uop qOIO April 20, 1999 On the Survival of Overconfident Traders in a Competitive Securities Market David Hirshleifer and Guo Ying Luo1 Abstract Recent research has proposed several ways in which overconfident traders can persist in competition with rational traders. This paper offers an additional reason: overconfident traders do better than purely rational traders at exploiting mispricing caused by liquidity or noise traders. We examine both the static profitability of overconfident versus rational trading strategies, and the dynamic evolution of a population of overconfident, rational and noise traders. Replication of overconfident and rational types is assumed to be increasing in the recent profitability of their strategies. The main result is that the long-run steady-state equilibrium always involves overconfident traders as a substantial positive fraction of the population. Keywords: Survivorship, Natural Selection, Overconfident Thaders, Noise traders 1David Hirshleifer: University of Michigan Business School, 701 Tappan Street, Ann Arbor, MI 48109-1234; Guo Ying Luo: Faculty of Management, Rutgers University, 94 Rockafeller Rd., Piscataway, NJ 08854-8054. Email: luo@business.rutgers.edu Tel: (732) 445-2996. We are grateful to Ivan Brick and Sherry Gifford for their helpful comments.

1 Introduction Several recent papers have argued that investor overconfidence or shifts in confidence offer a possible explanation for a range of anomalous empirical patterns in securities markets.1 An important general objection to such approaches is that rational traders ought to make profits at the expense of the irrational ones, so that irrationality should in the long run be eliminated as a significant factor in the market.2 This paper offers a new reason for the possible long-run survival of overconfident traders in competition with rational traders. The basic idea is that risk averse, overconfident traders trade more aggressively based on valid information than do rational traders. As a result, overconfident traders are better able to exploit risky profit opportunities created by the trades of liquiditymotivated traders or the mistakes of noise traders.3 Overconfident investors 1Odean (1998) examines investor overconfidence, overreactions, and the high volume of trade in securities markets. Daniel, Hirshleifer and Subrahmanyam (1998a, 1998b) examine the consistency of overconfidence and shifts in confidence with abnormal postevent returns in event studies, short horizon stock price momentum, long run reversal, short- versus long- horizon correlations between accounting performance and later stock price performance, and the relative ability of size, fundamental/price ratios, and risk measures to predict future returns. Odean (1998) and Daniel, Hirshleifer and Subrahmanyam (1998a) also examine conditions under which there will be excess or insufficient volatility of security returns relative to a rational benchmark. 2Luo (1998) provides a model of natural selection in which irrational traders lose money and the market evolves toward long run efficiency. Also, Figlewski (1978, 1982) finds that owing to wealth shifts among traders with diverse information, informational efficiency may or may not be achievable depending on the correlation of signals received by the traders and depending on the degree of traders' risk aversion. 3Apart from this informational benefit, overconfident investors who underestimate risk can potentially benefit from exploiting the risk premium on a positive net supply risky asset (i.e., investing heavily in the 'market portfolio'). This non-informational effect was noted previously in DeLong, Shleifer, Summers and Waldmann (1990), discussed below. We rule out this effect by assuming here that the risky security is in zero net supply. 1

trade aggressively both because they underestimate risk and because they overestimate the conditional expected value from their trading strategies. Since the information they exploit is valid, their more aggressive use of it (either long or short on the risky asset) causes them to earn higher expected profits (though lower expected utility). Their expected profits are limited by the fact that if there are too many overconfident traders, or if their confidence is too extreme, their trading pushes price against them excessively. Rational traders then profit by trading in opposition to overconfident traders. If trader types replicate according to the profitability of their strategies, we show that overconfident traders survive in the long run, and can even drive out rational traders completely. Several authors, beginning with De Long, Shleifer, Summers, and Waldmann (1990, 1991), have offered other distinct arguments as to why imperfectly rational traders, including overconfident ones, may survive in the long run.4 De Long et al (1991) examine traders who are overconfident in the sense that they underestimate risk. As a result of underestimating risk, these traders hold more of the risky asset (e.g., the market portfolio). Since the risky asset earns higher expected return, these traders can do well relative to rational traders. Our approach differs from De Long et al (1991) in the following respects. 'For example, noise traders themselves create a risk in the price that discourages rational traders from betting against them. Noise traders bear a disproportionate amount of the risk that they themselves create, and therefore may earn a correspondingly higher risk premium. In this sense they can "create their own space." De Long et al (1990) point out that, as a result, noise traders can earn higher expected profits than rational traders. Palomino (1996) shows that small market size can further enhance the survivorship of noise traders. 2

