Constrained Short Selling and the Probability of Informed Trade

Tyler R. Henry† University of Washington

This Draft: January 2005

Abstract

This paper considers the effect of private information on the returns to stocks with high levels of short interest. Using a structural trade model from the market microstructure literature, I model the impact of short selling constraints on the probability of information based trade (PIN). The model shows that PIN increases as the constraint on informed short selling is lowered. Empirically, I estimate PIN for a sample of high short interest stocks and form monthly portfolios based on PIN. After controlling for size, book-to-market, and momentum, I find that high PIN portfolios generally produce negative abnormal returns, while low PIN portfolios do not. The results support three main conclusions. First, some stocks with high levels of short interest do underperform. Second, the underperformance of high short interest stocks may be better explained by the probability of informed trading rather than by short interest levels alone. Finally, if PIN is a valid proxy for informed trading, then asymmetric information models offer new insights for the returns to high short interest stocks.

I would like to thank my committee members for discussions and guidance, including Jennifer Koski (Chair), Jonathan Karpoff, Ed Rice, and Lance Young. I also appreciate the valuable comments of Bart Danielsen, Wayne Ferson, Jaehoon Hahn, Andrew Karolyi, and seminar participants at the University of Washington, the 2004 FMA Annual Meetings, and the 2004 FMA Doctoral Consortium. I am responsible for any remaining errors.

† Ph.D. Student, Department of Finance and Business Economics, University of Washington, Box 353200, Seattle, WA 98195-3200. Email: [email protected]. I. Introduction

This paper offers a new approach to assess the information content of short interest by examining the relationship between informed trading and the returns to high short interest stocks.1 I use a measure of the probability of informed trading derived from a microstructure trade model [Easley, Kiefer, and O’Hara (1997b)] to help explain underperformance in stocks with high levels of short interest. I find that this measure is successful at predicting underperformance in highly shorted firms. In general, the results indicate that the underperformance of high short interest stocks may be limited to stocks with high levels of informed trading. Thus, a full understanding of the returns to these stocks may require the consideration of information asymmetries. This result is not surprising given both informed and uninformed motivations for short selling. This work contributes to two areas of the literature. First, it adds to the literature examining the information content of short interest. Traditionally, this line of research tests the relationship between the level of short interest in a stock and its future returns. Theory implies that short interest may be a negative information signal. Diamond and Verrecchia (1987) show that short selling is more likely to be carried out by informed traders with negative information due to the high costs associated with short sales. Figlewski (1981) argues that the actual level of short interest is a proxy for the unfulfilled demand to short sell, and therefore represents the amount of adverse information excluded from the market. The empirical prediction that emerges from these arguments is that stocks with high short interest will have low subsequent returns. Yet, the empirical results in this area have been mixed. Early papers did not find a strong or consistent relationship between short interest and subsequent returns, and claimed that a considerable amount of short selling was arbitrage or hedging related [Brent, Morse, and Stice (1990), Figlewski and Webb (1993), Senchack and Starks (1993), Woolridge and Dickinson (1994)]. Such trades are not motivated by information and should not have predictive power for future returns. Later papers found that stocks with high levels of short interest generate subsequent negative abnormal returns, and argued that short sellers do have information [Asquith

1 Short interest is the number of outstanding short positions for an individual stock. Short interest figures are collected by the stock exchanges and released to the public on a monthly basis. All references to short interest levels in this paper refer to the short interest ratio, which is the number of short positions divided by the number of shares outstanding. Further details are contained in Section IV. and Meulbroek (1996), Gintschel (2001), Desai et al. (2002)]. Recently, Asquith, Pathak, and Ritter (2004) have questioned the robustness of these findings. The approach in these recent papers is to form monthly portfolios based on the level of short interest in a stock and test for subsequent underperformance. Following Figlewski’s (1981) assumption that short interest is a proxy for negative information, they posit that negative abnormal returns should be increasing in the level of short interest. There are two concerns with this assumption. First, as pointed out by D’Avolio (2002) and Chen, Hong, and Stein (2002), variations in short interest levels across stocks could reflect differences in the transaction costs of short selling rather than differences in information. Secondly, this assumption disregards the role of short selling related to uninformed trading strategies. The second issue is where this paper offers additional guidance. Specifically, the contribution of this paper is to more finely characterize which high short interest stocks should have low subsequent returns. Because some short selling may be informed while some may be related to uninformed arbitrage or hedging strategies, the link between the level of short interest and the level of informed trading is uncertain. High levels of short interest may or may not be indicative of high levels of information. Accordingly, the short interest level for a stock is a noisy information signal. However, any information contained in high short interest levels is, presumably, negative. In other words, high short interest indicates the direction of information, conditional on the existence of information. Consequently, those high short interest stocks for which information exists should be the ones with greater underperformance. Identifying stocks with higher levels of informed trading may increase the precision of the information signal contained in short interest levels. For example, although Senchack and Starks (1993) find a weak relation between short interest and returns, they point out that an inability to purge noninformational short sales from the short interest numbers may confound any empirical tests, and remark that “the observed market reaction to short interest announcements may be underestimated due to noninformational short sales…” (p.186). Therefore, I borrow a measure from the market microstructure literature that identifies the level of informed trading in a stock. I use this measure, the probability of informed trading (PIN), to form portfolios in an effort to better forecast future underperformance. Thus, instead of testing the impact of short interest on future returns, I test the impact of PIN on future returns for a sample of stocks with high short interest.

2 Along these lines, this paper also contributes to the literature on microstructure models of asymmetric information. In a series of papers, Easley et al. (1996, 1997a, 1997b, 1998) develop a trade based measure of asymmetric information risk (PIN) and examine its empirical applications in a variety of contexts.2 If short sellers are informed, their presence in the market increases adverse selection costs for uninformed traders and the market maker. Their influence will be particularly relevant if the population of short sellers is, on average, more informed than the population of regular sellers or buyers. To address this issue, this paper considers the effect of short sale constraints on levels of information asymmetry. Because constraints affect informed and uninformed traders differently, these constraints have implications for the probability of informed trading. Furthermore, since I am interested in the returns to stocks with high amounts of short selling, the impact of these constraints on PIN may be important. In general, this paper unites these two distinct literatures. The approach is to reframe the short interest tests into an asymmetric information context. The objective is to determine if the signal content of short interest levels is improved when controlling for the probability of information based trade. I begin with a sequential trade model that is used to derive the empirical measure for the probability of informed trading [Easley, Kiefer and O’Hara (1997b)]. I extend this model to demonstrate how constraints placed on short selling will affect this probability. I find that as the fraction of informed traders who are constrained is lowered, the probability of informed trading increases. Under these conditions, the likelihood that short selling originates from informed traders is higher. The result is a link between informed short selling and the probability of informed trade. Thus, I use PIN to proxy for the amount of informed trading in an effort to identify stocks with informed short selling. Those stocks with higher amounts of informed short selling should have a stronger relation with future returns. Namely, I hypothesize that stocks with high levels of short interest and high levels of informed trading (high PINs) should have lower subsequent returns. I empirically test this hypothesis by forming size-neutral monthly portfolios on the basis of PIN for a sample of high short interest stocks. After controlling for the market premium and the effects of size, book-to-market, and momentum, I find that high PIN portfolios realize

2 For example, they use PIN to examine spread differences between active and infrequently traded stocks [Easley, Kiefer, O’Hara, and Paperman (1996)], to estimate the information content of orders in different markets [Easley, Kiefer, and O’Hara (1996)], to examine the information content of trade size [Easley, Kiefer, and O’Hara (1997b)] and to assess the informational role of financial analysts [Easley, Kiefer, and Paperman (1998)].

3 subsequent negative abnormal returns. These returns are both economically meaningful and statistically significant. For example, I find high PIN portfolios to have abnormal returns ranging from -0.876 percent to -1.047 percent per month. Except for firms in the lowest size quartile, low PIN portfolios do not statistically underperform. The results are generally robust when portfolio returns are value-weighted. I perform additional checks to see if PIN is serving as a proxy for some other underlying variable that may affect the returns of high short interest stocks. Namely, I test whether size-neutral portfolios formed on the basis of the bid-ask spread (another proxy for asymmetric information) or the level of short interest are able to predict future underperformance and find no evidence that they do. Overall, the results support three main conclusions. First, some stocks with high levels of short interest do underperform. Secondly, the underperformance of high short interest stocks may be better predicted by assessing the probability of informed trading, rather than considering short interest levels alone. This result suggests that PIN may be able to distinguish between stocks with a greater prevalence of informed versus uninformed short selling. Finally, if PIN is a valid proxy for the level of informed trading in a stock, then asymmetric information models may be important for empirical investigations of the returns to these stocks. The remainder of the paper proceeds as follows. Section II provides the motivation for an information-based approach. Section III lays out the original and extended version of the sequential trade model. Section IV describes the data and discusses estimation of PIN for both the original and extended model. Section V examines abnormal returns to portfolios formed on the basis of short interest and PIN. Section VI offers some robustness checks using spread-based portfolios and short interest-based portfolios. Section VII concludes.

II. Motivation

Researchers have long been interested in the relationship between short interest and stock returns, and this interest has experienced a resurgence in recent years. Theoretical models of overvaluation due to short sale constraints generally rely on two conditions: heterogeneous beliefs among investors and institutional constraints on short sale transactions. Miller (1977) describes a market with heterogeneous investors, where restrictions on short sales (not receiving the proceeds of a short sale) prevent willing investors from selling short. The result is an upward

4 bias in prices and low subsequent returns. Jarrow (1980) develops a general equilibrium model under heterogeneous expectations where short sale constraints may either increase or decrease the price of risky assets, depending on the economy.3 Diamond and Verrecchia (1987) model constrained short selling under rational expectations and suggest that short sellers will take positions in highly overvalued stocks. However, in their rational economy, agents will recognize the constraints present in the market and there will be no systematic overvaluation in equilibrium. Rubinstein (2004) provides a comprehensive survey of these models. Prior beliefs on the effect of short interest on asset prices depend on the information contained in short sales. Desai et al. (2002) present three prevailing views on the information content of short interest, all with different implications for pricing. Probably the most commonly held view is that high short interest is a negative signal since short selling is costly and more likely to originate from informed traders with adverse information [Diamond and Verrecchia (1987)]. The empirical prediction is that unexpected increases in short interest will lead to rapid downward price adjustment. An opposing argument popular on Wall Street maintains that high short interest is a positive signal that represents latent demand for shares. Since outstanding short positions will eventually be covered with a purchase, high levels of short interest may indicate future buying pressure.4 A third view is that short interest may contain no information content at all, especially if the bulk of short selling is part of uninformed arbitrage or hedging strategies [Brent, Morse, and Stice (1990)].5 In these cases, short positions are not motivated by information and should not affect asset values. Ultimately, the relation between short interest and subsequent returns should depend on trade motivation (and trader type). Not surprisingly, the empirical evidence on this relation has been mixed. Early studies failed to detect a strong relationship between short interest and subsequent returns and argued that most short selling is arbitrage or hedging motivated. Brent, Morse, and Stice (1990) find little evidence of speculative-based short selling and instead stress the prevalence of tax-based

3 Jarrow does point out that if investors have heterogeneous expectations on expected returns only, but agree on return covariance, prices will only be biased upward. 4 “Cushion Theory” is defined by Barron’s Dictionary of Finance and Investment Terms as the theory that a stock’s price must rise if many investors are taking short positions in it, because those positions must be covered by purchases of the stock. Also known as the “Short Interest Theory”. 5 Examples of such strategies include index arbitrage, risk (takeover) arbitrage, convertible bond arbitrage, pairs trading, and tax related short selling (shorting against the box). Additionally, some short selling may originate from underwriters trying to reduce price volatility in public offerings and specialists trying to offset temporary inventory positions.

5 and arbitrage-related short sales. Figlewski and Webb (1993) find increased short selling around option introductions as option writers hedge their position in the underlying stock. They do find small negative excess returns, however, in stocks without traded options.6 Senchack and Starks (1993) suggest that index futures arbitrage accounts for most short selling, but also report small negative abnormal returns in nonoptioned stocks around short interest announcements. Angel (1997) also documents a high incidence of short selling due to index arbitrage and program trading in TORQ data. The overriding conclusion from most of these studies is that a large fraction of short selling is uninformed. More recently, there has been evidence of economically significant underperformance in high short interest stocks. Asquith and Meulbroek (1996) find a strong and persistent negative relation between short interest ratios and subsequent returns for NYSE stocks, and conclude that short interest does indeed contain negative information. Desai et al. (2002) report similar results for a sample of Nasdaq stocks, and find the underperformance to be increasing in the level of short interest. Both of these recent papers include in their sample only stocks with high short interest ratios (greater than 2.5 percent). The authors point out that a randomly selected sample of stocks lacks the statistical power to identify a relationship between short interest and returns, since most stocks have little or no short interest. This sample choice may partially explain the inability of previous papers to detect a consistent relationship. Regardless, these two papers influenced the conventional wisdom on short interest and stock returns, contributing to a widely held belief that high short interest stocks underperform. For instance, if traders that short sell as part of a valuation strategy (e.g., they think the stock is overvalued) are likely to be informed, then their transactions may forecast low future returns. Empirical evidence consistent with the view that at least some short sellers are informed includes Dechow et al. (2001) and Christophe, Ferri, and Angel (2004). Dechow et al. show that short sellers position themselves in stocks with low book-to-market ratios, and that they are able to distinguish low ratios due to temporarily low fundamentals from low ratios due to temporarily high prices. This evidence is consistent with short sellers using the information in these ratios in anticipation of lower future returns. Using proprietary Nasdaq data, Christophe, Ferri, and Angel find that short selling increases substantially in a set of specific stocks prior to unfavorable earnings announcements.

6 Danielsen and Sorescu (2001) provide a theoretical link between option introductions and increases in short interest, and find that the negative abnormal returns that result from these option-related increases in short interest are predictable ex ante.

6 They assert that this may be evidence of informed short sellers targeting selected stocks. Thus, the trades of these informed short sellers should forecast low subsequent returns. Lately, the notion that high short interest stocks underperform has come under renewed scrutiny. Asquith, Pathak, and Ritter (2004) revisit the work of Asquith and Meulbroek (1996) and Desai et al. (2002), concluding that “the new conventional wisdom regarding short interest ratios and return predictability, as well as continual increases in short interest ratios over time, is premature and incomplete” (p. 2). They show that the underperformance of high short interest stocks is not robust when portfolio returns are value-weighted, despite negative abnormal returns in equal-weighted portfolios. They also show that the results may be driven by a small number of stocks with very high short interest ratios. Boehme, Danielsen, and Sorescu (2005) argue that the varied results of this literature stem from the failure of most studies to account for both of the theoretical requirements of overvaluation; constrained short selling and heterogeneous beliefs (e.g., differences in opinion). They report that only stocks with both high constraints and high dispersion of opinion are overvalued. Stocks lacking either condition are not overvalued. The results of the short interest literature offer two conclusions. First, short selling may be either informed or uninformed, and any relation between short interest and returns likely depends on this distinction. Second, asset pricing tests that do not control for these disparate motives may produce ambiguous empirical results. An ability to identify which stocks contain greater amounts of informed short selling may lead to clearer inferences about the information content of short interest. If the constraints on short sales restrict the private information of some investors from the trading process, these constraints affect asymmetric information levels. Additionally, this asymmetric information may impact prices. A method to estimate asymmetric information risk has been devised in the market microstructure literature in a series of papers by Easley et al. (1996, 1997a, 1997b). The measure derives from a sequential trade model with both informed and uninformed traders. Estimating the structural parameters of such a model provides an empirical measure of the probability of information based trade. Easley, Hvidkjaer, and O’Hara (2002, 2004) use this measure to show that information risk is priced in the cross-section of asset returns, and that size-neutral portfolios formed on the basis of PIN earn different returns. They argue for the empirical importance of asymmetric information for asset prices.

