1 THE JOURNAL OF FINANCE • VOL. LIII, NO . 6 • DECEMBER 1998 Volume, Volatility, Price, and Profit When All Traders Are Above Average TERRANCE ODEAN* ABSTRACT People are overconfident. Overconfidence affects financial markets. How depends on who in the market is overconfident and on how information is distributed. This paper examines markets in which price-taking traders, a strategic-trading insider, and risk-averse marketmakers are overconfident. Overconfidence increases ex- pected trading volume, increases market depth, and decreases the expected utility of overconfident traders. Its effect on volatility and price quality depend on who is overconfident. Overconfident traders can cause markets to underreact to the in- formation of rational traders. Markets also underreact to abstract, statistical, and highly relevant information, and they overreact to salient, anecdotal, and less rel- evant information. ODELS OF F INANCIAL MARKETS are often extended by incorporating the im- M perfections that we observe in real markets. For example, models may not consider transactions costs, an important feature of real markets; so Con- 1979 ! , Leland ~ 1985 ! , and others incorporate transactions costs stantinides ~ into their models. Just as the observed features of actual markets are incorporated into mod- els, so too are the observed traits of economic agents. In 1738 Daniel Ber- noulli noted that people behave as if they are risk averse. Prior to Bernoulli most scholars considered it normative behavior to value a gamble at its ex- pected value. Today, economic models usually assume agents are risk averse, though, for tractability, they are also modeled as risk neutral. In reality, people are not always risk averse or even risk neutral; millions of people engage in regular risk-seeking activity, such as buying lottery tickets. Kahne- * University of California, Davis. This paper is based on my dissertation at the University of California, Berkeley. I thank Brad Barber, Minder Cheng, Simon Gervais, David Hirshleifer, Bill Keirstead, Hayne Leland, Mark Rubinstein, Paul Ruud, Hersh Shefrin, René Stulz, Rich- ard Thaler, two anonymous referees, and seminar participants at UC Berkeley and at the Russell Sage Foundation Summer Institute in Behavioral Economics for their comments. I es- pecially thank Brett Trueman for his numerous suggestions and comments. Financial support from the Nasdaq Foundation and from the American Association of Individual Investors is gratefully acknowledged. 1887

2 The Journal of Finance 1888 1979 ! identify circumstances in which people behave in a man and Tversky ~ risk-seeking fashion. Most of the time, though, most people act risk averse, 1 and most economists model them so. This paper analyzes market models in which investors are rational in all 2 A substantial literature in cog- respects except how they value information. nitive psychology establishes that people are usually overconfident and, spe- cifically, that they are overconfident about the precision of their knowledge. As is the case with risk-aversion, there are well-known exceptions to the rule, but most of the time people are overconfident. Psychologists also find that people systematically underweight some types of information and over- weight others. The paper looks at what happens in financial markets when people are overconfident. Overconfidence is a characteristic of people, not of markets. It would be convenient if each person’s overconfidence had the same effect on markets. But this is not so. Some measures of the market, such as trad- ing volume, are affected similarly by the overconfidence of different market participants; other measures, such as market efficiency, are not. The effects of overconfidence depend on how information is distributed in a market and on who is overconfident. Because analyzing the overconfidence of only one type of trader presents an incomplete and perhaps misleading picture, I look at the overconfidence of different traders: price takers in markets where information is broadly disseminated, strategic-trading insiders in markets with concentrated information, and marketmakers. I also examine markets where information is costly. Three different models are employed to facili- tate this multifaceted analysis of overconfidence. These are modifications of ! and Hellwig ~ 1980 ! ~ ~ 1985 ! , and Gross- Diamond and Verrecchia 1981 , Kyle 1980 ! . man and Stiglitz ~ The main results presented are: • Trading volume increases when price takers, insiders, or marketmakers are overconfident. This is the most robust effect of overconfidence. An- ecdotal evidence suggests that in many markets trading volume is ex- Dow and Gorton ~ !! . Recent empirical studies ~ Odean ~ 1998a ! , 1997 cessive ~ !! indicate that overconfidence generates trad- 1998 ~ Statman and Thorley ing. From a modeling perspective, overconfidence can facilitate orderly trade even in the absence of noise traders. 1 Another place where observed behavior has found wide acceptance in economic models is in the discounted utility of future consumption. Nineteenth-century economists such as Senior, Jevons, and Böhm-Bawerk believed that, ideally, the present and the future should be treated equally; yet they observed that generally people value present consumption more highly than Loewenstein 1992 !! . Today, when it may affect the predictions of models, economists future ~ ~ usually assume that people discount the utility of future consumption. And people usually do discount the future—but not always. They will, for example, “bite the bullet” and get an un- pleasant experience over with, which they could otherwise delay. Loewenstein and Prelec ! ~ 1991 identify circumstances in which people demonstrate negative, rather than the usual positive, time preference. 2 In the first model presented here, investors also behave with less than full rationality by trading myopically.

3 1889 When All Traders Are Above Average • Overconfident traders can cause markets to underreact to the informa- tion of rational traders, leading to positive serially correlated returns. Returns are also positively serially correlated when traders under- weight new information and negatively serially correlated when they overweight it. The degree of this under- or overreaction depends on the fraction of all traders who under- or overweight the information. A re- view of the psychology literature on inference finds that people system- atically underweight abstract, statistical, and highly relevant information, and overweight salient, anecdotal, and extreme information. This may shed some light on why markets overreact in some circumstances, such , and underreact in oth- 1991 !! Ritter IPOs !~ ~ ~ as initial public offerings 1989, 1990 !! , ~ ers, such as earnings announcements ~ Bernard and Thomas ~ 1995 !! , dividend initiations and omissions ~ Michaely, Thaler, and Womack open-market share repurchases ~ Ikenberry, Lakonishok, and Vermaelen , and brokerage recommendations ~ Womack ~ 1996 !! . ~ 1995 !! • Overconfidence reduces traders’ expected utility. Overconfident traders hold underdiversified portfolios. When information is costly and traders are overconfident, informed traders fare worse than uninformed trad- ers. And, as Barber and Odean ~ 1998 ! find to be true for individual investors, those who trade more actively fare worse than those who trade less. Overconfidence may also cause investors to prefer active man- agement ~ 1992 !! despite evidence that ~ Lakonishok, Shleifer, and Vishny it subtracts value. • Overconfidence increases market depth. • Overconfident insiders improve price quality, but overconfident price takers worsen it. • Overconfident traders increase volatility, though overconfident market- makers may dampen this effect. Excess volatility in equity markets has Shiller 1981, 1989 ! , LeRoy and Porter been found by some researchers ~ ~ , though others have questioned these findings ~ Kleidon ~ 1986 ! , 1981 !! ~ 1986 !! Marsh and Merton ~ . The rest of the paper is organized as follows: Section I reviews related work. Section II describes some of the literature on overconfidence and on infer- ence and discusses why we should expect to find overconfidence in financial markets. Section III presents the models. Section IV discusses the results. And the final section concludes. Formal statements of the propositions, proofs, and the derivations of equilibria are presented in the Appendixes. Table I provides a summary of notation used in the models. I. Related Work A number of researchers have modeled economies in which traders hold mistaken distributional beliefs about the payoff of a risky asset. In Varian ! ~ 1989 traders’ priors have different means. Varian notes that the dispersion of posterior beliefs caused by differing distributional assumptions motivates trade. Harris and Raviv ~ 1993 ! investigate a multiperiod economy in which

4 The Journal of Finance 1890 ! ! ! 1 1 1 N I e 2 2 2 z v e h h h 1 1 1 i i 0,1 ti I S 0, 0, 0, ti 0 ti v 0 x l ~ ~ ~ # $ f f x W 5 1, ..., N N N k h 5 t I y 5 ; ; I i I I e; v z Costly Information Model ! ! ! 1 1 1 I e 2 2 2 z v e h h h 1 1 1 I 0, 0, 0, 1 0, v xx 11 ~ ~ ~ PP # $ 5 N N N k h 5 t I y 1 insider, ; ; I I I e; v z Insider Model 1 marketmaker ! ! 1 M 1 2 e N tm I 2 v e h h 1 1 1 Table I 0, 1 i i ~ ti t ti S 0, ti 0 ti I 0 x aa ~ # # $ v f P f x Notation x F W N 1, ..., 0, ..., 4 1, ..., N k g h 5 ; 5 5 5 ; ti I t i I y tm v m I e Price Takers Model ’s information set i ’s endowment of risky asset ’s demand for riskless asset ’s endowment of riskless asset ’s demand for risky asset ’s wealth Number of distinct signals Terminal value of risky asset Price of risky asset Per capita supply of risky asset Coefficient of absolute risk aversion Noise trader demand Signals Error term in signals i Time Overconfidence parameter Parameter underweighting priors Parameter underweighting signals of others Number of traders i Trader Fraction of traders who buy information i i i

5 1891 When All Traders Are Above Average risk-neutral traders disagree about how to interpret a public signal. The model of price-taking traders presented here differs from Harris and Raviv in that my traders are risk averse and disagree about the interpretation of private signals. Furthermore, the nature of this disagreement is grounded ! , risk-averse traders in psychological research. In Kandel and Pearson 1995 ~ disagree about both the mean and the variance of a public signal. In this case, the public signal may motivate increased trading even when it does not ~ 1990 show in an overlapping generations model change price. De Long et al. ! that oveconfident traders who misperceive the expected price of a risky as- set may have higher expected returns, though lower expected utilities, than 1986 suggests that overconfi- ~ rational traders in the same economy. Roll ! may motivate many corporate takeovers. Hirshleifer, Subrah- hubris ~ dence ! ! argue that overconfidence can promote herding manyam, and Titman ~ 1994 1978 ! in securities markets. Figlewski ~ finds that the inf luence of traders with different posterior beliefs on prices depends on wealth, risk aversion, 1978 also points out that a trader’s ~ and overall willingness to trade; Feiger ! ! find that inf luence on price depends on her wealth. Jaffe and Winkler ~ 1976 the probability of trading is a function of the precision of an investor’s information. ~ 1994 ! develop a model in which traders infer, from Shefrin and Statman past observations, the transition matrix governing changes in the dividend growth rate. In their model, some traders are true Bayesians; others make one of two common errors: They weight recent observations too heavily, thus underweighting prior information, or they commit a gambler’s fallacy, ex- pecting recent events to reverse so that short runs of realized events more closely resemble long-term probabilities. When all traders are rational, the market behaves as if it had a “single driver” and prices are efficient. Biased traders can introduce a “second driver,” thereby distorting prices and, over time, increasing volatility while decreasing market efficiency. ! ! ~ 1997 ! , and Wang ~ 1995 ~ look at overconfi- , Kyle and Wang Benos 1998 ! , but with two informed traders. In 1985 ~ dence in models based on Kyle Benos, traders are overconfident in their knowledge of the signals of others; they also can display extreme overconfidence in their own noisy signal, be- 1997 ! and Wang lieving it to be perfect. Kyle and Wang 1995 ! model over- ~ ~ confidence similarly to how it is modeled in this paper—that is, as an 3 Gervais and Odean overestimation of the precision of one’s own information. ! 1997 develop a multiperiod model in which a trader’s endogenously deter- ~ mined level of overconfidence changes dynamically as a result of his ten- dency to disproportionately attribute his success to his own ability. In Daniel, 1998 ! Hirshleifer, and Subrahmanyam ~ rational risk-averse traders trade with risk-neutral traders who overreact to private signals, properly weight public signals, and grow more overconfident with success. This results in return-event patterns that are consistent with many market anomalies. My paper differs from these others in that it examines how the effects of over- 3 I learned of Kyle and Wang’s work after developing the models in this paper.

