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Information in stock prices and trading volume Schneider, Jan 2006

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INFORMATION IN STOCK PRICES  ANDTRADING VOLUME  by  JAN  SCHNEIDER  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS  DOCTOR  FOR THE DEGREE  OF  OF  PHILOSOPHY  in  THE FACULTY OF GRADUATE  STUDIES  (Business A d m i n i s t r a t i o n )  T H E UNIVERSITY OF BRITISH  May  ©  2006  J a n Schneider, 2006  COLUMBIA  Abstract In this thesis I develop two theoretical models to analyze how investors can infer private information from market prices and aggregate trading volume. In the first chapter I provide a closed form solution for a rational expectations equilibrium where all investors infer information about the state of the economy from (1) private signals, (2) the market price and (3) aggregate trading volume. The main result of this model is that trading volume reveals the relative quality of the aggregate private information in the economy. Investors use volume to decide how they should weight the market price relative to their own private signals when they update their beliefs. In the second chapter, I assume that investors make individual mistakes when they infer information from the price. I show that in a heterogeneous information economy, bounded rationality on the individual level is observationally equivalent to a psychological bias on the aggregate level. If the investors are not able to infer perfectly the true state of the economy from the price, then the aggregate demand corresponds to the demand of a representative agent who is "underconfident". The underconfidence of the representative agent causes the price to adjust to new information too slowly.  ii  Table of Contents Abstract  ii  Table of Contents  iii  List of Figures  iv  CHAPTER I Introduction  1  CHAPTER II A Rational Expectations Equilibrium with Informative Trading Volume  18  2.1 Introduction  18  2.2 Setup of the Model  19  2.3 The Equilibrium  22  2.4 Properties of the Equilibrium  31  CHAPTER III Heterogeneous Beliefs and the Underconfident Representative Agent  34 3.1 Introduction  34  3.2 Homogeneous Information  35  3.3 Heterogenous Uncertainty about the Income of the Firm  35  3.4 The Underconfident Representative Agent  47  3.4 Comparative Statics  51  CHAPTER IV  Conclusion  61  Appendix Proofs  64  iii  List of Figures Figure 2.1  Time Line  20  Figure 2.2  Composition of Trading Volume  28  Figure 2.3  Trading Volume and the Dispersion of Signals  30  Figure 3.1  Price Reaction to New Information  52  Figure 3.2  Momentum Return  53  Figure 3.3  Time Line  54  Figure 3.3  Momentum Return with Reversals during the Formation Period .. 56  Figure 3.3  Momentum Return with Reversals during the Formation Period II 58  iv  Chapter 1 Introduction In afinancialmarket every investor is interested in the private information that other investors might possess. There are several ways to learn what other market participants know. In this thesis I examine how investors can infer private information from market prices and aggregate trading volume. I develop two theoretical models to address the following questions: (1) how can investors fully rationally infer information from prices and trading volume, and (2) how do the results of a world with perfectly rational investors change, if the rationality of investors is bounded, so that they are not fully able to infer all available private information? Grossman (1976) was one of the first authors who showed how investors can extract information about the future payoff of a security from its price. In the first part of this thesis I extend the Grossman model to a case where investors extract information from both the price and the trading volume. This model provides a closed form solution for a rational expectations equilibrium where all investors infer information about the state of the economy from (1) private signals, (2) the market price and (3) aggregate trading volume. 1  Several e m p i r i c a l studies s u p p o r t the idea that t r a d i n g v o l u m e contains information about (LMSW,  future returns.  For example,  Llorente, Michaely, Saar, a n d W a n g  2002) s o l v e a n d t e s t a m o d e l w h e r e t r a d i n g v o l u m e p r e d i c t s c h a n g e s i n  the autocorrelation of r e t u r n s .  1  However, the investors in L M S W ' s m o d e l r a t i o n a l l y  i g n o r e t r a d i n g v o l u m e w h e n t h e y u p d a t e t h e i r beliefs, s i n c e v o l u m e d o e s n o t c o n t a i n a n y i n f o r m a t i o n b e y o n d their o w n private signals a n d the m a r k e t price. T h e r e f o r e , i n L M S W , t r a d i n g v o l u m e provides i n f o r m a t i o n o n l y to a n outside observer of the economy, b u t not to investors w i t h i n the  economy.  M y m o d e l shows how investors w i t h i n the economy c a n learn f r o m t r a d i n g volu m e , a n d h o w v o l u m e i n f o r m a t i o n differs f r o m i n f o r m a t i o n c o n t a i n e d i n t h e p r i c e . T h e m a i n r e s u l t o f t h i s m o d e l is t h a t t r a d i n g v o l u m e r e v e a l s t h e r e l a t i v e q u a l i t y o f the aggregate private information i n the economy.  U n d e r low t r a d i n g v o l u m e ,  the  a g g r e g a t e i n f o r m a t i o n is m o r e p r e c i s e c o m p a r e d t o p r i v a t e s i g n a l s t h a n u n d e r h i g h trading volume.  Investors therefore use v o l u m e t o decide h o w t h e y s h o u l d weight  the m a r k e t price relative to their o w n private signals w h e n t h e y u p d a t e their beliefs.  W h e n t r a d i n g v o l u m e is l o w , i n v e s t o r s w e i g h t t h e m a r k e t p r i c e m o r e h e a v i l y .  C o n v e r s e l y , w h e n v o l u m e is h i g h , i n v e s t o r s w e i g h t t h e i r p r i v a t e s i g n a l s m o r e h e a v i l y . In o r d e r to show h o w investors infer i n f o r m a t i o n f r o m t r a d i n g v o l u m e , I develop a m o d e l where a large n u m b e r of s m a l l investors observe private noisy signals a future dividend.  In addition to their endowment  of i n f o r m a t i o n , t h e  are also e n d o w e d w i t h p r i v a t e c l a i m s to a r i s k y future l a b o r i n c o m e .  about  investors  T h e dividend  a n d t h e l a b o r i n c o m e a r e c o r r e l a t e d , so t h a t i n v e s t o r s h a v e t w o m o t i v e s for t r a d i n g :  ^ o r other evidence see also Gallant, Rossi, and Tauchen (1992), Campbell, Grossman, and Wang (1993), Conrad, Hameed, and Niden (1994), Lee and Swaminathan (2000), Gervais, Kaniel, and Mingelgrin (2001), and Connolly and Stivers (2003).  2  private i n f o r m a t i o n a n d risk sharing. P r i v a t e s i g n a l s a n d l a b o r e n d o w m e n t s a r e i d e n t i c a l l y d i s t r i b u t e d for a l l i n v e s t o r s , so t h a t a l l i n v e s t o r s o b s e r v e i n f o r m a t i o n o f i d e n t i c a l q u a l i t y . weight t h e i r signals e q u a l l y w h e n t h e y u p d a t e t h e i r beliefs.  Therefore,  investors  A s a result, the equi-  l i b r i u m price depends o n the average signal a n d the average exposure to the l a b o r risk.  Since investors are u n c e r t a i n a b o u t the average labor risk, t h e y are not able  to fully infer the average d i v i d e n d signal f r o m the price. In a d d i t i o n to the u n c e r t a i n t y a b o u t the aggregate  dividend information and  aggregate l a b o r risk, t h e investors are also u n c e r t a i n a b o u t t h e cross-investor correlation of the i n d i v i d u a l errors i n their private d i v i d e n d signals.  T h e correlation  o f t h e i n d i v i d u a l s i g n a l e r r o r s is i m p o r t a n t , s i n c e t h i s c o r r e l a t i o n d e t e r m i n e s q u a l i t y of the aggregate i n f o r m a t i o n relative to the private information.  the  For exam-  ple, if the signals are perfectly correlated, t h e n the average signal contains the same i n f o r m a t i o n as t h e i n d i v i d u a l s i g n a l s .  H o w e v e r , if investor specific s i g n a l errors are  uncorrelated, t h e n investors k n o w m o r e i n aggregate t h a n they k n o w individually. Investors therefore w i s h to k n o w the correlation of the i n d i v i d u a l signals, i n order t o assess t h e p r e c i s i o n o f t h e a v e r a g e s i g n a l i n t h e p r i c e . In a s y m m e t r i c e c o n o m y where private i n f o r m a t i o n a n d l a b o r risks are identically d i s t r i b u t e d for a l l i n v e s t o r s , t r a d i n g v o l u m e r e v e a l s t h e c o r r e l a t i o n o f s i g n a l s i n t h e following way: since investors weight their signals a n d e n d o w m e n t s identically w h e n t h e y calculate their d e m a n d s , the n u m b e r of shares t h a t a given investor b u y s or sells d e p e n d s o n l y o n t h e d i f f e r e n c e s b e t w e e n h i s p r i v a t e s i g n a l a n d e n d o w m e n t a n d the signal a n d endowment of the average investor i n the economy.  Therefore,  the  i n d i v i d u a l t r a d e s are f u n c t i o n s o n l y of the investor specific c o m p o n e n t s of signals and endowments.  H e n c e , if these c o m p o n e n t s are i n d e p e n d e n t across investors, a n d  3  if the number of investors in the economy is large, the per capita trading volume depends only on the distribution and not on the realization of these components. As a result, investors can infer the distribution of signals from trading volume. In particular, if investors are uncertain about the correlation of their private signals, then trading volume reveals this correlation. In traditional models of heterogeneous information, such as Grossman and Stiglitz (1980), investors form a weighted average of their private signals and the market price when they update their beliefs. In these models, all investors know the optimal weights for the price and the signals, since these weights are independent of the state of the economy. In my model, investors are uncertain how they should weight the market price relative to their own private information. Observing trading volume removes this uncertainty. Under high trading volume, the quality of the aggregate signal in the price does not exceed the quality of their private signals. Since the price contains additional noisefromthe aggregate labor income shock, investors weight their own signals more heavily than the price when volume is high. However, under low trading volume, the quality of the aggregate information exceeds the quality of the individual signals. Therefore, investors weight the price more heavily when trading volume is low. The idea that investors have a risk sharing and a private information motive for trading has been previously employed for example by Wang (1994) and LMSW (2002). However, in these models, there are only two agents that trade with each other. Therefore, trading volume does not provide any information for the investors beyond the information that they can infer from their own private signals and the market price. The technical difficulty that arises if investors are allowed to observe trading volume is that volume is a sum of absolute values and therefore not normally  4  distributed. Asset pricing models with heterogeneously informed investors usually rely on the properties of the normal distribution in order to be tractable. My model solves the problem that trading volume is not normally distributed by transforming a non-linear optimization problem into a problem that is linear conditional on the observation of trading volume. Several other authors have examined alternative approaches. For example, Bernardo and Judd (1996) show how to numerically solve a model where investors learn private information from trading volume. Their numerical approach has the advantage that it covers a large set of possible assumptions, however, a numerical approach does not provide the same clean economic intuition as an analytical solution. Blume, Easley, and O'Hara (BEH, 1994) provide a closed-form solution for a model where investors learn from past prices and past trading volume. Similar to my model, the investors in BEH face two dimensions of uncertainty: the realization and the quality of other investors' signals. However, in order to solve their model, BEH have to assume that investors are not fully rational: even though investors know the price at which they trade, they employ this price in order to update their beliefs only after they have completed their trade. BEH (page 160) comment on the difficulty of solving a rational expectations equilibrium where investors learn from trading volume: "Alternatively, there could be nonrevealing equilibria in which traders condition on price and volume. However, as volume is a sum of absolute values it cannot be normally distributed. So although such an equilibrium might exist there seems to be no hope of constructing it, and hence no hope of using a contemporaneous data approach to study volume." As I show in this paper, the case for a non-revealing equilibrium where investors condition their demands on prices and on volume is not completely hopeless.  