In this paper, we infer how the estimates of firm value by “optimists” and “pessimists” evolve in response to information shocks. Specifically, we examine returns and disagreement measures for portfolios of short-sale-constrained stocks that have experienced large gains or large losses. Our analysis suggests the presence of two groups, one of which overreacts to new information and remains biased over about 5 years, and a second group, which underreacts and whose expectations are unbiased after about 1 year. Our results have implications for the belief dynamics that underlie the momentum and long-term reversal effect.
Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.
Over the last several decades, financial economists have begun to carefully examine the consequences of disagreement or differences of opinion. In part, this attention has been motivated by large trading volumes in securities markets, which are difficult to explain without large levels of and changes in disagreement [see, e.g., Hong and Stein 2007]. In this paper, we measure how the distribution of beliefs about security values evolves over time in response to information shocks. That is, we examine the dynamics of disagreement.
When opinions differ regarding the value of a security, and investors can costlessly short-sell that security, the security price will reveal only the weighted mean of the distribution of beliefs, and nothing about the dispersion of agents’ beliefs. However, if a security becomes hard to borrow, then the price will reflect only the beliefs of the most optimistic agents [Miller 1977]. Moreover, as we will show here, using the cost of borrowing we can learn about the dispersion of beliefs.
For example, consider a constant absolute risk aversion [CARA]-normal framework with a single security that will pay an uncertain liquidating dividend |$\tilde D$| in one period. For simplicity, assume the risk-free rate is zero and that risk aversion is small [so that we can ignore risk premiums]. Most importantly, assume only a fixed number of shares are available for lending, and any shares lent and sold cannot be rehypothecated. If a frictionless market exists for lending these shares, the cost of borrowing will be the cost that matches this [inelastic] supply with demand from pessimists who wish to borrow the shares for the purpose of short selling.
In this setting, without disagreement, the price will equal the expected liquidating dividend [|$P = \mathbb{E}[\tilde D]$|], and both the short interest and the marginal lending fee will be zero. However, what if there is disagreement? First, assume there are two masses of investors A and B of equal measure and with equal risk aversion who disagree about the expected payoff. Specifically, assume |$\mathbb{E}_A[\tilde D] = \$30$| and |$\mathbb{E}_B[\tilde D] = \$20$|, where |$\mathbb{E}_i[\cdot]$| denotes the expectation of an investor of type |$i$|. If the number of shares available for borrowing is large, the security price will be the average of |$\mathbb{E}_A[\tilde D]$| and |$\mathbb{E}_B[\tilde D]$|: $25. Note also that if a “wisdom of crowds” effect is at work, meaning that the average across agents’ expectations is “rational” [i.e., that |$\mathbb{E}_{R}[\tilde D] = 0.5 \cdot \mathbb{E}_A[\tilde D]+ 0.5 \cdot \mathbb{E}_B[\tilde D]= \$25$|], then the expectation of the security price change over the period will be zero [since |$P=\mathbb{E}_{R}[\tilde D]$|].
However, according to the Miller [1977] intuition, when the number of shares available for borrowing is small the price will reflect primarily the views of the most optimistic agents. In this example, as the number of shares available for borrowing approaches zero, the security price will approach |$\mathbb{E}_A[\tilde D]= \$30$|. Also, the borrowing cost will approach the difference in the agents’ valuations [|$\mathbb{E}_A[\tilde D] - \mathbb{E}_B[\tilde D]= \$10$|]: the pessimistic agents in B will be willing to pay up to $10 to short a share they view as being overvalued by $10. Here, the rational expectation of the security price change will be |$-$|$5, so the optimistic agents in A earn a per-share expected payoff of |$-$|$5. However, the pessimistic agents [in B] who short the security also earn an expected payoff per share of |$-$|$5; they benefit from the $5/share expected price decline but are required to pay the $10/share borrow cost. Thus, an observer can unambiguously infer the beliefs of both optimists and pessimists from the price and the borrowing cost of the constrained stock.1
The objective of this paper is to extend this intuition to a multiperiod setting and bring it to the data. Specifically, we examine returns to portfolios of securities that have experienced either a large positive or a large negative price shock, presumably as a result of the arrival of new information about the security’s future payoffs. Moreover, motivated by the above toy model, we examine the returns of these portfolios, both when the securities in the portfolios are unconstrained, namely, when they can be borrowed at low cost, and when borrowing is costly. Consistent with the above example, this allows us to make inferences about how the distribution of beliefs of market participants evolves over time in response to these shocks.
As a prelude to our main empirical tests, we first examine the returns to high- and low-past-return portfolios of unconstrained firms.2 Specifically, we classify firms as “winners” or “losers” based on their returns over the preceding 1 year, excluding the last month. We then examine the cumulative abnormal returns [CARs] for a period of 5 years following the point in time at which the firms are classified as winners or losers. The abnormal returns are relative to the value-weighted [VW] U.S. equity market benchmark generated by Kenneth French and taken from his data library.
Panel A of Figure 1 plots the CARs for the past-winner and past-loser portfolios. It is not surprising that their CARs are roughly consistent with the momentum and reversal effects documented in the literature [see, e.g., Jegadeesh and Titman, 1993, 2001; DeBondt and Thaler, 1985]: for the first 6–12 months there is some continuation of the past return, and in years 2–5 we observe some reversal of the initial return, at least for the past-loser portfolio.
Figure 1
Returns following positive and negative price shocks
We calculate abnormal returns for each portfolio for each holding month |$k$| by regressing the time series of month-|$k$| excess returns on the capital asset pricing model [CAPM]-MktRF factor. Returns are then cumulated and plotted for the past winner and past loser portfolios [dashed lines] in panel A. The universe is all U.S. common stocks listed on the NYSE, AMEX, or NASDAQ in the sample from January 1927 to June 2020. Winners are defined as the firms whose returns from 12 months to 1 month before the portfolio-formation date were in the top 30|$\%$| of all firms, and the past losers are the firms in the bottom 30|$\%$|. The universe for the solid lines consists of short-sale constrained stocks, meaning they are in the bottom 30|$\%$| of institutional ownership and the top 30|$\%$| of short interest. For the constrained losers, we additionally impose the condition that they have not been in the constrained-winner portfolio within the past 5 years, to isolate the long-run effects of winners and losers [see Section ]. The time period for constrained stocks is May 1980 to June 2020. We add preformation abnormal returns in panel C. To focus on the differences after 12 months, we center CARs on |${t}=12$| in panel D. We obtain 95|$\%$| confidence intervals using bootstrapping [details in Internet Appendix B.III].
In panel B of Figure 1, we first rescale the axes. Note that the dotted lines that represent the CARs of the unconstrained past-winner and loser portfolios are the same as in panel A, but rescaled. We add to panel B the CARs for high- and low-past-return portfolios, but which are formed using only stocks that are hard-to-borrow based on proxies we will describe below. The two sets of CAR plots suggest that limited predictability exhibited by the unconstrained past winners and losers becomes both far stronger and strikingly asymmetric. Specifically, as of the formation date the constrained past winners start to earn negative abnormal returns and continue doing so for 5 years postformation. The portfolio earns a compounded market-adjusted return of |$-13\%$| [|$t=-4.96$|] in the first 12 months postformation, and also earns a statistically significant negative alpha in each of the next 4 years, earning a compounded market-adjusted return over these 4 years of |$-42\%$| [|$t=-7.39$|].3
Like the portfolio of constrained past winners, the portfolio of constrained past losers also earns a strongly negative compounded market-adjusted return over the first 12 months postformation, specifically a return of |$-19\%$|. However, in contrast with the portfolio of constrained past winners, the negative abnormal returns do not persist: the abnormal return of this portfolio in each of the next 4 years postformation is never statistically different from zero, and over the full 4-year period [from 2–5 years postformation] the portfolios’ compounded market-adjusted return is |$5\%$| [|$t=0.44$|]. Panel C extends the plot in panel B to include the abnormal returns in the formation period. Note the relative magnitudes of the preformation and the postformation returns, particularly for the constrained-winner portfolio.
The difference in mispricing persistence between constrained past winners and losers is our key empirical finding, and we confirm this finding using a variety of different statistical tests [e.g., panel D centers postformation CARs on |$t=12$| and shows bootstrapped confidence intervals]. Section reports the details of the empirical tests. We find similar differences in mispricing persistence out-of-sample in time [NYSE/AMEX stocks in the 1980s] and out-of-sample in regions [i.e., international firms]. Internet Appendix E reports the results of these tests.
Based on this difference in mispricing persistence and on the intuition coming from the toy model, one might be tempted to conclude that the disagreement persistence simply remains high for the constrained past winners out to 5 years postformation, while the disagreement is resolved more quickly for the constrained past losers.