7 Existing tests of the performance of high short interest stocks are undertaken without regard for asymmetric information. Given the different motivations for short selling, it is not immediately clear whether high short interest stocks will necessarily have higher levels of informed trading. In other words, short interest alone may not be a successful indicator of future underperformance. If the PIN measure described above is an accurate measure of asymmetric information, it may help to identify which highly shorted stocks have higher amounts of informed trading. Combining these two measures may increase the precision of the information signal. Specifically, high short interest indicates negative information conditional on the existence of information, and high PIN indicates the existence of information. Thus, among a sample of high short interest firms, high PIN stocks should have larger negative abnormal returns than low PIN stocks. To measure the probability of informed trading, I apply the discrete time sequential trade model of Easley, Kiefer, and O’Hara (1997b). 7 Before estimating the PIN for my sample of short interest stocks, I provide some intuition on how PIN will be affected by constrained short selling. To do this, I extend the model to allow for nonzero costs of short selling. This model extension is the subject of the next section.

III. The Model

The sequential trade model of Easley, Kiefer, and O’Hara (1997b) generally assumes that traders can short sell with zero cost. Diamond and Verrecchia (1987) develop a trade model that shows the effect of constrained short selling on asset prices. Because I am interested in the effect of PIN on the returns to high short interest stocks, and PIN is estimated from a model with costless short selling, some intuition about the impact of short selling constraints on PIN may be useful. Accordingly, I introduce the short selling constraints of Diamond and Verrecchia (1987) into the model of Easley, Kiefer, and O’Hara (1997b). The result is a relationship between these constraints and PIN. Ultimately, this leads to predictions about the effect of PIN on the returns to high short interest stocks. Measuring this effect is the goal of this paper.

7 There is a corresponding continuous time model with similar design. My choice to use the discrete time model is motivated by the allowance for no-trade outcomes in the discrete model. Short selling constraints may lead to an increased incidence of no-trade outcomes, which is one of the main points of Diamond and Verrecchia (1987). Since these are modeled in the discrete form model, it is the more suitable choice.

8

A. The Original Model: Zero Short Sale Constraints

Easley and O’Hara (1992) construct a discrete time sequential trade model that depicts the learning problem of a market maker as he observes the sequencing of trades during the day.8 In the model, one asset is traded throughout the day, and has a random value, V, at the close of trading. There are three types of agents in the model, informed traders, uninformed traders, and a market maker. The informed and uninformed traders transact with the market maker by either buying or selling one unit of the asset. There are three states in the model, a high signal state, a low signal state, and a zero signal state. At the beginning of trading each day, an information event occurs with probabilityα . An information event is represented as a signal, Ψ , of the value, V, and is observed only by the informed traders. If an information event occurs, the signal is low ( Ψ = L ) with probability δ and high ( Ψ=H ) with probability1−δ . If a high signal is observed, the asset value is denoted as V , and a low signal implies an asset value of V , where V is strictly greater than V .

If no information event occurs, then no signal is observed ( Ψ = 0 ) and the asset retains its

unconditional value, VV*(1)=+−δ δ V. Information events occur only once a day (at the beginning of trading), and then trades occur throughout the day. If a signal has been observed, informed traders arrive with probability µ and uninformed traders arrive with probability 1− µ . Given the arrival of uninformed traders, I assume that they are buyers with probability ½ and sellers with probability ½.9 The uninformed traders actually transact with probability ε . Figure 1, taken from Easley, Kiefer, and O’Hara (1997b), illustrates the decision tree that describes the trading process. The informed traders are assumed to be risk neutral and observe private information about the asset’s value. Therefore, the informed traders will buy the asset if they observe a high

8 The model is slightly modified in Easley, Kiefer and O’Hara (1997b), which is the direct model I propose to extend. Since the models are the same in spirit, I will refer to both papers intermittently; the earlier paper contains most of the theoretical development and the later paper builds much of the empirical application. Where there are slight differences in the models, I follow the assumptions of Easley, Kiefer, and O’Hara (1997b). All notation, assumptions and trading motivations described in this section follow from this later paper. 9 This assumption is consistent with Easley, Kiefer and O’Hara (1997b). In Easley and O’Hara (1992), this probability is not restricted to ½, and is denoted by γ . The implications derived going forward are unaffected by this assumption.

9 signal and the market maker’s ask quote is below V , and will sell the asset upon receiving a low signal if the market maker’s bid quote is above V . Informed traders always trade when an information event occurs and do not enter the market when no signal occurs. (This former feature will be altered in the extended model.) The uninformed traders do not observe the signal, and at least some of them trade for exogenous liquidity reasons such as portfolio rebalancing or consumption needs.10 Uninformed traders are allowed to have price sensitive demand schedules, thus electing not to trade at certain quoted prices. The market maker is assumed to be risk neutral and competitive and his quotes are set such that his expected profit conditional on a trade at each quote is zero. The market maker always loses when transacting with informed traders, and sets his spread so that he can make up these losses by trading with uninformed agents. There are no inventory or order processing costs in this model, and the bid-ask spread exists solely as a consequence of asymmetric information. Given his losses to informed traders, the market maker is naturally interested in determining when informed traders are present in the market. Specifically, if the market maker believes the pool of traders contains a large fraction of informed agents, as a protective measure he may widen his spreads to increase his profits from transacting with the uninformed traders. Timing throughout the day is denoted by discrete intervals tT= 1,2,..., . By design, only one trade can occur per interval. A key feature of this model is the importance of no-trade outcomes, where no transaction takes place during a given trading interval. At each time t there are three possible outcomes, a buy, a sell, or a no-trade outcome. The action space at each time

interval is denoted by QBSNtttt∈[,, ], where B represents a buy, S a sell (regular or short), and N a no-trade outcome. The no-trade outcome is important in that the frequency of no-trades may inform the market maker about the probability that an information event occurred at the beginning of the trading day. As the market maker observes many no-trade outcomes, he gradually reinforces his belief that no signal has occurred, and informed traders have not entered the market.

B. The Extended Model: Nonzero Short Sale Constraints

10 If informed traders are profiting at the expense of the uninformed, then the uninformed that are trading for speculative reasons would be better off not trading at all. Therefore, to avoid an equilibrium in which no-trade occurs, Milgrom and Stokey (1982) show that some uninformed traders must transact for exogenous liquidity reasons.

10

To address the significant legal and economic burdens imposed on short sellers, the costs they face can be formally modeled into the trading process described in the previous section. As noted earlier, in the original model informed traders always transact when a signal has been observed. This feature implicitly assumes that when a low signal is observed, agents wishing to sell the asset either already own it, or they can costlessly borrow a share to sell short without constraint. Imposing constraints on short selling may prevent the transactions of informed traders with a low signal. Additionally, as pointed out by Diamond and Verrecchia (1987), the costs of short selling may alter the pool of informed versus uninformed traders who transact with a short sale, shaping the market maker’s beliefs about the probability of a low or high state. The model is extended to allow for the likely scenario that a trader who wishes to sell a share (either an informed or an uninformed trader) does not own the share in question, and may face costs associated with borrowing a share to short. Both informed and uninformed traders must be able to short sell in order to avoid an equilibrium where a short sale instantly reveals the true state. Constrained short selling is modeled following the convention of Diamond and Verrecchia (1987) (“DV”), where different investors are subject to different constraints. A trader wishing to sell the asset owns the share with probability h , and must borrow the share with probability 1− h . DV describe three categories of constraints. All traders fall into one of the three categories. First, some traders are assumed to have no constraints associated with short selling. These traders have access to the full proceeds of the short sale for immediate and full reinvestment, and are exempt from other costs associated with short selling. For example, option market makers can earn interest on the proceeds of their short sales [Danielsen and Sorescu (2001)], and some traders are exempt from the uptick rule.11 The second category of constraint results in a short-restriction effect. In this case, the proceeds from a short sale are unavailable for reinvestment, or high borrowing costs prevent short sales by traders without a strong signal about the value. If uninformed traders transact for liquidity reasons, a restriction on the proceeds would prevent them from shorting. Although this short-restriction constraint could affect both trader types, it is modeled explicitly by DV to prevent only uninformed traders from shorting.

11 Some arbitrage trades, odd lot dealers and block positioners are exempt from the NYSE’s uptick rule. The specialist is not exempt from the uptick rule. See Angel (1997) or Worley (1990) for specific details.

11 The third category of constraint results in a short-prohibition effect, which prevents traders from being able to short sell altogether. This constraint may be imposed on traders for legal reasons (e.g., institutional traders prohibited from short selling), due to the absence of shares available for borrowing, or as a result of prohibitively high borrowing costs that will eliminate a profitable opportunity (e.g., stocks on special).12 For example, Almazan et al. (2004) document that among a sample of U.S. equity mutual funds from 1994 to 2000, 68.9% were formally restricted by their investment policies from short selling. Whatever the reason, this constraint prevents short selling by both informed and uninformed traders. One way to interpret this design is that the economic costs (as opposed to institutional or regulatory controls) have a prohibitive effect when the costs are high enough to negate potential profit from even a strong negative signal. This drives out both informed and uninformed investors. There is only a restrictive effect when the costs are high enough to eliminate only trades without a signal, driving out only uninformed traders. Naturally, while this is not an exact representation of short selling costs in financial markets, the formalized model structure is fairly consistent with practical details. For a comprehensive description of the institutional details of short selling, see the Appendix. DV model these constraints as the fraction of traders that are subject to the given constraint category, where category and trader type are independent (e.g., a trader’s type does not influence which category of constraints he falls into, but it does dictate how those constraints

affect his ability to short-sell). Following the notation of DV, c1 denotes the fraction of traders

who have no short sale costs, c2 denotes the fraction of traders subject to a proceeds restriction,

and c3 denotes the fraction of traders who are prohibited from shorting altogether. Since all traders fall into one of these categories, ccc123+ +=1. I assume that the market maker, the

uninformed and the informed traders have identical estimates of c1 , c2 , and c3 . Figure 2 outlines the extended decision tree with the additional constraints, where it is apparent that the event space has expanded considerably. The action space now has four elements, a buy, a regular sale, a short sale, or a no-trade outcome. The outcome from a given

time interval is given by QBRSSSNttttt∈[, , , ] where RS denotes a regular sale and SS denotes a

12 Stocks on special are those which receive a lower rebate rate on the proceeds of the short sale that are deposited as collateral than the baseline general collateral rate. These are stocks for which there is a high demand to borrow, and the lower rebate rate represents an increased cost of short selling. See the Appendix for details.

12 short sale. Empirically, however, we are unable to observe short selling in the order flow. So,

the observed action space remains unchanged from before, where QBSNtttt∈[,, ]. Note that the original model without the constraints [from Easley, Kiefer, and O’Hara (1997b)] is just a restricted version of this extended model (with constraints) in which h = 1. One important consequence of the additional constraints is that an informed trader who receives a low signal will be prevented from acting on this information if he falls into the pool of investors subject to

c3 . Furthermore, short sales by uninformed traders will be prevented for those that are subject to

c2 or c3 .

C. Constraints and the Probability of Informed Trade

The market maker, or any other uninformed trader, uses his estimates of the model parameters to assess the probability of informed trading. Under the original model, the structural parameters are α , the probability that an information event occurred, δ , the probability that an information event is a low signal, µ , the probability that a trader is informed, and ε , the probability that an uninformed trader will transact if selected to trade. The probability of informed trading, or PIN, is then calculated as the probability that a trade comes from an informed trader divided by the probability that a trade occurs:

αµ PIN = . (1) αµαµε+−(1 )

If we consider the extended model with short sale constraints from Figure 2, we have four additional parameters: h , the probability that a trader wishing to sell owns the stock, and the

fraction of traders subject to each of the three categories of short selling constraints, c1 , c2 , and

c3 . Under these conditions, the probability of informed trading is given by

αµ{ δ[+hhcc(1 − )( + )] + (1 −) δ } PIN* = 12 . (2) 11 αµδ{}[hhcc+− (1 )(12 + )] +− (1 δ ) +− (1 αµ ){}22 ε [ hhc +− (1 )( 1 )] + ε

13 Note that if h=1, PIN* collapses to the measure in equation (1). Since I hypothesize a relation between PIN and future returns to highly shorted stocks, I am first interested in the effect of the constraints on PIN. Particularly, I am interested in the

relation between PIN and c2 , since this is the constraint that discriminates between informed and uninformed traders. From equation (2), it is straightforward to show that PIN* is increasing in 13 the fraction of traders subject to c2 : ∂PIN * > 0 (3) ∂c2

The intuition is equally straightforward. As c2 increases, a larger proportion of traders are faced with only a restriction on short sales, and informed traders subject to c2 are able to short sell. This increases the pool of informed traders if a low signal is realized and thus increases the

probability of information based trade. Conversely, as c2 decreases (and thus, c3 increases) more traders are prohibited from shorting altogether, which will prevent more informed agents from trading on a low signal when they do not own the stock. The relationship in equation (3) establishes a link between PIN and the constraints associated with short selling. More importantly, it establishes a link between PIN and informed short selling. To the degree that informed trading in general is correlated with informed short selling in high short interest stocks, PIN may also proxy for the level of informed short selling. The ability to identify highly shorted stocks which correspondingly have high levels of informed trading may increase the power to detect a relation between short interest and future returns. Specifically, the model predicts that among high short interest stocks, those with high PINs should have relatively more informed short selling than those with low PINs. Consequently, the high PIN firms will display larger underperformance than the low PIN firms.

The focus is on PIN because it is reliably estimated, while c2 is not (as discussed in the next section). Furthermore, PIN is the variable of interest because it can be linked directly to informed trading, while the level of short interest cannot. If PIN can predict return performance in high short interest stocks, it stresses the importance of models of asymmetric information for the returns to highly shorted stocks. This result would complement the findings of Easley et al.

13 Note that since ccc123++=1, this differentiation holds c1 constant while allowing c3 to vary with c2 . The

result still holds if c3 is held constant and c1 varies with c2 . Proofs available from the author upon request.

14 (1996, 1997b, 1998). The objective for the remainder of the paper is to test for return differences in high short interest stocks based on PIN.

IV. Empirical Methodology

A. Data

To examine the effect of asymmetric information on the returns to high short interest stocks, I measure the PIN for each stock in my sample. There are two samples of data that I use in the study. First, there is an “initial sample” that I use for some diagnostic checks. Then, there is a “high short interest sample” which is used in the asset pricing tests and is a subset of the initial sample. The initial sample includes all NYSE stocks that have a short interest ratio greater than 2.5 percent in at least one month during my sample period, January 1995 to December 1999.14 The source of the data is the NYSE. The short interest ratio is defined as the number of currently outstanding short positions divided by the number of shares outstanding. In the remainder of the paper, all references to short interest refer to the short interest ratio. I use only NYSE stocks because the structural model more closely describes the price discovery process of a single market maker similar to that of the NYSE specialist. The decision to include only stocks with a short interest ratio greater than 2.5 percent is consistent with the recent literature. Admittedly, the 2.5 percent cutoff is somewhat arbitrary, but necessary in order to make comparisons to previous studies. As pointed out by Asquith and Meulbroek (1996) and Desai et al. (2002), it is difficult to detect a relation between short interest and returns from a randomly selected sample of short interest stocks since most stocks have low (or zero) short interest ratios. Furthermore, since my hypothesis applies to only high short interest stocks, some cutoff is necessary. I exclude from the short interest sample preferred shares, warrants, ADRs, convertible securities, and unit investment trusts. Shares outstanding and monthly return data are from CRSP. Daily trade and quote data used to estimate the structural parameters are taken from the NYSE TAQ database. First, I identify all firms that have short interest greater than 2.5 percent

14 One reason the sample ends in 1999 is that the procedure used to estimate the model parameters has computational difficulties with the larger trading volumes of recent years. At higher trading volumes, the estimation procedure fails to converge more frequently. Easley, Hvidkjaer, and O’Hara (2004) estimate PIN for a sample of stocks through 2001 and report increased instances of nonconvergence in the later years of their sample.

