past twelve month return (skipping the most recent month). 4 The value-weighted returns on these double-sorted portfolios are computed over the following month. The averages and t-statistics (in italics) of each portfolio return, as well as the differences in returns between momentum quintiles 5 and 1 within each R-squared quintile, are reported. 5 For robustness, we will also report, in Section 4.2, R 2 -based momentum profits based on independently sorted momentum portfolios and equal weighted portfolio returns. To control for the potential differences in firms size and BE/ME characteristics across quintiles, we also report characteristic-adjusted returns to account for the premia associated with size and BE/ME following the characteristics-matching procedure in Daniel, Grinblatt, Titman, and Wermers (1997). Each month all stocks in our sample are first sorted into size deciles, based on NYSE decile breakpoints, and then within each size decile further sorted into book-to-market deciles using NYSE breakpoints. Stocks are value-weighted within each of these 100 portfolios to form a set of 100 benchmark portfolios. To calculate the size and BE/ME hedged return for an individual stock, we subtract the return of the value-weighted benchmark portfolio to which that stock belongs from the return of that stock. The expected value of this excess return is zero if size and BE/ME completely describe the cross-section of expected returns. Panel A reports average returns of portfolios sorted by each stock s full-sample R-squared, RF 2 S, and the stock s cumulative raw return over the past year (skipping the most recent month), Ret (-12:-2). The left half of the panel presents results based on raw returns. In the lowest RF 2 S quintile, the average value-weighted raw return spread between past winners (momentum quintile 5) and past losers (momentum quintile 1) is 181 basis points per month with a t-stats of 6.65. This return spread falls steadily as R 2 F S R 2 F S increases. In the highest quintile, the return spread drops to an insignificant 51 basis points per month. The differences across R-Squared quintiles are highly significant: The test of the null hypothesis 4 We skip one month between the formation period and the holding period to minimize bid-ask bounce and other microstructure effect. 5 Throughout the paper, we follow Asness (1995), Fama and French (1996), and Grundy and Martin (2001) and focus mainly on momentum strategies with a one-month holding period, but report profits from momentum strategies with alternative holding periods in Section 4.3. 11
that the average momentum profit is the same across all five R-Squared quintiles produces an F-statistic of 3.40, indicating a rejection at better than the 1% level. 6 The negative and monotonic relation between momentum profit and return R-squared demonstrated here is clearly consistent with Hypothesis 1 in Section 2. The average characteristic-adjusted returns are reported in the right half of panel A. As expected, the size and book-to-market adjusted return for each double-sorted portfolio is considerably lower than the corresponding raw return. However, the average spread between momentum quintiles 5 and 1 within each RF 2 S group only decreases slightly. More importantly, the pattern of the momentum spread across different R 2 F S decreases from a significant 152 basis points in R 2 F S quintiles remain unchanged. It quintile 1 to an insignificant 42 basis points in R 2 F S quintile 5.7 This result suggests that the negative relationship between R- squared and momentum profit is not driven by insufficient controls for differences in size and book-to-market characteristics. We also report in Panel A the intercepts and the associated t-statistics from time series regressions of raw and characteristic-adjusted momentum spreads on the Fama and French (1993) three factor model, which employs the excess returns on the market portfolio and returns on two factor-mimicking portfolios (SMB and HML) designed to capture the size and book-to-market effects. The intercepts from the time series regressions are slightly larger than the corresponding return spreads, but the negative relation between momentum profitability and R-squared remain unchanged. For example, even after essentially adjusting returns twice using both the characteristic benchmark model and the Fama and French (1993) three factor model, recent past winners outperform losers by 176 basis points (t-stat=7.80) in the lowest ranked RF 2 S quintlie, and this number monotonically decreases to 68 basis points (t-stat=2.99) for the highest R 2 F S quintile. Panel B reports average returns on portfolios sorted first by RP 2 S, the R-squared estimated from weekly returns over the entire past sample, and then by each stock s cumulative raw return over the past year (skipping the most recent month), Ret (-12:-2). The pattern by 6 We can also easily reject the null that the momentum profit is identical across the two extreme quintiles (t-statistic=3.22). 7 The F-statistic for the null of constant characteristic-adjusted momentum spread across R-Squared quintiles equals 3.89 and is significant at better than the 1% level. 12
