The Price Impact of Borrowing and Short-Sale Constraints

Transcription

1 The Price Impact of Borrowing and Short-Sale Contraint Kathy Yuan September 1, 6 Abtract Thi tudy explore how trading contraint h a borrowing and hort-ale contraint affect aet price in the preence of aymmetric information. In a ymmetric information environment, borrowing and hort-ale contraint exacerbate downward and upward price movement, repectively. However, in the preence of information aymmetry, the price impact of each contraint i different. In an aymmetric information environment, price play an important role in haping uninformed invetor expectation. Uninformed invetor are uncertain whether trading contraint retrict informed invetor from tranmitting information to price, and thu they demand an information-diadvantaged premium in holding tock. Thi create a large price decline. Hence, information aymmetry combined with hort-ale contraint dampen the upward price movement while information aymmetry combined with borrowing contraint intenifie the downward price movement. The model alo generate the following empirical prediction: 1) price at extreme tail are le informative of the aet fundamental; ) bad new create greater return volatility than good new; 3) crahe are more likely than bubble; and 4) the kewne in return i more pronounced in tock with evere information aymmetry and with a greater percentage of contrained informed invetor. Journal of Economic Literature Claification Code: D8, D84, G14. Keyword: Rational Expectation Equilibrium, Nonlinear REE, Trading Contraint, Short-Sale Contraint, Borrowing Contraint, Aymmetric Information, Bubble, Crahe. mine. I thank Hung-Chi Chang and Aniket Gune for reearch aitance. Any error are

2 1 Introdtion In thi paper, we analyze the aet pricing implication of two prevalent trading contraint in the financial market: borrowing and hort-ale contraint. Borrowing and hort-ale contraint are regarded in the exiting literature a important market friction that may contribute to market abnormalitie h a udden crahe or contagion (Kyle and Xiong (1); Xiong (1)), a well a bubble (Harrion and Krep (1978); Hong and Stein (3); Scheinkman and Xiong (3)). Thi paper extend the reearch by exploring the ditinct effect of borrowing and hort-ale contraint on aet price when information i aymmetric. Extenive literature tudie the impact of each of thee trading contraint eparately. It ha been hown both theoretically and empirically that borrowing contraint may lead to underpricing and reult in fire ale while hortale contraint may caue overpricing. For example, Shleifer and Vihny (1997) how there i a limit of arbitrage when leveraged invetor face borrowing contraint. Xiong (1), Xiong and Kyle (1), Yuan (5), Gromb and Vayano (), and Liu and Longtaff (4) tudy borrowing contraint that are endogeneou in wealth/aet price. 1 Empirically, Coval and Stafford (5) find evidence for aet fire ale, uing market price of mutual fund tranaction caued by capital flow. For the overpricing impact of hort-ale contraint, the evidence i alo extenive. For intance, Allen, Morri, and Potlewaite (1993) how it i poible for hort-ale contraint to generate finite bubble. Other theoretical work include Miller (1977), Jarrow (198), Diamond and Verrecchia (1981), Harrion and Krep (1978), Hong and Stein (3), Scheinkman and Xiong (3), and Brunnermeier and Abreu (3). 1 Xiong (1), Xiong and Kyle (1), and Yuan (5) how borrowing-contrained arbitrager can have a price detablizing effect and can inde correlation in aet price. Gromb and Vayano () how that borrowing-contrained competitive arbitrager may take exce (or little) rik. Liu and Longtaff (4) how it i optimal for rik avere arbitrager to underinvet in arbitrage due to collateral contraint. Groman and Villa (199) how that rik-neutral agent facing a fixed amount of borrowing alter the optimal portfolio trategy even when the contraint i not binding. A number of empirical tudie have documented the overpricing impact of hort-ale contraint, including Seneca (1967), Seneca and Stark (1993), Figlewki (1981), Figlewki and Webb (1993), D Avolio (), Krihnamurthy (), Lamont and Stein (4), Ofek and Richardon (3), Geczy, Muto, and Reed (), and Aquith, Pathak, and Ritter (5). 1

3 However, mot of thee tudie do not explore contraint effect in an aymmetric information etting. Thoe that conider the aymmetric information etting include Diamond and Verrecchia (1981), Bai, Chang, and Wang (6), and Marin and Olivier (6) on the effect of hort-ale contraint, and Yuan (5) on the effect of borrowing contraint. 3 A central theme of thee tudie i that the aet price become le informative when invetor are contrained by either borrowing or hort-ale contraint. Mot of thee tudie predict binding contraint when price are low, which in turn ugget greater likelihood of a crah. In thi paper, we further the theoretical obervation of thee tudie and how that the price impact of hort-ale and borrowing contraint are different in the preence of information aymmetry. Thi finding i counter to reult in the exiting literature on trading contraint. Specifically, we find that hort-ale contraint are more likely to bind when price are high rather than low. Intuitively, thi correpond to the empirical phenomenon that informed invetor are hort-ale contrained when the high aet price i caued by a high level of noie demand. Intead of reulting in a bubble (a predicted in the exiting ymmetric information literature) or cauing a crah (a predicted in the exiting aymmetric information literature), hort-ale contraint when combined with information aymmetry dampen the upward price movement and thu make bubble difficult to form. By contrat, borrowing contraint are more likely to bind when price are low. Empirically. thi capture the feature of margin contraint: A drop in the aet price caue a decreae in the collateral value and conequently the amount of the margin loan invetor can borrow. We find that the interaction of borrowing contraint with information aymmetry exacerbate the downward price movement, making crahe in aet price more likely. To tudy the price impact of borrowing and hort-ale contraint, we employ a tandard noiy rational expectation equilibrium (REE) model of aet price with both informed and uninformed invetor. Informed invetor receive a noiy ignal about the aet payoff while uninformed invetor oberve only the price, from which they extract the informed invetor private ignal. In thi model, there i a noiy demand/upply hock o that price do not fully reveal private information, imilar to that ued by Hellwig (198) 3 Additionally, the etting in Barlevy and Veronei (3) incorporate rik-neutral agent under both a hort-ale contraint and a wealth contraint, with the hock ditributed exponentially. However, their emphai i on the tail event on aet price (h a crie) rather than on the impact of trading contraint.

