Asymmetric Information, Short Sale Constraints, and Asset Prices Harold H. hang Graduate School of Industrial Administration Carnegie Mellon University Initial Draft: March 995 Last Revised: May 997 Correspondence Address: GSIA, Carnegie Mellon University, Pittsburgh, PA 53, (4) 68-8836. E-mail Address: huibingz@andrew.cmu.edu. I wish to thank Dean Corbae, Burton Hollield, Uday Rajan, Bryan Routledge, Kristian Rydqvist, Chester Spatt and participants at the Economics and Finance Seminar at Carnegie Mellon University for many helpful discussions and comments. The paper was presented at the 996 Econometric Society Summer Meetings in Iowa City. The nancial support provided through the BP America Research Chair is gratefully acknowledged. All remaining errors are mine.
Asymmetric Information, Short Sale Constraints, and Asset Prices Abstract This paper investigates the eects of short sale constraints on asset prices and trading volume in an asymmetric information rational expectations framework. We classify short sale constraints into two types: short restrictions and proceeds restrictions. The rst limits the amount that traders can short and the second restricts traders' access to the proceeds of their short sales. Both types of restrictions inuence traders with favorable and unfavorable signals on the future payo asymmetrically. While traders rationally adjust their expectations to accommodate the asymmetric impact of the restrictions, their adjustment, in general, can not prevent the decrease in informational eciency of a nancial market. Consequently, imposing short sale constraints can bias equilibrium asset prices and aect trading volume. Using numerical simulations, we nd that imposing short sale constraints can bias the price upward or downward depending on the realizations of the future payo and the liquidity supply, and unambiguously reduce trading volume.
. Introduction An important function of nancial markets is to aggregate the diverse investors' information into market prices. Whether a market price can accurately reect the distribution of investors' true valuation of an asset (often called the informational eciency of a nancial market) depends on the institutional structure of nancial markets and non-institutional market frictions due to incomplete information. Short sale constraints, which are important features of many nancial markets including U.S. markets, have both the institutional and non-institutional features. For example, in U.S. stock markets, mutual funds are prohibited by law from engaging in short selling activity. Investors are prohibited from selling short when the current price is lower than the price of the previous trade (down-tick). Short sales may also be restricted by brokers to prevent investors from engaging in excessive gambling. In addition, the proceeds of short sales are placed in escrow at zero interest and are not available to short sellers for immediate re-investment. All these make it possible for short sale constraints to inuence the eciency of information aggregation of nancial markets. A number of papers have studied the eect of short sale constraints on risky asset prices. For example, assuming common knowledge of information on future payos of risky assets, Heaton and Lucas (996) and hang (997) nd that imposing short sale constraints increases risky asset prices in an innite horizon economy with heterogeneous agents. In their environments, a higher price is required to clear the market when supply of risky assets is limited as a result of imposing short sale constraints. Under the assumption of exogenous heterogeneous expectations on the future payo of risky assets, Miller (977), Jarrow (980), and Figlewski (98) also analyze the eect of short sale constraint on risky asset prices. Miller (977) and Figlewski (98) nd that short sale constraints bias the asset price upward because the market underweighs the traders with unfavorable information due to the imposition of short sale constraints. Jarrow (980) shows that imposing short sale constraints can bias the equilibrium asset price either way depending upon the dispersion of estimation on the covariance matrix
of future asset payos among traders. Using a simple rational expectations model with certain asymmetric information and market structure, Diamond and errecchia (987) nd that short sale constraints do not bias asset prices upward instead they aect the adjustment of speed of prices to private information. In their model, the information structure is the simplest other than perfect information. They assume that there are informed traders who observe identical private information and uninformed traders who observe only public information. All market participants are risk neutral. They have also abstracted from the possible variations in the size of the trades established by traders. Specically, a trader is allowed to buy a single share, sell a single share, short sell a single share, or do nothing. In this paper, we extend the Diamond and errecchia study by adopting a richer information structure and introducing risk averse informed traders who are allowed to trade freely as long as a short sale constraint is satised. The model is similar to the ones in Grossman and Stiglitz (980) and Peiderer (984) in which the unrestricted costless short sale constraint assumption is relaxed. Specically, we assume that informed traders are subject to short sale constraints which are classied into two types. The rst type is the restriction on the amount that a trader can short. It is called the short restriction. The second type is the restriction on a short seller's access to the proceeds of his short sale. It is called the proceeds restriction. Short restrictions censor the valuation of traders with unfavorable information about future payo at certain level. When a trader receives a private signal on the future payo of the risky asset below the level, his valuation about the asset payo is replaced by that of the trader who voluntarily hold the amount of the risky asset equal to the amount specied in the short restriction. This conceals the true valuation of traders facing the binding short restriction. Proceeds restrictions force traders with moderate unfavorable signals to take no position to avoid the opportunity cost associated with short sales. Their valuation of the future payo of the risky asset are also replaced by that of the trader who voluntarily takes no position. Although traders taking no
