Overconfidence, public disclosure and long-lived information

Accepted ManuscriptOverconfidence,public disclosure and long-lived informationDeqing ZhouPII:S0165-1765(12)00354-0DOI:10.1016/j.econlet.2012.06.022Reference:ECOLET5482To appear in:Economics LettersReceived date:14February2012Revised date:28May2012Accepted date:8June2012Please cite this article as:Zhou,D.,Overconfidence,public disclosure and long-lived information.Economics Letters(2012),doi:10.1016/j.econlet.2012.06.022This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting,typesetting,and review of the resulting proof before it is published in its final form.Please note that during the production process errors may be discovered which could affect the content,and all legal disclaimers that apply to the journal pertain.Highlights:The single insider in this model is irrationally confident.He is required to publicly disclose his trades after the fact.We find the more confident insider trades less aggressively in the first trade. Half of the total information is revealed by the first trade.Overconfidence,Public Disclosure and Long-lived InformationDeqing Zhou∗a Department of Business Statistics and Econometrics,Guanghua School of Management,Peking University,100871,Chinab Center for Statistical Science,Peking University,Chinac School of Finance,Central University of Finance and Economics,100081,ChinaAbstractWe analyze a strategic trading model where an overconfident insider is required to pub-licly disclose his trades after the fact.Wefind the more confident insider is more concerned about the effect the initial trading has on the future.Keywords:Insider;Market makers;Overconfidence;Public disclosure;JEL classification:C72,D82,G141.IntroductionUsing an extension of the two periods frameworks of Kyle(1985)and Huddart et al. (2001),this work characterizes the optimal behavior of an irrationally confident insider who possesses a long-lived noisy private signal concerning the fundamental value of a risky asset under the mandatory disclosure requirement.Kyle(1985)examines the equilibrium strategy of a rational insider with no public disclosure.Huddart et al.(2001)extend Kyle(1985)to allow for public disclosure,but like Kyle(1985),they also assume the insider to be rational. 1Abundant evidences show,however,that most people most of the time are overconfidentin the sense that they overestimate the precision of their knowledge.Thus,we attempt to address the following question:what is the effect of the interaction between irrationality and trade disclosure on the equilibrium results such as the insider’s trading strategy,the insider’s profits,the market depth,the trading volume and the price efficiency?Wefind that,when the information is good enough or when the insider is moderate overconfident,a linear and mixed equilibrium exists.The more confident insider trades less aggressively and camouflages his information with less noise in thefirst period.As a result, half of the total information is revealed by thefirst trade,no matter how confident the insider is.This is in sharp contrast with the no disclosure cases,such as Wang(1998)and Zhou(2011)where enhancing the trader’s confidence degree improves the market efficiency in revealing private information.In Section2we discuss the structure of our model.Section3derives the linear and mixed equilibrium and presents the properties of our model’s equilibrium.Section4concludes.2.The modelWe confirm to the notation of Huddart et al.(2001).A risky asset is traded between a single insider,multiple market makers and multiple noise traders.The asset’s fundamental value v is normally distributed with prior mean p0and prior varianceσ2v,in statistical notation,v∼N(p0,σ2v).An imperfect signal concerning the fundamental value,denoted as s,satisfiess=v+ǫ,in whichǫ∼N(0,σ2ǫ)andǫis independent of v.The insider observes in advance the realization of the signal s,while market makers have no such observation.Assume the insider believes thats=v+kǫ,in which k−1represents the insider’s confidence degree.When k<1,the insider is over-confident,believing that this signal is more precise than it actually is,when k>1,he is underconfident and when k=1he is rational.The market makers are rational.Consider a two periods model with the periods indexed by n∈{1,2}.In the nth period, the insider submits an order x n to maximize his expected profits,based on the information available to him and the belief mentioned above.Meanwhile,noise traders together submit an exogenous order u n∼N(0,σ2u).Assume u1,u2,v,ǫare mutually independent.Prior tothe nth trade,market makers observe the total orderflow x n+u n(but they cannot observe2x n or u n individually)and