Momentum and Asymmetric Information

Transcription

1 Momentum and Asymmetric Information Tian Liang Cornell University January 7, 2006 I would like to thank David Easley, Maureen O Hara and Gideon Saar for very helpful discussions and suggestions. Please address correspondence to Tian Liang, Department of Economics, Cornell University, 447 Uris Hall, Ithaca, NY Tel: tl227@cornell.edu.

2 Momentum and Asymmetric Information January 7, 2006 Abstract This paper examines the implications of asymmetric information for price evolution and investor behavior under a rational expectations framework. I present a simple asymmetric information-based asset-pricing model to show that private information and its revelation can generate price momentum. To empirically test this implication of the model, I use the probability of information-based trade (PIN) from Easley et al. (997b) as a proxy for private information, and nd that rms with high PINs do have larger magnitudes of momentum e ect even after controlling for size. The abnormal returns are both economically and statistically signi cant, and cannot be explained by the Fama-French factors. JEL classi cation: D80, G4 Keywords: momentum e ect, asymmetric information, rational expectations, probability of information-based trade

3 Introduction Financial academics have long recognized that stock returns appear to exhibit continuation, or momentum. However, there is little agreement about an explanation for this phenomenon. As noted by Brav and Heaton (2002), researchers have created two competing theories of nancial anomalies: "behavioral" theories built on investor irrationality and "rational structural uncertainty" theories built on incomplete information about the structure of the economic environment. Many studies based on behavioral theories show that private information plays a role. For example, Daniel, Hirshleifer and Subrahmanyam s (998) parsimonious theory states that investors overreact to private information signals and underreact to public information signals, which can result in short-term momentum and long-term reversal. Hong and Stein (999) also study investor behavior to analyze the anomaly, modeling the interaction of traders who underreact to private information and traders who partially exploit this underreaction. In contrast with the amount of research on behavioral theories, fewer studies have considered the relation between private information and the momentum phenomenon basing on rational structural uncertainty theories. The objective of this study is to examine the implications of asymmetric information for price evolution and investor behavior under a rational framework. This paper proposes a simple rational expectations model (Grossman and Stiglitz (980)) in which information asymmetry can generate positively correlated stock returns. The basic intuition of the results goes as follows. Assume that there is some news about the true value of a stock. Without loss of generality, consider the case where the news is good, that is, the expected value of the stock is high. Suppose at rst only a fraction of the investors (informed traders) see the news, while the uninformed cannot see it but can rationally infer how it will a ect the equilibrium price. The stock price will increase because the good news is partially re ected in the price. Meanwhile, the informed and uninformed traders hold di erent beliefs about the stock s expected return. Speci cally, the informed traders evaluation of the asset is higher than the uninformed. This is because the informed see the news directly, while the 2

4 uninformed only see the combination of the news and the noise, and hence are less a ected by the news. As a result, the informed perceive the stock to be underpriced and want to buy more of the stock while the uninformed traders want to sell. As time goes by, the information is no longer private. The uninformed investors now see the news and realize that they have underestimated the asset value, so they demand more of the asset and the price can increase further. The informed, on the other hand, have the same belief as before and nd themselves holding too many assets given the higher price. So selling is optimal for them. In the case where the news about the asset is bad, traders behave the opposite and the stock price can decrease further. Thus the revelation of the private information can drive price momentum. When the news comes, the informed traders trade on the information, while the uninformed underreact to the news and actually trade against the information. When the private information becomes publicly available, the informed traders reverse their trading direction and act as contrarians, while the uninformed act as trend chasers. The asset returns generated this way are positively correlated under certain constraints on the uncertainties of the asset returns and the signal. To empirically test the relation between private information and the momentum phenomenon, I use the probability of information-based trade (PIN) from Easley et al. (997b) as a measure of information asymmetry. If private information is a source of momentum, then the momentum e ect should be greater among rms with high PINs. Using NYSE data for the period of 984 to 200, I nd strong evidence that rms with high PINs have high momentum premium even after controlling for size. The abnormal returns are both economically and statistically signi cant, and cannot be explained by the Fama-French factors. This is largely consistent with the model s implication. The modeling of this paper is related to Brown and Jennings (989), Grundy and McNichols (989), Kim and Verrecchia (99), Hirshleifer, Subrahmanyam, and Titman (994), and He and Wang (995), all of which present multiple period rational expectations models with asymmetric information. On the relationship between asymmetric information and 3

5 momentum e ect, Holden and Subrahmanyam (2002) show that with sequential endogenous information acquisition, asset returns are positively autocorrelated. In a similar framework, Strobl (2003) shows that changing information asymmetry may explain disposition and momentum e ects. While these models maintain a strictly hierarchical information structure over time, the model in the present paper focuses on the e ect of dynamic information revelation. On the study of the PIN measure, Easley, Hvidkjaer, and O Hara (2004) show that PIN is robust to momentum e ects and the momentum factor. They nd that the inclusion of momentum reduces the unexplained return of a portfolio that is long in high PIN stocks and short in low PIN stocks, but it does not eliminate it. This is consistent with the results of this paper. Private information can drive price momentum, so they are closely related; but momentum cannot explain the PIN-based portfolio returns because it is the consequence, not the cause of private information. The article continues as follows. Section 2 presents the theoretical model, Section 3 empirically tests the implication of the model, and Section 4 summarizes the paper. 2 The Rational Expectations Models Consider a four-period model in which traders choose their portfolios at time 0, and 2, and the assets pay o at time 3. There is a bond yielding a gross return R. For simplicity, suppose R is always equal to, and the supply of the bond is completely elastic. Thus the bond price is always. There is also a risky asset, or stock. End-of-period value of the stock is given by v, where v s N(m; =). The per capita supply of the stock at time t is random, with ~x t s N(x t ; = t ). Stock supplies across periods are independent, and are independent of all other random variables. The aggregate supply of the stock at time t equals P t i=0 ~x i. The random supplies prevent the equilibrium from being fully revealing. The stock price at time t is denoted by p t. Traders potentially receive a signal, s s N(v; =) about the stock. At time 0 no trader 4

