Are Under- and Over-reaction the Same Matter? A Price Inertia based Account

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1 Are Under- and Over-reacion he Same Maer? A Price Ineria based Accoun Shengle Lin and Sephen Rasseni Economic Science Insiue, Chapman Universiy, Orange, CA 92866, USA Laes Version: Nov, 2008 Absrac. Theories on under- and over-reacion in asse prices fall ino hree ypes: (1) hey are respecively driven by differen psychological facors; (2) hey are driven by differen ypes of invesors; and (3) hey reflec un-modeled risk. We design an asse marke where informaion arrives sequenially over ime and is revealed asymmerically o invesors. None of he hree hypoheses is suppored by our daa: (1) Invesors do no respond differenly o public informaion and privae informaion, and hey do no behave in ways ha are claimed by muliple psychological models; (2) no groups of invesors are idenified o drive under- or over-reacion in paricular; (3) price deviaion from expeced payoff canno be jusified by risk merics. We find ha prices reac insufficienly o news surprises, possibly because of cauious conservaism on he par of invesors and under-reacing drifs ounumber overreacing reversals subsanially. Conrary o common beliefs, we find ha over-reacion is caused by slow adjusmen of prices o surprises, similar o he cause of under-reacion. I is he degree of price ineria ha drives he relaive frequencies of under- and over-reacion. We propose a simple price ineria heory of under- and over-reacion: when informaion arrives sequenially over ime, he marke is characerized by a slow convergence oward inrinsic value; when news surprises are of he same signs, prices falls behind newly updaed inrinsic values, manifesing under-reacing drifs; when news surprises change signs, prices again do no adjus quick enough o cach up wih he new inrinsic values, manifesing a emporal paern of overreacing reversals. JEL classificaion: C92; D53; G14 KEYWORDS: Experimenal finance, under-reacion, overreacion, behavior, price ineria, risk aversion Financial Suppor was provided by he Inernaional Foundaion for Experimenal Economics and Economic Science Insiue of Chapman Universiy. We hank Vernon Smih, David Porer and paricipans a Brownbag alk series of Economic Science Insiue. The paper benefied from commens a numerous seminars and meeings in America. Correspondence o:

2 Are Under- and Over-reacion he same maer? A Price Ineria based Accoun Absrac. Theories on under- and over-reacion in asse prices fall ino hree ypes: (1) hey are respecively driven by differen psychological facors; (2) hey are driven by differen ypes of invesors; and (3) hey reflec un-modeled risk. We design an asse marke where informaion arrives sequenially over ime and is revealed asymmerically o invesors. None of he hree hypoheses is suppored by our daa: (1) Invesors do no respond differenly o public informaion and privae informaion, and hey do no behave in ways ha are claimed by muliple psychological models; (2) no groups of invesors are idenified o drive under- or over-reacion in paricular; (3) price deviaion from expeced payoff canno be jusified by risk merics. We find ha prices reac insufficienly o news surprises, possibly because of cauious conservaism on he par of invesors and under-reacing drifs ounumber overreacing reversals subsanially. Conrary o common beliefs, we find ha over-reacion is caused by slow adjusmen of prices o surprises, similar o he cause of under-reacion. I is he degree of price ineria ha drives he relaive frequencies of under- and over-reacion. We propose a simple price ineria heory of under- and over-reacion: when informaion arrives sequenially over ime, he marke is characerized by a slow convergence oward inrinsic value; when news surprises are of he same signs, prices falls behind newly updaed inrinsic values, manifesing under-reacing drifs; when news surprises change signs, prices again do no adjus quick enough o cach up wih he new inrinsic values, manifesing a emporal paern of overreacing reversals. JEL classificaion: C92; D53; G14 KEYWORDS: Experimenal finance, under-reacion, overreacion, behavior, price ineria, risk aversion 2

3 1. Inroducion Numerous heories agains he efficien marke hypohesis (Fama, 1978), have aemped o reconcile empirical findings concerning under and over-reacion of asse prices o significan informaion evens. The purpose of his sudy is o es how robusly hese compeing heories hold up in laboraory markes where he emporal and spaial disribuion of informaion can be precisely conrolled. Experience wih asse markes boh in he field and he laboraory suggess ha human invesors behave in a more complicaed and sraegic manner han in any ad hoc models or simulaions so far consruced. Empirical resuls on asse price responses o news evens seem o suppor sysemaic under-reacion in which average pos-even reurns hold he same sign as pre-even dae reurns (Jagadeesh and Timan, 1993) 1. Meanwhile, numerous sudies conversely noe ha invesors can over-reac causing price movemen feauring disproporional price changes followed by subsequen reversions (DeBond and Thaler, 1985) 2. The anomalies lieraure ends o sugges ha in he shor o medium run, reurns appear o exhibi coninuaion, while over-reacing reurn reversals are more likely o occur in he long run. In eiher case, price adjusmens are considered o be biased. The anomalies lieraure has assembled abundan couner-evidence o challenge marke efficiency hypoheses, bu no provided a unifying alernaive heory. Fama (1998) poins ou ha even hough marke efficiency is a fauly descripion of price formaion, he alernaive hypoheses are vague and rarely esed: his he deems as unaccepable. In oher words, any alernaive mus also be esed o prove ha i ouperforms marke efficiency heory s predicion ha he expeced value of abnormal reurns is zero, and ha deviaion from zero in eiher direcion can solely be aribued o chance. In his sudy, we address hree disinc lines of approaches. The firs approach advocaes ha he price reacion o news evens is deermined by psychological beliefs of an aggregae agen, including he overconfidence and self-aribuion model by Daniel, 1 Jagadeesh and Timan (1993) argue ha under-reacion is he cause for he success of momenum invesing. Bernard and Thomas (1990), and many ohers, have documened drif afer earnings surprises for up o 12 monhs afer he iniial surprise. Ikenberry, Lakonishok, and Vermaelen (1994) conend ha he marke phase of under-reacion is an imporan moive for sock repurchase. Ikenberry and Ramnah (2002) found evidence ha he paern even persiss in sock splis, which is he simples corporae even. Abarbanell and Lehavy (2003) found ha analyss forecasing errors are correlaed wih he reporing choice a corporaion makes in announcing is unexpeced accruals. Mikhail, Walher, and Willis (2003) demonsraed ha he analyss under-reacion o corporae announcemens ends o decrease wih he years of experience ha hey follow he firm. 2 DeBond and Thaler (1985) find ha loser porfolios experience excepionally large January reurn as lae as five years afer he iniial porfolio formaion. DeBond and Thaler (1987) find ha here exiss mean reversion in sock reurns, and excess reurn o losers could be explained by a biased expecaion of he fuure which over weighs recen informaion and under weighs he long erm average. DeBond and Thaler (1990) find ha securiy analyss forecass are overopimisic and exhibi frequen revisions. Dreman and Lufkin (2000) argue ha he profiabiliy of conrarian sraegies resuls from he exisence of over-reacion and ha psychological facors may explain he origin of over-reacion. 3

4 Hirshleifer and Subrahmanyam (1998, hereafer DHS), he conservaism and represenaive bias by Barberis, Shleifer and Vishny (1998, hereafer BSV) and he disposiion effec model by Frazzini (2006). The second approach advocaes ha he ypes of reacions are driven by differen groups of invesors, feauring he informaion ransmission model wih heerogeneous agens by Hong and Sein (1998, hereafer HS). These wo approaches boh regard under- and over-reacion as wo disinc phenomena. The hird approach feaures Fama s (1998) risk-based explanaion, arguing ha underand over-reacion is purely a reflecion of risk ha are no capured by asse pricing models. Our laboraory es uilizes he model presened in HS (1999) o generae he asse valuaion process experienced by invesors. The asse pays a liquidaion value o is owner when ime reaches mauriy wih is book value uni is subjec o a series of surprises during is lifeime and informaion in each period available for he informed o compue he expeced liquidaion value, i.e. inrinsic value. Common knowledge amongs invesors exiss abou he underlying process and he ulimae sae-dependen payoff, and also concerning a periodic srucure of informaion release. Public informaion disseminaes insananeously hroughou he populaion, while privae informaion reaches only a subgroup of insiders. Informaion is boh dispersed over ime and asymmerically hroughou he invesor populaion, generaing a process of HS (1999) informaion diffusion. We idenify under- and over-reacion wih he advanage of knowing perfecly he inrinsic values. Our mehodology reveals under-reacion as he predominan shor-erm and long-erm regulariy in mos markes. To demonsrae he necessiy of knowing he inrinsic value, we also use he radiional reurn sign characerizaion o sor our daa and confirm he worry ha observed anomalies can be creaed by researchers ignoran of inrinsic value. When aking inrinsic value ino accoun, we find ha he radiional coninuaion/reversal of reurns signs does no consisenly indicae rue under/overreacion, as i may easily be misinerpreed absen he knowledge of inrinsic value. Our experimenal resuls demonsrae ha: (1) Daa do no suppor psychological based explanaions, eiher over-confidence model or disposiion effec model; (2) Underreacion and overreacion are no caused by differen groups of invesors. In paricular, HS' informaion ransmission model canno explain he drif paern of prices; (3) Risk canno accoun for drifs; overpricing or under-pricing canno be capured by risk-reurn relaionship; (4) Adjusmen o surprises is robusly insufficien and displays ineria. The inferred aggregae subjecive probabiliies displayed by hese markes indicae ha biased judgmens end o persis and ha he markes exhibi belief coninuaion. The conservaism accoun of BSV is he sole model ha bes governs he ineria paerns among all esable heories. In addiion, we find ha over-reacion is also caused by cauiously conservaive adjusmens. No heory has ever linked cauiousness wih overreacing behaviors. We 4

