閤 Insiders Outsiders. and Market Breakdowns OR。 Utpal Bhattacharya; Matthew Spiegel The Review of Financial Studies, Vol. 4, No. 2.(1991), pp. 255-282 Stable url: http://inksistor.org/sici?sic0893-9454%281991%0294%3a2%3c255%3aioamb%03e2.0.co%3b2-h The Review of Financial Studies is currently published by Oxford University Press Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyouhaveobtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the jsTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at www.istor.org/journals/oup.html Each copy of any part of a JSTOR transmission must contain the same copyright notice that ap on the screen or printed page of such transmission STOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support @jstor. org http:/www.jstor.org Fri mar 161201:042007
Insiders, Outsiders, and Market Breakdowns Utpal Bhattacharya; Matthew Spiegel The Review of Financial Studies, Vol. 4, No. 2. (1991), pp. 255-282. Stable URL: http://links.jstor.org/sici?sici=0893-9454%281991%294%3A2%3C255%3AIOAMB%3E2.0.CO%3B2-H The Review of Financial Studies is currently published by Oxford University Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/oup.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Fri Mar 16 12:01:04 2007
Insiders. Outsiders, and Market Breakdowns Utpal Bhattacharya University of Iowa Matthew Spiegel Columbia University A simple classical walrasian framework is pro posed for tbe study of manipulation among asym metrically informed risk-averse traders in finan cial markets, and it is used to analyze tbe occurrence ofa market breakdown in toe trading system. Such a phenomenon occurs wben toe out siders refuse to trade with the insiders because toe informational motive for trade of tbe insider out. weighs ber hedging motive. We demonstrate tbe robustness ofour results by proving tbat toe mar. ket collapse condition extends not only to tbe lin ear strategy function, but to tbe wbole class offea sible nonlinear strategy functions. Implications for insider-trading regulation are sketched. This article finds in closed form the entire set of noisy rational expectations equilibria for a model of an xchange economy characterized by asymmetric information and strategic behavior. The model has the following features. There is an informed risk-averse monopolist and a continuum of competitive risk- averse uninformed traders. The basic structure of the model includes strategic behavior by the informed Walrasian price formatie quests to Matthew Spiegel, Columbia University, Graduate School of Busi ness, 414 Uris Hall, New York, NY 10027 The Review of financial Studies 1991 Volume 4, number 2, Pp. 255-284 c 1991 The Review of Financial Studies 0893-9454/91/$1.50
Tie Review of financial Studies/v 4n 2 199 hich traders submit demand functions, an endogenous motive for trade due to the random endowment of the insider and rational expectations by the uninformed traders. Although subsets of these features have been included in various models in the literature, the current article is the first to have all of them moreover. it is the first to investigate the entire set of continuous equilibria, including non linear ones, within a noisy rational expectations Walrasian environ ent involving only risk-averse individuals The primary focus of the article is to examine the conditions that lead to a collapse of trade in a financial market. Glosten and milgrom (1985), Glosten(1989), and Leach and Madhavan(1989) found that a risk-neutral individual will not voluntarily act as a market-maker if he is at a severe informational disadvantage relative to some other traders. In this article, we extend their result to the case of the risk averse individual trading in a Walrasian market. It is shown that when this trader is too uninformed he will not trade with the more informed; the linear equilibrium fails, as do all of the continuous nonlinear equilibria. he introduction of risk-averse agents with nonzero expected aggregate endowments has three advantages. First, both conditions are realistic, in that aggregate security holdings are strictly positive and investors do worry about the volatility of their positions. Second the conditions ensure that there is a hedging as well as an informa tional motive for trade. 