First, we model overconfidence as overestimation of the precision of private information signals. We therefore derive beliefs endogenously about the payoff of the risky asset.5 Second, we model prices endogenously. We would often expect irrational traders who trade in a certain direction (e.g., buying a hot internet start-up) to push th e unfavorably to themselves. This influence on price tends to reduce the long-run expected profits to irrational trading. For example, on Friday November 13, 1998, in the first hours of trading after the initial public offering of TheGlobe.com, the price quickly leaped from the offer price of $9 to a price of $97, reflecting enthusiasm on the part of individual investors. By the end of the day the price had fallen by about 1/3, so many investors who bought in the aftermarket took heavy losses. Given the possible of adverse effects on price, it is interesting to see whether irrational traders can survive despite having an influence on price. Third, the results of De Long et al (1991) are driven mainly by a noninformational effect, that overconfident individuals who underestimate risk tilt their portfolios heavily toward the 'market' (high risk/high return) security. In our paper, the high profits of the overconfident arise from the overreaction in their assessments of mean, so that these investors exploit their information more aggressively in either a long or short direction.6 In 5Overconfidence in our sense implies underestimation of risk, consistent with their assumption. It also implies incorrect conditional means. Their assumption of a noise component of trades is implicitly consistent with misassessment of conditional means. However, by deriving beliefs endogenously, our model goes further by constraining the relation between errors in mean assessments and underestimation of risk. In our model the misperceptions of both first and second moments are determined endogenously through Bayes rule. 6This effect is reinforced by their underestimation of risk, but would work even if they 3

so doing, they gain profits by exploiting the mispricing created by liquidity/noise traders. In our model overconfidence is profitable even if the risky security is in zero net supply; it is not a matter of investing more heavily in the market portfolio, but of exploiting information more intensely. Kyle and Wang (1997) provide a distinct reason for the survival of overconfident traders based on imperfect competition in securities markets. An informed trader who knows he is trading against an overconfident informed opponent chokes back on his trades, to the benefit of the overconfident trader.7 An informed trader knows that the price execution in the direction indicated by his signal will be less favorable by virtue of the fact that the overconfident informed trader will trade aggressively based on the same signal. Being perceived as overconfident is in effect like being a Stackelberg leader, which generates oligopoly profits. Benos (1999) develops this theme to examine cases of imperfectly correlated signals. Fischer and Verrecchia (forthcoming) examine 'heuristic' traders who, owing to overconfidence or base-rate underweighting, overreact to new signals. In their paper as well, overreaction creates a 'first mover' advantage for heuristic traders owing to imperfect competition. In all three papers, holding constant the behavior of other traders, trading more aggressively reduces expected profits; the only benefit of overconfidence comes from being perceived as such by other informed traders. Furthermore, in all three papers, the effects described are did not underestimate risk. In contrast, underestimation of risk is essential to the result of De Long et al (1991). 7Wang (1997) extends the Kyle and Wang (1997) framework to a dynamic setting to show that overconfident traders can survive in the long run. 4