positions may have dierent valuation on the true future payo, they are all treated the same in aggregating the private information to form the equilibrium asset price. Thus both the short and proceeds restrictions reduce the informational eciency of a nancial market. The reduction in the informational eciency discussed above is partially oset by the increase in the informational eciency as a result of the interaction of the short and proceeds restrictions. Imposing the proceeds restriction reduces the amount of short sales of traders who sell short. This in turn reduces the proportion of traders facing binding short restrictions. As a result, we gain some informational eciency. However, the informational ineciency from censoring trader's true valuation tend to outweigh the eciency gain and the overall informational eciency decreases. Consequently, imposing short sale constraints may aect equilibrium asset prices and trading volume. Our study indicates that while equilibrium always exists for this type of model under the assumption of unlimited borrowing and unrestricted costless short sales, the existence of equilibrium becomes an issue when the latter assumption is relaxed. We show that a rational expectations equilibrium exists under certain conditions. Furthermore, in contrast to the linear equilibrium asset price function obtained for the case without short sale constraints, the equilibrium asset price with short sale constraints becomes a nonlinear function of the future asset payo and the liquidity supply of the risky asset. Because the nonlinearity in the price function complicates the analytical study of comparative statics, we resort to numerical simulation analysis. For parameter values consistent with existing studies, our numerical simulations reveal the following results. First, imposing short sale constraints can bias equilibrium asset prices upward or downward depending on the liquidity supply and the realization of future payo. When the liquidity supply is low, short sale constraints bias the price upward; when the liquidity supply is high, the constraints bias the price downward. This is due to higher rate of price change with respect to the change in the liquidity supply with the presence of the constraints than without. The higher rate of price change is reected by the 3
steeper equilibrium price and liquidity supply relationship for dierent levels of payo realizations. Both the short and proceeds restrictions bias the prices in the same direction. Second, trading volume are signicantly reduced when short sale constraints are imposed. This is due to both restricted short selling and traders taking no positions. Trading volume are in general higher when the payo is either low or high than when the payo is close to its mean. Third, when there is only the short restrictions, the proportion of traders facing the binding restriction increases as the payo increases. This can be attributed to the higher equilibrium price induced by imposing the restriction which makes short selling more protable. When the proceeds restrictions are also imposed, the proportion of traders facing binding short restrictions decreases as a result of the interaction of the two restrictions. However, a large proportion of traders takes no position which reduces the overall informational eciency of the nancial market. The rest of the paper is organized as follows. Section presents the model. Section 3 analyzes the eects of short restriction on asset prices and trading volume. Section 4 extends the model to include the proceeds restriction. Section 5 discusses the numerical results. Section 6 provides the concluding remarks.. The Model The model described here is similar to the models analyzed by Grossman and Stiglitz (980) and Peiderer (984). There exist a continuum of informed traders represented by i [0; ], and a few liquidity traders. Both types of traders live for two periods. Informed traders have the same initial wealth W 0 which they completely invest in two nancial assets: a riskless asset and a risky asset in the rst period. They then consume all the proceeds in the second period. The riskless asset earns a xed gross return R. The risky asset, however, can be bought at price P per share and its payo tomorrow, denoted F, is a random variable. We assume that the future payo F is normally 4
distributed with mean F and variance F. The liquidity traders only supply to or demand from the risky asset market. Their supply or demand, denoted X s, follow a normal distribution with mean zero and variance x. The information structure is characterized as follows. Every trader observes the market price P. In addition to that, each informed trader receives a private signal, denoted Y i, on the future asset payo F. The private signal, however, has a noisy component denoted by i. Thus, the relationship between the private signal and the future payo can be represented as Y i = F + i ; for i [0; ]: We assume that the noisy component i has a normal distribution with mean 0 and variance and it is independent of the future payo F and invariant across traders. We also assume that the noisy components are uncorrelated across individuals. Each trader's private signal on future payo is therefore normally distributed with mean F and variance +. F Each informed trader has preferences over the end of period wealth characterized by the constant absolute risk aversion (CARA) utility function. The advantage of adopting CARA utility function is that under normality assumption of future payos the utility maximization problem reduces to the mean-variance problem as long as the conditional distribution of the payo remains a normal distribution. Let I i = fy i ; P g be the information set of the trader i [0; ] and be his absolute risk aversion coecient which is assumed to be the same across traders. If there were no short sale constraints, the objective of the trader i [0; ] would be to choose a portfolio to maximize E(W i ji i )? ar(w i ji i ); for i [0; ]; where W i = (W 0?P x i )R+F x i and x i are trader i's second period wealth and his risky asset holding, respectively. Next we introduce short sale constraints. In the following two sections, we rst impose the short restriction but allow short sellers full access to the proceeds of their 5