4 and Groman and Stiglitz (198). In addition, ome informed invetor face borrowing and hort-ale contraint. Conitent with the margin requirement and hort-ale contraint oberved in the financial market, we pecify both contraint a retriction on invetor demand and the borrowing contraint i price dependent. We find that, when informed invetor are not contrained, the aet price i informative ince the uncontrained trading tranmit their ignal to the aet price. However, when a mall advere hock to the fundamental lower the price, informed invetor may become borrowing contrained. In thi cae, their ability to trade on their private information i limited, reulting a noiy price. Uninformed invetor now cannot eparate liquidity elling (or noie aet elling) from informed invetor information-baed elling. Thu, they find it increaingly difficult to extract the informed ignal from the falling price, and will bail out when the price fall. Thi behavior reult in a demand that i price inelatic or poibly backward-bending, which exacerbate the downward price movement. The inelatic uninformed demand inde everal feedback effect. For example, the falling aet price tighten informed invetor borrowing contraint, leading to greater volatility and poibly price multiplicity. Converely, we find that, when a mall poitive hock to the fundamental increae the price, informed invetor may be contrained out of the market due to hort-ale retriction. Again, in thi cenario, informed invetor private information i not embedded in the market clearing price, reulting a noiy price. Uninformed invetor are le willing to purchae the aet ince they cannot ditinguih noie demand from information-baed buying. Due to advere election concern, thi additional uncertainty about informed invetor contraint caue uninformed invetor to demand an informationdiadvantaged premium to hold the aet. Their demand become more elatic a the price increae, inding a dampening effect. Hence, large upward price movement become le likely. Additionally, in thi economy, uninformed invetor do not oberve informed invetor contraint tatu and can only infer from the price the probabilitie of informed invetor being retricted by trading contraint. Thi introde another ource of uncertainty, ince price informativene change with different informed invetor contrained tatu. In the preence of both trading contraint, the informed invetor contraint tatu varie more than the cae when informed invetor face only one contraint. It could range from uncontrained, to contrained by either or both trading 3

5 contraint. Therefore, the perceived uncertainty to uninformed invetor i greater. Conequently, the kewne in return and exce volatility are mh more pronounced, demontrating the importance of conidering both market friction in undertanding propertie of aet price. Overall, our analyi how that the interaction between trading contraint and information aymmetry generate a different impact on aet price depending on the type of contraint. 4 Thi finding i in contrat to the reult in the exiting literature howing both borrowing and hort-ale contraint lead to crah in an aymmetric information environment. Furthermore, our finding generate everal empirical implication: 1) price at extreme tail are le informative of the fundamental; ) bad new create greater return volatility than good new; 3) crahe are more likely than bubble; and 4) the kewne in return i more pronounced in tock with evere information aymmetry and with a greater percentage of contrained informed invetor In addition to our finding regarding information aymmetry and contraint effect, our paper make everal technical contribution to the literature. Firt, we how that the non-linear REE olution preented in Yuan (5) 5 can be generalized to any aymmetric information etting if the information trture i hierarchical. In particular, we generalize the non-linear REE olution method to a CARA-normal etting under both borrowing and hort-ale contraint. Second, thi paper i among the firt to tudy the aet pricing implication of hort-ale contraint veru borrowing contraint in an aymmetric 4 There are everal trand of the literature that tudy the return volatility aymmetric repone to new. For example, Black (1976), Chritie (198), Gloten, Jagannathan, and Runkle (1993), Braun, Nelon, and Sunier (1995), among other, label thi phenomenon a the leverage effect. Pindyck (1984), French, Schwert, and Stambaugh (1987), Campbell and Hentchel (199), Bekaert and Wu (), and Wu (1) propoe the volatility feedback effect a an alternative explanation. Detemple (1986), Feldman (1986), David (1997), and Veronei (1999) argue for rational learning and tochatic uncertainty among invetor a an explanation. Finally, Barberi, Shleifer, and Vihny (1998), Barberi, Huang, and Santo (1), and McQueen and Vorknick (4) offer explanation baed on the perpective that invetor are expoed to certain behavioral biae. The explanation in thee model differ from our in that information aymmetry play a central role in our explanation. Further, prediction from our model are information aymmetry baed and hence are different a well. 5 Yuan (5) tudie a borrowing-contrained economy with mean-variance invetor and normally ditributed hock 4

6 information etting. In thi repect, we complement the finding in Yuan (5), who tudie the aet pricing implication of borrowing contraint, Bai, Chang, and Wang (6) and Marin and Olivier (6), who tudy the aet pricing implication of hort-ale contraint. Our tudy capture different market phenomenon from the latter two tudie on hort-ale contraint. The difference are due to choice of model etup. Intead of independent noie trading, thee two tudie introde noie trading through informed invetor hedging need on their non-tradable aet. Thi modeling difference caue everal ignificant difference in reult. For example, hort-ale contraint are likely to bind when price are high in our tudy, which capture the phenomenon that informed invetor are hort-ale contrained when the high aet price i caued by a high level of noie demand, a cenario imilar to the tech bubble. A decreae in price informativene in thi cae lower the likelihood of bubble but will not caue crahe. By contrat, in Bai, Chang, and Wang (6) and Marin and Olivier (6), hort-ale contraint are likely to bind when aet price are low. Thi i becaue informed invetor are endowed with exce nontraded riky aet. To hedge thi un-traded rik, they have to hort-ell the traded aet that i poitively correlated with the non-traded aet. Conequently, the harp drop of price informativene due to hort-ale contraint caue a crah in the price of the traded aet. Therefore, they capture a different et of market condition. Furthermore, the ource of uncertainty in our tudy i alo different from that identified in thee two tudie. In Bai, Chang, and Wang (6) and Marin and Olivier (6), at a given price, informed invetor demand can be inferred and o i their contraint tatu. By comparion, in our tudy, informed invetor contraint tatu cannot be inferred with certainty ince the high price could be caued either by a high realization of private ignal or by a high level of noie trading. Thi introde an additional ource of perceived uncertainty to uninformed invetor and caue equilibrium price more kewed and more volatile. The remainder of the paper i trtured a follow. In Section, rational expectation equilibrium (REE) model for an economy with aymmetric information and trading contraint are developed. In Section 3, we preent the equilibrium olution and analyze the propertie of equilibrium price. Section 4 conclude. All proof are preented in the Appendix. 5