6 sees the signal. At time, the signal is seen by only fraction of the traders, so it corresponds to private information. At time 2 the signal is revealed to everyone, so the private information becomes public information. All traders know the distributions of all random variables. The investors all have CARA utility functions with coe cient of risk aversion > 0. The investors have an endowment of money that they can use to buy bonds and stocks. 2. The Static Model I rst introduce a static rational expectations model in which traders do not take the time 2 price change into account when they make time decisions. That is, at time they do not know that the private information will be revealed to public at some time in the future before the assets pay o. In this case the traders are rational, but have incomplete information about the market. This assumption will be loosened in the dynamic model introduced in Section Equilibrium Each investor chooses his demands for bond b t and for stock x t at time t (t = 0; ; 2) to maximize his expected utility at the nal period. Determining the market equilibria at time 0 and 2 is straightforward, because in these periods all investors information and beliefs are the same. At time 0, traders cannot see any signal, so their belief about the distribution of the nal stock payo is simply the unconditional one: v s N(m; =). Given the well-known properties of CARA utility under normal distributions, any investor s demand at time 0 is x 0 = E(v) p 0 V ar(v) = m p 0 : () Equating the per capita demand given in () and the per capita supply ~x 0 yields the time 0 equilibrium price p 0 = m ~x 0: (2) 5

7 At time 2, the signal s regarding the stock s payo is fully revealed to everybody. Again all traders hold the same belief about v s distribution. Using Bayes rule, it is easy to show that the distribution of v conditioning on s is vjs s N m s ;. (3) Because any trader s demand for the stock at time 2 is given by x 2 = E(vjs) p 2 V ar(vjs), market clearing implies the time 2 equilibrium demand and price to be x 2 = ~x 0 ~x ~x 2 (4) p 2 = m s (~x 0 ~x ~x 2 ) (5) where (~x 0 ~x ~x 2 ) is the aggregate stock supply at time 2. Now let us consider time. Traders at time see di erent information regarding the stock, so they do not all have the same beliefs. The informed traders observe the private signal s, therefore their belief is the same as the one at time 2 as given in (3). Determining the belief of the uninformed traders is more complex. While these traders do not observe the signal, they do know that there is a signal, they know its distribution, and they rationally infer how it will a ect the demands of the informed traders and thus the equilibrium price. To learn from the price, these traders must conjecture a form for the price function, and in a rational expectations equilibrium this conjecture must be correct. Suppose the uninformed conjecture the price function p = am bs c~x dx e~x 0 (6) 6

8 where a, b, c, d and e are coe cients to be determined in the equilibrium. Following O Hara (2003), for convenience the observable random variable is de ned to be = p e~x 0 am (c d)x b = s c b (~x x ): Note that although ~x 0 is not directly observable, p 0 will fully reveal its value to public at time 0, so ~x 0 is known at time. Calculation shows that conditioning on v, is distributed as N(v; ), where = c 2 : b Thus, given the conjecture in equation (5), the belief of the uninformed traders regarding the stock is normal vj s N m ; : (7) Proposition characterizes the nature of the equilibrium at time. Proposition There exists a partially revealing equilibrium at time in which p = am bs c~x dx e~x 0 where a = z ; b = ( ( ) ) ; c = z " d = ( ) z ; e = z ; = h ( ) z 2 i ; # ; z = ( ) : Proof. See the Appendix. The equilibrium depicted above is partially revealing in the sense de ned by Grossman and Stiglitz (980). The uninformed cannot learn the informed traders information from the price, but they can draw inferences from the price about the information. These inferences 7

9 will be correct, but not complete. Thus, compared to the informed, the uninformed are more likely to underreact to the information as discussed in the following section Trading Behavior At time, while the informed traders receive the signal regarding the asset s risk and return precisely, the uninformed only see the combination e ect of the signal and the random supply at time. In other words, for the uninformed, the information strength is attenuated by the random supply. Speci cally, if the news is good, the informed traders evaluation of the stock return will be higher than that of the uninformed, because the information they receive is relatively more accurate and thus has a stronger impact on their belief. The uninformed, on the other hand, cannot rely upon their inferences that much. Therefore their new belief about the distribution of the stock payo is less di erent from before. If the news is bad, conversely, the informed s evaluation of the stock will be lower than that of the uninformed for the same reason. Proposition 2 formally summarizes the above argument. Proposition 2 On average, the informed traders expectation of the stock payo v is higher than that of the uninformed when there is good news; the converse is true when there is bad news, i.e. E[E(vjs) E(vj)js > m] > 0; E[E(vjs) E(vj)js < m] < 0: Proof. See the Appendix. For simplicity from now on assume that x t = 0 (t = 0; ; and 2). Normalizing the expected stock supply to zero can be done without loss of generality. Introducing a positive mean supply would cause the unconditional risk premium to be non-zero, but would not alter our basic results. Now let us consider how traders portfolios di er in this asymmetric information world. As pointed out before, at time 0, traders all have the same demand for the stock as a result 8