short sales. We then extend the model to incorporate the proceeds restrictions. 3. Short Restrictions In reality, short restrictions can be created by law, for example, mutual funds are prohibited from engaging in short selling, or created by brokers, for example brokers may restrict a client's short selling activity by setting a limit if they suspect that the client may engage in excessive gambling. For now, we allow a short seller full access to the proceeds of his short sale. By this we mean that when an investor short sells the risky asset the funds generated by his short sale are immediately available for reinvestment in the riskless asset. This may not be possible for most of the traders but it may happen to large traders who are sometimes able to arrange more favorable terms for short sales, including the payment of some interests on the proceeds. For simplicity, we assume that every trader faces the same short restriction denoted?a x (A x 0). Let x i be trader i's holding of the risky asset. His objective is characterized as follows: where W i = (W 0? P x i )R + F x i. max x i E(W i ji i )? ar(w i ji i ); () such that x i?a x ; for i [0; ]; () It can be shown that the optimal risky asset holding for a trader who does not face a binding short restriction is given by: x i = E(F ji i )? P R (3) where = ar(f ji i ) is the conditional variance of future asset payo. Equation (3) indicates that if a trader expects the future payo [E(F ji i )] to be higher than the opportunity cost of holding a share of risky asset (P R), he will buy the asset. If the trader expects the future payo to be lower than the opportunity cost, he will short sell the asset. We can also conclude that for a given asset price a trader's 6
optimal risky asset holding is an increasing function of his expectation of future payo of the risky asset. To establish the direct linkage between a trader's risky asset holding and his private information, we make the following assumption on the relationship between an informed trader's expectation of the future payo and his private signal on the future payo. Assumption A trader's expectation of the future payo of the risky asset is an increasing function of his private signal on the future payo. This assumption says that if a trader receives a favorable signal on the future payo of the risky asset, his expectation of the future payo will be high. On the other hand, if a trader receives an unfavorable signal on the future payo of the risky asset, his expectation of the future payo will be low. This is consistent with our equilibrium result introduced later. Under this assumption, for a given asset price a trader's optimal risky asset holding is also an increasing function of his private signal on the future payo. This makes it possible to identify the level of private signal on the future payo which separates binding traders from nonbinding traders. For the convenience of exposition, we introduce the following denition. Denition A trader is the marginal nonbinding trader if his optimal risky asset holding is equal to the short restriction?a x. It is important that the marginal nonbinding trader voluntarily holds?a x amount of the risky asset rather than being restricted to do so. Let Y a be the marginal nonbinding trader's private signal on the future asset payo. We have the following result on the property of Y a. Lemma For a given asset price P and a nite short restriction?a x, under Assumption, there always exists a private signal level Y a such that a trader faces binding short restriction if his private signal is lower than Y a and he faces a nonbinding one if his private signal is higher than Y a. 7
Proof: see Appendix. Therefore, for any trader i, we can represent his risky asset holding as follows: x i = ( E(F ji i )?P R if Y i Y a ;?A x otherwise: (4) The market clearing condition for the risky asset can then be characterized by: (Y i <Y a ) (?A x )di + (Y i Y a ) We can rewrite the above market clearing condition as follows: (Y i Y a ) E(F ji i )? P R di = X s : (5) E(F ji i ) di + P R [? F Y (Y a )]? A x F Y (Y a ) = X s : (6) where F Y (Y a ) is the normal cumulative distribution function of private signal Y i evaluated at the marginal nonbinding trader's private signal level Y a. With the presence of liquidity traders and short restrictions, the existence of equilibrium becomes a complex problem. For a given short restriction, the equilibrium depends on the realization of liquidity supply (or demand). If liquidity traders demand more than what informed traders are allowed to sell short, i.e., X s <?A x, the risky asset market will not be cleared. To resolve this problem, we introduce the market maker into the economy. The role of the market maker is to ration the liquidity trader's demand for the risky asset at some nite price and make this information public. Specically, since the market maker observes the demand and supply, when X s <?A x, it rations the liquidity trader's demand at?a x and sets the price at the minimum price under which the liquidity demand equals?a x, i.e., minfp : X s =?A x g. 3 This gives rise to a rationing equilibrium minfp : X s =?A x g when X s <?A x. Alternatively, if the liquidity traders demand less than what informed traders are allowed to short or liquidity traders are net suppliers, then there will be some informed traders who face nonbinding short restrictions [? F Y (Y a ) > 0]. In this case, there exists an rational expectations equilibrium. Let P e denote the equilibrium risky asset price. 8