7 The Model.1 An Economy with Information Aymmetry The following model i an extenion of the Groman and Stiglitz model (198) in one apect: it include borrowing and hort-ale contrained invetor. In thi model, there are two date, time and time 1. At time, invetor trade competitively in the market baed on their private information. At time 1, payoff from the aet are realized and conumption occur. The model aume an underlying probability pace, (Ω, F, Q), on which all random variable are defined. A tate of nature i denoted by ω Ω. It i alo aumed that all random variable belong to a linear pace, N, of joint normally-ditributed random variable on Ω. In our model, there i one rik-free and one riky aet. The rik-free aet pay R unit, while the riky aet pay v unit of the ingle conumption good. Taking the rik-free aet to be the numeraire, we let P be the price vector for the riky aet. Invetor k divide hi initial wealth, W,k, between the rik-free and riky aet. We let D k be the riky aet holding by agent k. Thu, invetor k final wealth i given by: W 1,k = W,k R + D k (v RP ). (1) In thi model, each invetor maximize the expected utility of conumption baed on hi or her own information et. We aume that, for agent k, the utility function exhibit contant abolute rik averion, i.e., E [ e w 1,k /ρ ], where E i the expectation operator, conditional on invetor information at time. Again, to implify notation, we aume that all invetor have the ame rik averion parameter, ρ. Generalization of thi concept to heterogenou rik averion parameter i hown in Admati (1985). We aume that invetor are competitive and form a continuum with meaure 1. Invetor are of one of two clae: informed or uninformed. 6 Prior to trading, informed invetor receive private information related to the payoff of the riky aet. The ignal,, i a noiy ignal of the aet final payoff, v, given a follow: = v + ɛ, () 6 We denote the meaure of informed invetor a w i and the meaure of uninformed invetor a w ui, where w i + w ui = 1. We denote the meaure of uncontrained informed invetor a wi and the meaure of contrained informed invetor a wi c, where wc i +w i = w i. 6

8 where ɛ repreent the noie of the ignal and i independent of v. For informed invetor, the information et conit of the equilibrium price vector and the realization of a private information ignal,, which i correlated with v. By contrat, the uninformed invetor information et conit of only the equilibrium price. 7 Another ingredient of the model i the exitence of noie in the form of a random upply of the riky aet, m. Thu, the no-trade theorem doe not apply (Milgrom and Stokey 198). In addition, we aume v, m, and ɛ are mutually independent and jointly normally-ditributed with mean of, m, and variance of Σ v, Σ m, Σ, repectively. We aume that the rik-free aet i the numeraire aet and R = 1.. Short-Sale and Borrowing Contraint A unique feature of our model i the introdtion of a hort-ale contraint and a price-dependent borrowing contraint on informed invetor demand for the riky aet. Thee contraint are empirically relevant. The hort-ale contraint i typically oberved when the aet price i high relative to the fundamental while the borrowing contraint arie when the tock price i low relative to the fundamental. 8 Incluion of thee contraint in the model i eential for an in-depth undertanding of the propertie of aet price ditribution. We incorporate the contraint into the model by auming that only a fraction of informed invetor (w c i ) face hort-ale and borrowing contraint. 9 The following definition provide a decription of the hort-ale contraint. Definition 1 (Short-Sale Contraint) Informed invetor are hort-ale contrained when their demand i bounded from below by d, a contant. Note that when d =, definition 1 i the hort-ale contraint commonly 7 We denote informed agent by i, uninformed agent by ui and generic agent by k. 8 When the aet price i low, borrowing-contrained informed invetor cannot jutify a holding poition on a beaten-down tock to outide lender. Their borrowing capacity i tied to aet value. Hence, borrowing contraint can be modeled a a retriction on informed invetor demand that depend on aet price. 9 The model aume that only a fraction of informed invetor are contrained, for the ake of generality. Contraint on uninformed invetor in thi type of problem are normally immaterial ince they do not affect the inference problem of uninformed invetor. 7

9 oberved for retail invetor. Next, following Yuan (5), we capture the borrowing contraint by the following linear trture. Definition (Borrowing Contraint) Informed invetor are borrowing contrained if their tock demand i bounded from above by n(p ) = ap + b, 1 where a > and a < w i ρ(τ v + τ )/w c i The Equilibrium Concept Thi ection define the equilibrium concept for the above-pecified contrained economy. It i baed on the rational expectation model developed by Groman (1976) and Hellwig (198). The following i a tandard equilibrium definition. Definition 3 A contrained REE in a contrained economy i a price vector, P, and allocation function, D, h that: P i (, m) meaurable. For an uncontrained agent k, D k arg max Dk R n E(U(W k) F k ). F k i agent k information et. For a hort-ale contrained agent i, D c i arg max Di de(u(w i ) F i ). F i i agent i information et. For a borrowing-contrained agent i, D c i arg max Di ap +be(u(w i ) F i ). F i i agent i information et. 1 The financial contraint on informed invetor demand i tylized but realitic. For example, invetor often etablih margin account with dealer. Let u aume the invetor ha a margin account for the riky aet and the margin requirement i 3%. At the trading date, an invetor wealth conit of a poition (long or hort) in the riky aet (Q hare) and a poition (long or hort with a value of A) in the rikfree aet (W = QP + A). He can leverage up uing the margin account (7%W ). The upper bound of hi poition in the riky aet i (1 + 7%)Q + 7%A/P, which i endogenou in price. Thu, our definition can be conidered a a linearized verion of thi contraint. 11 We ue τ to denote the preciion of a random variable, that i, the invere of the variance; later, we ue 1 to denote indicator function. The firt retriction on a i to enure that it i a borrowing contraint. The econd i to enure that the demand curve of contrained and uncontrained informed invetor combined remain downward-loping with repect to P o that the reult of poible multiple equilibria i not trivial. We ue bc and c to denote borrowing and hort-ale contraint, repectively, and bc and c to denote the correponding complement. 8