15 in at least one month during the sample period. For my initial sample, I collect trade and quote data for these firms for all months that data are available. For example, a firm could have short interest greater than 2.5 percent for only 20 of the 60 months during the sample period, but I collect TAQ data and estimate the probability of informed trading for that stock for all 60 months. Initially, I have 73,940 firm-month observations of firms that have short interest greater than 2.5 percent during at least one month. I retain this initial sample for diagnostic checks on my estimation procedure, which I discuss below. For the asset pricing tests, the “high short interest sample” includes only firm-months where a firm has short interest greater than 2.5 percent in the previous month. From the example above, only the 20 months out of the entire 60 for which the short interest is actually greater than 2.5 percent would be included in the high short interest sample. After combining this sample with the return data from CRSP and the trade and quote data from TAQ I am left with a high short interest sample containing 20,618 firm- month observations. Table 1 includes descriptive statistics for the high short interest firms. Panel A contains mean monthly statistics for the firm-month observations in the high short interest sample (N=20,618), by year. These observations are those that will be used to form portfolios in later sections. The high short interest sample contains 1,436 unique firms. Naturally, some firms jump in and out of the high short interest sample as short interest changes from month to month. Panel B contains statistics for the same firms in the high short interest sample during months in which their short interest is less than 2.5 percent. On average, a firm is in the total sample (e.g., Panel A plus Panel B) for an average of 43.8 months and in the high short interest sample (Panel A) for an average of 14.6 months, or 36.9 percent of the time. Some obvious patterns emerge from these two panels. First, the average short interest ratio is much larger for these firms when they meet the 2.5 percent inclusion criteria. Over the sample period, the average short interest for these firms when they are included in the high short interest sample is 6.39 percent, but drops to 1.05 percent when they fall out of the high sample. Average price and market value of equity are smaller when these firms are included in the high short interest sample, and their subsequent returns are lower. For example, the average subsequent month’s raw return is 0.85 percent when firms are in the high short interest sample and 1.52 percent when they are not. Panel C combines the observations from Panel A and Panel B, providing a summary for these firms during all months.

16 For comparison to Panel A, Panel D shows statistics for all firm-months (N=90,987), where a firm’s short interest is between zero and 2.5 percent in a given month. Some of the firms in this panel never enter the high short interest sample (e.g., are not included in Panel A, B, or C). From this panel it is apparent that firms in the high short interest sample are smaller and trade at lower prices. Asquith and Meulbroek (1996) find that their sample of highly shorted NYSE and AMEX firms are smaller than a randomly matched sample, but Desai et al. (2002) find that their sample of highly shorted Nasdaq firms is larger than average. Rt+1 , the subsequent monthly raw return, averages 1.28 percent for the low short interest stocks over the entire 60 month period.

B. Estimating the Probability of Informed Trading

To estimate the structural parameters of the trade model requires knowledge only of the daily

number of buys, sells, and no-trade outcomes, (Bdd,,SN d) . Since the econometrician cannot observe short selling in the order flow, sales and short sales are pooled together in the estimation procedure. In the following discussion, I focus on the original model of Easley, Kiefer, and O’Hara (1997b) for ease of explanation, and discuss estimation of the extended model (with constrained short selling) later. The likelihood function of observing the number of buys, sells, and no-trade outcomes on a given day is Pr{()BSN , , |θ} where θ = {}αδ,,, µε . Assuming independence across trading days, the likelihood of observing the history of buys, sells and no-trades across the estimation period is found by taking the product across days such that

D PrBSN , ,D |θ = Pr BSN , , |θ (4) {()dd dd =1 } ∏ {}() dd d d =1

where D is the number of trading days in the estimation period.15 Easley, Kiefer, and O’Hara (1997b) show that this is computed by maximizing the following likelihood function

15 Easley, Kiefer, and O’Hara (1997b) test the assumption of independence across trading days and are unable to reject the assumed independence.

17 BSN++ D ⎧⎫BS D ⎪⎪⎛⎞µµ ⎛⎞ ⎛⎞1 BS+ N ∑∑log⎨⎬αδ (1−++ )⎜⎟ 1 αδ ⎜⎟ 1 ++− (1 α )⎜⎟ + log{}x [] (1 −− µ )(1 ε ) (5) d =1 ⎩⎭⎪⎪⎝⎠xx ⎝⎠ ⎝⎠1− µ d =1

1 where x =−()ε µ . Once the parameters have been estimated, the probability of informed 2 trading is calculated from equation (1). Trade direction is determined using the Lee and Ready (1991) algorithm where a trade that occurs above (below) the prevailing quote midpoint is classified as buyer-initiated (seller- initiated).16 A trade that occurs at the midpoint is buyer-initiated (seller-initiated) if the last trade was an uptick (downtick). I allow for a five-second lag in matching quotes to trades as is common in the literature to account for potential delays in trade reporting [see Lee and Ready (1991)].17 I include only trades and quotes that occur during normal trading hours on the NYSE and exclude quotes and trades from the regional exchanges. To calculate the number of no-trade outcomes in a day, I first calculate the average number of daily trades for a given stock and divide by the total number of seconds in a trading day. This time interval is then used as the no-trade period, and if this amount of time passes without a trade, a no-trade outcome is counted (if twice this time interval passes, I count two no- trade outcomes, and so on until a trade takes place). I compute this time interval for each stock individually and for each month in the sample. This approach accounts for differences in trade frequency across stocks and time varying trade volume, as opposed to assigning an arbitrary length of time for all stocks and months. While this seems like a logical approach to tabulating no-trade outcomes, there are other approaches. Chung, Li, and McInish (2004) experiment with different methods of calculating this interval, and find their estimation results to be robust to different specifications. I estimate the structural parameters for each stock on a monthly basis, by maximizing the likelihood function in equation (5) with the daily number of buys, sells and no-trades over the month. The parameter estimates then give a monthly measure of the probability of informed

16 Several studies test the effectiveness of the Lee and Ready algorithm. Odders-White (2000) finds the algorithm to be 85 percent accurate for NYSE TORQ data, while Lee and Radhakrishna (2000) find 93 percent accuracy for trades that can be unambiguously classified. Ellis, Michaely, and O’Hara (2000) report 81 percent accuracy for Nasdaq data. Despite its deficiencies, the algorithm is reasonably accurate and remains the standard in the literature. 17 Bessembinder (2003) and Peterson and Sirri (2003) find that the accuracy of the trade direction algorithm is improved when no timing lag is implemented. I keep the lag to be consistent with other papers that estimate PIN using TAQ data. My results are not sensitive to this convention.

18 trading from equation (1). Additionally, I calculate the monthly average of the daily trade- weighted quoted half-spread and relative half-spread. The relative half-spread is one half of the quoted spread divided by the spread midpoint. In the structural trade model, there are no inventory or order processing costs. Thus, the spread measures are alternative proxies for information asymmetry. Easley et al.(1996), Easley, O’Hara, and Paperman (1998), and Chung and Li (2003) all find a significant positive relationship between PIN and the adverse selection component from various spread decompositions. Heidle and Huang (2002) report an association between changes in spreads and changes in PIN for firms that undergo an exchange listing relocation. Table 2 shows the time series average of the cross-sectional correlations of the variables of interest in the study. The interrelations between PIN and the other variables are of primary interest. PIN is weakly positively correlated to the quoted half-spread, but displays stronger positive correlation to the relative spread (0.248). Additionally, correlation between PIN and short interest is positive but weak (0.068), suggesting the relationship between the two variables may not be very strong. This finding is not surprising given the previous discussion about various motives for short selling. That these two variables are not strongly correlated further motivates the approach in this paper to use PIN instead of the short interest level as a predictor of returns in highly shorted stocks. PIN is strongly negatively correlated with both firm size (-0.148) and the daily number of trades (-0.172), consistent with the findings of Easley et al.(1996) that smaller firms and low volume stocks have higher levels of private information. Lastly, PIN is negatively correlated with subsequent excess returns. This relation is in contrast to the results of Easley, Hvidkjaer, and O’Hara (2004) who find positive correlation, but is expected given my sample of only high short interest stocks. This highlights one of many key differences in the two studies; my sample generally includes firms where an information event is more likely to be negative. One concern with estimating the model at a monthly frequency is the precision of the parameter estimates. Most existing studies that use PIN estimate over longer time intervals to address this concern. Easley et al. use a minimum of 60 trading days in most of their studies, but Easley, Hvidkjaer and O’Hara (2002) discuss the possible value of estimating over monthly horizons. Vega (2004) estimates PIN over 40 trading days, and Fu (2002) over 45 days. I choose the shorter monthly interval in order to capture changes in monthly short interest positions. Short interest levels are collected by the exchanges on the fifteenth day of every

19 month (if it is a business day) and represent short positions outstanding as of the settlement date. The public announcement date occurs a few days later, usually between the 19th and 24th day of the month. So, if the 15th is a trading day, the short interest level reported that month represents the amount of short interest outstanding three trading days prior.18 I then begin my estimation period for month t two trading days prior to the 15th, and end it three trading days before the next month’s collection date. Therefore, my monthly PIN estimate corresponds with the period over which the short interest amount changed from month t-1 to month t. Figure 3 provides a graphical description of the intervals over which the variables are measured.

Settlement Trade Settlement Announcement Date Date Date (t+3) Date

Jan 1 Jan 12 Jan 15 Feb 12 Feb 15 Feb 22 Mar 1 Mar 31

SI SI t-1 t

PINt Rt+1

Figure 3: Timeline of Variable Measurement The above timeline gives an example of the measurement intervals for the variables used in the study. The above example shows the timing of February short interest. February’s short interest is collected for trades that settle as of February 15th. This represents all outstanding short positions as of February 12th, since settlement is t+3. Therefore, the PIN that accompanies February short interest is measured over the interval between January’s short interest trade date (January 12th) and February’s short interest trade date (February 12th). The public announcement of February short interest occurs on February 22nd. Portfolios are formed at the end of February and subsequent performance is calculated with the March monthly return.

C. Estimation Diagnostics: Monthly versus Quarterly Intervals

Estimating PIN at a monthly frequency usually gives me between 20 and 23 trading days per month with which to maximize the likelihood function. For diagnostic checks, I estimate the

18 Settlement was reduced to t+3 from t+5 in June, 1995. This is during my sample, and I account for this change accordingly. See Asquith, Pathak, and Ritter (2004) for more details concerning the timing of short interest collection dates and public announcement dates.

20 parameters for both monthly and quarterly frequencies on my initial trade and quote sample. The goal of these diagnostic checks is to verify that PIN can be reasonably estimated over monthly intervals. Estimating over a quarterly horizon provides the 60 days of trade data that is more common in existing studies that estimate PIN. At a monthly frequency, the likelihood function converges for 71,702 of 73,940 firm-months, giving a convergence rate of 96.97 percent.19 If the estimation interval is increased to a quarterly frequency, convergence improves only marginally to 98.53 percent. Table 3 Panel A shows the mean parameter values across all firms in the sample. In Panel A1, I estimate the parameters for each firm each month. I then average the monthly estimates across the three months in a quarter to arrive at a quarterly average of monthly parameter estimates. This average is then compared to the parameter estimates obtained from the actual quarterly estimation interval. All four parameter means ()α,δµε,, are very similar, and more importantly, the mean PIN using monthly estimates is 0.163, and the mean PIN using quarterly estimates is 0.162. Additionally, the monthly averaged PIN and the quarterly PIN have a correlation of 0.910. Estimating PIN at a monthly frequency therefore appears to give very similar results as estimating over quarterly intervals. Panel A2 compares the actual monthly parameter estimates (as opposed to a quarterly average of the monthly estimates), to the corresponding quarterly estimate. Note that the means are still very close across estimation interval (monthly PIN equals 0.161, quarterly PIN equals 0.160), and as expected the standard deviations for monthly estimates are larger. Also, the correlations, while all still strongly positive, are lower than in Panel A1. Specifically, the correlation between the monthly and the corresponding quarterly PIN drops to 0.709. It appears then, that estimating at a monthly frequency does pick up some variation in PIN across the quarter. This variation is desirable if the objective is to match changes in the probability of informed trading to monthly changes in short interest. Panel B in Table 3 provides some evidence on the precision of the parameter estimates. The winsorized mean standard error across all firms, and also the mean percentage of parameter estimates that achieve significance at a 10 percent level are included.20 The trader arrival

19 As stated earlier, this initial sample includes the data from all months for any firm that falls into the high short interest sample at any point in my sample period. I choose this larger sample for diagnostic checks because firms move in and out of the high short interest sample over the period, and I need the PIN for all months that data are available for accurate comparisons. 20 The mean standard error is winsorized at 0.5 percent of the tails to reduce the sensitivity of the mean to a few extreme values.

21 parameters, ε and µ , are much more precisely estimated than the information parameters α and

δ , as evidenced by lower standard errors and higher percentage of significant parameters. As pointed out by Easley, Kiefer, and O’Hara (1997b), since trader selection occurs with each trade outcome but the information events occur only once a day, we expect the precision of ε and µ to be superior. Precision of δ is certainly the worst of the four parameters, since only 47.41 percent of the estimates are significant. The relatively poor performance of δ is also not surprising given that all days are used in the estimate of α , but only days with an information event are used in the estimation of δ (Easley, Hvidkjaer, and O’Hara (2002), fn. 15). Despite the worse precision of α and δ , the standard errors are in line with those reported for mid and low volume stocks in Easley et al.(1996). At a quarterly estimation interval, the precision improves considerably for estimates of α and δ and marginally for estimates of ε and µ . Certainly, there is a tradeoff between precision and the frequency with which the parameters are estimated. Hasbrouck (2004) comments that PIN is likely to be well-identified even if the structural parameters used to derive it are not. Venter and de Jongh (2004) show that any biases in the estimates of α and µ , the two parameters that drive PIN, are usually in the opposite direction and cancel each other out. This cancellation has the effect of reducing the potential bias in PIN. Since PIN is my variable of interest, that it is probably well-specified is reassuring. Though I sacrifice some precision in estimating the likelihood function over a shorter interval, I am comfortable with this level of precision in order to capture monthly fluctuations in the probability of informed trading. Lastly, Panel C of Table 3 lists the number of boundary solutions (parameter estimates equal to 0 or 1) for each parameter. Boundary solutions usually result from a persistent buy-sell imbalance in the data. At a monthly frequency, 20.8 percent (7.3 percent) of the estimates for δ achieve a zero (one) boundary solution. Consistent with the above evidence, the estimation of δ is certainly the most problematic to estimate. In the original Easley, Kiefer, and O’Hara (1997b) model without short sale constraints, δ does not enter into the calculation of PIN in equation (1). In the extended model, however, δ does affect PIN, as seen in equation (2).