Interestingly, this asymmetry between constrained past losers and winners in return persistence is not mirrored in disagreement persistence. Panel A in Figure 2 plots the lending fee [from Markit] for our constrained past-winner and past-loser portfolios over the period from August 2004 through June 2020 for which we have borrow-cost data. Note that borrowing costs for the firms in our portfolios are high at portfolio formation and subsequently decline but remain elevated for about 5 years following both positive and negative shocks.4
Figure 2
Fees and disagreement following positive and negative price shocks
Panel A plots the indicative lending fee from Markit in event time for the constrained winners and losers, as well as portfolios containing matched unconstrained stocks [see Section ]. The Markit sample goes from August 2004 to June 2020. Panel B plots earnings forecast dispersion for portfolios independently sorted on past return [30|$\%$| breakpoints as above] and the 1-year change in dispersion of analyst earnings forecasts [|$\Delta FD$|] from IBES [quintiles, see Section ] among unconstrained stocks. 95|$\%$| confidence intervals are based on Newey-West standard errors.
This pattern of long persistence in disagreement implied by the lending fees similarly crops up for unconstrained winners and losers if we look at disagreement proxies other than lending fees. Panel B shows the average dispersion of analysts’ earnings forecasts for unconstrained winners and losers that are additionally in the top quintile of changes in forecast dispersion over the formation period.5 We observe an increase in analyst disagreement preformation and a gradual decrease postformation, lasting about 5 years.
In summary, we establish three empirical facts about short-sale constrained firms: [1] strong negative abnormal returns following positive price shocks that persist for about 5 years, [2] strong negative abnormal returns following negative price shocks that persist for about 1 year, and [3] strong disagreement following both positive and negative price shocks that persists for about 5 years. What do these empirical findings imply for the impulse responses of optimists’ and pessimists’ beliefs about firm values? Figure 3 illustrates the implications using stylized plots of impulse responses. The plots are generated from the model we will develop in Section . First, the strong, persistent negative abnormal returns following a positive price shock suggest the beliefs of the most optimistic agents are too optimistic at time 0 and decay towards rational beliefs over a roughly 5-year period [the upper green line in panel A of Figure 3]. These optimistic agents appear to overreact to positive information that is part of the shock.
Figure 3
Dynamics of beliefs and prices
Panel A plots belief [i.e., the expectation of the single liquidating dividend of the risky asset] paths for overreacting [green] and underreacting [pink] agents, for positive and negative information shocks [at time |$t=0$|]. The dotted lines represent rational expectations beliefs for those same shocks. The dashed blue line [labeled disagreement] plots the difference between the overreacting and underreacting agents’ beliefs. Panel B plots the resultant prices in unconstrained [dashed blue] and fully short-sale constrained [solid green and pink] markets. The dashed gray line represents the opinions of the sidelined agents. See Internet Appendix A for the details of the model and the parametrization that generates these belief and price paths.
Second, the shorter-lived negative abnormal returns following the negative price shocks imply that, following these shocks, the beliefs of the most optimistic agents [the lower pink line in panel A of Figure 3] are also too optimistic, suggesting these agents initially underreact to the new negative information, but that this underreaction is resolved after only 1 year.
Finally, the empirical finding that disagreement persists for about 5 years suggests the distance between the beliefs of optimists and pessimists decays toward zero over a period of about 5 years. In panel A of Figure 3, the blue line plots the level of disagreement coming out of our model, which is roughly consistent with what we see in panels A and B of Figure 2. Combined with the optimists’ beliefs inferred from prices that we plotted in Figure 3, panel B, the level of disagreement allows us to identify the pessimists’ beliefs, which are just the optimists’ beliefs minus the level of disagreement at that time.
Figure 3, panel A, shows graphically that the implied impulse responses of the agents’ beliefs to the positive and negative shocks are symmetric, in the sense that the optimist’s/pessimist’s beliefs following a positive shock are the mirror image of the pessimist’s/optimist’s beliefs following a negative shock.
What are the implications for the impulse response in prices that would be observed for unconstrained stocks in response to positive and negative price shocks? Again, relying loosely on the intuition from the simple two-type model above, the price would be the average of the beliefs, which would be the dashed blue line in panel B of Figure 3. These price paths are consistent with the large literature on momentum and subsequent long-term reversals, as discussed extensively in the behavioral literature and shown in panel A of Figure 1.
One type of shock that we have not yet discussed is one that is not accompanied by an increase in disagreement. In practice, such unambiguous news shocks certainly exist, and firms that have experienced these shocks will enter both winner and loser portfolios. However, for our baseline analysis of short-sale constrained stocks, the fact that we are considering only high short interest securities should filter out most low-disagreement shocks. To the extent that our mechanism partly drives the momentum and long-term reversal effects for unconstrained stocks, these effects should be more muted for low-disagreement shocks. Consistent with this hypothesis, compelling evidence indicates that momentum is stronger in high-disagreement [Lee and Swaminathan 2000; Zhang 2006; Verardo 2009], and in low-quality or difficult-to-value information environments [Daniel and Titman 1999; Hong, Lim, and Stein 2000; Cohen and Lou 2012].
In summary, this paper looks at constrained stocks to learn more about the dynamics of disagreement after large information shocks. High-energy physics provides a useful analogy to our exercise here: physics researchers examine the behavior of fundamental particles at high energies to test alternative models of particle interactions. By examining matter in extreme settings, they can pull apart the underlying components of these particles, leading to a deeper understanding of the structure that results in the behavior the we observe in more standard [low-energy] settings. Here, we examine “extreme” [constrained] securities in order to infer the distribution of beliefs that would otherwise be aggregated into the price. Results for the set of constrained stocks allow us to look at our existing models for unconstrained stocks from a new and different perspective that challenges existing models of momentum and long-term reversal, as these models are unable to explain the patterns for constrained and unconstrained stocks simultaneously. In Section , we briefly relate the empirical results to existing models and propose a new dynamic heterogeneous-agent model featuring overconfidence [Daniel, Hirshleifer, and Subrahmanyam 1998] and slow information diffusion [Hong and Stein 1999] that is able to both explain this asymmetry in mispricing persistence among short-sale constrained stocks, and to match momentum and long-term reversal effects for unconstrained stocks.
1. Related Literature
Miller [1977] argues that when investors disagree about the expected return on a security, and when short selling of that security is restricted, the security will be overpriced. Subsequent empirical research has explored this argument in detail. Consistent with the divergence-of-opinion part of Miller’s argument, firms for which the dispersion of analysts’ forecasts is high earn low future stock returns [Danielsen and Sorescu 2001, Diether, Malloy, and Scherbina 2002]. Moreover, the returns are still lower when disagreement is large and when short-sale constraits are binding [Boehme, Danielsen, and Sorescu 2006]. These negative abnormal returns are concentrated around earnings announcements, consistent with the idea that earnings announcements at least partly resolve disagreement [Berkman et al. 2009]. Shocks in the lending market have predictive power for future returns [Asquith, Pathak, and Ritter 2005, Cohen, Diether, and Malloy 2007]. Finally, some evidence suggests long-short anomaly strategy profitability tends to be concentrated in the short leg, and in stocks that are expensive to short [Hirshleifer, Teoh, and Yu 2011; Stambaugh, Yu, and Yuan 2012; Drechsler and Drechsler 2016].6
D’Avolio [2002] and Geczy, Musto, and Reed [2002] are early papers that examine the stock-lending market using proprietary borrow-cost data. A major takeaway of these studies is that, at least historically, all but a small percentage of common stocks can be borrowed at low cost for short selling purposes. Kolasinski, Reed, and Ringgenberg [2013] argue the lending supply curve is almost perfectly elastic at low levels of demand, but above a certain level, it becomes highly inelastic. They argue that this structure is a result of some shareholders being willing to make their shares available for borrowing at any cost, but where, once this supply is exhausted, the search costs associated with finding additional shares increases rapidly.
Data on borrowing costs are not always available and may not reflect the full set of costs associated with borrowing shares. However, a large literature has explored proxies for borrowing costs. We follow Asquith, Pathak, and Ritter [2005] who show that a combination of low institutional ownership and high short interest generally results in high borrowing costs. Our analysis of Markit reported lending fees confirms this finding. For a shorter subsample, we can calculate the Markit indicative and average lending fees and find that, among low institutional ownership firms, such fees are about three to five times higher for firms that also have high short interest at the same time, relative to firms with low short interest [see panel N of Table C.7 in the Internet Appendix]. By contrast, using residual institutional ownership [Nagel 2005] leads to portfolios of firms for which the fee is substantially smaller, on average, even when focusing on high-short-interest firms [see Table C.8, panel N, in the Internet Appendix]. The combined use of low institutional ownership and high short interest is also consistent with the model we present in Section , with the model in Blocher, Reed, and Van Wesep [2013], and with the empirical results reported in Asquith, Pathak, and Ritter [2005] and Cohen, Diether, and Malloy [2007].
To our knowledge, no one in the literature has simultaneously analyzed past returns and disagreement proxies, as we do here, with the goal of analyzing the evolution of the dynamic response of disagreement to large information shocks. However, a few studies have reported longer-term returns of short-sale constrained stocks [Chen, Hong, and Stein 2002; Nagel 2005; Lamont 2012; Weitzner 2020]. The results of these studies are broadly consistent with our hypotheses. Specifically, Nagel [2005] documents short-term overreaction to good cash flow news and short-term underreaction to bad cash flow news among stocks with low residual institutional ownership and that prices do not reverse over a 3-year period postformation.