Proposition For a given short sale restriction (?A x ), if the liquidity traders demand more than the informed traders are allowed to sell short, there is a rationing equilibrium given by minfp e : X s =?A x g. Alternatively, if X s?a x, there exists a rational expectations equilibrium of the form P e = g(f? F? X s ); (7) where g(:) is given by the following inverse function g? (P ) = f(p R? F )[? F Y (Y a )] + A x F Y (Y a )? f Y (Y a ) g; (8) +? F Y (Y a ) and = f F + + [? F Y (Y a )] g? ; = [? F Y (Y a )] ; = [? F Y (Y a )] 3 ; x 4 x and f Y (Y a ) is the normal probability density function of the private signal evaluated at Y a. Proof: see Appendix. A related conclusion on the equilibrium asset price without short restrictions can be drawn from this result. We can think of the model without short restrictions as a special case of our model after we loose up the short restriction such that the probability of a given trader facing a binding constraint is zero. We summarize this in the following corollary. Corollary If the probability of traders facing a binding short restriction is zero, i.e., F Y (Y a ) = 0, there exists an equilibrium asset price that can be represented as a linear function of the future asset payo and the liquidity supply of the asset, in particular, we have: where and are given by P e = F R + (F? F ) + X s ; = R ( + ); =? x R ( + ); and = ( x F + + )? : x 9
Proof: see Appendix. This is consistent with the nding of many other theoretical studies on the equilibrium asset prices in the asymmetric information rational expectations models such as Grossman and Stiglitz (980), Peiderer (984), Wang (994), among others. Proposition allows us to write the equilibrium asset price as follows: P e = F R + + R[? F Y (Y a )] (F? F )? ( + ) R[? F Y (Y a )] X s (9)? F Y (Y a ) R[? F Y (Y a )] A x + f Y (Y a )? F Y (Y a ) : A rst glance of the price function might mislead you to conclude that P e is linear in F and X s. It is in fact a nonlinear function. The reason is that the marginal nonbinding trader's signal level (Y a ) is a function of the equilibrium asset price. Hence, the proportion of traders who face binding short restriction, F Y (Y a ), is also a function of the equilibrium asset price. This makes the equilibrium asset price a nonlinear function of the payo and the liquidity supply but in an implicit form. While the nonlinearity in the price function makes it hard to conduct analytical comparative statics, we have the following observations. First, the conditional variance of the future payo is higher with short restrictions than without. This is made clear by comparing the conditional variance given in Proposition for F Y (Y a ) > 0 versus F Y (Y a ) = 0. Intuitively, this is because the information revealed by the equilibrium asset price only reects the true valuation of traders who face the nonbinding short restriction and these traders account for [?F Y (Y a )] of total informed traders. Second, the coecients of both (F? F ) and X s are now functions of [? F Y (Y a )], which makes the changes in the coecients ambiguous. For a given pair of (F? F ; X s ), the imposition of short restrictions may bias the price upward or downward depending on which factor dominates. We will discuss the quantitative eect of short restrictions on the price using numerical simulations in Section 5. Third, the upward biasedness is 0
partially oset by subtracting F Y (Y a ) R[? F Y (Y a )] A x from the price P e. This term has not been given enough attention in the previous studies [Miller (977), Jarrow (980), and Figlewski (98)] because they assume that A x = 0. In fact, the eect of the term on the equilibrium price becomes larger as more traders face binding short restrictions. The presence of this term in the price function indicates that investors realize that traders receiving private signals below Y a face the binding short restriction and their short sales amount to F Y (Y a )A x shares of the risky asset. They, therefore, adjust their expectations to accommodate this change. Finally, the last term, f Y (Y a )? F Y (Y a ) ; tends to bias the equilibrium asset price upward. This is the term that has been emphasized by the previous studies [Miller (977), Jarrow (980), and Figlewski (98)]. The biasedness increases as more traders face binding short restrictions. This term reects the fact that short restrictions conceal the information on the optimal asset demand of traders who face the binding short restriction. After having discussed the eects of short restrictions on the equilibrium asset price, we now turn to the trading volume. Since traders in the model only change their asset holdings once, assuming that traders' initial risky asset holdings are zero, the trading volume, denoted ol, can then be dened as 4 ol = i[0;] jxi jdi + jx s j (0) Let Y n be the private signal of the trader who voluntarily takes no position in the risky asset market, that is, his optimal risky asset holding is zero. Under Assumption, a trader i is a buyer if his private signal on the future payo is higher than Y n and he is a short seller if his private signal on the future payo is lower than Y n. The total trading volume then consists of four components: short sales of binding informed
traders, short sales of nonbinding informed traders, purchases of informed traders, and liquidity supply, i.e. ol = A xf Y (Y a )? + (Y n Y i ) (Y a Y i Y n ) E(F ji i )? P R di + jx s j: E(F ji i )? P R di () The short sales of the nonbinding informed traders and the purchases of informed traders, the second and the third term on the right hand side, can further be solved using the results provided in Proposition. We have (Y a Y i Y n ) E(F ji i )?P R di () = + f[ F? P R +? g (P )? F Y (Y a ) ][F Y (Y n )? F Y (Y a )] q F + a (Y? F ) ( p [e? + F )? e? (Y n? F ) ( + F ) ]g [? FY (Y a )] (Y n Y i ) E(F ji i )?P R di (3) = + f[ F? P R +? g (P )? F Y (Y a ) ][? F Y (Y n )] q F + p [? FY (Y a )] n (Y? F ) ( e? + F ) g If the equilibrium asset price were held unchanged, imposing a short restriction would reduce trading volume because traders with binding short restrictions were unable to short as much as they would have had the short restriction not been imposed. Since the equilibrium asset price changes when a short restriction is imposed, the eect on trading volume is dicult to determine analytically. In section 5, we use numerical simulations to analyze the eects of short restrictions on trading volume. Next, we extend our model to include proceeds restrictions.
