Journal of Financial Economics

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1 Author' peronal copy Journal of Financial Economic 04 (0) Content lit available at ScienceDirect Journal of Financial Economic journal homepage: Chaing noie $ Brock Mendel, Andrei Shleifer Department of Economic, Harvard Univerity, M9 Littauer Center, Cambridge, MA 038, United State article info Article hitory: Received May 00 Received in revied form September 00 Accepted ctober 00 Available online 5 March 0 JEL claification: D84 G G4 abtract We preent a imple model in which rational but uninformed trader occaionally chae noie a if it were information, thereby amplifying entiment hock and moving price away from fundamental value. n the model, noie trader can have an impact on market equilibrium diproportionate to their ize in the market. The model offer a partial explanation for the urpriingly low market price of financial rik in the pring of 007. & 0 Elevier B.V. All right reerved. Keyword: nider Sentiment nformed trading. ntroduction n the pring of 007, financial market, and in particular market for fixed income ecuritie, were extraordinarily calm. Corporate bond pread were remarkably low, a were the price of credit default wap (CDS) on financial firm (ee Fig. ). Thi tranquility ended in the ummer of 007, a the problem with ubprime mortgage precipitated a equence of event leading to a major financial crii. The price of rik eventually roe to the highet level in decade. $ We would like to acknowledge helpful comment from Malcolm Baker, Pedro Bordalo, John Campbell, Adam Clark-Joeph, Robin Greenwood, Sam Hanon, Joh Schwartztein, Jeremy Stein, Jeffrey Wurgler, an anonymou referee, and participant of the Harvard Finance Lunch. Brock Mendal would like to thank the ational Science Foundation for financial upport. Correponding author. Tel.: ; fax: addree: jbmendel@fa.harvard.edu (B. Mendel), ahleifer@harvard.edu (A. Shleifer). t i obviou from the tranquility in the pring of 007 that financial market, and in particular, derivative market, did not anticipate the crii. What make thi fact particularly intereting i that mot of the participant in thee market are ophiticated invetor. Unlike, ay, in the nternet bubble, thi pricing wa unlikely to be driven by the ma of demand by unophiticated invetor. Could the oberved tranquility of market in the pring of 007 have reulted from the trading behavior of ophiticated invetor that maked the potential bad new? n thi paper, we ugget that the anwer i ye. We propoe a very imple model, extending Groman and Stiglitz (980), which focue on the interaction of different type of invetor in a market, the vat majority of whom are rational, and how how thi interaction can utain incorrect price. The baic idea i to conider three type of invetor: a mall number of invetor, called nider, who poe valuable information and trade completely rationally, a mall number of oie trader who are vulnerable to entiment hock and trade on thoe, and the vat majority of utider, who poe no information but X/$ - ee front matter & 0 Elevier B.V. All right reerved. doi:0.06/j.jfineco

2 Author' peronal copy 304 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) Fig.. Credit default wap (CDS) pread //07 /3/07. Thee are contructed a a weighted average of CDS pread for individual firm in the Banking and Financial ervice ector. learn from price and trade rationally. All the nider have the ame information, and all the oie trader face the ame entiment hock. The focu of the paper i the trading by the ilent majority of utider, and it effect on price. The problem facing an utider i difficult. n the one hand, he want to follow the nider who know omething, but ince he oberve only price, would like to chae price increae caued by nider trading on valuable information. n the other hand, he want to bet againt the oie trader who are influenced by entiment, but again ince he oberve only price, would like to ell into a riing market and be a contrarian. Which one of thee motive dominate? n particular, i it poible for thi rational utider to get confued and to chae noie a if it were information? We how that in market with ufficiently few oie trader, the anwer i ye, and utider occaionally end up chaing entiment, thereby uppreing the poible impact of informed trading on price. They do o becaue, in thoe circumtance, they believe that price movement reflect information even though they reflect noie. The compoition of a market can be depicted graphically on a triangle, a in Fig.. The x-axi repreent the proportion of market participant who are nider, denoted by, and the y-axi repreent the proportion who are oie trader, denoted by. The remaining proportion i utider, denoted by, o that the three hare um to one. Both nider and utider are ophiticated, but the former are better informed. n the Fig.. Market compoition. The mae of nider, oie trader, and utider um to one. The Region of interet i the area near the origin but off the axe. market of interet, we think of mot trader a utider: rational and ophiticated, but not well-informed. Thi correpond to point near the origin in thi triangle, labeled Region of interet. We can think of the evidence in Fig. a an outcome in a market in our Region of interet. Specifically, the corporate bond and CDS market are dominated by

