Asset Price Dynamics When Traders Care About Reputation Amil Dasupta and Andrea Prat London School of Economics and CEPR October 2004; Revised November 2005 Abstract What are the equilibrium features of a dynamic nancial market where traders care about their reputation for ability? We modify a standard sequential tradin model to study a nancial market with career concerns. We show that this market cannot be informationally e cient: there is no equilibrium in which prices convere to the true value, even after an in nite sequence of trades. This ndin, which stands in sharp contrast with the results for standard nancial markets, is due to the fact that our traders face an endoenous incentive to behave in a conformist manner. We show that there exist equilibria where career-concerned aents trade in a conformist manner when prices have risen or fallen sharply. We also show that each asset carries an endoenous reputational bene t or cost, which may lead to systematic mispricin if asset supply is not in nitely elastic. Keywords: Financial Equilibrium; Career Concerns; Information Cascades; Mispricin. An earlier version of this paper was circulated under the title "Reputation and Asset Prices: A Theory of Information Cascades and Systematic Mispricin". We thank Patrick Bolton, James Dow, Satoshi Kowanishi, Stephen Morris, Hyun Shin, and Dimitri Vayanos for helpful discussions, and seminar participants at the CeFiMS, CEPR-Gerzensee meetins, Heidelber, Helsinki, IIES-Stockholm, LSE, Munich, NYU, Norweian School of Economics, Nurember, Pompeu Fabra, Warwick Business School, and Yale for comments. Dasupta thanks the ESPSRC for nancial support via rant GR/S83975/0. E-mail: a.dasupta@lse.ac.uk, a.prat@lse.ac.uk
Introduction The substantial increase in the institutional ownership of corporate equity around the world in recent decades has underscored the importance of studyin the e ects of institutional trade on asset prices. Institutions, and their employees, may be uided by incentives not fully captured by standard models in nance. For example, consider the case of US mutual funds which make up a sini cant proportion of institutional investors in US equity markets. An important body of empirical work hihlihts the fact that mutual funds (e.. Chevalier and Ellison [2]) and their employees (Chevalier and Ellison [3]) both face career concerns: they are interested in enhancin their reputation with their respective principals and sometimes indule in perverse actions (e.. excessive risk takin) in order to achieve this. Given the importance of institutions in equity markets, it is plausible to expect that such behaviour may a ect equilibrium quantities in these markets. What are the equilibrium features of a market in which a lare proportion of traders care about their reputation? While a rowin body of literature examines the e ects of aency con icts on asset pricin, the explicit modelin of reputation in nancial markets is in its infancy. 2 Dasupta and Prat [5] present a two-period micro-founded model of career concerns in nancial markets to examine the e ect of reputation in enhancin tradin volume. However, that analysis is done for a static market: each asset is traded only once. In this paper, in contrast, we study a multi-period sequential trade market in which some traders care about their reputations. We show that the dynamic properties of this market are very di erent from those of standard markets.. Summary of Results We present the most parsimonious model that captures the essence of our aruments. Much of our model is standard. We present a T -period sequential trade market for a sinle (Arrow) asset where all transactions occur via uninformed market makers who are risk neutral and competitive (followin Glosten and Milrom [20] and Kyle [2]) and quote bid and ask prices to re ect the informational content of order ow. In addition there is a lare roup of liquiditydriven noise traders who trade for exoenous reasons that are unrelated to the liquidation On the New York Stock Exchane the percentae of outstandin corporate equity held by institutional investors has increased from 7.2% in 950 to 49.8% in 2002 (NYSE Factbook 2003). For OECD countries as a whole, institutional ownership now accounts for around 30% of corporate equity (Nielsen [25]). Allen [2] presents persuasive aruments for the importance of nancial institutions to asset pricin. 2 For example, Allen and Gorton [4] and Dow and Gorton [9] examine the asset pricin implications of non-reputational aency con icts. Reputational concerns are implicit in the contractual forms assumed in the eneral equilibrium models of Cuoco and Kaniel [4] and Vayanos [38]. 2
value of the asset. Our only innovation is that we introduce a lare roup of reputationally-concerned traders (whom we call fund manaers), who trade on behalf of other (inactive) investors. These traders receive a payo that depends both on the direct pro ts they produce and on the reputation that they earn with their principals. 3 These fund manaers can be of two types (smart or dumb) and receive informative sinals about the asset liquidation value, where the precision depends on their (unknown) type. In each tradin round either a randomly selected fund manaer or a noise trader interact with the market maker. The asset payo is realized at time T and all payments are made. At time T, every fund manaer is evaluated on the basis of all available information, with the exception of the aent s private sinal. This implies that each investor can observe the liquidation of the asset and the portfolio choice of his own aent. 4 This assumption is plausible for relatively sophisticated investors, such as corporate pension plans, investment banks, insurance companies, and hede fund clients. It may instead be an unrealistic requirement for retail mutual fund investors, who typically have limited knowlede of their fund s portfolio composition. 5 We present the followin results.. We bein with an impossibility result. We show that, in this market of career-concerned traders, prices never convere to true liquidation value, even after an in nite sequence of trades. If fund manaers trade accordin to their private sinal, the price evolves to incorporate such private information. Over time, the price should convere to the true liquidation value. However, as the uncertainty over the liquidation value is resolved, two thins happen. First, the fund manaers have less opportunity to make tradin pro ts because the price is close to the liquidation value. The expected pro t for a fund manaer who trades accordin to his sinal is always positive, but it tends to zero as the price becomes more precise. Second, takin a contrarian position (e.. sellin when the price has been oin up) starts to carry a reputational cost: with hih probability, the trade will turn out to be incorrect and the fund manaer will look dumb in the eyes of (rational) principals. Because of the combination of these two e ects, when the price becomes su ciently precise fund manaers bein to behave in a conformist way: their trade stops re ectin their private information. From then on, 3 The principals may be line manaers at mutual fund companies with oversiht over the particular fund manaer s activities, or, directly, the investors who have placed their funds with the company. 4 For all our core results, it is irrelevant whether the investor observes the portfolio choice of other fund manaer besides his. 5 See Prat [28] for a discussion of the role of portfolio disclosure in deleated portfolio manaement with career concerns. 3
there is no information areation whatsoever and the price stays constant. 2. We then report a positive result. We prove that there exist an equilibrium where fund manaers trade sincerely in the beinnin, but, as soon as the price hits an upper threshold or a lower threshold, they bein poolin. We characterize the conditions that uarantee the existence of such truncated sincere equilibria, and construct examples to illustrate their properties. We also show that as career concerns become increasinly important to fund manaers, these truncated sincere equilibria are the most informative equilibria that can exist. 3. We show that, at any iven price, there may be a di erence between the expected total value of the asset for a reular trader and for a trader with career concerns. The di erence arises because manaers ain in reputation from buyin underpriced assets and sellin overpriced ones. We refer to this di erence as the reputational bene t or cost carried by the asset at that price. If the asset supply is not in nitely elastic, the reputational bene t or cost will a ect the equilibrium price. The transaction price can di er from this period s public expectation of liquidation value. Thus, the existence of such a reputational concerns can lead to the systematic mispricin of assets in nancial markets. We characterize the nature of such systematic mispricin and show that current mispricin depends on past trades and prices. 4. Finally, we examine a number of natural extensions. The baseline model was presented with arbitrary preferences over individual reputation and with binary sinals and asset liquidation values. In these extensions, we provide micro-foundations for the reputational preferences of fund manaers, and eneralize the model to richer payo and sinal spaces to show that our main results are robust to these chanes. Further, in our baseline model we assumed that fund manaers cared about their absolute reputations and were unaware of their type. We extend the model to demonstrate that as lon as self-knowlede is not very accurate, our core results remain unchaned. We also allow for rewards based on relative reputation and show that sincere tradin cannot be sustained in equilibrium. Our results enerate testable implications. As summarized above, our model suests that the incentives of career-concerned fund manaers can lead to the systematic mispricin of assets in nancial markets when asset supply is not perfectly elastic, persistently bouht (sold) assets will develop a reputational premium (discount), and trade above (below) fair prices. This has implications for lon-term returns: Assets that have been persistently bouht (sold) by fund manaers are likely to experience neative (positive) corrections when uncertainty is resolved, thus achievin low (hih) lon-term returns. 4