which the momentum profit varies across different R 2 P S quintiles is very similar to that of R2 F S. The value-weighted return spread decreases monotonically from a significant 173 basis points per month for the lowest R-squared quintile to an insignificant 56 basis points per month for the highest R-squared quintile. The size and book-to-market adjusted return spread also decreases monotonically from 146 basis points per month to an insignificant 39 basis points per month. Finally, the Fama-French three-factor intercepts display a similar trend across quintiles. Panel C reports results for double sorted portfolios based on RP 2 1, the R-squared estimated from weekly returns over the past year, and each stock s cumulative raw return over the past year (skipping the most recent month), Ret (-12:-2). The pattern by which momentum profit varies across RP 2 1 quintiles is again similar to those of R2 F S and R2 P S. The value-weighted return spreads decrease monotonically from 140 basis points per month for the lowest R- squared quintile to 65 basis points per month for the highest R-squared quintile. The size and book-to-market adjusted value-weighted return spread decreases from 118 basis points per month to 42 basis points per month. As expected, the differences in momentum spread across R 2 P 1 quintiles are slightly smaller due to the measurement errors in the R2 P 1 variable. Overall, the results in Table 2 strongly support a negative and monotonic relationship between firms return R-squared and price momentum. Such a relationship is consistent with our Hypothesis 1 that stocks with lower R-squared should exhibit more price momentum. Since R-squared is inversely related to the amount of private information incorporated into stock prices through the trading of overconfident investors, our findings suggest that price momentum increases with the amount of overconfident investors private information. We will provide further robustness checks of this negative relation between R-squared and momentum in the following subsections, and then directly analyze the link between R-squared and price informativeness in Section 5. 4.2 Robustness of R 2 -Sorted Price Momentum We have performed numerous additional tests and find the relation between R 2 and price momentum to be robust. For brevity s sake, we select a few representative ones and report them in Table 3. Results from other robustness checks are available upon request. 13
Panel A of Table 3 report results for sorting stocks using adj.rf 2 S. They are very similar to what we ve seen in the previous section: There is again a negative and monotonic relationship between R-squared and the momentum profit (using both raw returns and characteristics adjusted returns). This negative relation is actually slightly stronger than that associated with the unadjusted R 2 F S. In unreported results, we have also examined adj.r2 P S and adj.r2 P 1 sorted momentum profits, and found similar results to Tables 2B and 2C which employ unadjusted R-squared measures. Panel B of Table 3 performs independent sorts using R 2 F S and Ret (-12:-2), instead of the sequential sorting procedure employed in previous tests. We observe that the same negative and monotonic relation between R-squared and momentum spread still holds with the independent-sorting procedure. We have also performed the independent-sorting procedure for other R-squared measures, RP 2 S and R2 P 1, and found similar results. These results confirm that the negative relationship between R-squared and momentum profits is not sensitive to the sorting procedure we use. Panel C reports equal-weighted average monthly returns (both raw returns and characteristics adjusted returns) for portfolios sorted by R 2 F S and Ret (-12:-2). To prevent microstructure effects that are usually associated with small, low-priced stocks from having undue influences on the equal-weighted portfolio returns, we exclude stocks with prices less than $5 at the end of the portfolio formation period. We find that equal-weighted momentum profit again decreases monotonically with RF 2 S. Similar results are obtained using other R-squared measures, and therefore are not reported, for brevity s sake. Panel D reports value-weighted returns on portfolios sorted by R 2 F S and cumulative raw return over the past six months (skipping the most recent month), Ret (-6:-2). We observe a similar negative and monotonic relation between R-squared and momentum profit to the result using past twelve-month returns as the sorting variable. In summary, the results in Table 3 demonstrate that the negative relationship between R-squared and price momentum is robust when we employ different R-squared measures, different stock sorting procedures, and different portfolio return specifications. 14
4.3 Long-Run Reversal of Momentum Profits If price momentum is driven by investors overreaction to their private information, we would also expect to see price reversal in the long run when prices converge to their fundamental values. Table 4 reports the long-run performance of the R 2 -sorted momentum portfolios for various horizons. Panel A repeats Table 2A as the short-term performance benchmark with a one-month holding period. Panels B, C, D and E correspond to holding periods of six months, one year, years 2 and 3, and years 4 and 5 after portfolio formation, respectively. We report value-weighted average raw and characteristic-adjusted returns of the momentum portfolios and the spreads in returns between past winners and losers within each R 2 quintile, along with the associated t-statistics. Panel B shows that, for the first 6 months, the negative and monotonic relation between R-squared and momentum profit persists. The momentum profits decrease from 114 basis points per month for the lowest R-squared quintile to 44 basis points per month for the highest R-squared quintile. The size and book-to-market adjusted momentum profits follow a similar pattern, decreasing from 100 basis points per month in the lowest R-squared quintile to 40 basis points per month in the highest R-squared quintile. Panel C demonstrates that there is still a negative relation between R-squared and momentum profit for the first year after the formation of the momentum portfolios. But the magnitude of momentum profit and the differences in profit across the R-squared quintile are substantially smaller: In the lowest R-squared quintile, a momentum strategy of buying the winners and shorting the losers yields an average profit of only 48 basis points per month; the profit declines to an insignificant 24 basis points for the highest R-squared quintile. The size and book-to-market adjusted momentum profits also exhibit a similar pattern and magnitude. Panels D reports average returns for the momentum portfolios from one year to three years after the formation period. Across all R-squared quintiles, the momentum strategies produce negative average spreads (both raw and characteristics adjusted returns), and most of them are statistically significant. Panel E shows that when we evaluate the profitability of the momentum strategies over a holding period that is even further away from the formation period, that is, from three years to five years after portfolio formation, we also observe similar 15
