Underreaction,Momentum Trading,and Overreaction 2153 momentum traders is sufficiently small,since this in turn ensures that is sufficiently small.Moreover,detailed experimentation suggests that a unique covariance-stationary equilibrium does in fact exist for a large range of the parameter space.18 In general,it is difficult to solve the model in closed form,and we have to resort to a computational algorithm to find the fixed point.For an arbitrary set of parameter values,we always begin our numerical search for the fixed point at j=1.Given this restriction,we can show that the condition <1 is both necessary and sufficient for covariance-stationarity.We also start with a small value of risk tolerance and an initial guess for d of zero.The solutions in this region of the parameter space are well behaved.Using these solutions,we then move to other regions of the parameter space.This pro- cedure ensures that if there are multiple covariance-stationary equilibria, we would always pick the one with the smallest value of We also have a number of sensible checks for when we have moved outside the covariance- stationary region of the parameter space.These are described in Appen- dix A. Even without doing any computations,we can make several observations about the nature of equilibrium.First,we have the following lemma. LEMMA 1:In any covariance-stationary equilibrium,b>0.That is,momen- tum traders must rationally behave as trend-chasers. The lemma is proved in Appendix A,but it is trivially easy to see why =0 cannot be an equilibrium.Suppose to the contrary it is.Then prices are given as in the all-newswatcher case in equation(1).And in this case, cov(P+i-P,AP:-1)>0,so that equation (7)tells us that >0,estab- lishing a contradiction. We are now in a position to make some qualitative statements about the dynamics of prices.First,let us consider the impulse response of prices to news shocks.The thought experiment here is as follows.At time t,there is a one-unit positive innovation e+1 that begins to diffuse among news- watchers.There are no further news shocks from that point on.What does the price path look like? The answer can be seen by decomposing the price at any time into two components:that attributable to newswatchers,and that attributable to mo- mentum traders.Newswatchers'aggregate estimate of Dr rises from time t to time t +z-1,by which time they have completely incorporated the news shock into their forecasts.Thus,by time t+z-1 the price is just right in the absence of any order flow from momentum traders.But with >0,any positive news shock must generate an initially positive impulse to momentum- 1s Our experiments suggest that we only run into existence problems when both the risk tolerance and the information-diffusion parameter z simultaneously become very large-even an infinite value of y poses no problem so long as z is not too big.The intuition will become clearer when we do the comparative statics,but loosely speaking,the problem is that as z gets large,momentum trading becomes more profitable.Combined with high risk tolerance,this can make momentum traders behave so aggressively that our<1 condition is violated
momentum traders is sufficiently small, since this in turn ensures that 6f6 is sufficiently small. Moreover, detailed experimentation suggests that a unique covariance-stationary equilibrium does in fact exist for a large range of the parameter space.18 In general, it is difficult to solve the model in closed form, and we have to resort to a computational algorithm to find the fixed point. For an arbitrary set of parameter values, we always begin our numerical search for the fixed point at j 5 1. Given this restriction, we can show that the condition 6f6 , 1 is both necessary and sufficient for covariance-stationarity. We also start with a small value of risk tolerance and an initial guess for f of zero. The solutions in this region of the parameter space are well behaved. Using these solutions, we then move to other regions of the parameter space. This procedure ensures that if there are multiple covariance-stationary equilibria, we would always pick the one with the smallest value of f. We also have a number of sensible checks for when we have moved outside the covariancestationary region of the parameter space. These are described in Appendix A. Even without doing any computations, we can make several observations about the nature of equilibrium. First, we have the following lemma. LEMMA 1: In any covariance-stationary equilibrium, f . 0. That is, momentum traders must rationally behave as trend-chasers. The lemma is proved in Appendix A, but it is trivially easy to see why f 5 0 cannot be an equilibrium. Suppose to the contrary it is. Then prices are given as in the all-newswatcher case in equation ~1!. And in this case, cov~Pt1j 2 Pt,DPt21! . 0, so that equation ~7! tells us that f . 