2 Third, these assumptions enable us to focus on the risk premia commanded by the holders of the risky security and to study how it varies with the insider's position. Noisy rational expectations models with risk-neutral agents cannot be used to exam ine this issue, but models such as Grossman and Stiglitz(1980) Hellwig(1980), Bray(1981), Admati(1985), and Kyle(1989), which employ only risk-averse agents in a Walrasian environment, are capa ble of such an analysis. The primary difference between the present article and this literature is that none of the above models examine risk premia outside the linear equilibria, while the present analysis is extended to the full set of equilibria The model presented in this article combines the assumptions of the perfect and imperfect competition frameworks employed in the ted to our article, in that the er, they only solve for and analyze use exogenously define which are no tions of the equilibrium linear equilibrium odels, in that it is the only pricing rule considered The inclusion of risk- neutral agents destroys the risk-shan cept employed in the classical Further, evidence from Loderer and Sheehans(1989) study indicates that insiders 256
siders, Outsiders, and Market Breakdowns noisy rational expectations literature. In the models of perfect com petition, individuals believe they can trade any amount without alter ng the security's price. These studies include Grossman(1977) rossman and Stiglitz(1980), Verrechia(1982), Glosten and Milgrom (1985), Allen(1987), and Ausubel (1990). In contrast, models of imperfect competition employ traders who believe their transactions influence prices, as illustrated by Kyle(1985, 1989)and Caballe(1989) The present article, like the models of Grinblatt and Ross(1985) and Laffont and Maskin(1990), falls in between, with a large monopolistic insider and competitive outsiders. What differentiates our article from the last two is that all of our traders are risk-averse. In this respect he model is a partially revealing version of Kihlstrom and Postle waite's(1983) fully revealing model of a monopolist and a competitor With some exceptions, the above cited articles prevent prices from coming fully revealing through the use of noise traders or, equiv alently, an auctioneer who sets the aggregate demand to a random number 3 In contrast, in the models of Bray (1981), Ausubel (1990) Gale and Hellwig(1988), Glosten(1989), and Laffont and Maskin (1990), prices are not fully revealing because the insider has an informational advantage in two dimensions. In the latter specification, is possible to form equilibria by having each individual remain at his initial endowment The use of noise traders or an auctioneer who sets demand to a random number, however, prevents this. Under either assumption, trade must always occur; either trade is necessary to balance out the demands of the noise traders, or it is required to meet the auctioneer's quantity constraint As the primary focus of this article is to obtain and analyze the no-trade equilibrium, we have pted for the latter modeling techni The organization of the article is as follows. Our model is presented in Section 1. In Section 2, we solve for the equilibrium and present its general characteristics; it is in this section that the market break down condition is presented In Section 3, the market risk premium and the transmission of information are considered. In Section 4, we investigate the special case of the linear equilibrium and detail its properties. We conclude in Section 5 1. Model The model analyzes a two- date exchange economy. At date 0, endow ments are distributed, players submit demand functions, and the Wal rasian auctioneer then finds a price-quantity pair to equate supply Gale and Hellwig(1988) discuss the different roles of"noise traders, "and find that their diverse es make the results very difhcult to interpret
Tbe Review of Financial Studies/v 4 n 2 1991 with demand. 4 There are no restrictions on the demand schedule an individual must submit, except that it must be a continuous function If the auctioneer cannot find a price that equates supply and demand he market collapses and no trade occurs. Whenever two or more market-clearing price-quantity pairs exist, the auctioneer selects one at random and then allocates quantities to satisfy everyone's demand at the designated price. At date 1, all uncertainty is resolved, and the players receive their final payoffs and engage in consumption L The economy contains two sets of agents (M and C), whose indi dual members have exponential utility with risk-aversion parameter 8. The set M consists of a single monopolistic informed investor who we take to be a woman while Contains a continuum of small traders who we reter to as men The investors trade in two securities, both of which pay off in the economy's single consumption good. The first security is a riskless bond, the price of which at dates 0 and 1 is set equal to 1. This estriction is simply a normalization, since all consumption takes place on date 1. The second security is a risky stock, whose date-0 price is P(determined by market-clearing), and whose date-1 valu (Pi) is