5  In the first part of this thesis I assume that all investors are fully rational, and that they are able to optimally analyze prices and volume. However, in reality prices aggregate information in a complicated way, and extracting this information is a difficult task. In the second part of this thesis I assume that the computational skills of the investors are limited: they make individual mistakes when they infer information from the price. I show that in a heterogeneous information economy where investors' rationality is bounded prices react too slowly to new information. A large number of empirical studies document that prices potentially underreact to new information. For example, Ball and Brown (1968), Bernard and Thomas (1990) and others find that firms reporting unexpectedly high earnings outperform firms reporting unexpectedly low earnings. Givoly and Lakonishok (1979), and Stickel (1991) document similar drifts after analyst forecast revisions. Jegadeesh and Titman (1993) rank stocks according to past returns and find that past winners outperform past losers. The debate whether these findings violate market efficiency is ongoing. Some authors argue that return continuations can be explained with changes in firm's risks. Other authors argue that prices underreact to new information because investors are psychologically biased. For example, Barberis, Shleifer, and Vishny (BSV, 1998) link underreaction to the conservatism bias. The conservatism bias has been identified in experiments by Edwards (1968). Individuals who are subject to this bias tend to underweight new information when they update their priors. BSV show that, if the representative agent in the economy suffers from the conservatism bias, then prices will adjust to new information slowly.  2  2  Other psychologically motivated explanations include over/confidence as for example in Daniel,  6  In this paper I show that even if the representative agents appears to be psychologically biased, this is not necessarily true for the individual investors. In a heterogeneous information economy where the computational skills of the investors are limited, so that they are not perfectly able to infer each other's information from the price, the aggregate demand is equivalent to the demand of a representative agent who underestimates the quality of his information. This underconfidence of the representative agent arises from the fact that the investors know more in aggregate than they know individually, and that they fail to infer perfectly the aggregate knowledge from the price. In that way, bounded rationality on the individual level is equivalent to a psychological bias on the aggregate level. Several empirical studies support the notion that heterogeneous beliefs play an important role in the momentum phenomenon. For example, Verardo (2002) finds that profits from momentum strategies increase with the dispersion in analyst forecasts. Hong, Lim, and Stein (2000) find that momentum strategies work better among small stocks with low analyst coverage. Zhang (2005) finds that the post earnings announcement drift, the drift after analyst forecast revisions, and the profits from price momentum strategies all increase with various proxies for heterogeneous beliefs. To see how heterogenous information causes return continuations consider the following example. There is a firm that will pay out an uncertain future dividend. The investors receive a signal about this dividend of the form "dividend + noise". Assume for the moment that this signal is public, and that all investors interpret the signal in the same way. If this signal becomes more precise as the dividend payout Hirshleifer, Subrahmanyam (1998), and the disposition effect as for example i n Frazzini (2005) and Grinblatt and H a n (2005).  7  date approaches, the investors will increase the weight of the signal, as they update their beliefs. This updating process has two effects on the serial correlation of the returns. First, the "true" component of the signal produces a positive effect, since this value will be slowly incorporated into the price. Second, the noise component produces a negative effect, since, as the signal becomes more precise, the investors will reverse that part of their initial reaction to the signal that was due to the noise. If the economy is in a steady state, and the risk that the average investors has to bear does not change over time, these two effects will offset each other exactly. Hence, uncertain information does not produce serially correlated returns, if the investors are homogenously informed. Assume now that the signal contains an investor specific noise component, so that the investors axe heterogeneously informed. Assume that the rationality of the investors is bounded, and that the investors are not able to infer perfectly all the private information from the price. Since the individual signals are noisy, the investors will assign only small weights to these signals when they update their priors. However, the individual noise components will at least partially cancel each other out, when one aggregates the demands. Therefore, the aggregation of demands will reduce the negative noise effect on the correlation of returns, without reducing sufficiently the positive effect resulting from the true information component of the signals. The aggregate demand will be equivalent to the demand of a representative agent who is underconfident. As a result, prices adjust to new information too slowly. Several other authors have developed heterogeneous information models to explain momentum. In these models, returns are positively autocorrelated because investors receive information sequentially. For example, in Hong and Stein (1999) in-  8  formation about a liquidating dividend spreads slowly through a group of "newswatchers". Since information spreads slowly, and since the newswatchers are not able to infer informationfromthe price, the price adjusts slowly to each piece of new information. In Holden and Subrahmanyam (2002) some investors receive certain pieces of information before other investors. In their model, noise traders prevent the price from revealing all the information. Holden and Subrahmanyam state on page 4: "Thus, our consideration of the sequential nature of information acquisition is the key to generating positive serial correlation within a rational expectations model." In this paper, I show that the sequential information flow is not a necessary condition for return continuations. Instead, the key to momentum is simply the disagreement about future payoffs. I show in this paper how heterogeneous beliefs combined with bounded rationality leads to aggregate underreaction to new information. If the price underreacts to new information, momentum traders will rationally chase trends, as for example in Hong and Stein (1999). I do not examine the effect of momentum traders on prices and returns in this paper. In the real world, we would expect that the actions of momentum traders diminish the abnormal profits from momentum strategies. It is therefore surprising that, for example, Grundy and Martin (2001) find that momentum strategies can produce risk adjusted returns of more than one percent per month. However, these momentum profits are profits before trading costs. The evidence for the profitability of momentum strategies after trading costs is less clear. For example, Lesmond, Schill, and Zhou (2004) find that momentum strategies are not profitable after trading costs. Korajczyk and Sadka (2004) construct a liquidity weighted momentum strategy in order to minimize trading costs. Taking price impact trading costs into account, they estimate that this strategy earns positive  9  abnormal profits for an investment of up to $5 billion. In this paper, I do not ad3  dress the question to which degree momentum traders should arbitrage momentum profits away. Instead, my focus is to show a compelling source of momentum.  3  A s of December 1999. At that time, the total market capitalization of the N Y S E was $11.7  trillion. 10  Bibliography [1] Admati, Anat R., 1985, A noisy rational expectations equilibrium for multiasset securities markets,  53, 629-658.  Econometrica,  [2] Admati, Anat R., and Paul Pfleiderer, 1988, A theory of intraday patterns: volume and price variability,  Review of Financial  Studies,  1, 3-40.  [3] Back, Kerry, Cao, C. Henry, and Gregory A. Willard, 2000, Imperfect competition among informed traders,  Journal  of Finance,  55, 2117-2155.  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[57] Stickel, Scott, 1991, Common stock returns surrounding earnings forecast revisions: more puzzling evidence, The Accounting Review, 66, 402-416. [58] Verardo, Michela, 2002, Heterogeneous beliefs and momentum profits, working paper [59] Verrecchia , Robert E., 1982, Information acquisition in a noisy rational expectations economy, Econometrica, 50, 1425-1430. [60] Wang, Jiang, 1993, A model of intertemporal asset prices under asymmetric information, Review of Economic Studies, 60, 249-282.  [61] Wang, Jiang, 1994, A model of competitive stock trading volume, Journal of Political Economy, 102, 127-168.  [62] Zhang, X . Frank, 2005, Information uncertainty and stock returns, Journal of Finance, forthcoming.  17  Chapter 2 A Rational Expectations Equilibrium with Informative Trading Volume 2.1  Introduction  In this chapter I develop a rational expectations equilibrium where investors infer private information from the market price and from trading volume. A l l investors in this model observe private noisy signals about a future dividend and they are also endowed with claims to a risky future labor income. Investors have therefore two motives for trading: private information and risk sharing. Since investors are uncertain about the economy wide exposure to the labor risk, they are not able to fully infer the aggregate private information from the price. In addition to the uncertainty about the aggregate dividend information and aggregate labor risk, the investors are also uncertain about the cross-investor corre18  lation of the individual errors in their private dividend signals. The correlation of the individual signal errors is important, since this correlation determines the quality of the aggregate information relative to the private information. In the equilibrium, investors learn the correlation of signals from trading volume, and they learn the average signal from the price.  