3 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) utider, with mall but poitive mae of oie trader and informed trader. n the pring of 007, the oie trader were very calm (and hence very willing to ell inurance), and the majority of ophiticated but uninformed invetor took the low price of rik a evidence that the world wa indeed afe. A a conequence, they were alo willing to ell inurance. Even if there were informed invetor in thi market who aw the rik of a calamity and were buying inurance, their demand wa contrained by their rik-bearing capacity (Lewi, 00). Thi demand wa then inufficient to raie the price of rik ignificantly becaue the utider owned mot capital and believed that the low price of rik reflected good new. Starting in the ummer of 007, public new about fundamental revealed that the low price of rik wa not jutified. Becaue rik wa not correctly priced before, the reaction of the price to new wa ubtantial, a Fig. how. We examine the reponivene of price to entiment when almot all invetor are ophiticated. f S i the entiment hock and p i the price of the aet, our argument require that =@S be large. ne might think that thi will not be true in a market with only a few oie trader, becaue uch a market will behave almot like a market with no oie trader at all. We how how thi intuition can fail. Under plauible condition, =@S can be very large in our Region of interet. Thi implie that the mall ma of oie trader can have a diproportionately large impact on market price. Even with a modet oie trader hock, price can diverge harply from fundamental value in a market dominated by ophiticated trader. Thi counterintuitive reult hold becaue the utider, in their attempt to chae the nider, occaionally chae oie trader intead. n market with a high enough average information-to-noie ratio, each utider demand curve i upward loping. Since there i a large ma of thee trader, they exert trong preure on price in the direction where they oberve movement. Without oie trader, the fully revealing equilibrium of Groman and Stiglitz (976) prevail. utider oberve the price and infer the predictable component of the fundamental value: they fully trut the price. With the firt marginal ma of oie trader, price till reflect mainly information and utider till heavily rely on the price in their expectation formation. Their expectation till move almot one-for-one with movement in price and thu almot one-for-one with noie. They trade on thi information and hence amplify price movement due to noie. We conider three metric for tability and efficiency of the market. The firt i an ex pot meaure of the reponivene of price to the oie trader hock, =@S. The econd i an ex ante meaure of the variance of the price, conditional on the nider information, VarðpjnidenformationÞ. The third i the informativene of the pricing ytem, a defined by Groman and Stiglitz, corrðvalue,pþ. According to thi lat metric, additional nider make the market more efficient, We thank an anonymou referee for uggeting thi expoition. A table market i not very reponive to noie and ha low level of noie-induced variance. on average. However, we are epecially intereted in the firt two metric, becaue they peak to the quetion of how market can deviate from efficiency even when mot trader are ophiticated and oie trader hock are modet. We conider thee metric eparately to ditinguih between ex ante and ex pot tability. f we ee a ingle outcome in which the market price i apparently far away from fundamental value like credit default wap in the pring of 007 and want to undertand how and why thi happened, it i helpful to know that =@S i very large o a moderate entiment hock could plauibly caue a ignificant mipricing. We tudy =@S to undertand the magnitude of mipricing in individual realization of the model. We are alo intereted in the long-run propertie of market behavior. The model i tatic, but we can think intuitively about market behavior over time a many repeated outcome from the model. For thi, the important metric are moment of the data: the informativene of the pricing ytem, VarðpÞ, or VarðpjnidenformationÞ. High informativene or low conditional variance are indicative of a market that behave well, on average, over time. The literature on trading in financial market between better- and le-informed invetor i huge, o we can only refer to ome of the tudie. Groman and Stiglitz (980) conider a model with only rational invetor and demontrate that, when acquiring information i cotly, there cannot be a market equilibrium in which price fully reflect fundamental value. Becaue we are intereted in a different quetion than Groman and Stiglitz, we do not conider the aggregation of information from differentially informed rational trader. Rather, we focu on the effort of uninformed rational trader to piggyback on the trading of the informed one. Kyle (985) conider market with informed invetor and oie trader, but alo an uninformed but rational invetor who, in hi cae, i a market maker. Kyle i intereted in market microtructure, and hence focue on the behavior of a monopolitic rik-neutral market maker, a etting appropriate for hi objective. We in contrat are intereted in the market interaction of mall competitive invetor, and hence have a different model and different reult. Wang (993) preent a dynamic trading model with differentially informed invetor, and how that le-informed invetor can rationally behave like price chaer. Hi model incorporate effect imilar to our, but doe not focu on the extreme enitivity of price to noie in the Region of interet. Kogan, Ro, Wang, and Weterfield (006) examine the connection between oie trader urvival and their impact on market equilibrium. They find that the two are not a tightly linked a naive intuition would ugget. A oie trader wealth goe to zero over time, their price impact can decline much more lowly. Their reult are imilar to our in that they find oie trader impact can be diproportionate to their wealth, although their mechanim focue on the type of trading irrational trader engage in, rather than interaction effect. Barlevy and Veronei (003) conider a model with rik-neutral utider trading with oie trader and nider, optimally extracting information from the price of an aet. n their model, the utider