In empirical work motivated by these results, Dasupta, Prat, and Verardo [6] use data from the SEC 3-F lins of US portfolio manaers from 983 to 2004 to examine net trades between career-concerned institutional traders and reular investors. They nd that the persistence of institutional tradin has stron power in predictin the cross-section of stock returns at lon horizons. Stocks that have been persistently bouht (sold) by institutional investors over three or more quarters sini cantly underperform (overperform) the market. A stratey of buyin stocks that institutions sell for ve or more consecutive quarters and shortin ones that they buy for identical periods yields sini cant excess returns. Usin returns from such a portfolio, Dasupta, Prat, and Verardo estimate intercepts from timeseries reressions usin the CAPM model, the Fama-French three-factor model, and the fourfactor model that includes the Carhart momentum factor. Excess returns (alphas) rane from around 5% to % per year dependin on the chosen pricin model and the holdin period. Evidence from this paper also indicates that a vast majority of institutional traders exhibit conformist tradin patterns. Further supportin evidence can be found in the work of Dennis and Strickland [8]. These authors have examined the relationship between price reactions and institutional ownership for a cross section of stocks on sinle tradin days when the market return is hih (in absolute value). They nd that the manitude of a rm s return on such days is increasin in the extent of institutional ownership. In particular, they nd that the percentae of ownership by mutual funds, the class of institutions for which career concerns are best documented, to be specially important in drivin their results. They interpret this as evidence of positive feedback herdin by mutual funds. Our theoretical results provide a clean parallel to these empirical ndins. We show theoretically that fund manaers trade in a conformist manner precisely when prices have risen or fallen sharply, creatin a reputational value to buyin or sellin respectively. 6.2 Related Literature This paper brins toether two in uential strands of the literature. The rst strand concerns the theory of dynamic nancial markets with asymmetrically informed traders (Glosten and Milrom [20] and Kyle [2]). The second strand focusses on the analysis of career concerns in sequential investment decision-makin (Scharfstein and Stein [30]). Models in the rst strand consider a full- eded nancial market with endoenously determined prices but do not allow traders to have career concerns. Models in the second strand do the exact opposite: they analyze the role of reputational concerns in a partial equilibrium settin, where prices 6 See Sias [34] for a recent survey and reconciliation of the rowin literature on momentum tradin and herdin by institutions. 5
are exoenously xed. In the rst strand, Glosten and Milrom [20] have shown that in dynamic nancial markets the price must tend to the true liquidation value in the lon term. More recently, Avery and Zemsky [8] have shown that statistical information cascades à là Banerjee [9] and Bikhchandani, Hirshleifer, and Welch [0] are impossible in such a market. 7 After every investment decision, the price adjusts to re ect the expected value of the asset based on information revealed by past trades. Thus, traders with private information stand to make a pro t by tradin accordin to their sinals. But by doin so, they release additional private information into the public domain. In the lon run, the market achieves informational e ciency. Our work shows that the addition of career concerns chanes thins dramatically. The presence of a reputational motive can make the market informationally ine cient and can enerate mispricin. 8 Other authors (Lee [22] and Chari and Kehoe []) have arued that information cascades can occur when prices are endoenous. 9 However, their aruments hine on a market breakdown: trade stops altoether. Instead, in our model cascades occur in a nancial market with trade. In the second strand, Scharfstein and Stein [30] have shown that manaers who care about their reputation for ability may choose to inore relevant private information and instead mimic past investment decisions of other manaers. 0 This is because a manaer who possesses contrarian information (for instance he observes a neative sinal for an asset that has experienced price rowth) jeopardizes his reputation if he decides to trade accordin 7 An information cascade is an equilibrium event in which information ets trapped, and aents actions no loner reveal any of their valuable private information. 8 Avery and Zemsky [8] also show that if uncertainty is multidimensional (when "event uncertainty" compounds the usual value uncertainty) a weaker form of conformism, which they term herd behavior, may occur in nancial markets. However, in their model cascades are absent and prices always convere to the true liquidation value (see Avery and Zemsky Proposition 2). In contrast, we show that cascades occur even in the absence of event uncertainty, due to the presence of career concerns. There is almost universal areement in the literature on the meanin of a cascade, which is the de nition we have used (see footnote 7). However, there is little areement on the de nition of the term herds (for example, substantively di erent de nitions are used by Avery and Zemsky [8], Smith and Sorensen [35], and Chari and Kehoe []). In the interest of clarity, throuhout this paper we shall restrict attention to cascades only. Recently, Park and Sabourian [27] have explored eneralizations of the necessary conditions for herds in Avery and Zemsky s model. As in Avery and Zemsky, however, cascades cannot arise in their model. 9 In Lee s [22] model the existence of a transaction cost to tradin can prevent traders with relatively inaccurate sinals from tradin, thus trappin private information in an illiquid market. In Chari and Kehoe [], traders have the option of exitin the market (by makin an outside investment) and may in equilibrium nd it optimal to exit before further information arrives, thus, aain, trappin private information. 0 Other more recent papers in this strand include, for example, Avery and Chevalier [7], Prenderast and Stole [29], and Trueman [37]. 6
to his sinal. Scharfstein and Stein s analysis is carried out in partial equilibrium: Prices play no informational role in such an analysis. We show that conformist behavior arises in market equilibrium due to the reputational concerns of nancial traders. Such conformism endoenously prevents prices from playin their full role in informational revelation. By studyin career concerns in nancial equilibrium, we are able to study the e ects of micro-founded reputation-driven conformism on nancial market quantities (prices, informational e ciency, trades, and pro ts), which leads to relevant predictions on observable market variables, as discussed above. The rest of the paper is oranized as follows. In the next section we present the model. Section 3 discusses the impossibility of full information areation. Section 4 characterizes some of the equilibria of this ame, and proves that information cascades can arise in equilibrium. Section 5 studies the dynamics of the reputational bene t or cost of the asset and shows that, if the market makin sector has market power, the price can incorporate a reputational premium and systematic mispricin can arise. Extensions are examined in section 6. Section 7 concludes. 2 The Model The economy lasts T discrete periods: ; 2; :::T. Trade can occur in periods ; 2:::T. The market trades an Arrow security, which has equiprobable liquidation value v = 0 or ; which is revealed at time T. In practice, the asset could be a bond with maturity date T with a serious possibility of default. It could also be the common stock of a company which is expected to make an announcement of reat importance (earnins, merer, etc.) at time T : all traders know that the announcement will occur but they may have di erent information on the content of the announcement. There are a lare number of fund manaers and noise traders. At each period t 2 f; 2; :::; T either a fund manaer or a noise trader enters the market with probabilities and 2 (0; ) respectively. The traders interact with a risk-neutral competitive market maker, and can issue market orders (a t ) to buy (a t = ) one unit or sell (a t = 0) one unit of the asset. The market maker sets ask (p a t ) and bid (p b t) prices equal to expected value of v conditional on order history. Denote the history of observed orders at the beinnin of period t (not includin the order at time t) by h t. Let p t = E(vjh t ), p a t = E(vjh t ;buy), p b t = E(vjh t ;sell). A mixed stratey for the manaer at time t is denoted by t (h t ; s t ), the The partial equilibrium reputational model that is closest to ours is Ottaviani and Sorensen [26]. The connection will be discussed in detail in Section 3. 7