only important if the mass of informed traders, and especially overconfident traders, is high enough to influence prices significantly. In contrast with the commitment approach of these papers, in our model traders are perfectly competitive. Traders observe the market price and take it as given. Thus, a trader does not limit the size of his position out of fear that an overconfident informed trader will trade intensely in the same direction. The benefit to overconfidence in our model is that overconfident traders are willing to take on more risk, and hence better exploit the mispricing generated by the trades of 'noise' or liquidity traders. Unlike the commitment approach discussed above, in our model this benefit applies even if there is only a very small measure of informed traders. In other words, the profits arise not from the commitment to be aggressive (and the desirable effects of such commitment upon the behavior of other traders), but directly from the aggressiveness of the trading strategy.8 Gervais and Odean (1997) and Daniel, Hirshleifer, and Subrahmanyam (1998a) model traders who learn to be overconfident. In these settings, traders are biased in the way they learn about their own abilities, which creates and sustains overconfidence. In this approach, traders who are lucky enough to become rich also become more confident. Such an effect tends to maintain the importance of overconfidence in a dynamic steady state even if overconfident traders lose money. Our approach differs in that we do not allow a trader's confidence to grow over time; overconfident traders can thrive 8Benos (1999) describes his model (the same applies to the other two papers as well) as follows: "Our result comes from a first mover advantage, not from excessive risk-taking." In our model, the result comes from the combination of misperception of means and aggressive risk-taking, not from a first mover advantage. 5

profitably. Section 2 of the paper describes the basic model with three types of agents (rational, overconfident and liquidity/noise traders) who invest in a risk-free and a risky asset. Section 3 models the long-run survival of overconfidence in an evolutionary process where trader types replicate according to their expected profits. Section 4 describes results, and Section 5 gives concluding remarks. 2 The Static Model Consider a one-period competitive market consisting of two types of securities, a risk-free security with a constant payoff equal to one, and a risky security with a payoff equal to 0, where 0 is a normally distributed random variable with mean ( and variance 4r. There are three types of agents: rational traders, overconfident traders, and liquidity/noise traders. Both rational and overconfident traders receive a signal with respect to the payoff of the risky asset, denoted as s, where s = 0+e, and where e is normally distributed with E(e) = 0 and Var(e) = a2 and is independent of 0. The distribution of 9 is known to both rational and overconfident traders. The rational traders correctly perceive the distribution of e (i.e., Er(E) = 0 and Var(e) = a2 where the subscript r indicates a rational trader). Overconfident traders believe that the that thevariance of the residual error (r true residual error variance (i.e., Var(e) = ad < cr, where the subscript c indicates an overconfident trader). We assume that cr2 > 0, i.e., overconfident traders recongnize that their signal is imperfect. 6

Both rational and overconfident traders choose a portfolio to maximize expected utility of wealth (denoted as wi for i -= r, c) at the end of the period, based on their interpretation of the signal, s. Trader i's utility function is assumed to be exponential, U(wi) = -e-awi, a > 0, where a is the coefficient of absolute risk aversion. Together with normality, this implies a mean-variance expected utility function. The wealth at the end of the time period for each trader is the sum of the initial wealth (denoted as w) and the gain derived from the two types of assets. Since the payoff of the risk-free asset is always one, for trader i (i = r, c), wi = w + Xi(O - p), where Xi is trader i's demand for the risky asset and p is its price. Therefore, for i = r, c, trader i's optimal demand function Xi is the solution to max Ei(wi Is) - Vari(wi s ), s.t. Wi = w + Xi(O - p). Since 0 and s are independent and normally distributed, 22 Ei(0s)=0+ a02 (-o+~) and Vard(9is)= — 22 where a? = a^?2 for i = r and? 2= ac for i - c. This further implies that trader i's demand function is Xi = +?'(~ -+ )-P (1) avi where 2% + 20 tand vi = ra2 +2 Note that '7c > Yrr and vc < vr, hence, JXc > jXr\. In other words, overconfident traders' higher conditional mean and lower conditional variance about the risky asset's payoff result in taking a larger long or short position. 7