10 of the information symmetry. At time, by Proposition 2 traders hold di erent beliefs about the nal payo depending upon the value of the private information signal. If there is good news, on average the informed have higher expectations of the asset return than the uninformed do, so they are more likely to nd the price lower than expected, and thus want to buy the stock. Meanwhile, the uninformed want to sell. This follows because they are more likely to nd price higher than expected due to their low expectation. If there is bad news, on the other hand, the same reasoning applies and the informed sell as the uninformed buy. In other words, when information asymmetry exists, the informed traders trade on the information, while the uninformed underreact to the news and actually trade against the information. The next proposition con rms this intuition. Proposition 3 On average, at time the informed traders buy the stock while the uninformed sell when there is good news; the converse is true when there is bad news, i.e. E x I x I 0js > m > 0; E x U x U 0 js > m < 0; E x I x I 0js < m < 0; E x U x U 0 js < m > 0; where x I t and x U t are informed and uninformed traders demands at time t, respectively. Proof. See the Appendix. When the private information becomes public at time 2, the informed traders information regarding the asset return remains the same, and so does their belief. The uninformed observe the signal directly and use it to update their belief. The updated belief must be the same as that of the informed traders, because now all traders have the information. As implied by Proposition 2, on average, the uninformed traders expectation of the asset return increases if the news is good and decreases if it is bad. This change in their expectation re ects the uninformed traders adjustment corresponding to the more precise information. They realize their underreaction in the previous period and want to modify their portfolios accordingly. In 9

11 particular, the uninformed, in general, buy the stock due to their higher expectation when there is good news and sell due to their lower expectation when there is bad news. The informed, conversely, sell when there is good news and buy when there is bad news. This follows because their portfolios have already been fully adjusted to the signal at time. As the uninformed buys drive up the time 2 price, the informed will nd themselves holding too many stocks given the new price and want to reduce their holdings. Similarly, if the time 2 price is driven down by the uninformed sells, the informed traders demand for the stock rises. Therefore, at time 2 the informed act as contrarians while the uninformed are trend chasers. Proposition 4 completes this analysis. Proposition 4 On average, at time 2 the informed traders sell the stock while the uninformed buy when there is good news; the converse is true when there is bad news, i.e. E x I 2 x I js > m < 0; E x U 2 x U js > m > 0; E x I 2 x I js < m > 0; E x U 2 x U js < m < 0: Proof. See the Appendix. As a summary, at the time the news arises, the informed traders take advantage of their private information by trading on the information, while the uninformed underreact to the news in the sense that they insu ciently adjust their beliefs and trade against the information; when the private information becomes publicly available, the informed traders reverse their trading direction and act as contrarians, while the uninformed act as trend chasers Price Momentum The fact that the information regarding the asset s return is only partially re ected in the price at time and then fully re ected in the time 2 price suggests that the revelation of private information can cause price continuation, or momentum. On the other hand, the 0

12 additional risk premium due to the random supply at time reverses by time 2, causing a price reversal. If the e ect of information revelation dominates, the serial correlation in returns is positive. Proposition 5 Let 4p, 4p 2 denote the stock price changes at time and 2, respectively, i.e. 4p = p p 0 ; 4 p 2 = p 2 p. Then Cov(4p ; 4p 2 ) > 0 if and only if > 0: 0 Or equivalently, 4p and 4p 2 are positively correlated if and only if V ar(v)v ar(~x 0 ) > V ar(sjv)v ar(~x ): Proof. See the Appendix. Proposition 6 shows that when i) the impact of the signal is strong either because the fraction of informed traders is large, or the stock payo is highly volatile, or the signal is precise; ii) the reversal e ect of the time supply is small because the uncertainty of ~x is relatively low, the revelation of information will generate price momentum. 2.2 A Dynamic Model In the static model, traders are rational under the assumption that they have incomplete information about the market, that is, at time they do not know that the private information will be revealed to public at some time in the future before the assets pay o. In reality, however, people may correctly predict that the information will move gradually across traders and nally become common knowledge. Therefore information di usion and revelation may not be surprising as in the static model. This suggests us to consider a dynamic model in

13 which all traders are fully aware of the fact that once any private information arrives, it will soon become public. From now on, in addition to the earlier assumptions, I will also assume that m = 0 and t = (t = 0; ; and 2). The assumption m = 0 will shift the level of demands and prices, but it will not alter their relationships. An increase in demand is still an increase; a decrease in price is still a decrease. More importantly, since m is a constant, no change will be introduced to the correlations of the price changes, which are of prime interest in this paper. The assumption t = costs us the ability to detangle the e ects of supply shock uncertainties in di erent periods, but greatly simpli es the calculation. Computer simulations (not reported here) show that our main results remain the same if we consider heteroskedasticity Equilibrium Just as in the static model, each investor chooses his demand for stock x t at time t (t = 0; ; 2) to maximize his expected utility at the nal period, time 3. We can start from time 2 and use backward induction to solve for the optimal portfolio series. At time 2, the information is fully revealed, so all traders hold the same belief and face the same maximization problem, which is the same as the one in the static model at time 2. Therefore the equilibrium is characterized by x 2 = ~x 0 ~x ~x 2 (8) p 2 = s (~x 0 ~x ~x 2 ) (9) where (~x 0 ~x ~x 2 ) is the aggregate stock supply at time 2. Now consider time. At time an investor s problem is Max E Max x x2 E 2 U [x (p 2 p ) x 2 (v p 2 )] : 2