4. Proceeds Restrictions While it is possible for large traders to negotiate with brokers on terms of short sales including some interest payment on the proceeds of their short sales, for most traders the proceeds of their short sales are placed in escrow at zero interest and are not available for immediate re-investment. This creates the second type of restriction of short sales: proceeds restrictions. Because of the proceeds restriction, short sellers incur the loss of interest on the proceeds of their short sales. This discourages traders with moderate unfavorable signals on the future payo from short selling. For simplicity we assume that proceeds restrictions apply to every trader no matter how large he is. A trader's objective is to maximize the end of period wealth given in equation () and subject to the short restriction represented by equation () and the proceeds restriction which does not have direct eects on traders who buy assets. Thus, for asset buyers, the optimal asset holding given in equation (3) still holds. For short sellers, however, the proceeds restriction does have eects. If a trader short sells (?x i ) shares of the risky asset, he does not earn interest on the proceeds of his short sale. We therefore subtract P (?x i )(R? ) from (W 0? P x i )R + F x i, which gives the new end of period wealth W i = W 0 R? P x i + F x i. If a trader sells short, his risky asset holding is characterized as x i = ( E(F ji i )?P E(F jii)?p if?a x ;?A x otherwise: (4) It is important to notice that traders with moderate unfavorable signals on the future asset payo may change their position from short selling to doing nothing. The following result provides the condition that a trader will take no position with the proceeds restriction. Lemma A trader takes no position on the risky asset market if his expectation of the future asset payo satises the following condition: P E(F ji i ) P R: 3
Proof: see Appendix. Under Assumption, for a given price, a trader's position is completely determined by his private signal. We can then nd the private signal levels that divide informed traders into the following four dierent groups: binding short sellers, nonbinding short sellers, traders taking no positions, and buyers. In the previous section, we have dened the marginal nonbinding trader which allows us to identify the binding and nonbinding traders of the short restriction. We now introduce one more denition which will allow us to identify traders taking no positions. Denition A trader is the upper bound no position taking trader if his expectation of the future asset payo is equal to PR; A trader is the lower bound no position taking trader if his expectation of the future asset payo is equal to P. Let Y s be the lower bound no position taking trader's private signal and Y n be the upper bound no position taking trader's private signal. Under Assumption, Y a lies below Y s and Y s lies below Y n. If trader i's private signal on the future payo lies below the marginal nonbinding trader's private signal, Y i < Y a, he faces binding short restriction. If the trader's private signal lies between the marginal nonbinding trader's private signal and the lower bound no position taking trader's private signal, he sells short. If his private signal lies between the lower bound no position taking trader's private signal and the upper bound no position taking trader's private signal, he takes no position. Otherwise, the trader buys the risky asset. We can then express trader i's risky asset holding as: 8?A x if Y i < Y a ; >< E(F ji i )?P if Y x i = a Y i < Y s ; 0 if Y s Y i Y n ; >: E(F ji i )?P R if Y i > Y n : (5) The market clearing condition can be written as follows: (Y i <Y a ) (?A x ) di + (Y a Y i <Y s ) E(F ji i )? P di (6) 4
+ (Y s Y i Y n ) 0di + (Y i >Y n ) E(F ji i )? P R di = X s : Combining terms and rearranging yield: (Y a Y i <Y s ) E(F ji i )di + (Y i >Y n ) E(F ji i )di? P [F Y (Y s )? F Y (Y a )] (7)? P R[? F Y (Y n )]? A x F Y (Y a ) = X s : Similar to the case with short restrictions only, when liquidity traders demand more than what informed traders are allowed to sell short, the market maker steps in to impose the ration and set the price to the minimum price under which the liquidity trader's demand equals?a x. When liquidity traders demand less than what informed traders are allowed to short or liquidity traders are net suppliers of risky assets, for any nite price, there will be some informed traders want to buy risky assets because private signals on the future payo of the risky asset are normally distributed, therefore?f Y (Y n ) > 0. Since F Y (Y s )?F Y (Y a ) 0, we have?f Y (Y n )+F Y (Y s )?F Y (Y a ) > 0. We thus introduce the following result on the equilibrium for the model with both short restrictions and proceeds restrictions. Proposition For a given short restriction (?A x ), if the liquidity traders demand more than what the informed traders are allowed to sell short (X s <?A x ), there exists a rationing equilibrium given by minfp e : X s =?A x g. If X s?a x, then there exists a rational expectations equilibrium of the form P e = g(f? F? X s ); where g(:) is given by the following inverse function g? (P ) = + f(p R? F )[? F Y (Y n ) + F Y (Y s )? F Y (Y a )] + A x F Y (Y a )? [f Y (Y n )? f Y (Y s ) + f Y (Y a )]? F Y (Y n ) + F Y (Y s )? F Y (Y a )? P (R? )[F Y (Y s )? F Y (Y a )]g; (8) 5
and = f F + + [? F Y (Y n ) + F Y (Y s )? F Y (Y a )] g? ; x = [? F Y (Y n ) + F Y (Y s )? F Y (Y a )] ; = [? F Y (Y n ) + F Y (Y s )? F Y (Y a )] 3 4 x ; and F Y (:) and f Y (:) are the normal cumulative distribution function and the normal probability density function of the private signal, respectively. Proof: see Appendix. To see the net eects on the equilibrium asset price of the proceeds restriction, we relax the short restriction until the probability of traders facing a binding one is zero, i.e., F Y (Y a ) = 0. The equilibrium asset price is obtained by substituting F Y (Y a ) = 0 into the above equations. On the other hand, if the proceeds restriction is not binding for any trader, we should get the equilibrium result reported in the last section in which only the short restriction is imposed. Indeed, this is true if we set the gross rate of return of the riskless asset (R) to. When R =, the net rate of return of the riskless asset is 0, thus, there is no cost associated with the proceeds restriction. In that case, the proceeds restriction is not binding for any trader. To see the eects of both restrictions on the equilibrium asset price, we use the result in Proposition and rearrange terms to arrive at P e =? F R(? ) + ( + )(F? F ) R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )](? ) ( + ) X s R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )](? ) (9)? + F Y (Y a )A x R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )](? ) [f Y (Y n )? f Y (Y s ) + f Y (Y a )] R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )](? ) ; 6