6 The Journal of Finance 1892 confidence depend on who, in a market, is overconfident and on how infor- mation in that market is disseminated; it also relates market under- and overreactions to the psychological literature on inference. II. Overconfidence A. The Case for Overconfidence Studies of the calibration of subjective probabilities find that people tend ~ ! , to overestimate the precision of their knowledge Alpert and Raiffa 1982 ~ ; see Lichtenstein, Fischhoff, and 1977 ~ Fischhoff, Slovic and Lichtenstein ! ! ~ for a review of the calibration literature Phillips ! 1982 . Such overconfi- dence has been observed in many professional fields. Clinical psychologists !! , physicians and nurses, ~ Christensen-Szalanski and Bushy- Oskamp ~ 1965 ~ ! , Baumann, Deber, and Thompson ~ 1991 !! , investment bankers 1981 ~ head 1972 !! , engineers ~ Kidd ~ ~ !! , entrepreneurs ~ Cooper, Staël von Holstein ~ 1970 1988 , lawyers ~ Wagenaar and Keren ~ 1986 !! , nego- Woo, and Dunkelberg ~ !! ~ !! , and managers ~ Russo and Schoemaker Neale and Bazerman tiators ~ 1990 have all been observed to exhibit overconfidence in their judgments. ~ 1992 !! . ~ and Yates ~ 1990 ! ! ! ~ For further discussion, see Lichtenstein et al. 1982 The best established finding in the calibration literature is that people tend to be overconfident in answering questions of moderate to extreme dif- Fischhoff et al. ~ 1977 ! , Lichtenstein et al. ~ 1982 ! , Yates ~ 1990 ficulty , Grif- ~ ! ~ !! . Exceptions to overconfidence in calibration are that fin and Tversky 1992 people tend to be underconfident when answering easy questions, and they tend to be well calibrated when predictability is high and when performing repetitive tasks with fast, clear feedback. For example, expert bridge players 1987 !! , race-track bettors ~ Dowie ~ 1976 ! , Hausch, Ziemba, and Ru- ~ ~ Keren !! 1984 ~ Murphy and Winkler ~ 1981 !! tend to be binstein ~ and meteorologists well calibrated. Miscalibration is only one manifestation of overconfidence. Researchers also find that people overestimate their ability to do well on tasks and these overestimates increase with the personal importance of the task 1935 !! . ~ Frank ~ People are also unrealistically optimistic about future events. They expect Weinstein ~ good things to happen to them more often than to their peers ! ; Kunda ~ 1987 !! . They are even unrealistically optimistic about pure ~ 1980 4 ~ 1951 ! , Irwin ~ 1953 ! , Langer and Roth ~ 1975 !! . Marks chance events ~ Greenwald 1980 !! . ~ People have unrealistically positive self-evaluations ~ Most individuals see themselves as better than the average person and most Taylor and Brown individuals see themselves better than others see them ~ 4 ~ ! reports evidence that participants in foreign exchange markets are more opti- Ito 1990 mistic about how exchange rate moves will affect them than how they will affect others. Over two years the Japan Center for International Finance conducted a bimonthly survey of foreign 0 exchange experts in forty-four companies. Each was asked for point estimates of future yen dollar exchange rates. Those experts in import-oriented companies expected the yen to appre- which would favor their company ! ;, those in export-oriented companies expected the yen ciate ~ which would favor their company ! . ~ to fall

7 1893 When All Traders Are Above Average !! . They rate their abilities and their prospects higher than those of ~ 1988 their peers. For example, when a sample of U.S. students—average age 22— assessed their own driving safety, 82 percent judged themselves to be in the 5 top 30 percent of the group !! . ~ ~ 1981 Svenson And 81 percent of 2994 new business owners thought their business had a 70 percent or better chance of succeeding but only 39 percent thought that any business like theirs would be this likely to succeed !! . People overestimate their own ~ ~ 1988 Cooper et al. contributions to past positive outcomes, recalling information related to their ! ~ successes more easily than that related to their failures. Fischhoff 1982 writes that “they even misremember their own predictions so as to exagger- ate in hindsight what they knew in foresight.” And when people expect a certain outcome and the outcome then occurs, they often overestimate the degree to which they were instrumental in bringing it about ~ Miller and 1975 !! ~ 1988 ! argue that exaggerated beliefs in Ross ~ . Taylor and Brown one’s abilities and unrealistic optimism may lead to “higher motivation, greater persistence, more effective performance, and ultimately, greater success.” These beliefs can also lead to biased judgments. In this paper overconfidence is modeled as a belief that a trader’s infor- mation is more precise than it actually is. As a consequence, traders’ poste- rior beliefs are too precise—a result directly supported by the calibration literature cited above. How heavily information is weighted depends not only on overconfidence but also on the nature of the information. Because the overconfident traders in these models believe their information to be more precise than it is, they weight it too heavily when updating their Bayesian posteriors. Relying on these posteriors, they take actions that affect mar- kets. The models can also be used to analyze the effects of overweighting or when updating posteriors ! for reasons in addi- underweighting information ~ e.g., see Proposition 5 ~ . To understand such reasons, tion to overconfidence ! it is useful to brief ly review the psychological literature on inference. B. Inference Psychologists find that, when making judgments and decisions, people overweight salient information i.e., information that stands out and cap- ~ !! ~ ! Kahneman and Tversky ~ 1980 1973 . People also tures attention !~ , Grether give too much consideration to how extreme information is and not enough Griffin and Tversky ~ 1992 !! ; they “often behave as though to its validity ~ information is to be trusted regardless of its source, and make equally strong or confident inferences, regardless of the information’s predictive valu e... Whether the information is accurate and fully reliable or alternatively out- y... Fiske and of-date, inaccurate, and based on hearsay ma matter little” ~ !! . They overweight information that is consistent with their ~ Taylor 1991 existing beliefs, are prone to gather information that supports these beliefs, Lord, Ross, and Lepper ~ 1979 ! , and readily dismiss information that does not ~ 5 A modest 51 percent of a group of older Swedish students—average age 33—placed them- selves in the top 30 percent of their group.

8 The Journal of Finance 1894 1980 ! , Fiske and Taylor ~ 1991 !! . They are more confident Nisbett and Ross ~ Clark and Rutter ~ !! and weigh in opinions based on vivid information ~ 1985 cases, scenarios, and salient examples more heavily than relevant, abstract, Kahneman and Tversky ! , Bar- statistical, and base-rate information ~ ~ 1973 1980 , Hamill, Wilson, and Nisbett ! ! , Nisbett and Ross ~ ~ ! , 1980 ~ Hillel 1980 ! , Taylor and Thompson ~ 1982 ! , Tversky and Bar-Hillel and Fischhoff ~ 1981 !! Kahneman ~ 1982 . In addition to underweighting base-rate information, peo- Tversky and Kahneman ple underestimate the importance of sample size ~ ~ ~ 1972 !! and of regression to the mean, that ! , Kahneman and Tversky 1971 is, the tendency of extreme outcomes to be followed by outcomes closer to the Kahneman and Tversky ~ 1973 !! . population mean ~ In general then, we might expect people to overreact to less relevant, more e.g., an extreme event, a prominent news attention-grabbing information ~ ! while underreacting to impor- article with strong human interest, a rumor 6 In particular, we might expect people to under- tant abstract information. estimate the importance of single statistics that summarize a large sample . ~ e.g., corporate earnings of relevant data ! C. Information In the following models, traders update their beliefs about the terminal I , on the basis of three sources of information: a value of a risky asset, v private signal, their inferences from market price regarding the signals of others, and common prior beliefs. The overconfidence literature indicates that people believe their knowledge is more precise than it really is, rate their own abilities too highly when compared to others, and are excessively optimistic. To be consistent with these patterns, traders in the model must that are too precise, value I hold posterior beliefs about the distribution of v their own information more than others’ information, and expect higher util- ity than is warranted. In the models, traders overweight their private sig- nals and, therefore, their posteriors are too precise, their own information is valued more than that of others, and they overestimate their expected utility. For most of the propositions in this paper to be true, it is sufficient that 1 ! traders ~ 2 ! overweight their ~ hold posterior beliefs that are too precise and 7 Both conditions are satisfied if own information relative to that of others. each trader overweights his own signal. These conditions may be further amplified when traders underweight their common prior beliefs or under- 6 Reacting to how extreme information is rather than how reliable its source is can have dramatic consequences. On April 11, 1997, The Financial Times of London reported fraud in connection with an offshore fund called the Czech Value Fund, referring to the fund by the abbreviation CVF. Four days later Castle Convertible fund, a small closed-end fund with a diversified portfolio of convertible stocks and bonds trading on the AMEX under the ticker symbol CVF, plummeted 32 percent in twenty-two minutes. Trading was halted. After the Cas- tle managers assured the exchange that they had no news, trading resumed at close to its preplunge price. Apparently some investors reacted to word of extreme problems rather than to ~ New York Times , April 20, 1997, p. C1, byline Floyd Norris ! . the reliability of that word 7 Propositions 2 and 9 require additionally that traders value new information relative to prior information.

9 1895 When All Traders Are Above Average weight the signals of others when updating beliefs. Common priors incorpo- rate previous information about security returns’ behavior and thus constitute base-rate data that are likely to be underweighted when updating. The sig- nals of others constitute a large sample. Because large sample inferences are usually undervalued, it is likely that if traders err in valuing the signals of others, they will undervalue these. Most of the propositions are proven only for the situation where each trader overweights his own signals. Often, cor- ollaries can be proven for when traders underweight their priors or under- weight the signals of others. In the interest of parsimony, these corollaries are not stated formally or proven, though the intuition involved is at times 8 discussed. The calibration literature discussed above tells us that people overesti- mate the precision of their information. Overconfidence in one’s information is not the only type of overconfidence we might expect to find in the market. Traders could, instead, be overconfident about the way they interpret infor- mation rather than about the information itself. For example, traders of a stock might look at signals such as trading momentum, price 0 earnings ratio, or forecasts of industry trends. These are examples of public information that is available to any trader but is valued differently by different traders. Thus, a Graham-and-Dodd style fundamental investor might be aware of 0 recent changes in a stock’s momentum but consider its price earnings ratio to be a more important signal; a technical trader who follows momentum might believe otherwise. Each is overconfident in his style of analysis and the signal he utilizes. At the same time, each is aware of the beliefs, and 9 Given people’s tendency to reject perhaps even the signals, of the other. !! 1991 ~ , the Fiske and Taylor information that does not fit their beliefs ~ differing opinions of others are likely to be undervalued. In the models, traders who believe that their information is more precise than it actually is anticipate greater future utility than it is reasonable to anticipate. In this way these models capture some of the spirit of excessive 8 When traders in these models underweight new information, the opinions of others, or prior information, the means of their posteriors deviate from the posterior means rational trad- ers would form. As discussed above, these deviations are consistent with how people process different types of information. However, underweighting any of these three sources of informa- tion causes traders to underestimate the precision of their posteriors. Such underconfidence is not consistent with generally observed behavior. Even when they discount valid information, . Weakly held posteriors do 1979 people usually maintain strongly held beliefs ~ e.g., Lord et al. ~ !! not motivate the results in this paper and, when they arise, should not be considered realistic implications of the models. 9 ~ Even sophisticated investors may agree to disagree. The Washington Post January 7, 1992, ! p. C2, byline Allan Sloan reports that, during the same time period, the nation’s most prom- inent long-term investor, Warren Buffett, and its most prominent short sellers, the Feshbach brothers, held, respectively, long and short positions worth hundreds of millions of dollars in Wells Fargo Bank. Buffett controls the investments of Berkshire Hathaway Inc.; the Feshbachs ~ run an investment fund. ! Ostensibly, Buffett and the Feshbachs disagreed about how much the bank would be hurt by its weak loan portfolio. They also differed in their investment horizons. Despite being right about the loans, the Feshbachs lost $50 million when they had to close their positions. As of January 1992, Buffett was about even.

10 The Journal of Finance 1896 optimism which psychologists have documented. However, optimism is not limited to an inf lated opinion of the precision of unbiased signals. A trader ~ signal or theory. misinterpreted might also have false confidence in a biased ! D. Overconfidence in Financial Markets Why might we expect those trading in financial markets to be overconfi- are overconfident. The ex- dent? The foremost reason is that people usually ceptions to overconfidence mentioned in Section II.A generally do not apply to financial markets. Most of those who buy and sell financial assets try to choose assets that will have higher returns than similar assets. This is a difficult task and it is precisely in such difficult tasks that people exhibit the greatest overconfidence. Not only novices exhibit overconfidence. Griffin and Tversky ~ 1992 ! write that when predictability is very low, as in the stock market, experts may even be more prone to overconfidence than novices, because experts have theories and models which ~ e.g., of market behavior ! 10 they tend to overweight. Securities markets are difficult and slow places in which to calibrate one’s confidence. Learning is fastest when feedback is quick and clear, but in se- curities markets the feedback is often slow and noisy. There may even be a trade-off between speed and clarity of feedback whereby short-term traders get quicker, but noisier, feedback, and long-term traders receive clearer feed- back but must wait for it. The problem of noisy feedback can be exacerbated ~ 1985 ! by the endogeneity of the evaluation period. Shefrin and Statman 1998b ! confirms that investors prefer to sell winners propose and Odean ~ and hold losers. If investors judge their original purchase decisions on the basis of the returns realized, rather than those accrued, then, by holding losers, they will judge themselves to have made fewer poor decisions. Fur- thermore, the feedback from losses will be delayed more than that from gains, further facilitating positive self-evaluations. Selection bias may cause those participating actively in financial markets to be more overconfident than the general population. People vary in ability and those who believe they have more ability to trade may be more likely to seek jobs as traders or to trade actively on their own account. If people are uncertain judges of their own ability, then we might expect financial mar- kets to be populated by those with the most ability and by those who most overestimate their ability. Survivorship bias can also lead to overconfidence by market participants. Unsuccessful traders may lose their jobs or choose to drop out of the market; unsuccessful traders who survive will, on average, control less wealth than successful traders. If traders overestimate the degree to which they were Miller and Ross ~ responsible for their own successes—as people do in general ! , Langer and Roth ~ ~ ! ; Nisbett and Ross ~ 1980 !! —successful trad- 1975 1975 ers may grow overconfident and more wealth will be controlled by overcon- 10 This observation may not apply to experts who adhere to computer-based quantitative ~ see Dawes, Faust, and Meehl 1989 !! . models ~

11 1897 When All Traders Are Above Average 1997 ! this self-enhancing bias causes fident traders. In Gervais and Odean ~ wealthy traders, who are in no danger of being driven from the marketplace, to be overconfident. It is not that overconfidence makes them wealthy, but the process of becoming wealthy contributes to their overconfidence. An old Wall Street adage, “Don’t confuse brains with a bull market,” warns traders of the danger of becoming overconfident during a market rally; no doubt this warning is given for good reason. 11 It This paper finds overconfident traders have lower expected utility. does not necessarily follow that the overconfident traders lose their wealth and leave the marketplace. An overconfident trader makes biased judgments that may lead to lower returns. However, an overconfident risk-averse trader also chooses a riskier portfolio than he would otherwise hold and may be rewarded for risk-bearing with greater expected returns. It is possible that the profits of greater risk tolerance will more than compensate for the losses of biased judgments. Thus, as a group, overconfident traders could have higher expected returns, though lower expected utility, than properly cali- . brated traders, as is the case in De Long et al. ~ 1990 ! III. The Interaction between Overconfidence and Market Structure A. Price Takers Throughout this paper, expectations taken using the distributions that ! . ~ e.g., var traders believe to be correct are indicated by a subscript “b” b Expectations taken using the distributions that are actually correct are in- . In equilibrium, overconfident traders ! e.g., var ~ dicated by a subscript “a” a believe that they are acting optimally, and so they do not depart from the equilibrium. The traders could, in actuality, improve their expected utilities by acting differently, so the equilibria achieved here are not rational expec- tations equilibria. The model of price-taking traders is based on Diamond and Verrecchia 1981 ! ~ . A riskless asset and one risky asset are exchanged in three ~ and Hellwig ! 1980 1, t 5 2, and t 5 3. Consumption takes place only rounds of trading at times t 5 4, at which time the riskless asset pays 1 unit per share and each share 5 t at 2 1 ! . The riskless interest rate is as- ; N of the risky asset pays S v , h I v , where I v ~ v investors ~ i 5 1,..., N ! . As a modeling convenience sumed to be 0. There are N r ` we analyze the limit economy where N . Thus each investor correctly as- sumes that his own demand does not affect prices. At t 5 0 each trader has an of the risky asset. In trading round of the riskless asset and x f endowment of 0 i i 0 and x . t , trader i ’s demands for the riskless asset and the risky asset are f ti ti is the per capita supply of the risky asset; it is fixed, known to all, and un- S x 1981 and Hellwig ~ 1980 ! ~ changing. This differs from Diamond and Verrecchia ! where a stochastic supply of the risky asset provides an exogenous source of is the price of the risky asset in trading rounds 1, 2, and 3. Trader i ’s P noise. t 11 When objectively measured, expected utility is lower for overconfident traders. However, overconfident traders believe that they are maximizing expected utility.