0, establishing a contradiction. We are now in a position to make some qualitative statements about the dynamics of prices. First, let us consider the impulse response of prices to news shocks. The thought experiment here is as follows. At time t, there is a one-unit positive innovation et1z21 that begins to diffuse among newswatchers. There are no further news shocks from that point on. What does the price path look like? The answer can be seen by decomposing the price at any time into two components: that attributable to newswatchers, and that attributable to momentum traders. Newswatchers’ aggregate estimate of DT rises from time t to time t 1 z 2 1, by which time they have completely incorporated the news shock into their forecasts. Thus, by time t 1 z 2 1 the price is just right in the absence of any order flow from momentum traders. But with f . 0, any positive news shock must generate an initially positive impulse to momentum- 18 Our experiments suggest that we only run into existence problems when both the risk tolerance and the information-diffusion parameter z simultaneously become very large—even an infinite value of g poses no problem so long as z is not too big. The intuition will become clearer when we do the comparative statics, but loosely speaking, the problem is that as z gets large, momentum trading becomes more profitable. Combined with high risk tolerance, this can make momentum traders behave so aggressively that our 6f6 , 1 condition is violated. Underreaction, Momentum Trading, and Overreaction 2153
2154 The Journal of Finance trader order flow.Moreover,the cumulative order flow must be increasing until at least time t+j,since none of the momentum trades stimulated by the shock begin to be unwound until t+j+1.This sort of reasoning leads to the conclusions stated in the following proposition. PRoposrTIoN 1:In any covariance-stationary equilibrium,given a positive one- unit shockethat first begins to diffuse among newswatchers at time t: (i)There is always overreaction,in the sense that the cumulative impulse response of prices peaks at a value that is strictly greater than one. (ii)If the momentum traders'horizon j satisfies j=z-1,the cumulative impulse response peaks at t +j and then begins to decline,eventually converging to one. (iii)Ifj<z-1,the cumulative impulse response peaks no earlier than t+j,and eventually converges to one. In addition to the impulse response function,it is also interesting to con- sider the autocovariances of prices at various horizons.We can develop some rough intuition about these autocovariances by considering the limiting case where the risk tolerance of the momentum traders y goes to infinity.In this case,equation(7)implies that the equilibrium must have the property that cov(P+i-P,AP:-1)=0.Expanding this expression,we can write cov(AP+1,△P-1)+cov(△P+2,△P-)+.…+cov(△P+j,△P-1)=0. (8) Equation(8)allows us to state the following proposition. ProposrTIoN 2:In any covariance-stationary equilibrium,if price changes are positively correlated at short horizons(e.g.,if cov(AP1,AP)>0),then with risk-neutral momentum traders they are negatively correlated at a horizon no longer thanj+1-in other words,it must be that cov(AP+i,AP1)<0for some i≤j It is useful to explore the differences between Propositions 1 and 2 in some detail,since at first glance it might appear that they are somewhat contra- dictory.On the one hand,Proposition 1 says that in response to good news there is continued upward momentum in prices for at least j periods,and possibly more(ifj<z-1).On the other hand,Proposition 2 suggests that price changes begin to be reversed within j+1 periods,and quite possibly sooner than that. The two propositions can be reconciled by noting that the former is a con- ditional statement-that is,it talks about the path of prices from time t onward,conditional on there having been a news shock at time t.Thus Prop- osition 1 implies that if a trader somehow knows for sure that there is a news shock at time t,he could make a strictly positive expected profit by buying at this time and holding until time t +j.One might term such a strategy "buying early in the momentum cycle"-that is,buying immedi- ately on the heels of news arrival.But of course,such a strategy is not available to the momentum traders in our model because they cannot con- dition directly on the e's