14 Substituting from the equilibrium values of x 2, p 2 yields MaxE U (p 2 p )x x 2 x 2 2. (0) Traders at time see di erent information regarding the stock and hence do not all have the same beliefs. As in the static model, to learn from the price the uninformed traders must conjecture a form for the price function, which must be correct in a rational expectations equilibrium. Suppose the uninformed conjecture the price function p = bs c~x e~x 0 () where b, c, and e are coe cients to be determined in the equilibrium. Then upon seeing the equilibrium price and the signal, the informed traders know the precise value of ~x, while the uninformed do not. So to the informed traders, the only random variable in the objective function is ~x 2, while to the uninformed there are two, ~x 2 and s. (Note that ~x is also a random variable to the uninformed, but it is perfectly correlated with s:) The equilibrium can be solved using the standard result on expectations of exponential functions of multivariate normal random variables (for example, see Bray(98) or Brunnermeier (200)), which is given as Lemma 0 in the Appendix. Proposition 6 There exists a partially revealing equilibrium at time in which p = bs c~x e~x 0 where b = c = (' ) ; z '(' ) ; z e = ('2 ) ; z 3

15 where = ( ) 2 ; ' = c b = ( ) 2 ; z = '3 ( )(' ): Proof. See the Appendix Trading Behavior Because now traders are aware of the information revelation at time 2, they need to use their expectations of the time 2 price to make time decisions. Again let denote the random variable observable by the uninformed traders at time. The following proposition, which is parallel to Proposition 2 in the static model, describes traders expectations of price changes. Proposition 7 On average, the informed traders expectation of the time 2 price p 2 is higher than those of the uninformed when there is good news; the converse is true when there is bad news, i.e. E [E(p 2 js) E(p 2 j)js > 0] > 0; E [E(p 2 js) E(p 2 j)js < 0] < 0: Moreover, when there is good news, on average the informed traders expect the price to rise at time 2 while the uninformed traders expect it to fall; when there is bad news, on average the informed traders expect the price to fall at time 2 while the uninformed traders expect it to rise, i.e. E [E(p 2 p js)js > 0] > 0; E [E(p 2 p j)js > 0] < 0; E [E(p 2 p js)js < 0] < 0; E [E(p 2 p j)js < 0] > 0: Proof. See the Appendix. 4

16 The intuition of the above proposition is the same as before. At time, although the uninformed traders can draw inferences from the price and correctly estimate the sign of the signal using Bayes rule, they tend to underestimate the size of it because they can only observe the combination e ect of the signal and the random supply. Therefore, on average the uninformed traders believe that the asset is overpriced when the news is good and underpriced when the news is bad, and will be corrected in the next period. The informed traders, on the other hand, think the price does not yet fully re ect the information of the signal because of the uninformed traders beliefs. They expect the price to go in the same direction in the next period when the uninformed traders see the signal and adjust their beliefs accordingly. So the dynamic structure only changes the magnitudes of the information e ects on traders beliefs, not the signs. Because trading behaviors are determined by traders expectations, one would expect the traders to act the similar way as in the static model. Proposition 8 con rms this intuition and summarizes the trading behaviors of the informed and uninformed traders at time and 2, respectively. Proposition 8 On average, when there is good news, at time the informed traders buy the stock while the uninformed sell, and at time 2 the informed traders sell the stock while the uninformed buy; the converse is true when there is bad news, i.e. E x I x I 0js > 0 > 0; E x U x U 0 js > 0 < 0; E x I 2 x I js > 0 < 0; E x U 2 x U js > 0 > 0; E x I x I 0js < 0 < 0; E x U x U 0 js < 0 > 0; E x I 2 x I js < 0 > 0; E x U 2 x U js < 0 < 0: where x I t and x U t are informed and uninformed traders demands at time t, respectively. 5

17 Proof. See the Appendix Price Momentum As analyzed in the static model, the trading behaviors suggest that the revelation of private information can cause price momentum. Proposition 9 gives the condition under which the price changes are positively correlated. Proposition 9 Let 4p, 4p 2 denote the stock price changes at time and 2, respectively, i.e. 4p = p p 0 ; 4 p 2 = p 2 p. Then Cov(4p ; 4p 2 ) > 0 if and only if > 0: Proof. See the Appendix. The e ects of the fraction of informed traders and the uncertainty of random supplies are more complex than in the static model because of the dynamic structure. Nevertheless we can see clearly from Proposition 9 that if the stock payo is highly volatile (i.e. is small) or the signal is precise (i.e. is large), price momentum would be generated by the revelation of private information. I will test this implication empirically in the next section. 3 Empirical Tests Because private information is not directly observable and measurable, I use the probability of information-based trade (PIN) from Easley et al. (997b) as a proxy for it to empirically examine the relation between private information and the momentum phenomenon. If private information does drive price momentum, then the momentum e ect should be stronger among high PIN stocks. This section proves that it is true for NYSE stocks during the 6

18 period of 983 to 200. Section 3. brie y describes the PIN model. Section 3.2 introduces the data and methodology. Sections 3.3 to 3.7 present the empirical results for PIN-based price momentum strategies. Section 3.3 con rms the price momentum strategy for the sample of rms. Section 3.4 introduces PIN-based price momentum portfolios and examines their predictive power for cross-sectional returns over intermediate horizons. Section 3.5 controls for size and repeats the analysis in 3.4. Section 3.6 reports the subperiod results. Section 3.7 presents results from Fama-French three-factor regression. 3. The PIN Model Following Easley, Hvidkjaer, and O Hara s (2002) structural microstructure model, there are three types of people in the market: informed traders, uninformed traders (or liquidity traders) and competitive market makers. Trade occurs over t = ; :::; T discrete trading days and, within each trading day, trade occurs in continuous time. Information events occur at (and only at) the beginning of the day with probability. If an information event occurs, it is either bad news with probability, or good news with probability. Uninformed traders trade for liquidity reasons. Throughout the day, uninformed buyers arrive with Poisson intensity " b ; uninformed sellers, with intensity " s. Informed traders trade for speculative reasons. If an information event has occurred, informed traders buy if it is good news and sell if it is bad, with Poisson arrival intensity. The market maker sets prices to buy or sell at each time during the day, and then executes orders as they arrive. He observes the trade (either a buy or a sell), and uses this information to update his beliefs. As is standard in microstructure models, the market maker sets trading prices such that his expected losses to informed traders just o set his expected gains from trading with uninformed traders. Easley et al. (996a, 997a, 997b) show that given a particular sequence of trades, these structural models can be estimated via maximum likelihood, providing a method for determining the probability of information-based trading in a given stock. Speci cally, the 7