where is given by: = (R? )[F Y (Y s )? F Y (Y a )] R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )] : The following changes have been introduced. First, [? F Y (Y a )] is replaced by [? F Y (Y n ) + F Y (Y s )? F Y (Y a )]. Notice that [F Y (Y n )? F Y (Y s )] represents the proportion of traders who are now taking on positions because of the proceeds restriction. This indicates that traders have adjusted their expectation to accommodate the eect of the proceeds restriction on traders' behavior. Since? F Y (Y n ) + F Y (Y s )? F Y (Y a ) > 0, the conditional variance of the future payo of the risky asset is higher with the short sale constraint than without. This is made clear by examining the conditional variance formula given in Proposition. Second, imposing the proceeds restriction introduces the term below [f Y (Y n )? f Y (Y s )] R[? F Y (Y n ) + F Y (Y s )? F Y (Y a )](? ) which tends to further bias the equilibrium asset price upward. This is emphasized in the previous studies on the eects of the proceeds restriction on the equilibrium asset price. Third, the changes in the equilibrium asset price have been multiplied by [=(? )], which is also the consequence of the proceeds restriction. All the changes, however, make the changes in coecients of (F? F ) and X s even more ambiguous. Thus, the overall eects of the two restrictions on the equilibrium asset price are ambiguous. We resort to numerical simulations to provide quantitative results on the eect on prices of imposing short and proceeds restrictions. Next, we discuss the eect of the proceeds restriction on informational eciency of a nancial market. When the proceeds restriction is introduced, it has the following two eects on informational eciency. First, some of the traders who originally sell short now take no positions. When traders have access to the proceeds of their short sales they short even with moderate unfavorable signals on the future payo of the risky asset because they can re-invest the proceeds in the riskless asset and earn riskfree gross rate of return R. However, the proceeds restriction makes the proceeds of short 7
sales unavailable to these traders. In this case, they would rather take no positions than sell short the risky asset which earns an expected rate of return that does not compensate the risk they have to take. When these traders take no positions, their true valuation of the future payo of the risky asset can not be revealed. What the market aggregates is the valuation of the trader who voluntarily takes no position. Second, traders who still short reduce their short sales with the presence of proceeds restriction. This is because that denying their access to the proceeds of short sales makes their short sales less protable. However, this behavior may not aect the market eciency because in a rational expectations environment traders realize this reduction in short selling and will compensate that by adjusting their expectations correspondingly. On the other hand, because short sellers reduce their short sales, there may be fewer traders facing the binding short restriction than before which may contribute to the eciency improvement. This may explain why Diamond and errecchia (987) found that proceeds restriction increases informational eciency of a nancial market. Next, we analyze the eect of the proceeds restrictions on trading volume of the risky asset. Assuming that traders' initial risky asset holdings are zero, we can dene trading volume for this case in the same manner as what we did in last section (see Equation 0). The above denition can then be written as follows taking into account that traders with private signals lying in the interval [Y s ; Y n ] take no positions in the risky asset market: ol = A xf Y (Y a )? + (Y n Y i ) (Y a Y i Y s ) E(F ji i )? P E(F ji i )? P R di + jx s j: di (0) Applying the result in Proposition, we can write the second and the third term below: (Y a Y i Y s ) E(F ji i )?P di () 8
= + f[ F? P + g? (P ) ][F Y (Y s )? F Y (Y a )] q F + p [e? (Y a? F ) ( + F )? e? (Y s? F ) ( + F ) ]g; (Y n Y i ) E(F ji i )?P R di () = + f[ F? P R +? g (P ) ][? F Y (Y n )] q F + p e? (Y n? F ) ( + F ) g; where =? F Y (Y n ) + F Y (Y s )? F Y (Y a ). Since the proceeds restriction forces some traders to take no positions and reduces the short sales of traders who go short in the risky asset market, we may nd the overall trading volume decrease when the proceeds restriction is introduced. However, traders who buy the risky asset are also aected by the proceeds restriction indirectly via the asset price. Since the overall eect on the asset price is ambiguous, the eect on the trading volume associated with asset buyers are too ambiguous, which also makes the change in the total trading volume ambiguous. We will use numerical simulations to analyze the eects of the two restrictions in the next section. 5. Numerical Simulations In this section we discuss the eects of short restrictions and/or proceeds restrictions on equilibrium asset prices and trading volume using numerical simulations. For dierent sets of structural parameters, 5 we solve for the equilibrium asset prices and trading volume for three models: the benchmark model without short sale constraints, the model with a short restriction but no proceeds restrictions, and the model with both short restrictions and proceeds restrictions. 5. Parameter alues 9