4 Author' peronal copy 306 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) have a non-monotonic demand curve, leading the relationhip between price and fundamental to be S-haped. Thi induce a dicontinuity in price when the fundamental fall below a certain level, which Barlevy and Veronei interpret a a crah. Their mechanim i different from our, but their market tructure i imilar. n Stein (987), rational peculation can impoe an externality on trader trying to make inference from price, and conequently detabilize price. n Calvo (00), rational uninformed invetor optimally extract information from price affected by informed invetor. ntead of being confounded by the preence of oie trader, the confound he conider i occaional liquidity hock to the informed trader forcing them to withdraw from the market. The uninformed trader miinterpret thi a a negative hock to fundamental and drive down price. ur paper i alo related to the literature on noie trading. DeLong, Shleifer, Summer and Waldmann (990a) model the interaction between rational peculator, who would correpond to the utider in our model, and oie trader. With no nider in that model, trading by peculator unambiguouly tabilize price. n DeLong, Shleifer, Summer and Waldmann (990b), arbitrageur buy in anticipation of poitive-feedback trading by the oie trader, and thu detabilize price. 3 Allen and Gale (99) preent a model of tock price manipulation by a large invetor, who buy and thu timulate demand by uninformed invetor trying to infer information from price movement. Roi and Tinn (00) ue the Kyle (985) framework to model poitive-feedback trading by rational uninformed invetor trying to learn from price. Their model ha everal period and a different etup than our, but they are trying to get at ome related idea on how uninformed but rational peculator balance their deire to follow nider and to bet againt oie trader. Stein (009) conider arbitrageur trading againt a tatitical regularity (under-reaction) cauing a new type of market inefficiency in the proce of trading away profit opportunitie on the old type. He how that price can ometime be further away from fundamental value than they are without the arbitrageur. n both hi approach and our, rational trader try to puh price toward their rational expectation of fundamental value, but in our approach the expectation of fundamental value derive from both a private ignal and obervation of the price, wherea hi trader oberve the price and a tatitical regularity they can take advantage of. The ret of the paper proceed a follow. n Section we formally preent and olve the model. Section 3 3 We are auming that all the trader are price-taker, but in the limiting cae when we are thinking of literal nider, it i worth thinking about the poibility of price-manipulation. n the two-period model we conider, the nider problem turn into roughly that faced by the Kyle (985) nider o he trade le aggreively to make monopoly profit. To get more intricate price-manipulation behavior, we would need a longer-horizon dynamic model a in DeLong, Shleifer, Summer and Waldmann (990b). t i difficult to imagine equilibria in which nider ytematically manipulate price to take advantage of the utider upward-loping demand curve. n uch a trategy profile, it will generally be optimal for the utider to deviate by ubmitting a downward-loping demand curve. examine the lope of an utider demand curve. Section 4 analyze the implication of the demand curve for market equilibrium. Section 5 conider meaure of market tability and efficiency beide the enitivity of market price to entiment. Section 6 conclude. All proof and derivation are in the appendice.. The model There i a market for a riky aet in upply trading at price p. There are two period. Trading occur in period, then the aet pay off it fundamental value V in period. 4 The fundamental value i the um of three term. Firt i the unconditional expectation m. 5 Second i a hock n which i realized in period. n i ormally ditributed with mean zero and variance. Finally, there i a hock n to fundamental value which i not realized until the econd period. n i alo ditributed ormally with mean zero and variance. The fundamental value i then given by V ¼ m n n : ðþ n addition to thi riky aet, there i a rikle aet in elatic upply with return r. There are three type of agent participating in thi market: a ma of oie trader, of nider/informed trader, and of utider/uninformed ophiticated trader. We normalize ¼. n period, the nider tradergetaignalabouttheterminationvalueoftheaet. That i, each nider oberve the ame n. The oie trader do not learn from price and have a biaed belief about the fundamental value of the aet, given by a hock to their level of entiment, the random variable S. 6 S i ditributed normally with mean zero and 4 We think of period a a length of time over which all the agent trade anonymouly and repeatedly until the market ettle into equilibrium. n any uch ituation, nider and oie trader will make initial trade. f their demand curve are upward loping, the utider will trade in the direction of the ubequent price movement, and the equential behavior of trade may reemble that in a model of rational herding (ee Bikhchandani, Hirhleifer and Welch, 99; Froot, Scharftein and Stein, 99). We do not model thee dynamic interaction. 5 A Malcolm Baker pointed out, we could imagine a ituation in which utider have more variable in their information et (uch a rating on tructured finance), jut fewer than the nider. We could then write the value V a V ¼ mn 0 n n, where n 0 i obervable by all the agent. But thi i equivalent to a renormalization of the contant m to mn 0. Up to redefinition, any other variable we include in the model which are common knowledge become part of m. f intead, the ignal n 0 i obervable only to the nider and the utider, the equivalence i more complicated becaue we need to add a conditional mean of n 0 to the oie trader hock S. The reult are eentially unchanged. We are therefore weeping under the rug the majority of the information available to the trader by putting it into the contant m. Any information revelation or fundamental reearch that occurred before period i important economically, but doe not bear on the interaction we conider. 6 An alternative approach to modeling the idea that ophiticated trader react to noie i dipered information. n thoe model, trategic complementaritie caue trader to partially coordinate baed on a noiy public ignal uch a a price, o noie in the public ignal can be ubtantially magnified a each trader react to the other action. See Allen, Morri, and Shin (006), Angeleto and Pavan (007), Angeleto and La (009), and Haan and Merten (00), among other. n particular, Merten (008) find that with dipered information, mall ditortion in belief can render arbitrage infeaible.