19 likelihood function induced by this model for the total number of buys and sells on a single trading day is L(jB; S) = ( )e " "B b b B! e "s "S s S! e " "B b b B! e ("s) ( " s) S S! ( )e (" b) ( " b) B e "s "S s B! S!, where B and S represent total buy trades and sell trades for the day respectively, and = (; ; " b ; " s ; ) is the parameter vector. The model assumes that each day the arrivals of an information event and trades, conditional on information events, are drawn from identical and independent distributions. Thus the likelihood function for T days is a product of the above likelihood over days. Following Easley, Hvidkjaer and O Hara (2004), the log likelihood function, after dropping a constant term and rearranging, can be written as L(j(B t ; S t ) T t=) = TX [ " b " s M t (ln x b ln x s ) B t ln( " b ) S t ln( " s )] t= TX ln[( )e x t= St Mt s x Mt b e x Bt Mt b x Mt where M t = min(b t ; S t ) max(b t ; S t )=2, x s = "s " s, x b = s ( )x " b " b. St Mt Bt Mt s x b ], Given the estimates of the model s structural parameters, the probability that the opening trade is information-based, PIN, is PIN = " b " s, which is used as a measure of private information. 8

20 3.2 Data and Methodology I use the same yearly PIN estimates for New York Stock Exchange stocks as in Easley, Hvidkjaer and O Hara (2004). Speci cally, the PIN model is estimated for a sample of all ordinary common stocks listed on NYSE for the years 983 to 200. Buy and sell trades are classi ed according to the Lee-Ready algorithm (see Lee and Ready [99]). I also exclude any rm that was a prime, a closed-end fund, a real estate investment trust (REIT), an American Depository Receipt (ADR), or a foreign company is excluded. A stock in any year in which it did not have at least 60 days with quotes or trades to ensure the reliability of the PIN estimation. Returns for each stock are taken from the CRSP monthly return les. Following the algorithm of Lee and Swaminathan (2000), at the beginning of each month, from January 984 to December 200, all eligible stocks are ranked independently on the basis of past returns and PIN. The stocks are then assigned to one of 0 portfolios based on returns over the previous J months (J=3, 6, 9, or 2) and one of three portfolios based on the PIN measures over the same period. 2 The intersections resulting from the two independent rankings give rise to 30 price momentum-pin portfolios. This paper focuses on the monthly returns of extreme winner and loser deciles over the next K months (K=3, 6, 9, or 2). Previous research has shown that size is an important determinant of excess returns, and Easley, Hvidkjaer, and O Hara (2004) show that PIN and size are highly negatively correlated. So to isolate the e ects of PIN, I also sort stocks into 0 size deciles and apply the above methodology within each decile. As in Jegadeesh and Titman (993), the monthly return for a K-month holding period is based on an equal-weighted average of portfolio returns from strategies implemented in the current month and the previous K- months. For example, the monthly return for a three-month holding period is based on an equal-weighted average of portfolio returns from The yearly PIN estimates are available at /data.htm. I gracefully thank Hvidkjaer, Easely and O Hara for providing PIN estimates. 2 Ideally the PIN measures should be estimated over rolling J-month windows to match the returns, but this kind of estimation is extremely time-consuming. Moreover, the results derived from yearly PIN estimates are signi cant enough to show the positive correlation between PIN and price momentum. 9

21 this month s strategy, last month s strategy, and the strategy from two months ago. This is equivalent to revising the weights of one-third of the portfolio each month and carrying over the rest from the previous month. 3.3 Price Momentum This subsection con rms the price momentum for the sample of rms. Table I summarizes results from several price momentum portfolio strategies. Each January, stocks are ranked and grouped into decile portfolios on the basis of their returns over the previous three, six, nine, and 2 months. The results are reported for the bottom decile portfolio of extreme losers (R), the top decile of extreme winners (R0), and one intermediate portfolio (R5). The other intermediate portfolio results are consistent with ndings in prior papers (Jegadeesh and Titman (993)) and are omitted for simplicity of presentation. For each portfolio, Table I reports the equal-weighted average monthly returns over the next K months (K = 3, 6, 9, 2). In addition, for each portfolio formation period (J) and holding period (K), it reports the mean return from a dollar-neutral strategy of buying the extreme winners and selling the extreme losers (R0 R). These results con rm the presence of price momentum in the sample. For example, with a six-month portfolio formation period (J = 6), past winners gain an average of :45 percent per month over the next nine months (K = 9). Past losers gain an average of only 0:49 percent per month over the same time period. The di erence between R0 and R is 0:96 percent per month. The di erence in average monthly returns between R0 and R is signi cantly positive in all (J, K) combinations. In sum, Table I con rms prior ndings on price momentum. 3.4 PIN-Based Price Momentum This subsection introduces PIN-based price momentum portfolios and examines their predictive power for cross-sectional returns over intermediate horizons. Table II reports returns to portfolios formed on the basis of a two-way sort between price momentum and PIN mea- 20