The total demand for all liquidity/noise traders is x, where E(x) = 0 and Var(x) = ac. Within the subset of the population consisting of rational and overconfident traders, let A denote the proportion of rational traders, and 1 - A the proportion of overconfident traders. Assuming the supply of risky assets is zero, the market clearing condition for the risky asset is AXr + (1 - A)Xc = -x. (2) Substituting equation (1) into equation (2), the equilibrium price of the risky asset can be solved as G rAVc + (1 - A)Ocvr + avrVcx AVc + (1 - A)Vr where = 8 + ni(O - 0 + e) for i =r, c. Note that E(p) = 0. Furthermore, the expected profit for trader i can be calculated as irq(A) = E(Xi(O - p)) for i = r, c. Specifically, for the rational trader -7rr(A) 2 T [(A- 1)2 (o2 ~-4)2 + a2a ((T2)2(a +?2)] a [Ac r?+ (1 - A)aa + aaf and for the overconfident trader W [ A - 1) (a - ac)2 + a2a 2a?2 (e + a?2)] 7rc(A a [Aa' + (1 i- -. +.a 2 * c n a [Aor92O2 + ( 1- A)tJ2f2 + 2'c0r2]' 8

3 The Dynamic Model To examine the long-run survival of overconfident traders, we embed the static model of Section 2 within an evolutionary process in which traders types replicate differentially over time according to the profitability of their strategies. The natural selection process works as follows. If the expected profit for trader i is greater (less) than trader j, for i, j E {r, c} and j, then in the next time period the proportion of trader i increases (decreases). If the expected profits for both sets of traders are equal, then the proportion of traders remains the same in the following period. Specifically, let the fraction of the population of rational traders follow the following dynamics:9 At+ = At + f/(r (At) - rt(At); At), (3) where f(.) maps from [0,1] -+ [0,1] is a continuous function with the following properties: (2) f(.) = 0 if 7rr(At)- r(A,)= 0, and At E (0, 1), (ii) f() < 0 if 7r.r(At) - 7c(,At) < 0 and At > 0, (iii) f(.) 0 if lim [Trr(At) - rc(At)]: 0, At-,O+ (iv) f() > 0 if i7r(At) - 7rc(At) > 0 and At < 1, (v) f(.) 0 if lim [7r,(At) -7rc(At)] 0,,\t-*lThe above class of dynamics is general enough to encompass standard ones such as replicator dynamics and many other types of selection dynamics used -- --,~ — /,,,. ^afd^A.) ^^)' ' ' - 9 C., ( -- 9It can be shown that 7r,(At) - 'c.(At) = -- ~2 fc(Fl)("-,:2):,2a2a2 (2..2( )1 9

in evolutionary game theory (see Taylor and Jonker (1978), Weibull (1995)). It is similar to that used by Luo (1999). The dynamic equilibrium is defined to be either an interior fixed point (case (i) above) or a corner solution (case (iii) or case (v) above) of the above dynamics. We denote the dynamic equilibrium as A. 4 Results This section relates the long-run proportion of surviving overconfident traders to the underlying parameters of the model, such as the degree of overconfidence, noise volatility, and the volatility of underlying security payoffs. Proposition 1 For all positive parameter values (a, a, a2, C, Q), there is a unique dynamic equilibrium. Regardless of the initial fraction of overconfident traders At, the market always converges to this dynamic equilibrium. The equilibrium has the following properties: (i) If a2ocra2Co2 < o, - cr, then the dynamic equilibrium is the interior fixed point where A = 1- a-. (ii) If a.2aoar o o2 _-, the dynamic equilibrium is the corner point where A = 0 (all overconfident traders). (iii) The dynamic equilibrium cannot be the corner point where A = 1. (iv) The higher is the volatility of the underlying security payoff (a)} the higher is the proportion of overconfident traders in the equilibrium. (v) The more volatile is liquidity/noise trading (the higher is a2), the higher is the proportion of overconfident traders in the equilibrium. 10

(vi) The greater the confidence of the overconfident traders, the lower is the proportion that survive in the equilibrium. Proof. See the Appendix U We now comment on the results in order: (i) If a2c2aac2a < cra - a2, then for all positive parameters, no matter where At starts in the interval (01), it evolves into the interval (ior1), it efixedvolves inoit he interior fixed point where the rational traders and the overconfident traders coexist. Thus, so long as there is some liquidity/noise trading, and overconfidence is not too severe, the overconfident traders will persist in the long run. Intuitively, overconfident traders place greater weight on the signal optimistically, and therefore take a bigger (or more risky) position and better exploit the misvaluation created by liquidity/noise traders than do rational traders. Consequently, overconfident traders survive in the long run. However, if there are too many overconfident traders, the prices would be pushed against themseves excessively and rational traders would gain at the expense of overconfident traders by trading in the opposite direction to the overconfident traders. Hence, the rational traders survive in the long run as well. (ii) If a2ca~aJ >~ a? - _a, for all positive parameter values, rational traders are driven out of the market and only overconfident traders survive in the long run. This inequality shows that overconfident traders could be so favored by high risk aversion (high a), high volatility of liquidity/noise trading (high a?) or high volatility in the underlying asset value (high aj,) that rational traders are driven out of the market completely. 11