10 The market clearing condition i atified by: w i D i +w c i D c i +w ui D ui = m, where D i i uncontrained informed invetor demand, D c i i contrained informed agent demand, and D ui i uninformed agent demand. 3 Aet Price with Borrowing and Short- Sale Contraint In thi ection, we firt tart with the equilibrium olution() for an economy with borrowing contraint and hort-ale contraint, extending the reult outlined in Yuan (5). We next contrat the propertie of equilibrium price under the different contraint cenario and invetigate the correponding price impact. 3.1 Equilibrium under Borrowing and Short-Sale Contraint In an economy with borrowing and hort-ale contraint, if the price i low (or high) enough relative to the private ignal held by informed invetor, informed invetor can be borrowing (or hort-ale) contrained out of the market. A a reult, the price informativene could vary depending on the price level. Since the aet price i in the invetor information et, the varying price informativene make it difficult to olve for the invetor inference problem. Thi i epecially true conidering when informed invetor are not contrained by either hort-ale or borrowing contraint, the ditribution of their private ignal i doubly truncated (i.e., to the left and to the right). A a firt tep toward olving for the equilibrium, we examine informed invetor inference and optimization problem. Note that, in thi etting, the information trture i hierarchical. Specifically, informed invetor private ignal trictly dominate the price a a ignal for the fundamental. Thi mean, for informed invetor, the aet price i a redundant ignal and can be ignored. Conequently, their inference problem can be worked out in cloed-form. Further, for a given ignal and a given price, we can olve their demand explicitly, a expreed in the following reult. 9

11 Lemma 1 Informed invetor demand i repreented by: d c d c p P + d c if < κ c ; P > (d b)/a d D i (, P ) = d p P if κ c < < κ bc 1 P + κ bc ; P > (d b)/a d c d c pp if P (d b)/a d bc d bc p P + d bc if > κ bc 1 P + κ bc ; P > (d b)/a. (3) Next, we conider a fictitiou economy with only informed invetor and an aet upply given by ˆm(P ) = m w ui D ui (P ). In thi fictitiou economy, we have t c = t bc = ˆm(P ) dc + d c p P, t d c = ˆm(P ) + d p P, d ˆm(P ) dbc + d bc p P, t d bc c = ˆm(P ) + dc pp, d c which are obervable to uninformed invetor ince D ui (P ) i in their information et. They can compute ˆm(P ) for a given D ui (P ). Thee are the ufficient tatitic for the information in P in the repective region where informed invetor are either hort-ale contrained, uncontrained, borrowing contrained, or totally contrained out of the market. Given the information conveyed in P, we can olve for uninformed invetor optimal demand. Once we obtain a olution for D ui (P ), the reult in the following propoition provide a imple procedure to olve for equilibrium price. Propoition 1 In thi borrowing and hort-ale contrained economy, informed invetor aggregate demand, D i (, P ), i characterized by equation (3). Uninformed invetor demand, D ui (P ), i uniquely characterized by: D ui (P ) = arg max Dui R ne(u(w ui) P, D ui ); (4) the equilibrium price P (, m) i an element of the et of P that atifie: (d c ˆm(P ) + d c ) /d c p if < κ c ; P > (d b)/a (d P = ˆm(P )) /d p if κ c < < κ bc 1 P + κ bc ; P > (d b)/a (d ( c ˆm(P )) /d c p if P (d b)/a d bc + d bc ˆm(P ) ) /d c p if > κ bc 1 P + κ bc ; P > (d b)/a. (5) 1

12 Thi reult outline a procedure to olve for the equilibrium. Specifically, note that the right ide of equation (4) depend implicitly on D ui (P ). Therefore, uninformed invetor optimal demand i a fixed point of equation (4), which, we how later, i unique. After obtaining D ui (P ), we firt compute ˆm(P ). Next, we conider the fictitiou economy with only informed invetor and an aet upply given by ˆm(P ). We then olve for informed invetor aggregate demand. Finally, given ˆm(P ), we olve equation (5) to find the market clearing price. Thi non-linear REE olution technique generalize the reult outlined in Yuan (5). A long a the information trture i hierarchical, that i, informed invetor inference problem i independent of that of uninformed invetor, thi olution technique can be applied to any aymmetric information etting. The following reult expree the complicated algebraic equation, which D ui (P ) i a olution of. Corollary 1 When P (d b)/a, the following equation expree uninformed invetor optimal demand: D ui (P ) = (τ p τ )τ v P τ + τ p w ui /wi ρ + mτ p. wi τ + w ui τ p When P (d b)/a, uninformed invetor demand for the riky aet i the unique fixed point of the following algebraic equation: ( ) ( ( )) e tbc 1 D ui+t bc D ui P rbc t bc 1 Φ ( 1 + t bc D ui 1 Φ t bc 3 + t bc 4 D ui ) t bc 3 + t bc 5 D ui t bc 4 φ ( ) t bc 3 + t bc 1 Φ(t 4 D ui + bc 3 +t bc 4 D ui) 5 φ ( ) t bc 3 + t bc 5 D ui + etc 1 D ui+t c D ui P rc Φ (t c 3 + t c 5 D ui ) ( (t c 1 + t c D ui ) Φ (t c 3 + t c 4 D ui ) +t c 4 φ (t c 3 + t c 4 D ui ) Φ (t c 3 +tc + et 1 D ui+t D ui (1 P rbc P r c ) Φ (t 3 + t 6 D ui ) Φ (t 5 + t 6 D ui ) (t 1 + t D ui ) (Φ (t 3 + t 4 D ui ) Φ (t 5 + t 1 Φ(t bc 3 +tbc 5 D ui) tbc 4 D ui) Φ(t c 3 +tc 5 D ui) tc 4 D ui )) +t 4 φ (t 3 + t 4 D ui ) t 4 φ (t 5 + t 4 D ui ) (Φ(t 3 +t 4 D ui) Φ(t 5 +t 4 D ui))t 6 (φ(t 3 +t 6 D ui) φ(t 5 +t Φ(t 3 +t 6 D ui) Φ(t 5 +t All contant are defined in the appendix. 6 D ui) 5 φ (t c 3 + t c 5 D ui ) 6 D ui)) ) =. (6) 11