trader order flow. Moreover, the cumulative order flow must be increasing until at least time t 1 j, since none of the momentum trades stimulated by the shock begin to be unwound until t 1 j 1 1. This sort of reasoning leads to the conclusions stated in the following proposition. PROPOSITION 1: In any covariance-stationary equilibrium, given a positive oneunit shock et1z21 that first begins to diffuse among newswatchers at time t: (i) There is always overreaction, in the sense that the cumulative impulse response of prices peaks at a value that is strictly greater than one. (ii) If the momentum traders’ horizon j satisfies j $ z 2 1, the cumulative impulse response peaks at t 1 j and then begins to decline, eventually converging to one. (iii) If j , z 2 1, the cumulative impulse response peaks no earlier than t 1 j, and eventually converges to one. In addition to the impulse response function, it is also interesting to consider the autocovariances of prices at various horizons. We can develop some rough intuition about these autocovariances by considering the limiting case where the risk tolerance of the momentum traders g goes to infinity. In this case, equation ~7! implies that the equilibrium must have the property that cov~Pt1j 2 Pt,DPt21! 5 0. Expanding this expression, we can write cov~DPt11,DPt21! 1 cov~DPt12,DPt21!1{{{1cov~DPt1j ,DPt21! 5 0. ~8! Equation ~8! allows us to state the following proposition. PROPOSITION 2: In any covariance-stationary equilibrium, if price changes are positively correlated at short horizons (e.g., if cov~DPt11,DPt21! . 0!, then with risk-neutral momentum traders they are negatively correlated at a horizon no longer than j 11—in other words, it must be that cov~DPt1i,DPt21! , 0 for some i # j. It is useful to explore the differences between Propositions 1 and 2 in some detail, since at first glance it might appear that they are somewhat contradictory. On the one hand, Proposition 1 says that in response to good news there is continued upward momentum in prices for at least j periods, and possibly more ~if j , z 2 1!. On the other hand, Proposition 2 suggests that price changes begin to be reversed within j 1 1 periods, and quite possibly sooner than that. The two propositions can be reconciled by noting that the former is a conditional statement—that is, it talks about the path of prices from time t onward, conditional on there having been a news shock at time t. Thus Proposition 1 implies that if a trader somehow knows for sure that there is a news shock at time t, he could make a strictly positive expected profit by buying at this time and holding until time t 1 j. One might term such a strategy “buying early in the momentum cycle”—that is, buying immediately on the heels of news arrival. But of course, such a strategy is not available to the momentum traders in our model because they cannot condition directly on the e’s. 2154 The Journal of Finance
Underreaction,Momentum Trading,and Overreaction 2155 In contrast,Proposition 2 is an unconditional statement about the auto- covariance of prices.It flows from the requirement that if a trader buys at time t in response to an unconditional price increase at time t-1,and then holds until t +j,he makes zero profits on average.This zero-profit require- ment in turn must hold when momentum traders are risk-neutral,because the unconditional strategy is available to them. There is a simple reason why an unconditional strategy of buying follow- ing any price increase does not work as well as the conditional strategy of buying only following directly observed good news:Not all price increases are news-driven.In particular,a trader who buys based on a price increase observed at time t runs the following risk.It may be "late"in the momentum cycle,in the sense that there has not been any good news for the last several periods.Say the last good news hit at t-i.If this is the case,the price increase at time t is just due to a late round of momentum buying.And those earlier momentum purchases kicked off by the news at t-i will begin to unwind in the very near future (specifically,at t-i+j+1),causing the trader to experience losses well before the end of his trading horizon. This discussion highlights the key spillover effect that drives our results. A momentum trader who is fortunate enough to buy shortly after the arrival of good news imposes a negative externality on those that follow him.He does so by creating a further price increase that the next generation par- tially misinterprets as more good news.This causes the next generation to buy,and so on.At some point,the buying has gone too far,and the price overshoots the level warranted by the original news.Given the inability of momentum traders to condition directly on the e's,everybody in the chain is behaving as rationally as possible,but the externality creates an apparently irrational outcome in the market as a whole. D.Winners and Losers A natural question is whether the bounded rationality of either the news- watchers or the momentum traders causes them to systematically lose money In general,both groups can earn positive expected returns as long as the net supply Q of the asset is positive.Consider first the case where Q=0.In this case,it can be shown that the momentum traders earn positive returns,as long as their risk tolerance is finite.Because with Q=0,this is a zero-sum game,it must therefore be that the newswatchers lose money.The one ex- ception is when momentum traders are risk-neutral,and both groups break even.19 When Q>0,the game becomes positive-sum,as there is a return to risk- sharing that can be divided between the two groups.Thus,even though the newswatchers may effectively lose some money on a trading basis to the momentum traders.this can be more than offset by their returns from risk- 19 This result is related to the fact that newswatchers have time-inconsistent strategies,so that in formulating their demands they ignore the fact that they will be transacting with mo- mentum traders who will be trying to take advantage of them.Thus,in some sense,the news- watchers are more irrational than the momentum traders in this model