5 hypohesize ha over-reacion is jus a by-produc of cauiousness as well, and under- and over-reacion are caegorized differenly simply because of he difference in hisorical price posiions relaive o he inrinsic value. This is no so surprising, since mispricing, eiher over or under, would always be deermined by he pah in he inrinsic value. By carefully re-examining he overreacion cases in our daa, we confirm ha overreacion, eiher changing from under-pricing o over-pricing or changing from over-pricing o under-pricing, is mosly caused by prices falling behind he abrup changes in he inrinsic value. To es he price ineria heory more generally, we run numeric simulaions using parameers derived from he laboraory daa. Wihou inroducing anyhing oher han shor-erm insufficien adjusmen, a subsanial level of overreacion (persisen reurn reversal) emerges: he source for his apparen overreacion is also ha prices do no converge fas enough o rack abrup changes in inrinsic value, resuling in a urning in price direcion. The more inadequae he adjusmens o he changes in inrinsic value are, he more under-reacion paerns will prevail. We confirm ha shor erm insufficien adjusmen can be he sole driving facor in deermining he under- and over-reacion regulariies. Boh insances are governed by how responsively he marke prices adjus o informaion evens, raher han driven by conrasing behavioral rais or he ineracion of heerogeneous marke paricipans. We conclude ha informaional efficiency canno be fully described by any single heory surveyed in his paper. We argue ha boh shor-run reurn coninuaions (driven by slow adjusmen) and long-run reurn reversals (possibly he slow re-adjusmens o he new inrinsic value) can be summarized as manifesaions of ineria, and ha he heoreical dichoomy beween under-reacion sudies and over-reacion sudies can be inherenly misleading. The res of he paper is organized as following. Secion 2 surveys some prominen exising heoreical models ha aemp o explain empirical daa. Secion 3 specifies he experimenal design we devised for he laboraory markes and saes he esable hypoheses relaing o he various heoreical models. Secion 4 presens our daa summary. Secion 5 evaluaes he validiies of he five compeing models and presens our simulaion resuls. Secion 6 proposes a price ineria based heory of under- and overreacion. Secion 7 summarizes our findings briefly. 2. Theories on Under- and Over-reacion A number of heoreical models have aemped o provide parial or unified explanaions for eiher under or over-reacion, or boh regulariies. The firs approach feaures he defense of he efficien marke hypohesis. 2.1 Psychological Explanaions The firs approach feaures various models of invesor psychology. DHS (1998) develop a model of invesor overconfidence, examining how invesors assimilae new informaion. The uninformed invesors are no subjec o judgmen bias 5

6 and prices are deermined by he informed invesors. The model argues ha informed invesors are overconfiden abou he informaion hey privaely own, and as a resul, overreac o his ype of informaion. In addiion, a second characerisic named biased self-aribuion reinforces heir over-confidence whenever public informaion is in agreemen wih heir privae signal. When public informaion is no in accordance wih heir privae signal, biased self-aribuion leads o dismissal of he informaion as noise. The model suggess ha he marke under-reacs o public informaion and over-reacs o privae informaion so he responses produce shor-erm coninuaion of reurns and longerm reversals as public informaion evenually overwhelm he behavioral biases. BSV (1998) proposed a model known as a conservaism bias 1 (aribued o Edwards (1968)) which essenially means ha individuals end o underweigh new informaion when updaing heir prior beliefs. Conservaism bias leads invesors o updae heir beliefs very slowly in he face of new evidence. Represenaiveness bias, on he oher hand, leads o he formaion of biased esimaes based on a small sample of observaions. In he BSV model, he earnings follow a random walk bu invesors falsely perceive ha here are wo earning regimes. In regime A, earnings are mean-revering and in regime B, earnings are rending. If invesors believe regime A holds, price underreacs o changes in earnings because invesors misakenly hink he change is likely o be emporary; if invesors believe regime B holds, hey incorrecly exrapolae he rend and he price over-reacs. Overall, invesors form prior beliefs abou fuure prospecs and he prior beliefs carry over o fuure price formaion. Frazzini (2006) proposes ha he presence of disposiion effec 2, iniiaed by Shefrin and Saman (1985), will depress prices following good news as invesors rush o sell o lock in paper gains, and will hal prices from falling following bad news as invesors are relucan o sell absen a premium. The former would lead o higher subsequen reurns while he laer would lead o lower subsequen reurns. Therefore, he disposiion effec posis ha a pas capial gain or loss will shape he subsequen price formaion, a unique explanaion for asse price under-reacion. 2.2 Heerogeneous Invesors The second heoreical approach focuses on he marke microsrucure. Hong and Sein (HS, 1999) hypohesize ha he marke conains wo groups of invesors who rade based on differen sraegies. Informed invesors base heir rades on privae signals abou fuure cash flows while echnical invesors base heir decisions on a limied hisory of prices. Informaion obained by informed invesors is ransmied slowly ino he marke, leading o an under-reacion paern in reurns. Technical invesors rely on he pas hisory of prices and exrapolae he rend oo far, reinforcing momenum and pushing price away from inrinsic value which leads o an evenual reversal in reurns. No 1 See Richards Heuer, Jr., Psychology of Inelligence Analysis, Cenral Inelligence Agency, He wroe: As a general rule, people are oo slow o change an esablished view, as opposed o being o willing o change. The human mind is conservaive. I resiss change. 2 This is a behavioral endency on he par of invesors o hold ono heir losing socks o a greaer exen han hey hold ono heir winners. (Shefrin and Saman, 1985). O dean (1998) and Grinbla and Han (2005) modeled or documened evidences ha invesors 2 indeed rush o sell afer capial gains and are relucan o sell afer capial loss. 6

7 assumpion on he limiaions of psychology is made. HS posi ha informaion aggregaion failure leads markes o under-reac, and consequenly, his under-reacion will be more pronounced when informaion updaes have higher asymmery (Hong, Lim and Sein, 2001). On oher hand, HS model implies ha prices will exrapolae rends and move o overshooing levels due o momenum rading aciivies. 2.3 Risk based Explanaion Fama (1998) presens hree reasons for keeping marke efficiency as a viable working model. Firs, he lieraure does no presen a random sample of evens and pays more aenion o splashy resuls. Second, some apparen anomalies arise due o risk premium ha are no capured by exising asse pricing models. The empirical evidence concerning observed invesor mis-reacions can be he resul of poorly specified risk facor models, wih some effecs disappearing enirely afer accouning for size and booko-marke raio effecs (Fama and French, 1996). Third, Fama (1998) surveys he anomalies lieraure and finds an even spli beween evidence for over-reacion and under-reacion and conends ha long erm efficiency is likely o hold up. The argumen poses powerful skepicism upon he lieraure as here has long been absen a robus asse-pricing model. Empirical ess of hese heories face muliple challenges in daa collecion. Firsly, he observed anomalies may well reflec changes in he fundamenals ha are never known for cerain bu a bes inferred. Secondly, he absence of a robus risk pricing model may depic he marke s raional response o changes in risk facors as improper reacions, an issue Fama (1998) referred o as he bad-model problem. And hirdly, i is hard o draw a disincion beween public informaion and privae informaion, as informaion leakage o insiders before public declaraion canno easily be deeced. A laboraory conrolled asse marke is able o fully saisfy hese difficul daa requiremens and offers a new approach o disenangling he impasse. Smih (1982) formalizes he noion of an economic sysem by specifying hree consiuen componens: environmen, insiuion, and behavior. In he laboraory all environmenal parameers and insiuional rules are perfecly conrolled, hus providing a perspecive ha is impossible o obain in he field. 3. Experimenal Design Our design adops he HS (1999) informaion diffusion model in a muli-period exchange economy where informaion updaes are emporally dispersed and asymmerically held. Boh he DHS and BSV models have a similar muli-period srucure allowing long erm behavior o emerge. Invesors rade from period 1 unil period 10 and receive a liquidaion value per asse held afer period 10 rading is complee. In each period eiher a specific subse of invesors or all invesors receive informaion concerning he asses liquidaion value. There are no supply shocks in he marke ha may preven equilibrium prices from fully revealing he invesors privae informaion. The srucure of his economy is common knowledge. 7

8 3.1. The Random Walk Model Each invesor is endowed wih a porfolio of cash and a risky asse. The cash accoun may be regarded as a riskless bond, and for simpliciy we normalize is ineres rae o zero. The risky asse in he marke pays a liquidaion value of D T a he end of period T, which is imperfecly known o invesors a all imes prior o he end of periodt. The liquidaion value is deermined by wo componens: a known saring book value, D 1, plus a series of news surprises, each direcly affecing he curren book value, wih one surprise beween each consecuive pair of rading periods. Therefore, he book value of he asse as we proceed hrough ime follows a symmeric binomial random walk: D = D + ε, = 2,3, T (1) 1 L, Thus, for any period (afer period 1), he book value is: D, i= 2 = D1 + ε i, = 2,3, L T (2) In HS, ε is randomly generaed by a mean zero 1 normal disribuion. The disribuion for each surprise in our design is simplified o a binomial draw: + u, p = 0.5 ε = = 2,3, K, T (3) d, (1 p) = 0.5 The surprises are independenly and idenically disribued. If a posiive surprise occurs, a definie amoun of money, u will be added o he asse s curren book value; if a negaive surprise occurs, a definie amoun of money, d will be subraced from he asse s curren book value. There is a 50/50 chance of each oucome 2. Jus before he final periodt, he final surprise is revealed and he liquidaion value is ascerained. We se T = 10, so each session lass for 10 rading periods. The above model creaes a binomial process in he inrinsic asse value. The following descripive saisics can be obained. 1 Our parameers for all marke sessions sum o a mean zero disribuion. In individual marke sessions, he parameers can have negaive, posiive or zero mean. 2 The conen of informaion released o invesors can be srucured eiher o change he probabiliies relaed o he sae-dependen payoffs, or o change he payoffs (book value) direcly. Generally, values are perceived as addiive while probabiliies are no. For his reason, we deliberaely avoid he use of probabiliies and use direc moneary payoff adjusmens. 8