13 The following corollarie characterize the equilibrium for an economy with only borrowing or only hort-ale contraint, repectively. Both are pecial cae of propoition 1. Corollary In an economy when ome informed invetor face borrowing contraint, informed invetor aggregate demand, D i (, P ), i characterized by: D i (, P ) = { d d p P if < κ bc 1 P + κ bc d bc d bc p P + d bc if κ bc 1 P + κ bc. (7) Uninformed invetor demand for the riky aet i the unique fixed point of the following algebraic equation: ( ) ( ( )) t bc = etbc 1 D ui +t bc 1 + t bc D ui 1 Φ t bc 3 + t bc 4 D ui D ui P rbc 1 Φ ( ) t t bc 3 + t bc bc 4 φ ( ) t bc 3 + t bc 4 D ui 5 D ui + 1 Φ(tbc 3 +tbc 4 D ui) 1 Φ(t bc 3 +tbc 5 D ui) tbc 5 φ ( ) t bc 3 + t bc 5 D ui (t + et 1 D ui+t 1 + t D ui ) Φ (t 3 + t 4 D ui ) D ui (1 P rbc ) +t 4 φ (t 3 + t 4 D ui ). (8) Φ (t 3 + t 6 D ui ) Φ (t 3 +t 4 D ui)t 6 φ (t 3 +t Φ(t 3 +t 6 D ui) 6 D ui) the equilibrium price P (, m) i an element of the et of P that atifie: { (d P = ( ˆm(P )) /d p if < κ bc 1 P + κ bc d bc + d bc ˆm(P ) ) /d bc p if κ bc 1 P + κ bc (9) Corollary 3 In an economy when ome informed invetor face hort-ale contraint, informed invetor aggregate demand, D i (, P ), i characterized by:. D i (, P ) = { d c d c p P + d c if < κ c d d p P if κ c. (1) 1

14 Uninformed invetor demand for the riky aet i the unique fixed point of the following algebraic equation: = etc 1 D ui+t c D ui P rc Φ (t c 3 + t c 5 D ui ) + et 1 D ui+t D ui (1 P rc ) 1 Φ (t 5 + t 6 D ui ) ( (t c 1 + t c D ui ) Φ (t c 3 + t c 4 D ui ) +t c 4 φ (t c 3 + t c 4 D ui ) Φ (t c 3 +tc 4 D ui) Φ(t c 3 +tc 5 D ui) tc (t 1 + t D ui ) (1 Φ (t 5 + t t 4 φ (t 5 + t 4 D ui ) + (1 Φ(t 5 +t 4 D ui))t 6 φ(t 5 +t 1 Φ(t 5 +t 6 D ui) 6 D ui) 5 φ (t c 3 + t c 5 D ui ) 4 D ui )) the equilibrium price P (, m) i an element of the et of P that atifie: ). (11) P = { (d c ˆm(P ) + d c ) /d c p (d ˆm(P )) /d p if < κ c if κ c. (1) The following corollary decribe uninformed invetor inference in thee contrained economie. Corollary 4 For a given P, E[v P = dbc + dbc V ar[v P = dbc + dbc E[v P = dc + dc V ar[v P = dc + dc ˆm(P ), (, m) {bc}] E[v P = d ˆm(P ), (, m) {}], d bc p ˆm(P ), (, m) {bc}] V ar[v P = d ˆm(P ), (, m) {}] d bc p ˆm(P ), (, m) {c}] E[v P = d ˆm(P ), (, m) {}] d c p ˆm(P ), (, m) {c}] V ar[v P = d ˆm(P ), (, m) {}], d c p where E and V ar denote the conditional mean and variance, repectively. Thi corollary ugget that, when ome informed invetor are borrowing or hort-ale contrained out of the market, aet price are le informative and hence, to uninformed invetor, the perceived aet volatility i higher conditional on the aet price, which i a mh noiier public ignal. However, when informed invetor are borrowing contrained, uninformed invetor upect the aet price would be higher if informed invetor were able to borrow. Thu, the perceived aet value i higher when informed 13 d p d p d p d p

15 invetor are borrowing-contrained. The effect on the perceived volatility indicate that uninformed invetor are le willing to purchae the riky aet a the price fall. The effect on conditional expectation indicate that uninformed invetor are more willing to accommodate the ditreed elling of informed invetor a the price fall. Thee two countervailing effect may create a backward-bending region in uninformed invetor demand, making the riky aet a Giffen good for uninformed invetor. Converely, when informed invetor are hort-ale contrained, uninformed invetor upect the aet price would be lower if informed invetor were able to hort-ell. Thu, the perceived aet value i lower when informed invetor are hort-ale contrained. Both effect on the perceived volatility and the conditional expectation indicate that uninformed invetor would rede their demand dratically when informed invetor are hortale contrained. We examine thee comparative tatic reult in detail in the next ection. 3. The Price Impact In thi ection, we illutrate the equilibrium propertie uing numerical example. The example are choen to reflect reaonable parameter, where the riky aet i a tock. We tart with a numerical example where 15% of invetor are informed. Thi percentage correpond to the amount of total market capitalization held by intitutional invetor other than penion fund and inurance companie. We further aume that, a majority (in thi cae, 14%) of informed invetor face poible borrowing and hort-ale contraint. For implicity, we normalize invetor rik tolerance to A key parameter of our model i the quality of the information ignal received by informed invetor. In our example, we aume that informed invetor receive a high-quality ignal: the ignal-to-noie ratio i. The parameter, a, b, and d are choen o that there exit price region where informed invetor are borrowing or hort-ale contrained or uncontrained, repectively. To perform a comparative tatic analyi on the price impact of trading contraint, we vary the percentage of informed invetor who face trading contraint. 14

16 3..1 Symmetric Impact on Perceived Volatilitie When informed invetor are contrained by either borrowing or hort-ale contraint, they are unable to ubmit their optimal demand for the riky aet and, hence, the market price i le informative of their private ignal. Thi decreaed price informativene create greater perceived uncertainty for uninformed invetor, who rely on the market price a a public ignal for the fundamental value of the aet. Thi reliance i evident in Figure 1, where the conditional variance i ignificantly higher when price are relatively high (i.e., when hort-ale contraint are poibly binding) or when price are relatively low (i.e., when borrowing contraint are poibly binding). Thi lead to our firt obervation a=6,b=33,d= 9,m bar =6,v bar =3,τ m =.5,τ =5,τ v =.5,ρ=1.65,w =.1,w c i i =.14 Borrowing Contraint Short Sale Contraint Borrowing and Short Sale Contraint 4 3 BC SC BCSC GS GS GS Conditional Variance Conditional Variance Conditional Variance Equilibrium Price 1 1 Equilibrium Price 1 1 Equilibrium Price Figure 1: Conditional Variance. The dah-dotted line in the graph repreent uninformed invetor conditional variance of v, the fundamental value of the aet, in an economy without any trading contraint. The olid line in the graph repreent conditional variance in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hortale contraint, repectively. 15