probability that he chooses a t = iven history h t and sinal s t. The fund manaer can be of two types: 2 fb; with Pr ( = ) =. The type is independent of v. If at time t a fund manaer appears, he receives a sinal s t 2 f0; with distribution Pr (s t = vjv; ) = ; where 2 b < : Fund manaers do not know their type. Noise traders buy or sell a unit with equal probability independent of v. The returns obtained by the trader at time t is denoted t and is de ned by: ( t a t ; p a t ; p b v p a t if a t = t; v = p b t v if a t = 0 If a fund manaer traded at time t, his actions are observed at time T. Principals (e.., line manaers in the fund manaement rm) form a posterior belief about the manaer s type based upon the manaers actions, which we de ne to be ^ t (a t ; h t ; v) = Pr ( t = ja t ; h t ; v) Suppose the fund manaer at time t receives utility u(a t ; p a t ; p b t; v; h t ) = ( t ) + ( )r (^ t ) ; where 2 (0; ) is a parameter, and the functions and r measure the direct payo and reputational payo and are increasin and piecewise continuous in the relevant aruments. 2 We implicitly assume that the fund manaer in t observes the whole history of trades and prices up to his time, h t, (alon with the current price and his private sinal) and that principals observe h T (and the liquidation value v). The assumption that fund manaers and principals are able to observe the history of trades is not indispensable. Our main results would hold even if the fund manaer in t could only observe the price p t in t and his private sinal, and the principals could only observe the trade of their fund manaer and the liquidation value (indeed, in the equilibria we characterize, strateies only depend on these variables). We can show that in this settin, in contrast to well-known prior results, that the market cannot be fully informationally e cient even after an ini nite sequence of trades. 2 In section 6, we provide microfoundations for such a utility function. 8
3 The Impossibility of Full Revelation We rst present an example and then discuss the eneral result. 3. An Example Let and r be linear. The manaer maximizes t + ( ) ^ t. A sincere equilibrium is one in which fund manaers play accordin to their sinals: a t = s t for all t at which a fund manaer is active. If =, there is a sincere equilibrium (Glosten and Milrom [20], Avery and Zemsky [8]). Suppose instead that <. Proposition There is no sincere equilibrium. The arument is intuitive and we present it in detail here. Suppose there is a sincere equilibrium. Suppose that at t the price is p t and the manaer plays a t = s t. Let ^v t = Pr (v = js t = ; h t ) = Pr (s t = jv = ) p t = Pr (s t = jh t ) ^v 0 t = Pr (v = js t = 0; h t ) = Pr (s t = 0jv = ) p t = Pr (s t = 0jh t ) p t + ( p t ) ( ) p t ( p t ) + p t ( ) p t where = + ( ) b. The bid-ask prices are p a t = b tp t + b t ^v t p b t = s tp t + ( s t) ^v t 0 where b t = Pr(noise traderjbuy order, h t ), is the probability assined by the market maker that he faces a noise trader upon receivin a buy order and observin the history of trades. Similarly, s t=pr(noise traderjsell order, h t ). Straihtforward calculations show that b t = s t = 2 2 + ( )[p t + ( p t )( )] 2 2 + ( )[p t( ) + ( p t )] Suppose the current price is p t and the manaer in t observes s t = 0. If he buys, his expected tradin pro t is ^v 0 t p a t, while if he sells it is p b t ^v 0 t. Thus the di erence between the expected pro t of buyin and sellin is = 2^v t 0 p a t p b t: 9
The reputational payo s in a sincere equilibrium are: De ne ^ (a t = ; v = ) = ^ (a t = 0; v = ) = ^ (a t = ; v = 0) = ^ (a t = 0; v = 0) = ^ = : The expected reputational bene t of choosin a t = instead of a t = 0 when s t = 0 is r = Pr (v = js t = 0; h t ) (^ (a t = ; v = ) ^ (a t = 0; v = )) + Pr (v = 0js t = 0; h t ) (^ (a t = ; v = 0) ^ (a t = 0; v = 0)) = ^v 0 t + ^v t 0 = 2^v t 0 ^ It is a best response to play a t = 0 instead of a t = when s t = 0 when + ( ) r < 0: Let the price rise to one. Notice that as p t!, ^v 0 t! and ^vt!. That is, reardless of whether the manaer has received the hih or the low sinal, the accumulated information in prices convinces him that the expected liquidation value is. Also notice that b t and s t are bounded, and thus as p t!, it must also be true that p a t! and p b t!. Now, takin limits we have lim = lim p t! p 2^vt 0 p a t p b t = 2 = 0 t! lim r = lim p t! p 2^vt 0 ^ = (2 )^ > 0 t! Hence, for p t hih enouh it is a best response for a fund manaer with s t = 0 to play a t =. Thus, there can be no sincere equilibrium. This example sheds liht on why this market may fail to areate information e ciently. But provin that there is no sincere equilibrium is not su cient. There may be informative equilibria that are not fully sincere. To show the inevitability of information cascades, we need to arue that there exists no equilibrium that leads to fully revealin prices even after an in nite sequence of trades by informed fund manaers. This is done in the followin section. 0
3.2 The General Result Let us revert to the eneral problem. The expected utlitity of the t-period fund manaer is E(u(a t ; p a t ; p b t; v; h t )js t ) = X Pr (vjh t ; s t ) a t ; p a t ; p b t; v + ( )r (^ (a t ; v; h t )) De ne v=0; E(u(p a t ; p b t; v; h t )js t ) = E(u(; p a t ; p b t; v; h t )js t ) E(u(0; p a t ; p b t; v; h t )js t ) = X Pr (vjh t ; s t ) p a t ; p b t; v + ( )r (^ (v; h t )) v=0; where and p a t ; p b t; v = (v p a t ) (p b t v) r (^ (v; h t )) = r(^ (; v; h t )) r(^ (0; v; h t )) We restrict attention to non-perverse equilibria. We say that an equilibrium is perverse if for some period t and some history h t which occurs with positive probability on the equilibrium path, the fund manaer at t is more likely to buy if he has a neative sinal than if he has a positive sinal ( t (h t ; s t = 0) > t (h t ; s t = )). Thus, we are not excludin perverse behavior o the equilibrium path. Note that perverse equilibria are extremely implausible in a nancial context because perverse behavior after a history h t implies that the bid-ask spread is strictly neative conditional on h t : We are now ready to state our main result. There exists no non-perverse equilibrium in which the price p t converes to the true liquidation value v as t!. In particular, there exists an upper and a lower threshold, such that if the price crosses one of these bounds, trade is entirely devoid of information for the rest of the ame: Proposition 2 In any non-perverse equilibrium there exists (p; p) 2 (0; ) 2 such that if p t > p or p t < p then the actions of fund manaers from period t onwards cannot provide information about their private sinals. The proposition is proved in several steps. In any putative informative non-perverse equilibrium, we rst show that as the price approaches either 0 or the expected tradin pro t oes to zero. Second, we analyze the reputational incentives in such a putative equilibrium. As the price oes to one, the fund manaer faces a positive and non-in nitesimal expected reputational bene t if he chooses to buy rather than to sell. Conversely, when the price oes to zero, he faces a positive and non-in nitesimal bene t if he sells rather than buyin. Puttin