9 The expeced value for each surprise is: E( ε ) = pu (1 p) d = 0.5( u d), = 2,3, L, T, T = 10 (4) The variance for each surprise is: 2 2 VAR( ε ) = ( u + d) p(1 p) = 0.25( u + d), = 2,3, L, T, T = 10 (5) By subsiuing (3) ino (2), we derive he inrinsic value as of period as: E( D = D T ) = E( D ) + E( ε i ) i= ( T )( u d), T = 2,3, L, T, T = 10 (6) Since he surprises are i.i.d, by subsiuing (4) ino (2) we derive he variance for he liquidaion value as of period as: VAR( D T ) = VAR( D ) + ( T ) VAR( ε ) = 0.25( T )( u + d) 2, = 2,3, L, T, i T = 10 (7) By varying he relaive sizes of posiive surprise, + u, and negaive surprise, d, we consruced hree disinc environmens: (a) he bearish environmen, where he random disurbance is dominaed by he relaive size of he negaive surprises and he book value ends o fall; (b) he neural environmen, where he random disurbance consiss of posiive and negaive surprises ha balance each oher and he book value ends o remain consan over ime; and (c) he bullish environmen, where he random disurbance is dominaed by he relaive size of he posiive surprises and he book value is expeced o rise. To mainain he equaliy of he variance (7) of inrinsic value a each sage across all hree environmens, he spread beween posiive surprise and negaive surprise is kep consan a u + d = 80. Table I gives he parameer deails for each environmen. Before he marke opens, invesors have been informed of which news environmen hey will face. I is common knowledge ha each invesor knows he disribuion of he surprises and how he informaion updaes will be provided o he invesors. One surprise occurs beween each pair of successive rading periods. The acual realizaions and he corresponding asse value pahs are shown in Figure 1. The random draws for each successive shock were predeermined by flipping a coin (Pah 1, lef panel in Figure 1) and were used for he firs 12 sessions. The same se of realizaions was reversed for he second 12 sessions (Pah 2, righ pane in Figure 1). The purpose of using a limied number of pahs (only 2, one he converse of he oher) in all sessions is o keep cross session daa mos comparable. The paired comparison will allow us o examine variables of concern wihou inroducing oher disurbances due o variaion in sequence of draws. 9

10 Informed invesors can learn abou wo imporan values. One is he curren book value, D (doed line in Figure 1), a which he asse value sands a he given poin of ime; he oher is he inrinsic value, E ( DT ) (solid line in Figure 1). The compuaion of inrinsic value (expeced liquidaion value) is illusraed in he insrucions (Appendix C). As periods pass, he inrinsic value is gradually revealed wih less uncerainy: ha is, he possible range of he final liquidaion value shrinks as successive surprises are realized and he book value converges o he liquidaion value Informaion Updaes: Public Informaion and Privae Informaion The HS model argues ha when no all invesors are privy o informaion, he fully revealing equilibrium (Grossman, 1976) fails because of bounded raionaliy on he par of invesors. The DHS heory of overconfidence and biased self-aribuion argues ha price moves derived from privae informaion arrival are on average parially reversed over a longer ime horizon, while price moves in reacion o he arrival of public informaion are posiively correlaed wih laer price changes. To address he elemens of he HS and DHS models, we differeniae he level of privacy in he informaion updaes. In he public informaion case, every invesor in he markeplace simulaneously receives he same updae on change in he value of he asse. In he privae informaion case, only 4 ou of 9 invesors receive updaed informaion on changes in asse value, while he res are informed abou heir environmen (bearish, neural, and bullish) bu receive no direc updaes on book value changes. Common knowledge exiss abou he level of publiciy in informaion announcemens. Boh HS and DHS models are much more complicaed han our design is. We reduce heir complexiy o a subsanive exen, allowing us o pose a srong-form es on heir hypoheses. Environmen simpliciy in laboraory sudies generally ensures less noise, higher measuremen accuracy, and sronger resuls Trading Session Parameers Trading session parameers are given in Table II. Caginalp, Porer and Smih (2001) find ha he level of cash in he economy is an imporan facor in price formaion. Therefore, hrough he iniial endowmens we conrol he raio of cash o iniial inrinsic value across he hree environmens. Liquidiy level is kep a 50% across all 24 sessions ha were run. (Approximaely 50% cash relaive o he marke s enire worh.) Each rading session has exacly 9 invesors in he markeplace and each invesor paricipaes in only one rading session Implemenaion 10

11 Our rading sessions are conduced using he sandard double aucion wih open book (NYSE syle). The rading plaform is an Inerne based, coninuous ime rading insiuion, which allows invesors o rade in real ime during each of he 10 marke periods. Invesors receive iniial endowmens of cash and asses. They can pos bids and asks, and accep he bes posed bid or ask in he marke. Share and cash balances are auomaically updaed afer each rade. Before he session begins, deailed insrucions are provided o invesors. They are given sufficien ime o read hrough he insrucions and he marke monior answers quesions whenever an invesor needs clarificaion. A variey of numerical examples are presened in he insrucions. Afer finishing he insrucions, invesors mus ake a quiz consising of 10 quesions o ensure ha hey undersand he environmen in which hey will rade. They are asked o review heir insrucions and repea missed quesions unil hey answer hem correcly. Finally, before he session sars, invesors are given a hree-minue pracice rading period o familiarize hemselves wih he mechanics of execuing rades in he acual real ime on-line marke Hypoheses The 3 differen approaches lised in he Secion 2 will have disinc implicaions for our daa Psychological Explanaions DHS: Invesors respond more aggressively o privae informaion han o public informaion. DHS (1998) assumes ha prices are se by he informed invesors and he role of he uninformed is minimal. Their heory of overconfidence and biased self-aribuion assumes ha invesors view hemselves as more capable of evaluaing securiies han hey acually are. As a resul, he model argues ha price moves driven by privae informaion arrival are on average parially preserved in he long run, while price moves generaed by he arrival of public informaion are posiively correlaed wih laer price changes. We should expec ha invesors will give more weigh o informaion ha is only available o hem, and his would imply ha hey will respond more srongly o privae informaion han o public informaion. Since self-aribuion bias exers is effec hrough overconfidence bias, we will only es he laer. To evaluae his, we compare he responsiveness o new informaion in wo paired markes via a proxy adjusmen raio: adj _ raio j, = E P j, j, ( D P T j, 1 ) P j, 1 = ΔE ΔP j, j, ( D T ) (8) where ΔE j, ( DT ) measures how disan he previous period mean price is away from he new inrinsic value in marke j, and Δ P j, measures he acual change in mean price across he wo adjacen periods. DHS would imply ha, in wo paired markes, he following relaionship mus be found: 11

12 adj _ raio > raio (9) j, adj _ privae k, public where j and k subscrip a pair of markes in he same news environmen such ha marke j differs from marke k only in ha informaion is privae in he former. BSV: invesors under-weigh news surprises, and beliefs exhibi posiive auocorrelaion. BSV (1998) assumes ha prices are driven by a single represenaive agen and ha ha agen exhibis he cogniive biases of conservaism and represenaiveness. Under he influence of conservaism, invesors end o underweigh he arrival of new evidence when updaing heir beliefs, so heir pas beliefs end o persis. Represenaive bias 1 suggess ha invesors form subjecive beliefs based on a limied number of observaions ha ignore exising informaion concerning he objecive probabiliy disribuion governing he asse valuaion process. If hese effecs are in play, we should expec ha in each period new informaion will be inadequaely manifesed in rading prices, ha is, he adjusmen raio mus be significanly smaller han 1. To es his, we run he following regression: ΔP = β ΔE ( D = β ΔE ( D T T ) + γ ΔVAR ) + φ + ε + ε (10) where β is he esimaion for adjusmen raio and φ is a consan ha represens he change in he variance of inrinsic value across wo adjacen periods (see Equaion (7)). BSV predics ha β < 1. In addiion, if belief persisence exiss, we would expec he subjecive probabiliies 2, as indicaed by marke price levels ( pˆ and pˆ 1 ) encasing a posiive surprise, o be serially correlaed. COV ( pˆ, pˆ 1 ) > 0 (11) FRAZZINI: price will respond less when boh he capial gain sign is posiive (negaive) and he surprise sign is posiive (negaive). The adjusmen is more complee when he wo signs are opposing. Frazzini (2006) bases his predicions on he disposiion effec. The essenial argumen is ha when an asse is rading a a gain, a posiive surprise will resul in less 1 We do no offer a direc es of represenaive bias, as our design specifies he exac value pahs (i.i.d), raher han a mixure of rending and revering regimes. 2 The objecive probabiliy mus always be 0.5 for a posiive surprise, while he subjecive probabiliy for a posiive surprise can be inferred from he curren book value, curren price level and he number of remaining shocks. 12

13 adjusmen han a negaive surprise because invesors will rush o sell o lock in paper gains; and similarly, when asse is rading a a loss, a negaive surprise will resul in less adjusmen han a posiive surprise because invesors will hold ono he asse and refuse o sell absen a premium. Thus Frazzini (2006) summarizes ha when he signs are opposing he adjusmen is complee, eiher because invesors rading a a loss would be willing o accep he laes gain, or because invesors rading a a gain would be willing o accep he laes loss. In our sessions, measuring he capial gain or loss is a simple ask. According o Frazzini we would expec o observe ha price adjusmens following surprises would be less complee when he informaion conen and he capial gain overhang have he same sign: Δadj (12) _ raio j, g i > 0 < Δadj _ raiok, g i < 0 where g is he proxy for capial gains and i is he news surprise a period Heerogeneous Invesors HS: Pricing difference across wo comparable periods in paired reamens can be aribued o informaion asymmery. HS (1999) posi ha under-reacing anomalies may be generaed when informaion ravels slowly. If informaion were scaered asymmerically across differen invesors, hose who rely on informaion accuracy for decision making would only ac based on he informaion hey are personally privy o; consequenly, he process of informaion aggregaion would fail. In he HS model of informaion aggregaion failure, he marke price equals o he weighed average of informaion implied values, discouned by he level of risk each group faces. In he privae (asymmeric) informaion reamen, j, ha price would be: P privae j, 1[ 1 T 1 2 T = w E ( D ) θ VAR ] + w [ E ( D ) θ VAR ] (13) where [ E1 ( DT ) θ VAR1 ] is he valuaion of he uninformed group, [ E ( DT ) θ VAR ] is he valuaion of he informed group, and w1 and w2 are he weighs for each group. Because he uninformed group is compleely blocked from informaion updaing for he enire ime span of he rading horizon and hey fail o infer from marke price, heir valuaion in every period would be he same as heir valuaion in period 1; for he informed group, heir valuaion would be updaed whenever new informaion arrives. Similar o he assumpions in he HS model, he agens would ac as a represenaive agen and he risk preference parameer, θ, is assumed o be homogenous. In he public informaion reamen k, each invesor is equally informed, herefore, he equilibrium price is: 13