12 The Journal of Finance 1898 wealth is 5 f W 1 P . There is no x x ,for t 5 1, 2, and 3, and W v I 5 f 1 i ti 4 ti 3 i t 3 ti i 1. Prior to trading at t 2 and, t signal prior to the first round of trading at 5 5 5 i private signals, I y receives one of M again, prior to trading at t 5 3, trader ti 1 2 I v 1 e I , where ~ 0, h e I ; ! and N e are mutually inde- ,..., I e e I , I e I ,..., e 31 3 M 21 tm M tm 2 N ~ pendent. Each signal is received by the same number of traders. is as- M N P M . ! Y sumed to be a multiple of 5 0 N 5 y y 0 M is the average ( ( 1 ti tm m 5 1 5 i t . signal at time t signals in any time period is mo- The assumption that there are , N M tivated by the observation that when the number of traders is large there are likely to be fewer pieces of information about an asset than there are traders. Each trader knows that 0 M 2 1 other traders are receiving the same N two signals as she is. She believes the precision of these two signals to be , 2 $ 1. She believes the precision of the other 2 M k 2 signals to be k h e g h # 1. All traders believe that the precision of I v is h , g , h # 1; that is, h v e traders underestimate, or correctly estimate, the precision of their prior T T 5 $% , F P . 5 @ y y y P @ # P , and F # 5 F information. Let i 2 i 2 3 i i 2 2 3 i 1 3 i 2 represents the information available to trader ~ in addition to i F Thus, ti t . Note that a trader’s posterior is more precise than ! prior beliefs at time 1 ~k 1 2 that of a rational trader if, after receiving both of her signals, h h v !g! h ~ M 2 1 Mh 2 . 1 h $ v e e ! , thus traders have constant ab- 2 2 aW Trader ’s utility function is ~ i exp it CARA with a risk-aversion coefficient of ~ . Traders are a solute risk aversion ! assumed to be myopic, that is, they look only one period ahead when solving t 1, 2, 3, trader i solves their trading problem. Thus, at times 5 max E @ 2 exp ~ 2 a ~ W 1 ~ . f ! 6 F 1 # subject to P x x ! 1 f P # ti ti t t 2 1 i ti i t 2 1 i 1 1 t t x ti The traders in this model correctly conjecture that they do not affect prices, thus the only effect of assuming myopia is to eliminate hedging demands 1989 . As others, including Singleton ~ 1987 ! and ~ !! see Brown and Jennings ~ ! , have found, this simplifies the analysis. ~ Brown and Jennings 1989 When solving their maximization problems, traders conjecture that prices are linear functions of the average signals: P P 2 1 a ~ ! 5 a a 1 Y Y P 3 32 31 3 2 33 P P ~ 1 a ! 5 3 a . Y 2 2 21 22 The conjectures are identical for all traders and the coefficients determine an equilibrium in which the conjectures are fulfilled. Equilibrium is ob- tained because traders believe that they are behaving optimally even though, in fact, they are not. This equilibrium and the proofs for this section are presented in Appendix A.

13 1899 When All Traders Are Above Average There is no exogenous noise in this model. The purpose of noise is often to keep traders from using price and aggregate demand to make perfect infer- ences about the information of others. If rational traders with common pri- ors infer the same aggregate signal, they form identical posterior beliefs, and, if their endowments and preferences are also identical, they will not trade. If preferences and endowments differ, trading may occur but it might not occur in response to information, and this runs contrary to what we 12 The absence of exogenous noise in this model demon- observe in markets. strates that, with overconfidence, orderly trading can take place in response ~ ! has a similar 1989 to information even when no noise is present. ~ Varian ! Each trader can result when traders disagree about the mean of the prior. infer the aggregate signal, but each values his portion of the aggregate dif- ferently, arrives at a different posterior belief, and is willing to trade. In this model, traders can perfectly infer the aggregate signal from price. In practice, traders do not usually make this perfect inference. The certainty in the model would be dispelled if randomly trading noise traders were added to the economy. However, this certainty results not so much from the lack of noise trading as from the conventional assumption that traders are able to know the preferences of all other traders, to know the distributions of all random variables in the economy ~ though here these are distorted by over- confidence , and to make perfect inferences from their information. In ad- ! dition to knowing each other’s preferences, when traders are not risk neutral and do not have constant absolute risk aversion, they must also know each ! points out, 1986 other’s wealth to infer the signals behind trades. As Arrow ~ the information gathering and computational demands put on traders in models such as this would, in a more realistic setting, “imply an ability at information processing and calculation that is far beyond the feasible and cannot be well justified as the result of learning and adaptation.” It may be noise ~ ! that the principal source of noise in markets is not that a few traders do not attempt to optimize their utility, but that most traders are not certain how best to do so. In its lack of exogenous noise, this model is similar to that of Grossman ~ S y 1976 ! . But in Grossman’s model, a trader can infer the aggregate signal when deter- y from price and, having done so, can ignore his private signal i mining his demand. As Beja ! observes, this creates a paradox in which ~ 1976 fully informative prices arise from an aggregate demand function that is without information because if prices are fully informative, traders have no incentive to consider their private signals when formulating their demand. When traders are overconfident, they can still infer the average signal from price, but they do not ignore their own signal when determining their de- mand. Each trader considers his signal to be superior to those of others, and because the average signal weights all traders’ signals equally, it is not a sufficient statistic to determine an individual trader’s demand. 12 1989 ! for a discussion of no-trade theorems. See Varian ~

14 The Journal of Finance 1900 1978 ! model where price-taking This model is also related to Figlewski’s ~ traders with different posterior beliefs interact. Figlewski’s model does not have an exogenous noise source. To avoid the no-trade dilemma, he assumes traders are unable to infer the information of others from price. Were these traders overconfident, this assumption could be eased and results similar to ! 1976 ~ , risk- the ones presented here would follow. In Jaffe and Winkler neutral informed traders decide to trade after observing a risk-neutral mar- ketmaker’s bid and ask. The marketmaker can expect to lose to all rational investors, and so this market is unstable. Jaffe and Winkler suggest that the introduction of liquidity traders or traders who misperceive their ability— such as the overconfident traders modeled here—could stabilize this market. As discussed above, overconfidence causes traders to have differing pos- terior beliefs. The more overconfident traders are, the more differing these beliefs. This leads to the first proposition. 1: ROPOSITION When traders are price takers, expected volume increases as P . $ 2 overconfidence increases (if M ! In all of the propositions, expectations are taken over the true probability distributions. Here we see that as overconfidence increases, traders increas- ingly weight their own signals more heavily than they weight those of others when calculating their posterior beliefs. Their posterior beliefs are therefore 13 There is one exception to more dispersed and more trading takes place. 5 ~ effectively public ! signal received by this pattern. If M 1 there is only one all traders. And because all traders overvalue that signal equally, their be- liefs remain homogeneous and no trade takes place though price may change. If traders varied in their overconfidence in the public signal, they would ~ ! Expected volume also increases when traders underweight common trade. priors or the signals of others. ROPOSITION When traders are price takers, volatility of prices increases as 2: P overconfidence increases. When traders are overconfident, each overvalues his own personal signal. This results in the aggregate signal being overvalued relative to the common prior , a , ~ and ~ 3 !! where the coefficients a in the pricing functions equations 2 ~ ! 22 32 are increasing in k . Overweighting the error in the aggregate signal a and 33 h increases the volatility of prices. Decreasing has the same effect and de- creasing g lowers the weight on the aggregate signal and lowers volatility. An- other consequence of biased expectations is that they increase the variance of P 2 I v ! . Using this vari- the difference between price and underlying value, var ~ ance as a measure of the quality of prices we have the following proposition. ROPOSITION When traders are price takers, overconfidence worsens the qual- 3: P ity of prices. 13 1994 ! , volume is determined ~ In a dynamic setting, such as that of Shefrin and Statman see Karpoff not simply by differences in beliefs but by the rate of change of those differences ~ ~ 1986 !! .

15 1901 When All Traders Are Above Average We will see in the next model that when a strategic insider is overconfident, overconfidence can improve the quality of prices. These two models differ in that the next model has noise traders, but, more important, they differ in how information is distributed. Here all traders receive a signal, in the next model information is concentrated in the hands of a single insider. Even if noise is added to the current model, overconfidence will continue to distort M 5 prices, not improve them. This is most easily seen when 1, that is, when there is one public signal. If the signal is public, noisy demand will obviously not affect traders’ information, but overconfidence will continue to distort traders’ posterior expectations and, thereby, prices. Price quality also wors- ens when traders underweight common priors or the signals of others. Distorted expectations reduce expected utilities. When traders are overcon- fident, their expected utility is lower than when their probabilities are prop- erly calibrated. This is hardly surprising because traders choose their actions in order to optimize expected utility, and when they are overconfident, this at- Similarly, expected utility also tempt to optimize is based on incorrect beliefs. ~ g decrease. ! And so we have Proposition 4. h declines as and 4: When price-taking traders are overconfident, their expected ROPOSITION P utility is lower than if their beliefs are properly calibrated. There are no noise traders to exploit in this model, so the aggregate ex- pected returns from trading must be zero. Overconfidence decreases ex- pected utilities because it results in nonoptimal risk sharing. Overconfident traders hold underdiversified portfolios. Those who receive the highest sig- nals hold too much of the risky asset and too little of the risk-free asset; others hold too little of the risky asset and too much of the risk-free asset ! . Of course ~ given their preferences and the true distributions of signals 14 each trader believes that she is optimally positioned. To model overconfidence, I assume that traders overestimate the precision of their private signals. Doing so leads traders to hold differing beliefs and to overestimate the precision of their own posterior beliefs. Diverse posterior beliefs that are held too strongly are sufficient to promote excessive trading, increase volatility, distort prices, and reduce expected utility. For time-series results though, how posteriors are constructed matters. Assuming that peo- ple always overweight private signals implies that they always overweight new information. But, as discussed in Section II.B, people do not always overweight new information. They usually overrespond to salient informa- tion and underrespond to abstract information. They underweight valuable information and overweight irrelevant information. To examine time-series implications of the model, we therefore look at both over- and underweighted signals. For price-taking traders, overvaluing new information leads to price reversals, undervaluing it leads to price trends. 14 M 1, expected utilities will not be affected ~ although 5 If traders are overconfident and . Beliefs will be homogeneous, albeit mistaken, and traders will hold the ! prices will change same optimal portfolios they would hold if they valued their information correctly.