Our empirical approach is based on the premise that large disagreement shocks coincide with large price shocks. The approach has a natural link to theories of momentum [see, e.g., the model in Hong and Stein 2007]. In Section , we point out that our empirical results are inconsistent with existing explanations [for a survey, see Subrahmanyam 2018] and with straightforward extensions of these explanations. We further present the intuition of a new model based on established behavioral concepts that provides a first explanation of the new evidence, and we relate this model to existing theoretical approaches.
Our paper further speaks to the ongoing debate on whether bubbles are empirically identifiable. The empirical challenge in identifying asset pricing bubbles has been the lack of observability of the fundamental value, which leads to the joint hypothesis problem [Fama 1970]. Recent work by Greenwood, Shleifer, and You [2019] shows sharp price increases of industries, along with certain characteristics of this run-up, help forecast the probability of crashes and thereby help identify and time a bubble. Our analysis of constrained winners adds to this strand of literature. Specifically for individual stocks, price run-ups paired with indications of limits to arbitrage strongly forecast low future returns. Our theoretical and empirical approach can be interpreted as a methodology for identifying individual stock bubbles, and determining the decay rates of these bubbles.
2. Data
We collect daily and monthly returns, market capitalizations, and trading volumes from the Center for Research in Security Prices [CRSP]. Our sample consists of all common ordinary NYSE, AMEX, and NASDAQ stocks from May 1980 through June 2020.7
We form portfolios of these individual stocks at the start of each month |$t$| based on three firm-specific variables. The first sorting variable is each firm’s cumulative past return from month |$t-12$| through month |$t-2$|. Note that this cumulative return is the same as the measure of momentum used in Carhart [1997], Fama and French [2008], and numerous other studies.
The second sorting variable is the institutional ownership ratio [IOR]. Our measure of IOR is constructed using Thomson-Reuters Institutional 13-F filings through June 2013, and on WRDS-collected SEC data after June 2013.8 We calculate IOR as the ratio of the number of shares held by institutions, scaled by the number of shares outstanding from CRSP, lagged by one month.9
Our third sorting variable is the short-interest ratio [SIR], which we calculate as short interest scaled by the number of shares outstanding [from CRSP]. Short interest comes from two sources: from April 1980 to May 1988, and after June 2003, we get short-interest data at the security level from Compustat.10 From June 1988 through June 2003, our short-interest data come directly from NYSE, AMEX and NASDAQ.11 However, prior to June 2003, if short-interest data from an exchange are missing for a given firm-month, we use short interest as reported by Compustat if available. After June 2003, if Compustat short-interest data are missing for a given firm-month, we use data from the exchanges if available.12
We merge quarterly book-equity data from Compustat with CRSP data using the linking table provided by CRSP. To calculate the monthly updated book-to-market ratio, we divide the most recently observed book value by the sum of the most recent market equity of all equity securities [PERMNOs] associated with the company [PERMCO]. We assume that the book value of quarter |$q$| can be observed by investors at the time of the earnings announcement for quarter |$q$|.
Our data on stock lending fees come from IHS Markit. These data start in August 2004.13 We use the indicative fee [a proxy for marginal costs] and simple average fee [equal-weighted average of all contracts for a particular security] averaged within a month to assess the cost of short selling.
Analyst forecasts of fiscal-year-end earnings are from IBES. We use the summary file unadjusted for stock splits, to avoid the bias induced by ex post split adjustment, as pointed out by Diether, Malloy, and Scherbina [2002]. Forecast dispersion [|$FD$|] is the standard deviation of fiscal year-end forecasts normalized by the absolute value of the mean forecast. We eliminate values for which the mean forecast is between |$-$|$0.1 and |$+$|$0.1 per share to avoid scaling by small numbers.14
3. Empirical Results
In this section, we analyze the short- and long-term price dynamics of short-sale-constrained stocks following large information shocks, with the goal of characterizing the evolution of beliefs following such shocks. Once again, the idea is that the price of a constrained security will reveal the beliefs of the optimistic agents. For the constrained winners, the beliefs will be those of the agents who became most optimistic following a positive information shock in the formation period, that is, those who overreacted to the information. In contrast, from the prices of the constrained losers, we can infer the beliefs of the agents who underreacted to the [negative] information shock in the formation period.
To select stocks for which short-sale constraints are binding, we follow Asquith, Pathak, and Ritter [2005] and independently sort on IOR and SIR. This explicitly accounts for the supply and demand sides of the shorting market [see also Cohen, Diether, and Malloy 2007]. IOR has been shown to be closely related to lending supply [see, e.g., D’Avolio 2002]. Assuming, for example, that IOR is a direct proxy for easily available lending supply, and it is at 10|$\%$|, an SIR of 10|$\%$| would indicate that easily available supply is exhausted and short selling is likely constrained.15 IOR and SIR are available from 1980, giving us an approximately 40-year sample with which to measure long-term returns.
Finally, to identify the shocks that drive disagreement, at the start of each month |$t$| we sort on each stock’s cumulative return from month |$t\!-\!12$| through month |$t\!-\!2$|.16 These returns incorporate all information that cause valuation changes, including the effect of the disagreement shocks. For all three sorts, namely, past return, SIR, and IOR, the breakpoints are the 30th and the 70th percentile. We use independent sorts to maximize independent variation in all three variables. This 3|$\times$|3|$\times$|3 sort provides us with 27 portfolios. Each portfolio is value weighted, both to avoid liquidity-related biases associated with equal-weighted portfolios [Asparouhova, Bessembinder, and Kalcheva 2013] and to ensure the effects that we document are not driven by extremely low market-capitalization stocks. We label as constrained the set of stocks that are simultaneously in the low IOR and the high SIR portfolio. We designate the firms with the highest past returns as the past winners [|$W$|], and those with the lowest past returns as the past losers [|$L$|]. The firms that are simultaneously constrained and either past losers or past winners are labeled as constrained winners [|$W^*$|] and constrained losers [|$L^*$|], respectively.
Furthermore, we want to make sure we do not confound results for our constrained-loser portfolio by unintentionally blending in former constrained winners that are in the process of falling.17 In the constrained-loser portfolio, we therefore only include those constrained losers that were not constrained winners at any point during the preceding 5 years.18 This subset of losers better reflects the return patterns of short-sale constrained stocks with initially negative news.19
3.1 Characteristics
The left two columns of Table 1, labeled |$W^*$| and |$L^*$|, report summary statistics for the constrained winner and loser portfolios, respectively.20 On average each month, 49 stocks are classified as constrained winners and 36 as constrained losers.21 The representative constrained winner stock has a market capitalization of $3 billion. Constrained losers are about half as big, on average. The formation-period returns are large in magnitude for the stocks in both portfolios: the average firm in the |$W^*$| portfolio almost doubles in size over the formation period, and the |$L^*$| portfolio almost halves.
Table 1
Characteristics of constrained and matched portfolios
|$W^{*}$||$L^{*}$||$W^{*,m}$||$L^{*,m}$|Number of stocks49364936Average market equity [B$]3.001.473.111.22Formation-period return [|$\%$|]82.36–47.3771.37–36.07Institutional ownership [IOR, |$\%$|]16.9117.5576.3374.40Change in IOR over preceding year [PP]1.18–5.069.120.95Short interest [SIR, |$\%$|]6.526.415.366.33Change in SIR over preceding year [PP]2.281.050.691.07Book-to-market ratio0.300.910.350.90Idiosyncratic volatility [|$\%$|, daily]3.043.952.112.69Turnover [|$\%$|]32.6428.2224.3623.51Change in turnover over preceding year [PP]15.951.865.161.65Ind. fee [|$\%$|]6.987.610.651.11Change in ind. fee over preceding year [PP]1.693.07–0.240.38Simple avg. fee [SAF, |$\%$|]5.266.590.431.08Change in SAF over preceding year [PP]0.673.16–0.220.39SIRIO [|$\%$|]100.3173.306.477.79Option volatility spread [|$\%$|]–5.47–6.34–0.96–1.04
|$W^{*}$||$L^{*}$||$W^{*,m}$||$L^{*,m}$|Number of stocks49364936Average market equity [B$]3.001.473.111.22Formation-period return [|$\%$|]82.36–47.3771.37–36.07Institutional ownership [IOR, |$\%$|]16.9117.5576.3374.40Change in IOR over preceding year [PP]1.18–5.069.120.95Short interest [SIR, |$\%$|]6.526.415.366.33Change in SIR over preceding year [PP]2.281.050.691.07Book-to-market ratio0.300.910.350.90Idiosyncratic volatility [|$\%$|, daily]3.043.952.112.69Turnover [|$\%$|]32.6428.2224.3623.51Change in turnover over preceding year [PP]15.951.865.161.65Ind. fee [|$\%$|]6.987.610.651.11Change in ind. fee over preceding year [PP]1.693.07–0.240.38Simple avg. fee [SAF, |$\%$|]5.266.590.431.08Change in SAF over preceding year [PP]0.673.16–0.220.39SIRIO [|$\%$|]100.3173.306.477.79Option volatility spread [|$\%$|]–5.47–6.34–0.96–1.04
This table shows time-series averages of value-weighted mean characteristics of the portfolios in the month of portfolio formation. Shown are the average number of stocks, the average market equity [in billion U.S. dollars], return from month t-12 to the end of month t-2 [in |$\%$|], level of short interest 2 weeks prior to formation [in |$\%$|], and change from 11.5 months ago to 2 weeks ago [in PP], institutional ownership [in |$\%$| of number of shares outstanding] and its change over the preceding year [in PP], the ratio of book equity of the most recently observed fiscal year to last month’s market equity, the average standard deviation of daily idiosyncratic returns in each portfolio [daily, in |$\%$|] over the month prior to formation [Ang et al. 2006], levels [in |$\%$|] and changes [in PP] over the preceding 12 months in monthly turnover, the level [in |$\%$|, annualized] and change [in PP, over the preceding 12 months] in the Markit indicative as well as simple average loan fee, the ratio of short interest to institutional ownership [SIRIO] as in Drechsler and Drechsler [2016] [in |$\%$|], and last, the open-interest weighted average of differences in implied volatilities between matched put and call option pairs at month-end [in |$\%$|], as in Cremers and Weinbaum [2010]. The sample period is April 1985 [to account for the 5-year lookback period for losers that were not constrained winners before] to June 2020, except for Markit data, which are available from August 2004. For a comparison with the broader universe of stocks, see the averages for the remaining portfolios displayed in Table C.7 in Internet Appendix C.V.