5 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) variance S. Every oie trader ha the ame realization S. S i independent of all fundamental. 7 utider are rational and optimally interpret the price ignal they oberve. All agent have contant abolute rik-averion utility with parameter g. We begin by deriving the period- demand curve directly from utility maximization. Each agent i begin with wealth W i and chooe demand D i to maximize E i ½ e gðd iv ðw i D i pþrþ Š: Maximizing thi expreion i equivalent to minimizing minu thi expreion, which i in turn equivalent to minimizing the log of that. Auming for the moment that V i normally ditributed conditional on agent i information et, the firt-order condition immediately give the demand curve: D i ¼ E i½vš pr g i ðvþ, ðþ where E i ½Š denote the expectation with repect to agent i information et and i ðvþ denote the variance of V conditional on agent i information et. For the nider, thi become D ¼ m n pr : ð3þ g For the utider, thi become D ¼ me½ n jpš pr, ð4þ g where E½ n jpš and are endogenou. i given by ¼ Varð n jpþ : Finally, the demand for the oie trader i given by D ¼ ms pr, ð6þ g where i the variance perceived by the oie trader. Since the oie trader do not oberve a ignal or ue the price to update their information et, their perceived variance i the ame 8 a the ex ante variance ¼ð Þ. With all thi in hand, we can proceed to olve the model. mpoing 7 n the CDS market in 007, we would argue that the price of rik diverged from fundamental, caued by a miperception about the rikine of the underlying, which implie a miperception about the expected return on the derivative. 8 n the model, we give each trader the ame rik averion and exogenouly pecify the oie trader perceived variance a.we could imagine ituation in which the level of rik averion varied acro type or in which the oie trader incorrect belief extended beyond the firt moment of the aet value. Making thee tranformation turn out to be equivalent to further altering the compoition of the market. Conider replacing the ma of the utider (or oie trader or nider, repectively) with their ma divided by their rik-averion parameter and call thee C, C, C. Let C be the um of thee term. For all purpoe we conider, thi market i equivalent to one with homogeneou rik averion equal to one and =C unit of the aet. Since the upply of the aet affect only the equilibrium rik premium, which we do not conider, thi i equivalent to varying the compoition of the market, putting more weight on the le rik-avere participant. Changing the variance perceived by the oie trader work identically, becaue thi variance enter their demand only multiplicatively with their rik averion. ð5þ market clearing and rearranging give g m ð Þ pr ð Þ E½ n jpš¼ ð Þ S n : ð7þ We can olve the ignal extraction problem to find the expectation 9 of n given p. t i given by E½ n jpš¼ ignal, ð8þ ð Þ S where the ignal i proportional to the difference between the left-hand ide of (7) and it unconditional expectation. A complete derivation i given in Appendix A. n equilibrium, the conditional expectation and variance are given by E½ n jpš¼ ð Þ S pr ð Þ m ð Þ g, ð9þ ¼ S 4 ð Þ S 4 : ð0þ Plugging thi back into the market clearing equation and olving for the price give p ¼ r ðm A Þ ABrgð Þ S ABrg n, ðþ where we have defined A and B a A ¼ g g gð Þ, ðþ B ¼ : ð3þ ð Þ S n (), r appear in each term becaue it i the rikle dicount factor. A i a factor decribing the aggregate rik-bearing capacity of the market, the invere of which correpond to the rik-premium agent demand in equilibrium in the firt term. The econd term i the impact of the oie trader entiment hock on the market price. The coefficient here i =@S and will be the ubject of ome examination. The third i the impact of the aggregate information about fundamental value on the price. f the market reemble the oie trader-free benchmark, the coefficient on S hould be cloe to zero and the coefficient on n hould be cloe to r. 9 We how in Appendix A that there are ituation in which the utider may have a higher expectation of fundamental value than either the nider or the oie trader. Thi i another ene in which market can be conidered untable.

6 Author' peronal copy 308 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) The ultimate object of interet are how completely the fundamental information n and the entiment S are incorporated into the price of the aet. We can write the impact of the fundamental information n and entiment hock S a ¼, ¼ ABrgð Þ : ð5þ t i difficult to evaluate thee expreion analytically. n thoroughly tudied pecial cae, there are either only nider and utider or only oie trader and utider. n the former cae, the coefficient on n doe turn out to be r, while in the latter cae the coefficient on S decreae toward zero a the rik-bearing capacity of the ophiticated trader increae. Thee are ign of a table market that price aet effectively. From thee obervation, the natural intuition to build would be that adding more ophiticated invetor, and in particular, adding more informed ophiticated invetor, puhe the coefficient on S toward zero and decreae the market volatility. Similarly, intuition might ugget that a mall necearily implie a mall coefficient on S, ooie trader hock do not get factored into the price of the aet. A we how in Section 5, neither of thee intuition hold for market in the Region of interet. The reaon for thi i that price in thi model are driven primarily by the trading behavior of the utider, who have mot of the rik-bearing capacity and hence ability to move price in thi model. utider are trying to chae information, but may occaionally end up chaing noie. Their effort to chae information make them more aggreive when they think there i more information in the market, which mean that adding nider to the market might detabilize price. Thee effort to chae information alo lead them to chae noie in ome circumtance by mitake, which might alo have a detabilizing influence. n the analyi below, we eek to develop thi logic. To thi end, we focu on evaluating =@S in the Region of interet. n Appendix B, we prove the following lemma: ¼ rð Þ utiderdemandcurveslope, r g ð6þ where the utider Demand Curve Slope i defined =. Lemma make it clear that the lope of an utider demand curve i crucial for tability of financial market, a proxied for by =@S. ur firt tep, then, i to examine thi lope. 3. The lope of the utider demand curve n the cae of interet, utider compoe mot of the market. A uggeted by Lemma, their demand curve and it lope in particular are then important to undertanding to ee how the market behave. The lope of an utider demand curve (after ome rearrangement) i given by dd dp ¼ r g ð Þ S ð ð Þ Þ ð Þ S : ð7þ We can undertand the demand curve better by looking at it three multiplicand eparately. The third term i the eaiet to interpret, a it determine the ign of the lope. Specifically, r r gð Þ S g i the lope of the aggregate nider demand curve time the variance of their ignal minu the lope of the aggregate oie trader demand curve time the variance of their ignal. f the oie trader are noiier than the nider are inide, then the demand curve will be downward loping. The middle term r gð Þ i the lope of the aggregate oie trader demand curve. When thi lope i mall, the oie trader demand i highly