17 Obervation 1 Borrowing and hort-ale contraint have a ymmetric impact on the perceived volatility of the riky aet for uninformed invetor: When contraint are binding, the perceived volatility i higher. Furthermore, the graph in Figure 1 how that uninformed invetor perceive greatet aet volatility when they are uncertain whether informed invetor are contrained or not. Thi indicate that the informed invetor contraint tatu i another ource of uncertainty in thi economy. Thi finding leading to the following obervation. Obervation The perceived volatility of the riky aet i higher becaue (1) informed invetor are contrained from tranmitting their private ignal to price, and () there i an additional ource of uncertainty: the contraint tatu of informed invetor. The graph in Figure how that the conditional variance i higher when a maller percentage of informed invetor face trading contraint. Thi reult i ummarized in the following obervation. Obervation 3 The decreae in price informativene i maller when fewer informed invetor are ubject to trading contraint. 3.. Aymmetric Impact on Conditional Expectation Although the impact of trading contraint on conditional variance i ymmetric, their impact on conditional expectation i not. When informed invetor are borrowing contrained (that i, when price are relatively low), uninformed invetor rationally infer that informed invetor hold a better ignal than the price otherwie reveal and thu update their belief of the value of the riky aet upward. By contrat, when informed invetor poibly hort-ale contrained (that i, when price are relatively high), uninformed invetor rationally infer that informed invetor hold a wore ignal than the price otherwie reflect and thu revie their belief of the aet value downward. Thi reult i hown in the graph in Figure 3 and i tated in the following obervation. Obervation 4 The binding borrowing contraint caue uninformed invetor to revie their expectation of the value of the riky aet upward in the low price region, while the binding hort-ale contraint caue them to revie their expectation downward in the high price region. 16

18 a=6,b=33,d= 9,m =6,v =3,τ =.5,τ =5,τ =.5,ρ=1.65 bar bar m v Borrowing Contraint Short Sale Contraint Borrowing and Short Sale Contraint w =1% c w =1% c w =1% c w c =7% w c =7% w c =7% GS 1. 1 GS GS Conditional Expectation Conditional Expectation.8.6 Conditional Expectation Equilibrium Price 1 1 Equilibrium Price 1 1 Equilibrium Price Figure : Comparative Static on Conditional Variance. The dotted line in the graph repreent uninformed invetor conditional variance of v, the fundamental value of the aet, in an economy without any trading contraint. The olid and the dahed line in the graph repreent conditional variance in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively. The olid (dahed) line correpond to the economie where 1% (7%) of invetor are informed and contrained. Furthermore, thi reviion of uninformed invetor belief depend on the lo of price informativene created by trading contraint. When the public ignal or aet price, i le informative, uninformed invetor rely on their prior belief to infer the value of the riky aet. The graph in Figure 4 how that the expected mean i le elatic with repect to price when a greater percentage of informed invetor are contrained. Obervation 5 When more informed invetor face trading contraint, uninformed invetor rely more on their prior belief rather than noiy price to form their expectation of the value of the riky aet. 17

19 1 1 a=6,b=33,d= 9,m =6,v =3,τ =.5,τ =5,τ =.5,ρ=1.65,w bar bar m v =.1,w c i i =.14 Borrowing Contraint BC GS 1 1 Short Sale Contraint SC G S Borrowing and Short Sale Contraint 1 BCSC GS Conditional Expectation 6 4 Conditional Expectation 6 4 Conditional Expectation Equilibrium Price Equilibrium Price Equilibrium Price Figure 3: Conditional Expectation. The dah-dotted line in the graph repreent uninformed invetor conditional expectation of v, the fundamental value of the aet, in an economy without any trading contraint. The olid line in the graph repreent their conditional expectation in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively Aymmetric Optimal Stock Holding by Uninformed Invetor We have hown that borrowing and hort-ale contraint have a ignificant impact on uninformed invetor inference about the fundamental value of the riky aet. Relative to the cae without any trading contraint, uninformed invetor in a contrained economy have higher perceived uncertainty, a lower expectation of the aet value when the price i relatively high, and a higher expectation when the price i relatively low. The impact of trading contraint on the uninformed invetor optimal tock holding i ummarized below. Obervation 6 Compared to the cae without any trading contraint, unin- 18

20 a=6,b=33,d= 9,m bar =6,v bar =3,τ m =.5,τ =5,τ v =.5,ρ= Borrowing Contraint w c =1% w c =7% GS 1 1 Short Sale Contraint w c =1% w c =7% GS Borrowing and Short Sale Contraint 1 w c =1% w c =7% 1 GS Conditional Expectation 4 Conditional Expectation 4 Conditional Expectation Equilibrium Price Equilibrium Price Equilibrium Price Figure 4: Comparative Static on Conditional Expectation.The dotted line in the graph repreent uninformed invetor conditional expectation of v, the fundamental value of the aet, in an economy without any trading contraint. The olid and the dahed line in the graph repreent conditional expectation in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively. The olid (dahed) line correpond to the economie where 1% (7%) of invetor are informed and contrained. formed invetor demand in a contrained economy i maller when the price i high (i.e., when informed invetor are hort-ale contrained), and may turn backward when the price i low (i.e., when informed invetor are borrowing contrained). Figure 5 graph example of uninformed invetor demand with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively. Note that the backward-bending region occur when price are low, or, when borrowing contraint are more likely to bind. The intuition for thee reult i a follow. In the high-price region, an increae in price reflect a higher hort-ale contraint probability. The reultant 19

21 decreae in price informativene lead uninformed invetor to revie downward their belief of the aet value and upward their perceived uncertainty. Hence, uninformed invetor rede their demand dratically a price increae, that i, their demand i downward-loping but more price elatic. However, in the low-price region, a falling price alo rede price informativene and caue uninformed invetor to rede their demand, reulting a price in-elatic demand or even a backward-bending demand curve. 8 6 a=6,b=33,d= 9,m =6,v =3,τ =.5,τ =5,τ =.5,ρ=1.65,w =.1,w c bar bar m v i i =.14 Borrowing Contraint Short Sale Contraint Borrowing and Short Sale Contraint 8 8 BC G S 6 SC GS 6 BCSC GS Equilibrium Price Equilibrium Price Equilibrium Price Uninformed Demand Uninformed Demand Uninformed Demand Figure 5: Uninformed Invetor Demand. The dah-dotted line in the graph repreent the demand for the riky aet by uninformed invetor in an economy without any trading contraint. The olid line in the graph repreent their demand in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively Aymmetric Price Impact: Bubble and Crahe In the previou ection, we have howed that, with borrowing contraint, uninformed invetor demand could turn backward when the aet price i low. Thi poibility could reult in a backward-bending market exce demand curve (total demand minu the market fixed upply of the riky aet).