Table 1
Characteristics of constrained and matched portfolios
|$W^{*}$||$L^{*}$||$W^{*,m}$||$L^{*,m}$|Number of stocks49364936Average market equity [B$]3.001.473.111.22Formation-period return [|$\%$|]82.36–47.3771.37–36.07Institutional ownership [IOR, |$\%$|]16.9117.5576.3374.40Change in IOR over preceding year [PP]1.18–5.069.120.95Short interest [SIR, |$\%$|]6.526.415.366.33Change in SIR over preceding year [PP]2.281.050.691.07Book-to-market ratio0.300.910.350.90Idiosyncratic volatility [|$\%$|, daily]3.043.952.112.69Turnover [|$\%$|]32.6428.2224.3623.51Change in turnover over preceding year [PP]15.951.865.161.65Ind. fee [|$\%$|]6.987.610.651.11Change in ind. fee over preceding year [PP]1.693.07–0.240.38Simple avg. fee [SAF, |$\%$|]5.266.590.431.08Change in SAF over preceding year [PP]0.673.16–0.220.39SIRIO [|$\%$|]100.3173.306.477.79Option volatility spread [|$\%$|]–5.47–6.34–0.96–1.04
|$W^{*}$||$L^{*}$||$W^{*,m}$||$L^{*,m}$|Number of stocks49364936Average market equity [B$]3.001.473.111.22Formation-period return [|$\%$|]82.36–47.3771.37–36.07Institutional ownership [IOR, |$\%$|]16.9117.5576.3374.40Change in IOR over preceding year [PP]1.18–5.069.120.95Short interest [SIR, |$\%$|]6.526.415.366.33Change in SIR over preceding year [PP]2.281.050.691.07Book-to-market ratio0.300.910.350.90Idiosyncratic volatility [|$\%$|, daily]3.043.952.112.69Turnover [|$\%$|]32.6428.2224.3623.51Change in turnover over preceding year [PP]15.951.865.161.65Ind. fee [|$\%$|]6.987.610.651.11Change in ind. fee over preceding year [PP]1.693.07–0.240.38Simple avg. fee [SAF, |$\%$|]5.266.590.431.08Change in SAF over preceding year [PP]0.673.16–0.220.39SIRIO [|$\%$|]100.3173.306.477.79Option volatility spread [|$\%$|]–5.47–6.34–0.96–1.04
This table shows time-series averages of value-weighted mean characteristics of the portfolios in the month of portfolio formation. Shown are the average number of stocks, the average market equity [in billion U.S. dollars], return from month t-12 to the end of month t-2 [in |$\%$|], level of short interest 2 weeks prior to formation [in |$\%$|], and change from 11.5 months ago to 2 weeks ago [in PP], institutional ownership [in |$\%$| of number of shares outstanding] and its change over the preceding year [in PP], the ratio of book equity of the most recently observed fiscal year to last month’s market equity, the average standard deviation of daily idiosyncratic returns in each portfolio [daily, in |$\%$|] over the month prior to formation [Ang et al. 2006], levels [in |$\%$|] and changes [in PP] over the preceding 12 months in monthly turnover, the level [in |$\%$|, annualized] and change [in PP, over the preceding 12 months] in the Markit indicative as well as simple average loan fee, the ratio of short interest to institutional ownership [SIRIO] as in Drechsler and Drechsler [2016] [in |$\%$|], and last, the open-interest weighted average of differences in implied volatilities between matched put and call option pairs at month-end [in |$\%$|], as in Cremers and Weinbaum [2010]. The sample period is April 1985 [to account for the 5-year lookback period for losers that were not constrained winners before] to June 2020, except for Markit data, which are available from August 2004. For a comparison with the broader universe of stocks, see the averages for the remaining portfolios displayed in Table C.7 in Internet Appendix C.V.
SIR is close to IOR for the stocks in the constrained portfolios, indicating a good chance of these stocks being hard to borrow. On average, the constrained stocks in both |$W^*$| and |$L^*$| have high idiosyncratic volatilities and high turnovers, consistent with disagreement among traders.
To check whether we have accurately identified stocks with binding short-sale constraints, we calculate the level and 12-month changes of the Markit indicative and simple average loan fee over the time period for which we have these data. The fees in the constrained portfolios are large, and they have generally increased leading up to the formation date, suggesting a high and increased level of disagreement.22
Because the Markit data are only available from 2004, we calculate two additional measures for the full sample period going back to 1980. The first one is SIRIO, that is, the number of shares sold short [short interest], divided by the number of shares held by institutions [institutional ownership], following Drechsler and Drechsler [2016].
These summary statistics also suggest this combination of low IOR and high SIR successfully identifies constrained firms. On average, constrained winners exhibit a SIRIO of 100.31|$\%$|, which likely pushes them above the point of cheap lending and makes short selling these stocks highly expensive.
For the firms and dates for which options data are available, we calculate the option volatility spread [Cremers and Weinbaum 2010], defined as the difference in annualized implied volatilities between at-the-money, 1- to 12-month-to-maturity put and call options. Note that if the volatility spread is negative, put-call parity will be violated in the sense that the price of an actual share will be higher than the price of a synthetic share [constructed using a bond, put, and call]. Such a violation would indicate that arbitraging this put-call parity violation is costly, likely because borrowing and shorting the underlying stock is costly [Ofek, Richardson, and Whitelaw 2004]. Note both constrained portfolios have large negative volatility spreads, consistent with the other data, suggesting that shorting the stocks in the |$W^*$| and |$L^*$| portfolios is costly.
3.2 Strategy performance over different horizons
Next, we will examine trading strategy returns at horizons from 1 to 5 years after portfolio formation. Our 1-year trading strategy, like that of Jegadeesh and Titman [1993], purchases at the start of each month $1 worth of the new [value-weighted] |$W_t^*$| or |$L_t^*$| portfolio, holds that portfolio for the following |$T=12$| months, and then sells it. Thus the short-horizon [S, 1 year] strategy |$W^{*}_{\rm S}$|, at the start of month |$t$| just after trading, consists of $1 of the newest |$W_t^*$| portfolio, plus some amount of each of the last eleven |$W^*$| portfolios formed in months |$t\!-\!11$| through |$t\!-\!1$|. In contrast to Jegadeesh and Titman [1993], we weight each of these additional 11 portfolios by their cumulated dollar value, meaning we do not rebalance the invested amount until they are sold at |$T$| [here, |$T\!=\!12$|] months.23 Thus, the construction of our one-year strategy is consistent with the calendar-time portfolio approach advocated by Fama [1998]. Note that we can assess the performance of these strategies with standard asset pricing tests applied to the monthly strategy returns, and these tests will summarize the portfolios’ forecastable performance for the first year postformation, much like the event-study plots in Figure 1, but without the econometric issues typically associated with studies of CARs [Barber and Lyon 1996; Fama 1998].
The first three columns of Table 2 display summary statistics for the returns of the 1-year strategies [the remaining columns present results for matched-firm strategies, which we discuss in Section ]. Panel A shows the raw average monthly excess returns, the number of months in the sample [T], the average number of unique stocks in the strategy in any given month [AvgN], and the realized annualized strategy Sharpe ratio [SR].