22 D. Estimating the Extended Model

In this section, I examine both the reliability and the usefulness of estimating the structural parameters using the extended version of the model. Since my sample consists of stocks with high amounts of short selling, the parameter estimates from the model that incorporates short sale constraints may be more informative. To empirically estimate the extended model, a new likelihood function needs to be derived. The analog likelihood function to equation (5) for this more general model is given by

B S N ⎧ ⎛⎞µ ⎛⎞µ[(1)()]hhcc+− + ⎛⎞µ(1− hc ) ] ⎫ ⎪αδ(1−++ ) 1 αδ 1 +12 1 + 3 ⎪ D ⎜⎟ ⎜⎟⎜⎟1 ⎪ ⎝⎠xxhhc⎝⎠[+− (1 )123 ]⎝⎠ (1 −µεε )[(1 − ) +2 (1 − hcc )( + )] ⎪ log ∑ ⎨ BSN++ ⎬ d =1 ⎪ ⎛⎞1 ⎪ ⎪+−(1α ) ⎜⎟ ⎪ ⎩ ⎝⎠1− µ ⎭ D N ++−−−+−+logxhBS+ [ (1 hc ) ] S (1µεε )[(1 )1 (1 hcc )( )] ∑ {}123[]2 d =1

(6) 1 where x =−()ε µ .21 Essentially, I am asking the estimation procedure to fit three additional 2 free parameters (h, c,2 c3 ) without the benefit of additional data (e.g., empirically observed short sales). I maximize this likelihood function monthly for my sample of high short interest stocks. In many cases, the estimate for h converges to the boundary condition of one, which is equivalent to the original model. Table 4 contains the results of this estimation procedure. Panel A shows the mean of the average parameter estimates across all stocks in the sample in both the original and extended model. The parameter estimates are very similar. The exception is δ , the probability that an information event is bad news, which equals 0.354 in the original case and 0.494 in the extended case. The PINs are nearly identical, equaling 0.157 and 0.156 in the original and extended model, respectively. While this seems promising at first glance, the standard errors for the parameter estimates in the extended model take on extremely high values in many cases, casting doubt on the reliability of the estimates. Additionally, the estimates for c1 , c2 , and c3 are very

21 Derivation available from author upon request.

23 sensitive to the starting values selected in the maximization. In the case shown in Table 4, the starting value for each constraint was 1/3, so the estimates did not migrate too far from the starting values. Panel B of Table 4 provides correlation coefficients between original and extended parameter estimates. I point out from this table that the PINs from the two models have a very strong correlation of 0.811. A likelihood ratio test was able to reject the restriction that h=1 in only about 20 percent of cases. So, the restriction is usually not binding, and the extended likelihood function does not provide us with much additional information. One explanation for this is that we simply cannot estimate the additional parameters without the benefit of more finely partitioned data. Unfortunately, daily short sale data is not publicly available, which limits the empirical application of the extended model.22, 23 Despite its empirical shortcomings, the extended model provides the intuition about the effect of short sale constraints on the probability of informed trading. This in turn guides the empirical predictions about the relation between PIN and future returns. This intuition is what motivates the asset pricing tests in the next section. Going forward, the remaining tests in the paper use the parameters estimated from the original model and the likelihood function in equation (5), since I have more confidence in the reliability of its estimates. Moreover, I am convinced that the PINs estimated from the two models are very similar. In the asset pricing tests of the next section, PIN estimates are used to form monthly portfolios of high short interest stocks.

V. Portfolio Abnormal Returns

22 One exception is the NYSE TORQ data, where short sale orders are identified in the data. Matching the order placement data to actual transactions is not a trivial exercise. Angel (1997) looks at the specialist’s quote revisions around short sale order placement using this data, which covers three months from November 1990 to January 1991. 23 If we are able to empirically observe daily short sales, then the likelihood function changes to accommodate this extra partitioning of the data, and is given by: SS N BRSSSN+++ D ⎧ BRS ⎫ ⎪ ⎛⎞µµ ⎛⎞⎛⎞µ(1−+hc )(12 c ) ⎛⎞µ(1− hc )3 ] ⎛⎞1 ⎪ log⎨αδ (1−++ )⎜⎟ 1 αδ ⎜⎟ 1 +⎜⎟ 1 +⎜⎟ 1 + +− (1α )⎜⎟ ⎬ ∑ xxxhc(1−−[−+−+− ) (1µεε ) (1 )1 (1 hcc )( )] 1 µ d =1 ⎪⎩ ⎝⎠ ⎝⎠⎝⎠123⎝⎠2 ⎝⎠ ⎭⎪ D BRSSSRS++ SS N +−−[−+logxh[ (1 hc ) ] (1µεε ) (1 )1 (1−+hc)( c )] ∑ {}1 ⎣⎦⎡⎤2 23 d =1

1 where x =−2 εµ (1 ) Derivation available from author upon request.

24 Empirical tests of the underperformance of high short interest stocks generally involve an examination of the relation between short interest levels and subsequent monthly returns. Since the empirical predictions of Diamond and Verrecchia (1987) describe rapid price adjustments to unexpected changes in short interest around the announcement date, these are not direct tests of their model. Such a test requires a model of expected short interest.24 Instead, the literature has evolved to tests of overvaluation, which are more in the spirit of Miller (1977).25 In a sense though, these are indirect tests of Diamond and Verrecchia’s predictions. For example, evidence of negative abnormal returns in the month subsequent to announcement would contradict Diamond and Verrecchia’s hypothesis of an immediate adjustment. The most recent of these empirical tests is Asquith, Pathak, and Ritter (2004). They show that underperformance of high short interest stocks is not robust when portfolio returns are value- weighted (VW). They also show that underperformance in portfolios of stocks with short interest greater than 2.5 percent is driven by a small number of stocks with very high short interest. For example, abnormally low returns in a portfolio with short interest greater than 2.5 percent may be driven by stocks with short interest in the 99th percentile. When portfolios are “truncated” to create independent samples, this underperformance is not robust. For instance, they find a truncated portfolio with short interest between 2.5 percent and 5 percent does not display the underperformance that a portfolio with short interest greater than 2.5 percent. In general, they argue that evidence on the underperformance of high short interest stocks is more ambiguous than previously thought. This sentiment exactly describes my motivation for suggesting a new approach to detecting any existing underperformance. Before examining this new approach, I test for abnormal performance in portfolios formed on the basis of short interest only. This is important for two reasons. First, given the different sample periods across studies, I want to ensure that I generate results similar to past studies when using their methods. Secondly, since I am advocating a refined approach, I need to contrast the results of my approach to that of previous authors. Consistent with the literature, I

24 Senchack and Starks (1993) use such a model to test the predictions of Diamond and Verrecchia (1987). They find small negative abnormal returns around the announcement date for stocks with increases in unexpected short interest, but discuss the limitations of their model of unexpected short interest and the confounding effects of noninformational short sales. 25 Note that Miller’s (1977) theory produces hypotheses related to the level of short sale constraints and overvaluation. It is worth reiterating that the goal of this paper is not to measure the level of the short sale constraint. In other words, my hypotheses apply specifically to stocks with high levels of short interest, which is not necessarily synonymous with high costs of short selling.

25 apply the calendar-time portfolio approach advocated by Fama (1998) and Mitchell and Stafford (2000). I regress the time series of the monthly portfolio returns on the three Fama and French (1993) factors and a momentum factor [Carhart (1997)]. The excess return for a portfolio is regressed on the four factors in the following equation:

Rpt,,−=+ R f t a bRMRF t + sSMB t + hHML t + mMOM t +ε t, (7)

where Rpt,,− R f t is the monthly portfolio return in excess of the one-month Treasury bill, RMRF is the market factor, SMB is the size factor, HML is the book-to-market factor, and MOM is the momentum factor.26 These return factors are described in detail in Fama and French (1993) and Carhart (1997). The intercept, a, is the measure of abnormal performance. The various portfolios contain different subsets of high short interest stocks. If high short interest portfolios underperform, a should be negative and significantly different from zero.

A. Overlapping Short Interest Portfolios

Monthly high short interest portfolios are constructed using both absolute and relative levels of short interest. The first set of tests I perform are on overlapping portfolios in the sense that a firm can be included in more than one portfolio in a given month. Each month, four portfolios are formed using an absolute cutoff criteria, where a firm enters a portfolio if it’s short interest in the previous month is greater than 2.5 percent, 5 percent, 7.5 percent, or 10 percent, respectively. The firms in the 10 percent portfolio will then be in the 2.5 percent, 5 percent, and 7.5 percent portfolios as well. Additionally, two portfolios are formed using a relative cutoff, in which a firm enters a portfolio if it’s previous month’s short interest was in the 95th or 99th percentile for that month. These criteria are chosen for comparability to previous studies [Asquith and Meulbroek (1996), Desai et al. (2002), Asquith, Pathak, and Ritter (2004)]. The portfolios are rebalanced monthly and a firm exits the portfolio if it’s short interest drops below the cutoff level. Note that these are not distinct portfolios and their returns are not independent. Table 5

26 I am grateful to Kenneth French for making the return factors publicly available on his website. Construction of the return factors is also discussed in great detail on the website.

26 contains OLS coefficient estimates and t-statistics for the regressions in equation (7). Panel A shows that all six portfolios have negative intercepts, five of which are significant, when the portfolio returns are equally weighted. The intercept for the 99th percentile portfolio, despite having a large value (-1.038), is not significant. This is likely due to the small number of firms that are included in this portfolio (about 16 per month). The negative abnormal returns generally become larger in magnitude (more negative) with increasing levels of short interest. The 2.5 percent portfolio has an intercept of -0.702 (t-statistic of -4.27), suggesting monthly underperformance of -0.702 percent. The 95th percentile portfolio displays negative abnormal performance of -0.969 percent per month (t-statistic of -3.28). In general, these results indicate underperformance of high short interest portfolios on an equal-weighted basis. Desai et al. (2002) find abnormal performance in equal-weighted portfolios of Nasdaq stocks (June 1988- December 1994) of -0.76 percent per month in their 2.5 percent short interest sample, and -1.13 percent per month in a 10 percent short interest sample. Asquith, Pathak, and Ritter (2004) find underperformance of -0.65 percent per month in a 2.5 percent short interest sample of NYSE- Amex firms (June 1988-2002). Regressions are also run using value-weighted portfolio returns, the results of which are in Panel B of Table 5. Fama (1998) finds that abnormal returns for EW portfolios are often not robust when portfolio returns are value-weighted. If model misspecification is greater for smaller stocks, giving more weight to larger firms helps mitigate some of the misspecification. This point is especially salient here, since many high short interest stocks in my sample are smaller firms. On a value-weighted basis, the intercepts are generally much smaller than for the EW portfolios, and the significance is greatly reduced or eliminated in most cases. The exception is the 5 percent portfolio with an intercept of -0.806 (t-statistic of -2.5). The 2.5 percent and 95th percentile portfolio remain marginally significant, though with much smaller intercepts (t-statistics of -1.99 and -1.84, respectively). The remaining portfolios are not significant. Thus, on a value-weighted basis, much of the underperformance displayed by EW portfolios is absent, suggesting the abnormal returns may be driven by smaller stocks, or the asset pricing model is misspecified for small firms.

B. Independent Short Interest Portfolios

27 Following the lead of Asquith, Pathak, and Ritter (2004), I also form independent monthly portfolios by truncating the short interest levels used in portfolio selection. In this case, a given firm is included in only one of the portfolios. This redesigned sorting procedure should help to determine if the results in Table 5 are driven by a small number of stocks with extremely high short interest ratios. For example, perhaps the underperformance in the 5 percent portfolio is driven by stocks with short interest in the 99th percentile. Each month, five portfolios are formed based on the previous month’s short interest. For example, the [2.5% - 5%) portfolio includes all firms with a short interest ratio greater than or equal to 2.5 percent and less than 5 percent. The other portfolios, [5% - 7.5%), [7.5% - 10%), [10% - 99th percentile), and ( ≥ 99th percentile), are similarly defined.27 Table 6 reports OLS coefficient estimates and t-statistics from these regressions. Results for the equal-weighted portfolios in Panel A still show evidence of underperformance, with three of the portfolios having significant intercepts. The [7.5% - 10%) portfolio is marginally significant with a t-statistic of -1.87. To test the hypothesis that the regression intercepts are all jointly zero, I calculate the F- statistic of Gibbons, Ross, and Shanken (1989). The GRS F-statistic has a value of 3.4572, with a corresponding p-value of 0.0091. Therefore, the null hypothesis that all intercepts are jointly zero is strongly rejected. Panel B shows the results from independent VW portfolios. The intercepts for the [2.5% - 5%) and the [7.5% - 10%) portfolios are no longer significant. The [5% - 7.5%) portfolio, however, still displays monthly underperformance of -0.914 percent (t-statistic of -2.19). For these regressions, the GRS F-statistic is only 1.2978, with a p-value of 0.2795. Thus, in this case, the hypothesis that all portfolio intercepts are equal to zero cannot be rejected. I conclude from this section that portfolios of high short interest stocks generally underperform on an equal-weighted basis, for both overlapping and independent portfolios, during my sample period. When returns are value-weighted, much of this underperformance disappears, yet some portfolios still have negative abnormal returns. Notably, this underperformance is not in the most heavily shorted stocks. In other words, negative abnormal returns are not increasing in the level of short interest on a value-weighted basis. Perhaps then, the short interest level alone is not a suitable proxy for overvaluation. In other words, the

27 There is no [10% - 95th percentile) portfolio since the 95th percentile cutoff is less than 10 percent in most months. However, the 99th percentile exceeds 10 percent in all months.

28 relationship between short interest levels and future returns does appear to be ambiguous. Additionally, the GRS F-statistic calls into question the validity of these intercepts. An alternate approach, and the one I examine next, is to identify which highly shorted firms have the highest probabilities of informed trading. If investors with negative private information are able to trade on their signal, and we can identify in which stocks they are most likely to be trading, we may be better able to predict underperformance in highly shorted stocks. Such a result would support the use of PIN with short interest levels as a cleaner proxy for overvaluation in highly shorted stocks.

C. PIN-Based Portfolio Construction

Since the constraints on short selling affect the ability of privately informed agents to trade, a measure of asymmetric information may help to explain returns to highly shorted stocks. The probability of informed trading, or PIN, derived from the trading model above is such a measure of asymmetric information. So, an estimate of PIN may increase the precision of the short interest signal. Recall that PIN increases as more informed traders are able to short sell ( c2 increases). Easley, Hvidkjaer, and O’Hara (2004) show that size-neutral zero investment portfolios formed on the basis of PIN produce positive abnormal returns. In this section, I examine whether or not highly shorted stocks with greater levels of private information have larger abnormal returns. Since my sample contains only high short interest stocks, any information should be negative and I expect these realized abnormal returns to be negative. My hypothesis is that high PIN portfolios will have larger underperformance than low PIN portfolios. To test this hypothesis, I construct portfolios that are sorted on firm size and PIN. Sorting by size is necessary to isolate the effect of PIN, since PIN and size are strongly negatively correlated (Table 2). Each month, I sort all high short interest stocks into quartiles based on the previous month’s market capitalization. Within each size quartile, I sort into terciles based on the prior month’s PIN. This process gives me 12 portfolios every month. Sorting by PIN within quartile allows me to have an approximately equal number of stocks in each portfolio. Given my small sample size, ensuring there are a reasonable number of firms in each portfolio is especially important.