14 P = E ( D ) θ VAR public k, T (14) By subracing (10) from (11), he following is derived: P privae j, P public k, = w1[ E1( DT ) E ( DT )] + θ w1 ( VAR VAR1 ) (15) = diff _ info + θ diff _ VAR j, k, j, k, The firs iem diff _ info j, k, gives he level informaion asymmery across he wo groups, while he second iem diff _ VAR j, k, gives he difference in he level of risk (variabiliy) ha each group faces. HS inerpreaion of overreacion as a consequence of momenum rading can be direcly verified by examining price ime series, ha is, if momenum aciviies fuels overreacion, overacing levels of prices can be direcly observed Risk Explanaions FAMA: Asse price reflecs risk, ha is, he higher he risk is, he higher he expeced reurn required by invesors o hold he asse. Fama (1998) argues ha he under- and over-reacion anomalies are chance resuls, apparen long-erm over-reacion o informaion is abou as common as under-reacion, and pos-even coninuaion of pre-even abnormal reurns is abou as frequen as poseven reversal. He claims ha apparen anomalies arise from he mehod of daa analysis, and mos long-erm reurn anomalies end o disappear wih reasonable changes in echnique; in oher words, if risk were accuraely measured, over-reacion and underreacion would be equally frequen and pricing would be unbiased. In our sessions, because here is only a single risky asse, he only source of uncerainy comes from ha asse s liquidaion value. If he aggregae marke has a meanvariance uiliy ype, he variance of he liquidaion value as a proxy for risk. The following is a naïve risk reurn relaionship model 1 : E( r ) = λ VAR + ε (16) where he expeced reurn is E ( r ) = [ E ( DT ) P ] / P, λ is risk premium raio, VAR is he variance of liquidaing value a period (see Equalion (7)), and ε accouns for deviaions due o chance. 1 Mos models discussed in his paper, excep Fama s marke efficiency model, are grounded in he srucure of a one asse marke; herefore, we did no use a muli-asse marke. Evidence on menal accouning (Rockenbach, 2004) indicaes ha mos invesors end o lump all risky asses ino a single menal accoun, producing a siuaion ha would degenerae ino managing a single risky asse marke. 14

15 4. Experimenal Resuls 4.1. Transacion Price Time Series and Drif The ransacion prices of various marke sessions evolve oward he inrinsic value in diverse manners. We observe all sors of paerns including upward momenum, downward momenum, U-shaped and humped-shaped ime series. Incidences where ransacion prices occur ouside he range of possible inrinsic values are rare. The ime series of ransacion prices in all 24 marke sessions are shown in Figure 2. In he bearish environmen, as shown in Panel 1-A (Session 3, 6, 13, and 16) and Panel 2-A (Session 5, 10, 15, and 22), prices converge o inrinsic value from above. In he neural environmen, as shown in Panel 1-B (Session 1, 11, 17, and 19) and Panel 2-B (Session 4, 9, 21, and 23) in Figure 2, he ransacions sar slighly below inrinsic value and quickly rack i in laer periods. In he bullish environmen, as in Panel 1-C (Session 2, 8, 4, and 20) and Panel 2-C (Session 7, 12, 16, and 24), prices converge o inrinsic value from below. Under sandard von-neumann-morgensern uiliies, agens price a symmeric payoff disribuion below is expeced value if hey are risk averse, and price a symmeric payoff disribuion above is expeced value if hey are risk seeking. However, we observe an inconsisency in ha agens seem o be risk averse in pricing in he bullish environmen and risk seeking in pricing in he bearish environmen, even hough he level of risk (variance of he inrinsic value) is he same for a paricular period across all environmens. The long-dash curve in Figure 2 is he Lowess 1 fied line of he ransacion prices, a compuaional mehod developed o assess scaer plos by robus locally weighed fiing. As shown in Figure 2, in he bearish and bullish environmen, he marke prices show clear paerns of downward and upward drif respecively. If we define under-reacion as he coninuaion of reurn signs, boh cases srongly demonsrae his ype of under-reacion drif. Under he neural environmen, he paerns are mixed bu wih smaller average deviaions; mos price rends are relaively fla, while some show a sligh upward drif and some have a hump-shaped curve. The 12 sessions on he lef side of Figure 2 are sessions where informaion updaes are publicly announced, while he 12 sessions on he righ side are sessions where informaion updaes are privaely revealed o a subse of invesors. The drifing paerns are similar across he wo informaion condiions. (We will discuss his maer in more deail below.) 4.2. The Prevalence of Under-reacion The Lowess fied curves offer an inuiive represenaion of under-reacion paerns in which reurns hold he same sign from he news release a he session opening o he liquidaing poin. 1 Lowess was inroduced in Visual and Compuaional Consideraions in Smoohing Scaer plos by Locally Weighed Fiing, W. S. Cleveland, Compuer Science and Saisics: Elevenh Annual Symposium on he Inerface, Norh Carolina Sae Universiy, Raleigh, Norh Carolina, 1978, pages

16 Laboraory sudies provide he advanage of being able o direcly assess boh he signs and he magniudes of reacion, raher han having o solely rely on prices o infer deviaions from inrinsic value. The average deviaions of ransacion prices from inrinsic value in every rading period across all marke sessions are ploed in Figure 3. Deviaions are essenially equivalen o expeced reurns. As shown in Figure 3, in he bearish environmen we observe large overpricing (negaive expeced reurns) relaive o he inrinsic value in almos all markes, wih higher deviaions in iniial periods. In he bullish environmen, we observe large underpricing (posiive expeced reurns) relaive o he inrinsic value in 7 ou of 8 sessions, wih all sessions being more severely underpriced iniially. In he neural environmen, a roughly even spli is found beween under-pricing and overpricing. The drifing paerns in bullish and bearish sessions epiomize he definiion of under-reacion, while he neural sessions appear o show reurn reversals in numerous cases even as hey flaen ou on average. To consruc a beer classificaion rule of soring underreaion and overreacion for each period, we consider evaluae he emporal posiion of ransacion prices relaive o inrinsic values. Below we define under-reacion using a price migraion rule wih boh a shor and long-erm lens. The long-erm idenificaion is judged by he price posiion relaive o he inrinsic value over he session s enire duraion. In all sessions, before beginning all invesors are informed abou he (approximaely normal) disribuion of he asse liquidaion value. Suppose he prices sar on one side of his inrinsic value, move away from pas price (average price in all pas periods) and migrae o he oher side of he inrinsic value, hen he pricing of he rading period is inerpreed as over-reacion; if he prices keep proximae o pas price and always fall shor of reaching he inrinsic value, he pricing of a rading period is inerpreed as under-reacion. (See he lef panel in Figure 4.) We precisely define he shor-erm reacion for each rading period such ha he price migraion is examined wihin he wo-period ime window enveloping an informaion even (See righ panel in Figure 4): E ( DT ) P, if P 1 > E 1( DT ) shor _ magniude= (17) P E ( DT ), if P 1 < E 1( DT ) where P 1 is he average ransacion price in period -1 and he sign of he deviaion is dependen upon he posiion of price relaive o inrinsic value in he previous period. We precisely define he long-erm reacion magniude in he same fashion excep ha he price migraion is examined across all ranspired periods: E ( DT ) P, if P > E1... 1( DT ) long _ magniude = (18) P E ( DT ), if P < E1... 1( DT ) 16

17 1 where P is he average ransacion price in he curren period, P1... is he average ransacion price over all previous periods, E D ) is he inrinsic value as of period, E1... 1( DT ) is he average inrinsic value over all previous periods; and he sign of he deviaion is dependen upon he posiion of average pas price relaive o average pas inrinsic value. In boh cases, if he reacion magniude is negaive, he pricing will be inerpreed as under-reacion, and if he reacion magniude is posiive, he pricing will be inerpreed as over-reacion. under reacion, if magniude < 0 reacion _ ype = (19) over reacion, if magniude > 0 The disribuion of reacion magniude is shown in Figure 5. Among 238 rading periods in 24 sessions, in he long run, 71.8% of he periods have curren rading prices saying on he same side of inrinsic value as previous average price, while 28.2% of he periods display migraion o he oher side of inrinsic value. In he shor run, 82.8% of he periods have curren rading prices saying on he same side of inrinsic value as he previous period price, while 17.2% of he periods display migraion o he oher side. We noe ha boh disribuions are lef skewed wih a mean less han zero, indicaing ha under-reacions dominae no only in frequency bu in magniude oo. This confirms he empirical findings suggesing dominance of under-reacion in he shor run, and also rejecs Fama s (1998) claim ha under-reacion and over-reacion cancel ou on average in he long run. In addiion, we evaluae our mehodology agains he radiional reurn sign characerizaions. We measure reurn as he price percenage change across wo adjacen periods, and classify he periods where pre-even-period reurn and pos-even-period reurn have he same signs as under-reacion, and opposing signs as over-reacion. We found 113 periods of under-reacion and 80 periods of over-reacion. For he long erm reurn, we classify he periods where pre-even cumulaive reurn 1 and poseven reurn have he same signs as under-reacion, and opposing signs as overreacion. We found 93 periods of under-reacion and 83 periods of over-reacion. However, an implici assumpion is made ha he coninuaion of reurn signs indicaes inadequae pas adjusmen, while he reversal of reurn signs indicaes excessive pas adjusmen. The problem lies in ha we generally do no have he knowledge of wheher: (a) he magniude of he previous adjusmen is sufficien or excessive; (b) he previous price is converging o or diverging from he inrinsic value. A glimpse a he ime series pahs shown above suggess ha hese worries are no superfluous. For insance, in Session 21, he urning in reurn sign a period 5 is an ( T 1 Fama (1998) argues ha long-erm reurn should be compued by average abnormal reurn (AAR) or cumulaive abnormal reurn (CAR) as buy-and-hold reurn (BAHR) ends o have higher errors. We use CAR in our analysis. 17