2.2  Setup of the model  The economy is populated by a countable set of investors. I will refer to an individual investor as investor i, i = 1,2,  There are two time periods, t = 0 and t = 1.  Figure 2.1 shows a picture of the time line. At time t = 0, investor i is endowed with N = N + 7ii t  units of a non-traded asset. At time t = 1 the investors receive a payoff of Y for each unit Ni they are endowed with at time t — 0. The total non-traded income of investor i at time t = 1 is therefore given by NY.  I will refer to NY  as the labor  income of investor i, even though other interpretations are possible. In addition to the labor income, the investors also receive income from their investments in the financial market. The financial market consists of two assets: a risk free bond and a risky firm. One dollar invested in the bond at time t = 0 pays one dollar at time t = 1. Investors can buy or sell an unlimited amount of the bond. Investors can trade shares of the firm at time t = 0 at the equilibrium price P. At time t = 1 the firm pays a liquidating dividend D for each share the investors hold at time t = 0 after trading. At time t = 0 the investors observe private noisy signals  19  labor  endowments  payoffs  Ni = N + m  NiY  demands  payoffs  r i s k y asset  XiD private  information  signals  a b o u t r i s k y asset  bi = D + r} + ei  public information  price  a b o u t r i s k y asset  P volume  V F i g u r e 2.1:  Time Line.  This  figure  shows endowments, demands,  payoffs,  private  information and public information.  a b o u t t h i s d i v i d e n d . T h e s i g n a l o f i n v e s t o r i is g i v e n b y  Di = D + rj + €i w h e r e r\ a n d e$ a r e e r r o r t e r m s .  T h e correlation of the  how m u c h the investors disagree about  t h e f u t u r e payoff.  states of the w o r l d regarding this correlation. e r r o r s is l o w .  In this state, the  c o r r e l a t i o n o f e r r o r s is h i g h . error t e r m  =  T h e r e are two  possible  I n state L , t h e c o r r e l a t i o n of s i g n a l  are i n d e p e n d e n t across investors. I n state H  In this state, all i n d i v i d u a l signals c o n t a i n the  e, so t h a t t h e e r r o r s a r e p e r f e c t l y  all investors observe the  across investors determines  s a m e s i g n a l D + 77 + e.  the same  correlated across investors a n d T h e investors might  have some  information about this correlation, however they d o not know the realization of the correlation w i t h certainty.  F o r any given  20  realization  of the  correlation state  the  r a n d o m variables  D, N, Y, rj, {ei}^ ,  { n j g  0  m e a n zero a n d variances a , 2  D  a%, a , Y  c , , a\  are j o i n t l y n o r m a l l y d i s t r i b u t e d w i t h  0  a n d CT .  2  2 1  e x c e p t for t h e c o r r e l a t i o n o f t h e e, i n s t a t e H, correlated with L e t Xi  Cov[D, Y] = ODY  A l l variables are u n c o r r e l a t e d  a n d e x c e p t for D a n d Y,  w h i c h are  > 0.  b e t h e t i m e t = 0 d e m a n d for t h e r i s k y asset o f i n v e s t o r i.  1 X=  lim  T  Let  h  yXi i=l  b e t h e p e r c a p i t a d e m a n d , p r o v i d e d t h a t t h i s l i m i t e x i s t s . A n e q u i l i b r i u m is g i v e n by a price P that  satisfies X  w i t h p r o b a b i l i t y one.  = supply per capita  (2.1)  T o s i m p l i f y t h e n o t a t i o n I w i l l set t h e s u p p l y e q u a l t o z e r o ,  a n d I a s s u m e t h a t a l l i n v e s t o r s o w n z e r o s h a r e s p r i o r t o t h e t r a d i n g d a t e t = 0. T h e a s s u m p t i o n of zero s u p p l y means that all dividends that investors w h o h o l d long positions of the f i r m receive are p a i d b y investors w h o h o l d short positions.  This  a s s u m p t i o n w i l l r e m o v e a c o n s t a n t from t h e e q u i l i b r i u m p r i c e , b u t i t w i l l n o t affect any  of the results. Since investors h o l d zero shares before t r a d i n g , the n u m b e r of shares t h a t investor  i t r a d e s is g i v e n b y \Xi\.  Let 1  V=  lim  h  Y\Xi\  T  i=l 1  Since all variables are normally distributed, dividends and labor income can be negative.  It  is possible to choose means and variances so that the probability for negative payoffs w i l l be arbitrarily small. However, since all investors know these distributions, the choice of the mean does not affect the trading volume. In order to simplify the notation I set therefore all the means equal to zero.  21  be the (double counted) per capita trading volume, provided that this limit exists. Investor i chooses his demand X by maximizing t  E  - w, T  e  P  where p is the coefficient of absolute risk aversion, Wi is the future wealth, and T is the information set of investor i. This information set is given by Ti =  {Di,Ni,P,V}.  Investors can therefore condition their demand on their private dividend signals, their private labor endowments, the equilibrium price, and the equilibrium trading volume. Note that investors know Ni, their own exposure to the labor risk, but they do not observe N, the economy wide exposure to this risk. This assumption will prevent the equilibrium price from completely revealing all private information.  2.3  The equilibrium  Let 1  h  V h-too h —'  D= lim -  Z  D.  i=l  be the aggregate information about the future dividend. Then we have D + 77  in state L  D + n + e in state H If the correlation of signal errors ti is low (state L), then the individual signal errors cancel each other out in the average signal. Only in this case investors are able to learn information about the future dividend from the price. If the correlation of 22  signals is high (state H), all investors observe the aggregate signal JJ = D + r\ + e directly, so that the price does not provide any additional information about D. Investors are not only uncertain about the realization of the average signal D, but they are also uncertain wether the world is in state L or in state H. In order to find an equilibrium for this economy, I willfirstanalyze the case where investors know the correlation of signals. Lemma  (Correlation of signals is high).  1  Assume  the world is in state H,  assume all investors know that the world is in state H.  and  Then there exists an equi-  librium price of the form  P = $£(D  where  and  + V + e) -$%N,  (2.2)  are constants. If the price is given by (2.2), then the  coefficients  are given by  ®Z  =  2 ,  2^  2  ^" = P  a T l d  2  2 •  2'  and the demand of investor i is given by H  X  =  _^DY . n  and trading volume is given by  V 7T OfD Lemma (1) shows that, if all investors know that the world is in state H, the price P is a linear function of the aggregate signal D + 77 + e and the exposure to the aggregate labor risk N. The price decreases with N, since the labor payoff Y and the dividend D are positively correlated. 23  Since all investors hold zero shares before they start trading, the demand Xf is equivalent to the number of shares that investor i buys or sells. Note that these equilibrium trades can also be written as *>'--%(»<-»)  In state i / , investor i sells shares if his private labor risk exposure N is higher than the average risk exposure N. The equilibrium demands are independent of the dividend signals D since all investors observe the same signal Di = D + r) + e. t  Since the equilibrium demands depend only on investor specific components n  it  and since these components are independent across investors, the per capita trading volume is given by the unconditional expectation of the absolute number of shares that any given investor trades. Since the n are normally distributed, we have t  V« = nXi\ = ^ [ X i ]  =  ^ - ^ a  n  As a result, volume is constant. Hence, if all investors know that the world is in state H, then investors ignore trading volume when they choose their demands. Lemma  2  (Correlation of signals is low). Assume the world is in state L, and as-  sume all investors know that the world is in state L. Then there exists an equilibrium price of the form P = $ {D  + )-$ N,  L  where $£> and  are constants.  §  L D  (2.3)  L  D  V  N  If the price is given by (2.3), then we have  < $g < 1  and  > 0  and the demand of investor i is given by  Xi = ty €i D  24  ^ Ui, N  wheretyp> 0 and  > 0, and trading volume is give by  Lemma (2) shows that, if all investors know that the world is in state L, the price P is a linear function of the aggregate signal D + n and the exposure to the aggregate labor risk N. The investor specific signal errors  are not part of the  price, since they cancel each other out in the aggregate demand. Note that the equilibrium trades X± can also be written as Xl =tfD(Di - (D + r,))  -  * (N N  t  -  JV).  As opposed to state H, investors in state L do not only trade to share risk, but also because they are heterogeneously informed. In this case, the equilibrium demands depend on the differences between the individual signals Di and the average signal D+r] and the individual labor endowments N and the average endowment N. Since investors are uncertain about the aggregate endowment N, the price does not fully reveal the aggregate dividend signal. In both correlation states, L and H, the demands Xi depend only on the differences between individual and average signals and endowments, since information and labor risks are identically distributed across investors. If signals and endowments are identically distributed, then investors weight their signals and endowments identically when they form their demands. As a result, the price depends on the averages whereas the equilibrium demands depend on the differences of signals and endowments. Therefore, the price is a function only of the common components D and N, and the demands are functions only of the investor specific components e* (in state H) and n». As a result, for a given state of signal dispersion, equilibrium demands and the price are independent. 25  T h e o r e m  1 (Informative trading volume).  in state L, and by (2.3)  L  are uncertain  exists an equilibrium  price is given by (2.2)  in state H, then we have  V Hence, if investors  If the equilibrium  < V. H  about the correlation  where trading volume reveals this  of signal errors, then there correlation.  Note that investors cannot learn the correlation of signals by comparing their private signals A to the price P, since the price depends partially on the unknown aggregate labor risk exposure N. However, as Theorem 1 shows, investors can infer the dispersion of beliefs in the economy from trading volume. Given this information, the investors use their private signals and the price to estimate the future dividend. In that way, observing trading volume helps the investors to separate two sources of uncertainty: uncertainty about the realization of the aggregate information, and uncertainty about the quality of the aggregate information.  