29 Table 7 gives the results of these portfolio sorts, including the average number of stocks in each monthly portfolio, the average size, PIN, short interest, quoted and relative half-spread, and the estimated parameters from the structural trading model. These figures illustrate the effectiveness of the sorts and the characteristics of the different portfolios. Within each size quartile, I check for differences in means across the low and high PIN portfolios with a nonparametric Wilcoxon rank sum test (Z-statistic). The sorts appear to be fairly successful in controlling for firm size, and there is significant variation in PIN within each size quartile. Admittedly, sorting into a larger number of portfolios would better control for size, though at the expense of reduced power. On average, I am left with less than 30 firms per portfolio each month. Both the average portfolio PIN and the difference between the low and high PIN decrease with higher size quartiles, confirming the negative relation between size and PIN. For example, the high PIN portfolio for small stocks (portfolio 13) has an average monthly PIN of 0.2884 while the high PIN portfolio for large stocks (portfolio 43) has an average PIN of 0.1860. Within the low size quartile, the difference between the low and high PIN portfolios is 0.190. This difference falls to 0.116 in the high size quartile. Thus, there is less variation in PIN for larger firms. Easley, Hvidkjaer, and O’Hara (2004) report a similar finding. Notably, the variation of average short interest across PIN portfolios is quite low. The Wilcoxon test fails to reject a difference in mean short interest between the low PIN and high PIN portfolio for all but the highest size group. Furthermore, the average short interest for the high PIN portfolios is lower than that of the corresponding low PIN portfolio in the first two size quartiles (though not statistically different). Initially at least, this indicates that the level of informed trading is not necessarily increasing in the level of short interest, and further motivates a PIN-based test. As expected, both the quoted spread and the relative spread are decreasing in size and increasing in PIN. Wider spreads are expected in stocks with higher levels of asymmetric information. The structural parameters from the trading model display most of the expected relationships as well. The probability than an information event occurred (α ), is increasing in PIN. This parameter drives the difference in PIN across terciles. Furthermore, the highest average α is in the low size, high PIN portfolio (portfolio 13). Interestingly, the probability that new information contains a low signal (δ ) is increasing in PIN in the lower three size quartiles. The difference in means for δ in the lower two size quartiles is significant, with differences of 0.099 and 0.064, respectively. Normally, in a random sample of stocks we would

30 not have any theoretical prior for a difference in the probability than new information was bad news [Easley et al.(1996)]. Since I am looking only at firms with large amounts of short selling, this finding here is not surprising. Overall, the portfolio sorts give wide variation in PIN within size quartiles with little variation in short interest, allowing for a clean test of the effect of PIN on abnormal returns. Next, I examine whether or not PIN can predict differences in returns.

D. Regressions of PIN-Based Portfolios

If my hypothesis is correct, high PIN portfolios of high short interest stocks should produce negative abnormal returns after controlling for effects related to size, book-to-market, and momentum. Furthermore, high PIN portfolios should underperform low PIN portfolios. Table 8 shows estimated coefficients and t-statistics for regressions of monthly portfolio excess returns on the four return factors in equation (7). The intercept is the measure of monthly abnormal return. Panel A shows results for equal-weighted portfolios. The high PIN portfolio in three of the four size quartiles produces significantly negative intercepts, ranging from -0.832 to -0.912 percent per month. Thus, PIN is able to predict economically significant underperformance for even large size equal-weighted portfolios in my sample. Also, within these three size quartiles, the low PIN portfolios do not statistically underperform. Within the low size quartile, the low and middle PIN portfolios produce abnormal returns, but the high PIN portfolio does not. The low and middle PIN portfolios have large intercepts of -1.232 and -1.546 per month, respectively. With the exception of the low size, high PIN portfolio, the results from these regressions support my hypothesis. Specifically, an increase in negative returns as we move from low PIN to high PIN portfolios, and the absence of significantly abnormal returns for the low PIN portfolios in the top three size quartiles are consistent with my hypothesis. The absence of underperformance in the low size, high PIN portfolio, however, is inconsistent with the remaining results. The GRS F-statistic strongly rejects the null that all 12 intercepts are jointly equal to zero, with a value of 2.746 (p-value = 0.00724). The underperformance of high PIN portfolios is generally robust to value-weighted returns as well. Panel B of Table 8 shows the coefficient estimates from these regressions. Most of the results from Panel A hold. The exception is that the high PIN portfolio in the high size

31 quartile is no longer significant. Easley, Hvidkjaer, and O’Hara (2004) also do not find abnormal returns in large size, PIN-based portfolios. The underperformance of the high PIN portfolios in size quartiles two and three is even stronger after value-weighting, with abnormal returns of -0.997 percent and -1.045 percent per month. The increase in negative performance as we move from low to high PIN portfolios still generally holds. Again, the exception is the low size, high PIN portfolio. The GRS F-test again rejects the null hypothesis that all 12 intercepts are jointly equal to zero (p-value = 0.0156). The results from Table 8 indicate that PIN is reasonably successful in predicting negative abnormal returns in high short interest stocks, even for value-weighted portfolios.28 Consistent with my hypothesis, high PIN portfolios generally have larger negative abnormal returns than low PIN portfolios when controlling for size, book-to-market, and momentum. The abnormal returns are economically meaningful. On a value-weighted basis, high size portfolios do not underperform even with high average PINs. This result represents a new finding for the short interest literature, and is a key result of this paper. Specifically, the relationship between the level of short interest and subsequent returns is unclear, and may be responsible for the ambiguous findings cited by Asquith, Pathak, and Ritter (2004). On the other hand, a measure of the probability of informed trade from the microstructure literature seems to give predictions about underperformance in high short interest stocks. Additionally, this relationship between PIN and informed short selling has theoretical grounding. The results of this section also complement the literature that examines the link between information and asset returns [Easley, Hvidkjaer, and O’Hara (2002, 2004)]. If PIN is a successful measure for asymmetric information, then applying such a measure seems to be a useful framework for investigating the returns to high short interest stocks. The underperformance generated by the portfolios in Table 8 indirectly contradict the prediction of Diamond and Verrecchia, that price adjustments to announced changes in short interest occur almost immediately. As stated earlier, a direct test of their model would require an event study around the announcement date and a model of expected short interest [Senchack and Starks (1993)]. The adjustment to the information content of short interest appears to lag at least

28 Similar regressions were run on portfolios formed from independent sorts of size and PIN. The results were qualitatively similar, except for the portfolios at the extremes (low size/low PIN, high size/high PIN) where there were much fewer stocks in the portfolio. As a result, all remaining sorts in the paper are done within size quartile to ensure adequate power.

32 into the next month. On the other hand, the results support Diamond and Verrecchia’s contention that informed traders will take short positions in the most overvalued stocks, and that these stocks will have negative subsequent returns.

VI. Robustness Checks

It is possible that PIN is serving as a proxy for some other underlying variable. Likely candidates include the spread and the short interest ratio. The former seems more likely given the evidence cited earlier that PIN is strongly related to the adverse selection component of the spread [Easley et al.(1996), Easley, O’Hara, and Paperman (1998), and Chung and Li (2003)]. Also, Table 7 shows that the relative half-spread increases with PIN in our portfolio sorts, as expected. Short interest on the other hand, does not necessarily increase with PIN in our portfolio sorts, but seems to be a plausible alternative. I repeat the sorting procedure from the previous section, but perform the secondary sort on the basis of either relative spread or short interest. I sort on relative spread and short interest ratio within each size quartile. Table 9 contains results for the sort on size and relative half-spread. Panel A shows the mean firm size for each of the 12 portfolios, and Panel B shows the mean relative half-spread. Within size quartile, firm size is decreasing in the spread as expected. Panel C shows the estimated intercepts and corresponding t-statistics from the factor regressions. Portfolio returns are value-weighted. Within each size quartile, there is no consistent pattern in the size of the intercepts as we move from low spread to high spread. In three of the four size quartiles, the low spread intercept is larger than the high spread intercept. However, in three of the four quartiles, the middle spread portfolio has the largest intercept, and all three are statistically significant. In size quartile three, the magnitude of the intercepts are decreasing in the spread, contrary to what intuition would suggest. Two portfolios have large negative abnormal returns, around -1.36 percent per month. One of these, however, is a low spread portfolio. In general, while some portfolios formed on the basis of the relative spread do underperform, no consistent relationship is evident. My prior expectation was that the spread may pick up a lot of the same information as PIN, but that does not seem to be the case here. One explanation for this is that PIN is intended purely as a measure of asymmetric information, while the spread may, in practice, contain elements of inventory and order processing costs. This is consistent with George, Kaul,

33 and Nimalendran (1991) and Huang and Stoll (1997) who suggest that the bid-ask spread is composed mainly of inventory and order processing costs. Furthermore, Easley, Hvidkjaer, and O’Hara (2002) found cross-sectional returns to be explained by PIN, but not the bid-ask spread. Overall, this strengthens the interpretation of PIN as a proxy for information risk. Table 10 displays results for the sort on size and short interest. Based on the results of Table 7, I do not expect short interest to be able to reliably predict underperformance, and this is confirmed from Panel C. Similar to the sort on spread, there is no obvious pattern in underperformance as we increase short interest within each size quartile. Only one of the high SI portfolios has a significant negative intercept (the low size quartile, a = -1.487), but all four of the middle SI portfolio intercepts are significant. Also, the magnitudes of two of these four middle SI intercepts are the largest within their respective size quartile. This finding seems to contradict the notion that high SI portfolios should underperform low SI portfolios, and the ability of short interest as a predictor of underperformance in size-neutral portfolios is absent. Lastly, the GRS F-test gives a value of 1.7868, with a p-value of 0.0807. Thus, the hypothesis that the 12 intercepts are jointly zero is rejected only at a higher significance level than those of the previous portfolio sorts. The results from this table shed particular light on the mixed empirical results of other studies on the returns of short interest stocks. The evidence supports the notion that the level of short interest is not the best indicator of future underperformance. Instead, some measure of informed trading seems to be superior.

VII. Concluding Remarks

This paper advocates an alternate approach to testing the information content of short interest. I estimate the probability of informed trading as a proxy for information risk in a sample of high short interest stocks to see if differences in PIN help to explain different returns for size-neutral portfolios. In general, the results support the use of PIN as a measure of informed trading in highly shorted stocks. I find that high PIN stocks have significant negative abnormal returns. Low PIN stocks, on the other hand, generally do not underperform. The PIN for these stocks is related to the level of short sale constraints, implying that PIN increases as the proportion of informed traders that are constrained decreases. Further, the differences in PIN can be motivated by different reasons for short selling, such as arbitrage versus information-based short selling.

34 There are several opportunities for further study. PIN is one among many possible proxies for informed trading. There are several other measures of adverse selection that could be applied to high short interest stocks to confirm the relation between private information and future returns. Natural candidates include the adverse selection component of the spread [Glosten and Harris (1988), George, Kaul, and Nimalendran (1991), Huang and Stoll (1997)] and a measure of price impact [Hasbrouck (1991), Brennan and Subrahmanyam (1996)] . This is especially important given different findings in the literature concerning the relationship among these different measures of adverse selection. For example, Chung and Li (2003) show a strong relationship between PIN and the adverse selection component of the spread, while Chung, Li, and McInish (2004) find that price impact and PIN are significantly positively related. Odders- White and Ready (2005) find that a variety of adverse selction measures, including PIN, are all related to changes in credit ratings. However, of all their measures, PIN has the weakest correlation to the remaining measures. Dennis and Weston (2001) also find low correlation between PIN and other proxies for informed trading (and in some cases, negative correlation). In short, confirming the results of this paper with alternate measures of informed trading would strengthen the interpretation of PIN as an appropirate proxy. Easley, Hvidkjaer, and O’Hara (2004) create a traded factor based on PIN and incorporate this into a factor asset pricing model. They find some evidence that such a factor helps explain returns to portfolios formed on the basis of size and PIN. A natural extension of this study would be to apply such a factor to high short interest stocks, but would require a longer time series of data. This remains a task for future research. Lastly, this study does not incorporate trading costs into the analysis, so it remains to be seen whether or not such a trading strategy is profitable after costs. The answer to this question depends partly on how long-lived the information in the high PIN portfolios are. For example, by altering the selection criteria, and allowing a stock to remain in the portfolio even after it’s PIN falls below a certain cutoff value, we could test whether or not the abnormal returns are persistent. Such an approach is applied in Asquith, Pathak, and Ritter (2004), and Desai et al.(2000). Not surprisingly, they find different results. This presents an opportunity to examine if high PIN stocks also have long-lived information.

35 Appendix

Institutional Details of Short selling

The short sale of a stock occurs when an investor sells shares that he does not own. The mechanics of a short sale involve borrowing the shares from another investor, such as a brokerage house (e.g., from investors’ margin accounts) or an institutional investor, selling them on the open market, and then buying back shares at a later date and returning them to the original owner to close out the position. A short transaction is profitable when the price at which the shares were repurchased is lower than the price at which they were sold. The risk profile of a short position is quite different from a long position, where the maximum gain from a short position is 100% (if the stock falls to a price of zero) and the maximum loss is unlimited. The opposite is true for a long position, where the maximum loss is 100% and the maximum gain is unlimited. Because of the inherent risks in short selling, and its potential as a vehicle for market manipulation, short selling is heavily regulated in U.S. and foreign markets, and is even outlawed in some foreign markets.

Short sale Constraints

In U.S. stock markets, there are several legal and institutional restrictions on short sales, making them generally more costly than regular sales. These restrictions lead to constrained short selling, either because an investor is legally prohibited from shorting or because the restrictions make short selling prohibitively costly. Many institutional investors, such as mutual funds and pension funds, are prohibited or severely restricted from short selling due to regulatory “prudent investor rules” that govern the operation of these funds. Furthermore, corporate insiders are prohibited from short selling pursuant to the Securities Exchange Act of 1934. In addition to outright regulatory prohibitions on short selling for certain investors, there are other restrictions, both institutional and economic, that add to the overall costs of short selling. In order to short-sell a security, an investor must first borrow the shares from an investor that owns them, usually through a broker. If shares are located, the proceeds of the sale are usually not available to the short-seller and are used as collateral on the margin account set up to

36 carry out the transaction. The Federal Reserve’s Regulation T requires total margin collateral amounting to 150% of the current market value of the security [12 CFR 220.12]. So in addition to the proceeds of the short sale not being available for reinvestment (they remain in the margin account for collateral), the investor is required to deposit cash collateral for the remaining 50%.29 Whether or not the short seller receives interest on the proceeds of the short sale is another matter and may depend on trader identity. Three parties are involved in a short transaction, the lender of shares, the borrower of shares (the short-seller) and an intermediary who brokers the deal. If the lender of securities is an institutional account, they usually have access to the proceeds of the short sale for reinvestment, and pay the broker of the deal interest on the loaned proceeds. The difference between the rate paid the broker and the rebate rate given to the short- seller is the broker’s spread on the deal. When the lender of shares is a retail margin account, the investor usually is not aware that their shares have been lent out, and receive no benefit. When the borrower (the short-seller) is a large institutional account, they are usually able to negotiate the receipt of interest on the proceeds, though the “rebate rate” they receive is usually below the market rate. Small retail investors often receive no interest on the proceeds of a short sale. The higher the demand for borrowing a security the lower the rebate rate received by the short-seller, and in some cases, the lender may even demand a premium. Stocks for which the rebate rate has fallen below the baseline general collateral rate are said to “be on special”.30 The investor may be subject to additional maintenance margin calls if the price moves against his short position. Moreover, the short-seller must reimburse the security lender for any dividends or distributions made while the short position is open. As pointed out by Dechow et al. (2001), payment of dividends to the lender is a real cost to the short-seller if the drop in share price around the ex- dividend date is less than the amount of the dividend that needs to be reimbursed. The restrictions associated with the timing of short sales represent additional costs. Pursuant to Rule 10a-1 under the Securities and Exchange Act of 1934, short sales are subject to an uptick rule on the NYSE. The rule requires that all short sale transactions take place on an uptick or a zero uptick, limiting the ability of short sellers to get quick execution of their orders and subjecting them to further price risk. Short selling in Nasdaq stocks is restricted by a “bid-

29 Cash collateral can include the market value of long positions in the investor’s account or interest-bearing Treasury securities of which the investor receives the interest. 30 For a detailed description and analysis of the equity lending market and stocks on special see D’Avolio (2002), Geczy, Musto and Reed (2002) and Reed (2002).