18 accurae reflecion of he urn in he inrinsic value; and in Session 4, he coninuaion of reurn around period 5 is acually a movemen away from inrinsic value. Pure reurn-sign based classificaion is problemaic because a change in reurn sign is no necessarily he resul of previous excessive adjusmen and a saionary reurn sign is no necessarily he resul of previous inadequae adjusmen; raher, he behavior of a pos even reurn can simply be a correc adjusmen o he new saus quo. 4.3 Sources of Overreacion The plos in he neural environmen in Figure 3 hin an imporan poenial cause for over-reacion. Mos over-reacions appear in he neural environmen, ye Panel 1-B and 2-B in Figure 2 show ha i is he change in he inrinsic value raher han he change in he price ha generaes he paerns. The migraion of price from one side of inrinsic value o he oher side is considered as overreacion; however, he migraion can be eiher caused by he excessive adjusmen in price or he change in he inrinsic value coupled wih sluggish price adjusmen. We conduc analysis on 41 periods of overreacion ou of a oal of 238 periods. Table III abulaes all he overreacion periods ino four scenarios: (a) price migraes from below o above and he news surprise is posiive; (b) price migraes from below o above and he news surprise is negaive; (c) price migraes from above o below and he news surprise is posiive; d) price migraes from above o below and he news surprise is negaive. Case (a) and (c) indicae ha he price adjusmen no only correcs he previous period deviaion bu also absorb fully he news surprise, herefore, boh cases relaes o excessive adjusmen version of overreacion. Case (b) and (d) indicaes ha he price adjusmen is insufficien such ha i does no absorb fully he news surprise, herefore, boh cases relaes o insufficien adjusmen. The resuls in Table III sugges ha overreacion (migraion o he oher side of inrinsic value) is mosly generaed by sluggish adjusmen (34 periods), raher han excessive adjusmen (only 7 periods). Indeed, we will examine he overall magniude of he sluggish adjusmen in Secion 5.1.2; indeed, we will show ha sluggish adjusmen prevails in our daa and he average adjusmen o news surprise is only 38%. (Refer o Table V) 5. Verifying he Compeing Models Secion 4 has demonsraed he dominance of under-reacion phenomenon across all sessions, especially under he bullish and bearish environmens. In his Secion, we will evaluae more specifically he unique predicions of he five compeing models. 18

19 5.1. Psychological Explanaions DHS DHS (1998) predics ha agens will respond more acively o privaely held informaion. Tha means ha when informaion updaes reach only an insider group, due o overconfidence hose insiders will rade more aggressively and lead he marke o over-reacion. As specified in he hypoheses, DHS has sraighforward implicaions for he rae of adjusmen. We derive he difference in adjusmen rae by pairing he adjusmen rae in a paricular period of he privae informaion reamen wih he same period of he public informaion reamen: diff _ rae j, k, adj _ raio j, privae adj _ raio j, public = (20) DHS suggess ha he erm should be sricly posiive while HS suggess ha i should be sricly negaive. Figure 6 provides he hisogram for he paired difference in adjusmen raes. The hisogram indicaes ha he disribuion is hardly disinguishable from he zero mean symmeric normal disribuion, and we canno rejec ha hypohesis given he Suden es generaes a p-value of Table IV liss he average urnover, bid-ask spread, mean price, median price and closing price for all 24 sessions, using each rading period as an observaion. Across he wo informaion condiions, none of hese measures differ subsanially excep volume. I is noeworhy ha here is considerable reducion in volume when informaion is privaely revealed. We noe ha he reducion is mosly derived from uninformed invesors who involved in many fewer ransacions han heir counerpars in he public informaion reamen BSV Model BSV argues ha once agens form beliefs hey will become relucan o change his prior and will end o under weigh he arrival of new informaion. In Figure 7, we graph he adjusmen rae disribuion and show i is lepokuric and posiively skewed. In 60.7% of he cases price adjuss in he correc direcion bu insufficienly, in 26.4% of he cases price adjuss in he opposie direcion of he surprises, and in 12.9% of he cases price adjuss in excess of he inrinsic value change. Insufficien adjusmen is he dominan phenomenon. Saring from he iniial deviaion from inrinsic value unil evenual convergence, he sessions adjus average valuaion hesianly on he pah o liquidaion. As prices are he manifesaion of agens beliefs, he ineria in prices can be undersood as ineria in willingness o change beliefs. Nex, we run regressions on how swifly prices on average adjus o inrinsic value changes. The regressions in Table V show ha he average adjusmen o surprises o inrinsic value is around 38%. The effec is robus using mean, median and closing prices, 19

20 and he coefficiens are significan a 0.01 level. In addiion, he dummy variable privae k does no have a significan effec on he rae of adjusmen. The BSV model also implies ha agens end o ignore new informaion and exhibi belief persisence. To demonsrae he belief coninuaion argumen of BSV model, we derive he session s aggregae subjecive probabiliy esimaes from observed prices. The naural probabiliy for a Head draw is always 0.5. However, he subjecive probabiliy of agens in he sessions can hardly be he same as 0.5 because prices persisenly deviae from inrinsic value. In every rading period, given he curren level of price, we can derive he subjecive probabiliy (of a Head draw). This offers a perspecive ha allows us o probe he invesors subjecive beliefs 1. We denoe a session s aggregae belief on he probabiliy of Head draw in period as pˆ. Because curren period price reflecs he marke s average belief on he probabiliy of a Head draw, he following equaliy should hold: P = E ( D ) = D + ( T )[ pˆ u (1 pˆ ) d] (21) T P is he average rading price in period. Figure 8 shows he evoluion of subjecive probabiliy across adjacen periods in he 24 sessions. Mos poins in he scaer plo reside in he 1 s and 3 rd quadrans, indicaing ha an underesimaed subjecive probabiliy (negaive deviaion) in one period is likely o be followed by an underesimaed subjecive probabiliy (negaive deviaion) in he nex period, oo. This is in accordance wih BSV s predicion of belief persisence Disposiion Effec The disposiion effec (Shefrin and Saman, 1985; O dean, 1998; and Grinbla and Han, 2001) has been given much aenion in recen empirical research. This heory predics ha invesors will rush o sell afer a capial gain and are relucan o sell afer a capial loss. Frazzini (2006) proposes ha he presence of disposiion invesors will depress prices following good news as hey rush o sell o lock in paper gains, and will hal prices from falling following bad news as hey are relucan o sell absen a premium. Table VI summarizes he predicions of Frazzini (2006). The key variable for esing he disposiion effec is he measure of capial gains. Similar o Frazzini (2006), we measure he reference price as he volume weighed hisoric average price: 1 Measuring implied probabiliy offers us a unified way o look a deviaions across differen sessions. I has an imporan advanage over measuring deviaion direcly. Because 1 uni of deviaion in he firs period ends o be less significan han 1 uni of deviaion in he laes period, implied probabiliy provides a convenien way of weighing deviaions across periods. 20

21 1 Pi Volume i i= 1 RP = 1 (22) Volume i= 1 i Then we compue capial gain overhang as: g P RP P = = (23) P RP P The resuls in Table VII clearly rejec Frazzini (2006). We observe more adjusmen in cases where news conen and capial gains hold he same sign, conrary o he disposiion effec predicion. In addiion, he number of asks in he marke seems o remain relaively unchanged across various gain condiions Heerogeneous Invesors: HS Hong and Sein s informaion diffusion model suggess ha informaion aggregaion failure arising from informaion asymmery will keep he marke price away from full informaion implied values. As specified in Equaion (9) in our hypoheses, he price in a given period of a privae informaion session and in he same period of a public informaion session should reflec he informaion asymmery and associaed difference in uncerainy amongs invesors. Noe ha for each pair of comparable periods, every facor is kep he same excep for how many invesors are old he conen of surprises. Table VIII runs regression analysis on Equaion (9) and provide no suppor for he relaionship beween informaion asymmery and price difference. Tha is, he average valuaion of informed raders and uninformed raders does no capure he emporal developmen of prices. This resul comes as no surprise, as Figure 2 shows ha he price series in he wo reamens are exremely alike. We do no observe ha coninuaion of rends leads price o migrae o he oher side of inrinsic value. In mos sessions, he markes spend he whole experimen session in converging o he inrinsic value. (See o Figure 3) I is clearly shown ha under-reacion prevails while overreacion reversals happen mosly when he inrinsic value evoluion changes is iniial cause. Hence, here is no reasonable basis o assume over-reacion in our daa is caused by a separae se of momenum raders Risk based Explanaion: Fama Fama (1998) argues ha on average over-reacion and under-reacion should cancel ou. The hisograms in Figure 5 indicae ha under-reacion dominaes over-reacion boh in magniude and in frequency. Empirical findings of anomalies can arise from imperfec evaluaion of risk facors, bu we can precisely measure he price deviaions in our daa agains curren inrinsic 21