Theorem 1 shows that trading volume in state H is higher than trading volume in state L. This result might seem surprising, since in state L investors have two reasons for trading, risk sharing and private information, whereas in state H investors only trade to share risk. To understand why volume increases with the correlation of signals, note that, given CARA utility and normal distribution, the general demand functions of the investors are given by v  E[D - P\T \ - pCov[D - P, NiYlft] t  The demand of investor % depends on the conditional expectation of the future investment payoff, the variance of this payoff, and the covariance of the investment 26  payoff with the labor payoff. As Appendix 4 shows, in state L, the conditional expectation of the future dividend is a linear combination of the private dividend signal, the private labor risk, and the price: E[D\Fi) = ip bi + i> Ni + ippP, D  (2.5)  N  where tpo > 0, tpN > 0, and ipp > 0. Investors increase their expectations of D with their private labor risks JVj, since they use N to estimate the aggregate labor risk t  N in the price, and since the price decreases with this risk. As Appendix 4 shows, plugging (2.5) into (2.4) gives the state L equilibrium demands  X  i  ~ pVar[D|^]  al  ^  Ui  Hence, the equilibrium demands in state L can be written as the sum of two components: the first component is due to the fact that investors observe heterogeneous signals, and the second component is given by the equilibrium demand Xf in Lemma 1  (1). By (2.6) we can write trading volume in state L as  7T VpVar[£>|^];  Ge  ir\al  pVarpI^]  As appendix 4 shows, we have  a  2 D  pVar[D|^]"  Hence, private information affects trading volume in two opposing ways. On the one hand, private information increases volume since investors trade based on the differences of their dividend signals. On the other hand, private information reduces volume, since the fact that investors use their private labor risks to estimate the aggregate risk reduces the trading that is due to differences in labor risk endowments. 27  trading due to differences in dividend signals  reduction of risk sharing due to private information A  =v  7  trading due to differences in labor endowments  ^ J  Y total trading volume in state H  Figure 2.2: Composition of Trading Volume. If the world is in state H, so that investors only trade to share risk, then trading volume is given by V . The private H  signals in state L have two effects on V : (1) private information reduces V by H  H  A, since investors use their private labor endowments to estimate the aggregate endowment, and (2) the private signals induce the information trading C. The total trading volume under heterogeneous information is given by V = \ A B + C . V L  2  2  L  is less than the sum of B and C, since these two trading motives partially cancel each other out for the average investor in the economy.  28  As Theorem 1 shows, the total effect of private information on trading volume is always negative. The reason for this negative effect is that for some investors in the economy the demand due to private signals will partially offset the demand due to risk sharing. Therefore, the number of shares that the average investor in the economy trades is less than the sum of his risk sharing demand and his private information demand. Figure 2.2 shows a geometric interpretation of the two components of trading volume volume under heterogeneous information and the relation of these components to trading volume under homogeneous information. I have shown so far how trading volume depends on two extreme cases: the error terms  are perfectly correlated or they are not correlated at all. In order to  examine how trading volume depends on the correlation of error terms in general, assume the dividend signal of investor i is given by D  =  i  D  +  r)-rVuel  +  y / l - u  e, 2i  CT = 2  a\2.  =  a  2  (2.7)  where 0 < u < 1, t\ is a common error term and e^i are investor specific error terms, which are independent and identically distributed across investors. The case u> = 0 corresponds to state L, and the case u — 1 corresponds to state H in the theorem. Since o\x = of . the total variance of the signal does not depend on u. Hence, the 2  specification of the dividend signal in (2.7) allows to examine the effect of a change in the correlation of signal errors, independent of the effect of a change in the total error variance. Figure 2.3 shows the equilibrium trading volume, assuming that all investor know u. As Figure 2.3 shows, trading volume decreases with the dispersion of signals.  29  Volume  (low correlation)  (perfect correlation)  Figure 2.3: T r a d i n g V o l u m e a n d the D i s p e r s i o n o f Signals. This figure shows how trading volume in a symmetric economy with a large number of small investors depends on the correlation of the investor specific signal errors, holding the total variance of the error terms constant. The private signal of investor i is given by Di = D+n+i/u) ei + y/1 — uje^i, where w and €\ are common error terms and ej is an investor specific error term, and <r  2  cr  2  x  =  = a\. The total variance of the signal errors is given by a^+wa^ + ^—^a^. = a^+a . 2  2i  The remaining parameters are given by p = a  2  30  D  = UDY =  =  n — !•  a  2.4 2.4.1  Properties of the Equilibrium U p d a t i n g o f beliefs  C o r o l l a r y 1. If the equilibrium is given by the Theorem 1, the expected future dividend conditional on the information of investor i is given by E[D\F ] = t  if trading volume is high, and E[D\Ti} =  + ip N + ibpP, N  t  if trading volume is low, and we have tpp > 0, ipx > 0, and 0 < ipp < ipp. Corollary 1 shows that investors use the price to update their beliefs only when trading volume is low. Under high trading volume, the aggregate signal is identical to the individual signals. Since the uncertain aggregate labor risk iV prevents the price from revealing the aggregate signal, investors completely ignore the price when they update their beliefs. However, under low trading volume, the quality of the aggregate information exceeds the quality of the private information. Therefore, if trading volume is low, investors reduce the weights on their private signals and weight the price more heavily. In this case, the investors also use their own labor risk Ni in order to estimate the aggregate risk N in the price.  31  2.4.2  Trading strategies  C o r o l l a r y 2. //  the equilibrium is given by the Theorem 1, the demand of investor  i is given by  X?  =  VZDi-V%Ni-*$P  where all coefficients are greater than zero and we havetyp> $p. Corollary 2 shows that, after controlling for the signals A and the labor endowment N, investors always trade against the price. However, investors trade less aggressively against the price when trading volume is low than when trading volume is high. The reason for this behavior is that, under low trading volume, the price does not fully adjust to the aggregate information in the economy. Investors interpret therefore high prices partially as good news and low prices partially as bad news.  2.4.3  Volume predicts the future risk premium  I have so far assumed that the risky asset has a zero net supply. This assumption removes a risk premium from the return, since the average investor does not require a risk premium if he does not hold any shares. The following result shows the relation between volume and the risk premium, if the asset has a positive supply. Corollary 3. Assume the per capita supply of the risky asset is given by S > 0, and assume that all investors hold S shares prior to trading. Let P§=o> be the equilibrium price in Theorem 1. Then we have P = P| L  - $ S L  =0  S  and 32  P = PjLo - <&f S, H  s  e  { A H}  and E[D - P\V ] = $ S S  S  S  and E[D - P\V ] > E[D - P\V ] > 0. H  L  Corollary 3 shows that the price decreases with the supply of the risky asset. The expected future return is given by the risk premium that investors require in order to hold the per capita supply S. Since the quality of the aggregate information decreases with trading volume, investors require a higher risk premium when trading volume is high. Hence, the expected future return increases with trading volume.  33  Chapter 3 Heterogeneous Beliefs and the Underconfident Representative Agent 3.1  Introduction  In the first part of this thesis I have assumed that all investors are fully rational, and that they are able to optimally analyze prices and volume. However, in reality prices aggregate information in a complicated way, and extracting this information is a difficult task. In this chapter I assume that the computational skills of the investors are limited: they make individual mistakes when they infer information from the price. I show that in a heterogeneous information economy where investors' rationality is bounded prices react too slowly to new information. The remainder of this chapter is organized as follows. Section 3.2 describes the model for the base case where all investors are homogeneously informed. Section 34  3.3 adds heterogeneous beliefs. In section 3.4 I compare the aggregate demand to the demand of a representative agent. In section 3.5 I examine how the size of the momentum profits depends on the parameters of the model.  3.2  Homogeneous Information  There are two assets in the economy, a risk-free asset and a risky asset. Investors can trade these assets at the trading dates t = 0,1,2,  One dollar invested in the  risk-free asset at time t pays 1 + r dollar at time t + 1. Let R = ^ price of one dollar at time t +  be the time t  Investors can buy or sell an unlimited amount of  the risk-free asset. A s s u m p t i o n 1 (Risky asset). Investors can trade shares of a single risky asset at the equilibrium price P . At each trading date t, investors receive a dividend D for t  t  each share they are holding at time t — 1. The value D is given by the process t  D = A - i + <k t  where Do = do, d  ~ ./V~(0, cr,), and E[d d ] = 0 for t ^ s. 2  t  t  s  The dividends are normally distributed with an unconditional mean of zero. The assumption of normal distribution allows to solve for the equilibrium price explicitly. Setting the unconditional expectation equal to zero simplifies the notation, but does not affect the results of the model. Assume for the moment that the investors know the realizations of future dividends with certainty. Then the price of the asset is given by the present value of all future dividends  35  I will call Pf the full information value of the risky asset. Let [ii,^] be the time interval starting on trading day t\ and ending on trading day t . Assume an investor 2  buys one share of the asset at time ti and sell it at t . Let T\ = t — h be the length 2  2  of this interval. Then Ti  A  = Yl  t+s  + R Pt* -Pt,  RSL>  t l t 2  (3-2)  Tl  s=l  is the cumulative dollar excess return that this investor receives. Plugging the full information value of (3.1) into (3.2), we get  3=1  Hence, under full information returns are serially uncorrected.  1  Next, assume investors do not know the realizations of future dividends. At each trading date, investors know only the realizations of all past dividends up to the current dividend D . In addition, the investors receive noisy public signals about t  each future dividend innovation. The time t signal about  dt+  s  is given by  dt+a,t — dt+s + Vt+s,t s Vt+s,t =  where all e  (+Si  y~] t+aj) £  j are independent normally distributed random variables with mean  zero and variance of. The first subscript of 1  d + ,t t  s  and n  refers to the dividend  t+ait  Returns axe only serially uncorrelated, if one calculates returns as in (3.2). Without taking  dividends and the discount rate into account, we have, for example, Pf \ — Pf = 2~Is=i (l+r^*- ' 1  +  and Cov[Pf - P , P F  +1  t  F t +2  - P  ] = ,  F t +1  36  7  d (1  +  r )  ( 2+  r )  -  innovation d + , and the second subscript refers to the trading date at which the t  s  investor observes the signal. The time t information about d + is given by the true t  s  dividend innovation d + plus s independent noise terms. At each trading date, one t  s  noise term is removed from the signals, so that the signals become more precise, as the payout date of the corresponding dividend approaches. In that way, the investors know more about dividends that thefirmwill pay in the near future than dividends in the far future. For example, the time t = 0 signal about the dividend innovation at time t = 3 is given by ^3,0 =  d + r]3 , 3  773,0  t0  = e i + 3 i  e  3]2  + e . 