37 test” implemented by the SEC in 1994 (as opposed to a tick test), whereby a short sale may occur only if the execution price is not below the previous inside bid (but doesn’t require an actual trade at a higher price).31 There are also tax consequences that make short transactions more costly than long transactions, since profits from short sales are taxed at the short-term capital gains rate, regardless of the length of the holding period.32 Short sellers also expose themselves to a form of recall risk, known as a “short squeeze”. The terms of the loan by which the short-seller borrows shares permits the original owner to demand the shares be returned at any time, subjecting the short-seller to ‘buy-in’ risk. The short- seller will try and find other shares to borrow, but if he is unable to do so, must purchase shares in the open market at the prevailing market price. As a consequence, short sellers usually prefer highly liquid securities where the short squeeze risk is lower. Additionally, short sellers may try to determine the identity of the original owner of the borrowed shares to minimize this risk, favoring buy-and-hold investors (such as index funds) over active investors. Lastly, short selling strategies that require long time horizons (such as waiting for the rest of the market to realize a security is overvalued) expose the short-seller to holding period costs. These include not receiving market interest rates on their collateral, and the risk of being squeezed into unwinding their position before realizing a profit. While most investors are faced with these restrictions on short selling, certain investors enjoy exemptions from some of these restrictions, thus reducing their costs of shorting. Nasdaq market makers, for example, are exempt from NASD Rule 3370(b) which requires most investors to check a “hard to borrow” list of stocks before shorting, and make an affirmative determination that the shares will be deliverable ( known as “a locate”) for stocks that are on the list.33 Thus, market makers are able to carry out a short transaction when others are not able to since they are exempt from having to locate shares. Nasdaq market makers are also currently exempt from the bid test governing the timing of short sales. NYSE specialists, on the other hand, are not exempt from the exchange’s tick test, though there are certain exemptions for some

31 Currently, the SEC is proposing a two-year experiment which creates uniform rules across U.S. markets by using the bid test in all markets. The proposal would also drop all short sale restrictions on certain highly liquid stocks and eliminate certain existing exemptions from short sale restrictions on Nasdaq market makers. 32 Given the changes in the tax code over the last 30 years, this cost has also changed through time. If the short-term and long-term capital gains rate for a given investor are equal, this tax consequence becomes irrelevant. 33 See Evans et al.(2002) for a detailed description of the market maker’s exemption from locating shares.

38 index arbitrage trades, odd lot dealers, or block positioners.34 Option market makers often receive interest on the proceeds of their short sales, thus reducing their costs.

34 NYSE Rule 440(b) specifies that exchange specialists are not exempt from the uptick rule, though the exchange has the ability to exempt the specialist if it chooses. Most index arbitrage trades are not exempt from the uptick rule, though some are. See Angel (1997) or Worley (1990) for specific details.

39 References

Almazan, A., K. C. Brown, M. Carlson, and D. A. Chapman, 2004, Why constrain your mutual fund manager?, Journal of Financial Economics 73, 289-321.

Angel, J. J., 1997, Short Selling on the NYSE, Working Paper, Georgetown University.

Asquith, P., P. Pathak and J. Ritter, 2004, Short Interest and Stock Returns, Working Paper, MIT, Harvard University, and University of Florida.

Asquith, P., and L. Meulbroek, 1996, An empirical investigation of short interest, Working Paper 96-012, Harvard Business School.

Bessembinder, H., 2003, Issues in assessing trade execution costs, Journal of Financial Markets 6, 233-257.

Boehme, R.D., B.R. Danielsen, and S.M. Sorescu, 2005, Short-Sale Constraints and Overvaluation, Forthcoming, Journal of Financial and Quantitative Analysis.

Brent, A., D. Morse, and E. K. Stice, 1990, Short Interest - Explanations and Tests, Journal of Financial and Quantitative Analysis 25, 273-289.

Carhart, M. M., 1997, On persistence in mutual fund performance, Journal of Finance 52, 57-82.

Chen, J., Hong, H. and Stein, J.C., 2002, Breadth of Ownership and Stock Returns, Journal of Financial Economics 66, 171-205.

Christophe, S. E., M. G. Ferri, and J. J. Angel, 2004, Short selling Prior to Earnings Announcements, Journal of Finance 59, 1845-1875.

Chung, K., M. Li, 2003, Adverse-Selection Costs and the Probability of Information-Based Trading, The Financial Review 38, 257-272.

Chung, K., M. Li, and T. McInish, 2004, Information-based trading, price impact of trades, and trade autocorrelation, Forthcoming, Journal of Banking and Finance..

D'Avolio, G., 2002, The market for borrowing stock, Journal of Financial Economics 66, 271- 306.

Danielsen, B. R., and S. M. Sorescu, 2001, Why do option introductions depress stock prices? A study of diminishing short sale constraints, Journal of Financial and Quantitative Analysis 36, 451-484.

Dechow, P. M., A. P. Hutton, L. Meulbroek, and R. G. Sloan, 2001, Short-sellers, fundamental analysis, and stock returns, Journal of Financial Economics 61, 77-106. Dennis, P. J. and J. P. Weston, 2001, Who’s Informed? An Analysis of Stock Ownership and Informed Trading, Working Paper, University of Virginia (McIntire) and Rice University.

Desai, H., K. Ramesh, S. R. Thiagarajan, and B. V. Balachandran, 2002, An investigation of the informational role of short interest in the Nasdaq market, Journal of Finance 57, 2263- 2287.

Diamond, D. W., and R. E. Verrecchia, 1987, Constraints on short selling and asset price adjustment to private information, Journal of Financial Economics 18, 277-311.

Easley, D., S. Hvidkjaer, and M. O'Hara, 2004, Factoring Information into Returns, Working Paper, Cornell University and University of Maryland.

Easley, D., S. Hvidkjaer, and M. O'Hara, 2002, Is information risk a determinant of asset returns?, Journal of Finance 57, 2185-2221.

Easley, D., M. Kiefer, and M. O'Hara, 1996, Cream-skimming or profit sharing? The curious role of purchased order flow, Journal of Finance 51, 811-833.

Easley, D., M. Kiefer, and M. O'Hara, 1997a, The information content of the trading process, Journal of Empirical Finance 4, 159-186.

Easley, D., N. M. Kiefer, and M. O’Hara, 1997b, One day in the life of a very common stock, Review of Financial Studies 10, 805-835.

Easley, D., N. M. Kiefer, M. O’Hara, and J. B. Paperman, 1996, Liquidity, information, and infrequently traded stocks, Journal of Finance 51, 1405-1436.

Easley, D., and M. O’Hara, 1992, Time and the Process of Security Price Adjustment, Journal of Finance 47, 577-605.

Easley, D., M. O'Hara, and J. B. Paperman, 1998, Financial analysts and information-based trade, Journal of Financial Markets 1, 175-201.

Ellis, K., R. Michaely, and M. O'Hara, 2000, The accuracy of trade classification rules: Evidence from Nasdaq, Journal of Financial and Quantitative Analysis 35, 529-551.

Evans, R. B., C. C. Geczy, D. K. Musto, and A. V. Reed, 2002, Impediments to short selling and option prices, Working paper, University of Pennsylvania.

Fama, E. F., 1998, Market efficiency, long-term returns, and behavioral finance, Journal of Financial Economics 49, 283-306.

Fama, E. F., and K. R. French, 1993, Common Risk-Factors in the Returns on Stocks and Bonds, Journal of Financial Economics 33, 3-56.

41

Fama, E. F., and K. R. French, 1996, Multifactor explanations of asset pricing anomalies, Journal of Finance 51, 55-84.

Figlewski, S., 1981, Informational Effects of Restrictions on Short Sales: Some Empirical Evidence, Journal of Financial and Quantitative Analysis 16, 463-476.

Figlewski, S., and G. P. Webb, 1993, Options, Short Sales, and Market Completeness, Journal of Finance 48, 761-777.

Fu, H., 2002, Information Asymmetry and the Market Reaction to Equity Carve-Outs, Working Paper, University of Minnesota.

Geczy, C. C., D. K. Musto, and A. V. Reed, 2002, Stocks are special too: an analysis of the equity lending market, Journal of Financial Economics 66, 241-269.

George, T. H., G. Kaul, and M. Nimalendran, 1991, Estimation of the Bid-Ask Spread and its Components: A New Approach, Review of Financial Studies 4, 623-656.

Gibbons, M.R., S.A. Ross, and J. Shanken, 1989, A test of the efficiency of a given portfolio, Econometrica 57, 1121-1152.

Gintschel, A., 2001, Short interest on Nasdaq, Working Paper, Emory University.

Glosten, L. R. and L. E. Harris, 1988, Estimating the Components of the Bid-Ask Spread, Journal of Financial Economics 21, 123-142.

Glosten, L. R., and P. R. Milgrom, 1985, Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders, Journal of Financial Economics 14, 71-100.

Hasbrouck, J., 1991, Measuring the Information Content of Stock Trades, Journal of Finance 46, 179-207.

Hasbrouck, J., 2004, Empirical Market Microstructure: Economics and Statistical Perspectives on the Dynamics of Trade in Securities Markets, Manuscript, NYU.

Heidle, H. G., and R. D. Huang, 2002, Information-Based Trading in Dealer and Auction Markets: An Analysis of Exchange Listings, Journal of Financial and Quantitative Analysis 37, 391-424.

Huang, R. D. and H. R. Stoll, 1997, The Components of the Bid-Ask Spread: A General Approach, Review of Financial Studies 10, 995-1034.

Jarrow, R., 1980, Heterogeneous Expectations, Restrictions on Short Sales, and Equilibrium Asset Prices, Journal of Finance 35, 1105-1113.

42 Lee, C. M. C., and B. Radhakrishna, 2000, Inferring investor behavior: evidence from TORQ Data, Journal of Financial Markets 3, 83-111.

Lee, C. M. C., and M. J. Ready, 1991, Inferring Trade Direction from Intraday Data, Journal of Finance 46, 733-746.

Miller, E. M., 1977, Risk, Uncertainty, and Divergence of Opinion, Journal of Finance 32, 1151- 1168.

Mitchell, M. L., and E. Stafford, 2000, Managerial decisions and long-term stock price performance, Journal of Business 73, 287-329.

Odders-White, E. R., 2000, On the Occurrence and Consequences of Inaccurate Trade Classification, Journal of Financial Markets 3, 259-286.

Odders-White, E. R., and M. J. Ready, 2005, Credit Ratings and Stock Liquidity, Forthcoming, Review of Financial Studies.

Peterson, M., and E. Sirri, 2003, Evaluation of the biases in execution cost estimation using trade and quote data, Journal of Financial Markets 6, 259-280.

Reed, A. V., 2001, Costly short selling and stock price adjustment to earnings announcements, Working paper, University of Pennsylvania.

Rubinstein, M., 2004, Great moments in financial economics: III. Short sales and stock prices, Journal of Investment Management 2.

Senchack, A. J., and L. T. Starks, 1993, Short-Sale Restrictions and Market Reaction to Short- Interest Announcements, Journal of Financial and Quantitative Analysis 28, 177-194.

Vega, C., 2004, Are Investors Overconfident?, Working Paper, University of Rochester.

Venter, J. H. and D. de Jongh, 2002, Extending the EKOP Model to Estimate the Probability of Informed Trading, Working Paper, North-West University.

Woolridge, J.R., and A. Dickinson, 1994, Short Selling and Common Stock Prices, Financial Analysts Journal 50, 30-38.

Worley, D., 1990, The regulation of short sales: The long and the short of it., Brooklyn Law Review 55, 1255-1299

43 Figure 1: Decision Tree Diagram of the Trading Process without Short-Sale Constraints

Taken from Easley, Kiefer and O'Hara (1997b)

Informed Sell µ Low signal (Ψ=L) ε Uninformed Sell Information event δ 1/2 Uninformed No-Trade occurs 1 - µ 1 - ε (Ψ=H,L) ε Uninformed Buy 1/2 α Uninformed No-Trade 1 - δ µ Informed Buy 1 - ε

High signal ε Uninformed Sell (Ψ=H) 1 - µ 1/2 Uninformed No-Trade 1 - ε 1 - α ε Uninformed Buy 1/2 Information Uninformed No-Trade event does 1 - ε not occur (Ψ=0)

ε Uninformed Sell

1/2 Uninformed No-Trade 1 - ε ε Uninformed Buy 1/2 Uninformed No-Trade 1 - ε Figure 2: Decision Tree Diagram of the Trading Process with Short-Sale Constraints

Informed SS c1 c2 Informed SS 1 -h c3 Uninformed SS Informed NT c1 µ c2 Uninformed No-Trade Low signal h Informed RS 1-h c3 (Ψ=L) ε Uninformed No-Trade Information event δ 1/2 h Uninformed RS occurs 1 - µ 1 - ε Uninformed NT (Ψ=H,L) ε 1/2 Uninformed Buy Uninformed SS α Uninformed NT c1 1 - δ µ Informed Buy 1 - ε c2 Uninformed No-Trade 1-h c3 High signal ε Uninformed No-Trade (Ψ=H) 1 - µ 1/2 h Uninformed RS 1 - ε Uninformed No-Trade 1 - α ε 1/2 Uninformed Buy Information Uninformed No-Trade event does 1 - ε not occur (Ψ=0) Uninformed SS c1 c2 Uninformed No-Trade 1-h c3 ε Uninformed No-Trade

1/2 h Uninformed RS 1 - ε Uninformed No-Trade ε 1/2 Uninformed Buy

1 - ε Uninformed No-Trade Table 1 Descriptive Statistics for High Short Interest Firms

Mean monthly statistics for all firms that fall into the high short interest sample for at least one month during the sample period, January 1995 – December 1999, by year. Panel A includes only firm-months where short interest is greater than or equal to 2.5%. Panel B contains only firm months where short interest is less than 2.5%, and Panel C includes all months. For comparison, Panel D contains statistics for all firm-months (not just high short interest firms) where short interest is less than 2.5%. No. Firms per Month is the monthly average number of firms in the sample. Short interest ratio is the monthly short interest divided by the number of shares outstanding. Size is the market value of equity, in millions of dollars. Rt+1 is the subsequent month’s return and HPR(t-12,t-1) is the prior 12 months’ holding period return.