22 value uncerainy. In our rading sessions here is only one source of uncerainy: he variance of he inrinsic value which is a linear funcion of he number of pending surprises. The risk associaed wih he inrinsic value uncerainy, as measured by is variance, declines over ime. If agens are risk averse (seeking) in valuing he symmeric inrinsic value disribuion, he required rae of reurn should be posiive (negaive) o enice rade. We do no observe such consisency. Figure 9 shows ha he aggregae risk-reurn compensaion raio (he long-dash line) is indisinguishable from zero. However, hough invesors are exposed o he same level of uncerainy in he same period in all he environmens, hey consisenly require a premium o final claims in he bullish environmen, a zero premium in he neural environmen, and srikingly, a discoun in he bearish environmen. The higher he risk, he more pronounced is his effec. If he variance of inrinsic value is no a perfec proxy for risk, a weaker es would be ha since for any given period each uni held is exposed o he same surprise disribuion, wih he same level of variabiliy, he compensaions across differen environmens should, on average, be equal o each oher. E ( r bullish) E( r neural) = E( r bearish), = 1,2, L T = (24), Table IX indicaes ha he compensaions in each period are no equal across he hree news environmens, bu here exiss a sysemaic difference. Under he bearish environmen an average negaive expeced reurn of -15.7% is observed, while under he bullish environmen an average posiive expeced reurn of 7.3% is observed. In neural environmen, he overall average expeced reurn is 0. A sage-wise cross secion comparison reveals his resul even more convincingly, as he expeced reurn is always larges for he bullish environmen and smalles for he bearish environmen. This subsanively rejecs Equaion (20) and seriously quesions wheher under-reacion (overreacion) can be explained by risk aiudes. 6. A Theory of Price Ineria: Numeric Simulaions In Secion 4.3, a enaive conclusion is reached ha sluggish adjusmen is responsible for he emergence of overreacion. I remains unverified wheher price ineria can be he sole driver of he relaive frequencies of under- and over-reacion. This Secion will simulae, using parameers obained from he daa, o demonsrae he speed of adjusmen is he key facor in generaing under- and over-reacion. Tha is, when he marke response o news surprise is sluggish, boh regulariies will emerge wih he former ounumbering he laer; when he marke response o news surprise is excessive, boh regulariies will again emerge bu he laer ounumbering he former. The only inpu variable o be conrolled is he speed of adjusmen and we will analyze he resuls respecively. 22

23 6.1 Simulaion Seup The analysis has indenified insufficien adjusmen as a prevailing phenomenon under all condiions. We now urn o a simulaion experimen o examine how he speed of shor-erm adjusmen migh impac he under- and over-reacion paerns we observe. The se-up of he simulaion is exacly he same as laid ou in he experimen design. To drive he simulaion we derive wo parameer values from he daa: he saring price 2 and he adjusmen raio. Our daa imply ha: P 1 ~ N(507.6, ) and 2 adj _ raio ~ N(0.27, 0.81 ). The purpose is o vary he disribuion of adjusmen raio and evaluae how he under- and over-reacion paerns change accordingly. We choose hree alernae 2 disribuions for adjusmen raio: insufficien adjusmen, adj _ raio ~ N(0.27, 0.81 ), 2 which was observed in our daa, appropriae adjusmen, adj _ raio ~ N(1, 0.81 ), which migh be observed if price adjusmens were accurae on average, and excessive 2 adjusmen, adj _ raio ~ N(1.73, 0.81 ), which migh be observed if price adjusmens were aggressive on average. In addiion, we use boh he shor-erm and long-erm perspecives inroduced earlier o analyze he reacion magniudes. 6.2 Simulaion Resuls and Discussion The simulaion resuls sugges ha he adjusmen raio is he key facor driving he reacion disribuions. When he average adjusmen raio is below 1, under-reacion dominaes (Figure 10, Panel A-1 and B-1); when he average adjusmen raio is 1, underreacion is as frequen as over-reacion (Figure 10, Panel A-2 and B-2); when he average adjusmen raio is greaer han 1, over-reacion dominaes (Figure 10, Panel A-3 and B- 3). These resuls hold for boh shor and long-erm perspecive (See Table X). There is a sligh endency for over-reacion o grow when swiching o long-erm perspecive, which is consisen wih empirical resuls ha over-reacion end o show up more in he long erm. The simulaion reveals an imporan message: over-reacion and under-reacion are boh by-producs of he adjusmen responsiveness in he environmen. Overreacion is also creaed by ineria because he price adjusmen does no cach up wih he speed of change in he inrinsic value. Wheher he pricing a a poin should be classified as underor over-reacion largely depends on he pas, whereas he poin-of-ime prices in he wo labeling siuaions could have been exacly he same. Noe ha a any poin of ime, price is slowly correcing is error. In our sessions, hey simulaneously derive in he proporions observed from he same cause: slow adjusmen. 7. Conclusion How marke prices incorporae he arrival of new informaion has long been a puzzle o heoriss and empirical researchers. In his paper, we designed and colleced daa from conrolled laboraory markes ha replicae a version of he HS (1999) informaion 23

24 diffusion model. We srucured he informaion arrivals such ha hey were emporally dispersed and asymmerically held across individual invesors. The muli-period exchange economy allowed us o evaluae observed price dynamics agains he predicions of various compeing models. The rading price ime series daa manifess under-reacing drifs in boh bullish and bearish environmens. The bullish environmen was usually inerpreed by invesors as no as good enough (under-pricing persised), while he bearish environmen was inerpreed as no bad enough (over-pricing persised). This gives he impression ha agens are risk seeking in bearish environmens and risk averse in bullish environmens, even hough a each period, we designed he risk exposure o be idenical across hese environmens. We found he risk-reurn premium relaionship does no hold in aribuing abnormal reurns o chance. Informaion asymmery had no significan affec on price adjusmens, boh over he shor and long erms. The level of informaion asymmery did no explain he price difference across wo comparable rading periods. This quesions he HS argumens ha he failure of informaion aggregaion may be a reason for under-reacion and ha asymmeric informaion slows down he adjusmen process oward inrinsic value. The DHS overconfidence model predicions are no confirmed in our privae informaion reamen, as privaely informed invesors were no found o lead he markes o overreacion. Furhermore, capial gains and losses did no correlae wih price adjusmen magniude in he fashion specified by he disposiion effec model. An analysis of he sessions subjecive probabiliies indicaed ha biased judgmens end o persis and exhibi belief coninuaion. The ineria paerns discovered were mos consisen wih he conservaism accoun of BSV among exising heories. We idenified boh he shor-run and long-run under and over-reacion disribuions wih he advanage of knowing perfecly he inrinsic asse values. Under-reacion is he predominan regulariy in all markes in he shor run, while under-reacion dominaes he long erm price behavior in boh bullish and bearish environmens, bu dissipaes in he neural environmen. We showed ha he common echnique of reurn sign characerizaion wih ignorance of inrinsic value or accurae esimaion echniques can produce misleading conclusions. Over-reacing ypes of price adjusmens were no necessarily he markes excessive adjusmen o earlier informaion. Tha is, he iniial adjusmen migh be appropriae, while laer sage price reversals were he marke s readjusmen o a new inrinsic value, wih ineria generaing a prolonged period of opposie reurn signs. Sizable over-reacion is presen in he clearly under-reacion predominaed markes as well. This leads us o hypohesize: are under-reacion and over-reacion caused by he same maer? A re-examinaion of our resuls and a simulaion wih he parameers exraced from daa sugges ha consciously conservaive adjusmens alone provide an accoun for boh under- and over-reacion regulariies. The empirical and heoreical dichoomy of reaing he wo anomalies migh be quesionable. Our simulaion suggess ha simple ineria is he key facor in deermining he disribuion of reacion magniudes, and ha slow adjusmen is he reason for boh under- and over-reacion regulariies observed. 24

25 No single heory presened in his paper can oally summarize he informaional efficiency shown in he daa. Invesors exhibied belief coninuaion and accordingly prices displayed ineria. Invesors seemed o place considerable weigh on heir previous beliefs (pas price) when forming valuaions abou uncerain fuure payoffs. Boh shorrun reurn coninuaions (driven by slow adjusmen) and long-run reurn reversals (possibly re-adjusmens o evoluionary urns in inrinsic value) can be explained by price ineria (belief coninuaion). 25

26 Appendix Appendix A: Asse wih a random walk inrinsic value and informaion diffusion Hong and Sein (1999) define an asse which pays a single dividend a he liquidaing dae. T This liquidaing dividend is D T = D0 + j = ε 1 j. A each period, invesors rade claims on he risky asse. The asse pays a single liquidaing dividend a imet. ε s are he dividend 2 innovaions and are assumed i.i.d, normal random variables ε ~ N (0, σ ). To incorporae he noion ha informaion moves gradually across he "newswacher populaion", hey divide his populaion ino z equal-sized groups and every dividend innovaion, ε, can be decomposed ino z independen sub-innovaionsε, wih j 2 ε = + L +, each wih variance ε / z. j, i ε j,1 ε j,2 ε j, z The iming of informaion release is as follows: a, news abou ε +z 1, begins o spread, and newswacher group 1 observes ε +z 1, 1, group 2 observes ε +z 1,2,..., group z observes ε + z 1, z. A, each sub-innovaion of ε +z 1, has been seen by a fracion 1/z of he oal populaion. A +1, he groups roae, group 1 now observes ε +z 1, 2, group 2 observes ε +z 1, 3,..., group z observes ε + z 1, 1. A +1 he informaion ε +z 1 has spread furher, and each sub-innovaion of ε +z 1, has been seen by a fracion 2/z of he oal populaion. Roaion coninues unil ime + z 1, a which poin every one of he z groups has direcly observed each of he sub-innovaions ha comprise ε + z 1. ε + z 1, has become oally public by ime + z 1. The parameer z can be hough of as a proxy for he rae of informaion flow. Higher values of z imply slower informaion diffusion 1. The model predics ha because newswachers fail o infer inrinsic value from marke prices, hey rely on heir privae informaion in deciding heir demand and consequenly, under-reacion is more severe when informaion diffuses more slowly. Noe ha, hough he auhors claim he diffusion primarily occurs hrough disseminaion across he populaion, i is in fac mixed crosspopulaion and over-ime diffusion. The reason is simple: he faser ha raders o ge know fuure dividend innovaions, he earlier hey learn abou heir fuure siuaions. The price a ime should equal o: j, i P = D + [( z 1) ε ( z 2) ε ε + z 1 ] / z θ Q 1 Higher z, on he one hand, means more numbers of raders; and on he oher hand, i means he diffusion of informaion for a longer ime horizon. Therefore, higher z no only indicaes slower diffusion across he invesor public, bu also more fundamenal uncerainy ha spans over ime. 26