In contrast, Proposition 2 is an unconditional statement about the autocovariance of prices. It flows from the requirement that if a trader buys at time t in response to an unconditional price increase at time t 2 1, and then holds until t 1 j, he makes zero profits on average. This zero-profit requirement in turn must hold when momentum traders are risk-neutral, because the unconditional strategy is available to them. There is a simple reason why an unconditional strategy of buying following any price increase does not work as well as the conditional strategy of buying only following directly observed good news: Not all price increases are news-driven. In particular, a trader who buys based on a price increase observed at time t runs the following risk. It may be “late” in the momentum cycle, in the sense that there has not been any good news for the last several periods. Say the last good news hit at t 2 i. If this is the case, the price increase at time t is just due to a late round of momentum buying. And those earlier momentum purchases kicked off by the news at t 2 i will begin to unwind in the very near future ~specifically, at t 2 i 1 j 1 1!, causing the trader to experience losses well before the end of his trading horizon. This discussion highlights the key spillover effect that drives our results. A momentum trader who is fortunate enough to buy shortly after the arrival of good news imposes a negative externality on those that follow him. He does so by creating a further price increase that the next generation partially misinterprets as more good news. This causes the next generation to buy, and so on. At some point, the buying has gone too far, and the price overshoots the level warranted by the original news. Given the inability of momentum traders to condition directly on the e’s, everybody in the chain is behaving as rationally as possible, but the externality creates an apparently irrational outcome in the market as a whole. D. Winners and Losers A natural question is whether the bounded rationality of either the newswatchers or the momentum traders causes them to systematically lose money. In general, both groups can earn positive expected returns as long as the net supply Q of the asset is positive. Consider first the case where Q 5 0. In this case, it can be shown that the momentum traders earn positive returns, as long as their risk tolerance is finite. Because with Q 5 0, this is a zero-sum game, it must therefore be that the newswatchers lose money. The one exception is when momentum traders are risk-neutral, and both groups break even.19 When Q . 0, the game becomes positive-sum, as there is a return to risksharing that can be divided between the two groups. Thus, even though the newswatchers may effectively lose some money on a trading basis to the momentum traders, this can be more than offset by their returns from risk- 19 This result is related to the fact that newswatchers have time-inconsistent strategies, so that in formulating their demands they ignore the fact that they will be transacting with momentum traders who will be trying to take advantage of them. Thus, in some sense, the newswatchers are more irrational than the momentum traders in this model. Underreaction, Momentum Trading, and Overreaction 2155
2156 The Journal of Finance sharing,and they can make a net profit.Again,in the limit where the mo- mentum traders become risk-neutral,both groups break even.The logic is similar to that with Q=0,because risk-neutrality on the part of momentum traders dissipates all the risk-sharing profits,restoring the zero-sum nature of the game. E.Numerical Comparative Statics In order to develop a better feeling for the properties of the model,we perform a variety of numerical comparative statics exercises.20 For each set of parameter values,we calculate the following five numbers:(i)the equi- librium value of ;(ii)the unconditional standard deviation of monthly re- turns AP;(iii)the standard deviation of the pricing error relative to a rational expectations benchmark,(P-P);(iv)the cumulative impulse response of prices to a one-unit e shock;and (v)the autocorrelations of returns.The detailed calculations are shown in Appendix B;here we use plots of the impulse responses to convey the broad intuition. We begin in Figure 1 by investigating the effects of changing the momen- tum traders'horizon j.We hold the information-diffusion parameter z fixed at 12 months,and set the standard deviation of the