toether the pro t motive and the reputational motive, we conclude that if the price is hih enouh or low enouh the fund manaer will inore his private information. From then on, the market is stuck in an information cascade. No additional private information is revealed. But this means that we could not have been in an informative non-perverse equilibrium. The proof is involved and is presented in detail in the appendix. Our result bears a connection to Ottaviani and Sorensen [26], who provide a eneral analysis of reputational cheap talk and show that full information transmission is enerically impossible. Of particular interest to the present paper is their Proposition 9, where they consider a sequence of experts providin reports on a common state of the world and they show that informational herdin must occur. Our result is di erent because: (a) we show the necessity of informational cascades (while Ottaviani and Sorensen prove that herdin must occur but they cannot exclude that the true value is revealed in the limit); (b) our experts have a pro t motive as well as a reputational motive; and (c) most importantly, our model is embedded in a nancial market. 4 Price Dynamics: Information Cascades in Equilibrium We have just demonstrated a neative result: non-perverse equilibria never result in the lonrun converence of prices to true value. A natural question remains: What equilibria can arise in this sequential trade market with reputationally sensitive traders? We deal with this question here. To simplify our construction of equilibria, we assume that the functions and r are linear, thouh the spirit of our aruments extend to more eneral functions. In all other respects we use the eneral speci cation introduced for the main result. We de ne a truncated sincere equilibrium as follows. There exists p 2 0; 2 such that:. A manaer with s t = buys if p t p and sells if p t < p; 2. A manaer with s t = 0 buys if p t > p and sells if p t p. Notice that a crucial property of the truncated sincere equilibrium is that at hih and low prices fund manaer trade without reard to their private information. Havin thus earlier ruled out equilibria where prices convere to true value after even an in nite sequence of trades, we shall now explicitly demonstrate that conformist behaviour can arise in equilibrium, ivin rise to an information cascade. We can prove: Proposition 3 Given (; b ; ; ), there exists a > 0 such that for all < a truncated sincere equilibrium exists where p is the unique p t 2 (0; ) that solves (p t ) + ( ) r (p t ) = 0; 2
where (p t ) and r (p t ) are de ned under sincere strateies. In addition, for small enouh, there exists no non-perverse equilibrium in which tradin reveals information when p t =2 [p; p]. The formal proof is involved and is presented in the appendix. We note here only that the proof is constructed under the followin natural o -equilibrium beliefs: if a fund manaer trades in a contrarian way in a reion of p t where equilibrium strateies dictate conformism, he is assumed to have traded sincerely. Here, we construct an example based on proposition 3 which provides some intuition behind the proof of the result. 3 Suppose that = 0:25, = 0:5, = 0:4, = and b = 0:5. The expected total payo di erence between buyin and sellin (u ) depends on p t and on the equilibrium strateies followed, t and t 0. Usin the formulas presented above, it is not hard to see that the expected payo di erence between buyin and sellin in the sincere part of the equilibrium, u (p t ; ; 0); crosses 0 exactly once in p t 2 [0; ] at p ' 0:44. In order to compute a truncated sincere equilibrium, we must also compute u (p t ; 0; 0) and u (p t ; ; ), that is the payo di erences in the lower and upper conformism reions. Fiure shows, from the point of view of a manaer who observes s t =, the expected payo di erences for the relevant reions, that is u (p t ; 0; 0) in the conformist-sell reion p t 2 [0; 0:44), u (p t ; ; 0) in the sincere reion (thick line) p t 2 [0:44; 0:856], and u (p t ; ; ) in the conformist-buy reion p t 2 (0:856; ]. The key point is that the total bene t is positive if and only if p p, indicatin that the best response of a manaer with s t = is indeed to buy if and only if p p. A few remarks are in order.. The raph displays a small positive jump at p ' 0:44. This represents the switch in payo di erences between the conformist part of the equilibrium and the sincere part. The jump derives from the combination of chanes both in and in r. In eneral, the excess pro ts from buyin vs sellin will be hiher in the conformist section of the equilibrium than in the sincere part. This is because separation worsens the bid-ask spread (which is zero in the conformist part). Thus, in makin the transition from conformist play at p < 0:44 to sincere play at p 0:44 the fund manaer faces a neative impact on pro ts. On the other hand, the reputational di erence between buyin and sellin will be lower under conformist strateies at p ' 0:44 than under sincere strateies. This is because 3 If is hih, the most informative equilibrium has a more complex structure. There exist reions in which a contrarian fund manaer plays a mixed stratey. However, there are still two price thresholds such that no information is revealed outside of them. 3
0.375 0.25 0.25 0 0 0.25 0.5 0.75 0.25 p 0.25 Fiure : Excess payo to buyin in a truncated sincere equilibrium (s t = ) separation enhances the reputational bene t of ex post correct trades. Thus, in makin the transition from conformist play at p < 0:44 to sincere play at p 0:44 the fund manaer faces a positive impact on reputational rewards. The small property hihlihted in Proposition 3 uarantees that the reputational ains will overwhelm the loss in pro ts, thus uaranteein an equilibrium switch from conformist to sincere play at p ' 0:44 when career concerns are su ciently important. 2. Note that this example illustrates that in practice, does not need to be particularly small for a truncated sincere equilibrium to exist and for conformism to arise in equilibrium Proposition 3 identi ed su cient conditions only. 3. When p = p the total bene t line displays a neative jump due to the combination of neative e ects on pro t and reputation. However, it is easy to check that the both functions must still be positive. If p is hih, a manaer with s t = has a two-fold reason to buy rather than sell. 5 Reputational Premium and Systematic Mispricin From the viewpoint of fund manaers the expected total value of the asset can di er from expected liquidation value. The di erence captures the expected reputational bene t or cost 4
that the fund manaer incurs if he buys or sells the asset. Intuitively, an asset that has seen recent price rises (falls) as a result of persistent buyin (sellin) is likely to be underpriced (overpriced): buyin (sellin) such an asset may enhance the manaer s reputation. w t s t w t s t Consider a fund manaer at time t who observes sinal s t and forms expectation ^v t s t. Let be the price at which the manaer is indi erent between buyin and sellin. We refer to as the fund manaer s expected total value of the asset. It is the solution of ^v t s t w t s t + ( ) ^v t st ^ t (a t = ; v = ) + ^v t s t ^ t (a t = ; v = 0) = w t s t ^v t s t + ( ) ^v t st ^ t (a t = 0; v = ) + ^v t s t ^ t (a t = 0; v = 0) If the fund manaer has no career concerns ( = ), we simply have that ws t t = ^v s t t : the expected total value is just the expected liquidation value. However, if < there may be a wede between the two values, which we indicate with t s t = w t s t ^v t s t = 2 ^v t s t ^ t (a t = ; v = ) ^ t (a t = 0; v = ) + ^v t s t ^ t (a t = ; v = 0) ^ t (a t = 0; v = 0) The value t s t depends on the manaer s private sinal. It is interestin to look at the averae value of t s t for all manaers. Recall that the ex ante probability that the manaer receives sinal s t = is p t + ( ) ( p t ). Then, let! : t = (p t + ( ) ( p t )) t + ( p t ( ) ( p t )) t 0 = p t ^ t (a t = ; v = ) ^ t (a t = 0; v = )! + 2 ( p t ) ^ t (a t = ; v = 0) ^ t (a t = 0; v = 0) We refer to t as the reputational bene t or cost of the asset for the manaer in t. We can then see that in a truncated sincere equilibrium the reputational payo is 8 >< 2 p t + ( p t ) if p 2 [0; p] t = 2 (2p t ) if p 2 [p; p] >: p t + ( p t ) if p 2 [ p; ] 2 It is then easy to show that: Proposition 4 In a truncated sincere equilibrium, the reputational bene t is positive if and only if p t 2. It is strictly increasin in p t except possibly at the truncation points. There is a systematic di erence between the valuation of the asset for traders with career concerns and traders without. The di erence is increasin in the price, except possibly at a 5
0.75 0.5 0.25 0 0 0.25 0.5 0.75 0.25 p 0.5 0.75 Fiure 2: Reputational bene t in a truncated sincere equilibrium countable number of points. This point is well illustrated by plottin the above function for the example computed in the previous section. This is shown in Fiure 2. So far, this di erence in valuation has not a ected the equilibrium price. This is because, by utilizin the Glosten-Milrom set-up, we have implicitly assumed that the price is always equal to the valuation of the side of the market that has no career concerns. Is this assumption reasonable? In practice, we should expect a ood proportion of traders (if not the majority) to have career concerns. If those traders s valuations of the asset are systematically di erent from the objective expected value of the asset, we should observe systematic mispricin. Ideally, we should use a formal model of the non-career concerned side of the market that takes into account credit constraints and enerates an imperfectly elastic demand function. However, this di cult task is clearly outside the scope of this paper. Instead, we limit ourselves to studyin what is perhaps the polar opposite of the Glosten-Milrom case: we assume that in every period there is only one trader without career concerns who can sell the asset. While this is a stark and very unrealistic way of capturin the idea that the reputational bene t translates into a price premium, we believe that the underlyin intuition is valid for any microstructure model where the side of the market without career concerns does not have a demand that is in nitely more elastic than the side with career concerns. Speci cally, we assume that in every period t the fund manaer faces one short-lived 6