27 where θ is a funcion of newswachers risk aversion and he variance of heε s. Glosen (1985) posis he opposie, ha is, he price a ime should equal o: P = D + z 1 θ Q Noneheless, he auhors argue ha invesors will fail o infer inrinsic value from prices and such equilibrium will be impossible. HS has wo essenial feaures. Firs, as ime passes and more value innovaions are realized he uncerainy of he liquidaing dividend declines. Second, here is informaion asymmery among he invesor public, wih some invesors holding a more accurae esimae of he inrinsic value. The firs feaure is readily feasible in he laboraory as long as we keep he ime horizon finie. The second feaure, asymmery, is more difficul o implemen in he laboraory exacly as specified by Hong and Sein (1999), because roaion is an awkward scenario and roaing informaion concerning sub-innovaions are no an inuiive concep o explain o invesors. To make he scenario readily digesible for invesors, we keep he firs HS feaure, delee he sub-innovaion feaure, and simplify he asymmery among invesor public by having a fixed group of invesors ha receive privae informaion during he rading periods. As a design conrol his privae informaion reamen is paralleled wih a public informaion reamen where all invesors are equally informed simulaneously. This simplificaion enables us o implemen he laboraory rading sessions wih much less cogniive cos on he par of invesors and a much more inuiive explanaion for he privileged few, while reaining he essenial informaion diffusion boh across ime and he populaion. Appendix B: Daa Daa is available a hp://esi2.chapman.edu/sandler/24mk.x Daa dicionary is provided below: Info: 0="public", 1="privae" Environmen: -1="bad"; 0="neural"; 1="good" Acion 1="buy", 2="sell", 3="bid", 4="ask", 5="cancel buy", 6="cancel sell" Appendix C: Experimen Insrucions Public Informaion, Bullish Environmen: hp://esi2.chapman.edu/sandler/rw_inc/page1.hml Public Informaion, Neural Environmen: hp://esi2.chapman.edu/sandler/rw_con/page1.hml Public Informaion, Bearish Environmen: hp://esi2.chapman.edu/sandler/rw_des/page1.hml Privae Informaion, Bullish Environmen, Informed Group: 27

28 hp://esi2.chapman.edu/sandler/privae/rw_inc/page1.hml Privae Informaion, Bullish Environmen, Uninformed Group: hp://esi2.chapman.edu/sandler/dark/rw_inc/page1.hml Privae Informaion, Neural Environmen, Informed Group: hp://esi2.chapman.edu/sandler/privae/rw_con/page1.hml Privae Informaion, Neural Environmen, Uninformed Group: hp://esi2.chapman.edu/sandler/dark/rw_con/page1.hml Privae Informaion, Bearish Environmen, Informed Group: hp://esi2.chapman.edu/sandler/privae/rw_des/page1.hml Privae Informaion, Bearish Environmen, Uninformed Group: hp://esi2.chapman.edu/sandler/dark/rw_des/page1.hml 28

29 References: Abarbanell, Jeffery S., and Reuven Lehavy, 2003, Role of repored earnings in explaining apparen bias and over/under-reacion in analyss' earnings forecass, Journal of Accouning and Economics 36, Barberis, Nicholas C., Andrei Shleifer, and Rober W.Vishny, 1998, A model of invesor senimen, Journal of Financial Economics 49, Bernard, Vicor L., and Jacob K. Thomas, 1989, Pos-earnings-announcemen drif: Delayed price response or risk premium? Journal of Accouning Research 27, Gunduz, Caginalp, David P. Porer, and Vernon L. Smih, 2001, Financial bubbles: Excess cash, momenum, and incomplee informaion, Journal of Psychology and Financial Markes 2(2), Daniel, Ken D., David A. Hirshleifer and Avanidhar Subrahmanyam. 1998, A heory of overconfidence, self-aribuion, and securiy marke under- and over-reacions, Journal of Finance 53, DeBond, Werner F.M., and Richard H. Thaler, 1985, Does he sock marke overreac? Journal of Finance 40, DeBond, Werner F.M., and Richard H. Thaler, 1987, Furher evidence on invesor overreacion and sock marke seasonaliy, Journal of Finance 42, DeBond, Werner F.M., and Richard H. Thaler, 1990, Do securiy analyss overreac? American Economic Review 80(2), Dreman, David N. and Eric A. Lufkin, 2000, Invesor over-reacion: Evidence ha is basis is psychological, The Journal of Psychology and Financial Markes 1(1), Edwards, Ward, Conservaism in human informaion processing. In: Kleinmuz, B. (Ed.), Formal Represenaion of Human Judgmen. Wiley, New York. Fama, Eugene F, 1998, Marke efficiency, long-erm reurns and behavioral finance, Journal of Financial Economics 49, Fama, Eugene F, 1970, Efficien capial markes: A review of heory and empirical work, Journal of Finance 25, Frazzini, Andrea, 2006, The disposiion effec and under-reacion o news, Journal of Finance 62, Grinbla, Mark, and Bin Han, 2005, Prospec heory, menal accouning, and momenum, Journal of Financial Economics 78, Glosen, Lawrence R. and Paul R. Milgrom, 1985, Bid, ask and ransacion prices in a specialis marke wih heerogeneously informed raders, Journal of Financial Economics 14, Grossman, S. 1976, On he efficiency of compeiive sock markes where raders have diverse informaion, Journal of Finance 31, Hong, Harrison, and Jeremy C. Sein, A Unified Theory of Under-reacion, Momenum Trading and Over-reacion in Asse Markes. Journal of Finance, 54, Hong, Harrison, Terrence Lim, and Jeremy C. Sein. 2000, bad news ravels slowly: Size, analys coverage, and he profiabiliy of momenum sraegies, Journal of Finance 55, Ikenberry, David L., Josef Lakonishok, and Theo Vermaelen, 1995, Marke under-reacion o open marke share repurchases, Journal of Financial Economics 39, Ikenberry, David L., and Sundaresh Ramnah, 2001, Under-reacion o self-seleced news evens: The case of sock splis, Review of Financial Sudies 15, Jegadeesh, Narasimhan, and Sheridan Timan,1993, Reurns o buying winners and selling losers: Implicaions for sock marke efficiency, Journal of Finance 48,

30 Mikhail, Michael B., Beverly R. Walher, and Richard H. Willis, 2001, The effec of experience on securiy analys under-reacion and pos-earnings-announcemen drif, Journal of Accouning and Economics 35, O'dean, Terrance. 1998, Are invesors relucan o realize heir losses? Journal of Finance 53 (5), Rockenbach, Beina, 2004, The behavioral relevance of menal accouning for he pricing of financial opions, Journal of Economic Behavior and Organizaion 53, Shefrin, Hersh M., and Meir Saman, 1985, The disposiion o sell winners oo early and ride losers oo long: Theory and evidence, Journal of Finance 40, Smih, Vernon L. 1982, Microeconomic sysems as an experimenal science, American Economic Review72 (5),

31 Tables: Table I Table I. Parameerizaion for he News Surprises If he random draw is Head, book value will change by +u; if he random draw is Tail, book value will change by d. Since he spread beween posiive and negaive surprises is kep consan across hree environmens, u + d =80, he variance of he final liquidaion value a each period is kep he same independen of he environmen. Surprise (Toal: 9) Environmen Bearish Neural Bullish Head (p = 0.5) Tail (1 - p = 0.5) + u d Table II Table II. Session Parameers Each session has 3 groups of 3 idenically endowed invesors who each receive a group specific porfolio of cash and shares. Liquidiy raio = oal cash / expeced oal share worh. News Session Invesor Endowmens Opening Info. Invesor Environ ID Release Group 1 Group 2 Group 3 Expeced Grouping men Value Pah1 Pah2 Cash Shares Cash Shares Cash Shares Public 3, 6 13,18 9(3,3,3) Bearish Privae 5,10 15,22 9(3,3,3) Liquidiy Raio Neural Bullish Public 1,11 17,19 9(3,3,3) Privae 4,9 21,23 9(3,3,3) Public 2,8 14,20 9(3,3,3) Privae 7,12 16,24 9(3,3,3)

32 Table III Table III. Sources of Overreacion: Insufficien adjusmen v.s. excessive adjusmen Overreacion is classified when price migraes o he oher side of inrinsic value across wo adjacen periods. 41 of 238 periods are classified as overreacion periods in he daa. In 34 ou of he 41 overreacion periods, prices do no cach up wih he new inrinsic value and hus migrae o he oher side; in 7 ou of he 41 overreacion periods, prices adjusmen excessively and migrae o he oher side. Deviaion from Inrinsic Value Surprise in Curren Period Previous Period Curren Period Table IV Table IV. Comparison on Oher Measures across Informaion Condiion The numbers in parenhesis are sandard deviaions. Each pair consiss of he measures in a period of he privae informaion sessions and he counerpar in he public informaion sessions. All oher condiions remain he same excep for he informaion privacy. The suden es hypohesizes ha he paired difference holds a mean of zero. Volume Bid-Ask Spread Mean Price Median Price Closing Price Public 12.9 (10.7) (129.3) (111.8) (112.2) (121.5) Privae 8.4 (9.7) (158.9) (116.2) (116.6) (123.4) Suden Paired Comparison p-value p=0.049 ** p=0.08 p=0.787 p=0.84 p=