16 The Journal of Finance 1902 5: When price-taking traders overvalue (undervalue) new infor- P ROPOSITION mation, price changes exhibit negative (positive) serial correlation. Any prediction based on this proposition requires an analysis of the type of information traders are receiving. Note that the serial correlation of returns and of price changes will have the same sign. Up to this point, all of the traders in this model are overconfident. What would happen if some traders were rational? In general, rational traders would mitigate but not eliminate the effects of the overconfident traders ~ just as rational traders do not eliminate the effects of trader errors in De or Shefrin and Statman ~ Long et al. !! . In markets such as the ! 1990 ~ 1994 ! points one modeled here, traders vote with their dollars. As Figlewski ~ 1978 out, “a trader with superior information but little wealth may have his in- formation undervalued in the market price.” Due to the assumption of con- stant absolute risk aversion, wealthy traders in the model trade no more than poor ones and so the impact on price of traders with particular view- points depends here on their numbers, not wealth. The mere presence of rational traders does not drive price to its rational value. To change price, traders must be willing to trade. Willingness to trade generally depends on strength of beliefs, risk tolerance, and wealth. Though possibly endowed with ~ superior information, rational traders may trust their beliefs no more and than overconfident traders. Their wealth and risk tolerance possibly less ! may not exceed those of others. Introducing rational traders into the model per trader ! , volatility, and the inefficiency of prices. reduces trading volume ~ The expected utility of rational traders is greater than that of overconfident traders. Introducing additional overconfident traders who are less overcon- fident than the existing ones has similar, though less extreme, results. In the preceding proposition, whether price changes are negatively or pos- itively correlated depends on whether traders overvalue or undervalue new information. In the following proposition, rational traders are added to the ! . When rational traders trade with economy as described in Appendix A ~ # ! , the in- ~g overconfident traders who undervalue the signals of others 1 formation of rational traders will be underrepresented in price. Thus prices may trend. 6: When rational traders trade with overconfident traders who ROPOSITION P (sufficiently) undervalue the signals of others, price changes will be positively serially correlated. Positive serial correlated price changes are most likely when the precision of the rational traders’ signal is high and when overconfident traders signifi- cantly undervalue the signals of others. The specific region where price changes are positively serially correlated is identified in the proof of Prop- Appendix A ! . osition 6 ~ B. An Insider ~ 1985 ! . Other than nota- This model of insider trading is based on Kyle tional differences, the only changes made to Kyle’s original model are that

17 1903 When All Traders Are Above Average the insider’s private signal of the terminal value is noisy and that the in- sider is overconfident. This is a one-period model in which a risk-neutral, privately informed ! and irrational noise traders submit market orders to a ~ trader the insider risk-neutral marketmaker. There are two assets in the economy, a riskless asset and one risky asset. The riskless interest rate is assumed to be 0. The 2 1 S v , h I N ~ v terminal value of the risky asset is ; v S ! . is assumed to equal 0; v this simplifies notation without affecting the propositions. Prior to trading, I v 1 I e . I e is normally dis- a risk-neutral insider receives a private signal I y 5 . The insider believes the precision tributed with mean zero and precision h e of h e to be k I , where 1, and the precision of $ to be h h , where I h # 1. k v v e , the insider demands After observing submits a market order for ! x units I y ~ z of the risky asset. Without regard for price or value, noise traders demand I 2 1 ! . The marketmaker observes ; z 0, h I N units of the risky asset, where ~ z ! and sets the price ~ P z . The marketmaker cor- x only the total demand I 1 is rectly assumes that the precision of I e h I is h and that the precision of . v e v The propositions do not change if the marketmaker, like the insider, be- ~ After trading, the risky asset pays . ! lieves the precision of the prior to be h h v its terminal value I v . The insider and the marketmaker know the true dis- tribution of and are aware of each other’s beliefs about the precisions of I v I z I . and y The insider conjectures that the marketmaker’s price-setting function is a x z , 1 linear function of I . 1 x 1 I z ! ~ ~ 4 ! H P L 5 x ~ I v 2 P ! , conditional on his He chooses x to maximize his expected profit, , and given his beliefs about the distributions of I v , I y , and I z and the signal, I y ! , that the mar- 1985 conjectured price function. It is assumed, as in Kyle ~ ketmaker earns zero expected profits. The marketmaker conjectures that the insider’s demand function is a linear function of I y , A 1 x I y . ~ 5 ! 5 B I conditional on total demand ~ x 1 She sets price to be the expected value of v z ! v , I y , and I I and the conjectured z I , given her beliefs about the distributions of demand function. In Kyle’s original model, a linear equilibrium always exists in which the conjectured price and demand functions are fulfilled. Given the assump- tions of overconfidence made here, a linear equilibrium exists whenever h 1 2 h The equilibrium and the proofs for this section are ~ . k h . h k v e v The intuition behind the equilibrium condition is ! presented in Appendix B. v condi- the following. The marketmaker sets price to be the expectation of I tional on the order f low she observes and on her conjecture about the insid- er’s demand function. The insider is trying to maximize his profit. His profit increases if he trades more with the same profit margin or if he trades the same amount with a larger margin. If the insider increases his demand, the

18 The Journal of Finance 1904 marketmaker shifts the price and thus lowers the insider’s expected profit margin. Equilibrium exists at the demand-price pair where the insider be- lieves that, if he increases his demand, the negative effect of the lower ex- pected profit margin will just offset the gains of greater trading and, if he lowers demand, the losses from trading less will just offset gains from a higher expected profit margin. If the insider and the marketmaker disagree too much about the relative precisions of the prior and the private signal, , the there is no equilibrium; for any given insider demand function ~ ! A,B such that the insider will ! H,L ~ marketmaker will choose a pricing function ~ i.e., greater B ! . prefer a yet steeper demand function ! model, the insider can only inf luence price through his As in Kyle’s ~ 1985 demand. This assumption is particularly critical when overconfidence is in- troduced to the model. If the insider could credibly reveal his private signal to the marketmaker, then, due to the different weights each attaches to the prior and to the signal, the insider and the marketmaker would have dif- ferent posterior beliefs about the expected value of the terminal payoff. And because they are both risk neutral, they would each be willing to trade an infinite amount. Infinite trading is a possible problem whenever risk- neutral traders value common information differently. In Harris and Raviv 1993 ! , risk-neutral traders attribute different density functions to a public ~ ~ signal Harris and Raviv avoid infinite trading by assuming a fixed number . Jaffe and Wink- ! of shares are available and that short sales are not allowed 1976 ! avoid infinite trading by assuming only one asset share can be ler ~ exchanged. The willingness to trade infinitely is inherent in risk neutrality, not in overconfidence. Risk neutrality is assumed here for tractability. is decreasing instead h All of the propositions in this section are true when of increasing. k Expected volume increases as the insider’s overconfidence 7: ROPOSITION P increases. Expected volume is measured as the expected value of the sum of the abso- lute values of insider demand and noise trader demand. When the insider is , I overconfident, he believes that he has received a stronger private signal, y than is actually the case. In calculating his posterior expectation of the final value of the risky asset, he overweights his signal and derives a posterior expectation farther from the prior than he should. Based on this posterior belief, he trades more aggressively than is optimal, thus increasing expected volume. P 8: ROPOSITION Market depth increases as the insider’s overconfidence increases. P Volatility of prices increases as the insider’s overconfidence ROPOSITION 9: increases. P ROPOSITION 10: The quality of prices improves as the insider’s overconfidence increases.

19 1905 When All Traders Are Above Average I v ! Overconfidence causes prices to be more sensitive to changes in value ~ I y , and less sensitive to changes in informed and in the insider’s signal ~ ! x ! ~ I z ! . The marketmaker sets price to be ~ demand I and noise trader demand conditional on observed orderf low and her conjecture the expectation of v I about the insider’s demand function. She realizes that the insider is over- confident and that he will trade more in response to any given signal than he would if he were rational. She therefore moves price less in response to 1 changes in total order f low z ! than she would if the insider were ratio- ~ I x I nal. That is, she f lattens her supply curve, thereby increasing market depth which is measured as the inverse of the derivative of price with respect to ~ . Because the overconfident insider trades more in response to order f low ! any given signal than he would if he were rational, his expected trading increases relative to that of noise traders. Therefore the signal-to-noise ratio in total order f low increases and the marketmaker is able to make better inferences about the insider’s signal. This enables her to form a more accu- rate posterior expectation of v and to set a price that is, on average, closer to I I v . This improves the quality of prices, which is measured, as in the previous ! and value model, as the variance of the difference of price I v ! . Because ~ P ~ y , the price she sets I the marketmaker can better infer the insider’s signal, y than if the insider were rational. varies more in response to changes in I volatility ! . From a different perspective, This increases the variance of price ~ although the marketmaker has f lattened her supply curve, thus dampening given level of expected order f low, the increased order f low volatility for any generated by the overconfident insider more than offsets this dampening, and results in increased volatility. Thus both market depth and volatility rise with overconfidence. 11: The expected profits of the insider decrease as his overconfi- ROPOSITION P dence increases. The insider’s expected profits, E ~ I ~ 2 P !! , are equivalent to his expected x v a ~ to buy or utility because he is risk neutral. The insider submits a demand that is optimal given his beliefs about the distributions of I v and I y to sell ! ,a demand that he believes will maximize his expected profits. He is mistaken about the precision of his knowledge, but conditional on his beliefs he be- haves optimally. The demand he submits is not, however, the same demand he would submit were he not overconfident, and it is not optimal given the and I y . Therefore the insider’s expected profits are v I true distributions of 15 lower than they would be if he were not overconfident. 15 1997 ! Kyle and Wang ~ show that under particular circumstances when both a rational insider and an overconfident insider trade with a marketmaker, the overconfident insider may earn greater profits than the rational insider. The overconfident insider earns greater profits by “precommitting” to trading more than his share in a Cournot equilibrium. For this result to hold, traders must trade on correlated information, have sufficient resources and risk tolerance to trade up to the Cournot equilibrium, know each other’s overconfidence, and trade with a third party ~ ! . Furthermore, if one trader can trade before the other, the e.g., the marketmaker result may not hold.

20 The Journal of Finance 1906 This model includes an overconfident insider, noise traders, and a rational marketmaker who expects to earn zero profits. Whatever profits the over- confident insider gives up are passed on to the noise traders in the form of lower losses. Were the rational marketmaker not constrained by assumption to earn zero profits, she would benefit from the insider’s overconfidence. and the next one ~ ! This model require a source of uncertain demand for the risky asset so that the insider’s information is not perfectly deducible from total demand. Noise traders who trade randomly and without regard to price ~ 1985 ~ , though they may lack perfect real world analogues, pro- as in Kyle !! vide an analytically tractable source of uncertain demand. Overconfident, risk-averse, price-taking traders with private signals, such as the traders described in the previous section, could also provide uncertain demand in a market. In that case, if the insider were not too overconfident, he would profit at the expense of the overconfident price takers. Unfortunately, re- placing noise traders with overconfident price takers greatly complicates the model. C. Marketmakers and Costly Information The next model examines the behavior of overconfident marketmakers. It also offers an explanation for why active money managers underperform pas- sive money managers: Active managers may be overconfident in their ability to beat the market and spend too much time and money trying to do so. . Risk-averse traders The model is based on Grossman and Stiglitz ~ 1980 ! decide whether or not to pay for costly information about the terminal value of the risky asset; those who buy information receive a common signal; and a single round of trading takes place. The participants in this trading are ! , traders who do not buy ~ the traders who buy information informed traders uninformed traders , and noise traders who buy or sell with- the information ~ ! 16 As in the previous models, a riskless asset and out regard to price or value. one risky asset are traded; the riskless interest rate is assumed to be 0; each 2 1 ! . Traders believe the N ~ S share of the risky asset pays , h I v , where I v ; v v # h , where 1; that is, they undervalue the common h h to be v I precision of v N i 5 1,..., N ! . As a modeling convenience we ~ investors prior. There are ` . Thus each investor correctly as- r N analyze the limit economy where sumes that his own demand does not affect prices. Each trader has an en- N 0 of the riskless asset and x ! x of the risky asset. S x 5 ~ dowment of f ( i N 0 0 i 0 i is the average endowment. As a notational convenience it is assumed that 5 0 and S v 5 0. Prior to trading, traders choose whether or not to pay cost S x 2 1 h I v in order to receive a signal I e , where I e; N ~ 0, y c I 5 1 ! . Noise trader e 16 In this section “traders” refers to informed traders and to uninformed traders but not to noise traders who are referred to explicitly as “noise traders.” As in the insider model, noise traders could be replaced with overconfident price takers, such as those discussed in Sec- tion III.A. Overconfidence would motivate trading and the model’s results would not change significantly. However replacing noise traders with overconfident price takers greatly compli- cates the equilibrium without adding much intuition.

21 1907 When All Traders Are Above Average 2 1 trader is I z , where I z ; N ~ 0, h demand per nonnoise ! ! . Thus 2 I z is the ~ z * supply of the risky asset per trader at the time of trading. In equilibrium, l is the fraction of traders who choose to become informed. All traders, even those who remain uninformed, are overconfident about , where k $ 1. In the k the signal, which they believe to have precision h e previous models, traders were overconfident about their own signals but not those of others. Here everyone believes the information is better than it is, but some decide the cost is still too high. It is as if all money managers overestimate their ability to manage money actively, but some decide the costs of doing so are too high and so, despite their overconfidence, choose to 17 manage passively. In real markets one would expect traders to hold a spec- trum of beliefs about the value of costly information. Those who were more overconfident about the information would be more likely to buy it. One could alternatively specify in this model that those traders who do not buy 18 the signal value it rationally. i ’s demand for the risky asset is x and for the risk-free asset is Trader 1 i . So his final wealth is W i x 5 ’s utility function is . Trader f I v 1 f i i 1 1 i i 1 1 W U ~ ! exp ~ 2 aW ! 52 is the common coefficient of absolute risk , where a i i 1 1 aversion. He maximizes his expected utility by choosing whether or not to become informed, and then, conditional on his information, by choosing his optimal demand subject to the budget constraint. That is, if he is informed, he solves 6 ~ E , @ 2 exp ~ 2 aW f 1 ! 6 I ! # subject to x P x P 1 f # y max I 0 I 1 I 1 0 I b I 1 x I 1 and if he is uninformed he solves max 7 E ~ @ 2 exp ~ 2 aW , f ! 6 P # subject to x 1 P P 1 f ! x # U 1 U U 1 U 0 U 1 b 0 x 1 U 17 In practice, some practitioners of passive investing tout their own skills as superior active managers. For example, Barclays Global Investment Advisors, the largest manager of index funds, has a Global Advanced Active Group that actively manages more than $70 billion. And George Sauter who oversees $61 billion in stock-index mutual funds at Vanguard Group also The Wall Street Jour- actively manages Vanguard Horizon Fund Aggressive Growth Portfolio ~ ! , February 25, 1997, p. C1, byline Robert McGough nal . 18 Assuming that traders who do not purchase the signal value it correctly will result in a range of possible equilibria rather than a single equilibrium point. At one end of the range the same fraction of traders becomes informed as when all traders are rational. Here the rational uninformed traders believe that the fraction of traders who are informed is optimal, and the overconfident informed traders believe that the fraction of traders who are informed is too small. Traders in neither group believe they would benefit from changing groups. At the other end of the range, the same fraction of traders becomes informed as when all traders are over- confident. Here the rational uninformed traders believe that the fraction of traders who are informed is greater than the optimum, and the overconfident informed traders believe that the fraction of traders who are informed is optimal.