(iii) As long as liquidity/noise trading is present, A = 1 cannot be an equilibrium, since overconfident traders can better exploit noise than rational traders and survive in the market. However, we have shown (proof available on request) that if liquidity/noise trading vanishes (a -= 0) or overconfidence is extreme (a2 = 0), then overconfident traders are driven out of the market. (iv) The higher is the volatility coming from the volatility of the underlying security payoff, the larger is the proportion of surviving overconfident traders. As volatility of the underlying security payoff increases, rational traders are not able to infer as clearly that a high price indicates overvaluation. This increases the riskiness for them of a contrarian strategy of selling when price is high and buying when price is low. The perceived risk of this strategy is reduced by observation of the private information signal, but this perceived risk reduction is greater for the overconfident. As a result, higher volatility of underlying security payoff increases the relative expected profitability for the overconfident. Figure 1 plots the increasing relationship between the long-run proportion of surviving overconfident traders and cra. (v) As the volatility of liquidity/noise trading increases, the equilibrium proportion of overconfident traders increases as well. Noise creates misvaluation, which offers greater profit opportunities for overconfident traders than for fully rational ones. Figure 2 plots the increasing relationship between the long-run proportion of surviving overconfident traders and ux. (vi) If overconfident traders are too confident, their trading becomes too aggressive, and the equilibrium proportion of surviving overconfident traders decreases. Figure 3 plots the increasing relationship between the long-run proportion of surviving overconfident traders and the perceived error variance 12

a2(an inverse measure of overconfidence). 5 Conclusion Recent research has proposed several ways in which overconfident traders can persist despite competition from rational traders. This paper offers an additional reason: overconfident traders do better than purely rational traders at exploiting misvaluation caused by liquidity or noise trading. Using a model of a perfectly competitive asset market involving rational traders, overconfident traders and liquidity/noise traders, we examine both the static profitability of overconfident versus rational trading strategies, and the dynamic evolution of the population of traders. Different investor types are assumed to become more prevalent when their strategies are more profitable. In some cases there is an interior equilibrium with both rational and overconfident traders. If the degree of risk aversion, the volatility of liquidity/noise trading or the volatility of the underlying security payoff becomes sufficiently large, rational traders are driven out of the market and only overconfident traders survive. The higher the noise volatility and the higher the volatility of the underlying security payoff, the larger is the proportion of surviving overconfident traders. The more intense is their confidence, the lower is the proportion of surviving overconfident traders. Finally, our main result is that unless the degree of overconfidence is infinite, the long-run steady-state equilibrium always involves overconfident traders surviving as a positive fraction of the population. 13

Appendix Proof of Proposition 1: a2 or2ocr2,2 (i) Solving 7rr(A) = 7rc(A) results in an interior fixed point A = 1 - c If a aar2a < a2 - o2, then for all positive parameter values, A E (0, 1). Furthermore, since for any At < A, 7rr(At) > -7c(At) and for any At > A, 7rr,(A) < 7rc(A), using the dynamics defined in equation (3), no matter where the At starts in the interval (0,1), the market converges to this interior fixed point. (ii) For the corner solution corresponding to A 0= to be an equilibrium, using the definition of the dynamics, it requires lim (7r(At) - 7rr(At)) < 0. This is At —O+ true under the parameter restrictions that a2a~a~aS > a2 cr. Furthermore, since for any At > 0, 7rr(At) < r(At), using the dynamics defined in equation (3), no matter where the At starts in the interval (0,1), the market converges to this equilibrium where A = 0. (iii) For the corner solution A = 1 to be an equilibrium, the dynamics require that lim (7rr(At) - 7rc(At)) > 0. For all positive parameters, At- I + Therefore, the corner solution corresponding to A = 1 cannot be an equilibrium. (iv) At the interior fixed point, = < 0. Hence, (iv) follows. (v) When a2a22Crae < cr- a2 the dynamic equilibrium has the interior fixed point A= 1 -. Since c- 9 < 0, (v) follows. (vi) At the interior fixed point, = - < 0. Hence, (vi) follows. a(~ = (o_) 14