To conduct the numerical analysis, we need to choose parameter values for trader's risk aversion, the rate of return of the riskless asset, and the distributions of the payo of risky asset, the private signal, and the liquidity supply. Following Wang (994), we choose the absolute risk aversion coecient to be.0 for all traders. The rate of return of the riskless asset is chosen to be 0.4 percent which is consistent with the real returns on Treasury bills. The payo of the risky asset is calibrated to the value weighted stock returns of market portfolio. The mean payo is set at 8 percent and the variance is chosen to be 0.04 (the corresponding standard deviation is 0 percent per year). For the variance of the private signal and the variance of the liquidity supply, we use the values suggested in Gennotte and Leland (990). The variance of the signal is set at 0.4 and the variance of the liquidity supply is chosen to be 0.0007. In addition to the above parameters, we also need to specify the short sale limit. Since no short sale has been assumed in many studies, we might want to set the short sale limit to 0. However, when the short sale limit is 0, traders are not allowed to short at all. As discussed in Propositions and, in this case, a rational expectations equilibrium is only possible when the liquidity traders are net suppliers of the risky asset on the market, in other words, there are positive liquidity supply. Otherwise, the market will be in a rationing equilibrium set by the market maker. To allow liquidity traders to demand risky assets from the market in the model, we choose positive short sale limit values. We have experimented with a range of short sale limit values and the results are qualitatively similar. The reported results are for the short sale limit equal to 0.5. 5. Numerical Algorithm For a given set of parameter values, we use the following numerical algorithm to nd equilibrium asset prices, proportions of traders in various groups, and trading volume. We rst simulate a number of realizations for the asset payo and the liquidity supply (one hundred realizations are simulated for each variable). For every combination of the asset payo and liquidity supply (there are ten thousand such combinations), we 0
then solve for the equilibrium asset prices determined implicitly in Propositions and. This involves the following steps: () Choose some initial signal levels Y a, Y s, and Y n (Y a < Y s < Y n ) and solve for,,, using the results in Propositions and, and the risky asset prices (P ) using Equation 9 for the model with short restrictions only and Equation 9 for the model with both short and proceeds restrictions; () Update the signal levels Y a, Y s, and Y n using the parameter values, the risky asset price computed above and the following equations: In the model with short restrictions only: E(F jy a ; P )? P R =?A x ; (3) In the model with both short and proceeds restrictions: E(F jy a ; P )? P =?A x (4) E(F jy s ; P ) = P (5) E(F jy n ; P ) = P R (6) (3) Replace the initial Y a, Y s, and Y n by their updated values and re-compute,,, and the risky asset prices (P ). Iterate the above steps until Y a, Y s, Y n, and P converge according to some prespecied criteria. The proportions of traders in various groups [identied by F Y (Y a ), F Y (Y s ), and F Y (Y n )] are determined simultaneously as the equilibrium prices. (4) Use the converged results given above to solve for the equilibrium trading volume using Equations,, and 3 for the model with short restrictions only and Equations 0,, and for the model with both short and proceeds restrictions. 5.3 Discussions of Numerical Results To demonstrate the relationship between the important endogenous variables such as equilibrium asset prices, trading volume, and proportions of traders facing binding
short and/or proceeds restrictions and the exogenous variables such as the payo and the liquidity supply, we plot the endogenous variable against one exogenous variable at a time while holding the other xed at dierent levels (low and high). 5.3. Equilibrium Asset Prices and Short Sale Constraints We rst discuss the eect of short sale constraints on the equilibrium asset prices. The top two panels of Figure show the relationship between the equilibrium asset price and the future payo of the risky asset. The panel on the left corresponds to a low liquidity supply or liquidity demand (X s =?0:0394) and the one on the right corresponds to a high liquidity supply (X s = 0:057). Both panels show that the equilibrium asset prices are positively related to the future payo with or without short sale constraints. In both panels the equilibrium asset prices are linear functions of future payo without short sale constraints, which are consistent with the theory. When the short restriction and/or the proceeds restriction are imposed, the following changes have been introduced. First, the relationship becomes nonlinear. This is consistent with our theoretical nding reported in the last two sections. Second, short sale constraints can bias the equilibrium asset prices upward or downward depending on the liquidity supply. When the liquidity supply is low, the short sale constraints bias the price upward. When the liquidity supply is high, the short sale constraints bias the price downward. Intuitively, when the liquidity supply is low, it puts upward pressure on the price. With the presence of the short sale constraint the upward pressure on the asset price is even higher because imposing the short sale constraint limits individual trader's supply of the risky asset. Liquidity traders have to oer a higher price to induce traders facing nonbinding short restriction to short more, traders taking no positions to nd it protable to short, and even buyers to become sellers. This is reected as a higher coecient for the liquidity supply in the pricing function with the short sale constraint than without (to be discussed in detail later). Since liquidity supply negatively aects the equilibrium price, a high coecient coupled with a low liquidity supply will yield a higher price than