3)3  At time t = 1, this signal becomes = cfo +  ^3,1  %,i =  £3,1 +  £3,2-  The information of the investors about d is therefore more precise at t = 1 than at 3  t = 0. Note that the signal tZ  3i  i  encompasses the information contained in the signal  ci o. An investor who observes ci i does not need ci o to forecast future dividends, 3]  3]  since d fl is distributed as d 3  3ii  3]  plus noise. The time t information about all future  dividends is therefore given by the vector It = { A , d +i,t, dt , d ,t, • • •}• t  +2tt  t+3  (3.4)  I omit past dividends in (3.4), since past dividends are irrelevant information for an investor who knows the current dividend. Given the information in (3.4), each investor has to decide how many shares to buy of the risky asset. The economy is populated by a countable set of investors. An individual investor is indexed by i € {1,2,3,...}. Let X be the number of shares that investor i holds at time t. it  37  Assumption 2  (Investors). Investor i chooses X  by maximizing  it  E  -exp{-pW }  it  it+1  ,  where E [•} is the expectations operator conditional on the information of investor i it  at time t, p > 0 is the parameter of risk-aversion, and Wu is the wealth of investor i at time t.  For simplicity I assume that investors are myopic. This assumption allows to derive analytical results. It is straight forward to extend the model to the case where all investors maximize the utility of life-time consumption. I have analyzed this case numerically and found that the qualitative results match the results under a myopic investment policy. If the time t + 1 wealth of investors i is normally distributed, conditional on his information set at time t, the demand for the risky asset of investor i is given by Xit —  E [P it  t+1  + D ]-{l t+1  Var [P  P  it  t+1  + r)P  + A+i]  t  (3.5)  In order to find an equilibrium, we have to aggregate the demand of all investors. Let (3.6)  be the average demand per investor. Definition 1  (Equilibrium). An equilibrium price is given by a real-valued price  process P such that t  (a) X = average supply, t  38  (b) there exists real numbers M, <& , $Dt, ®dts such that |$ | < M, \$Dt\ < M, t  (  \$dts\ < M, and  P  t  = $ + $DtD t  +  t  ®dtsdt . + s  s=l In general there will exist more than one price process that satisfies condition (a). Condition (b) rules out non-linear equilibria and equilibria that contain ra2  tional "bubble-components". The requirement that the equilibrium price depends only on the current dividend D and not on past dividends is not restrictive, since t  the investors do not need past dividends to forecast future dividends. The supply of the risky asset will determine the risk premium. Since, in this model, the riskiness of the asset does not change over time, the risk premium will be a constant. Hence, the risk premium will not affect the correlation of returns. Therefore, in the remainder of the paper I will set the average supply equal to zero. This assumption will simplify the notation by removing the risk premium from the returns, but it will not affect any result of the model. Lemma 3  (Equilibrium with noisy public signals). If the information of the in(3.4), then there exists a unique equilibrium  vestors about future dividends is given by price given by -.  Pt = Et P  oo  dt+s + Vt+s,t  F t  where  A,-s  Cov[d ,d + ] t+g  t  s  1  Var[d ] t+S  and 2  Pf is given by (3.1).  Whenever applicable, equations hold with probability one.  39  Lemma 3 shows that the equilibrium price equals the expectation of the full information value of the risky asset, conditional on the public information available at time t. The equilibrium price does not depend on the parameter of risk aversion, since all investors are homogenous, and the supply of the risky asset equals zero. Therefore, the demand of each investor equals zero in the equilibrium. The full information value of the asset is the present value of all future dividends. Since investors do not bear any risk in the equilibrium, the price must be given by the expected future payoff of the asset. If there would a positive supply of the risky asset, the price in Lemma 3 would be given by the expected fundamental value of the asset plus a constant depending on the risk aversion. If the price is given by Lemma 3, we can write the price as -  Pt.  Pf  +  OO  -}~2(A -l)R -'d s  3=  full information value  "s  t+s  +  OO  -^A  R -V ,t ,  (3.7)  S  S J  s  S=l  1  uncertainty  noise  The price consists of three parts: the full information value of the asset given by (3.1), an uncertainty part, and the signal noise. From Lemma 3 we have 0 < A < 1. s  The uncertainty part decreases therefore the effect of the dividends on the price, relative to the full information value. The weights of the dividends for the price in Lemma 3 are smaller than the weights for the full information value, since the investors receive uncertain signals about future dividends. The uncertainty increases 2  with the ratio of signal noise variance to dividend variance s%. If the investors d  observe undisturbed signals, the uncertainty effect in (3.7) disappears. Lemma 3 shows that A decreases monotonically with s. The uncertainty effect is therefore s  stronger for dividends in the far future than dividends in the near future. The uncertainty effect is stronger for dividends in the far future, since the investors receive more information about the dividends as the dividend payout date approaches. 40  Let PF be the uncertainty part and P  N t  be the noise part of the price in (3.7),  so that we have P = P + PF + P».  (3.8)  F  t  Recall from (3.2) that A  t  is the cumulative excess return of the asset during the  t l t 2  time interval [£i,£ ]- If the price is given by the full information value Pf, this 2  return is given by the fundamental return A £ . If the price is given by (3.8), the t2  uncertainty part PF and the noise part P^ will cause the return to deviate from the full information return. These deviations are given by  =  ( T T ^ ^ - ^ -  (- ) 3  9b  The return A^ is the dollar excess return that is due to the dividend uncertainty, t2  and A^ is the dollar excess return that is due to the noise. Plugging the price in t2  (3.8) into the definition of the return in (3.2), we get A  = A £ + A£ + A £ .  M 2  t2  t2  2  The total return is the sum of the full information return, the uncertainty return, and the noise return. The full information return A £ equals zero by (3.3). Since (2  the uncertainty return is not correlated with the noise return, we have Cov[A , A ] = Cov [ A g ht2  t2t3  V  v  uncertainty effect  ] +Cov '  . *  noise effect  (3.10)  '  Hence the covariance of the returns during the two time intervals [ti, £2] and [£2, h] is the sum of two effects: the uncertainty effect and the noise effect. 41  L e m m a 4 (Correlation of returns with noisy public information). / / the price is given by Lemma 3, then we have  Cov[A» ,A» ] t2  Cov [ A , A ] M2  t2t3  = Gov [ A J k ,  t3  + Cov [A^ , 2  <  0,  =  0.  Lemma 4 shows that the uncertainty effect is positive, the noise effect is negative, and the two effects offset each other exactly. The uncertainty effect is positive for the following reason. Assume the firm will pay a positive dividend at some trading date in the future. Since the price reacts to this information slowly, the positive future payoff will increase the return during the first period [ti,t ] and during the second 2  period [t , £3]. In this way, the slow reaction of the price to news has a positive effect 2  on the correlation of returns. To see why the noise effect is negative, assume a signal associated with a particular future dividend contains a positive noise component. On each trading date, the investors increase the weight that they put on this signals when they update their beliefs. Therefore, the positive noise component will initially increase the price and produce a positive return, since the investors can not distinguish between noise and true information. However, at some future trading date, the investors will learn that this particular part of the signal was just noise. A positive noise component today will therefore decrease the price at some point in the future. In that way, the signal noise has a negative effect on the correlation of returns. Lemma 4 shows that the uncertainty effect and the noise effect offset each other exactly. This result makes intuitively sense. Since the riskiness of the firm stays constant over time, changes of the price can only be due to changes in the information, 42  and not to changes of the risk premium. But the price at any given time reflects all publicly available information. Future price changes are therefore caused only by the arrival of new information. Since new information is not correlated with today's information, past returns cannot have any predictive power for future returns.  3.3  Heterogenous Uncertainty about the Income of the Firm  I will now compare the economy with homogeneous beliefs of the previous section to an economy with heterogeneous beliefs. Assume Investor i receives at time t the information lit = {D , t  dit+ij, d 2,ti dit+3,t, • • •}, it+  (3-11)  where s dit+ ,t — dt+s + Y^ s  ^•it+sj >  and the noise terms  e  j  it+Si  are independent across the investors. The only differ-  ence between the homogeneous information structure in (3.11) and the information given in (3.4) is that now the investors observe signals with individual noise terms instead of common noise terms. There are two possible interpretations for the individual noise terms. One is that the investors receive private information about future dividends. The other interpretation is that the investors observe the same public information, but understand this information in different ways. In this case, d+ t  s  can be interpreted as the true information contained in the public signal, and  the individual noise terms can be interpreted as the mistakes that the individual investors make when they analyze the public signal. 43  In either case, if the investors receive heterogenous signals they will have an incentive to use the price as an additional source of information. If the investors are perfectly rational, the result of Grossman (1976) will apply, and the price will fully reveal all information. In this case, the only possible equilibrium price is the full information value of the risky asset. The assumption of perfect rationality is strong, since it requires that the investors completely understand the underlying model of the economy, and are able to infer information from the price without an error. Instead, I will now assume that the rationality of the investors is bounded. When the investors use the price to infer information, they make individual mistakes. In addition, I will for simplicity assume that the investors only use the current price and not past prices as a source of information. This assumption is not as restrictive as it might seem, since, if all investors are rational, past prices are redundant information as the result in Lemma 5 will show. Note that, since investors are myopic, their demands depend only on their beliefs about the next periods' payoff of the asset. Let V = P -E[P \I }. it  We can interpret  t  t  (3.12)  it  as the unexpected part of the price Pt, after accounting for the  information contained in 1^. If In and P are jointly normally distributed, then it t  follows form the properties of the multivariate normal distribution that I# and are independent, and that E[D  t+1  + P \I , P ] = E[D t+1  it  t  + P |I,] +  C W [  t+1  Therefore, we can interpret I# and  t+1  ^ { ^] r  P  P t | I < t 1  ^-  (- ) 3  13  in the following way: lu contains the non-price  related information that investor i at time t uses to forecast the future payoff, and ^u contains the additional information that the investor obtains when he observes the 44  price. Since 1^ and ^ are independent, the additional information in  is strictly  incremented to the non-price related information in I . I will now assume that the it  investors make individual mistakes when they infer this additional information from the price. The information that investor i infers from the price at time t is given by *wt = tfat + e*« where the individual noise terms e  (3-14)  are independent normally distributed with  mt  mean zero and variance cr . The demand for the risky asset in (3.5) becomes then 2  X  a  E[P  =  t+1  + D \I ,y }-(l t+1  pVar[Pt  it  +1  + r)P  2it  +  t  D \I ,^2it] t+1  it  Note that by (3.13) the demand function in (3.15) is fully rational, if cr^ = 0. 