Panel A: High SI Firms: Months when SI ≥ 2.5% (N=20,618) No. Firms Short Interest Year Size Price R HPR per Month Ratio t+1 (12,1)tt−−

1995 265.8 0.0630 1993.00 27.59 0.0208 0.1746 1996 310.0 0.0617 2248.14 29.23 0.0151 0.3317 1997 405.3 0.0639 2755.60 30.80 0.0122 0.2571 1998 411.7 0.0663 3043.81 29.23 -0.0042 0.1193 1999 325.3 0.0646 3802.17 28.55 -0.0012 0.0342

Total 343.6 0.0639 2768.54 29.08 0.0085 0.1834

Panel B: High SI Firms: Months when SI < 2.5% (N=39,902) No. Firms Short Interest Year Size Price R HPR per Month Ratio t+1 (12,1)tt−−

1995 688.1 0.0091 2797.14 64.46 0.0249 0.1461 1996 706.6 0.0101 3294.23 52.62 0.0201 0.2819 1997 664.8 0.0111 4031.72 36.71 0.0195 0.2888 1998 620.7 0.0112 5156.08 35.00 0.0031 0.1994 1999 645.1 0.0109 6113.61 31.60 0.0085 0.0685

Total 665.0 0.0105 4278.56 44.08 0.0152 0.1970

Panel C: High SI Firms All Months (N=60,520) No. Firms Short Interest Year Size Price R HPR per Month Ratio t+1 (12,1)tt−−

1995 953.9 0.0241 2572.66 54.14 0.0238 0.1544 1996 1016.6 0.0258 2975.78 45.74 0.0186 0.2969 1997 1070.1 0.0311 3544.03 34.47 0.0168 0.2762 1998 1032.3 0.0332 4311.39 32.69 0.0001 0.1670 1999 970.4 0.0289 5338.37 30.60 0.0053 0.0571

Total 1008.7 0.0286 3748.45 39.53 0.0129 0.1903 Table 1 (continued)

Panel D : All Firms: SI < 2.5% (N=90,987)

No. Firms Short Interest Year Size Price R HPR per Month Ratio t+1 (12,1)tt−−

1995 1440.2 0.0064 2888.69 46.13 0.0216 0.1226 1996 1506.3 0.0069 3457.67 52.71 0.0185 0.2437 1997 1529.6 0.0076 4228.84 62.29 0.0192 0.2761 1998 1542.2 0.0077 5191.33 75.97 0.0013 0.1899 1999 1564.1 0.0075 6149.63 73.37 0.0036 0.0358

Total 1516.5 0.0072 4383.23 62.09 0.0128 0.1736 Table 2 Cross-Sectional Correlations

Time-series average of the monthly cross-sectional correlation between variables. SI is the monthly short interest ratio, Spr ($) is the monthly average of the trade-weighted daily quoted half-spread, Spr (%) is the monthly average of the trade-weighted daily relative half-spread. PIN is the probability of informed αµ trading, calculated as . α is the probability than an information event occurred, δ is the αµαµε+−(1 ) probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, and µ is the probability that a trader is informed. Size is the market value of

equity, No. Trades is the monthly average of the daily number of trades, and Rt+1 is the subsequent month’s return.

No. SI Spr ($) Spr (%) PIN α δ ε µ Size R Trades t+1 SI 1 Spread ($) 0.069 1 Spread (%) 0.076 -0.003 1

PIN 0.068 0.078 0.248 1 α -0.005 -0.017 0.026 0.735 1 δ -0.065 -0.083 0.176 0.107 0.142 1 ε -0.067 -0.138 -0.212 -0.574 -0.315 -0.107 1 µ 0.035 0.078 0.198 -0.153 -0.605 0.012 0.037 1

Size -0.115 -0.124 -0.345 -0.148 0.075 -0.126 0.194 -0.220 1 No. Trades -0.007 -0.192 -0.345 -0.172 0.058 -0.193 0.285 -0.255 0.660 1

Rt+1 0.000 0.003 -0.031 -0.010 0.015 -0.011 0.001 -0.026 0.030 0.013 1

Table 3 Parameter Estimate Diagnostics: Monthly versus Quarterly Estimation Interval

This table shows the results of diagnostic checks on parameters estimated from maximizing the following likelihood function:

BS BSN++ D D ⎧⎫⎛⎞µµ ⎛⎞ ⎛⎞1 BS+ N ∑∑log⎨⎬αδ (1−++ )⎜⎟ 1 αδ ⎜⎟ 1 ++− (1 α )⎜⎟ + log{}x [] (1 −− µ )(1 ε ) . α is the probability than an information event d =1 ⎩⎭⎝⎠xx ⎝⎠ ⎝⎠1 − µ d =1 occurred, δ is the probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, and µ is the probability that a trader is informed. B and S are the daily number of buys and sells, respectively, and αµ N is the number of no-trade outcomes. PIN is the probability of informed trading, calculated as . Panel A1 αµαµε+−(1 ) compares monthly parameter estimates to quarterly estimates, and shows the mean, standard deviation, and correlation between these estimates. Panel A2 uses monthly parameter estimates and averages across the quarter, and compares this quarterly average to the actual quarterly estimate. Panel B shows the average standard error for each parameter when estimated over a monthly or quarterly interval, and shows the percentage of parameter estimates that achieve significance at a 10 percent level. Panel C shows the number (and percent) of boundary solutions (parameter estimate equal to zero or one) achieved by the maximum likelihood estimation for monthly versus quarterly estimation intervals.

Panel A: Parameter Summary Statistics

Parameter α δ ε µ PIN Estimation Interval: Month Quarter Month Quarter Month Quarter Month Quarter Month Quarter

A1: Quarterly Average of Monthly Estimates vs. Quarterly Estimates

Mean 0.451 0.428 0.397 0.393 0.549 0.552 0.252 0.241 0.163 0.162 StdDev 0.164 0.167 0.246 0.261 0.033 0.031 0.080 0.088 0.071 0.068 Corr 0.712 0.938 0.943 0.619 0.910

A2: Monthly Estimates vs. Quarterly Estimates

Mean 0.448 0.427 0.396 0.393 0.550 0.553 0.252 0.241 0.161 0.160 StdDev 0.243 0.170 0.318 0.260 0.053 0.032 0.121 0.092 0.092 0.069 Corr 0.481 0.720 0.590 0.401 0.709

Panel B: Precision of Estimated Parameters

Parameter α δ ε µ

Estimation Interval: Month Quarter Month Quarter Month Quarter Month Quarter

Standard Error 0.1560 0.0922 0.1296 0.0975 0.0197 0.0105 0.0526 0.0312 % Significant @ 10% 75.18% 95.25% 47.41% 79.20% 100% 100% 95.68% 99.19%

Table 3 (continued)

Panel C: Number of Corner Solutions

Parameter α δ ε µ PIN Corner Value 0 1 0 1 0 1 0 1 0 1

Monthly 4036 3074 14911 5266 0 0 90 0 4126 0 5.63% 4.29% 20.80% 7.34% 0.00% 0.00% 0.13% 0.00% 5.75% 0.00%

Quarterly 438 176 1153 244 0 0 11 0 449 0 2.00% 0.80% 5.25% 1.11% 0.00% 0.00% 0.05% 0.00% 2.05% 0.00% Table 4 Parameter Estimates for the Unrestricted Model

Panel A shows the mean and standard deviation of the parameters estimated by maximizing the following likelihood function,

B S N BSN++ D ⎡ µ µ[(1)()]hhcc+− + µ(1− hc ) ] 1 ⎤ ⎛⎞ ⎛⎞12 ⎛⎞3 ⎛⎞ logαδ (1−++ ) 1 αδ 1 + 1 + +− (1α ) ∑ ⎢ ⎜⎟ ⎜⎟⎜⎟1 ⎜⎟⎥ d =1 ⎝⎠xxhhc[+− (1 ) ] (1 −µεε )[(1 − ) + (1 − hcc )( + )] 1 − µ ⎣⎢ ⎝⎠1 ⎝⎠2 23 ⎝⎠⎦⎥

D N ++−−−+−+logxhBS+ [ (1 hc ) ] S (1µεε )[(1 )1 (1 hcc )( )] ∑ {}123[]2 d =1

These estimates represent the parameters from the unrestricted structural model where short-sale constraints are nonzero. These estimates are compared to the restriction that h=1, or that short-sale constraints are nonzero. α is the probability than an information event occurred, δ is the probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, µ is the probability that a trader is informed, h is the probability that a potential seller owns the stock,

and c1 , c2 , and c3 represent the fraction of traders subject to the various categories of short-sale constraints. Panel B gives the correlation between the restricted and unrestricted parameter estimates.

Panel A: Restricted versus Unrestricted Parameter Estimates

α δ ε µ h c1 c2 c3 PIN

Mean 0.4317 0.3542 0.5521 0.2511 - - - - 0.1572 Restricted (h=1) Std Dev 0.1333 0.2090 0.0324 0.0733 - - - - 0.0630

Mean 0.4215 0.4936 0.6017 0.2654 0.7769 0.3466 0.3819 0.2715 0.1563 Unrestricted Std Dev 0.1361 0.1906 0.0484 0.0881 0.1476 0.0623 0.0949 0.0999 0.0706

Panel B: Correlation Coefficients

Restricted Model (h=1) Unrestricted Model

α δ ε µ PIN α δ ε µ h c1 c2 c3 PIN* α 1 δ 0.163 1 ε -0.345 -0.151 1 µ -0.446 0.172 -0.243 1 PIN 0.671 0.190 -0.774 0.163 1

α 0.605 0.471 -0.292 -0.097 0.479 1 δ 0.207 0.824 -0.079 0.080 0.136 0.436 1 ε 0.077 -0.316 0.521 -0.148 -0.206 -0.179 0.056 1 µ -0.148 0.077 -0.281 0.573 0.271 -0.360 0.112 0.071 1 h -0.282 0.168 0.149 -0.017 -0.267 -0.103 -0.228 -0.686 -0.193 1 c1 -0.032 -0.111 0.145 -0.160 -0.159 -0.086 -0.007 0.132 -0.136 -0.207 1 c2 -0.001 -0.036 -0.019 -0.030 -0.010 -0.055 0.056 0.172 -0.077 -0.285 -0.247 1 c3 0.021 0.104 -0.072 0.128 0.109 0.106 -0.049 -0.246 0.158 0.400 -0.390 -0.796 1 PIN* 0.485 0.463 -0.683 0.306 0.811 0.737 0.377 -0.318 0.182 -0.200 -0.216 0.013 0.123 1 Table 5 Four Factor Coefficient Estimates for EW and VW Overlapping Portfolios of High SI Firms

Regression coefficients from a time-series regression of SI monthly portfolio returns (in excess of one-month T-bill rate) on the three Fama and French (1993) factors and a fourth momentum factor from Carhart (1997). The intercept from the following equation is the measure of the portfolio monthly abnormal return:

Rp,,tft−=+ R a bRMRFt + sSMB t + hHML t + mMOM tt +ε , where Rp,,tft− R is the monthly excess portfolio return, RMRFt is the market factor, SMBt is the size factor, HMLt is the book-to- market factor and MOM t is the momentum factor. The 2.5% SI sample contains all firms that had a short interest ratio greater than or equal to 2.5% the previous month. The 5%, 7.5%, and 10% portfolios are similarly defined. The 95th percentile portfolio contains all firms with SI in the 95th percentile of all NYSE firms the previous month. Likewise for the 99th percentile portfolio. Each regression contains 60 monthly observations over the sample period January 1995 – December 1999. Portfolio returns are equal weighted in Panel A and value weighted in Panel B. The t-statistics are in parentheses. Panel C contains descriptive summary statistics for each short interest sample.

Panel A: Equal Weighted

SI Sample N Intercept RMRF SMB HML MOM Adj. R2

-0.702 1.195 0.564 0.356 -0.341 ≥ 2.5% 20618 0.96 (-4.27) (27.47) (11.64) (4.82) (-7.14)

-0.818 1.237 0.730 0.366 -0.332 ≥ 5% 9296 0.92 (-3.40) (19.46) (10.31) (3.39) (-4.75)

-0.816 1.242 0.820 0.416 -0.354 ≥ 7.5% 5171 0.88 (-2.60) (14.99) (8.88) (2.96) (-3.89)

-0.820 1.177 0.871 0.254 -0.285 ≥ 10% 3167 0.83 (-2.12) (11.50) (7.64) (1.46) (-2.54)

≥ 95th -0.969 1.269 0.816 0.407 -0.347 5292 0.90 percentile (-3.28) (16.23) (9.36) (3.07) (-4.04)

≥ 99th -1.038 1.043 0.870 -0.085 -0.199 996 0.55 percentile (-1.36) (5.19) (3.88) (-0.25) (-0.90)

Table 5 (continued)

Panel B: Value Weighted

SI Sample N Intercept RMRF SMB HML MOM Adj. R2

-0.439 1.083 0.123 -0.054 -0.006 ≥ 2.5% 20618 0.91 (-1.99) (18.56) (1.88) (-0.54) (-0.10)

-0.806 1.114 0.281 0.048 -0.109 ≥ 5% 9296 0.83 (-2.50) (13.08) (2.96) (0.33) (-1.16)

-0.446 1.128 0.454 0.192 -0.140 ≥ 7.5% 5171 0.78 (-1.16) (11.08) (4.00) (1.11) (-1.25)

-0.486 1.192 0.441 0.156 -0.086 ≥ 10% 3167 0.76 (-1.13) (10.45) (3.47) (0.81) (-0.69)

≥ 95th -0.624 1.184 0.544 0.189 -0.131 5292 0.84 percentile (-1.84) (13.21) (5.44) (1.24) (-1.33)

≥ 99th -0.578 0.838 0.239 -0.504 -0.249 996 0.50 percentile (-0.84) (4.59) (1.18) (-1.63) (-1.24)

Panel C: Characteristics of Short Interest Samples

SI Sample nfirms Size PIN SI Spr ($) Spr (%) α δ ε µ ≥ 2.5% 343.6 2768.54 0.1536 0.064 0.0808 0.0076 0.4310 0.3527 0.5532 0.2452 ≥ 5% 154.9 1999.13 0.1588 0.099 0.0814 0.0080 0.4317 0.3251 0.5500 0.2471 ≥ 7.5% 86.2 1431.98 0.1626 0.130 0.0815 0.0083 0.4314 0.3188 0.5479 0.2500 ≥ 10% 52.8 1203.18 0.1640 0.159 0.0825 0.0085 0.4296 0.3121 0.5463 0.2511 ≥ 95th percentile 88.2 1441.87 0.1621 0.129 0.0811 0.0084 0.4308 0.3193 0.5483 0.2501 ≥ 99th percentile 16.6 915.43 0.1666 0.232 0.0872 0.0084 0.4268 0.3019 0.5425 0.2545

Notes: Nfirms is the monthly average number of firms in a given portfolio, Size is the market value of equity, PIN is the probability of informed trading, SI is the average short interest ratio, Spr ($) is the monthly average of the trade-weighted daily quoted half- spread, Spr (%) is the monthly average of the trade-weighted daily relative half-spread, α is the probability than an information event occurred, δ is the probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, µ is the probability that a trader is informed. Table 6 Four Factor Coefficient Estimates for EW and VW Independent Portfolios of High SI Firms

Regression coefficients from a time-series regression of distinct SI monthly portfolio returns (in excess of one-month T-bill rate) on the three Fama and French (1993) factors and a fourth momentum factor from Carhart (1997). The intercept from the following equation is the measure of the portfolio monthly abnormal return:

Rp,,tft−=+ R a bRMRFt + sSMB t + hHML t + mMOM tt +ε , where Rp,,tft− R is the monthly excess portfolio return, RMRFt is the market factor, SMBt is the size factor, HMLt is the book-to- market factor and MOM t is the momentum factor. The portfolios are formed using independent SI samples. The [2.5% - 5%) SI sample contains all firms that had a short interest ratio greater than or equal to 2.5% and less than 5% the previous month. The other portfolios are similarly defined. Each regression contains 60 monthly observations over the sample period January 1995 – December 1999. Portfolio returns are equal weighted in Panel A and value weighted in Panel B. The t-statistics are in parentheses. GRS is the F-test of Gibbons, Ross and Shanken (1989) and the corresponding p-value. Panel C contains descriptive summary statistics for each short interest sample.