33 Table V Table V. Feasible GLS Regressions on Adjusmen Rae The number in parenhesis is he z-saisic. *** sands for 0.01 significance level, and ** sands for 0.05 significance level. The coefficiens are esimaed wih generalized leas squares, wih heeroskedasiciy and panel-specific AR(1) auocorrelaion. ΔP k, represens he change in price in session k from period -1 o period. ΔE k, (D T ) represens he difference beween curren inrinsic value and pas period average (median or closing) price. Panel A: Descripive Saisics N Mean Sd. Dev. Min Max mean ΔP k, median closing ΔE k, (D T ) Panel B: Regression Resuls Independen Variable mean median closing ΔE k, (D T ) 0.38 (8.48 *** ) 0.34 (7.12 *** ) 0.42 (5.13 *** ) privae k 1.78 (0.39) 0.54 (0.11) 2.66 (0.33) environmen k (=neural) (3.08 *** ) (2.78 *** ) 7.74 (0.77) environmen k (=bullish) (5.17 ***) (4.72 *** ) (2.57 ** ) consan ΔP k, 33

34 Table VI Table VI. Disposiion Effec Predicion: Average Adjusmen raio abulaed by Capial Gain and Surprise Sign The predicion is based on Frazzini s (2006) inerpreaion of disposiion effec. Surprise Capial Gain + + Incomplee posiive adjusmen; More offers o sell; Complee negaive adjusmen; Complee Posiive adjusmen; Incomplee negaive adjusmen; Fewer offers o sell Table VII Table VI I. Acual Adjusmen Raio and Number of Asks (.) shows sandard deviaion; [.] shows # of observaions. If he disposiion effec holds, For Panel A, he shaded areas should show less drif han non-shaded area; for Panel B, he upper-lef shaded area should have more offers o sell and he boom-righ should have fewer offers o sell. Panel A: Average Adjusmen Panel B: Average Number of Asks Surprise Surprise Overhang + Overhang (2.3) (4.6) (8.3) (8.6) (0.7) (0.5) (8.1) (6.0) 34

35 Table VIII Table VIII. Feasible GLS Regressions on he Effec of Informaion Asymmery on Prices Noe: The number in parenhesis is he z-saisics. *** sands for 0.01 significance level. The coefficiens are esimaed wih generalized leas square, wih heeroskedasiciy and panel-specific AR(1) auocorrelaion. diff_price j,k, is he price difference (measured in mean, median and closing price) a period beween public informaion session j and privae informaion session k; diff_info j,k, is he difference in informaion implied inrinsic value a period beween he informed invesors in session j and he uninformed invesors in session k; diff_var j,k, is he risk exposure difference a period beween he informed invesors in session j and he uninformed invesors in session k. Oher condiions are exacly he same for each paired observaion. Panel A: Descripive Saisics N Mean Sd. Dev. Min Max mean diff_price j,k, median closing diff_info j,k, diff_var j,k, Panel B: Regression Resuls Independen Variable diff_price j,k, mean median Closing diff_info j,k, 0.18 (0.66) 0.17 (0.59) 0.34 (1.01) diff_var j,k, (-0.36) (0.09) (-0.57) environmen k (=neural) (-1.06) (-0.99) (-1.74) environmen k (=bullish) 5.18 (0.35) 9.8 (0.65) -5.6 (0.32) consan

36 Table IX Table IX. Expeced Reurn for Trading Periods Expeced reurn is compued knowing he expeced inrinsic value a each period. There are 8 observaions in each cell. A each period, he risk exposed for holding a uni of asse is he same across environmen. Laer periods face less uncerainy. Bullish environmens are associaed wih posiive expeced reurns while bearish environmens are associaed wih negaive expeced reurns. Period Mean Bullish 28.00% 25.00% 20.10% 21.60% 19.70% 14.30% 10.20% 10.00% 6.40% 15.70% Neural -9.00% -3.80% -0.70% 0.40% 2.20% 4.00% 1.60% 3.30% 2.40% 0.00% Bearish % % % -8.00% -6.20% -6.10% -2.70% -2.80% -0.20% -7.30% Table X Table X. The Relaive Frequencies of Under- and Over-reacion in Simulaion Resuls The simulaion resuls sugges ha he raio of adjusmen responsiveness is he driving deerminans of he relaive frequencies (number of rading periods) for under- and over-reacion. When adjusmen displays ineria, under-reacion prevails. Consisen wih empirical resuls, over-reacion end o arise more in he long erm perspecive. Adjusmen Responsiveness Raio Disribuion Shor Term Perspecive Under-reacion Long Term Perspecive Over-reacion Under-reacion Over-reacion Insufficien N (0.27, ) 67.8% 32.2% 62.6% 37.4% Appropriae N (1, ) 49.8% 50.2% 49.9% 50.1% Excessive N (1.73, ) 32.6% 67.4% 45.9% 54.1% 36

37 Figures Figure 1 Panel A: Bearish Environmen, u = 20, d = 60 µ =360 σ =180 Panel B: Neural Environmen, u = 40, d = 40 µ =540 σ =180 Panel C: Bullish Environmen, u = 20, d = 60 µ =720 σ =180 Book Value Lef: Pah 1 Righ: Pah 2 Expeced Value Saring Value Figure 1. Asse book value random walk pahs. Pah 1 shows he se of draws randomly seleced before any sessions were run; Pah 2 shows he se of draws reversed (+ ) from Pah 1.All sessions used eiher he Pah 1 or Pah 2 se of draws. In all news regimes he asse has a saring book value of 540 cens, and he iniial sandard deviaions for he liquidaion value disribuions are 120 for all hree environmens. The only difference lies wih he relaive sizes of he poenial upward and downward surprises, causing heir expeced liquidaion (inrinsic) values o differ. The liquidaion value disribuion is approximaely normal if he number of rading periods is large enough. As rading periods approach liquidaion, he sandard deviaion of inrinsic value declines. The doed line shows he level of book value, while solid line gives he inrinsic (expeced liquidaing) value. In he neural environmen, he wo values compleely overlap because he expecaion of he surprise is zero. 37

38 Figure 2 Panel 1-A: Bearish Environmen + Public Panel 2-A: Bearish Environmen + Privae Informaion Panel 1-B: Neural Environmen + Public Informaion Panel 2-B: Neural Environmen + Privae Informaion Panel 1-C: Bullish Environmen + Public Informaion Panel 2-C: Bullish Environmen + Privae Informaion Figure 2.Transacion price ime series. The lef side consiss of 12 sessions wih public informaion, while he righ side consiss of 12 sessions wih privae informaion. The dashed curve is he Lowess leas square fied line ha bes capures he curvaure of he ransacion price ime series. 38

39 Figure 3 A: Bearish Environmen B: Neural Environmen C: Bullish Environmen Figure 3. Deviaions of average price per period from inrinsic value. The red horizonal lines indicae a deviaion of zero. The solid lines represen public informaion markes, while he dashed lines represen privae informaion markes. Deviaions also represen expeced reurns. The bearish environmen exhibis consisen downward drif oward inrinsic value, he bullish environmen exhibis consisen upward drif, while he neural environmen exhibis fla or reversals in reurn series. Figure 4 Shor Term Classificaion Long Term Classificaion Figure 4. Idenificaion of under- and over-reacion wih price migraion rule. In boh graphs, he deviaions are measured as he absolue difference beween curren period average price, P, and curren inrinsic value, E (D T ). In he shor erm classificaion, he sign of he deviaion is negaive if P is on he same side of E (D T ) as P -1 is compared o E -1 (D T ),. In he long erm classificaion, he sign of he deviaion is negaive if P is on he same side of E (D T ) as he mean of all previous prices P 1-1 is compared o he mean of all previous inrinsic values E 1-1 (D T ). 39

40 Figure 5 Figure 5. Under-reacion magniude hisograms. Boh shor erm (197:41) and long erm (160:78) hisograms sugges a dominance of under-reacion. This is disinc from pure reurn sign based idenificaions (113:80 for shor erm reurn and 93:83 for long erm reurn.) There is a endency for overreacion o rise over a longer erm perspecive. Figure 6 Figure 6. Paired difference in adjusmen raio. Each pair compares he adjusmen rae in a period of he privae informaion sessions and is counerpar in he public informaion sessions. DHS predics he disribuion o have a negaive mean, while he above hisogram reveals a close o zero mean, symmeric disribuion. 40

41 Figure 7 Figure 7. Adjusmen raio hisogram. Adjusmen raio is compued as he raio beween he price adjusmen magniude and he change in he inrinsic value. A raio beween zero and 1 is considered as insufficien adjusmen; a raio larger han 1 is considered as excessive adjusmen; and a raio smaller han zero is considered as opposie adjusmen. Figure 8 Figure 8. Subjecive probabiliy and belief coninuaion. Subjecive probabiliy is compued as he probabiliy of a Head draw in each new period ha could suppor he ongoing price. The objecive probabiliy is always 0.5 in any period. Any probabiliy falling ouside he inerval [0,1] represens a price ha is impossible o be jusified by any of he possible liquidaion values. 41

42 Figure 9 Bearish Environmen Neural Environmen Bullish Environmen Fied Line for All Figure 9. Risk-Reurn Plo. The fied line for E, ) = k, ( r k λ VAR + ε has a slope of 0.00, wih z=- 0.21, p> z =0.83, which indicaes ha he coefficien is indisinguishable from zero. For any level of risk, he expeced reurns for he bullish environmen ends o be larger han for he neural environmen, which in urn ends o be larger han for he bearish environmen. 42

43 Figure 10 A-1 B-1 A-2 B-2 A-3 B-3 Figure 10. Simulaion. Panel A uilizes he shor-erm perspecive and Panel B uilizes he long-erm perspecive in defining under (over) reacion. For each block, 500 marke sessions are conduced. For he firs row, he mean adjusmen raio is 0.27; for he second row, he mean adjusmen raio is 1; and for he hird row, he mean adjusmen raio is The relaive frequencies of under- and over-reacion are primarily driven by he magniude of shor-erm adjusmens. 43

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