2  L e m m a 5 (Equilibrium with heterogeneous beliefs). If the demand of the investors is given by (3.15), then there exists an equilibrium price of the form  r  (3.16)  Dt + f^^R'-'dt+s s=l  If the price is given by (3.16), then we must have  and A < s  < 1 for A given by Lemma 3, and lim^^o P = Pf. s  t  Lemma 5 shows that the coefficients $ds have the same properties as the coefficients A in the case of homogenous information in Lemma 3. However, as opposed s  to Lemma 3, the price in Lemma 5 does not contain any signal noise. The price does not contain any signal noise, because the price depends on the aggregate demand and therefore on the aggregate information in the economy. Since the investor 45  specific noise terms in (3.11) cancel each other out in the aggregate demand, the equilibrium price cannot depend on these noise terms. We can write the price in Lemma 5 as Pt  1  +  0 0  s=l  full information value  uncertainty  The price consists of two parts: the full information value of the asset given by (3.1) and a part that is due to the uncertainty about future dividends. By Lemma 5, the uncertainty part decreases the effect of the dividends on the price, relative to the full information value. Since  > A , the uncertainty effect in the heterogeneous a  information economy is smaller than in the comparable economy with homogeneous information. The uncertainty under heterogeneous information is small, since the investors know more on aggregate in the heterogeneous information economy than in the homogeneous information economy, and since the individual investors can access the aggregate information by observing the price. If the individual mistakes that the investors make when they learn from the price are small, the price will be close to the full information value by Lemma 5. Lemma 6  (Correlation of returns with heterogenous beliefs). / / the price is given  by Lemma 5, then we have Cov [A , tlt2  where A^  t  A ] t2t3  = Cov [A? , A£ ] lta  t3  > 0,  is defined according to (3.9a).  Lemma 6 shows that the returns of the time periods [ii,^] and  are posi-  tively correlated. Recall from equation (3.10) that, if the investors receive homogenous information, the covariance of the returns is the sum of two effects: a positive 46  uncertainty effect and a negative noise effect. In the case of heterogenous information, there is no noise component in the price, since the investor specific noise components cancel each other out in the aggregate demand. Therefore, the only remaining effect on the covariance is the positive effect generated by the uncertainty return A¥, .. This uncertainty return is due to the fact that the price reacts slower t  to future dividends than the full information price. In that way, the slow adjustment of the price to future dividends produces positively correlated returns. Chordia and Shivakumar (2002) find that winners outperform losers not only during the post-formation period of a momentum trading strategy, but also during the pre-formation period. Chordia and Shivakumar interpret this finding as evidence that time varying expected returns cause momentum. Note, however, that the covariance in Lemma 6 works in both directions. If the returns during the formation period are positively correlated with the returns during the post-formation period, then the same will be true for the correlation between formation and pre-formation period. Thefindingsof Chordia and Shivakumar are therefore consistent with the idea that aggregate underreaction to new information causes momentum.  3.4  The Underconfident Representative Agent  I will now construct a representative agent for the economy of the previous section. Recall that the heterogeneously informed investors of Lemma 5 observe the signals lit = {Dt, dit+i,t, du+2t, • • •},  d  it+Sit  = d +s + Vit+s,t, t  (3-17)  where the r]it+ ,t are individual noise terms. If the information of the investors is 3  given by (3.17), then the aggregate demand does not contain any signal noise, since the individual noise terms cancel each other out. Since the aggregate demand does 47  not depend on the signal noise, a representative agent must be able to observe the undisturbed signal I = {D ,d ,d ,---}. t  t  t+1  (3.18)  t+2  As I will show in the following, the representative agent suffers from a psychological bias. He updates his priors too slowly, given the quality of his signals. Specifically, he calculates the distributions of future prices and dividends, as if he observed the noisy signal It = {A, d~t+\,u <k+2u •••},  dt  +S}t  =d  t+s  + VZ •  f]t+s,u  (3.19)  where Z is a positive constant, and the noise terms w +s,t in (3.19) have the same t  distribution as the r] ^ in (3.17). I will assume that the psychological bias of the it+a  representative agent is restricted to the way he uses his information in (3.18). When he calculates the distributions of future prices, he fully rationally takes the correct functional form of the equilibrium price into account. The representative agent is therefore aware of the fact that the equilibrium price does not contain any of the noise terms n . st  However, he does not use the price as a source of information,  since he is the only agent in the economy, and since the price can only contain information that he already knows. Even though he is aware that he has access to the full information in (3.18), his psychological bias drives him to update his priors according to (3.19). L e m m a 7 (Underconfident representative agent). Let  Z - l -  Cov[D + P i,Pt\Iit} Var[P \I } + ol  1  t+1  1 +r  t+  t  it  Then the equilibrium price in the representative agent economy is equivalent to the price in the economy populated by individual investors of Lemma 5, and we have  Z e [0,11, lim„2 ^, Z = 0, and lim 2 0  a  Z = 1. 48  Note that the variable Z in Lemma 7 does not depend on i, the index for the individual investor, since covariances of normally distributed random variables only depend on the distribution of the signal and not on the realization of the signal. Part (a) of Lemma 7 shows that the aggregate demand in the heterogeneous information economy of the previous section is equivalent to the demand of a representative agent, who observes the undisturbed signal I = {D , d i, d , • • • }, but updates t  t  t+  t+2  his priors as if he observed the noisy signal I from equation (3.19). In that sense, t  the representative agent is underconfident. He underestimates the quality of his information. The coefficient Z € [0,1] measures the degree of underconfidence. Recall from the previous section that I is the non-price related information of it  investor i at time t, and that $  2it  is the additional information that this investor  obtains when he observes the price. Similarly to (3.13), it follows from the properties of the normal distribution that E[D  t+1  + P \I , $«] = E[D t+1  it  t+1  + P \I ] + (1 + r)(l - Z)# t+1  it  it  (3.20)  (see Appendix C ) . The function Z determines therefore the degree to which the beliefs of the investors depend on the information obtained from the price. If  — >  0, the investors are fully rational, and the price related information will affect the expected future payoff with the factor 1 + r. If a\ — > oo, the investors are not able to infer any information from the price, and the price related information ty will it  not effect their beliefs at all. By equation (3.19) the representative agent is underconfident, if Var[\/Z -nt+sj} = Z -sal > 0, for s = 1,2,3,... . The representative agent is therefore underconfident if and only if both of the following two conditions are satisfied: the individual investors have 49  heterogeneously beliefs (a > 0), and the investors make individual mistakes when 2  they learn from the price (Z > 0). Comparison of equations (3.19) and (3.20) shows that the measure Z ties the ability of the investors to learn from the price to the underconfidence of the representative agent. The better the investors are able to learn from the price, the higher is the influence of the information coming from the price on the private beliefs, and the lower is the aggregate underconfidence. The relation of individual mistakes and aggregate underconfidence shows that bounded rationality on the individual level is equivalent to a psychological bias on the aggregate level. As a result of his underconfidence, the representative agent does not put enough weight on his signals, when he updates his beliefs. Thus, it seems as if the representative agent suffers from a conservatism bias, identified in experiments by Edwards (1968). Individuals who are subject to this bias tend to underweight new information when they update their priors. Barberis, Shleifer, and Vishny (1998) assume that a representative agent suffersfromthe conservatism bias to model the underreaction of the price to new information. Lemma 5 and Lemma 7 show that even if the representative agent is psychologically biased, this does not need to be the case for the individual investors. If the investors make individual mistakes when they infer informationfromthe price, then the representative agent will be underconfident because the investors rationally reduce the weights that they put on the price related information when they update their beliefs. The aggregate conservatism bias and the resulting underreaction occurs therefore naturally in a world with heterogeneous information and bounded rationality.  50  3.5  Comparative Statics  Figure 3.1 shows how the dividend coefficients of the price in Lemma 5 depend on the parameters of the economy. The top curve shows the dividend coefficients for the case, where all investors know the realizations of all future dividends (of = 0). Since investors discount dividends in the far future more than they discount dividends in the near future, the dividend coefficients are higher for dividends in the near future than for dividends in the far future. The remaining curves of Figure 3.1 show how the the dispersion of beliefs and the ability to learn from the price affects the discounting. The fact that the dividend coefficients under heterogeneous information are smaller than under full information shows that the price underreacts to new information about future dividends. The price adjusts to news about future dividends slower if the dispersion of beliefs among investors is higher (high value ofCT )or if the 2  investors are less able to infer information from the price (high value of of ). I will now demonstrate how the expected return of a momentum strategy depends on these parameters.  3.5.1  Momentum profits and length of the holding period  For the two consecutive time intervals  £ ] and [t , t ], I will refer to the first 2  2  3  interval as the formation period and to the second interval as the holding period. Consider the regression coefficient _  Cov[A ,A ] Var[A ] ' tlt2  ht2  where A  titj  t2t3  { 6  -  Z i )  is the cumulative excess return for the period [U, tj], as denned in (3.2).  