inelatic, o it i difficult to trade with them without changing the price ignificantly. Thi make it harder to trade againt oie trader irrationality, a ignificant ource of equilibrium profit for the utider. Limited ability to make profit from the oie trader dampen the utider willingne to trade, making hi demand curve le teep. When thi lope i large, the utider can gain a lot by trading againt oie trader. When thi lope i mall, the oie trader make it hard to trade againt them o the utider demand curve i le teep (Fig. 3). The firt term i harder to interpret: : r ð Þ r g S gð Þ The econd term in the denominator i time the lope of the aggregate oie trader demand curve time the variance of their hock. The firt term i the lope of the aggregate nider demand curve time the variance of their hock, time ð Þ, which i a term decribing the interaction between the nider and the utider trying to emulate them. We would like to undertand thi demand curve in term of three effect: the utider trying to trade againt the oie trader, trying to avoid advere election from better-informed nider, and trying to trade with nider when they have a trong ignal. We interpret the middle term a being olely a matter of trading againt oie trader. Thi make ene, a thi term doe not involve the nider and o cannot have anything to do with them. The expreion = appear in the utider demand curve both additively and multiplicatively. n the third term, it decribe the portion of information

7 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) Fig. 3. utider demand curve lope. The lope of the utider demand curve depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. due to nider, which increae the utider deire to trade with nider, driving up the lope. We therefore interpret thi term a a following-nider or poitivefeedback effect. The expreion alo appear in the denominator of the firt term, and it i large when nider are aggreive trader. The effect of a big term here i to make the lope flatter, regardle of it ign. When there i enough information in the market for the curve to be upward loping, a large = make it le upward loping. When there i not enough information in the market for the utider to have an upward-loping demand curve, thi term make their demand curve le downward loping. We interpret thi term a the advere election term. Whenever nider are aggreive, it make utider le aggreive becaue they are afraid of trading againt nider. We can directly evaluate the utider demand curve lope at ¼0 and ¼0 to ee how the utider behave in the imple cae: ¼ 0 ) dd dp ¼ r gð Þ, ð8þ know that trading againt oie trader i optimal becaue price contain no new information. With only nider to trade with, the demand curve i perfectly inelatic becaue price are fully revealing and everyone behave like an nider (no-trade theorem intuition applie). The eparating cae i eay to identify: Lemma. The lope of the utider demand curve i poitive if and only if ð= Þ 4ð=ð ÞÞ S 40. Lemma ay the utider demand curve i upward loping if the expectation of the proportion of a price move due to nider i greater than the proportion due to oie trader. n particular, for every market with a non-zero number of nider, there i a 40 uch that the utider demand curve i poitively loping whenever 0o o. Moreover, it can be hown that for a fixed poitive number of oie trader, more nider alway mean a higher lope of the demand curve. n the next ection we conider the implication of thi utider behavior on market equilibrium. ¼ 0 ) dd dp ¼ 0: ð9þ 4. Market equilibrium Thee are reauring. With only oie trader to trade againt, the utider demand i very elatic, ince they We are intereted in whether it i poible for =@S to be large in the Region of interet. We know that it i generally

8 Author' peronal copy 30 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) very mall on the axe becaue the ophiticated trader effectively trade againt the oie trader. The general expreion for =@S i difficult to analyze in the interior of the domain, o we take three alternative approache. Firt, we analyze the pecial cae that we do undertand well: market with either no oie trader or no nider. By undertanding thee market thoroughly, we can gain inight into the behavior of market with imilar compoition. Second, we perform local experiment: we ak how =@S change a we move infiniteimally away from one of our well-undertood cae. The market with no nider ha a very mall =@S, a doe the market with no oie trader. We ak how =@S change when we add the marginal nider or oie trader. Thee two experiment are depicted in Fig. 4. The final approach i numerical. We calculate =@S for a range of parameter value and acro the Region of interet to etablih that =@S can in fact achieve a maximum near the origin. 4.. The cae of the miing type To gain more inight into the market equilibrium, we evaluate the comparative tatic of price in the cae in which either oie trader, nider trader, or uninformed ophiticated trader are miing. Firt, uppoe oie trader are abent. When ¼0, note that ¼, o the expreion for the impact of information and entiment on price become r, ¼ 0: ðþ Thi i intuitive. With no noie coming from the oie trader, the uninformed invetor can perfectly back out the ignal n, o they behave a if they are informed. ow, etting ¼ 0 to get rid of the nider trader and noting that thi implie ¼, the comparative tatic ¼ ¼ r : ðþ ð3þ Again thi i an intuitively appealing reult. The utider know that any price movement i due to oie trader o chooe to trade againt it, but their ability to do o i limited by their rik-bearing capacity. Their collective rik-bearing capacity depend on their ma, which i pinned down here to be. Thu, the term diappear from the denominator. Finally, we can look at the ituation with only nider and oie trader, o ¼ ¼ rð Þ ð Þ, ð4þ ð Þ : ð5þ The intuition for thee reult i exactly a above. Thee reult make clear that the model we preent ubume a a pecial cae the previouly tudied model. Each of the three poible pairing ha been tudied eparately, and we are looking at what happen when all three type are preent. 4.. The firt noie trader Whentherearenooietraderinthemarket,weknow =@S i zero. The main contention of thi paper i that market with very mall number of oie trader need not behave qualitatively like market with none at all. To quantify thi claim, we can look at the difference in =@S when we go from ¼0 to 40. To keep the ize of the market contant, we perform thi experiment holding the number of nider contant and changing an utider into a oie trader. That i, d¼ d. Thi i a comparion of the equilibrium behavior of two different but imilarly compoed market. The tronget poible proof of our claim would be a dicontinuou jump. Thi doe not occur, but the next tronget proof would be a very high derivative at 0. n Appendix B we prove that thi i exactly what we ee: p=@s@j ¼ 0 ¼ =rð Þ. n p=@s@j ¼ 0 become arbitrarily large for mall. Fig. 4. Changing the market compoition. We conider what happen locally a we move from market with no oie trader (nider) to market with very few oie trader (nider). Since we generally think of the nider a being a mall population, thi propoition focue on the mot relevant part of the domain. n thi region, the firt marginal oie trader can have an enormou impact on market equilibrium depite being infiniteimally mall himelf.

