2148 The Journal of Finance horizon results-such as return reversals,the book-to-market effect,and the cashflow-to-price effect-can be largely subsumed within a three-factor model that they interpret as a variant of the APT or ICAPM.However,this position has been controversial,since there is little affirmative evidence that the Fama-French factors correspond to economically meaningful risks.Indeed, several recent papers demonstrate that the contrarian strategies that ex- ploit long-horizon overreaction are not significantly riskier than average.10 There seems to be more of a consensus that the short-horizon underreaction evidence cannot be explained in terms of risk.Bernard and Thomas(1989)re- ject risk as an explanation for post-earnings-announcement drift,and Fama and French(1996)remark that the continuation results of Jegadeesh and Tit- man(1993)constitute the "main embarrassment"for their three-factor model. IⅡ.The Model A.Price Formation with Newswatchers Only As mentioned above,our model features two classes of traders,newswatch- ers and momentum traders.We begin by describing how the model works when only the newswatchers are present.At every time t,the newswatchers trade claims on a risky asset.This asset pays a single liquidating dividend at some later time T.The ultimate value of this liquidating dividend can be written as:Dr=Do+,where all the e's are independently distrib- uted,mean-zero normal random variables with variance o2.Throughout,we consider the limiting case where T goes to infinity.This simplifies matters by allowing us to focus on steady-state trading strategies-that is,strat- egies that do not depend on how close we are to the terminal date.11 In order to capture the idea that information moves gradually across the news- watcher population,we divide this population into z equal-sized groups.We also assume that every dividend innovation e;can be decomposed into z in- dependent subinnovations,each with the same variance 2/:=+...+ef. The timing of information release is then as follows.At timet,news about e+-1 begins to spread.Specifically,at time t,newswatcher group 1 observes e+-1, group 2 observes e2-1,and so forth,through group z,which observes e+-1. Thus at time t each subinnovation ofe+has been seen by a fraction 1/z of the total population. Next,at time t +1,the groups "rotate,"so that group 1 now observes e-1,group 2 observes e1,and so forth,through group z,which now observes e-1.Thus at time t+1 the information has spread further,and 10 See Lakonishok et al.(1994)and MacKinlay (1995).Daniel and Titman(1997)directly dispute the idea that the book-to-market effect can be given a risk interpretation. 11 A somewhat more natural way to generate an infinite-horizon formulation might be to allow the asset to pay dividends every period.The only reason we push all the dividends out into the infinite future is for notational simplicity.In particular,when we consider the strat- egies of short-lived momentum traders below,our approach allows us to have these strategies depend only on momentum traders'forecasts of price changes,and we can ignore their forecasts of interim dividend payments.horizon results—such as return reversals, the book-to-market effect, and the cashflow-to-price effect—can be largely subsumed within a three-factor model that they interpret as a variant of the APT or ICAPM. However, this position has been controversial, since there is little affirmative evidence that the Fama–French factors correspond to economically meaningful risks. Indeed, several recent papers demonstrate that the contrarian strategies that ex￾ploit long-horizon overreaction are not significantly riskier than average.10 There seems to be more of a consensus that the short-horizon underreaction evidence cannot be explained in terms of risk. Bernard and Thomas ~1989! re￾ject risk as an explanation for post-earnings-announcement drift, and Fama and French ~1996! remark that the continuation results of Jegadeesh and Tit￾man ~1993! constitute the “main embarrassment” for their three-factor model. II. The Model A. Price Formation with Newswatchers Only As mentioned above, our model features two classes of traders, newswatch￾ers and momentum traders. We begin by describing how the model works when only the newswatchers are present. At every time t, the newswatchers trade claims on a risky asset. This asset pays a single liquidating dividend at some later time T. The ultimate value of this liquidating dividend can be written as: DT 5 D0 1 (j50 T ej , where all the e’s are independently distrib￾uted, mean-zero normal random variables with variance s2. Throughout, we consider the limiting case where T goes to infinity. This simplifies matters by allowing us to focus on steady-state trading strategies—that is, strat￾egies that do not depend on how close we are to the terminal date.11 In order to capture the idea that information moves gradually across the news￾watcher population, we divide this population into z equal-sized groups. We also assume that every dividend innovation ej can be decomposed into z in￾dependent subinnovations, each with the same variance s2 0z: ej 5 ej 11{{{1ej z . The timing of information release is then as follows. At time t, news about et1z21 begins to spread. Specifically, at time t, newswatcher group 1 observes et1z21 1 , group 2 observes et1z21 2 , and so forth, through group z, which observes et1z21 z . Thus at time t each subinnovation of et1z21 has been seen by a fraction 10z of the total population. Next, at time t 1 1, the groups “rotate,” so that group 1 now observes et1z21 2 , group 2 observes et1z21 3 , and so forth, through group z, which now observes et1z21 1 . Thus at time t 1 1 the information has spread further, and 10 See Lakonishok et al. ~1994! and MacKinlay ~1995!. Daniel and Titman ~1997! directly dispute the idea that the book-to-market effect can be given a risk interpretation. 11 A somewhat more natural way to generate an infinite-horizon formulation might be to allow the asset to pay dividends every period. The only reason we push all the dividends out into the infinite future is for notational simplicity. In particular, when we consider the strat￾egies of short-lived momentum traders below, our approach allows us to have these strategies depend only on momentum traders’ forecasts of price changes, and we can ignore their forecasts of interim dividend payments. 2148 The Journal of Finance