monopolistic trader who has an asset to sell and a lare number of buyers. The opposite case (many sellers, one buyer) would be similar. The monopolistic seller operates only in period t. If he does not sell the asset at t, he will keep it until T when he will receive the liquidation value v. The monopolistic seller does not know the liquidation value, and infers it from the market by observin past order ow and also from the current period order ow. We face a further modelin choice. If there are noise traders who submit market orders, a monopolistic seller can set an in nitely hih price and make in nite pro ts. We could assume that noise traders submit limit orders (with stochastic limits), but we choose to simplify the analysis by assumin that there are no noise traders. The results can be taken as the limit of a model in which there is a vanishin probability of a noise trader who submits a limit order. We assume that the price chared by the seller cannot o above. If the price is reater than, new assets will be issued. A characterization of the equilibrium set is di cult. The monopolistic seller faces an informed buyer (the type of the buyer is s t ) and must choose whether to exclude the low type (s t = 0). This, in turn, re ects on investors beliefs and makes the analysis extremely complex. We o er a partial characterization of equilibrium, restricted to the set of prices for which play is sincere. To do this, we de ne the averae price P t as the expected price at which a transaction occurs at time t, that is P t = Pr (buyjp t ) p a t + Pr (selljp t ) p b t: Reinterpretin p t as the expectation of the liquidation value v of the asset conditional on publicly available information 4, we can prove the followin: Proposition 5 Suppose that in equilibrium, at a certain p t, the fund manaer trades sincerely. Then, if p t > 2 there is overpricin: Pt > p t. Proof. If the equilibrium is sincere, the expected bene t of a manaer with s t = of buyin rather than sellin is (p) + ( ) r (p) where (p) = 2^v (p) p a (p) p b (p) ; r (p) = (2^v (p) ) 4 The last period transaction price will no loner be identical to such expectations with a monopolistic market maker. ; 7
The bid price is set competitively at p b (p) = ^v 0 (p). The monopolistic seller sets p a (p) so that the manaer s expected bene t is zero: (2^v (p) p a (p) ^v 0 (p)) + ( ) (2^v (p) ) If p t > 2, the second addend is strictly positive. Hence, we must have p a (p) + ^v 0 (p) > 2^v (p) : But, because ^v (p) > ^v 0 (p), we have p a (p) > ^v (p). Hence, = 0: P t = Pr (s t = ) p a t + Pr (s t = 0) ^v 0 (p) > Pr (s t = ) ^v (p) + Pr (s t = 0) ^v 0 (p) = p In the appendix, we provide an example to illustrate overpricin. In what follows, we consider variations on the baseline model and the robustness of our main result. 6 Extensions 6. A More General Set-Up The baseline model was presented with binary states and sinals. Here, we eneralize our analysis to include many states and sinals. Let V = fv ; v 2 ; :::; v N be a discrete space of states and [s min ; s max ] be a continuous space of sinals. Denote v by v min and v N by v max. Let k (v) be the prior probability mass function of v (which is independent of ) and assume that k (v) > 0 for all v. A fund manaers of type receives a sinal distributed accordin to f (sjv), with the followin properties: A Full support: f (sjv) > 0 for all s and v. A2 Monotone Likelihood Ratio Property (MLRP): For every pair fs 00 ; s 0 and fv 00 ; v 0, such that s 00 > s 0 and v 00 > v 0, f (s 00 jv 00 ) f (s 0 jv 00 ) > f (s 00 jv 0 ) f (s 0 jv 0 ) : A3 Garblin: Let f (s) P v f (sjv) k (v) for every s. De ne f b (sjv) = f (sjv) + ( ) f (s) ; where 2 (0; ) is a parameter that captures the informative of a bad manaer s sinal compared to a ood manaer s. 8
The rst assumption (A) is crucial. It implies that the sinal is never fully informative: for all s and v: Pr (vjs) <. If a manaer knows he has the truth, he would follow his sinal even if all his predecessors had traded in the opposite direction. Since the sinal space is no loner binary, it is now important to allow for the possibility that the manaer does not trade. Let a t = 2 denote the decision not to trade. First, note that if there are no career concerns ( = ) there exists a fully informative equilibrium (see Avery and Zemsky [8]). When instead career concerns are present, we shall see that full information revelation is impossible. As before, we focus on non-perverse equilibria. Let (a t jh t ; s t ) denote the probability that a manaer who observes s t after history h t selects action a t. We say that an equilibrium is non-perverse if (jh t ; s t ) and (0jh t ; s t ) are respectively non-decreasin and non-increasin in s t for all h t. Proposition 6 There exists no non-perverse equilibrium in which lim t! p t = v for all v. The proof is in the appendix. Here we provide some intuition for the result. Given the MLRP, it is easy to see that the manaer would be willin to randomize between actions at a maximum of two sinals. Call these s H and s L, so that any non-perverse equilibrium must be characterized the the followin strateies: buy if s > s H, sell if s < s L, and do not trade if s 2 (s L ; s H ). If the equilibrium is to reveal information, both s H and s L cannot be at the boundary. Suppose s L > s min. In what follows, if s H = s max ; simply replace s H by s L, and substitute buyin with not tradin. Then, a manaer who buys will reveal that his sinal s s H and the manaer who sells will reveal that his sinal s s L. In a non-perverse equilibrium, as p t! v min, from the manaer s perspective it becomes inevitable that the ex post evaluation will be carried out in the state v = v min. But at such a state, revealin oneself to have received a sinal in the rane [s min ; s L ] is strictly better for one s reputation than revealin oneself to have received a sinal outside that rane, due to the strict MLRP. Since pro ts are irrelevant in the limit as prices convere to true value, sellin becomes optimal reardless of sinal, and the manaer does not follow the proposed equilibrium strateies. 6.2 Self Knowlede The baseline analysis was carried out under the assumption that the manaer did not know his type. What happens if the fund manaers have some self-knowlede? We now allow the fund manaer to receive two sinals: the now familiar s t and a new sinal z t, with Pr (z t = j) =. 9
If they know their types perfectly (! ), one can show that information cascades will not occur. A hih-type manaer will trade sincerely even if most people before him have made trades of the opposite sin. However, if we assume that fund manaers only have some limited information about their types (! 2 ), we still have a failure in information areation: Proposition 7 If self-knowlede is not too accurate, there exists no equilibrium in which a fund manaer with z t = plays sincerely. 5 As usual, the proof is iven in the appendix. However, the intuition is simple. As lon as the received sinal about his type leaves some residual uncertainty, as public information about the state becomes su ciently precise, a manaer with a ood type-sinal and a contrarian value-sinal becomes convinced that he has simply received two incorrect sinals. Thus, just as in the baseline case, it becomes optimal for him to inore his sinals and behave in a conformist manner. 6.3 Relative Reputation Suppose that the reputational component of the fund manaer s payo depends on her relative reputation. The payo is now ( t ) + ( )r t (^ ; :::; ^ T ) : We assume that r is still continuous and di erentiable in its components and that, for fund manaer t, @r t @^ t > 0 @r t @^ 0 for 6= t: This formulation encompasses the standard benchmarkin settin in which the reputational payo is an increasin (and perhaps convex) function of the di erence between the reputation of a particular manaer and the averae reputation of all manaers:! r t (^ ; :::; ^ T ) = R ^ t P T = ^ T Proposition 8 There is no sincere equilibrium. 5 We could actually show the stroner result (analoous to Proposition 2) that there exists no informative equilibrium. : 20