22 sures. To create this table, I sort all sample rms at the beginning of each month based on their returns over the past J months and divide them into 0 portfolios (R to R0). I then independently sort these same rms based on their PIN and divide them into three PIN portfolios (P to P 3). P represents the lowest PIN portfolio, and P 3 represents the highest PIN portfolio. Table values represent the average monthly return over the next K months (K = 3, 6, 9, 2). Several key results emerge from Table II. First, among the winner (R0) portfolios, high PIN stocks generally do better than low PIN stocks over the next 2 months. This is seen in the consistently positive returns to the portfolios at the intersections of R0 and P 3 P. For example, with a six-month portfolio formation period and three-month holding period (J = 6; K = 3), high PIN winners (R0P 3) outperform low PIN winners (R0P ) by 0:80 percent per month. Similar results can be found in every (J, K) cell. Apparently, winner stocks that have high PINs in the recent past tend to outperform winner stocks with low PINs. The nding that high PIN rms earn higher expected returns is consistent with Easley, Hvidkjaer, and O Hara s (2002). In that paper, this nding is interpreted as evidence that high PIN rms command a greater information risk premium. However, Table II also contains evidence that seems to con ict with the Easley et al. s conclusion. In all (J, K) cells, returns to the portfolios at the intersections of R and P 3 P are negative. This implies that among the loser (R) portfolios, high PIN stocks generally do worse than low PIN stocks over the next 2 months. For example, with a six-month portfolio formation period and three-month holding period (J = 6; K = 3), low PIN losers (RP ) outperform high PIN losers (RP 3) by 0:93 percent per month. As I will show in Section 3.5, the nding that high PIN rms earn lower expected returns is mainly driven by large stocks. This is consistent with Easley et al. (2004) s suggestion that PIN only a ects expected returns among small stocks. The most important nding arises when we focus on the bottom row of Table II. The bottom row of each cell shows the return to a dollar-neutral price momentum strategy 2

23 (R0 R). It is clear that R0 R returns are higher for high PIN rms than for low PIN rms. For example, for J = 6 and K = 6, the price momentum spread is :47 percent for high PIN rms and only 0:20 percent for low PIN rms. The di erence of :27 percent per month is both economically and statistically signi cant. The other cells illustrate qualitatively the same e ect. The price momentum premium is clearly higher for high PIN (presumably greater private information) rms. In sum, Table II shows that over the next 2 months, price momentum is more pronounced among high PIN stocks, which strongly supports our asset pricing model s implication that private information generates price momentum. 3.5 PIN-Based Price Momentum: Controlling for Size Previous research has shown that size is an important determinant of excess returns, and Easley, Hvidkjaer, and O Hara (2004) show that PIN and size are highly negatively correlated. So to isolate the e ects of PIN, this subsection examines the relation between PIN and the momentum e ect after controlling for size. Speci cally, at the beginning of each month all available stocks are sorted into 0 size deciles based on market capitalization at the end of the prior year, and within each size decile, three portfolios are formed based on the PIN measures. 3 P represents the lowest PIN portfolio, and P 3 represents the highest PIN portfolio. The stocks are also independently sorted based on past J month returns and divided into 0 portfolios. R represents the loser portfolio, and R0 represents the winner portfolio. Within each size decile, the stocks at the intersection of the sort of PINs and the sort of past returns are grouped together to form portfolios. Figure I summarizes how the winner and loser stocks based on previous three months returns are distributed over the size deciles. Table III reports returns to portfolios formed on the basis of last three months returns and PINs for size deciles (smallest), 3, 5, 8 and 0 (largest). Results for other 3 As pointed out by Easley, Hvidkjaer, and O Hara (2004), due to the strong negative correlation between size and PIN, independent sorts into size and PIN portfolios provide too few rms in the <large size, high PIN> and <small size, low PIN> cells to be useful. 22

24 formation periods and size deciles are similar and are omitted for simplicity of presentation. Figure I shows that both extremely winners and losers have heavy tails on the left end. Small size stocks contain more winners and losers, but in a moderate way that in each size decile the number of winner and loser stocks are large enough to derive useful results. Table III results suggest that our earlier nding, that low PIN stocks outperform high PIN stocks among loser portfolios, is driven by large stocks. Among the smallest stocks (Panel A), for all K values, returns to the portfolios in P 3 P columns are positive. This implies that for small stocks, high PIN stocks generally do better than low PIN stocks over the next 2 months among no matter winner or loser portfolios. In contrast, among large stocks (Panel D and E), returns to the portfolios at the intersections of R and P 3 P are all negative and frequently signi cant. So for large stocks, among the loser (R) portfolios, high PIN stocks generally do worse than low PIN stocks over the next 2 months. This is consistent with Easley et al. (2004) s suggestion that PIN only a ects expected returns among small stocks. More importantly, the nding that price momentum is more pronounced among high PIN stocks is still valid in Table III. As before, the bottom row of each cell shows the return to a dollar-neutral price momentum strategy (R0 R). It is clear that R0 R returns are higher for high PIN rms than for low PIN rms in all panels. For example, in Panel B (size decile 3), when K = 6 the price momentum spread is 2:5 percent for high PIN rms and only 0:5 percent for low PIN rms. The di erence of 2:00 percent per month is both economically and statistically signi cant. The other cells in Panels A to D (size deciles, 3, 5, and 8) illustrate qualitatively the same e ect. In Panel E (size decile 0), the momentum returns show the same patterns, but the di erences are not statistically signi cant. This may be explained by the fact that PIN does not work well with large stocks. In sum, Table III shows that over the next 2 months, price momentum is stronger among high PIN stocks even after controlling for size. 23