22 The Journal of Finance 1908 5 I and i 5 U indicate prototypical informed and uninformed traders where i is the endogenously determined price of the risky asset. In equilib- and P rium all traders believe that the expected utility of the informed traders is equal to that of the uninformed. Because all traders believe the precision of and the precision of I v is h h , and because the equilibrium is deter- is y k h I v e mined by the traders’ beliefs, the equilibrium obtained is the same as would e was actu- occur in a model without overconfidence where the precision of ally k h and that of was h h I . Once again equilibrium holds because the v e v traders believe that they are behaving optimally, though, in fact, they are not. The equilibrium and the proofs for this section are presented in Appen- dix C. In the previous two models, expected utility drops as overconfidence in- creases. In this model, where traders are overconfident about a costly signal, it is those who buy the signal who are most hurt by their overconfidence. For many sets of the parameters specifying this economy (and ROPOSITION 12: P perhaps for all sets), when traders overvalue costly information, the expected utility of informed traders is lower than that of uninformed traders. When traders overestimate the value of the costly signal, too many of them are willing to buy it. Its benefits are therefore spread too thin, resulting in lower expected utilities for the informed traders. The proposition states only that this is true for many sets of the parameters that specify the economy. Explicit so- lutions for the expected utilities of the informed and uninformed traders are given in Appendix C. I evaluate these for a wide variety of parameter values and find that in every case the expected utility of the informed is less than that 19 1or h , 1 ~ and 0 , l , 1 ! . of the uninformed if k . i.e., 0 l , 1 ! , ~ When some traders buy information and others do not , individual informed traders trade, on average, more than individual un- This is the case even when there is no overconfidence. informed traders. ~ Uninformed traders as a group, however, may trade more than informed ! traders as a group when their numbers are sufficiently larger. When trad- ers are overconfident, the expected utility of informed traders is lower than that of uninformed traders, therefore it follows that the expected utility of those who, on average, trade more is lower than that of those who, on av- 1998 ! finding erage, trade less. This is consistent with Barber and Odean’s ~ that individual investors who turn over their common stocks at higher rates earn, on average, lower net returns. When some traders buy information and others do not, this model does not offer much intuition about how overconfidence affects total trading volume and volatility. Volume and volatility can increase, decrease, or remain un- changed as overconfidence increases. Even when there is no overconfidence, volume and volatility rise or fall in response to increases in other parameter 19 If the uninformed traders are rational rather than overconfident, they optimize correctly. In this case it is trivial to show that their expected utility is at least as high as that of informed traders. If it were not, they would become informed.

23 1909 When All Traders Are Above Average y . Volume values such as the coefficient of risk aversion and the precision of I and volatility vary in response to changes in the fraction of investors be- l , which is itself extremely sensitive to changes in the coming informed, various parameter values. These patterns appear to be idiosyncracies of the 20 model rather than generalizations about markets. When all traders are informed, they act as marketmakers who have some 21 information about the terminal value of the risky asset. The supply sched- ~ ~ I v6 I y ! 1 a var . There are two separate compo- Q I v6 I y ! 5 ule they set is: E P b b nents to this price: a var v6 I y ! Q is a response to noise trader demand ~ since I ~ b z ! and hedges traders in their capacity as marketmakers against in- 5 Q I y I y ! is a response to the signal I I and represents traders’ ~ v6 ventory risk. E b speculations about terminal value. If there were no signal, price would be completely determined by inventory risk; if there were a signal, but no noise trader demand, price would be completely determined by the signal. A de- h means that these marketmakers perceive themselves as having crease in less reliable prior information and therefore facing greater risk. Thus, lower h steepens the supply curve, decreasing market depth and increasing the inventory-risk component of volatility. On the other hand, when increases k these marketmakers see themselves as having more reliable information and their perception of the risk of holding inventory is diminished. An increase therefore f lattens the supply curve, increasing market depth and de- k in creasing the inventory-risk component of volatility. Overconfidence in- creases market depth because it lowers a marketmaker’s perceived risk. closer to ~ I v6 I y ! 0, thereby in- I y and farther from S v 5 Increasing k moves E b creasing the speculative component of volatility while decreasing its inven- tory risk component. When expected noise trader demand is low, the speculative component of volatility dominates and increasing increases vol- k atility. When expected noise trader demand is high, inventory-risk domi- decreases volatility. k nates and increasing Figure 1 graphs supply curves in two economies. The dashed line repre- 1 and the solid line an economy where k sents an economy where 2. All k 5 5 other parameter values are the same in both economies and all traders are informed. The supply curves are conditional on traders receiving a signal of 5 2 ~ one standard deviation above the mean signal ! . The solid line is f lat- I y 20 To understand why, in this model, expected volume can rise or fall with overconfidence it is helpful to look at boundary cases. When cost is so high that all traders remain uninformed * i.e., l ~ ! , the traders do not trade with each other and all trading is done between the 5 0 uninformed traders and the noise traders. Thus, expected volume equals the expected demand ~ of the noise traders i.e., 0 p h ! % . When overconfidence increases sufficiently, some traders, but 2 z not all, will become informed. Informed and uninformed traders will now trade with each other and they will also continue to fill the demand of the noise traders, so expected volume will rise. depending on ~ As overconfidence continues to rise, all traders may eventually become informed , in which case expected volume will fall back to the expected de- ! the other parameter values mand of noise traders. 21 Note that when all traders are informed, this model is analogous to a one-period version of the model in Section III.A with noise traders added and M 5 1.

24 The Journal of Finance 1910 Q ! y I v6 I ~ var a Figure 1. Supply curves when all traders are informed. ! y I v6 I ~ E 5 P 1 b b ~ where is price and Q is quantity, for economies in which k 5 2 P solid line ! and k 5 1 ~ dashed y line ! and where, for both economies, signal h 1, 5 I 2 has been received, 5 h 5 2, h 5 1, e v h 5 1. 0.25, a 5 2, c 5 0.09, and l 5 z 5 k ter, which means that market depth is greater when 2. The two supply curves cross at about Q 2. If demand is less than 2 and greater than 5 approximately 1.5, price will be closer to its unconditional expected value, 2 k 5 1 than when k 5 2. But when the magnitude of noise trader 0, when I demand is high ~ i.e., I z . z ,2 1.5 ! price will be closer to its expected 2or 1. When expected noise trader demand is k 5 2 than when k 5 value when low, demand will more often fall into the area where the magnitude ~ and volatility 5 k of price is smaller for ! 1. When expected noise trader demand is high, the economy with k 5 2 will have lower volatility. The following proposition summarizes the above discussion. P ROPOSITION 13: Market depth is increasing in the overconfidence of a risk- averse marketmaker. Volatility increases when expected noise trader demand is high and decreases when it is low. (Precise definitions of high and low expected noise trader demand are given in Appendix C.) IV. Discussion This paper examines the effects of overconfidence in a variety of market settings. These settings differ principally in how information is distributed and how prices are determined. For some market measures, such as trading volume, overconfidence has a similar effect in each setting. For others, such

25 1911 When All Traders Are Above Average as market efficiency, it does not. Which set of predictions is appropriate to a market depends on the informational structure and price setting mechanism of that market. For example, if, for a particular market, crucial information is first obtained by well-capitalized insiders and marketmakers are primar- ily concerned about trading against informed insiders, then the model of the Section III.B ~ ! overconfident insider is appropriate. However, if relevant in- formation is usually publicly disclosed and then interpreted differently by a large number of traders each of whom has little market impact, the over- Section III.A applies. The observation that over- ~ ! confident price-taker model confident traders will pay too much for information ~ Section III.C ! applies to markets in which traders choose between investing passively and expending resources on information and other costs of active trading. We find the fol- lowing effects of overconfidence on different market measures. ~ Sec- Overconfidence increases trading volume . Overconfident price takers tion III.A ! form differing posterior beliefs and trade speculatively with each other. Were these traders rational they would hold identical posteriors and trade only to initially balance their portfolios. Overconfident insiders ~ Sec- tion III.B also trade more aggressively than if they were rational. And, as ! ! , overconfident market- seen in the model of marketmakers ~ Section III.C makers set a f latter supply curve. A f latter supply curve encourages more trading when traders are price sensitive. Thus, in all three settings, over- confidence leads to greater trading volume. Though there is anecdotal evi- dence of excessive trading—for example, roughly one-quarter of the annual international trade and investment f low is traded each day in foreign ex- Dow and Gorton ~ 1997 change markets ; the average annual turnover rate ~ !! ~ on the New York Stock Exchange is currently greater than 60 percent NYSE ! —without an adequate model of what trading volume in Fact Book for 1996 rational markets should be, it is hard to prove that aggregate market vol- 1998a looks at the buying and selling activities of ume is excessive. Odean ! ~ individual investors at a discount brokerage. Such investors could quite rea- sonably believe that their trades have little price impact. On average, the stocks these investors buy subsequently underperform those they sell ~ gross , even when liquidity demands, risk management, and of transactions costs ! tax consequences are considered. As predicted by the model of price-taker overconfidence, these investors trade too much. However, overconfidence about the precision of private signals alone is not enough to explain why these investors make such poor trading decisions. In addition to overvaluing their information, these investors must also misinterpret it. Statman and Thorley ! 1998 ~ find that trading volume increases subsequent to market gains. If success in the market leads traders to become overconfident—as Gervais ~ ! find—these increases in volume may be driven by and Odean 1997 overconfidence. depends Whether overconfidence improves or worsens market efficiency on how information is distributed in the market. On the one hand, when information is distributed in small amounts to many traders or when it is publicly disclosed and then interpreted differently by many traders, over- confidence causes the aggregate signal to be overweighted ~ Section III.A ! .

26 The Journal of Finance 1912 the asset’s true value, I v , than would other- This leads to prices further from wise be the case. Though all available information is revealed in such a market, it is not optimally incorporated into price. On the other hand, when information is held exclusively by an insider and then inferred by a market- ! , overconfidence prompts the insider to maker from order f low Section III.B ~ reveal, through aggressive trading, more of his private information than he otherwise would, thereby enabling the marketmaker to set prices closer to the asset’s true value. If, however, the insider’s information is time sensitive and becomes public soon after he trades, this gain in efficiency is short- lived. Given the broad disclosure of information in U.S. equity markets and the brevity of gains in efficiency from overconfident insiders, we would ex- pect overconfidence, in net, to decrease efficiency in these markets. volatility ; marketmaker’s overconfi- Traders’ overconfidence increases dence may lower it. By overweighting the aggregate signal of the price tak- ers further from its true underlying ~ Section III.A ! , overconfidence drives price , and further from its unconditional mean, S v . This results in in- value, I v Sec- creased volatility. By prompting the insider to reveal more of his signal ~ closer to tion III.B , overconfidence enables the marketmaker to move price ! , and its unconditional value, S v . v the true underlying value, I further from This, too, increases volatility. In the first case, overconfidence increases vol- atility by distorting the prices implied by public, or broadly disseminated, information; in the second, overconfidence increases volatility by moving prices closer to the values implied by highly concentrated, private information. Pri- ~ Section III.C ! f latten their sup- vately informed, risk-averse marketmakers ply curves when they are overconfident, just as they would if they were less risk averse, because overconfidence leads them to perceive less risk in hold- ing inventory. Flattening the supply curve dampens volatility. The inf luence of a group of traders on price will depend on their numbers, wealth, risk tolerance, overconfidence, and information. In a market with many traders and few marketmakers it is unlikely that dampening of volatility by over- confident marketmakers will offset increases in volatility due to overconfi- Shiller dent traders. Some research suggests that market volatility is excessive ~ , LeRoy and Porter 1981 !! , but this is a difficult proposition to 1981, 1989 ~ ~ ! ~ 1986 ! , Kleidon ~ prove !! . Pontiff ~ 1997 ! finds excess ~ Marsh and Merton 1986 volatility for closed-end funds. market depth . When an insider is overconfident, Overconfidence increases for any given signal he trades more aggressively . The marketmaker ad- ~ ! ~ ! by increas- justs for this additional trading in response to the same signal Section III.B . Overconfident risk-averse marketmakers ~ ing market depth ! perceive that their estimate of the security’s true value is more precise than it is and that they face less risk by holding inventory. So they f latten their Section III.B ! . supply curves, which also increases market depth ~ . Overconfident traders do not prop- expected utilities Overconfidence lowers erly optimize their expected utilities, which are therefore lower than if the traders were rational. Overconfident traders hold underdiversified portfo- lios. When information is costly, those who choose to become informed trade more and fare worse than those who remain uninformed Section III.C ! .In ~