Table 2
Short-horizon [S, 1-year] performance of constrained and matched portfolios
|$W^{*}_{\rm S}$||$L^{*}_{\rm S}$||$W^{*}_{\rm S}$|-|$L^{*}_{\rm S}$||$W^{*,m}_{\rm S}$||$L^{*,m}_{\rm S}$||$W^{*,m}_{\rm S}$|-|$L^{*,m}_{\rm S}$||$W^{*}_{\rm S}$|-|$W^{*,m}_{\rm S}$||$L^{*}_{\rm S}$|-|$L^{*,m}_{\rm S}$||${\it DiD}$|A. Raw excess returnsAverage–0.18–0.900.730.940.830.11–1.12–1.740.62 [–0.54][–1.81][2.00][3.47][2.04][0.43][–4.38][–4.60][1.80]No. of months411411411411411411411411411AvgN188106 262203 SR–0.0760–0.30760.35280.50960.36760.0772–0.7663–0.87280.3160B. CAPM regressionsIntercept–1.07–1.890.820.14–0.100.24–1.21–1.790.58 [–4.28][–5.55][2.27][0.85][–0.36][0.98][–4.52][–4.67][1.70]MktRF1.371.51–0.141.231.43–0.200.140.080.06 [14.94][11.97][–1.02][30.36][12.76][–1.56][1.57][0.61][0.63]|$R^2$|.5783.4432.0081.7393.6643.0343.0145.0026.0014IR–0.7096–0.86370.40000.1458–0.07740.1739–0.8334–0.90000.2970C. Four-factor regressionsIntercept–0.95–1.370.420.090.21–0.12–1.04–1.580.54 [–4.07][–3.96][1.20][0.73][1.29][–0.65][–3.79][–3.94][1.50]MktRF1.171.160.011.151.22–0.070.02–0.050.08 [17.18][16.01][0.11][30.60][39.35][–1.44][0.28][–0.71][0.94]HML–0.31–0.10–0.21–0.040.46–0.50–0.28–0.560.29 [–2.73][–0.64][–1.27][–0.55][4.22][–4.60][–1.76][–2.62][2.00]SMB1.001.22–0.220.680.96–0.280.320.250.07 [11.63][8.09][–1.41][8.27][17.40][–3.15][2.68][1.57][0.45]MOM0.01–0.630.640.17–0.470.64–0.16–0.160.00 [0.09][–5.18][5.04][4.09][–4.38][5.85][–1.51][–0.85][0.02]|$R^2$|.7505.6365.1942.8645.8844.5551.0932.0801.0163IR–0.8177–0.77490.22810.13210.2711–0.1276–0.7470–0.82760.2788
|$W^{*}_{\rm S}$||$L^{*}_{\rm S}$||$W^{*}_{\rm S}$|-|$L^{*}_{\rm S}$||$W^{*,m}_{\rm S}$||$L^{*,m}_{\rm S}$||$W^{*,m}_{\rm S}$|-|$L^{*,m}_{\rm S}$||$W^{*}_{\rm S}$|-|$W^{*,m}_{\rm S}$||$L^{*}_{\rm S}$|-|$L^{*,m}_{\rm S}$||${\it DiD}$|A. Raw excess returnsAverage–0.18–0.900.730.940.830.11–1.12–1.740.62 [–0.54][–1.81][2.00][3.47][2.04][0.43][–4.38][–4.60][1.80]No. of months411411411411411411411411411AvgN188106 262203 SR–0.0760–0.30760.35280.50960.36760.0772–0.7663–0.87280.3160B. CAPM regressionsIntercept–1.07–1.890.820.14–0.100.24–1.21–1.790.58 [–4.28][–5.55][2.27][0.85][–0.36][0.98][–4.52][–4.67][1.70]MktRF1.371.51–0.141.231.43–0.200.140.080.06 [14.94][11.97][–1.02][30.36][12.76][–1.56][1.57][0.61][0.63]|$R^2$|.5783.4432.0081.7393.6643.0343.0145.0026.0014IR–0.7096–0.86370.40000.1458–0.07740.1739–0.8334–0.90000.2970C. Four-factor regressionsIntercept–0.95–1.370.420.090.21–0.12–1.04–1.580.54 [–4.07][–3.96][1.20][0.73][1.29][–0.65][–3.79][–3.94][1.50]MktRF1.171.160.011.151.22–0.070.02–0.050.08 [17.18][16.01][0.11][30.60][39.35][–1.44][0.28][–0.71][0.94]HML–0.31–0.10–0.21–0.040.46–0.50–0.28–0.560.29 [–2.73][–0.64][–1.27][–0.55][4.22][–4.60][–1.76][–2.62][2.00]SMB1.001.22–0.220.680.96–0.280.320.250.07 [11.63][8.09][–1.41][8.27][17.40][–3.15][2.68][1.57][0.45]MOM0.01–0.630.640.17–0.470.64–0.16–0.160.00 [0.09][–5.18][5.04][4.09][–4.38][5.85][–1.51][–0.85][0.02]|$R^2$|.7505.6365.1942.8645.8844.5551.0932.0801.0163IR–0.8177–0.77490.22810.13210.2711–0.1276–0.7470–0.82760.2788
This table shows average monthly excess returns [in |$\%$|, panel A] as well as results from CAPM [panel B] and Fama-French-Carhart four-factor regressions [panel C] for short-horizon [1 year] buy-and-hold strategies of constrained winners, constrained losers, and their matched counterparts, as described in the text, as well as their differences. |${\it DiD}$| is the difference-in-differences, namely, |$[W^{*}_{\rm S}-W^{*,m}_{\rm S}]-[L^{*}_{\rm S}-L^{*,m}_{\rm S}]$|. Newey and West [1987]|$t$|-statistics are shown in parentheses. AvgN is the average number of unique stocks in the portfolio. The row labeled SR displays the Sharpe ratios and IR the information ratios. The sample period is May 1980 to June 2020. The first return is calculated in April 1986; that is, the first time we invested 12 times in a row and had the chance to see whether a constrained loser had been a constrained winner over the previous 5 years.
Table 2
Short-horizon [S, 1-year] performance of constrained and matched portfolios
|$W^{*}_{\rm S}$||$L^{*}_{\rm S}$||$W^{*}_{\rm S}$|-|$L^{*}_{\rm S}$||$W^{*,m}_{\rm S}$||$L^{*,m}_{\rm S}$||$W^{*,m}_{\rm S}$|-|$L^{*,m}_{\rm S}$||$W^{*}_{\rm S}$|-|$W^{*,m}_{\rm S}$||$L^{*}_{\rm S}$|-|$L^{*,m}_{\rm S}$||${\it DiD}$|A. Raw excess returnsAverage–0.18–0.900.730.940.830.11–1.12–1.740.62 [–0.54][–1.81][2.00][3.47][2.04][0.43][–4.38][–4.60][1.80]No. of months411411411411411411411411411AvgN188106 262203 SR–0.0760–0.30760.35280.50960.36760.0772–0.7663–0.87280.3160B. CAPM regressionsIntercept–1.07–1.890.820.14–0.100.24–1.21–1.790.58 [–4.28][–5.55][2.27][0.85][–0.36][0.98][–4.52][–4.67][1.70]MktRF1.371.51–0.141.231.43–0.200.140.080.06 [14.94][11.97][–1.02][30.36][12.76][–1.56][1.57][0.61][0.63]|$R^2$|.5783.4432.0081.7393.6643.0343.0145.0026.0014IR–0.7096–0.86370.40000.1458–0.07740.1739–0.8334–0.90000.2970C. Four-factor regressionsIntercept–0.95–1.370.420.090.21–0.12–1.04–1.580.54 [–4.07][–3.96][1.20][0.73][1.29][–0.65][–3.79][–3.94][1.50]MktRF1.171.160.011.151.22–0.070.02–0.050.08 [17.18][16.01][0.11][30.60][39.35][–1.44][0.28][–0.71][0.94]HML–0.31–0.10–0.21–0.040.46–0.50–0.28–0.560.29 [–2.73][–0.64][–1.27][–0.55][4.22][–4.60][–1.76][–2.62][2.00]SMB1.001.22–0.220.680.96–0.280.320.250.07 [11.63][8.09][–1.41][8.27][17.40][–3.15][2.68][1.57][0.45]MOM0.01–0.630.640.17–0.470.64–0.16–0.160.00 [0.09][–5.18][5.04][4.09][–4.38][5.85][–1.51][–0.85][0.02]|$R^2$|.7505.6365.1942.8645.8844.5551.0932.0801.0163IR–0.8177–0.77490.22810.13210.2711–0.1276–0.7470–0.82760.2788
|$W^{*}_{\rm S}$||$L^{*}_{\rm S}$||$W^{*}_{\rm S}$|-|$L^{*}_{\rm S}$||$W^{*,m}_{\rm S}$||$L^{*,m}_{\rm S}$||$W^{*,m}_{\rm S}$|-|$L^{*,m}_{\rm S}$||$W^{*}_{\rm S}$|-|$W^{*,m}_{\rm S}$||$L^{*}_{\rm S}$|-|$L^{*,m}_{\rm S}$||${\it DiD}$|A. Raw excess returnsAverage–0.18–0.900.730.940.830.11–1.12–1.740.62 [–0.54][–1.81][2.00][3.47][2.04][0.43][–4.38][–4.60][1.80]No. of months411411411411411411411411411AvgN188106 262203 SR–0.0760–0.30760.35280.50960.36760.0772–0.7663–0.87280.3160B. CAPM regressionsIntercept–1.07–1.890.820.14–0.100.24–1.21–1.790.58 [–4.28][–5.55][2.27][0.85][–0.36][0.98][–4.52][–4.67][1.70]MktRF1.371.51–0.141.231.43–0.200.140.080.06 [14.94][11.97][–1.02][30.36][12.76][–1.56][1.57][0.61][0.63]|$R^2$|.5783.4432.0081.7393.6643.0343.0145.0026.0014IR–0.7096–0.86370.40000.1458–0.07740.1739–0.8334–0.90000.2970C. Four-factor regressionsIntercept–0.95–1.370.420.090.21–0.12–1.04–1.580.54 [–4.07][–3.96][1.20][0.73][1.29][–0.65][–3.79][–3.94][1.50]MktRF1.171.160.011.151.22–0.070.02–0.050.08 [17.18][16.01][0.11][30.60][39.35][–1.44][0.28][–0.71][0.94]HML–0.31–0.10–0.21–0.040.46–0.50–0.28–0.560.29 [–2.73][–0.64][–1.27][–0.55][4.22][–4.60][–1.76][–2.62][2.00]SMB1.001.22–0.220.680.96–0.280.320.250.07 [11.63][8.09][–1.41][8.27][17.40][–3.15][2.68][1.57][0.45]MOM0.01–0.630.640.17–0.470.64–0.16–0.160.00 [0.09][–5.18][5.04][4.09][–4.38][5.85][–1.51][–0.85][0.02]|$R^2$|.7505.6365.1942.8645.8844.5551.0932.0801.0163IR–0.8177–0.77490.22810.13210.2711–0.1276–0.7470–0.82760.2788