25 3.6 Robostness Tests Table IV con rms the PIN-Momentum patterns for three subperiods. The rst subperiod spans 984 to 989, the second subperiod covers 990 to 995, and the last subperiod covers 996 to 200. Table IV only reports the results for the three-month formation period, but results are similar for other formation periods. In all three subperiods, momentum is stronger among high PIN stocks. In fact, the result is strongest in the more recent subperiod. 3.7 Fama-French Factor Regression Test Table V provides evidence on the source of abnormal returns for the various PIN-based price momentum strategies in Section 3.4. It reports the results from time-series regression using monthly portfolio returns: r i r f = a i b i (r m r f ) s i SMB h i HML " i where r i is the monthly return for portfolio i; r f is the monthly return on one-month T-bill; r m is the value-weighted return on the NYSE/AMEX/Nasdaq market index; SMB and HML are the Fama-French small rm factor and book-to-market factor, respectively; b i, s i, h i are the corresponding factor loadings; and a i is the intercept coe cient. For parsimony, results are only reported for the portfolios with six-month formation and six-month holding periods (J = 6, K = 6). The rst cell on the left reports the estimated intercept coe cient; the subsequent cells report estimated coe cients for b i, s i, and h i, respectively. The last cell reports the adjusted R 2. The estimated intercept coe cients from these regressions (a i ) are interpretable as the risk-adjusted return of the portfolio relative to the three-factor model. Focusing on these intercepts, it is clear that the earlier results cannot be explained by the Fama-French factors. The intercepts corresponding to R0 R are positive across all three PIN categories. The return di erential between winners and losers remains much higher for high PIN (P 3) rms 24

26 than for low PIN (P ) rms. 4 Conclusion This paper examines the implications of asymmetric information for price evolution and investor behavior under a rational expectations framework. In the asymmetric informationbased asset-pricing model, at the arrival of the private information, informed and uninformed traders disagree on the expected return of the asset and thus hold di erent portfolios. The informed traders believe that the price does not fully re ect the news, and trade on the news. The uninformed ones, conversely, trade against the news. When the news is completely revealed, the uninformed traders adjust their beliefs accordingly and act as trend chasers, while the informed act as contrarians. Thus the asset price can move continuously towards the direction of the news. In other words, the revelation of the private information can generate price momentum. To empirically explore the role of private information in the prediction of cross-sectional stock returns, PIN is adopted as a proxy of information asymmetry. The price momentum premium is signi cantly higher for high PIN rms. This implies that information and momentum phenomenon are closely related, which is highly consistent with the model. A natural question to ask is whether one could make money with a PIN-based momentum strategy. As pointed out by Easley and O Hara (2004), assets with greater private information command a risk premium because asymmetric information creates a risk for uninformed traders. Therefore, the abnormal return to a PIN-based momentum strategy is associated with information risk, and a risk-averse investor needs to carefully examine the risk level to determine whether a high expected return from such a strategy would lead to a high expected utility. 25

27 Appendix Lemma 0 If X is a n-dimensional multivariate normal random variable with mean and variance-covariance matrix, A is a symmetric n n matrix, B is a n-dimensional vector and C is a scalar, then we have E[exp(X 0 AX B 0 X C)] = ji 2Aj 2 exp[c B 0 0 A 2 (B 2A)0 ( 2A) (B 2A)]; where I is the identity matrix. Proof of Proposition. It is su cient to show that there is an equilibrium price of the form given in the statement of the proposition. Equating mean per capita demand by informed and uninformed traders to per capita supply gives m s p ( ) m p ( ( ) ) = ~x 0 ~x So, p = m s( ( ) c ) ~x ( ) b c x ( ) ~x b 0 (2) ( ) Equating (7) and (5) gives the coe cient values in the statement of Proposition. Q.E.D. Proof of Proposition 2. By (3) and (6), = = E(vjs) E(vj) m s m ( ) ( )( ) (s m) ( ) (~x x ) 26

28 where Obviously " # 2 = ; = s > 0. Therefore (~x x ): E[E(vjs) E[E(vjs) E(vj)js > m] > E ( ) (~x x ) = 0; E(vj)js < m] < E ( ) (~x x ) = 0: Q.E.D. Proof of Proposition 3. For informed traders, x I 0 = x U 0 = ~x 0 ; x I = E(vjs) p V ar(vjs) = [m s (am bs c~x dx e~x 0 )( )] where the values of a, b, c, d and e are as given in Proposition. So x I x I 0 = [ b( )] (s m) ( ) (c~x dx e~x 0 ) ~x 0 : Therefore E x I x I 0js > m = [ b( )] E [s mjs > m] > 0 since x t = 0 (t = 0; ; and 2) and b( ) > 0. Similarly E x I x I 0js < m = [ b( )] E [s mjs < m] < 0: 27

29 For uninformed traders, x U = E(vj) p V ar(vj) = m s (~x x ) (am bs c~x dx e~x 0 )( ) So x U x U 0 = [b( ) ] (s m) (~x x ) ( ) (c~x dx ) ~x 0 : Therefore E x U x U 0 js > m = [b( ) ] E [s mjs > m] < 0 since x t = 0 (t = 0; ; and 2) and b( ) > 0. Similarly E x U x U 0 js < m = [b( ) ] E [s mjs < m] > 0: Q.E.D. Proof of Proposition 4. For informed traders, x I is as given in the proof of Proposition 3, x I 2 = x U 2 = ~x 0 ~x ~x 2. So x I 2 x I = (~x 0 ~x ~x 2 ) ( ) (c~x dx e~x 0 ) [ b( )] (s m): Therefore E x I 2 x I js > m = [ b( )] E [s mjs > m] < 0 28