The coefficient \I> measures the ability to forecast future price differences based on  51  lit*** 100  Figure 3.1: Price Reaction to New Information. Thisfigureshows the dividends coefficients of the price given by Lemma 5. The remaining parameters are given by p = a\ = \, r = 0.004. the observations of past price differences. Since the unconditional expected returns are zero in this model, we have E  Ht  tit -  2 3  2  (3.22)  Therefore, we can interpret \t as the cumulative expected dollar return for a buyand-hold strategy with time horizon T = £3 — t , if the asset has increased by 2  2  one dollar during the period [ti,^]- Figure 3.2 shows how this momentum return increases monotonically with the length of the holding period. The dependence of the momentum return on the remaining parameters is less clear. Even though, as Figure 3.1 shows, the parameters of the economy affect the underreaction of the price to new information about future dividends in an intuitively obvious way, this effect is not as obvious for the momentum return. The dependence of the momentum return on the parameters is complicated, because the underreaction affects current and future prices, and the momentum return depends on the difference between 52  0.3  0.11  36  Figure 3.2: Momentum Return. return  =  ^^j^^f^  This figure shows how the momentum  depends on length  of the holding period T . The 2  momentum return \I/ • Ti is the expected dollar return for a buy-and-hold strategy with time horizon T , if the asset has increased by one dollar during 2  the formation period. The remaining parameters are given by Ti = 6, a\ =  0.1, ol = 10 , o\ = p = 1, r = 0.004. 5  these prices. Therefore, instead of focussing on the dependence of the momentum return on the parameter values, I will now show how the behavior of the asset during the formation period affects the holding period return.  3.5.2  Momentum profits and reversals during the formation period  I will now divide the formation period [ti, t \ into two separate time intervals [t\ , t\b] 2  and  [ti6,i2]-  a  Figure 3.3 shows a picture of the time line. In order to be able to 53  formation period H ha  T\  holding period  1  1  hb  £2  h£3  Tib  a  T2  Figure 3.3:  T i m e Line.  compare momentum returns for formation periods with different lengths, I multiply the expected holding period return ^ with the length of the corresponding formation period. If the length of the formation period is Ti, then the resulting return \I> - Ti is the expected dollar return for a buy-and-hold strategy with time horizon T , if the 2  asset has increased by an average of one dollar per trading date during the formation period. The top part of Figure 3.4 shows the expected returns from three separate regressions, where the dependant variable is either the return during the first part of the formation period, or the second part of the formation period, or the total formation period. For example, ^ Ti a  a  in the top part of the Figure is the expected cumulative  return for the holding period [ii,^], if the asset has had an average return of one dollar per trading date during the period [ti , a  Similarly, ^bTu is the expected  holding period return for an average return of one dollar during the period [t\ , tu,]. a  The return \&Ti is the expected return, if the average return was one dollar during the period [ti, ^2] • The bottom part of Figure 3.4 shows coefficients \I> and ^b from a  E  A  tlbt2  =tf A a  tlofu  + Vl> A . b  tl6t2  (3.23)  The sum ty Ti A + ^bTuB can be interpreted as the expected dollar return for a  a  the holding period  [£2,^3],  if the asset has increased by an average of A dollar per"  trading date during the first part of the formation period [£i ^lb]? and an average 05  54  of B dollar per trading date during the second part of the formation period [t\ , t\b]. a  The top part of Figure 3.4 shows that taken separately, the returns during both parts of the formation period are positively correlated with the return during the holding period. However, if one calculates expected returns as in (3.23), the bottom part of Figure 3.4 shows that the return during the first part of the formation period is negatively correlated with the holding period return, and that the return during the second part of the formation period is positively correlated with the holding period return. Assume the return of the asset during the second part of the formation period [tu,, t ] is positive. Then the expected holding period return will 2  be higher, if the return at the beginning of the formation period [ti , a  is negative,  than if the return at the beginning of the formation period is positive. In other words, the asset exhibits stronger momentum if the asset has experienced a recent reversal. In addition, it is also possible to predict future reversals. Assume, that the returns A  t l o t u  and A  t l f c t 2  are both positive, but that the return at the beginning  of the formation period A  t l a t l 6  the formation period A  . In that case, as Figure 3.4 shows, the expected holding  t l b t 2  is sufficiently higher than the return at the end of  period return will be negative, even though the total return for the formation period [£1,^2]  is positive. As Figure 3.5 shows, for certain parameter values it is possible to  predict the timing of the reversal during the holding period. To understand the source of the predictive power of reversals note that future dividends produce trends in the price, if the price reacts to news about dividends slowly. Assume now that the price of the risky asset has increased in the recent past. Then it is not clear whether this trend has just started or whether it is already at the end. Dividing the formation period into two subperiods helps to distinguish between these two situations. If the price has decreased during thefirstpart of the  55  F i g u r e 3.4: M o m e n t u m R e t u r n w i t h R e v e r s a l s d u r i n g t h e F o r m a t i o n  Period.  In thisfigure,the formation period [ti,t2] is divided into two time intervals  [hajtib] and [t\b,t2\- The return \f/Tj in the top and in the bottom part of thefigureis the expected dollar return for a for the holding period \t2,ta\, if the asset has increased by an average of one dollar per trading date during the formation period \t\,t^. The return ty Ti (^bT\b) in the top part of thefigureis the expected dollar return, if the a  a  asset has increased by an average of one dollar per trading date during the formation period [ti ,tib] ([tib,^])- In the bottom part of the figure, the sum 9 Ti A + ^bTuB is a  a  a  the expected dollar return, if the asset has increased by an average of A dollar per trading date during thefirstpart of the formation period [ti ,tib], and an average of B dollar a  per trading date during the second part of the formation period [t\ ,tib\- The remaining a  parameters are given by T  = Tib = 3, T i = T i + Tu, of = 0.1, a\ = 10 , a\ = p=\, 5  l a  a  r = 0.004.  56  m-m-m u mm m mm • • • > • • • • > • • > • • • • • • • •  36  57  F i g u r e 3.5: M o m e n t u m r e t u r n w i t h reversals d u r i n g the  formation  pe-  r i o d I I . In thisfigure,the formation period [t\,t2] is divided into two time intervals [tia,tib] and [ii&jtz]. The return i t T i in the top and in the bottom part of thefigureis the expected dollar return for a for the holding period  [£2,^3],  if the asset has increased by  an average of one dollar per trading date during the formation period [ti, £2]- The return ^a.T\ (&bT\b) in the top part of thefigureis the expected dollar return, if the asset has a  increased by an average of one dollar per trading date during the formation period [t\ , t\b] a  ([hb,^])- In the bottom part of thefigure,the sum  Ti A +  a  a  bTibB is the expected dol-  lar return, if the asset has increased by an average of A dollar per trading date during the first part of the formation period [ti ,tib], and an average of B dollar per trading date a  during the second part of the formation period [t\ ,t\b\. The remaining parameters are a  given by T  la  = T  lb  = 3, r = T x  + T , a = 0.01, 0% = 10 , a\ = p = 1, r = 0.004. 2  la  lb  5  t  58  59  formation period and increased during the second part, then it is more likely that the recent price trend is a new, and therefore the trend is more likely to continue.  60  Chapter 4 Conclusion In this thesis I examine how investors can infer private information from market prices and aggregate trading volume. I develop two theoretical models to address the following questions: (1) how can investors fully rationally infer information from prices and trading volume, and (2) how do the results of a world with perfectly rational investors change, if the rationality of investors is bounded, so that they are not fully able to infer all available private information? In the first part of this thesis, I provide a closed form solution for a rational expectations equilibrium where all investors infer information about the state of the economy from (1) private signals, (2) the market price and (3) aggregate trading volume. My model shows how investors can learn from trading volume, and how volume information differs from information contained in the price. The main result of this model is that trading volume reveals the relative quality of the aggregate private information in the economy. Under low trading volume, the aggregate information is more precise compared to private signals than under high trading volume. Investors therefore use volume to decide how they should weight the market price 61  relative to their own private signals when they update their beliefs. When trading volume is low, investors weight the market price more heavily. Conversely, when trading volume is high, investors weight their private signals more heavily. In this model, the price does not fully reveal all private information, since the price also depends on the unknown aggregate exposure to the labor risk. This combination of private information and unknown labor risk has two effects on the autocorrelation of returns. First, private information causes return continuations, since the price incorporates this information slowly. Second, the aggregate labor endowment causes return reversals, since this endowment influences the price temporarily without affecting the future payoff of the risky asset. The total effect in this model is always negative, so that returns are negatively autocorrelated, if investors are fully rational. In the simplified world of the model in the first chapter of this thesis, the price can only be partially revealing, if the price contains an additional source of noise, such as the uncertain aggregate labor risk exposure. Without this additional noise, fully rational investors can infer all private information from the price. In real financial markets, however, prices aggregate information in a complicated way, and extracting this information is a difficult task. In the second part of this thesis I assume therefore that the computational skills of the investors are limited: they make individual mistakes when they infer information from the price. If investors make individual mistakes when they analyze prices, then they are not able to infer all private information from the price, even if there is no additional source of noise that disturbs the price. Therefore, in a world with bounded rationality, private information can have a positive effect on the autocorrelation of returns, without the negative noise effect that is required if all investors are fully rational. Hence, if the  62  computational skills of the investors are limited, prices react to new information too slowly.  