Proof. Suppose a sincere equilibrium exists and consider the fund manaer in the last period, T. As this aent s action does not a ect the reputations of all other aents, the analysis is identical to the case without relative performance. Thus, this aent will not trade sincerely for extreme prices. One may think that bein evaluated on a relative basis will encourae contrarian behavior and lead to perfect information areation. This intuition comes from small numbers of manaers, while ours is an asymptotic result. If there are incentives to play in a contrarian manner, some manaers may indeed play in a contrarian manner early on in the ame. As for the pro t motive, the incentive for contrarian play will become smaller as time passes and information becomes more precise. In the lon term, the intuition of our baseline result is still valid. 6.4 Micro-founded Career Concerns The analysis to date has assumed that manaers attach positive value to bein able to impress their principals. While this assumption is consistent with empirical evidence as discussed in the introduction, we have not provided theoretical foundations for manaerial career concerns. There are many ways to do so. We present a particularly simple example. 6 There are two periods; i = ; 2;after which the world ends. In each period, an asset with liquidation value v i 2 f0; is traded. At the end of period i, v i is revealed to all. The payo s v and v 2 are iid. Within each period, there is a lon sequence of T tradin rounds. To ive a concrete example, there could be a bond in each period: one issued at 0 with maturity date T, one issued at T with maturity date 2T. The default probability of a bond is independently distributed. Alternatively, it could be the stock of the company that makes an earnins announcement at T and another one at 2T (however, one would need to ensure that the sinal a fund manaer receives in the rst period provides no information on the second period s announcement). Suppose there are N (<< T ) identical fund manaement companies, which exist in N identical but non-overlappin reions each bein the sole employer of fund manaers in their reion. Each such fund manaement company employs exactly one fund manaer, who must be from its own reion. We assume that the company acts in the best interest of its clients (investors), and thus maximizes tradin pro ts net of costs. Each reion has a lare number of potential fund manaers. In period i, each fund manaer receives a sinal about v i. The sinal structure within each period is as in the main model above. That is, manaers can be of type = or = b; and a manaer of type = 6 A fully micro-founded analysis with endoenous contractin is provided in Dasupta and Prat [5]. In that paper, career concerns are considered at the rm-wide level. 2
receives more accurate sinals about both v and v 2. The proportion of ood manaers in each reion is in the rst period and 2 in the second period. is known at the beinnin of period, while 2 is revealed at the beinnin of period 2: There are also many noise traders. Durin each period, and in each tradin round, either a noise trader or the fund manaer (who has not traded yet in that period) is chosen to trade with a risk neutral market competitive maker who sets prices to re ects information contained in order ow. If the noise trader is selected, he buys or sells one unit of the prevailin asset with equal probability. If a manaer is chosen he may buy or sell a unit of the asset as he chooses. Since T >> N, each manaer ets to trade once in each period with probability. 7 The selection process is symmetric within period and independent across periods. Hence, uninformed traders have the same probability of facin a noise trader or a fund manaer in every period. In period, the company employs a manaer, with whom it is randomly matched. The fund manaer receives a xed fee y for every period in which he is hired. His total fee is y if he only works in the rst period and 2y if he works in both periods. There are no lon-term contracts, but the manaer can also receive a share of the rst-period pro ts: b where b 2 [0; ]. At the beinnin of period 2, the company is able to observe the manaer s actions and their outcomes from period, and can choose to retain him or re him and be randomly matched with another manaer chosen from the reion. 8 Manaerial career concerns now arise endoenously. The second period has many continuation equilibria because the fund manaer is indi erent. We focus on the e cient equilibrium in which he buys when s t = and sells when s t = 0. The expected pro t in the second period is then an increasin function of the ability of the second-period manaer. In order to maximize pro ts, the company will retain the rstperiod manaer at beinnin of period 2 if b 2 and she will replace him with a randomly selected manaer if b < 2. The manaer in period therefore must maximize b (a ; p t ) + Pr(b(a ; v ) 2 )y Dividin the payo by b + y > 0, we have that the manaer maximizes in period (p t ) + ( )r(b(a ; v )) 7 For nite T and N, there is a small probability that not all fund manaers et to trade before time T: Formally, we could assume an exoenous retention rule for the remainin fund manaers (for instance, they all et red). As T!, the probability of facin such exoenous termination vanishes, which is the case we discuss here. 8 The analysis would be more complicated if the manaer received a share of the second-period pro t as well. However, Dasupta and Prat [5] show that, if contracts are endoenous, the fund manaer s compensation is independent of his return in the last period. 22
where = b b+y, and r(b(a ; v )) = Pr(b(a ; v ) 2 ), which is a special case of the payo s that we have considered throuhout the paper. 6.5 Informed Individual Traders To date we have restricted attention to a market in which uninformed market makers are faced with either noise traders or reputationally sensitive traders. A natural added member of the marketplace would be informed individual investors, who do not face career concerns. How would the results chane if informed individual investors operated in our baseline model? It is clear that informed individuals devoid of career concerns would trade sincerely, and thus, in the presence of such traders prices would eventually convere to true value. However, the basic intuition of the main result in unchaned in this case: once prices were close enouh to true value, career concerned institutional traders would bein to inore their own information. Thus converence to true value would take place much more slowly than in the case without fund manaers, and the extent of slowdown in converence would depend on the proportion of career concerned traders in the market. In addition, conformist tradin by institutional traders would still occur in the presence of informed individual traders, in keepin with empirical ndins (e.., Dennis and Strickland [8]). 7 Conclusion The central messae of this paper is that the presence of (even small amounts of) reputational concerns will prevent institutional traders from tradin sincerely when prices become close enouh to liquidation value. Such a tendency to nelect valuable private information is an endoenous (and pervasive) obstacle to the converence of prices to liquidation values in the lon run, and can be the basis of herd-like behavior by institutional traders, alon the lines already documented in the empirical literature. Further, we have arued that the presence of reputational incentives can drive a wede between the expected liquidation value of an asset and its total value to fund manaers. We have characterized the properties of such reputational bene ts, and presented an example of how such reputational premia can be incorporated into prices, thus leadin to the systematic mispricin of assets. While we have carefully related our central results on conformist tradin to existin theoretical explanations in the introduction, it remains for us to do the same for our explorations on mispricin. We now proceed to do so. While the classical no-trade aruments of Milrom and Stokey [23] and Tirole [36] preclude bubbles in markets with asymmetric information and rational aents in eneral, a number of papers construct examples of bubbles while examinin the role of hiher order beliefs in 23
asset pricin. Allen, Morris, and Postlewaite [5] build on the no-trade theorems to develop necessary conditions for the existence of bubbles, and provide examples of economies in which bubbles can exist. Morris, Postlewaite, and Shin [24] illustrate the connection between bubbles and hiher order uncertainty. Prices are biased statistics of true value in the recent work of Allen, Morris, and Shin [6]. There are also a lare number of papers in which bubbles arise and persist because some traders are irrational (e.. DeLon, Shleifer, Summers, and Waldmann [7], Shleifer and Vishny [32], Abreu and Brunnermeier [], and Scheinkman and Xion [3], amonst others). More closely related to our work, a few papers construct examples of bubbles based on aency con icts. Most notably, Allen and Gorton [4] develop a model in which prices can divere from fundamentals due to churnin by portfolio manaers. In their model, bad fund manaers buy bubble stocks at prices above their known liquidation value in the hope of resellin them before they die at even hiher prices to other bad fund manaers. Their behavior is the result of an option-like payo structure under which pro ts are shared with manaers but losses are not. Churnin thus creates the possibility of short-term speculative pro ts. A related principal-aent con ict leads to bubbles in Allen and Gale [3]. In contrast to both of these papers, in our example, mispricin arises without option-like payo s purely due to reputational concerns of nancial traders. 8 Appendix 8. Proof of Proposition 2 We proceed by provin two preliminary results. Lemma 9 There exists a function f : [0; ] [0; ]! [0; ] such that Pr(v = jh t ; s t ) = f(p t ; s t ) and f satis es the followin properties: (a) f(p t ; s t ) is strictly increasin and continuous in p t. (b) f(; s t ) = = f(0; s t ) (c) f(p t ; ) > f(p t ; 0) 24