This table shows average monthly excess returns [in |$\%$|, panel A] as well as results from CAPM [panel B] and Fama-French-Carhart four-factor regressions [panel C] for short-horizon [1 year] buy-and-hold strategies of constrained winners, constrained losers, and their matched counterparts, as described in the text, as well as their differences. |${\it DiD}$| is the difference-in-differences, namely, |$[W^{*}_{\rm S}-W^{*,m}_{\rm S}]-[L^{*}_{\rm S}-L^{*,m}_{\rm S}]$|. Newey and West [1987]|$t$|-statistics are shown in parentheses. AvgN is the average number of unique stocks in the portfolio. The row labeled SR displays the Sharpe ratios and IR the information ratios. The sample period is May 1980 to June 2020. The first return is calculated in April 1986; that is, the first time we invested 12 times in a row and had the chance to see whether a constrained loser had been a constrained winner over the previous 5 years.
The short-horizon constrained-loser [|$L^{*}_{\rm S}$|] and -winner [|$W^{*}_{\rm S}$|] strategies both exhibit negative excess returns [panel A] and negative alphas relative to CAPM and Fama-French-Carhart four-factor model benchmarks [panels B and C].24 These negative alphas suggest that, following both positive and negative information shocks, the marginal investors [i.e., the most optimistic agents] are too optimistic initially, but revise their beliefs downward over the first year following the information shock.
Panel B of Figure 1 suggests that the predictable negative abnormal returns of the constrained-winner [|$W^*$|] portfolios persist longer than the negative abnormal returns of the constrained-loser stocks [|$L^*$|]. To assess the statistical significance of the differences in persistence of the negative abnormal returns, we next focus on years 2 through 5 postformation. We now form long-horizon [L, years 2–5] strategies |$W^{*}_{\rm L}$| and |$L^{*}_{\rm L}$|, like the short-horizon [1 year] strategies described above, but which now hold the |$W^{*}$| and |$L^{*}$| portfolios formed in months |$t-59$| to |$t-12$|; that is, we skip the most recent year and hold the 48 portfolios from the preceding 4 years.25
Table 3 presents the results of this analysis. The average number of unique stocks held in the strategy is quite large now: on average, |$417$| unique stocks are classified as constrained winners in at least one month between 2 and 5 years prior to strategy formation.26 Panel A presents raw excess returns and Sharpe ratios, and panels B and C present CAPM and four-factor alphas, respectively, for the strategies. Strikingly, whereas the constrained-loser strategy has positive alphas over this period, the constrained-winner strategy significantly underperforms relative to both benchmarks; the monthly four-factor alpha is |$-0.62$||$\%$| [|$t = -4.93]$|.27 The difference in the monthly four-factor alphas of |$W^{*}_{\rm L}$| and |$L^{*}_{\rm L}$| is |$-0.88$||$\%$||$[t=-4.33]$|.28
Table 3
Long-horizon [L, 2- to 5-year] strategy performance for constrained and matched portfolios
|$W^{*}_{\rm L}$||$L^{*}_{\rm L}$||$W^{*}_{\rm L}$|-|$L^{*}_{\rm L}$||$W^{*,m}_{\rm L}$||$L^{*,m}_{\rm L}$||$W^{*,m}_{\rm L}$|-|$L^{*,m}_{\rm L}$||$W^{*}_{\rm L}$|-|$W^{*,m}_{\rm L}$||$L^{*}_{\rm L}$|-|$L^{*,m}_{\rm L}$||${\it DiD}$|A. Raw excess returnsAverage0.231.03–0.801.010.830.18–0.780.20–0.98 [0.66][2.81][–3.87][3.69][3.01][1.58][–4.88][1.01][–6.38]No. of months363363363363363363363363363AvgN417203 621461 SR0.10940.4849–0.60610.58480.46280.2941–0.72630.1694–0.7450B. CAPM regressionsIntercept–0.740.14–0.870.17–0.030.20–0.900.17–1.07 [–3.61][0.59][–4.34][1.30][–0.20][1.72][–5.39][0.89][–5.76]MktRF1.431.320.111.251.27–0.020.180.050.14 [19.51][17.57][1.26][29.01][30.90][–0.48][3.03][0.70][1.72]|$R^2$|.7016.6029.0115.8197.7893.0021.0458.0025.0166IR–0.62800.1039–0.66780.2255–0.03730.3192–0.86190.1427–0.8215C. Four-factor regressionsIntercept–0.620.26–0.880.14–0.050.19–0.760.31–1.07 [–4.93][1.20][–4.33][1.53][–0.52][2.20][–5.29][1.38][–5.24]MktRF1.251.130.131.171.150.010.08–0.030.11 [18.03][22.15][1.64][42.36][35.76][0.41][1.21][–0.45][1.39]HML–0.280.11–0.39–0.170.17–0.34–0.11–0.06–0.04 [–5.08][1.56][–4.99][–3.78][3.66][–7.28][–1.16][–0.60][–0.59]SMB0.740.88–0.140.510.79–0.280.220.090.14 [8.87][12.29][–1.73][13.06][18.48][–6.17][2.59][1.14][1.43]MOM–0.08–0.140.060.080.040.05–0.16–0.180.02 [–1.29][–2.44][0.64][3.55][1.84][2.11][–3.04][–2.18][0.16]|$R^2$|.8190.7352.0893.9159.9360.3511.1183.0410.0277IR–0.68480.2342–0.70080.2790–0.11460.3871–0.76060.2641–0.8277
|$W^{*}_{\rm L}$||$L^{*}_{\rm L}$||$W^{*}_{\rm L}$|-|$L^{*}_{\rm L}$||$W^{*,m}_{\rm L}$||$L^{*,m}_{\rm L}$||$W^{*,m}_{\rm L}$|-|$L^{*,m}_{\rm L}$||$W^{*}_{\rm L}$|-|$W^{*,m}_{\rm L}$||$L^{*}_{\rm L}$|-|$L^{*,m}_{\rm L}$||${\it DiD}$|A. Raw excess returnsAverage0.231.03–0.801.010.830.18–0.780.20–0.98 [0.66][2.81][–3.87][3.69][3.01][1.58][–4.88][1.01][–6.38]No. of months363363363363363363363363363AvgN417203 621461 SR0.10940.4849–0.60610.58480.46280.2941–0.72630.1694–0.7450B. CAPM regressionsIntercept–0.740.14–0.870.17–0.030.20–0.900.17–1.07 [–3.61][0.59][–4.34][1.30][–0.20][1.72][–5.39][0.89][–5.76]MktRF1.431.320.111.251.27–0.020.180.050.14 [19.51][17.57][1.26][29.01][30.90][–0.48][3.03][0.70][1.72]|$R^2$|.7016.6029.0115.8197.7893.0021.0458.0025.0166IR–0.62800.1039–0.66780.2255–0.03730.3192–0.86190.1427–0.8215C. Four-factor regressionsIntercept–0.620.26–0.880.14–0.050.19–0.760.31–1.07 [–4.93][1.20][–4.33][1.53][–0.52][2.20][–5.29][1.38][–5.24]MktRF1.251.130.131.171.150.010.08–0.030.11 [18.03][22.15][1.64][42.36][35.76][0.41][1.21][–0.45][1.39]HML–0.280.11–0.39–0.170.17–0.34–0.11–0.06–0.04 [–5.08][1.56][–4.99][–3.78][3.66][–7.28][–1.16][–0.60][–0.59]SMB0.740.88–0.140.510.79–0.280.220.090.14 [8.87][12.29][–1.73][13.06][18.48][–6.17][2.59][1.14][1.43]MOM–0.08–0.140.060.080.040.05–0.16–0.180.02 [–1.29][–2.44][0.64][3.55][1.84][2.11][–3.04][–2.18][0.16]|$R^2$|.8190.7352.0893.9159.9360.3511.1183.0410.0277IR–0.68480.2342–0.70080.2790–0.11460.3871–0.76060.2641–0.8277
See caption to Table 2. The only difference here is that we hold stocks that were allocated to one of the portfolios at some point during months |$\{t-60,...,t-13\}$| before formation. The first monthly return is calculated in April 1990, that is, the first time we invested 48 times in a row.