reversal patterns. The magnitude of the spreads reported in Panels D and E as well as in previous panels is consistent with the evidence in the literature that the profits of momentum strategies are concentrated in the first few months after portfolio formation, and that they tend to dissipate after 6 months, and eventually reverse at longer horizons. See, for example, Lee and Swaminathan (2000) and Jegadeesh and Titman (2001). Figure 1 plots the cumulative momentum profits for each of the five R-squared quintiles over the period from one month to five years after portfolio formation. Figure 1A plots cumulative raw average profits of the momentum strategy and Figure 1B plots characteristicsadjusted profits. The graphs confirm the findings in Table 4 that the momentum profits across all five R-squared quintles reverse at longer horizons. Across the R-squared quintiles, Panel D of Table 4 does not provide a clear pattern in the price reversal from year 2 to year 3. Panel E shows that the price reversal from year 4 to year 5 decreases monotonically from 28 basis points per month in the lowest R-squared quintile to 4 basis points in the highest R-squared quintile. However, the difference is not statistically significant. We also note that in Figure 1 the cumulative profits (raw or adjusted) from the portfolio formation to year 5 are negative for all R-squared quintiles, except the adjusted profits of quintile 4, indicating that in the long run prices over-correct the short-term momentum effect. 8 This pattern suggests that other mechanisms might also contribute to the long-run price reversal. Overall, although overreaction cannot explain the full magnitude of long-run reversal, the fact that momentum profits do reverse in the long run supports the hypothesis that price momentum can be generated by investors overreaction. 4.4 Controlling for Other Effects The relation between return R-squared and price momentum may also be induced by mechanisms other than investor overreaction to private information. Here, we include several variables in our analysis to control for these potential alternative effects. First, Hong and Stein (1999) propose an underreaction theory: When investors are heterogeneous, firm-specific information diffuses slowly across investors. Consequently, prices 8 Cooper, Gutierrez, and Hameed (2004) and Griffin, Ji, and Martin (2003) also find a similar pattern. 16
might underreact to information and display momentum if investors fail to extract useful information from market prices. Hong, Lim, and Stein (2000) provide some evidence for such a mechanism by using firm size and analyst coverage as proxies for the speed of information diffusion. They find that stocks with smaller size and less analyst coverage, presumably more prone to slower information diffusion, tend to have more pronounced price momentum. To account for the Hong and Stein s slow-information-diffusion effect, we employ the variables used by Hong, Lim, and Stein (2000): firm size and analyst coverage, as control variables in our analysis of the relation between R-squared and momentum. In addition, several other studies, e.g., Badrinath, Kale, and Noe (1995), and Hou (2004), suggest that another variable, institutional ownership, is also related to more visibility and faster information diffusion. Thus, we incorporate institutional ownership as well. It is also important to note that firm size, analyst coverage and institutional ownership are highly related to the availability of public information. For firms with larger size, more analyst coverage, and higher institutional ownership, stock prices tend to incorporate more public information. Consequently, overconfident investors who overreact to their private signals will have less of an impact on the price formation process. This potential tension between public information and private information has been pointed out by Daniel, Hirshleifer, and Subrahmanyam (1998). Related to this point, the model we provide in the Appendix predicts that more public information reduces the volatility amplification effect by overconfident investors, and leads to a smaller firm-specific return variation and a higher return R-squared for a firm s stock. Second, firms return R-squared is partially determined by their fundamental correlation with the market and their industries. To control for the effect of the fundamental correlation, we include firms fundamental R-squared of regressing firms earnings on aggregated marketand industry-earnings as a control variable. Third, Jiang, Lee, and Zhang (2004) and Zhang (2004) argue that if price momentum is driven by investors psychological biases, momentum would be more pronounced for firms with greater information uncertainty since the biases are likely to be more severe for these firms. Empirically, these studies use firm size, analyst coverage, dispersion in analyst forecasts, and return volatility as proxies for information uncertainty. We also include return volatility and 17