generated by three additive factors: P=pp+E+n. Throughout time, both securities have their supplies normalized to unity. At date 0, prior to trade, the informed investor knows the values of up and e and has a prior belief that n is normally distributed with mean zero and variance o 2. In addition, she is also endowed with 1 - So shares of the stock, 1 -Bo shares of the bond, and w shares of a nontraded asset. The nontraded endowment can be interpreted as human capital or any other illiquid asset. While our model takes the expected endowment of this asset to be zero, none of the basic results are altered if a positive mean is assumed The payoff of this nontraded endowment is correlated with the return of the risky stock; it, there fore, needs to be"hedged. " For simplicity, perfect correlation is assumed. Thus, at date 1. she earns an amount wP, from the nontraded endowment 5 At date 0, the uninformed traders know up and correctly believe that 6, n, and w are independently normally distributed with zero means and variances o 2, 02, and o 2, respectively. The uninformed traders are modeled as a continuum of individuals whose set has a finite measure. More precisely, if we let f(i represent the density of traders of"type"i, then it is assumed that 4 The market mechanism e d here is similar to wilson' s (1979) auction of shares s The assumption of perfect correlation between the payoffs of the nontraded asset and the risky asset makes our model equivalent to models where noise is introduced by making the insider's endow ment of the risky at date 1 unknown to everyone but he
foi di= n Let So(i) and B (i) represent piecewise continuous functions giving the initial stock and bond endowments of uninformed traders Then their total endowments are so(if(o di= So and B6()f(1)d=B The functions So(i) and Bo (i are common knowledge in the econ The functions S(i and B(i are the respective demand densities for the stock and bond by type i uninformed traders. The aggregate demand is represented by the variables S and B. Thus, s(ifci di= S and B(f(1)d=B. Notice that the informed has an informational advantage because she knows her endowment (w) and two of the factors generating the return of the risky asset (e and up); whereas the uninformed only know up. Hence, o2 is a measure of the small investors" informational disadvantage. A similar metric for "informational disadvantage' appears throughout most of the noisy rational expectations literature. 2. Characterization of Equilibrium The Bayesian-Nash equilibrium must satisfy two conditions. First, each of the uninformed investors must submit a demand schedule that maximizes his expected utility, subject to his budget constraint and the available information, including P. Second, knowing this demand schedule, the informed investor has to submit a demand function so as to maximize her utility subject to her wealth and infor mational endowments, which are exogenous. The problem is con siderably simplified by noting that there is just one non-price-taker and that supplies add to unity. Therefore, one can reduce the insider's 259
problem to that of picking a price P on the aggregate demand sched ule of the uninformed traders, S(P) The next fact is used extensively throughout the article Fact 1. If X, and X, bave a bivariate normal distribution, wbere ui 42, 0m, 02, and p are the unconditional means, standard deviations, and correlation of the two random variables, then the condition al distribution of x, given that X,= x2 is a normal distribution whose mean is E(X,I x2)=u,+ po, (x2-u2/02, and variance is Var(X1|x2)=(1-p2) It is now possible to analyze the problem of the informed monop alist 2.1 Problem of the informed monopolist Using the well-known properties of exponential utility functions and normal distributions, the monopolist wishes to maximize over P and B the following value function maxv=(1-S+u)(H+e)+(1-B) 0.50(1-S+u)2o (1) subject to her budget constraint [(1-S)-(1-S)]P+[(1-B)-(1-B0)]=0.(2) Since the insider is effectively selecting a price on the aggregate demand function generated by the uniforme optimization problem becomes very simple (2), one elimi- from (1), and then derives the first-o condition for a maximum of (1) with respect to Pas S(P-kp-e)+(S-S)+62S(1-S+t)=0.(3) A more convenient and informative representation of the above equa tion Is P=μp+a(P)+r, a(P)=(S0-5)/S′-62(1-S) r=∈-602 (5) Equation(4) brings into sharp focus a number of insights devel