27 1913 When All Traders Are Above Average practice, the cost of active managers’ information must be ref lected in their fees. Thus, this finding is consistent with many studies of the relative per- 22 ~ ~ uninformed ! money managers. informed formance of active and passive ! It is also consistent with the lower net returns earned by individual inves- Barber and Odean !! . tors whose portfolio turnover is high ~ ~ 1998 Overconfident traders who discount the opinions of others can cause mar- ! Section III.A kets to underreact to the information of rational traders ~ . Mar- kets also underreact when traders underweight their own new information or under- and overreact when they overweight it. The degree of overreaction depends on what fraction of all traders receives the information and on how ~ 1997 ! find willing these traders are to trade. ~ Bloomfield, Libby, and Nelson that traders in experimental markets undervalue the information of others; 1996 find that the impact of a signal on price, in an ! Bloomfield and Libby ~ experimental market, depends on what fraction of traders receive that sig- Underreactions occur when all traders undervalue a signal or when nal. ! only a small fraction of traders overvalue it, but others discount their opin- ion. Overreactions require that a significant fraction of active traders those ~ most willing to trade ! significantly overvalue a signal. Some documented market return anomalies indicate overreactions to pub- 23 1997 ! points out that if Fama ~ lic events, but most find underreactions. markets occasionally overreact and at other times underreact this could be due to simple chance. Like markets, people, too, sometimes overvalue infor- mation and at other times undervalue it. Though these valuation errors may appear due to chance, psychologists find that they are systematic. People typically overreact to salient, attention-grabbing information, overvalue cases, anecdotes, and extreme realizations, and overweight irrelevant data. They underreact to abstract statistical information, underestimate the impor- tance of sample size, and underweight relevant data. Markets appear to ref lect the same systematic biases as their participants. Reactions to announcements are considered underreactions when returns in periods following the announcement are of the same sign as returns on the day of the announcement. One of the most robust underreaction anom- ~ 1989, 1990 !! . alies is post-earnings announcement drift Bernard and Thomas ~ 22 1968 finds underperformance by mutual funds. Lakonishok ~ ! In an early study, Jensen ! document that as a group active equity managers consistently underperformed et al. ~ 1992 S&P 500 index funds over the period 1983 to 1989. They conclude that, after factoring in management fees, active management subtracts value. Using a variety of benchmarks and ~ 1993, 1994 ! find that, at least before fees, some benchmarkless tests, Grinblatt and Titman 1995 ! claims that such results are heavily fund managers earn abnormal returns. Malkiel ~ 1997 ~ also finds little evidence of skilled mutual fund inf luenced by survivorship bias. Carhart ! management. Lakonishok et al. ask why pension funds continue to give their money to active managers when index funds outperform active management. They suggest a number of reasons based on agency relationships. They also point out that the pension fund employees may be overconfident in their ability to pick superior money managers. 23 I wrote the following discussion of market underreactions nearly two and one-half years after the original draft of this paper ~ ! and subsequent to reading more recent November 1994 Barberis, Shleifer, and Vishny ~ working papers on this topic ! , Daniel et al. ~ 1998 ! , Fama ~ 1997 1997 !! . ~

28 The Journal of Finance 1914 Corporate earnings summarize the operations of a company into a single sta- tistic. This statistic is based on a large sample of information and is highly rel- evant to the value of the company. It is prototypical of the information that people typically undervalue: Abstract, relevant, and based on a large sample. Mar- Michaely et al. kets also underreact to dividends omissions and initiations ~ 1995 ~ . The decision to omit dividends is generally made reluctantly and in !! of div- ! response to significant corporate difficulties. The omission or initiation ~ idends is appreciated by investors, but it may not be fully appreciated because news contained in the omission ~ the bad ! has been con- ~ or good ! or initiation densed into a single event. We might expect a greater reaction when an omis- ! is accompanied by a well-publicized graphic portrayal of a sion ~ or initiation ! ~ company’s woes or good fortune . Like dividend initiations, open-market re- ~ !! are positive signals that abstract from a Ikenberry et al. purchases 1995 ~ wealth of more salient information. In addition to possibly signaling manage- ment’s sanguine outlook, the announcement of open-market repurchases states that the supply of shares in a company will be reduced. Investors who do not realize that firms face upward-sloping supply curves when they repurchase ~ 1992 !! , and that price is therefore likely to rise, may under- Bagwell shares ~ react to the announcement. Most of the documented long-run return patterns following information 1997 events are underreactions. Fama classifies the poor long-term perfor- ~ ! IPOs mance following initial public offerings ~ 1991 ! , Loughran and !~ ~ Ritter 1995 and seasoned equity offerings ~ SEOs !~ Loughran and Ritter ~ Ritter !! ! , Spiess and Aff leck-Graves ~ 1995 !! as “in the over-reaction camp.” ~ 1995 Using the definition of underreaction given above, however, SEOs would be classified as underreactions because the usually negative market reaction at the announcement of the SEO is followed by underperformance. Using the above definition, IPOs are unclassifiable because no market reaction is ob- servable following the announcement of an IPO. As discussed in Sec- tion III.C, price ref lects the opinions of those most willing to trade. Generally, if a minority of traders overreacts to information and the majority discounts the minority’s opinion or underreacts to the same information, price will underreact. Negative opinions are incorporated into stock prices when in- vestors sell securities they already own and when they sell short. The ma- jority of investors, however, are unwilling to sell short. The first day’s price for IPOs is therefore determined by the minority of investors that is most sanguine about a company’s prospects: those investors who subscribe to the IPO—some with the intention of f lipping it on the first day—and those who buy it on the first day. No one with a bad opinion of the company owns an IPO on the first day and it can be difficult to short the stock so soon. Thus, the high first-day price for an IPO may ref lect an overreaction by a minority of market participants to the optimistic stories and scenarios that accom- pany the IPO’s promotion. Markets often underreact to announcements of abstract, highly statistical, or highly relevant information. Earnings changes, dividend omissions, and brokerage recommendations are all examples of such information. However,

29 1915 When All Traders Are Above Average behind each of these events lie many concrete, salient stories: new products succeed, others fail, ad campaigns are waged, employees are fired, scandals emerge. Though the sum of these stories is underweighted, the individual ~ parts may, in fact, be overweighted. As Joseph Stalin put it, “The death of ! a single Russian soldier is a tragedy. A million deaths is a statistic.” If markets do systematically overreact, they may do so to highly publicized, 24 graphic news and to rumors. Though brokerage recommendations are delivered in their salient form only to customers, some recommendations are both widely disseminated and attention grabbing. ’s monthly “Dartboard” column The Wall Street Journal pits the recommendations of four analysts against the random selections of a dart. The analysts, whose portraits are featured, explain the reasons for ~ 1993 their picks. Many readers follow this contest, and Barber and Loeff ler ! show that the market overreacts to these recommendations. Similarly, the Wall market overreacts to recommendations made on the popular TV show ~ 1987 !! . Street Week ~ Pari Another signal to which we might expect overreactions is price change. Price change may be the most salient signal received by investors because, unlike other signals such as earnings, it directly, rather than indirectly, con- tributes to changes in their wealth. It is also the most publicized signal, instantly available on many computer screens, reported daily in newspapers and other media, and mailed monthly to investors in brokerage statements. Furthermore, many investors may overweight the predictive value of price 25 overex- changes; they may see deterministic patterns where none exist, ! , and put too much faith in techni- trapolate those that do ~ e.g., momentum cal trading rules—though not necessarily the same technical trading rules as each other. As we saw in the model of price-taking traders, the impact of a private signal depends on how many people receive that signal ~ and, as Figlewski ! ~ ! . The 1978 points out, on the wealth and risk tolerance of those traders impact of traders, even rational traders, depends on their numbers and on their willingness to trade. The mere presence of a few rational traders in a market does not guarantee that prices are efficient; rational traders may be no more willing or able to act on their beliefs than biased traders. It is markets with higher proportions of rational traders that will be more effi- cient. So, if information processing biases are more pronounced in individ- 24 Lehmann 1990 ! , Je- ~ There is some evidence of short-term mean-reversion in returns ~ gadeesh . Such reversions possibly could be due to overreaction to salient news stories, ~ 1990 !! De Bondt ~ but this has not been shown. Mean reversion has also been found at longer horizons 1985, 1987 !! . Returns tend to be positively serially correlated at intermediate ho- and Thaler ~ Jegadeesh and Titman ~ 1993 !! . Positively serially correlated prices could be the result of rizons ~ underreactions to important information such as earnings changes. They could also be the result of price momentum traders overreacting to price as a signal and purchasing stocks that have risen, thereby driving prices even higher ~ 1997 !! . as in Hong and Stein ~ 25 ~ 1985 ! show that people hold strong beliefs that random Gilovich, Vallone, and Tversky sequences are nonrandom.

30 The Journal of Finance 1916 uals than in institutional traders, it should come as no surprise that return see Fama anomalies are greatest for small firms ! for a review ! which ~ ~ 1997 are traded more heavily by individuals. V. Conclusion Overconfidence is costly to society. Overconfident traders do not share risk optimally, they expend too many resources on information acquisition, and they trade too much. These are dead weight losses. Overconfidence in- creases trading volume and market depth, but decreases the expected utility of overconfident traders. When information is costly, overconfident traders who actively pursue information fare less well than passive traders. Over- confident traders increase volatility, though overconfident marketmakers may dampen it. Price-taking traders, who are overconfident about their ability to interpret publicly disclosed information, reduce market efficiency; overcon- fident insiders temporarily increase it. When there are many overconfident traders, markets tend to underreact to the information of rational traders. Markets also underreact to abstract, statistical, and highly relevant infor- mation and overreact to salient, but less relevant, information. Like those who populate them, markets are predictable in their biases. Appendix A: Price Takers L An equilibrium exists in which the linear price conjectures, equa- 1: EMMA tions (2) and (3), lead to linear demand functions. The coefficients of the price conjectures are h h S v 2 a S x v a 5 , ~ A1 ! 31 g M 2 h h ~k 1 2 h 1 g! e v 2 h g! 1 g M ~k e a 5 5 a , ~ A2 ! 32 33 g M h g! h 1 h 2 ~k 1 2 e v a a ! 1 ~a F 1 a 5 ! S v 2 a S x var 6 ~ P 3 2 33 32 31 21 b !! g! 1 g M 2 k! h 1 a ~g ~h h h 1 ~k 1 g M 2 v ~a S 33 e 32 e v , ~ A3 ! 1 h 2 g! 1 h M ~k 1 g h e v h ~k 1 g M 2 g! e a 5 ! A4 ~ , 22 1 h g! h M ~k 1 g h 2 v e and 1 g 1 k M 2 k 2 x a a 1 a ! A5 5 1 a S v 2 ~ S . P 22 21 22 1 D S 2 M h kg h h e v

31 1917 When All Traders Are Above Average Proof: We first solve the equilibrium for the third round of trading. believes i F Trader has a multivariate normal distribution. We calculate 3 i the mean and the covariance matrix of this distribution which are T ! 5 @ S v S v a ! 1 a and S v a F 1 ~a # 1 a v ~ S E 33 31 22 21 i 3 32 b 1 a a a 1 a 1 1 a 22 22 33 32 32 1 1 1 h h k h k M h M k h h h h h h h v v e v e e v a 1 1 a a a 1 1 33 33 22 32 1 1 h k h h h h h h h h k h M e v v e v v , C5 2 2 a a a a 1 k! ~g M a 2 k 22 22 22 22 22 1 1 C 1 2 kg k h h h h M h M h h h v v e v e 3 4 a a a a a 1 a 1 33 33 32 32 33 32 1 1 C C 1 2 M M h h h k h h k h e v v e ! A6 ~ where 2 a 1 g k g M 1 a 22 33 C 1 a , 1 a 5 32 1 22 S D 2 h kg h h M h h e v v 2 2 2 ! k! 2 1 M g 1 a a !~g ~a ~a 33 33 31 1 32 5 1 ! A7 ~ . C 2 2 M h kg h h e v Let T 1 2 1 2 1 T 2 1 2 ~a 5 h [ ! ! . ~h h cov ! # , a ~ ~h h ! ! v I @~ h h 1 a !~h F A v v 22 b 3 v 32 33 v i Then, by the projection theorem, 2 1 E F !! F ! 5 S v 1 A C ~ I ~ v6 ~ E 2 F i 3 i 3 i 3 b b P P y ~ y v !~k 2 g! h S 1 ~ 1 Y h 1 h 1 ! !~g h M Y e 3 i 2 3 e i 2 v 5 A8 ~ ! ~k 2 g M 2 g! h h 1 1 h e v and 1 2 T 1 2 var ! ~h h 5 ! 2 F v6 A C I ~ A 3 v b 1 5 ! . ~ A9 1 g h 2 g! h h 1 2 ~k M e v