Panel A: Equal Weighted SI sample N Intercept RMRF SMB HML MOM Adj. R2

-0.583 1.153 0.432 0.341 -0.349 [2.5% - 5%) 11322 0.95 (-3.66) (27.39) (9.22) (4.77) (-7.55)

-0.840 1.232 0.619 0.314 -0.286 [5% - 7.5%) 4125 0.88 (-2.77) (15.40) (6.94) (2.31) (-3.25)

-0.686 1.328 0.718 0.612 -0.489 [7.5% - 10%) 2004 0.85 (-1.87) (13.71) (6.66) (3.72) (-4.59)

[10% - 99th -0.821 1.273 0.883 0.444 -0.322 2171 0.83 percentile) (-2.06) (12.08) (7.52) (2.49) (-2.78)

-1.038 1.043 0.870 -0.085 -0.199 ≥ 99th percentile 996 0.55 (-1.36) (5.19) (3.88) (-0.25) (-0.90) GRS: 3.4572 (p-value = 0.0091)

Panel B: Value Weighted SI sample N Intercept RMRF SMB HML MOM Adj. R2

-0.309 1.071 0.083 -0.057 0.042 [2.5% - 5%) 11322 0.87 (-1.16) (15.18) (1.06) (-0.48) (0.55)

-0.914 1.099 0.206 0.006 -0.102 [5% - 7.5%) 4125 0.74 (-2.19) (9.98) (1.68) (0.03) (-0.85)

-0.620 1.159 0.464 0.305 -0.250 [7.5% - 10%) 2004 0.61 (-1.08) (7.66) (2.75) (1.19) (-1.51)

[10% - 99th -0.550 1.292 0.537 0.360 -0.058 2171 0.74 percentile) (-1.18) (10.48) (3.91) (1.72) (-0.43)

-0.578 0.838 0.239 -0.504 -0.249 ≥ 99th percentile 996 0.50 (-0.84) (4.59) (1.18) (-1.63) (-1.24) GRS: 1.2978 (p-value = 0.2795) Table 6 (continued)

Panel C: Characteristics of Short Interest Samples

SI Sample nfirms Size PIN SI Spr ($) Spr (%) α δ ε µ [2.5% - 5%) 188.7 3387.67 0.1493 0.035 0.0803 0.0073 0.4306 0.3758 0.5558 0.2438 [5% - 7.5%) 68.8 2709.95 0.1542 0.061 0.0813 0.0076 0.4326 0.3340 0.5526 0.2433 [7.5% - 10%) 33.4 1802.10 0.1605 0.086 0.0801 0.0080 0.4339 0.3289 0.5502 0.2485 [10% - 99th percentile) 36.2 1331.75 0.1633 0.123 0.0800 0.0086 0.4318 0.3163 0.5480 0.2495 >= 99th percentile 16.6 915.43 0.1666 0.232 0.0872 0.0084 0.4268 0.3019 0.5425 0.2545

Notes: Nfirms is the monthly average number of firms in a given portfolio, Size is the market value of equity, PIN is the probability of informed trading, SI is the average short interest ratio, Spr ($) is the monthly average of the trade-weighted daily quoted half- spread, Spr (%) is the monthly average of the trade-weighted daily relative half-spread, α is the probability than an information event occurred, δ is the probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, µ is the probability that a trader is informed. Table 7 Statistics for Portfolios Sorted by Firm Size and PIN

This table describes the composition of the 12 portfolios created from monthly sorts of firm size and PIN. Firms are grouped first by size quartile, and within quartile, by PIN. For example, portfolio 23 contains the firms in the highest PIN group within the second size quartile. Nfirms is the monthly average number of firms in a given portfolio, Size is the market value of equity, PIN is the probability of informed trading, Spr ($) is the monthly average of the trade-weighted daily quoted half-spread, Spr (%) is the monthly average of the trade-weighted daily relative half-spread, α is the probability than an information event occurred, δ is the probability that an information signal is a low signal, ε is the probability that an uninformed traders transacts if selected to trade, µ is the probability that a trader is informed. The numbers are monthly averages, and for each size quartile a Wilcoxon Rank Sum test (Zstat) is performed to test for difference in means across the low and high PIN portfolios.

Firm Size PIN Portfolio nfirms Size PIN SI Spr ($) Spr (%) α δ ε µ

Low 11 28.6 221.80 0.0981 0.080 0.0863 0.0126 0.2586 0.3651 0.5663 0.2995 Low 12 28.9 222.16 0.1756 0.078 0.0867 0.0122 0.4415 0.3917 0.5396 0.2577 High 13 27.6 183.71 0.2884 0.079 0.0895 0.0135 0.6347 0.4644 0.4961 0.2796 13-11 -38.09 0.1903 0.000 0.0032 0.0009 0.3761 0.0993 -0.0703 -0.0199 Zstat (-6.03) (9.44) (-0.23) (1.36) (2.14) (9.44) (4.68) (-9.43) (-3.80)

21 28.4 662.48 0.0887 0.063 0.0817 0.0077 0.2351 0.3514 0.5715 0.2977 22 28.8 662.96 0.1595 0.067 0.0825 0.0079 0.4377 0.3659 0.5479 0.2300 23 27.4 626.49 0.2419 0.061 0.0836 0.0080 0.6052 0.4160 0.5162 0.2442 23-21 -35.99 0.1532 -0.002 0.0019 0.0003 0.3702 0.0645 -0.0553 -0.0535 Zstat (-2.37) (9.44) (-1.32) (0.70) (1.58) (9.44) (3.40) (-9.25) (-8.15)

31 28.5 1687.71 0.0725 0.060 0.0773 0.0056 0.2193 0.3566 0.5802 0.2946 32 28.9 1644.95 0.1422 0.063 0.0788 0.0058 0.4427 0.3279 0.5592 0.2043 33 27.5 1581.24 0.2163 0.061 0.0783 0.0061 0.6038 0.3705 0.5355 0.2211 33-31 -106.47 0.1438 0.001 0.0010 0.0005 0.3845 0.0139 -0.0448 -0.0735 Zstat (-2.83) (9.44) (1.03) (0.49) (2.21) (9.44) (0.71) (-9.32) (-9.11)

Low 41 28.2 9454.10 0.0698 0.048 0.0750 0.0038 0.2668 0.2966 0.5896 0.2345 High 42 28.7 8317.14 0.1234 0.051 0.0755 0.0040 0.4640 0.2569 0.5732 0.1704 High 43 27.3 8108.49 0.1860 0.053 0.0747 0.0043 0.6156 0.2650 0.5557 0.1889 43-41 -1345.61 0.1162 0.005 -0.0002 0.0004 0.3488 -0.0316 -0.0339 -0.0456 Zstat (-1.56) (9.44) (4.47) (0.20) (2.11) (9.44) (-2.26) (-8.66) (-8.15)

All Firms 338.8 2772.02 0.1543 0.064 0.0808 0.0076 0.4333 0.3521 0.5529 0.2436

Table 8 Four Factor Coefficient Estimates for EW and VW Portfolios Sorted by Size and PIN Each month, 12 portfolios are formed based on a sort of the previous month’s market value of equity and PIN. Firms are grouped first by size quartile, and then by PIN within each size quartile. For example, portfolio 23 contains firms in the second size quartile, and within that quartile, the highest PINs. Regression coefficients from a time-series regression of monthly portfolio returns (in excess of one-month T-bill rate) on the three Fama and French (1993) factors and a fourth momentum factor from Carhart (1997) are below. The intercept from the following equation is the measure of the portfolio monthly abnormal return:

Rp,,tft−=+ R a bRMRFt + sSMB t + hHML t + mMOM tt +ε ,

where RRp,,tft− is the monthly excess portfolio return, RMRFt is the market factor, SMBt is the size factor, HMLt is the book-to-

market factor and MOM t is the momentum factor. Each regression contains 60 monthly observations over the sample period January 1995 – December 1999. Portfolio returns are equal weighted in Panel A and value weighted in Panel B. The t-statistics are in parentheses. GRS is the F-test of Gibbons, Ross and Shanken (1989) and the corresponding p-value.

Panel A: Equal Weighted Panel B: Value Weighted

Firm Adj. Adj. PIN Portfolio Intercept RMRF SMB HML MOM Intercept RMRF SMB HML MOM Size R2 R2

-1.232 1.442 0.680 0.542 -0.789 -1.244 1.475 0.682 0.640 -0.779 Low 11 0.83 0.79 (-2.80) (12.42) (5.26) (2.75) (-6.18) (-2.46) (11.05) (4.58) (2.82) (-5.31)

-1.546 1.280 1.172 0.609 -0.255 -1.618 1.240 1.241 0.614 -0.211 Low 12 0.83 0.81 (-3.72) (11.64) (9.56) (3.26) (-2.11) (-3.65) (10.58) (9.50) (3.09) (-1.64)

-0.515 1.139 0.937 0.671 -0.381 -0.707 1.260 0.948 0.786 -0.340 High 13 0.74 0.75 (-1.09) (9.12) (6.73) (3.17) (-2.77) (-1.50) (10.11) (6.82) (3.71) (-2.48)

-0.296 0.964 0.543 -0.033 -0.467 -0.488 0.961 0.576 -0.044 -0.427 21 0.78 0.79 (-0.76) (9.38) (4.74) (-0.19) (-4.14) (-1.27) (9.45) (5.08) (-0.25) (-3.82)

-0.328 1.126 0.822 0.575 -0.315 -0.335 1.127 0.822 0.566 -0.291 22 0.77 0.78 (-0.79) (10.25) (6.71) (3.08) (-2.61) (-0.83) (10.54) (6.91) (3.12) (-2.48)

-0.832 1.148 0.772 0.397 -0.522 -0.876 1.117 0.781 0.390 -0.485 23 0.85 0.85 (-2.42) (12.62) (7.62) (2.57) (-5.22) (-2.61) (12.59) (7.90) (2.59) (-4.97)

-0.305 1.334 0.348 0.363 -0.568 -0.394 1.360 0.323 0.388 -0.564 31 0.85 0.84 (-0.85) (14.08) (3.30) (2.26) (-5.45) (-1.05) (13.75) (2.93) (2.31) (-5.18)

-0.653 1.219 0.605 0.384 -0.437 -0.698 1.210 0.581 0.429 -0.433 32 0.81 0.79 (-1.68) (11.88) (5.29) (2.21) (-3.88) (-1.74) (11.41) (4.92) (2.38) (-3.71)

-0.912 1.252 0.432 0.516 -0.107 -1.047 1.263 0.481 0.548 -0.065 33 0.78 0.80 (-2.38) (12.37) (3.83) (3.00) (-0.96) (-2.85) (13.00) (4.44) (3.32) (-0.61)

-0.436 1.070 0.152 0.080 -0.045 -0.096 0.954 0.057 -0.137 0.065 Low 41 0.76 0.69 (-1.16) (10.78) (1.37) (0.47) (-0.42) (-0.22) (8.34) (0.45) (-0.70) (0.52)

-0.334 1.193 0.099 0.058 -0.227 -0.467 1.160 -0.003 -0.092 0.158 High 42 0.90 0.76 (-1.38) (18.58) (1.38) (0.53) (-3.21) (-1.08) (10.17) (-0.02) (-0.47) (1.26)

-0.887 1.105 0.209 0.148 -0.008 -0.536 1.167 0.084 -0.152 0.079 High 43 0.82 0.78 (-2.83) (13.34) (2.26) (1.05) (-0.09) (-1.29) (10.60) (0.69) (-0.81) (0.65) GRS: 2.7460 (p-value = 0.0072) GRS: 2.4405 (p-value = 0.0156) Table 9 Portfolios Sorted by Size and Relative Half-Spread

Each month, 12 portfolios are formed based on a sort of the previous month’s firm size and relative half-spread. Firms are grouped first by size quartile, and then within each size quartile by relative spread. For example, portfolio 23 contains firms in the second size quartile, and within that quartile, the highest spread. Panel A and Panel B report the mean size and relative half-spread, respectively, in each portfolio. Panel C lists the intercept from a time-series regression of value-weighted monthly portfolio returns (in excess of one-month T-bill rate) on the three Fama and French (1993) factors and a fourth momentum factor from Carhart (1997). The intercept from the following equation is the measure of the portfolio monthly abnormal return:

Rp,,tft−=+ R a bRMRFt + sSMB t + hHML t + mMOM tt +ε , where Rp,,tft− R is the monthly excess portfolio return, RMRFt is the market factor, SMBt is the size factor, HMLt is the book-to- market factor and MOM t is the momentum factor. Each regression contains 60 monthly observations over the sample period January 1995 – December 1999. The t-statistics are in parentheses. GRS is the F-test of Gibbons, Ross and Shanken (1989) and the corresponding p-value.

Panel A: Portfolio Size Panel B: Portfolio Spread (%)

Low Spread High Spread Low Spread High Spread

Low Size 267.16 221.60 142.48 0.0075 0.0117 0.0192

708.60 654.33 604.96 0.0052 0.0073 0.0110

1782.96 1666.65 1504.67 0.0040 0.0054 0.0081

High 14387.42 6681.35 4594.92 0.0027 0.0039 0.0057 Size

Panel C: Intercept and t-statistic

Low Spread High Spread

-0.585 -1.362 -1.270 Low Size (-1.30) (-3.25) (-2.23)

-0.282 -0.824 -0.560 (-0.74) (-2.46) (-1.32)

-1.361 -0.822 0.248 (-3.56) (-2.19) (0.64)

-0.120 -0.263 -0.860 High Size (-0.31) (-0.83) (-2.01) GRS: 2.6507 (p-value = 0.0092) Table 10 Portfolios Sorted by Size and Short Interest Ratio

Each month, 12 portfolios are formed based on a sort of the previous month’s firm size and short interest ratio (SI). Firms are grouped first by size quartile, and then within each size quartile by SI. For example, portfolio 23 contains firms in the second size quartile, and within that quartile, the highest SI. Panel A and Panel B report the mean size and SI, respectively, in each portfolio. Panel C lists the intercept from a time-series regression of value-weighted monthly portfolio returns (in excess of one-month T-bill rate) on the three Fama and French (1993) factors and a fourth momentum factor from Carhart (1997). The intercept from the following equation is the measure of the portfolio monthly abnormal return:

Rp,,tft−=+ R a bRMRFt + sSMB t + hHML t + mMOM tt +ε , where Rp,,tft− R is the monthly excess portfolio return, RMRFt is the market factor, SMBt is the size factor, HMLt is the book-to- market factor and MOM t is the momentum factor. Each regression contains 60 monthly observations over the sample period January 1995 – December 1999. The t-statistics are in parentheses. GRS is the F-test of Gibbons, Ross and Shanken (1989) and the corresponding p-value.

Panel A: Portfolio Size Panel B: Portfolio SI

Low SI High SI Low SI High SI

Low Size 214.26 219.35 199.83 0.0322 0.0568 0.1515

651.50 652.25 666.32 0.0309 0.0498 0.1143

1669.10 1675.02 1614.40 0.0299 0.0470 0.1088

High Size 9261.89 9557.03 6869.55 0.0288 0.0412 0.0832

Panel C: Intercept and t-statistic

Low SI High SI

-1.069 -0.826 -1.487 Low Size (-2.83) (-1.96) (-2.58)

-0.301 -0.652 -0.723 (-0.95) (-1.99) (-1.54)

-0.433 -1.057 -0.677 (-1.28) (-2.80) (-1.67)

-0.176 -0.838 -0.222 High Size (-0.40) (-2.27) (-0.59) GRS: 1.7868 (p-value = 0.0806)