9 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) The driving force behind thi reult i the poitive lope of the utider demand curve. 0 At ¼0 the utider demand curve i flat. By Lemma, adding a ufficiently mall number of oie trader will make the utider demand curve trictly upward loping. With an upward-loping demand curve, the utider will trade with any price movement they oberve. When the oie trader doe tart trading, the utider chae him trading very aggreively, mitaking it for an nider trade. Thi caue the entiment S to be factored into the price much more trongly than it would if only the oie trader were trading on it. Subequent oie trader do not have nearly a big an effect becaue the utider demand curve flatten and eventually become downward loping a more and more oie trader join the market. everthele, thi propoition capture the fact that it doe not take many oie trader to get a noiy market. Thi big effect only come into play becaue the utider demand curve i upward loping at 0. Thi highlight the centrality of the preence of nider. Without them, thi lopewouldnotbepoitiveandtheeffectofnoiewould not be nearly o pronounced. Thi ugget that there may be circumtance in which adding nider can detabilize the market. We how exactly that in the next ection Detabilizing inider n a market with only oie trader and utider, the utider know any price movement to be caued by the oie trader, o they trade againt any price movement they oberve. Their demand curve are trongly downward loping. utider willingne to keep the oie trader from affecting market price i limited only by their rikbearing capacity. What happen when we tart adding nider? n a perfect world, two nice thing would happen. Firt, the nider information would be factored into the price perfectly. Second, the nider, who have a lower perceived variance and thu a higher rik-bearing capacity, would effectively trade againt any oie trader hock. To examine thi, we look at how =@S change when we add d nider. Holding the number of oie trader contant o that d¼ d, the experiment we are conidering i turning an utider into an nider. The derivative i hard to evaluate in general analytically, but can be igned locally near ¼0 becaue of the fact =@j ¼ 0 ¼ 0. n Appendix B we prove the following propoition: Propoition. For ufficiently mall level of S,, and, increaing the number of nider while decreaing the number of utider increae price intability, p=@s@40. ntead of decreaing the impact of oie trader, adding an nider increae it. Thi effect hold in 0 n general, changing the compoition of the market will alo affect the utider demand curve lope by changing in the denominator. n Appendix B we how that thi i irrelevant for thi particular experiment =@j ¼ 0 =@j ¼ 0 ¼ 0. Starting from the boundary, the firt oie trader or nider ha no firt-order effect on the variance perceived by the utider. particular in the Region of interet near the origin, where there are many utider. The intuition for thi reult i twofold. Firt, when nider join the market, the nformativene of price to the utider goe up quickly. n particular, if i large compared to S and the nider are more inide than the oie trader are noiy, the lope of the utider demand curve quickly hift upward. The firt marginal nider i not enough to make the demand curve lope up, but a the curve hift toward flatne, the utider top trading againt the oie trader, o the oie trader have a greater impact. Thi effect i magnified by the fact that in the Region of interet there are many utider all trading with the ame trategy. Second, after enough nider have been added to the market, the utider demand curve become upward loping. nce thi happen, they tart actively trading with any price movement they ee. Since they cannot ditinguih between price movement caued by nider and utider, they occaionally trade with the oie trader. Again, ince there are many of them, on thee occaion the market behave like there i a large ma of oie trader. Propoition how that in a neighborhood of ¼0, =@S i increaing in, but doe not tell u anything globally. We can numerically calculate thee derivative for a range of parameter value (Table 4). The figure make clear that the effect decribed are tronget when the oie trader are not very noiy. When the average quality of nider information i high compared to the average ize of the entiment hock S, the odd that any price movement i due to noie trading i low, o it i optimal mot of the time for the utider to trade with the price movement. n thee ituation, large entiment hock do not happen often, but even moderate Table =@S, ¼ 0:, S ¼ 0:, ¼. Price enitivity to noie =@S. The equilibrium enitivity of the aet price to noie hock S depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. ¼ ¼ Table =@S, ¼, S ¼ 0:, ¼. Price enitivity to noie =@S. The equilibrium enitivity of the aet price to noie hock S depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. ¼ ¼