a low coecient. Similarly, a high coecient coupled with a high liquidity supply will yield a lower asset price than a low coecient. This explains why we observe a lower price with the short sale constraint than without when the liquidity supply is high. Third, both the short and proceeds restrictions work in the same direction. They both either bias the equilibrium price upward in case of low liquidity supply or bias the price downward in case of high liquidity supply. Fourth, the biasedness induced by the short restriction increases as the future payo increases. The reason is that the proportion of traders who face binding short restriction increases as the future payo of the risky asset increases. This can be seen in the top panels of Figures 3 and 4 which will be discussed in detail later. The bottom panels of Figure show the relationship between the equilibrium asset prices and the liquidity supply. The left panel shows the price and liquidity supply relationship when the payo is low (F = 0:606). The right panel shows the price and liquidity supply relationship when the payo is high (F = :603). Both panels indicate that the equilibrium prices are negatively related to the liquidity supply. The equilibrium price in the case with unrestricted short sale is linear in the liquidity supply which is again consistent with the theoretical result of previous studies. The panels reveal that imposing short sale constraints makes the following dierences. First, the equilibrium asset price becomes a nonlinear function of the liquidity supply, which is consistent with our theoretical results. Second, imposing short sale constraints increases the rate of price change with respect to the change in liquidity supply, in other words, the coecient of the liquidity supply is higher in the pricing function. This is revealed by the steeper equilibrium price and liquidity supply relationship for both the low and high payo realizations. As a result, imposing short sale constraint can bias the price upward or downward. Specically, low liquidity supply or liquidity demand biases the price upward and high liquidity supply biases the price downward. The qualitative feature of this nding remains unchanged regardless of the level of realizations of the payo. However, the entire schedule is higher when the payo is high than when it is 3
low. Third, the short and proceeds restrictions bias the prices in the same direction. This result reinforces the nding reported above on the price and payo relationship. 5.3. Trading olume and Short Sale Constraints Next, we discuss the eect of short sale constraints on the trading volume of the risky asset which are presented in Figure. The top panels of Figure show the volume and payo relationship. The left panel corresponds to a low liquidity supply and the right panel corresponds to a high liquidity supply. Both panels show `U' shaped curves for the volume and payo relationship. This implies that trading is more active when the payo is either low or high than when it is close to the mean. The result is consistent with the empirical nding of Gallant, Rossi, and Tauchen (99). We also nd that both short restriction and the proceeds restriction reduce trading volume of the risky asset. As stated in the last two sections, both short and proceeds restrictions reduce short sales. This contributes to the decrease of the trading volume. The proceeds restriction further reduces trading volume by forcing traders with moderate unfavorable signals on the future payo taking no positions in the risky asset market. However, because the change in price can lead to changes in trading volume by inuencing both buyer and seller's asset holdings, the theoretical analysis in the last two sections do not provide denite answer to the eects of short sale constraints on trading volume. Here, our numerical simulations show that signicant decreases in volume are induced by short sale constraints. The bottom panels of Figure show the relationship between trading volume and the liquidity supply of the risky asset. The interesting observation is that trading volume increases as the liquidity trading increases. The only case that does not show this pattern is when both the short restriction and the proceeds restriction are imposed and the payo is low and liquidity traders demand risky assets. What happens in this case is that there is a large proportion of traders taking no positions which outweighs the increase in the trading volume due to increased liquidity demand. This can be clearly seen in the bottom left panel of Figure 3. The gap between the short 4
sellers and asset buyers is large and increases as liquidity traders change their positions from net suppliers to net buyers of the risky asset. 5.3.3 Proportions of arious Groups of Traders and Short Sale Constraints Finally, we examine the proportions of various groups of traders according to their risky asset holdings. The top panels of Figures 3 and 4 show the relationship between proportions of various groups of traders and the future payo of the risky asset. The left panel corresponds to a low liquidity supply and the right one corresponds to a high liquidity supply. When there is only the short restriction (Figure 3), the proportion of traders facing the binding restriction increases as the future payo increases. The nding is also invariant to dierent levels of liquidity supply. As we have mentioned before, when the payo increases, equilibrium asset price increases. Short selling becomes more protable and traders will short more ceteris paribus. This implies that more traders will face the binding short restriction. When the proceeds restriction is imposed (Figure 4), the proportion of traders facing binding short restriction decreases from 0{60% to 0.0-5%. This is consistent with our theoretical ndings reported in the last section which indicates that the interaction between the short restriction and proceeds restriction reduces the proportion of traders facing binding short restriction and induces some eciency gain. However, the gures also show that there are 50{70% traders now taking no positions [the gap between F Y (Y s ) and F Y (Y n )] as a result of imposing the proceeds restriction. Thus, the loss of informational eciency outweighs the gain. The combined eects of the short restriction and proceeds restriction are to reduce the informational eciency which is consistent with our theoretical nding. The bottom panels of Figures 3 and 4 show the relationship between proportions of various groups of traders and the liquidity supply. In comparison to the relationship between the various proportions and the future payo, the relationship between the various proportions and the liquidity supply is very moderate. In fact, when there is only the short restriction in the model (Figure 3), the proportion of traders facing binding restriction is almost invariant as a function of the liquidity supply for a given 5