30 since x t = 0 (t = 0; ; and 2) and b( ) > 0. Similarly E x I 2 x I js < m = [ b( )] E [s mjs < m] > 0: For uninformed traders, x U is as given in the proof of Proposition 3, thus x U 2 x U = (~x 0 ~x ~x 2 ) [b( ) ] (s m); (~x x ) ( ) (c~x dx ) So E x U 2 x U js > m = [b( ) ] E [s mjs > m] > 0 since x t = 0 (t = 0; ; and 2) and b( ) > 0. Similarly E x U 2 x U js < m = [b( ) ] E [s mjs < m] < 0: Q.E.D. Proof of Proposition 5. p 0, p and p 2 are given in equation (2), Propositioin and equation (4), respectively. So 4p = p p 0 = (a )m bs c~x dx e ~x 0 ; 4p 2 = p 2 p = e a m ~x 0 c b s ~x ~x 2; 29

31 where the values of a, b, c, d and e are as given in Proposition. Cov(4p ; 4p 2 ) = b e b V ar(s) e c c V ar(~x 0 ) V ar(~x ) Because we have V ar(s) = ; V ar(~x 0) = 0 ; V ar(~x ) = ; Cov(4p ; 4p 2 ) = 0 : Q.E.D. Proof of Proposition 6. At time any trader s objective function is given by (0), where x 2 and p 2 are given by equations (8) and (9), respectively. Because the informed traders can observe s and therefore precisely infer the value of ~x, the only unknown random variable for them in the objective function is ~x 2. Lemma 0 and straightforward calculation shows that an informed trader s demand at time is x I = 3 s ( ) 3 p ( ) 2 (~x 0 ~x ) where is given in the statement of Proposition 6. Now consider the uninformed traders. Because of their conjecture about p (), ~x and s are perfectly correlated for the uninformed. So there are two random variables in the objective function: ~x 2 and s. Substituting (8), (9) and s = p c~x e~x 0 ; b we can rewrite (0) as MaxE [U(X 0 AX B 0 X C)] x 30

32 where X = A = B = C = ~x 27 5 ; s 2( ) b c 7 5 ; b c b 2 c 2 3 x b x c b c ( e ~x 0 c e ~x0 c e c c p ~x0 c p c ; p x 2 c p e ) 2 ~x 0 : c Let denote the random variable observable by the uninformed traders: = p e~x 0 b = s c b ~x : To compute x we need the mean and the variance-covariance matrix (X) = (X) = E(~x 2j) 7 5 E(sj) V ar(~x 2j) : 0 V ar(sj) ~x 2 is independent of any observable variable at time, hence ~x 2 j s N(0; =). The the distribution of s is N, and the distribution of conditional on s is N(s; ), where 0; c 2 = : b 3

33 So the conditional distribution of s given is 0 B sj s ; C A : Now we can apply Lemma 0 and derive the expression of x U, an uninformed trader s demand of the stock at time, in terms of the to-be-determined price conjecture coe cients b, c, e and the known coe cients,,,, and. Then the market clearing condition x I ( )x U = ~x 0 ~x yields p in terms of the coe cients including b, c, e. Equating this expression of p and the right hand side of () gives the coe cient values in the statement of Proposition 6. Q.E.D. Proof of Proposition 7. From equation (9) we know p 2 = s (~x 0 ~x ~x 2 ) : So an informed trader s expectation of p 2 at time is simply E(p 2 js) = s (~x 0 ~x ) : For an uninformed trader, E(sj) is given in the proof of Proposition 6. Because j~x s c N ~x b ;, Bayes rule implies that E(~x j) = (s '~x ) = (s '~x ) ( ) : 32

34 So s (~x0 ~x ~x 2 ) E(p 2 j) = E j = [E(sj) ~x 0 E(~x j)] ( ) = ( ) ' 2 (s '~x (s '~x ) ) ~x 0 ( ) ( ) ~x 0 = (s '~x ( ) ' 2 ) ( ) where ' is given in the statement of Proposition 6 and is given in the proof of the proof of Proposition 6. Therefore E(p 2 js) E(p 2 j) = = ( ) ( ) ' 2 s ( ) '( ) ' ~x ( ) ' 2 ( ) ' 2 ( ) ' 2 s ( ) ~x '( ) ( ) ' 2 ' ( ) The coe cient of s is positive and ~x is independent of s with mean 0, so we have E [E(p 2 js) E(p 2 j)js > 0] > 0; E [E(p 2 js) E(p 2 j)js < 0] < 0: To prove the second part of the proposition, note that E(p 2 p js) = E(p 2 js) p : 33

35 Substituting coe cient values given in Proposition 6 yields E(p 2 p js) = = b s e ~x 0 c ( )'3 ( )'4 s ( )z ( )z ~x 0 ( ~x )3 [' ( )] ~x : ( )z Similarly E(p 2 p j) = E(p 2 j) p ( ) = ( ) ' 2 e ~x 0 ( ) b (s '~x ) = '2 fz ( )( ') [( ) ]g 2 ( ) 2 ( ) [( ) ' 2 ] [( ) ] z (s '~x ) ( )'4 ( )z ~x 0; The signs of the coe cients of s s lead to the corresponding conclusions in Proposition 7. Q.E.D. Proof of Proposition 8. The statement and the proof of 6 imply that x I = 3 s ( ) 3 p ( ) 2 (~x 0 ~x ) = 3 [ ( )b]s 3 ( )(c )~x 3 ( )(e )~x 0 ( ) ' = s ( ) '( 2 ) ~x z z ( ) '( 2 ) ~x 0 : z So x U = ' z s '2 z ~x ['2 ( )] ~x 0 : z 34