fundamental e shocks equal to 0.5 per month.Finally,we assume that the aggregate risk tolerance of the momentum traders,y,equals 1/3.21 We then experiment with values of j ranging from 6 to 18 months.As a baseline,focus first on the case where j=12 months.Consistent with Proposition 1,the impulse response function peaks 12 months after an e shock,reaching a value of 1.342.In other words, at the peak,prices overshoot the change in long-run fundamentals by 34.2 per- cent.After the peak,prices eventually converge back to 1.00,although not in a monotonic fashion-rather,there are a series of damped oscillations as the momentum-trading effects gradually wring themselves out. Now ask what happens as j is varied.As can be seen from Figure 1,the effects on the impulse response function are nonmonotonic.For example, with j=6,the impulse response peaks at 1.265,and withj=18,the peak reaches 1.252,neither as high as in the case where j=12.This nonmono- tonicity arises because of two competing effects.On the one hand,an in- crease in j means that there are more generations of momentum traders active in the market at any one time;hence their cumulative effect should be stronger,all else equal.On the other hand,the momentum traders ratio- nally recognize the dangers of having a longer horizon-there is a greater risk that they get caught trading late in the momentum cycle.As a result, they trade less aggressively,so that is decreasing in j. 20 Appendix A briefly discusses our computational methods. 21 Campbell,Grossman,and Wang(1993)suggest that this value of risk tolerance is about right for the market as a whole.Of course,for individual stocks,arbitrageurs may be more risk-tolerant,since they may not have to bear systematic risk.As we demonstrate below,our results on overreaction tend to become more pronounced when risk tolerance is increased
sharing, and they can make a net profit. Again, in the limit where the momentum traders become risk-neutral, both groups break even. The logic is similar to that with Q 5 0, because risk-neutrality on the part of momentum traders dissipates all the risk-sharing profits, restoring the zero-sum nature of the game. E. Numerical Comparative Statics In order to develop a better feeling for the properties of the model, we perform a variety of numerical comparative statics exercises.20 For each set of parameter values, we calculate the following five numbers: ~i! the equilibrium value of f; ~ii! the unconditional standard deviation of monthly returns DP; ~iii! the standard deviation of the pricing error relative to a rational expectations benchmark, ~Pt 2 Pt * !; ~iv! the cumulative impulse response of prices to a one-unit e shock; and ~v! the autocorrelations of returns. The detailed calculations are shown in Appendix B; here we use plots of the impulse responses to convey the broad intuition. We begin in Figure 1 by investigating the effects of changing the momentum traders’ horizon j. We hold the information-diffusion parameter z fixed at 12 months, and set the standard deviation of the fundamental e shocks equal to 0.5 per month. Finally, we assume that the aggregate risk tolerance of the momentum traders, g, equals 103.21 We then experiment with values of j ranging from 6 to 18 months. As a baseline, focus first on the case where j 5 12 months. Consistent with Proposition 1, the impulse response function peaks 12 months after an e shock, reaching a value of 1.342. In other words, at the peak, prices overshoot the change in long-run fundamentals by 34.2 percent. After the peak, prices eventually converge back to 1.00, although not in a monotonic fashion—rather, there are a series of damped oscillations as the momentum-trading effects gradually wring themselves out. Now ask what happens as j is varied. As can be seen from Figure 1, the effects on the impulse response function are nonmonotonic. For example, with j 5 6, the impulse response peaks at 1.265, and with j 5 18, the peak reaches 1.252, neither as high as in the case where j 5 12. This nonmonotonicity arises because of two competing effects. On the one hand, an increase in j means that there are more generations of momentum traders active in the market at any one time; hence their cumulative effect should be stronger, all else equal. On the other hand, the momentum traders rationally recognize the dangers of having a longer horizon—there is a greater risk that they get caught trading late in the momentum cycle. As a result, they trade less aggressively, so that f is decreasing in j. 20 Appendix A briefly discusses our computational methods. 21 Campbell, Grossman, and Wang ~1993! suggest that this value of risk tolerance is about right for the market as a whole. Of course, for individual stocks, arbitrageurs may be more risk-tolerant, since they may not have to bear systematic risk. As we demonstrate below, our results on overreaction tend to become more pronounced when risk tolerance is increased. 2156 The Journal of Finance