32 The Journal of Finance 1918 v is the same for all traders and so the subscript The conditional variance of I i is dropped in var 1976 ! . Following Grossman ~ v6 ! , we can solve equa- ~ F I 3 b ! and get demand function 1 ~ tion P v6 F ~ I ! 2 E i 3 b 3 ! A10 ~ . 5 x i 3 var ~ F a ! v6 I 3 b We calculate the average demand per trader and equate this to the per trader P supply, S x . Then solving for , we can match coefficients to those of the 3 conjectured second-period price function to obtain a a , and a as given , 33 31 32 and ! A2 ! . Thus the linear price conjectures are fulfilled in equations ~ A1 ~ 3. 5 t and equilibrium exists at t 5 2, we again use the projection theorem, To solve the equilibrium at calculating P E 6 F ~ ! i 2 b 3 1 g M 2 g! h 2 ~k v 1 h S S v 2 a S x h v e 5 g! ~k 1 g M 2 h h 1 h 2 e v 2 P 2 g!~ h ~k 1 g M ! 1 1 2 g M 2 g! h !! !~ h Y v 2 S v ~k 1 h S ~h ~k 2 g!~ y 2 i 2 e 2 e e v 1 ~h h ~k 1 g M 2 g! h ! !~h h h 1 2 ~k 1 g M 2 g! 1 v e e v A11 ~ ! and ~ 6 F P ! var b 3 2 2 2 2 M ! h gk ~h~g 1 k M 2 k! h 2 ! ~~k 2 g! h ~ M 2 1 ! 1 1 1 g! 2 M g ~k e e v . 5 2 2 gk 1 ~k 1 g M 2 g! h M !~h h 2 1 2 ~k 1 g M ~h g! h ! h e e v v ! ~ A12 ’s second round demand is Since traders are myopic, trader i P 2 ~ P ! 6 F E 2 i 3 b 2 ~ A13 . ! 5 x 2 i 6 F ~ ! a P var 2 3 b P Equating per trader demand and per trader supply, solving for , and match- 2 given in and a a ing coefficients gives us the equilibrium values for 22 21 A3 ! and ~ A4 ! . Using the unconditional expectation and variance equations ~ P as given in , we can follow the same steps as above to calculate P of 1 2 A5 ! . Q.E.D. equation ~

33 1919 When All Traders Are Above Average To simplify the exposition, Propositions 1 to 5 are proven for the case of 1 and h 1. Proposition 1 states that expected volume increases as g 5 5 overconfidence increases. This is restated formally and proven in terms of trading round 2 volume. It is also true for round 3 expected volume. k . 1 , and M $ 2 then ROPOSITION 1: If P N 6 x 6 x 2 i 1 i 2 E ! A14 ~ a ( S D N i 5 1 is an increasing function of k . ! , the per Proof: The first step of the proof is to calculate equation ~ A14 capita expected trading volume in trading round 2. Traders have negative exponential utility functions, which means that their demand for the risky asset does not depend on their wealth. They have the same prior beliefs about the distribution of . Because they have the same beliefs as well as the I v S x . 5 x same risk aversion, all traders have the same first period demand: i 1 ! and ~ A4 Coefficients from equations are substituted into equation ~ 3 ! ; ~ A3 ! 3 , ~ A11 ! , and ~ A12 ! are then substituted into equation ~ A13 ! which equations ~ ! A14 ! . The expectation operator is moved inside ~ is substituted into equation the summation and the denominator is moved outside the expectation. We N N have then the average expectation of identical half-normal distributions. Taking expectations and simplifying gives us N 2 6 x x 6 2 i i 1 E a ( D S N i 1 5 ~ M 2 1 ! h 2 e 5 ! M p 2 2 !k ~k 1 M ! 1 ~k 1 ~ 2 1 2 h h ! M e v . 3 2 2 !~ 1 2 k 1 k M M h a 1 ~~k 2 2 2 k 1 1 1 M 2 1 ! 1 2 k M !~~ ! h ~k ! 1 ! e v ~ A15 ! $ 2, and k Bearing in mind that 1, one can show that, in the given M . ~ ! with respect to k is posi- parameter range, the derivative of equation A15 A15 is increasing in k . Q.E.D. tive and so equation ! ~ Proposition 2 states that volatility increases when overconfidence increases and when traders undervalue their prior beliefs. Three alternative measures ~ P ! ! , var P ~ P 2 ! , and var . The proposition is true ~ P of volatility are var a a 2 2 a 3 3 for all three measures. It is proven for var ~ ! . P 3 a P If k $ 1 ROPOSITION var 2: ~ P . ! is an increasing function of k then 3 a

34 The Journal of Finance 1920 ~ A1 ! and ~ A2 ! into equa- Proof: Substituting coefficients from equations A3 ~ ! tion , we can calculate 2 h 2 ! ~k 1 M 2 1 ! M ~ h 1 2 h e e v P 5 ~ ~ ! A16 ! . var a 3 2 ~ ~k 1 M 2 1 !! h 2 1 h h M v e v with respect ~ A16 ! In the given parameter range, the derivative of equation is positive. Q.E.D. to k The quality of prices can be measured as var P I v ! for t 5 2 or 3. As this ~ 2 t a variance increases, the quality of prices worsens. Proposition 4 and its proof 5 t 5 2; the are given here in terms of t 3. The proposition is also true for proof is analogous. If k $ 1, var P ~ ROPOSITION . 2 I v ! 3: k is an increasing function of P 3 a A1 ! and ~ A2 Proof: into equa- ! Substituting coefficients from equations ~ ! , we can calculate 2 ~ tion 2 2 1 1 M M 1 2 h 1 ~~k 2 M ! 2 2 M 2 k ! h e v ! I 5 . v ~ ~ 2 ! A17 P var a 3 2 ~k 1 M ! h M ! 2 2 1 ~ h 1 e v ~ ! with respect In the given parameter range, the derivative of equation A17 to k is positive. Q.E.D. If M $ 2 traders’ expected utilities will be lower when k . 1 ROPOSITION 4: P k . 5 than when 1 In this model, traders can infer the aggregate signal. So if they Proof: 5 5 have perfectly calibrated probability beliefs 5 g i.e., 1 ! their poste- ~ k h rior beliefs will be identical. Because they have the same beliefs as well as CARA utility functions with the same risk aversion, perfectly calibrated trad- ~ S x ! ; this i.e., ers will hold the same amount of the risky asset in equilibrium maximizes their utility. This is the position to which traders, whether over- confident or perfectly calibrated, trade in the first round of trading where . Following steps similar to those used to ! x S 5 ~ i.e., there is no signal x i 1 A15 , we can calculate the expected net trading subsequent ~ obtain equation ! to the first round of trading. This is N 1 2 M ~ 6 6 x h ! 2 x ! 2 ~k 2 1 3 1 i i e . ! A18 ~ 5 E a ( D S ! M p N a 1 i 5 We see that when traders are perfectly calibrated, they do not trade in later rounds and continue to hold their optimal portfolio of the risky asset, . S x ~ k . 1 ! i.e., M $ 2, they are However, if traders are overconfident and if

35 1921 When All Traders Are Above Average expected to trade in later rounds, thus departing from their optimal portfo- lio and reducing their expected utility. Q.E.D. P ~~ P 5: Cov 2 P ROPOSITION ! , ~ P and 2 P k !! is a decreasing function of 1 2 2 a 3 5 P ~~ ! , ~ P 2 2 P 1. !! 5 0 when k P h 5 g 5 cov 1 2 a 2 3 Proof: P Noting that is a constant and substituting coefficients from equa- 1 , ~ A2 ! , ~ A3 ! , and tions A4 ! into equations ~ 2 ! and ~ 3 ! , we can calculate ~ A1 ! ~ P 2 P ~~ ! , ~ P P 2 !! cov 3 2 2 a 1 2 2 1 g M ! 2 ~k g M 2 1 !g 2 ~ M h!! h ~k 1 ~ e 52 ! , ~ A19 2 ~h g M ! h 2 ! 1 ~h h h 1 2 ~k 2 g 1 g M ! h ~k ! M g 1 e e v v A19 1 ~g 2 h! k ~ M ! is0if g which has the opposite sign of 2 . So equation 5 h 5 1, negative if k 2 g . M ~h 2 g! , and positive if k 2 g , M ~h 2 g! k 5 g . 5 5 1, the sign of expression ~ A19 ! is the opposite of k 2 1. Note that when h g ! k is negative. Q.E.D. A19 The derivative of equation ~ with respect to Adding rational traders to the economy greatly complicates the expression for covariance. For simplicity, and without altering the basic finding that returns may be positively serially correlated when overconfident traders trade with rational traders, the following proposition is proven for an econ- omy with 0 M rational traders in which trad- N overconfident traders and N is publicly revealed in period v I ers receive private signals in period 2 and 3; therefore P 5 v . As above, in period 2 each overconfident trader re- I 3 5 I v 1 I e ; rational traders receive ceives one of M y I possible signals, m 2 i 2 1 2 signal y and I 5 , where I r for ; N ~ 0, h 1 v I ! r I r e is independent of I r 2 2 m t 2 r 2 1 2 ; ratio- ! ! ; N ~ 0, ~g h I . Overconfident traders believe that M 1,..., m 5 r r t nal traders hold correct distributional beliefs about their signals and those of others. ROPOSITION Let h 5 0 . If rational traders trade with overconfident trad- 6: P ~ P h 2 P 2 , P ! . P M ! is positive if ~ 1 2 g!~ 1 1 g cov ers, then 1 2 3 r a 2 1 ~k g~ M 2 1 !!~k 1 g~ M 2 1 ! 2 M ! h 1 1 . e We can determine the equilibrium as was done in Lemma 1. Then Proof: we can calculate P ! 2 P ~ , P P 2 cov 2 2 3 1 a 1 2 M 1 1 g M ! h ~~ g!~ 2 ~k 1 g~ M 2 1 ! 1 1 !~k 1 g~ M 2 1 ! 2 M ! h ! e r 5 2 M g~ M 2 1 ! 1 1 ! h M ! 1 ~ 1 ~ g ~k ! h 1 1 ~ 1 1 M ! h ! 1 v r e ~ A20 !

36 The Journal of Finance 1922 1 2 g!~ 1 1 g M ! h which is positive when . ~k 1 g~ M 2 1 ! 1 1 !~k 1 ~ r M ! 2 M ! h 2 g~ 1 . Q.E.D. e Appendix B: An Insider h EMMA 1 2 h If 2: . k h k , an equilibrium exists in which the insider’s h L v e v linear price conjecture, equation (4), and the market-maker’s linear demand conjecture, equation (5), are fulfilled. In equilibrium the coefficients of equa- tions (4) and (5) are B1 ! 0, 5 ~ A h h k v e B2 ~ ! , B 5 ! ! h k h 2 ~k h 2 h h 1 v e z v ~ ! H 5 0, B3 1 h k 2 h h ~k 2 1 h k h ! h e z e v v 5 L ! . ~ B4 ! 1 2 h ! h h k ~h v v e x , that he believes will maximize Proof: The insider submits a demand, his expected profit. To do this he solves I ~ x ~ I v 2 P ! 6 y ! 5 max E ! ~ x ~ B5 v 2 ~ H 1 L ~ x 1 z !!! 6 y ! , ~ E max b b x x 4 ~ P . Taking first-order conditions has been substituted for ! where equation we have and solving for x H ~ I v6 y ! 2 E b ! B6 . ~ 5 x L 2 We can calculate y h k e E ~ v6 ! I 5 B7 y . ~ ! b h k 1 h h e v Substituting equation ! into equation ~ B6 ! we get B7 ~ h k 2 H e 1 y . ~ B8 ! 5 x 1 k L h ! 2 L ~h h 2 e v

37 1923 When All Traders Are Above Average And so, if the linear conjectures hold, h k 2 H e B 5 and ! B9 ~ . 5 A ~h L ! 1 h k L 2 2 h v e The marketmaker sets price equal to the expected value of I v given the order f low she observes. We can calculate h h 2 ABh Bh e z z e 5 ! . ~ I v6 x z I 1 x ~ 1 I z ! 1 5 P E a 2 2 1 1 h B ! h h ~ h B 1 h h ! 1 h h h ~ h v z e e e z e v v v ~ B10 ! So, if the conjectures hold, h h Bh 2 ABh e z z e and ! B11 ~ . 5 L 5 H 2 2 h 1 ! h h 1 B B h h h ~ h ~ 1 h h ! 1 h e e e z z v e v v v 1 through ~ B11 ! have four unknowns. When k h The four equations in ~ B9 ! e h 2 h h ! , they have one real nonnegative solution: equations ~ B1 $ through k v v B4 ! . Thus the conjectures are fulfilled and an equilibrium exists. Q.E.D. ~ ~ 6 x 6 1 6 I z 6 : ! Expected trading volume is E a P , then 1 , h # 1 , and k h 7: 1 2 h h If . k h k $ E is an ~ 6 x 6 1 6 I z 6 ! ROPOSITION a v v e increasing function of k . B3 ! and ~ B4 ! into equation ~ B8 ! , and sub- Proof: Substituting equations ~ ! x , we can calculate stituting equation ~ B8 for 2 ! h 1 h k~ 2 e v E 1 x I 6 5 1 6 6 ! 6 ! ~ z B12 ~ . a ! ! h ~k 2 k h h ! p 1 2 p h h h z v e z v , the derivative of equation ~ h . k h 1 h 2 B12 ! with respect to k is When h k v v e positive. Q.E.D. Market depth is measured as the inverse of the derivative of price with : x 1 I z !! ~ i.e., ~ respect to order f low 1 2 P $ ROPOSITION , h # 1 , and k h 8: 1 2 h h is . k h If , then ~ dP 0 d ~ x 1 I z !! k 1 v v e an increasing function of k .