63  Appendix Proof of Lemma 1 Assume the price is given by P = §l{D  (4.1)  + ri + e) -$%N  Let Ti = {D, N, P,V} be the information set of investor i. Then the demand of investor i is given by v  _ E[D - P\T] - pCov[D - P,  NY\Ti}  pVax[D-P\Ti]  '  -  [4  Z)  where  Cov[D-P,NiY\Fi]  =  Tl^t^^-  (  43c  )  Plugging the demand into the equilibrium condition in (2.1) and solving for the price and comparing coefficients with (4.1), we get 64  a n  d $N i Lemma 1. Plugging Q  these coefficients into (4.1), and (4.1) into (4.2), we get the equilibrium demand DY ^-nj.  a v  A, =  Since the  are independent across investors, we have  Proof of Lemma 2 Assume the price is given by P = &j>{D + ri)-^N  (4.5)  Then the demand of investor i is given by (4.2). Let T = {£>i,Ni,V} and T = {Di, N P, V}. To simplify the notation, I will write § h  D  for  and $  N  for  in  the following. Then we have E[D\Ti\ = ^_ } a  +  v  rni-ri  V»V>\n  +  g  ( J  A  (4.6a) (A K M  «)  = oJTaJTal  ( 4 6 b )  =TIV/'^  Cov[D,NiY\ti] E[P|  l+  ^ = v t ^ ]  $  65  +^  (- ) 4 6c  A  - ^ ^ T ^ ^  ( 4 6 d )  Let  Cov[£>, P\Tj) Vax[P|^i]  (4.7)  Then we have E[D-P\T}  = E[D\^ + z(p-E[P\^  -P  (4.8b)  Var[D|7i] = Var[D|^i]-ZCov[D,P|^;] Cov[D,Y\Fi)  CT +CT (l-Z$ )"  =  2  (4.8a)  2  C  (4.8c)  ' D ' " JJ  Plugging (4.8) into the demand of investor i in (4.2) and plugging these demands into the equilibrium condition in (2.1) and solving for the price we get P  =  1  1 a% + tf+ o-  t  Comparing coefficients with (4.5), we get o-l al + al + al{l-Z)  ol +  (4.9a)  a\{\-Z)  PO~DY —  o% + <T* + o*(l-Z)o*  N  °N + ol + a*(l-Z)  (4.9b)  Note that an equilibrium exists, if there exists a real number Z, such that the denominators in (4.9) are not equal to zero, and such that Z satisfies (4.7), where $ u and $JV are given by (4.9). Let g(Z) be the right hand side of (4.7). Then it 66  follows from (4.9) that 5 ( 0 ) > 0, and we have m  9 {  <>b<>?(°D + tf)°N  =  o* o*(o* + o*)o% + f?o*{o% + a\ + o*)o*(o% + o * ) '  )  D  D  hence 0 < 5(1) < 1. Define f(Z) = Z-g'Z).  Then we have / ( 0 ) < 0 and /(l) > 0.  Hence, since f(Z) is continuous for Z e  it follows from the intermediate value  [0,1],  theorem that there exists an equilibrium for some Z e  [0,1].  Next I will show that in any (linear) equilibrium we must have 0 < Z < 1. Assume the economy is in an equilibrium. From (4.6f) and (4.9a) we have  Co»[AP|i-,]  =  ^  (  i  -  ^  |  ^  Using (4.7) we get  Using (4.6e) and (4.7) again we get M. So, since <&D =^ 0 by (4.9a), we have 0 < Z < 1. Assume Z = 1. Then we have  2 = <£+o'  1  by (4.9a), and we must have <&JV = 0 by (4.10). But we can only have  $ J V = 0 if Z > 1 by (4.9b). Hence Z e (0,1). For the demand we have from (4.2) and (4.8) where  o% + l* + o* if ~ ^ * * ) D  = P  ° , DY  Ll  +  +a  )Z  l(l-Z® ))-Z$ ^—  a  *p = 1 - Z  D  N  D  (412a)  (4.12b) (4.12c)  67  Plugging (4.5) into (4.11) we get 1 pWs,i[D\Ti}  ($z> - V <b )D + % €i - (9 - y ® )N P  D  D  N  P  - Vm  N  N  (4.13)  From (4.9a) and (4.12) we have *rj-*p*D  =  , \  2  ,  (4  2  ~ ( 4 + o$)Z$ ) D  - (1 -  Z)$  al{al + al)Z 2\°~D  ol + aZ +  a + ai + ai n  =  a^l-Z)  c  (1 - Z)$  D  (1-Z)$D-(1-Z)$  D  =  (4.14)  0  Similarly,from(4.9b) and (4.12) we have  (al + a (l - W)) ol + al + al \ \ //  - (1 - Z)*  2  £  * r» f^: -z +  -  z)  N  N  a -r a N  n  -(1-Z)  'JV ^ n u  =  (4.15)  0  From (4.8b) and (4.9a) we have VarfDl^ ,] = a (l - $ ). Hence we getfrom(4.13), 7  2  D  D  (4.14), and (4.15) 1-Z  Since the  (4.16)  and rij are independent across investors, we have (4.17)  68  Proof of Theorem 1 B y (4.4) a n d (4.17) w e h a v e V  > V  H  if  L  ^ > * °~D  7  2 D  (  + * >  2  2  .  (4.18)  F r o m (4.8), ( 4 . 1 2 b ) , a n d (4.16) w e h a v e  Xi  =  ^ ei-^ ni D  N  'z$ -& N  pVar^l^j H e n c e w e h a v e f r o m (4.18) V  H  > V  2 D  if  L  ,2  \ 2  pVar[D|^J I  ®  a  /)Vai[D|^] I  c  D  W e have  pVar[D\Fi] M 2  =  p V a r ^ ]  2  *  2  ^  2  +  (z$> ^—  Z^-g^  N  - 2^1pVar[D|^] ) >  2  F r o m (4.8b) a n d (4.9b) w e h a v e 2 Z  $  2  " - T ? - 2 - ^fpV*i[D\Fi] N + n  <T  a  °~D  =  Z<S> ^-^-pa {l-<S> )  °~N  =  Z  +  ~n  N  DY  D  0  *JV-O—;—o -  h~;—o  -(1 - ^ ) $ J V H e n c e w e get  17  69  JV "t"  JV  n  CT  JV  CT  "r ° n  From (4.14) and (4.10) we have Vjjai - Z ( l - Z)$N  „2  N  (1 - Z) & al - Z ( l - Z ) $ N~2 2  'N  D  il-Z)  s2  JI  <N „2 aN  'N  (1-Z) = 0. Hence  Z& N  M = so V  H  aW  n  a% +  al(l-Z)  <0,  > V. L  Proof of Corollary 1 If signals are perfectly correlated, we have -E[-D|.Fj] =  g 2  2 ^ _3 .2 A +0  by (4.3a). In this  case, investors do not use the price to update their beliefs since they cannot directly observe the aggregate labor risk shock N. If the correlation of signals is low, we have E[D|^i]  = E[D|£] + z ( p - E [ P | £ ] ) =  2 • I2 • 2 { ? » - ( 4 + < f l S * £ ) A +  a  a  (1 - Z)$£A +  ^  'AT  > 0,  > 0, and 0 < ^ < if>%. 70  ^  rJVi + Z P  by (4.8a) and (4.14). So since $^ > 0, 0 < $ g < V$  +  and Z € (0,1), we have  Proof of Corollary 2 Let Vaxx, = Varf/}^], if the correlation of signals is low, and let Var# be the corresponding variance if the correlation of signals is high. Then we have \l/p = ^ i ^ - by (4.2). From (4.3b) and (4.8b) we have Var = Vai„ 3 ffi~,f" . g  L  =  )a^%% z)^P-  Z  >  So  D  s i n c e  *D  +<  Z)  So  e (0,1) and Z € (0,1), we have  Proof of Corollary 3 Assume the price is given by P =  + 77 + e) - $jfN -  when the dispersion of signals is low, and by P = $Z(D  +  ri)-§5N-$%S  when the dispersion of signals is high. Then it follows directly from Appendix A and B that  CT2 + C T ( 1 - Z $ ) 2  D  1-Z  hence 0 <  < $f.  71  Proof of Lemma 3 If an equilibrium exists, the price must be of the form CO  P =$ + t  +Y  t  (4.19)  $dtsd +s t  3=1  by Definition 1. Then we have A+i +  Pt+i  = $t+i +  + E ^t+is  (1 + $Dt) A + i  dt+l+3  ^t+l+SJ  (4.20)  3=1  The variance of the signals contained in the vector I is given by t  Var dt+ ,t  Var  s  dt+s + E  6 t + s  j  a  3=1  2 , +  d  2 sa . €  Let (4.21) Since all the elements of I are independent, we have t  E E  A +i d i  H+l+s  y~i  t+i+s,j  e  a  t +  s a.  j~  It  + S t  (4.22)  t,  (4.23)  A id +i+ js +  t  s  .3=1  So E [ D + i + Pt+i] i t  t  CO  =  $  t +  i +  (1 + § + i ) { D D t  t  + AA+^t)  +E s=l  $  t +  i +  (1 +  $rjt+i)(A + A d ) x  t+U  +E 3=1  72  ®dt+is  A  As+i -\  " W s - ^ d  *  Sa  A  2~ A + i s  d,t+l+s,t  Plugging this conditional expectation into (3.5), and (3.5) into the equilibrium condition in Definition 1, and solving for the price, we get i f  A  00  Pt =  \  dt+ls—Z dt+l+s,t I •  Comparison of coefficients with (4.19) gives 1 -(- r =  Y ^ ( i + $Dt+i),  $«tt+n = r17+-r( l + ^ + i ) > .  l A , 1 + r A _i a  If an equilibrium exists, then there exists a real number M such that |$ | < M, t  \®Dts\  ®Dt =  <  M  ,  and  and $  \$dts\ < M for  dfs  all t, s, by condition (b) of Definition 1. So $ = 0, t  =  Proof of Lemma 4 I will prove the Lemma for Cov A  t)t+  i,  A i t +  ) t +  . The case for general time intervals  2  follows by induction. From (3.7) and (3.8) we have  r ^ s=l  (1 + r) '  v  So -  i +  r  ^ + i  -  73  r  2_.  (i  +  r  )  , - i  s  dt+s-  So  Cov since A  s  > A  a + i  \u  u  _ d  A —A  1 ^  A  a  A  a  A _i — A  a+1  s  ^  a  n  . W e have  2^(i  r  5=1  +  r  )-i^+-.''  v  7  SO / A  0 0  ^ti*2 - (1 ) T x * a /  + r  ^ti  _ r  E  A  \  l As-1 — A ^  oo  \  s  _  .  (1 + r ) s=l ' v  S o w e get  <7  2  Cov  ^ i , "A+l 8=1  *  '  Hence  Cov At,t+i, A i t +  So  Cov  AN  ) t +  AN  2  A  A t+l,t+2  8=1  *  '  8=1  ^  '  _  _1_ sr^ A — A i  ~  ^2 2-^1 (\ _|_ ] 2 s - l s=l ^ '  =  0.  a  A  N  N  *-%t+l> ^t+l,t+2  s+  r  < 0.  74  A a  - A -, l  s  A  A  a  .  s  Proof of Lemma 5 Assume the price is given by oo  P = a> D + Y,*d.dt . t  D  t  (4.24)  +a  3=1  Since all elements of lu are independent and since Cov[d , d ] = 0 for t ^ s, we have t  s  Var[d \Iu] = sa A ,  (4.25)  2  t+s  s  where A„ is given by (4.21), and we have Cov[d , d \Iu] = 0 for t ^ s. So we get t  s  oo  E[P \l D ] t  iu  = $D +  t  D  t  J2®ds [dt+s\Iit], E  3=1  Var[P + € \I , D ] = £ $ Var[d \I ] 2  t  Pit  it  t  s  t+s  it  + a , 2  p  3=1 OO  E[Dt  +1  + Pt i\Iu,D } +  = {l + $ )(D  t  D  + E[d \Iu})  t  +Y®d3-iE[d \I ],  t+1  t+s  it  3=2 oo  Cov[D  t+1  + P i,P t+  t  + e \Iu, D ] = (1 + $ )<f> iVar[d \Iu\ + ^ Pit  t  D  d  t+1  § -i$ Var[dt \lu ds  ds  +s  3=2  Note that all of these conditional moments are finite, since v  _ Cov[D i-rPt i,Pt+epu\Iit} Var[Pt + e u\l ] t+  +  P  75  tt  \<tds\  < M. Let (  "  . ^  We have E[D  + P \I ,D ,P  t+1  t+1  it  t  +e ]  t  Pit  = E[D  + P i\Iiu D ] +Y(P + e  t+1  =  t+  t  t  Pit  t  t  it  - Y^ )E[dt \I ,  (1 + $ ( 1 - Y))D + (1 + $ D  - E[P \I , A ] )  D  dl  +1  it  A]  OO  + £(4>*-i  -  Y$ )E[d \I } ds  t+s  + Y(P + e ).  it  t  (4.27)  Pit  s=2  Plugging the conditional expectation into  (3.15),  using  (3.6), ( 4 . 2 2 ) ,  and the equi-  librium condition in Definition 1, we get  «- i + r - y 1  (1 +  - Y))D + (1 + $ t  D  - YQ^AKk+i + J ^ - i  -  Y$ ) ds  8=2  Comparison of coefficients with (4.24) gives $  = r  D  (4.28a)  *- " T r  H  - | ! t , - r )  ( 4 2 8 b )  Next I will show that 0 < Y < 1 + r, if a ^ > 0. From (4.28a) and (4.28b) we have 2  1  +$  D  =$  [y + A7 (l + r- Y)]. 1  dl  So, using (4.28c) we get oo  Cov[D  t+1  + P ,P t+1  t  + e \l D ) Pit  iu  t  =  + 5=1  76  + ~ r  Using (4.26) we get (l±r)E~ D ] - ——— 1 + Var[P + e it\\iu D \  A-^l Var[d \l ] ^ i ^ . r j IT I 2w =i(A» ~ ) d l. t+«l ^J  x  Cov[D i+Pt+i,Pt+£pit\Iit, t+  s  t  P  t+s  n  t  it  2  1  < J )  v / a r  S  t  £ f  s  So, applying (4.26) again, we get Y  (l + r ) S Z i ^ ' ^ H ^ M  _  ( 4 2 9 )  So 0 < Y < 1 + r, since A > 0. Let s  R  ^  1 +  r  y + ^-Hi  s  r-r)'  +  and s From (4.28c) we have 1+r  A  B,  **=—n  1.  T  #  -  =  Since 0 < 7 < 1 + r we have A < B < 1. So A < $ s  and  lims^oo  =  s  a  (  4  -  < 1, and  ds  °  3  )  > $ds+i>  0. From (4.29) we have lim 2 Y = 1 + r. Hence lim 2 ^ B s a  CT  0  liixw _ $ds = 1I will now show that an equilibrium exists. Let 0  L = K r  u, Then we havefrom(4.30) L <  1  r(l + r) ~ ' s  1  < U . Let B be the set of all infinite sequences s  of real numbers x = (xo,x\,...), such that x € [L, U ]. For $ = ($/0) $71) • • •) £ # s  s  let /,(*) be the right hand side of (4.30). Let f ( * ) = (/ ,/i,.. .)• Then f € B. 0  77  i  With each element x € B associate the norm ||x|| =  siux,.  6 x  |a; |. Then f($) is s  continuous. Since lim _ (l7 - L) = 0, it is easy to show that B is compact. So 1  s  00  s  B is complete. Also B is clearly convex. Hence it follows from Schauder's fix point theorem that there exist a <&* € B, such that f ($*) = <&*.  Proof of Lemma 6 If the price is given by Lemma 5, then we have P = Pf + P/ , where 7  t  1  0 0  P" =  -Y(^s-R - )d . s 1  t+s  3=1  So 11 v^v A , _ i  1 ^t,t+i  -  / 1  +  r  t + i  t  -  r  - A,  (1 + r) 3  r  1  So 1_ d  AU U ^t,t+l»^t+l,t+2j A  r  A - A ^ (l + r) s  a  -  ?  /  s  s+1  „ ~7J  s  A -! - A f >0, (l + r)» s  since A > A \. s  s+  Proof of Lemma 7 Assume the price is given by oo  P =$ D t  D  t  + J2$dsd s. t+  (4.31)  8=1 lr  The proof that B is compact is a simple extension of the proof that every k-cell is compact.  See for example Rudin, 1976, page 39, Theorem 2.40.  78  For Z given by Lemma 7, Y given by (4.29), and A given by (4.21), we have a  = sa Z  Var  2  £  3=s  = sa  l-  Y 1  +r  / y +4-1(1+ 1  r  -y)  +r  Let  Then we have 1 +r y +41(1+r-y)'  ( 4  -  3 2 )  Under the assumptions of Lemma 7, the conditional expectation for the representative agent is given by CO  E[D  t+1  + P |Ift] = (1 + ® ){D + Aidt+i) + ^ t+1  D  t  *d._i4<W  s=2  So we get Pt  1  +r  (1 +  Comparison of coefficients gives $£> =  + Aidj+i) + ^  ^-lAadj+a  s=2  and (4.33a)  r7  A, 1  +r  Now the result follows from (4.28), (4.32), and (4.33). 79  (4.33b)  

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