set the price as p n.After thefirst trade and before the second, the insider’s submission in thefirst period is publicly disclosed and market makers update the price to p∗1based on this information.In a semi-strong efficient market,p1=E[v|x1+u1],p∗1=E[v|x1],p2=E[v|p∗1,x2+u2].Useπn to denote the insider’s profit obtained from the nth trade,E k(·|·)and E(·|·)to denote respectively the conditional expectation under the insider’s belief and that under the rational belief.3.The equilibriumFirst,we present some preliminary results in Lemma1.Lemma1.The insider evaluates the asset’s fundamental value v asσ2vE k(v|s)=c k s,with c k=for c k c −11∈(0,2)2and any given p 0,there exists a Nash equilibrium characterized by the following constants:λ1=λ2=1 c k c −11−(c k c −11)2σu 2−c k c −11 Σ0121c k c −11Σ0,(8)σ2z = 12σ2u ,(9)Σ1=Σ0λ1,h 1=c k c −11−1,h ∗1=2(c k c −11−1)λ2,h 2=1−c k c −11.(12)Proof.See Appendix A. Unlike the rational belief cases examined by Huddart et al.(2001),Cao and Ma (2000)and Zhang (2004)where the insider trades only on the pricing error,an irrational insider trades on both the pricing error and the current price.Actually,an overconfident insider trades negatively on the current price in the first period and trades positively on it in the second period,whereas an underconfident insider trades reversely.In Wang (1998)and Zhou (2011)with no disclosure,enhancing the trader’s confidence degree improves the market efficiency in revealing private information,in our model with disclosure,however,each trade reveals half of the total information,3no matter how confident the insider is.From Theorem 1,it is straightforward to show that:Proposition 1.(i)β1decreases and β2increases with k −1.(ii)σz decreases and γincreases with k −1.σ2ǫ−σ2vc 1 x 2−η2p ∗1Remarks:(i)shows that a more confident insider trades less aggressively in thefirst period and more aggressively in the second.Intuitively,an overconfident insider is more concerned about the effect thefirst trading has on the second than the rational one since he overestimates his information advantage.(ii)presents that a more confident insider camouflages his information with less noise,and correspondingly,market makers adjust more to the insider’s announcement.Define the expected trading volume in the nth period as that in Admati and Pfleiderer (1988):1V ol(n)=2−2),V ol(2)increases with k.When k is samll enough such that c k c−11∈(2√aggressively in thefirst period and more aggressively in the second.In contrast to the case with no disclosure,we show that with disclosure the more confident insiderfinds it optimal to inject less noise to thefirst trade so that thefirst trade still reveals half of the total initial information.Appendix AProof of Lemma1.First,E k(v|s)=E(v|v+kǫ)=c k s.SinceCov(v−c1s,s)=0,Cov(v−c1s,u n)=0,for n=1,2,by the property of normally distributed variables,we know v−c1s is independent of s,u1, and u2.Note that in the equilibrium,p∗1,x1,x2are all measurable to the information set {s,u1,u2}.Hence,v−c1s is also independent of p∗1,x1,x2.This givesE[v−c1s|x1+u1]=E[v−c1s|x1]=E[v−c1s|p∗1,x2+u2]=E[v−c1s]=0,from which expressions for p1,p∗1and p2in Lemma1are verified.Proof of Theorem1.In the second period,given the market makers’pricing strategy(4), the insider’s profit is estimated by the insider asE k[π2|p∗1,s]=x2[c k s−λ2x2−(1+h2)p∗1],(13) therefore,1x2=[c k s−(1+h2)p∗1]2.(14)4λ2For identifying h2,Lemma1impliesE[p2−p∗1|x1]=0,6but from(4),we knowE[p2−p∗1|x1]=12λ2,η2=c k c−11−1β22Σ1+σ2u.(16)Combining(15)and(16)givesλ2=1 c k c−114 Σ1for c k c−11∈(0,2),andβ2= 2−c k c−11 σ2u4λ2(c k s−(1+h2)p∗1)2|s]by(14), =(c k s−(1+h1)p0−λ1x1)x1+12λ2]x1+[1−(2−c k c−11)γ2λ2(2−c k c−11)2(h∗1+1)γ]p0=0.7Since the insider is willing to paly a mixed strategy x1=β1(c1s−p0)+η1p0+z,where z∼N(0,σ2z),the insider must be indifferent across all values of x1,thus−2λ1+(2−c k c−11)2γ2=0,(19)2λ2−(1+h1)+1.(21)2Substituting(21)into(18)yields(2−c k c−11)γλ1=σu 2−(c k c−11)2,λ1and thus expressions forη1and h1in(11)are verified.8Application of the projection theorem for normally distributed variables givesβ1Σ0λ1=,(27)β21Σ0+σ2zΣ1=Σ0−β21Σ20σ2u.(29)c k c−11ThenΣ1=Σ0−c k c−11β21Σ20by(26),(2−c k c−11)σ2u=Σ0−4λ21σ2u222c k c−11 σ2u.Hence V arx1and V ol(1)increase with k.When n=2,V arx2=β22Σ1+η22V arp∗1=β22Σ1+η22(Σ0−Σ1),(30) where(30)results from the factΣ0=V ar(c1s−p∗1+p∗1)=Σ1+V arp∗1.9Substituting forβ2andη2from(8)and(12)into(30)yieldsV arx2=(c k c−11)2+4(c k c−11−1)2dc k c−11=2(−4+4c k c−11+(c k c−11)2)2−2),V arx2and V ol(2)increase with k.When c k c−11∈(2√2−c k c−112=2Zhou,D.Q.,2011.Overconfidence on public information.Economics Letters,112,239-242. 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