dispersion in analyst forecasts as additional variables in our analysis to control for the effect of information uncertainty. Finally, liquidity and other trading imperfection might also affect price momentum. Lee and Swaminathan (2000) find that trading volume predicts the magnitude and persistence of price momentum. To filter out such an effect, we incorporate firms share turnover rate into the list of our control variables. We also include an illiquidity measure proposed by Amihud (2002), the ratio between a firm s average daily absolute return and daily dollar trading volume. The higher the ratio is, the market is more illiquid, as a trade of a given size causes a greater price change. Table 5 reports summary information on our control variables (Size, Num Analyst, Inst, FRSQ, Tvol, Disp, Turnover, and Illiq) for firms with different levels of return R-squared. We sort firms annually based on their full-sample return R-squared, using NYSE breakpoints and then compute the average characteristics for different R-squared quintiles. Reported are the time series averages of the cross-section mean characteristics. As expected, Table 5 shows that firm size, analyst coverage, and institutional ownership all increase monotonically across the R-squared quintiles, consistent with these variables proxying for the amount of public information. The firm size (market capitalization) increases from 84.2 million for the lowest quintile to 4394 million for the highest quintile, the number of analysts covering a firm increases from 2.7 to 16.9, and the percentage of institutional ownership increases from 20.1% to 50.1%. Fundamental R-squared increases monotonically across the return R-squared quintiles. However, the range of fundamental R-squared (from 0.291 in the lowest R-squared quintile to 0.414 in the highest quintile) is much smaller than the range of return R-squared (from 0.058 to 0.373), suggesting that the variation in fundamental R-squared is too small to explain the variation in return R-squared. Table 5 also shows that across the return R-squared quintiles, return volatility decreases monotonically, while the dispersion in analyst forecasts tend to increase. For the two liquidity measures, Illiq decreases monotonically, while Turnover is smaller in the lowest R-squared quintile and is relatively flat for the other quintiles. To further explore the relation between price momentum and return R-squared, we con- 18
struct residual R-squared measures by regressing return R-squareds on our control variables, and then form momentum portfolios based on the residuals from the regression (ResR 2 ). Since R-squared is bounded between 0 and 1, we perform the following logistic transformation before running the regressions: ( ) R LR 2 2 ln 1 R 2 Note that LR 2 increases monotonically with R-squared. To construct residual R-squared, we employ seven regression specifications. Regression model 1 includes firm size as the only control variable, and the corresponding regression residuals are denoted by Res. R 2 (Size). In regression model 2, all three public information variables, firm size, analyst coverage and institutional ownership, are employed. We denote the residuals by Res. R 2 (Size, Analyst, Inst). In regression model 3, we also include the fundamental R-squared, and denote the residuals by Res. R 2 (Size, Analyst, Inst, F RSQ). Regression model 4 adds firms return volatility, and we denote the residuals by Res. R 2 (Size, Analyst, Inst, F RSQ, T vol). Regression model 5 also incorporates the dispersion of analyst forecasts, and we denote the residuals by Res. R 2 (Size, Analyst, Inst, F RSQ, T vol, Disp). Due to the limited availability of data on analyst coverage and institutional ownership, the residual R-squared measures from regression models 2-5 only cover the sample period from 1981 onwards. Regression models 6 and 7 incorporate share turnover and Amihud s illiquidity measure. To avoid the sample period being shortened by the availability of analyst coverage and institutional holding, we include firm size and share turnover in regression 6 and firm size and 19
the illiquidity measure in regression 7. We denote the residuals by Res. R 2 (Size, T urnover), and Res. R 2 (Size, Illiq). Finally, we interact the three residual R-squared measures with return over the past year (excluding the most recent month) to form double-sorted momentum portfolios. The average monthly raw returns and characteristics-adjusted returns for these portfolios are reported in Table 6. Panels A through G report results for each of the seven residual R-squared measures, respectively. The results from all seven panels share very similar patterns as the results in Panel A of Table 2, which reports the double-sorted momentum profits based on raw return R-squared and past one year return. The momentum profits, either raw or characteristics-adjusted, always decrease monotonically from low to high residual R-squared quintiles, regardless of the regression specification to calculate residual R-squared. For example, when only firm size is employed as the control variable to form the residual R-squared (Panel A), past winners outperform past losers by 192 basis points per month among firms in the lowest quintile of residual R-squared, and this spread shrinks to 55 basis points for firms in the highest quintile of residual R-squared. When we restrict our sample to the post-1981 period during which we control for analyst coverage, institutional ownership, and fundamental R-squared, return volatility and dispersion in analyst forecasts, in addition to firm size (Panel E), we observe the same pattern in momentum profits. The average momentum spread in the lowest residual R-squared quintile is 115 basis points per month, and it declines to an insignificant 21 basis points in the highest residual R-squared quintile. When we control for firm size and illiquidity in full sample (Panel G), the average momentum spread is 187 basis points per month in the lowest residual R-squared quintile, and it declines monotonically to 58 basis points per month in the highest residual R-squared quintile. In all cases we can reject the null hypothesis that the average momentum spread is the same across all five residual R-Squared quintiles; F-statistics are all significant at better than the 1% level. 20