Proof. Pr (v = jh t ; s t ) = Pr (s t; h t jv = ) Pr (v = ) Pr (s t ; h t ) = Pr (s tjv = ) Pr (v = ) Pr (h t jv = ) Pr (s t ; h t ) = Pr (s tjv = ) Pr (v = ) Pr (s t ; h t ) = Pr (s tjv = ) Pr(h t ) p t Pr (s t ; h t ) Pr (v = jh t ) Pr (h t ) Pr (v = ) Thus Pr (s t ; h t ) = Pr(h t )[p t Pr (s t jv = ) + ( p t ) Pr (s t jv = 0)] Pr (v = jh t ; s t ) = p t Pr (s t jv = ) p t Pr (s t jv = ) + ( p t ) Pr (s t jv = 0) = f(p t; s t ) Now parts (a) and (b) follow immediately. To see part (c) note that where = + ( f(p t ; ) = ) b. Similarly = p t p t + ( p t ) Pr(st=jv=0) Pr(s t=jv=) p t p t + ( p t ) f(p t ; 0) = p t p t + ( p t ) Since > b > 2, > 2. Thus < <. This then implies f(p t; ) > f(p t ; 0) which completes the proof of the lemma. The mixed stratey of manaer t in this market will enerally depend on both this history he observes and his sinal. We denote this by t s t (h t ). A mixed stratey equilibrium is a sequence f t s t (h t )t= T. For notational convenience, we often omit the history arument and we denote the t fund manaer s stratey as t 0 ; t 2 [0; ] 2. Consider a set of equilibrium strateies t 0 ; t 2 [0; ] 2. We can now compute the posteriors reardin fund manaers. The posterior belief is iven by ^ (a t ; v; h t ) = Pr (a t j = ; v; h t ) Pr (a t j = ; v; h t ) + Pr (a t j = b; v; h t ) ( ) Note that Pr (a t = j = ; v = ; h t ) = (h t ) + 0 (h t ) ( ) 25
and similarly for the other realizations of ^. Therefore, we can write ^ (a t = ; v = ; h t ) = (h t ) + 0 (h t ) ( ) (h t ) + 0 (h t ) ( ) ^ (a t = ; v = 0; h t ) = (h t ) ( ) + 0 (h t ) (h t ) ( ) + 0 (h t ) ^ (a t = 0; v = ; h t ) = ( (h t )) + ( 0 (h t )) ( ) ( (h t )) + ( 0 (h t )) ( ) ^ (a t = 0; v = 0; h t ) = ( (h t )) ( ) + ( 0 (h t )) ( (h t )) ( ) + ( 0 (h t )) We now show that in all non-perverse equilibria either the manaer with the hih sinal or the manaer with the low sinal play a pure stratey. Lemma 0 There are no mixed stratey equilibria in which 0 < t 0 < t < for any t. Proof. Consider a putative equilibrium in which > t 0 > 0, i.e. the aent at time t who receives sinal zero is exactly indi erent between buyin and sellin. We will show that in this equilibrium, it must be the case that the aent who receives sinal at time t must strictly prefer to buy rather than sell. Consider the expected direct payo di erence between buyin and sellin: P v=0; Pr (vjh t; s t ) p a t ; p b t; v. This can be written as f (p t ; s t ) ( p a t ) p b t + ( f (p t ; s t )) (0 p a t ) p b t 0 Since ( p a t ) p b t > 0 > (0 p a t ) p b t 0, and by Lemma 9 f (p t ; ) > f (p t ; 0), it is clear that X v=0; Pr (vjh t ; s t = ) p a t ; p b t; v > X v=0; Pr (vjh t ; s t = 0) p a t ; p b t; v Now consider the expected reputational payo di erence between buyin and sellin: This can be expressed as: X v=0; Pr (vjh t ; s t ) r(^ (v; h t )) f (p t ; s t ) [r(^ (; h t ; )) r(^ (0; h t ; ))] + ( f (p t ; s t )) [r(^ (; h t ; 0)) r(^ (0; h t ; 0))] Notice that ^ (; h t ; )) > ^ (0; h t ; )). To see why consider the expressions above. Suppose + 0 ( ) + 0 ( ) < ( ) + ( 0 ) ( ) ( ) + ( 0 ) ( ) 26
Alebraic manipulation shows that this implies that ( ) ( 0 ) < 0 which is a contradiction since > 0 and 0 > 0. Similarly it is also true that ^ (; h t ; 0) < ^ (0; h t ; 0). To see why, consider aain the expressions above. Suppose ( )+ 0 ( )+ 0 > ( )( )+( 0 ) ( )( )+( 0 ) This implies that ( ) ( 0 ) > 0 which is a contradiction. Thus, r(^ (; h t ; )) r(^ (0; h t ; )) > 0 > r(^ (; h t ; 0)) r(^ (0; h t ; 0)). Given Lemma 9, we know that f (p t ; ) > f (p t ; 0), and thus X Pr (vjh t ; s t = ) r(^ (v; h t )) > X Pr (vjh t ; s t = 0) r(^ (v; h t )) v=0; v=0; v=0; Finally, the aruments above imply that X Pr (vjh t ; s t = ) ( + ( )r) > X Pr (vjh t ; s t = 0) ( + ( )r) = 0 v=0; Thus, if 0 < 0 (h t ) <, then (h t ) =. An identical arument establishes that if 0 < (h t ) <, then 0 (h t ) = 0. This completes the proof of the lemma. Let f st (h t ) T t= be any non-perverse perfect Bayesian equilibrium of this ame: (h t ) > 0 (h t ) for all h t. Lemma For every " > 0 there exists p (") and p 2 (") such that (for all h t ): " for all p t > p 2 (") and " for all p t < p (") Proof. With terms de ned as above, rst consider the expected direct payo di erence between buyin and sellin. By a sliht abuse of notation, we can write: = X Pr (vjh t ; s t ) (v p a t ) p b t v v=0; = f(p t ; s t ) ( p a t ) p b t + ( f (p t ; s t )) (0 p a t ) p b t 0 Note that this function is bounded above by U = X Pr (vjh t ; s t ) ((v) ( v)) And it is bounded below by L = X v=0; = f(p t ; s t ) ( () ( )) v=0; Pr (vjh t ; s t ) ((v ) ( v)) = ( f(p t ; s t )) ( ( ) ()) 27
It is apparent that U > 0. From Lemma 9 we also conclude that U strictly increasin in p t and lim pt!0 U = 0. Similarly, L < 0, strictly decreasin in p t and lim pt! L = 0. This means that for every " > 0 there exists p (") and p 2 (") such that " for all p t > p 2 (") and " for all p t < p (") Second, consider the expected di erence in reputational payo s between buyin and sellin. Aain, abusin notation, we write: r = f(p t ; s t ) (r (^ (a t = ; v = ; h t )) r (^ (a t = 0; v = ; h t ))) +( f(p t ; s t )) (r (^ (a t = ; v = 0; h t )) r (^ (a t = 0; v = 0; h t ))) Lemma 2 For every number > 0 (but not too lare), there exist ~p (") and ~p 2 (") such that (for all h t ): r for all p t > ~p 2 (") and r for all p t < ~p (") : Proof. From Lemma 0 we know that there cannot be an equilibrium in which 0 < 0 (h t ) < (h t ) <. Thus, either 0 = 0 (h t ) < (h t ) < or 0 < 0 (h t ) < (h t ) =. If 0 = 0 (h t ) < (h t ) <, the di erence r (^ (a t = ; v = ; h t )) r (^ (a t = 0; v = ; h t )) reduces to r ( (h t )) + ( ) r ( (h t )) + ( ) ; which lies in the interval: r r () ; r r On the other hand if 0 < 0 (h t ) < (h t ) = ; then the di erence reduces to + 0 (h t )( ) r + 0 (h t )( ) r which lies in the interval: r () r ; r 28 r
Now, de nin and Ur = r L r = min r () r r > 0 ; r we note that in all non-perverse equilibria, the di erence r () r (^ (a t = ; v = ; h t )) r (^ (a t = 0; v = ; h t )) is bounded below by L r and bounded above by U r. Now consider the case where v = 0. With a procedure (that we omit) analoous to the one used above, we see that the di erence is bounded below by and it is bounded above by r (^ (a t = ; v = 0; h t )) r (^ (a t = 0; v = 0; h t )) U 0 r = max L 0 r = r r () Thus r is bounded above by r < 0; r ; r r () < 0: and bounded below by f(p t ; s t )U r + ( f(p t ; s t )L r + ( f(p t ; s t ))U 0 r f(p t ; s t ))L 0 r Finally, we note that U r = L 0 r > 0 and L r = U 0 r > 0. Thus, r is bounded above by U r = f(p t ; s t )Ur ( f(p t ; s t ))L r and bounded below by L r = f(p t ; s t )L r ( f(p t ; s t ))Ur Now, utilizin Lemma 9 above, we know that U r strictly increasin in p t and lim pt!0 U r = L r < 0. Similarly, L r, strictly decreasin in p t and lim pt! L r = L r > 0. Thus, for every number 0 < < L r there exists a price ep 2 () < such that for p t > ep 2 (); r. The part of the proof concernin ep () is analoous and it is omitted. 29
Fix any number 2 (0; L r). By Lemma 2, we know there exists ep 2 () < such that for p t > ep 2 (); r. Now consider another number, = ( ) (where 2 (0; )). By Lemma, we know that there exists p 2 () < such that if p t > p 2 (); then then. Thus for p t > max[ep 2 (); p 2 ()], u = + ( )r ( ) + ( ) = ( ) > 0 Thus, for any price reater than max[ep 2 (); p 2 ()] the fund manaer would always choose to buy. The case for sellin is symmetric. 8.2 Proof of Proposition 3 Suppose that the manaer plays accordin to () and (2). above. Given the symmetry of the ame and the proposed equilibrium, we restrict attention to p t 2 0; 2. It is easy to see that a manaer with s t = 0 always wants to sell. Thus we can set t 0 = 0 for the relevant price rane. To simplify notation, let t = denote the probability that a manaer with s t = buys. Further, we de ne: Pr (s t = jv = ; h t ) = Pr (s t = jh t ) = t p t + ( ) ( p t ) Under the equilibrium strateies, it is easy to compute the followin prices: and the followin beliefs, p a t = 2 + ( ) 2 + ( ) t p t p b t = 2 + ( ) ( ) 2 + ( ) ( t) p t; ^ (a t = ; v = ) = ^ (a t = 0; v = ) = ( ) + ( ) + ^ (a t = ; v = 0) = ^ (a t = 0; v = 0) = ( ) ( ) + ( ) ( ) + Notice that in computin ^ (a t = ; v = ) and ^ (a t = ; v = 0) we have not restricted attention to > 0. We are thus implicitly imposin the followin o -equilibrium beliefs: if a manaer trades in a contrarian manner in a reion where equilibrium strateies require conformism, then he is assumed to have played sincerely. The beliefs used here are natural, in that they would be the on-equilibrium beliefs that would occur if a small proportion of manaers always traded sincerely for exoenous reasons. 30
Now, consider a manaer with s t =. His expected tradin pro t from buyin instead of sell is (p t ; ) = 2^v t p a t p b t His expected reputational payo from buyin instead of sellin is r (p t ; ) = ^v t (^ (a t = ; v = ) ^ (a t = 0; v = )) + ( ^v ) (^ (a t = ; v = 0) ^ (a t = 0; v = 0)) : Finally, de ne the total di erential payo from buyin instead of sellin as u (p t ; ) = (p t ; ) + ( ) r (p t ; ) : In order to show the result, we need to show that for a iven (; b ; ; ), there exists a > 0 such that for all <, u (p t ; ) 0 for p t p and u (p t ; 0) 0 for p t p. The rst step is captured in the followin lemma: Lemma 3 Given (; b ; ; ), there exists a > 0 such that for all < ; for every, u is increasin in p t : Proof. Given (; b ; ), we have For every 2 [0; ], @ @p t r (p t ; ) = @ @p t ^v t (^ (a t = ; v = ) ^ (a t = 0; v = )) (^ (a t = ; v = 0) ^ (a t = 0; v = 0))! : (^ (a t = ; v = ) ^ (a t = 0; v = )) (^ (a t = ; v = 0) ^ (a t = 0; v = 0)) ( ) + ( ) ( ) + min ( ) + ( ) ( ) + = = is strictly positive. Also @ @p t ^v t = t (2 ) p t 2 t = ( ) 2 t But max pt2[0; ] t = 2 2 + ( ) 2 = 2. Hence @ @p t ^v t > 4 ( ) for all p t : 3