Table 3
Long-horizon [L, 2- to 5-year] strategy performance for constrained and matched portfolios
|$W^{*}_{\rm L}$||$L^{*}_{\rm L}$||$W^{*}_{\rm L}$|-|$L^{*}_{\rm L}$||$W^{*,m}_{\rm L}$||$L^{*,m}_{\rm L}$||$W^{*,m}_{\rm L}$|-|$L^{*,m}_{\rm L}$||$W^{*}_{\rm L}$|-|$W^{*,m}_{\rm L}$||$L^{*}_{\rm L}$|-|$L^{*,m}_{\rm L}$||${\it DiD}$|A. Raw excess returnsAverage0.231.03–0.801.010.830.18–0.780.20–0.98 [0.66][2.81][–3.87][3.69][3.01][1.58][–4.88][1.01][–6.38]No. of months363363363363363363363363363AvgN417203 621461 SR0.10940.4849–0.60610.58480.46280.2941–0.72630.1694–0.7450B. CAPM regressionsIntercept–0.740.14–0.870.17–0.030.20–0.900.17–1.07 [–3.61][0.59][–4.34][1.30][–0.20][1.72][–5.39][0.89][–5.76]MktRF1.431.320.111.251.27–0.020.180.050.14 [19.51][17.57][1.26][29.01][30.90][–0.48][3.03][0.70][1.72]|$R^2$|.7016.6029.0115.8197.7893.0021.0458.0025.0166IR–0.62800.1039–0.66780.2255–0.03730.3192–0.86190.1427–0.8215C. Four-factor regressionsIntercept–0.620.26–0.880.14–0.050.19–0.760.31–1.07 [–4.93][1.20][–4.33][1.53][–0.52][2.20][–5.29][1.38][–5.24]MktRF1.251.130.131.171.150.010.08–0.030.11 [18.03][22.15][1.64][42.36][35.76][0.41][1.21][–0.45][1.39]HML–0.280.11–0.39–0.170.17–0.34–0.11–0.06–0.04 [–5.08][1.56][–4.99][–3.78][3.66][–7.28][–1.16][–0.60][–0.59]SMB0.740.88–0.140.510.79–0.280.220.090.14 [8.87][12.29][–1.73][13.06][18.48][–6.17][2.59][1.14][1.43]MOM–0.08–0.140.060.080.040.05–0.16–0.180.02 [–1.29][–2.44][0.64][3.55][1.84][2.11][–3.04][–2.18][0.16]|$R^2$|.8190.7352.0893.9159.9360.3511.1183.0410.0277IR–0.68480.2342–0.70080.2790–0.11460.3871–0.76060.2641–0.8277
|$W^{*}_{\rm L}$||$L^{*}_{\rm L}$||$W^{*}_{\rm L}$|-|$L^{*}_{\rm L}$||$W^{*,m}_{\rm L}$||$L^{*,m}_{\rm L}$||$W^{*,m}_{\rm L}$|-|$L^{*,m}_{\rm L}$||$W^{*}_{\rm L}$|-|$W^{*,m}_{\rm L}$||$L^{*}_{\rm L}$|-|$L^{*,m}_{\rm L}$||${\it DiD}$|A. Raw excess returnsAverage0.231.03–0.801.010.830.18–0.780.20–0.98 [0.66][2.81][–3.87][3.69][3.01][1.58][–4.88][1.01][–6.38]No. of months363363363363363363363363363AvgN417203 621461 SR0.10940.4849–0.60610.58480.46280.2941–0.72630.1694–0.7450B. CAPM regressionsIntercept–0.740.14–0.870.17–0.030.20–0.900.17–1.07 [–3.61][0.59][–4.34][1.30][–0.20][1.72][–5.39][0.89][–5.76]MktRF1.431.320.111.251.27–0.020.180.050.14 [19.51][17.57][1.26][29.01][30.90][–0.48][3.03][0.70][1.72]|$R^2$|.7016.6029.0115.8197.7893.0021.0458.0025.0166IR–0.62800.1039–0.66780.2255–0.03730.3192–0.86190.1427–0.8215C. Four-factor regressionsIntercept–0.620.26–0.880.14–0.050.19–0.760.31–1.07 [–4.93][1.20][–4.33][1.53][–0.52][2.20][–5.29][1.38][–5.24]MktRF1.251.130.131.171.150.010.08–0.030.11 [18.03][22.15][1.64][42.36][35.76][0.41][1.21][–0.45][1.39]HML–0.280.11–0.39–0.170.17–0.34–0.11–0.06–0.04 [–5.08][1.56][–4.99][–3.78][3.66][–7.28][–1.16][–0.60][–0.59]SMB0.740.88–0.140.510.79–0.280.220.090.14 [8.87][12.29][–1.73][13.06][18.48][–6.17][2.59][1.14][1.43]MOM–0.08–0.140.060.080.040.05–0.16–0.180.02 [–1.29][–2.44][0.64][3.55][1.84][2.11][–3.04][–2.18][0.16]|$R^2$|.8190.7352.0893.9159.9360.3511.1183.0410.0277IR–0.68480.2342–0.70080.2790–0.11460.3871–0.76060.2641–0.8277
See caption to Table 2. The only difference here is that we hold stocks that were allocated to one of the portfolios at some point during months |$\{t-60,...,t-13\}$| before formation. The first monthly return is calculated in April 1990, that is, the first time we invested 48 times in a row.
These results suggest that, following a positive information shock, the most optimistic agents continue to revise their beliefs downward in years 2 through 5 postformation. By contrast, following a negative information shock, the beliefs of the optimistic agents are relatively stable after about 1 year; we see no evidence of belief revisions either upward or downward.
As a robustness check, in Internet Appendix F.I we repeat these tests using Fama and MacBeth [1973] regressions. This allows us to control for various other measures and generates consistent results.
Panel A of Figure 4 plots the cumulative returns to the |$W^{*}_{\rm S}$| and |$L^{*}_{\rm S}$| strategies, hedged with respect to the CAPM-market factor over the sample period.29 Both strategies fall consistently over the full sample period, suggesting the strategy underperformance is not concentrated in a particular subperiod. The magnitude of the underperformance results in a dramatic negative cumulative return: an initial investment of $1,000 into the 1-year hedged constrained-winner [constrained-loser] strategy is worth |$\$18.90$| [|$\$0.34$|] at the end of June 2020.
Figure 4
Performance of hedged constrained strategies over calendar time
This figure presents the investment value for a set of hedged portfolios. To calculate the portfolio value, we assume an investment at the beginning of the sample of $1,000. We also assume the exposure to the market is hedged. We calculate the hedging coefficient by running a full-sample regression of the strategy excess returns on the market excess returns. Then, using the full-sample regression coefficients, we subtract the returns of the [zero-investment] hedge-portfolio |$[\text{b}_{\text{Mkt}}[\text{R}_{\text{Mkt}}\text{-}\text{R}_{\text{f,t}}]]$| from the strategy excess returns and add the risk-free rate to generate the hedged strategy returns. Panel A plots the evolution of $1,000 invested in hedged calendar-time short-horizon [1 year] buy-and-hold constrained-winner and -loser [that were not constrained winners in the past 5 years] strategies. Panel B shows results for the corresponding long-horizon [2–5 years] strategies.
For comparison, panel B plots the cumulative returns for the market-hedged long-horizon strategies [|$W^{*}_{\rm L}$| and |$L^{*}_{\rm L}$|]. The cumulative returns over the sample period are strongly negative for the constrained-winner strategy, but are positive for the constrained-losers strategy. Moreover, the difference between the two strategies does not appear to be driven by a particular subsample.
As a further robustness check, we replicate our main findings on a sample of international equities, and for an out-of-sample time-period prior to the availability of short interest data for NASDAQ stocks [see Internet Appendices E.I and E.II]. In all cases, we see strong negative abnormal returns over the 1 year following portfolio formation for constrained losers. Over years 2–5 postformation, only the constrained winners underperform. Thus, the finding of differential performance-persistence appears robust.