22 Sh an example i hown in Figure 6, indicating price multiplicity and, hence, higher volatility in the low price region. Thi lead to the following obervation. Obervation 7 With borrowing contraint, crahe from a mall advere hock are more likely and volatility i higher when price are low. 8 6 a=6,b=33,d= 9,m bar =6,v bar =3,τ m =.5,τ =5,τ v =.5,ρ=1.65,w =.1,w c i i =.14 Borrowing Contraint Short Sale Contraint Borrowing and Short Sale Contraint 8 8 BC GS 6 SC GS 6 BCSC GS Equilibrium Price Equilibrium Price Equilibrium Price Total Demand Total Demand Total Demand Figure 6: Uninformed and Informed Invetor Demand. The dahdotted line in the graph repreent the demand for the riky aet by informed and uninformed invetor in an economy without any trading contraint. The olid line in the graph repreent their demand in an economy with borrowing contraint, hort-ale contraint, and both borrowing and hort-ale contraint, repectively. In all thee example, informed invetor hold a private ignal of.7. Intuitively, a mall advere hock to the fundamental could reult in a low aet price and conequently caue informed invetor to be borrowing contrained. Thee invetor may have to condt noie elling to meet their margin requirement. Thi may put further downward preure on the aet price. Furthermore, a hown in the previou reult, uninformed invetor would 1

23 not purchae the ditreed aet due to higher perceived uncertainty and advere election concern. The information aymmetry between informed and uninformed invetor multiplie the effect of borrowing contraint and exacerbate the downward price movement, cauing higher price volatility or even price multiplicity. By contrat, a mall poitive hock to the fundamental i le likely to build a bubble. Firt, a higher price may create hort-ale contraint, which do not put upward preure on the aet price. Second, a hown in Figure 5, uninformed invetor demand le in the high price region where price informativene i reded. Uninformed invetor, in thi cae, demand an information diadvantaged premium to hold the riky aet. Therefore, in the high price region, information aymmetry play a dampening effect on the upward price movement. Thi dicuion lead to our next obervation. Obervation 8 With hort-ale contraint, bubble are le likely a high price are le informative and uninformed invetor are likely to rede their demand dratically a the price increae. Yuan (5) ha hown the equilibrium price ha an aymmetric ditribution when ome informed invetor face borrowing contraint. One may conjecture that ymmetric trading contraint (i.e., hort-ale contraint bind when price are high and borrowing contraint bind when price are low) create a more ymmetric price ditribution. However, an examination of the likelihood of bubble and crahe in thi contrained economy yield the oppoite finding. The degree of aymmetry in the price ditribution i higher when both trading contraint are preent. The comparative tatic analye in the previou ection lead to our lat obervation. Obervation 9 In thi borrowing and hort-ale contrained economy, the aet price ditribution i negatively kewed. The kewne i more pronounced with evere information aymmetry and with a greater percentage of contrained informed invetor. 4 Concluding Remark Thi tudy explore the aet pricing implication of borrowing and hort-ale contraint in the preence of information aymmetry. Our analyi how

24 that price informativene varie with the price level in a contrained economy. We find that borrowing contraint and hort-ale contraint have different price impact in an aymmetric information environment. When informed invetor are contrained by either borrowing or hort-ale contraint, aet price become le informative. However, ince le informative price caue uninformed invetor to demand an information-diadvantaged premium, the downward price movement i exacerbated while the upward price movement i leened. Thi finding i contrary to the theoretical finding in the exiting literature but i in-line with empirical obervation. The reult of our tudy indicate it i important to conider the impact of an aymmetric information environment a well a market friction h a borrowing and hort-ale contraint for an in-depth undertanding of the propertie of aet return and volatilitie. 3

25 A Proof of Lemma 1 The contrained informed invetor optimization problem i: max E [U(v, D c Di c i, P ] + µ D c i Di c λ(di c ap b). (A1) The uncontrained informed invetor optimization problem i: max E [U(v, D Di i ), P ]. (A) Solving the above optimization, we find that the informed invetor demand f c : R R i: d c d c p P + d c if < κ c ; P > b/a d D i (, P ) = d p P if κ c < < κ bc 1 P + κ bc ; P > b/a d c d c pp if P b/a d bc d bc p P + d bc if > κ bc 1 P + κ bc ; P > b/a. We denote our inference contant a follow: d c = wi ρτ, d c p = wi ρ(τ + τ v ), d = (wi + wi c )ρτ, d p = (wi + wi c )ρ(τ + τ v ), d bc = wi ρτ, d bc p = wi ρ(τ + τ v ) wi c a, d c = wi ρτ, d c p = wi ρ(τ + τ v ), κ c 1 = (τ + τ v )/τ, d bc = wi c b, d c = wi c d, κ bc 1 =(τ + τ v + a/ρ)/τ, κ bc = b/(ρτ ), κ c = d/(ρτ ). B Proof of Propoition 1 Cae 1: When informed invetor are not contrained, the market clearing condition can be expreed a: where d d p P = ˆm(P ) and P = p p m ˆm, p = d /d p and p m = d m /d p. 4