10 Author' peronal copy 3 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) Table 3 =@S, ¼ 0:, ¼, S ¼. Price enitivity to noie =@S. The equilibrium enitivity of the aet price to noie hock S depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. ¼ ¼ Table 4 =@S, ¼, ¼, S ¼. Price enitivity to noie =@S. The equilibrium enitivity of the aet price to noie hock S depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. ¼ ¼ hock can become enormouly magnified, even more o than in market with only oie trader. n thee market, the enitivity of the price to the entiment hock S can far exceed both and, a hown in the lower left-hand corner of Fig. 5 and in Table. n particular, the figure how that =@S become large in market with quiet oie trader ( S ¼ 0:) and well-informed nider ( ¼ ). Thi illutrate the central point of the paper: in thee type of market, it i rational for utider to interpret any movement they ee a baed on information, and o trade with them. The ma of utider trade make the effect =@S very large (Table ). Quantitatively, we want to focu on the magnitude diplayed in the lower left-hand panel of Fig. 5 and Table. Analytically, we have p=@s@ p=@s@, but it i the level of =@S that i of fundamental interet. Among the parameter value conidered, the maximum value of for =@S i achieved with 7% oie trader and % nider in the market, in the market with ¼, S ¼ 0:. Thi greatly exceed the value of that would obtain if only oie trader participated in the market and 0.7 if there were no nider in the market. n the cae with mall mae of each type, the interaction of oie trader and nider caue the utider to occaionally chae noie aggreively, o that noie hock are greatly amplified. Becaue S i mall, the market Fig. 5. Senitivity of price to noie =@S. The equilibrium enitivity of the aet price to noie hock S depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively.

11 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) generally behave well, but occaionally the price of the aet diverge harply from fundamental value. n thi ene, the quetion i one of ex ante or ex pot tability. Ex ante, the additional nider make the Price ytem more informative (hown in Section 5.) and more table mot of the time. Ex pot and for pecific realization of S, the additional nider increae =@S and o increae the enitivity of the price to thee hock. Thi i a meaure of ex pot intability. t i tempting to make normative judgement about the effect of nider baed on thi detabilizing effect, but to do o would be premature. Adding nider doe increae the effect of the oie trader entiment, increaing market volatility at time, but it alo lead to fundamental information being factored into the price more effectively, leading to le volatility at time. Fig. 6 how the effect of the fundamental hock n on the period- price for different market configuration and parameter value. n all cae, more nider move the market toward more fully pricing their fundamental information. n thi repect, they are tabilizing the market. Recall that the cae in which nider are detabilizing are thoe in which S i mall compared to and the compoition of the market lie in the Region of interet. Thee are exactly the market where oie trader are generally very quiet. Mot of the time, oie trader get only a mall hock, nider information get factored into the price effectively, and the market behave well. t i only on a rare occaion (like the pring of 007, we argue) that the oie trader get a moderate or big hock and the market behave inefficiently becaue the rational utider trade along with the noie Demand covariance We would like to think that mot of the time, utider uccefully trade with the nider. The reult of the previou ection howed that when they fail to do o, they can fail rather dramatically. Here we how that, on average, they do indeed trade together. A reaonable way to meaure whether utider and nider trade together i the covariance of their demand. n Appendix B we prove the following propoition: Propoition 3. The utider, on average, trade with the nider. Specifically, CovðD,D ÞZ0. Thi i extremely intuitive. f the utider are rational, they mut be doing their bet to emulate the nider. f their demand did not poitively covary, it would be profitable for the utider to flip the lope of their demand curve. utider earn rational rik premia jut for holding the aet, a term we do not focu on. Propoition 3 implie that they alo make money, on average, by trading with the nider and againt the oie trader. oie trader ytematically loe money ince they alway trade againt the information contained in n. Their loe are mitigated only in thoe tate when S and n have the ame ign. Since the two hock are uncorrelated, thi occur only Fig. 6. Senitivity of price to information: =@n. The equilibrium enitivity of the aet price to information hock n depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively.