38 The Journal of Finance 1924 ~ B3 ! and ~ B4 ! into equation ~ 4 ! , and differ- Proof: Substituting equations x 1 z ! gives us entiating with respect to ~ I 1 2 dP 5 1 L . ~ B13 ! 0 D S ~ I z d ! x 1 for 0 , the derivative of 1 ! L with respect to k is Substituting equation ~ B4 L h 1 h h . Q.E.D. . k 2 h k positive when e v v Volatility is measured as the variance of price. If k $ 1 , h # 1 , and k h . 1 2 h h is an ROPOSITION k h ! 9: var P ~ , then P v a e v k . increasing function of ! and ~ B2 ! ~ ~ 5 ! , and equa- Proof: Substituting equations B1 into equation ! , ~ B3 ! , and ~ B4 ! into equation ~ 4 tions , we can calculate ~ 5 ! k h e ! B14 ~ . ! 5 ~ P var a ~k h h ! 1 h h 2 v v e B14 ! with respect to k is positive. Q.E.D. The derivative of equation ~ Quality of prices is measured as the variance of the difference between price and true underlying value. 2 k $ 1 , h # 1 , and k h If 1 10: h h ROPOSITION . k h is a , then var ! ~ P 2 I v P a v v e k decreasing function of . , and equa- and ~ B2 ! into equation ~ Substituting equations ! ~ B1 Proof: ! 5 P 2 I v ! ,wecan ~ B4 ! into equation ~ 4 ! , and then into var tions ~ 5 ! , ~ B3 ! , and ~ a calculate k h 1 h 2 h v e ~ B15 ! . ~ 5 ! v I 2 P var a 1 2 h h ! h h ~k v v e ~ B15 ! with respect to k is negative. Q.E.D. The derivative of equation 11: If k $ 1 , h # 1 , and k h is 1 ROPOSITION h h !! . k h P , then E 2 ~ x ~ I v 2 P a v v e a decreasing function of k .

39 1925 When All Traders Are Above Average ~ B1 ! through ~ B4 ! into equations ~ 4 ! and ~ 5 ! , Proof: Substituting equations and then into E ~ I v 2 P !! , we have x ~ a ! h k 2 h h 2 1 h ~k h k 1 v e e v E ~ I ~ . v ~ B16 5 !! ! P 2 x a ! h 2 ~k h h h h 1 ! z e v v h h , the derivative of . k 1 2 ~ B16 ! with respect to k is nega- h h k When v e v tive. Q.E.D. Appendix C: Marketmakers and Costly Information ~ 1996 !! The following expectation x be a ~ To f t is needed in this section. Let 2 , then and variance normally distributed random variable with mean m s 2 2 s ! 2 B 1 ~m 1 1 2 1 Bx 1 Ax C 2 e E ~ ! m 1 C 5 ! 2 1 . C1 ~ exp S D H J 2 2 2 s 2 2 2 A s 1 ~ ! 1 2 2 s A % These expectations can be easily calculated: h h k e e ! 5 v6 ! ~ C2 ~ , [m y I I I y [m v6 ,E I ~ I y I y ! 5 E a b b a h h h k 1 1 h h v e e v 1 1 var y ! 5 ~ ~ I ! C3 ~ , r [ r [ v6 , var 5 I I v6 I y ! i a b a b i k 1 h h 1 h h h v e e v 2 h 1 h ~ h ! k h k e e v e var S [ ! var ! , 5 5 ! ~m ~m C4 ~ . S [ b b a b a b 2 ~h h h ~h h ! 1 k h 1 ! k h h !~h e v v v e v m y I Because is normally distributed, is also normally distributed. Define b ! P var v6 ~ I r b ib . ~ C5 ! [ r S 2 ~ l var l! r ~ I v6 P ! 1 1 ib b L 3: EMMA There exists an equilibrium in which each informed trader’s de- mand for the risky asset is P 2 ! ~ I v6 I y E b ! C6 ~ , 5 x 1I v6 I y ! ~ var a I b

40 The Journal of Finance 1926 s demand for the risky asset is each uninformed trader ’ P I v6 P ! 2 ~ E b , ! C7 ~ 5 x 1U P ~ a v6 var I ! b price is m E ! P ~ I v6 b b S P 5 r 1 ~ 1 2 l! ~ ! C8 l , z I r 1 a S D S I ~ r v6 var ! P ib b and the fraction of traders who choose to become informed is 2ac 1 2 ! e ~ r ib 1 0if 0, 2 # S D S b 2ac 2ac 2ac 1 ! ~ e 2 2 1 ! h 1 2 r ~ e ! r e ~ r ib z ib ib * 5 l 2 1 [ if 2 1 0, , a D D S S F G 2ac 2 ! 2 ~ ! h S e a S r 1 ib z b b 2ac 2ac 5 ~ e ! 2 1 ! 1 2 h r e ~ ib z $ 1 . 2 1if D S 2 a r S ib b ! ~ C9 Proof: The derivation of this equilibrium roughly follows Grossman and 1980 ! and Demski and Feltham ~ Stiglitz ! . Solving equation ~ 6 ! gives ~ 1994 ~ !~ see Grossman ~ 1976 !! . Assume for the moment that, given us equation C6 , I is normally traders’ distributional beliefs and conditional on observing P v distributed. ! ~ Then We will see below that the assumption is self-fulfilling. gives us equation ~ ! . ~ 7 ! solving equation C7 Informed trader demand per trader times the fraction of traders who are in- formed plus uninformed trader demand per trader times the fraction of trad- ers who are uninformed must equal noise trader supply per trader. That is, E y ! P P 2 ~ v6 I I ! ~ I v6 P 2 E b b 2 l! ! C10 ~ 1 52 I z . ~ 1 l var v6 ~ I v6 P ! ~ I a I y var a ! b b C10 ! for P gives us equation ~ C8 ! . Let Solving equation ~ ar i b I 1 z[ D m z I C11 ! . ~ l C11 ! ~ ~ C8 ! gives us Substituting equation into equation l! 2 1 ~ r S l r S 5 P C G where z D [ 1 , C 1 1 1 ~ I v6 P ! r var ib b 2 ar S b ib 1 2 , and D [ G ~ [ ! C12 . h z 1 D S D 1 S l b

41 1927 When All Traders Are Above Average C8 ! holds, then P is a linear function of D z and the two are If equation ~ z D is a linear combination of two normally dis- informationally equivalent. tributed random variables and is normally distributed itself. Therefore, given I ! 5 ~ I v6 D z! is normally distributed, as was v6 ~ traders’ distributional beliefs, P assumed, and has mean and variance. 1 E 5 G D z , and var ~ I v6 D z! 5 D I z! 2 G S [ r ~ . ~ C13 ! v6 u 1 b b b 1 h h v * we observe that in equilibrium traders are indifferent be- l To solve for tween buying and not buying information; thus, E . U !# 5 E W @ U ~ @ ! !# ~ ~ C14 W 1U 1I b b P Bearing in mind that has the same information content as z D ,wecan calculate U @ W ! C15 !# 5 E ~ @ E l## @ E , @ 2 exp $ 2 aW z D % 6 I y # 6 ~ E b b b i 1 I b 1 2 a 5 E var exp ! 2 a E C16 @ W ~ l 6 I y , 1 # 2 z E D @ W # y 6 I J H GG F F b I I 1 1 b b b * 2 2 ! P 2 ~m b 1 2 a z f D ~ 1 x C17 ! P 2 c exp l , 2 E E 5 b b I 0 I 0 D S H J F F GG * 2 ar b i 2 ! 2 P l! , z D v6 I ~ E ~ r b i ac 1 e 2 P ~ 2 a exp f C18 1 x ! 5 E b i 0 i 0 S D H J F G ! 2 r ar u u r b i ac U E W @ C19 ~ ~ !# . ! e 5 U b 1 ! r u C17 ! is obtained from equation ~ C16 ! by substituting f Equation ~ 1 P 1 x 0 I 0 I x I v 2 P ! for W ~ , and taking expectations. To obtain , equation ~ C6 ! for x 1 i 1 iI I 2 , substitute ! ~ 2 P ! C1 , apply equation ! ~m we multiply out equation ~ C18 b ! C3 ~ C13 ! , and note that E ~ from and I v6 D z! 5 E ! ~m C14 6 D z! . From equations ~ ~ b b b C19 and ! ~ we have r b i ac ~ C20 ! . e 5 1 ! r u Substituting for r , and noting that and r in equation ~ C20 ! , solving for l u b i l cannot be negative or larger than 1 gives us equations ~ C9 ! . Q.E.D.

42 The Journal of Finance 1928 12: There exist values of h P ,h ROPOSITION ,h 1 , a, and c, such that if h 5 z e v * , 1 and k 5 1 , and 0 , l and ,orif k h . 1 @ U W E , then !# , 1 , ~ 1 I a 5 @ !# , where i 5 I, and i ~ U represent prototypical informed and W U E a 1 U uninformed traders. To calculate the actual expected utility of the informed trader, we Proof: e I take iterative expectations using the actual distributions of I v . Equa- and tion ~ C1 ! is used to solve each expectation. The result is: 6 W 5 E @ E @ ~ @ U ~ W ! !# I y # | D z , l## U @ E E 1 I a a a 1 a I 2 0 1 2 2 1 0 2 C 2 52 1 ~ 2 ~ 1 2 D C z!! var 2 ~ D z!! ~m var 6 a 4 b a 3 2 ~ c 2 f ! 3 exp 1 ~ aC $ x , z!!!% ! a var D ~ D z! 0 ~ 2 ~ 1 2 2 C ~ var I a 0 a 4 0 I 1 ~ ! C21 where 2 h ~h h ~h h k ! 1 1 k h ! e e v v h 1 k h 5 ! 2 , 1 ~h 1 C v e 2 D S h h 1 h 1 h k~ ! v e e v 2 2 !~h h ~k 2 k h e! 1 v ! C22 ~ , C 5 3 2 h h k~ ! 1 v e 2 2 G ~m 2 C z!! D 6 2 r C 1 var G ~ r i b 2 a b 2 i 1 a 2 2 C23 2 , ! ~ 1 C 5 C 1 4 2 z!~ 1 2 C r var ~m ~m z! 6 D z!! 2 var 2 6 D 2 var D ~m 6 b b 3 b a a i b a S a , var C24 ~ 5 ~ ~m . 6 D z! 5 S D ! 1 2 G 1 ! , and var S ~ D z! 5 G a 2 a b a a 2 1 S D a Similarly we can calculate the actual expected utility of the uninformed trader: E 6 ~ @ U !# 5 E l## @ E , @ E z @ U ~ W D | ! W I y # U a a a a 1 1 U 2 0 1 2 C 2 1 ~ 52 2 D ~ var z!! a 5 2 ~ 1 ~ aC 3 x ! 2 var ~ D z! 0 $ 2 ~ 1 2 2 C var ~ D z!!!% , ~ C25 ! af exp U I 1 0 5 a a 0 where 2 ! ! C C 2 2 C !~ G 2 G ~ G ~ 1 1 1 3 1 1 C26 5 ~ var , ~ I v6 D z! 2 ! C 5 a 2 r 2 r u u 2 2 h h S 1 S b b , and var ~ C27 ! . D 5 ~ I v6 2 5 z! G a 3 S D 1 h 1 D S a a v

43 1929 When All Traders Are Above Average C21 ! is less than equation ~ C25 ! for any parameter set for which If equation ~ * 1, 1, and 0 , l k 5 h . , k 1, h , 1, and for any parameter set for which 1, 5 * 1, then the proposition is true. I evaluate equations C21 ! and , ~ and 0 , l C21 ! to be less ~ C25 ! for a wide variety of parameter values and find equation ~ ! than equation ~ 25 whenever these parameter conditions are true, suggesting that the expected utility of informed traders is always less than than of un- informed traders if traders are overconfident or if they undervalue their prior information. Q.E.D. * l In the final proposition and are written as functions of the economy’s P * ~k , h , h .Asin , h !! , h c , a , c ! and P ~k , h , h , , h a , h , i.e., l ~ parameter values v z v z e e Proposition 10, market depth is measured as the inverse of the derivative of price with respect to order f low. P ROPOSITION . k 13: $ 1 , then, for any choice of parameter values Let k 1 2 * h ,h ,h , a, and c such that ,h ( k , h ,h l ,h ,a,c) 5 1 and ,h z 2 e e v z v * 5 h ,h ( ,h k ,h , ,a,c) , 1 l z e v 1 1 2 2 1 h dP , h ~k , h ! , a , c ! c , a , h , , h dP ~k , , h , h h , z v z e 2 e 1 v C28 ! ~ . S D D S z d d I z I and ~ P ~k , , h , h p , h 0 , h f , a , c !! . var 2 ~ P ~k % , h , h , , h ! , h 6 , a var c !! if E z ~ , I 6 a e v 1 a z e v 2 a z ! C29 ~ h P var , h , h ~ , h , , h p , a , c !! 5 var 0 ~ P ~k f , h , h 2 , ~k % , h 5 , a , c !! if E ! ~ 6 I z 6 e a e a 1 a z z 2 v v ! C30 ~ and ~ P ~k , , h , h p , h 0 , h f , a , c !! , var 2 ~ P ~k % , h , h var , h ! , h 6 , a , c !! if E z ~ 6 I . a e v 1 a z e v 2 a z ! C31 ~ where h~ h h 2 !~~k 1 1 k ! !h h h 1 k k 1 2 v e 2 1 e v C32 ~ . ! 5 f 2 ~k ~ 1 k 2 ! h h ! h a 1 e 2 1 v * * 5 1 into equation 1 C5 ! , and equation ~ C5 ! and l ~ 5 l Proof: Substituting , tak- C12 P 5 D z . Differentiating P with respect to I z , we find that ! into equation ~ ! gives us market depth: C3 ~ ing the inverse, and substituting from equation 2 1 k 1 dP h h h e v ! C33 ~ , 5 D S a I z d

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