Insiders Outsiders, and Market breakdowns oped in the noisy rational expectations literature. First, given P, the riable T is fully revealed. This tells us that although there may be a nonlinear term in the price, the price is still a linear function of the normally distributed information variable of the insider, T, and it reveals this. Importantly, this revelation would not occur if the aggre gate demand function submitted by the outsiders was a correspon dence. In the Appendix, we prove that it is not optimal for the out. siders to submit such a correspondence. Second, even though the outsiders know T, they cannot fully identify e, since w also enters the equation. The general public is therefore left uncertain as to whether the primary motivation for trading by the informed is"hedging"or nformational. Third, if e is the only variable that the informed has n informational advantage in(o,=0), the market clearing price P fully reveals e. Fourth, although the uninformed cannot disentangle , they can learn something about Pi from the offer price P. We now proceed to analyze how this learning takes place 2.2 Problem of the uninformed competitors Equation (5) tells us that t is a linear function of two normally dis- tributed random variables, implying that it is also a normally distrib- uted random variable. Simple calculations can then be used to show that E(T)=0 and var()=02=02+6如2AsP1=μp+∈+nby construction, one also knows that Pi is a normally distributed random variable with E(P)=μ and var(P1)=σ2+ p(P,T)=σ/2(02+σ2)]/2 Hence, by observing the equilibrium price P, and thereby T, the uninformed update their priors on P,. The posterior distribution of Pi given P, using fact 1, is now normally distributed with E(P1|P=E(P1|) =p+o{r-0]1σ2=μp+σ{P-(p+a(P)/a2 var(P1|P)=Ⅴar(P1|r)=(2+a2)-0:/σ2. (7) Equations (6)and(7) make precise the " learning procedure""of the uninformed. Equation(6) gives them the posterior mean of P1 as a function of P, while Equation (7) is used to update the variance Notice that the posterior precision on P is higher than the prior precision, and this improvement does not depend on P(assuming P exists) The above arguments show that the final period 1 wealth of the ith 261
Tbe Review of Financial Studies/v4n21991 uninformed trader, S(i)P,+ B(i), is normally distributed. Therefore one can use the assumption that their utility functions are negative exponential and rewrite their objective in a mean-variance frame vork as maxv=S()E(P|P)+B()-0.50s()var(f|P),(8) subject to the budget constraint [S()-S0()P+[B()-B()=0. The first-order condition of (8), subject to(9), is then used to deter mine the individual demand schedule of each uninformed investor The demand, it should be noted is the same for all the uninformed 2.3 Feasible aggregate demand curves Aggregating all the individual demands we get S()f( nE(P I P)-PI 8 Var(P I P) (10) Substituting for E(P I P)from(6) and for Var(P I P) from(7) in(10), we obtain a differential equation in S(P) K t k2st KSPt KS'st KsS=0 (11) where K1=S,K2=-1 G2-0282o2 K={1+1+K3+k32}≥ K5=-(2K3+6o2) The solution of(11) gives us all the possible aggregate demand curves that satisfy the first-order condition of the informed investor Her second-order condition [obtained by differentiating (3) with respect to P] further restricts this set, and provides the complete set of feasible aggregate demand curves. In order to find the set of equi libria, we analyze the problem in three steps. First, Proposition 1 establishes the set of solutions satisfying Equation(11). Second, Prop osition 2 uses the second-order condition to find restrictions that any 262
Insiders, Outsiders, and Market Breakdowns solution to the problem must satisfy. Third, Theorem 1 uses Propo- tions 1 and 2 to determine the set of feasible equilibrium aggregate demand functions of the uninformed Proposition I. Tbere exists a complex number C such that tbe set of feasible aggregate demand curves that satisfy Equation(11) can be [(K,+ K2)(K,P+ Ks)-K,K,+ K, KSI(K,+ K2S)K3/K2]=C, for any initial condition(S*, P*). In the present application, one can simplify tbe above equation by substituting out K, and k2to obtain (K3-1)(KP+K)-K4S+KKSI(S-S)8]=C.(12) Proof Differentiate(12) with respect to P to get back (11). For a detailed exposition of the "method of integrating factors"that was utilized to solve (11), refer to the appendix Q ED While there always exists a curve passing though any (S, P) pair, not all of them satisfy every equilibrium condition imposed by the economics of the problem. As the next proposition shows, the insi der's second-order conditions impose the intuitive restriction that an equilibrium aggregate demand curve cannot have any upward sloping region Proposition 2. In equilibrium, it is necessary and suficient for an ggregate demand curve to bave a sufficiently negative slope for all S, and satisfy(12). The exact requirement is dP/ds s-on-t war(P|P0 (13) (K3+K2)(K3P*+K5)-KK1+K3KS*≥0 (14) Conversely, if S*>So, then 263