26 Cae : When informed invetor are borrowing contrained, the market clearing condition can be expreed a: where d bc d bc p P + d bc = ˆm(P ) and P = p bc p bc m( ˆm d bc o ), p bc = d bc /d bc p and p bc m = d bc m/d bc p. Cae 3: When informed invetor are hort-ale contrained, the market clearing condition can be expreed a: where d c + d c d c p P = ˆm(P ) and P = p c p c m( ˆm d c o ), p c = d c /d c p and p c m = d c m/d c p. Cae 4: When informed invetor are both hort-ale and borrowing contrained, i.e., P (d b)/a, the market clearing condition can be expreed a: where d c d c pp = ˆm(P ) and P = p c p c m ˆm, p c = d c /d c p and p c m = d c m/d c p. Therefore, we obtain the reult hown in the propoition(). C Proof of Corollary 1 We firt define the following contant: κ c =κ c, τ = 1/(1/τ v + 1/τ ), κ bc = κ bc 1 P + κ bc, ( ) = ρ (wi + wi c) τ, = 1 ( ) 1 1 =, τ m τ P bc τ P c ρwi τ τ m θ c τ P c = κ c 1 / τ + τ p c (τ + τ p c), θbc = τ P bc κ bc 1 / τ + τ p bc (τ + τ p bc). τ P 5

27 We alo ue Φ( )(φ( )) to denote the cumulative (probability) ditribution function of a tandard normal variable, i.e., Φ( ) NORMCDF (, 1) and φ( ) NORMP DF (, 1). We then expre the probabilitie of informed invetor being repectively hort-ale contrained, borrowing contrained, and uncontrained in the following equation. P r c = P r bc = 1 Φ φ { (κ c 1 P +κc )} ( (κ bc 1 P + κ bc 1/ τ ( ) ( (κ c 1/ ) d = Φ τ 1/ τ ) ), P r = 1 (P r bc + P r c ). A pecified in the text, uninformed invetor divide their initial wealth, W, between riky and rik-free aet. We let D ui repreent the riky aet holding by uninformed invetor. Thu, the uninformed invetor final wealth i given by: W 1,ui = W,ui R + D ui (v RP ). The expected utility of an uninformed invetor, conditional on informed invetor being hort-ale contrained, can be expreed in the following form: E [ e w1/ρ P, κ c ] = E [e (W R+D ui (v RP ))/ρ P, κ c )] = e (W R D ui RP )/ρ E [ e Duiv/ρ P, κ c )]. Uing iterative expectation, we can write thi expreion a: E [ e D uiv/ρ P, κ c ] = E [E [ e D uiv/ρ ] P, κ c ]. Since f(y) N(, 1 ), x = y+ɛ x, f(ɛ x ) N(, 1 τ x 1 ), f(y x) N( x, ), τ τ x τ + τ x τ + τ x we obtain ), = E [ e D uiv/ρ ] = e D uiv/ρ f (v P, ) dv e D 1 1 uiv/ρ e 1/(τv + τ ) π 1! v τv+τ τ 1 τv+τ dv = e ( τ ρ(τv+τ) D D ui ui ρ ) (τv+τ). 6

28 Therefore, E [ e D uiv/ρ P, κ c ] = E [ e ( τ = e ( D ui ρ (τv+τ) ) E [e = e ( D ui ρ ) (τv+τ) = e ( D ui Φ Φ τ ρ(τv+τ) D ui ρ(τv+τ) Dui P, κ c ] D ui ρ ) (τv+τ) P, κ c ] e τ ρ(τv+τ) D ui f ( P ) (κ c 1 P +κc ) f ( P ) dd (κ c 1 P +κc ) ρ ) (τv+τ) e τ ρ(τv+τ) D τ P c d c «P P + m (1 w i )D ui dc ui τ+τ P c d c 1 τ (τ+τ P c ) ρ(τv+τ) D ui (κc ) ( τ P c τ +τ P c (κc ) τ P c τ +τ P c 1/ τ P c + τ d c P P + m (1 w i)d ui d c 1 τ d c (τ +τ P c) D ρ(τ v+τ ) ui 1/ τ P c + τ d c P P + m (1 w i)d ui d c d c 1 We ue g to denote, h to denote τ, (τ v +τ ) (τ v +τ ) gc 1 to denote, τ +τ P c and h c τ to denote P c. τ +τ P The expected utility of uninformed invetor c conditional on informed invetor being hort-ale contrained can then be implified a: E [ e w 1/ρ P, κ c ] D ui = e ρ g e (W R D ui RP )/ρ e h ρ D ui Φ (κc 1 P + κc dc ) hc g c Φ (κc 1 P + κc dc ) hc g c. h c d c P d c P + 1 d c P P + m (1 w i)d ui d c d c P P + m (1 w i)d ui d c d c ««m 1 w i d c D ui 1 ρ gc hd ui + hg c D ui /ρ /. ) / The expected utility of uninformed invetor condition on informed invetor 7

29 being borrowing contrained or uncontrained can be imilarly expreed. E [ e w 1/ρ P, κ bc 1 P + κ bc ] D ui = e ρ g e (W R D ui RP )/ρ e h ρ D ui ( 1 Φ (κbc 1 P + κbc ) d bc hbc P d bc h bc d bc P d bc P dbc P dbc d bc + 1 d bc d bc g bc + 1 d bc ««m 1 w i d bc D ui 1 ρ gbc hd ui m 1 w i d bc ) D ui + hg bc D ui /ρ / 1 Φ (κbc 1 P + κbc P dbc ( ) d bc hbc P d bc g bc d bc + 1 d bc m 1 w i d bc ) D ui, where g bc = 1/(τ + τ bc p ) and h bc = τ P bc/(τ + τ P bc). E [ e w1/ρ P, κ c < < κ bc 1 P + κ bc D ui = e ρ g e (W R D ui RP )/ρ e h ρ D ui Φ Φ (κbc 1 P +κbc (κc 1 P +κc d ) h P d P + 1 d g d ) h P d P + 1 d g ] h d P d P + 1 d m 1 w i d m 1 w i d D ui «+hg D ui /ρ ««m 1 w i d D ui 1 ρ g hd ui D ui «+hg D ui /ρ / Φ «(κbc 1 P d +κbc ) h P P + m (1 w i )D ui d g Φ «κc 1 P d +κc h P P + m (1 w i )D ui d g, where g = 1/(τ + τ P ) and h = τ P /(τ + τ P ). Since we aume the rik-free aet i the numeraire aet and R = 1, the firt-order condition of the optimal demand problem for the riky aet can 8