Therefore @ @p t r (p t ; ) 4 ( ) for all p t and : It is also easy that @ @p t (p t ; ) is bounded below on p t 2 0; 2. Hence, iven (; b ; ; ), there exists a low enouh such that @ @p t (p t ; ) + ( ) @ @p t r (p t ; ) > 0 for every p t 2 0; 2 and. For the remainder of the proof, consider <. Let p to be the unique solution of u (p t ; ) = 0. Clearly, u (p t ; ) 0 for all p t p. Note that, for = 0 u (p; ) = r (p; ) = (2^v (p) ) = 0 and thus ^v (p) = 2 We now arue that for = 0, u (p; 0) < 0. When = 0 u (p; 0) = r (p; 0) = ^v (p)( = 2 ( ) + ( ) + ( ^v (p))( ) = 2 ( + 2 )( ) 2) < 0 since + < 2. Then, by continuity in, for a iven (; b; ; ), there exists 2 > 0 such that for all < 2, u (p; 0) 0. But then, by Lemma 3, for such, u (p; 0) 0 for all p < p. Now set = min( ; 2 ) and the proof of existence is complete. For the non-existence of more informative equilibria, we need the followin additional result: Lemma 4 In the limit as! 0; for every 2 (0; ) and for all p < p, u (p; ) > u (p; ). Proof. Consider d d r (p; )j p=p = But note that ^v (p) d d ^ (a t = 0; v = ) ( ^v (p)) d d ^ (a t = 0; v = 0) d ( ) + d ( ) + d ( ) ( ) + d ( ) ( ) + = = 32 (( ) + ) 2 (( ) ( ) + ) 2
But ( ) + < ( ) ( ) + for 2 (0; ) implies that d d ^ (a t = 0; v = ) > d d ^ (a t = 0; v = 0) If! 0, ^v (p)! 2 and d d r (p; )j p=p > 0: But then u (p; ) > u (p; ) for every 2 (0; ). By lemma 3, for every and p < p, u (p; ) u (p; ). Suppose that p t < p and that there exists an informative nonperverse equilibrium. We know that in such a case a manaer with s t = 0 must sell for sure. But lemma 4 also says that u (p; ) < 0 for all and p < p. Then, a manaer with s t = would sell for sure as well. But then the equilibrium is not informative. This completes the proof of the result. 8.3 Proof of Proposition 6 First, it is easy to check that in a non-perverse equilibrium, iven h t, there are at most two realizations of s t for which the aent randomizes amon actions. A non-perverse equilibrium is characterized by a partition of the interval [s min ; s max ] into three reions. Let s L denote the threshold below which the fund manaer sells for sure and above which he does not trade. Let s H be the equivalent threshold between not tradin and buyin. It is clear that if this non-perverse equilibrium is to be informative (that is, allow for prices to convere to true value in the lon run), either s H < s max or s L > s min or both. We assume that s L > s min and show that in this equilibrium, it is impossible for p t! v min. The case for s H < s max and p t! v max is symmetric. The proof is by contradiction. Suppose that v = v min (the proof for a hih v is analoous) and that there exists an equilibrium in which p t! v min. It is clear that in a non-perverse equilibrium lim E [r(b(buy; v))] = r(b(buy; v min )); p t!v min that is, as prices convere to v min all fund manaers expectation of the reputational return from buyin converes to the ex post reputational return when the posterior is evaluated at v min. Consider a fund manaer with sinal s t 2 (s H ; 0). The equilibrium dictates that he buys. 33
When p t! v min, the expected reputation of a manaer who buys is Pr ( = js > s H ; v min ) = Pr (s > s Hj = ; v min ) Pr (s > s H jv min ) = = and similarly, the reputation of a manaer who sells is R s max s H f (sjv min ) ds R s max s H f (sjv min ) ds + ( ) R s max s H f b (sjv min ) ds R s max s H f (sjv min ) ds ( + ( )) R s max s H f (sjv min ) ds + ( ) ( ) R s max s H f (s) ds Pr ( = js < s L ; v min ) = R s L s min f (sjv min ) ds ( + ( )) R s L s min f (sjv min ) ds + ( ) ( ) R s L s min f (s) ds We shall show that Pr ( = js > s H ; v min ) < Pr ( = js < s L ; v min ). For this, it su ces to show that But R smax s H R smax s H f (sjv min ) ds R sl s min f (sjv min ) ds < f (s) ds R sl = s min f (s) ds where for all s 00 > s 0 and all v > v min, R smax s R H sl s min f (s) ds f (s) ds Pv k(v) R s max s H f (sjv) ds Pv k(v) R s L s min f (sjv) ds f (s 00 jv) ds f (s 0 jv) ds > f (s 00 jv min ) ds f (s 0 jv min ) ds by the MLRP. This then implies that the above is true for all s 00 2 (s H ; s max ] and s 0 2 [s min ; s L ) and thus Pr ( = js > s H ; v min ) < Pr ( = js < s L ; v min ). Note that this analysis could have been just as well carried out even if s H = s max. In this case, we would have considered a trader whose equilibrium strateies for s > s L would have been not to trade. An arument identical to the above would establish that as p t! v min, this trader would enjoyed a strict reputational ain by defectin and sellin. Finally, from our earlier aruments, it is clear that as p t! v min, expected tradin profits become in nitesimal. Thus, traders whose equilibrium strateies require buyin or not tradin would prefer to deviate and sell, and thus this cannot be an equilibrium. 8.4 Proof of Proposition 7 Suppose there exists a non-perverse equilibrium in which a fund manaer who observes z t = always plays a t = s t. 34
Let Consider a fund manaer with z t = and s t = 0 and suppose that the current price is p t. It is easy to see that ^v st;zt t = E [vjs t ; z t ; h t ] : lim p t! ^vst;zt t = for all s t ; z t : Hence, the expected bene t in terms of tradin pro t of playin a t = 0 instead of a t = for a fund manaer with z t = and s t = 0 oes to zero as price approaches : lim = 0: p t!0 The expected reputation bene t/cost of playin a t = 0 instead a t = for a fund manaer with z t = and s t = 0 is Thus, r = ^v 0; t ((^ t (a t = 0; v = ) ^ t (a t = ; v = ))) + ^v 0; t (^ t (a t = 0; v = 0) ^ t (a t = ; v = 0)) : lim r = ^ t (a t = 0; v = ) ^ t (a t = ; v = ) : p t! As p t!, a fund manaer with z t = and s t = 0 plays a t = 0 only if ^ t (a t = 0; v = ) ^ t (a t = ; v = ) : () In a non-perverse equilibrium in which a fund manaer with z t = plays a t = s t, as p t! ; beliefs have the followin bounds (based on the assumption that all aents with z t = b play a t = ): It is easy to see that and ^ t (a t = ; v = ) Pr ( = jnot (z t = and s t = 0) ; v = ) ^ t (a t = 0; v = ) Pr ( = jz t = ; s t = 0; v = ) : Pr ( = jnot (z t = and s t = 0) ; v = ) ( ) > Pr ( = jz t = b) = ( ) + ( ) ; Pr ( = jz t = ; s t = 0; v = ) = Inequality () is satis ed only if ( ) ( ) + ( b ) ( ) ( ) : ( ) ( ) + ( b ) ( ) ( ) ( ) ( ) + ( ) : 35
If! 2, the inequality reduces to which is false. ( ) ( ) + ( b ) ( ) ; 8.5 An Example of Mispricin In this section we provide an example of how the reputational concerns of traders can lead to systematic mispricin of assets when the supply of the asset is not in nitely elastic. The model underlyin this example is described in Section 5. We use the parameter values used earlier in the paper: = 0:5, = 0:4, = and b = 0:5, but we now let = 0 (rather than = 0:25). When the monopolistic seller sets the price, she faces three options: sell to the hih type only (s t = ), always sell, or never sell. If she never sells, her expected payo is simply p t. If she sells to the hih-type only, she sets the price p a such that a manaer with s t = is indi erent between buyin or sellin, which implies (note that the equilibrium is separatin) the condition: or (2^v p a ^v 0 ) + ( ) (2^v ) p a = 2^v ^v 0 + p = 2 2p + p 3 2p (2^v ) = 0; If p a < ^v, the seller prefers not sellin. This occurs when p < 0:535. If the price computed above were reater than, the seller would be constrained to settin p a =. This happens when p > 0:2729. The expected pro t of a seller who sells to the hih-type only is Pr (s t = ) p a + Pr (s t = 0) ^v 0 = (p + ( ) ( p)) p a + ( ) p Suppose instead that the seller sells to the low type as well. The price p a is then iven by the maximum price that a manaer with s t = 0 is willin to pay to buy in a poolin equilibrium: (^v 0 p a ) + ( ) ^v 0 + ( ^v 0 ) = 0; 36
that is p a pool = ^v 0 + p = 2 3 2p ^v 0 This price would be reater than when p > 0:8333. The seller prefers sellin to the both types only if 4 + ( ^v 0 ) (p + ( ) ( p)) + ( ) p p a pool ; which is true if p 0:78078. Note that it is easy to check that for 0:2729 < p < 0:78078, when the market maker would chare p a =, and wish to sell only to the hih type, it is not bene cial for the low type to deviate from the equilibrium strateies and buy. Thus the equilibrium is indeed sincere in the rane 0:535 < p < 0:78078. The averae price P t is now iven by 8 p if 0 p < 0:535 >< (p + ( ) ( p))p a + ( )p if 0:535 p < 0:2729 P = (p + ( ) ( p)) + ( )p if 0:2729 < p < 0:78078 p >: a pool if 0:78078 < p < 0:8333 if 0:8333 < p This function is plotted in Fiure 3. The thick line represents P t while the thin line represents the true expected value p t. Notice that there is overpricin at all prices, except when p is so low that there is a cascade in which the manaer never buys. References [] Dilip Abreu and Markus Brunnermeier. Bubbles and Crashes. Econometrica 7(): 73-204. 2003 [2] Franklin Allen. Do Financial Institutions Matter? Journal of Finance 56 (4): 65-75, 200. [3] Franklin Allen and Doulas Gale. Bubbles and Crises. Economic Journal, 0: 236-255. 2000. [4] Franklin Allen and Gary Gorton. Churnin Bubbles. Review of Economic Studies 60(4): 83-36. 993. 37
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