References [1] Benos, Alexandros, 1998, "Aggressiveness and Survival of Overconfident Traders," Journal of Financial Markets, 1(3-4), 353-383. [2] Daniel, Kent, David Hirshleifer and Avanidhar Subrahmanyam, 1998a, "Investor Psychology and Security Market Under- and Overreactions," Journal of Finance, LIII, 1839-1885. [3] Daniel, Kent, David Hirshleifer and Avanidhar Subrahmanyam, 1998b, "Investor Overconfidence, Covariance Risk, and Predictors of Securities Returns," University of Michigan Business School. [4] De Long, J. Bradford, Andrei Shleifer, Lawrence H. Summers, and Robert Waldmann, 1990, "Noise Trader Risk in Financial Markets," Journal of Political Economy, 99 (4), 703-738. [5] De Long, J. Bradford, Andrei Shleifer, Lawrence H. Summers, and Robert Waldmann, 1991, "The Survival of Noise Traders in Financial Markets," Journal of Business, 64 (1), 1-19. [6] Figlewski, Stephen, 1978, "Market 'Efficiency' in a Market with Heterogeneous Information," Journal of Political Economy, 86(4), 581-597. * I [7] Figlewski, Stephen, 1982, "Information Diversity and Market Behavior," Journal of Finance, 37, 87-102. [8] Fischer, Paul E. and Robert E. Verrecchia, forthcoming, "Public Information and Heuristic Trade," Journal of Accounting and Economics. 15

[9] Gervais, Simon and Terrance Odean, November 1998, "Learning to be Overconfident," Graduate School of Management, University of California at Davis. [10] Kyle, Albert and Albert Wang, 1997, "Speculation Duopoly with Agreement to Disagree: Can Overconfidence Survive the Market Test?" Journal of Finance, 52 (5), 2073-2090. [11] Luo, Guo Ying, 1998, "Market Efficiency and Natural Selection in a Commodity Futures Market," Review of Financial Studies, 11 (3), 647 -674. [12] Luo, Guo Ying, 1999, "The Evolution of Money as a Medium of Exchange," Journal of Economic Dynamics and Control, 23, 415-458. [13] Odean, Terrance, 1998, "Volume, Volatility, Price, and Profit when All Traders are Above Average," Journal of Finance, LIII, 1887-1934. [14] Palomino, Frederic, September 1996, "Noise Trading in Small Markets," Journal of Finance, 51 (4), 1537-50. [15] Taylor, P., and Jonker, L., 1978, "Evolutionarily Stable Strategies and Game Dynamics," Mathematical Biosciences, 40, 145-156. [16] Wang, F. Albert, 1997, "Investor Sentiment, Delegated Portfolio Management, and Market Evolution," Graduate School of Business, Columbia University. [17] Weibull, J.W., 1995, Evolutionary Game Theory, MIT Press, Cambridge. 16

Figure 1 The relationship between the surviving overconfident traders and volatility 1-A I 1-2=" ''2 ac - c 0 a 2cr 2 C 2 a^ 2 -,2 c - c '2 C7o Figure 2 The relationship between the surviving overconfident traders and noise 1-1A 1 a2a 2a 2a- 2 c 2 a2 a' -ac 0 a2Crj20'20'2 2 2 a, -a Or 2 ax Figure 3 The relationship between the surviving overconfident traders and overconfidence level 1-A. 1-=.- 2 2 2 /^ C /a - a 0 2 2 2 2 2 C a-r,.a