12 Author' peronal copy 34 B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) half the time, and only rarely will be large enough to offet their loe due to trading againt the nider. How can Propoition 3 and Propoition, be true at the ame time? Mot of the time the utider trade with the nider (thi i Propoition 3). They do not, however, trade on the ame ide of the market 00% of the time. n rare occaion (for the parameter value we are intereted in), the oie trader get a modetly big hock. Becaue of the ignal extraction at the heart of the model, the utider believe that thi i mot likely an nider hock, o trade with the oie trader. Thi mean that the effect of a moderate oie trader hock i big (Propoition, ), but only rarely i there a big enough oie trader hock to caue the utider to trade againt the nider. Propoition i a tatement about =@S and how it change a we vary the number of nider in the market. ow, we expect thi derivative to be non-negative regardle of the compoition of the market, becaue a light increae in S hift up the oie trader demand curve while leaving everyone ele unaffected. Fundamentally, Propoition 3 and, are about different thing. Propoition 3 i for each fixed et of parameter value (pecifically, the makeup of the market). Propoition and are anwering quetion about two imultaneou experiment: How much bigger would the price have been if S had been higher by ds? With that quetion anwered, how much bigger would the anwer to that quetion be if were higher by d? 5. ther meaure of market tability and efficiency 5.. Good variance, bad variance Another metric to meaure the impact oie trader have on market efficiency i the variance of the equilibrium aet price. That variance can be written a p ¼ : Thi variance naturally plit into two piece: variance caued by entiment hock, and variance caued by nider information being factored into the price. The latter i good variance, a it reduce volatility between time and. The remaining bad variance can be looked at a the variance perceived by the nider: varðpjn Þ¼ : ð7þ From thi equation, it i clear that analyzing varðpjn Þ i nearly equivalent to analyzing =@S. Holding S contant and varying other parameter, increae in =@S map one-to-one into increae in varðpjn Þ. A S converge to zero, =@S get large, but that effect i offet by the decreae in S. n Appendix C we how that the limit of varðpjn Þ a S converge to zero i zero, and that the convergence i aymptotically linear. Thi temper the trength of ome of our reult, but leave unchanged the concluion about how the market behavior varie a we vary the market compoition. We would like to think that in a market compoed of ophiticated invetor, adding nider would be tabilizing and would decreae the variance of the price. Propoition howed that =@S can increae, o it come a no urprie that the variance of price can alo be increaed by the addition of nider. n Appendix B we prove the following propoition: Propoition 4. For ufficiently mall and, changing a marginal utider into an nider increae both the variance of the price and the bad variance. The et of parameter for which the variance increae i identical to the et for which =@S increae in Propoition 4. By thi metric a well, adding nider to a market i detabilizing. The interaction between nider and utider in the preence of oie trader caue the oie trader hock to be integrated into the price more trongly, increaing the bad variance. t alo ha the effect of increaing the enitivity of the price to the nider information, increaing the good variance. aturally, thi lead to the quetion of which effect i tronger. A natural way to compare the trength of thee two effect i the nformativene of the priceytem,whichweconidernext. Fig. 7 how the tandard deviation of the price acro different market compoition for variou parameter value. n particular, it how that with quiet oie trader ( S mall), the market can be more volatile with mall number of oie trader than with more of them. With more oie trader, each utider demand curve become downward loping, o utider partially offet the variance caued by oie trader. 5.. nformativene of the price ytem Groman and Stiglitz (976) define the nformativene of the price ytem to be ðcorrðp,n ÞÞ. Thi i a ratio of information-to-noie in price and give a meaure of how well market perform their function of reflecting information known to agent. The nformativene in thi cae can be written p ¼ 4 S ð Þ : ð8þ From thi expreion two important propoition immediately follow: Propoition 5. Adding nider alway weakly increae the nformativene of the price ytem. Propoition 6. Adding oie trader alway decreae the nformativene of the price ytem. Thi effect become unboundedly large a and approach zero. Any increae in the number of nider increae the nformativene of the price ytem. Thi can be een a the combination of two effect. Firt, chaing behavior by the utider caue the bad variance to increae, which would tend to dampen the nformativene of the price ytem. At the ame time, thi chaing behavior i applied

13 Author' peronal copy B. Mendel, A. Shleifer / Journal of Financial Economic 04 (0) Fig. 7. Standard deviation of p. The equilibrium tandard deviation of the aet price in period two depend on the compoition of the market and the relative tandard deviation of the ignal S,n,n. Thee are S,,, repectively. to any information that the nider have. The utider chae the nider, and the good variance increae. Propoition 5 ay that the good variance increae by more than the bad variance. Propoition 6 conider an alternative experiment of adding oie trader (while removing utider). t i no urprie that additional oie trader decreae the nformativene of the price, but it i by no mean obviou that the effect can become unboundedly large a goe to zero. We have analyzed three way of meauring the tability and efficiency of the market, with an eye toward eeing whether a mall number of oie trader can have an effect. The principal concluion i that the preence of oie trader can in fact have a large influence on the market equilibrium. Ex ante, mall number of oie trader do little to diminih the nformativene of the price ytem, but can hugely increae the variance of the price in period. Thereultwefocumoreonitheurpriingone:expot, market with a mall number of oie trader can have large enitivitie to the oie trader hock, =@S. Thi, perhap, can explain the evidence in Fig Why =@S i important Given that there i at leat one metric which cleanly identifie the efficiency of the market, why bother with any other metric, in particular, =@S? The model i tylized and effectively tatic, but if we think of it a repeating itelf, the time erie behavior will be bet decribed by the variance and nformativene reult. t i only when trying to undertand pecific market realization that =@S i important. The ex ante metric how u that, on average, the oie trader may have a fairly mall effect in the Region of interet. t i no urprie that the price react to the oie trader hock. What i urpriing i that the enitivity of the price to the oie trader hock i not monotonically decreaing in the number of nider. n order to undertand particular intance of ophiticated market going awry, it i important to keep in mind that =@S i liable to be big exactly in the market in which we think oie trader are quietet. 6. Concluion We preented a imple model in which rational but uninformed trader occaionally chae noie a if it were information, thereby amplifying entiment hock and moving price away from fundamental value. The model offer a potential explanation for the urpriingly low market price of financial rik in the pring of 007. We fill a gap in the theoretical literature by howing condition under which oie trader can have an impact on market equilibrium diproportionate to their ize in the market. Explaining market outcome by calling on large number of oie trader or large entiment hock