3.3 Matching
In the preceding subsection, we documented that portfolios of constrained winners and constrained losers exhibit strong negative abnormal returns, and that these negative abnormal returns persist for approximately 5 years and 1 year, respectively. We hypothesize that this asymmetric persistence in predictability results from a combination of short-sale constraints and differential interpretation of information shocks among investors. Each of the firms in these two portfolios, on the portfolio formation date, has low IOR and high SIR. We argue that the low IOR is a proxy for a low lending supply [because noninstitutional holders are less likely to make their shares available for lending]. Then, when increased disagreement leads to an increased demand by pessimistic investors to borrow and [short] sell shares that likely exceeds the supply of easily borrowed shares, borrowing becoming costly. That is, the stocks become constrained.
However, another possibility is that, in sorting on IOR, SIR, and past return to form our portfolios, we are inadvertently selecting some other firm characteristic directly linked to return predictability. For example, the previous literature has argued that short interest is a proxy for informed demand and thus predicts future returns [Boehmer, Jones, and Zhang 2008; Boehmer, Huszar, and Jordan 2010; Engelberg, Reed, and Ringgenberg 2012; Rapach, Ringgenberg, and Zhou 2016]. Sorting on high short interest alone, in combination with high or low past return, could give the same results.
To examine this possibility, for each constrained portfolio, we create a matched portfolio that contains firms that are as close to identical as possible to the firms in our constrained portfolio on the dimensions of size, short interest, past return, and the logarithm of book-to-market, except these matched firms are not short-sale constrained [i.e., they have high institutional ownership]. Specifically, for each stock in the constrained-winner portfolio [|$W^*$|] and the constrained-loser portfolio [|$L^*$|], we run a matching procedure based on the Mahalanobis [1936] distance for these four metrics to find a statistical twin stock in a universe of unconstrained potential matches. We limit the unconstrained matching universe to stocks above the 70th percentile cutoff for institutional ownership to ensure that the matched firm is unconstrained. In addition, we impose the constraint that the matched firm must fall in the same past-return bucket and that, for the firms in the constrained-loser portfolio, the matched firm not be a constrained-winner stock at any time within the past 5 years [equivalent to the constraint we place on the actual constrained losers].30
The last two columns in Table 1 reveal that, not surprisingly, the value-weighted portfolios of matched stocks for |$W^*$| and |$L^*$|, that is, |$W^{*,m}$| and |$L^{*,m}$|, are similar along the matching dimensions of size, short interest, past return, and book-to-market. They are also similar along related dimensions [e.g., turnover and volatility]. However, they differ substantially on dimensions that proxy for short-sale constraints, such as SIRIO, volatility spread, and Markit loan fees. This finding suggests the matched firm-portfolios should be well suited to uncovering differences solely based on the fact that one set of firms is short-sale constrained, while the other is not.
Table 2 shows that short-horizon strategies based on matched-firm portfolios, |$W^{*,m}_{\rm S}$| and |$L^{*,m}_{\rm S}$|, exhibit no abnormal performance relative to a CAPM or four-factor model. Consistent with this and our earlier findings, the final three columns show the strategies based on constrained winners and losers dramatically underperform their matched-firm strategies. The final column of this table shows the difference-in-differences [ DiD] is statistically indistinguishable from zero.
The results are also consistent with the hypothesis that the short-sale constraints are responsible for the underperformance of the constrained winner portfolio in years 2–5 postformation. Table 3 shows that the long-horizon strategies based on matched firms, |$W^{*,m}_{\rm L}$| and |$L^{*,m}_{\rm L}$|, exhibit no abnormal performance relative to a CAPM. The DiD—testing whether the constrained portfolios underperform the matched firms by different amounts for the winners and losers—is now highly statistically significant,31 in all three panels.
Figure 5 adds more background to the matching approach. It shows the buy-and-hold performance of the constrained and matched portfolios on a year-by-year basis.32 Whereas the portfolios of constrained stocks exhibit distinct return predictability, the matched portfolios’ returns can be explained by the CAPM in each of the 6 years postformation.33 Strikingly but consistent with the asymmetry in persistence we document elsewhere, the constrained winners underperform significantly relative to the CAPM for each of the 5 years following formation [panel A], while the constrained losers only exhibit a significantly negative alpha in the first year [panel B]. This visual assessment is consistent with the time-series regressions presented throughout the paper.
Figure 5
CAPM alphas of constrained and matched 1-year strategies
The first set of points show the annualized CAPM alphas of strategies of value-weighted portfolios of constrained past winners [panel A] and losers [panel B], respectively, in years 1 through 6 postformation. The second set of points in panels A and B is the result of strategies of matched stock-portfolios, based on the Mahalanobis distance calculated on size, log[book-to-market], past return, and short interest. The whiskers represent 95|$\%$| confidence intervals based on Newey-West standard errors. For details, see Section .
3.4 Earnings announcements
This paper builds on the Miller [1977] insight that constrained securities with restricted short selling will be generally overpriced, since their prices will reflect the beliefs of the more optimistic investors. This implies that the arrival of new information will generally cause these optimists to revise their beliefs about firm value downward, resulting in a negative event return. One point in time when disagreement is likely to be resolved is when firms announce their earnings, which usually happens once per quarter.34 This means that, for overpriced securities, we should see more extreme negative returns immediately following earnings announcements than at other times.
Figure 6 plots the average cumulative abnormal returns [ACAR] around earnings announcements for the constrained winner and loser strategies and the corresponding matched-firm strategies explored in Sections and . Panel A plots the ACARs for the 1-year strategies |$W^{*}_{S}$| and |$L^{*}_{S}$| and their matched-firm counterparts, and panel B does the same for the 2- to 5-year strategies |$W^{*}_{L}$| and |$L^{*}_{L}$| and their corresponding matched-firm strategies.35 Consistent with the mispricing hypothesis and the return predictability discussed earlier, both the 1-year portfolios of constrained winners and losers significantly decline immediately following earnings announcements. For the 2- to 5-year strategies, the strong negative returns immediately following earnings announcements are present only for the constrained-winner-based strategies. This finding is again consistent with the hypothesis that overpricing is eliminated in the first year following portfolio formation for the |$L^{*}$| portfolios, but that the overpricing persists out to 5 years for the constrained winners, and that much of this mispricing is eliminated on earnings announcement dates.
Figure 6
CAR around earnings announcements
This figure shows cumulated abnormal returns of the constrained winners [|$W^{*}$|] and losers [|$L^{*}$|] as well as their unconstrained matched counterparts [|$W^{*,m}$| and |$L^{*,m}$|] around the day [D=0] of an earnings announcement that occurs in the quarter after portfolio formation [months t to t+2]. We include all stocks that were in the respective portfolio in months t-12 through t-1 [panel A, short horizon] and t-60 to t-13 [panel B, long horizon] and calculate their buy-and-hold weight from formation to each day plotted, by using the price change adjusted by the cumulative price adjustment factor [CFACPR in CRSP]. Abnormal returns are calculated by adjusting for beta times the CAPM-market-factor. For each stock, beta is estimated in a 1-year window of daily returns prior to the month in which the earnings announcement occurs. To construct the figure, we first center daily abnormal returns on the day of announcement [D=0]. They are then cumulated by stock [cumulative abnormal return, CAR] and averaged [ACAR, weighted by the buy-and-hold weight] by portfolio and day relative to announcement. See Internet Appendix B.V for details.
3.5 Lending fees
The market price of a constrained security will reflect the valuation of the agents who are most optimistic about the firm value. In a similar vein, the per-period marginal cost of short selling a security, at a given point in time, likely reflects the beliefs of the most pessimistic investors about how much the security price is likely to fall per period. That is, it is a direct measure of disagreement.
Here, we use the Markit indicative lending fee as a proxy for the marginal costs of short selling. This lending fee clearly will not capture all of the costs or risks associated with short selling, and therefore may not be a good proxy for the all-in cost of short selling. However, changes in the fee over time should be correlated with changes in the pessimistic agents’ willingness to pay to borrow shares for the purpose of shorting, and therefore with changes in disagreement between pessimists and optimists for constrained stocks.
Panel A of Figure 2 reveals that the fees increase leading up to portfolio formation and then fall slowly over about 5 years postformation, that is, at a symmetric rate for constrained winners and losers. On the surface, this finding might seem puzzling, given the evidence earlier in this Section that the abnormal return persistence for past-winners and losers is so different. However, it is consistent with a model in which, for the constrained losers between 2 and 5 years postformation, the optimists’ beliefs about the security value are unbiased, while the beliefs of the most pessimistic investors—those agents whose beliefs determine the borrowing costs—are too pessimistic initially but then fall over time. This model is discussed in the Introduction and is presented in Section .
We confirm the finding that the postformation decline in lending fees is symmetric across constrained winners and losers by regressing changes in portfolio lending fees on dummy variables that equal one when an observation corresponds to the first year postformation [|$I_{0