Probability of price crashes, rational speculative bubbles, and the cross-section of stock returns

Accepted Manuscript

Probability of Price Crashes, Rational Speculative Bubbles, and the Cross-Section of Stock Returns Jeewon Jang , Jangkoo Kang PII: DOI: Reference:

S0304-405X(18)30281-2 https://doi.org/10.1016/j.jfineco.2018.10.005 FINEC 2977

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Journal of Financial Economics

Received date: Revised date: Accepted date:

24 January 2017 30 October 2017 7 December 2017

Please cite this article as: Jeewon Jang , Jangkoo Kang , Probability of Price Crashes, Rational Speculative Bubbles, and the Cross-Section of Stock Returns, Journal of Financial Economics (2018), doi: https://doi.org/10.1016/j.jfineco.2018.10.005

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Probability of Price Crashes, Rational Speculative Bubbles, and the Cross-Section of Stock Returns* Jeewon Jang†

Abstract

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Jangkoo Kang‡

We estimate an ex ante probability of extreme negative returns (crashes) of individual stocks as a measure

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of potential overpricing and find that stocks with a high probability of crashes earn abnormally low returns. Stocks with high crash probability are overpriced regardless of the level of institutional ownership or variations in investor sentiment, and moreover, they exhibit increasing institutional demand until their prices reach the peak of overvaluation. We also find that institutional investors who overweight

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high crash probability stocks outperform the others, indicating that they have skill in timing bubbles and crashes of individual stocks. Our findings imply that sophisticated investors may not always trade against

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mispricing but time the correction of overpricing, and suggest that the crash effect we find could arise at

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least partially from rational speculative bubbles, not entirely from sentiment-driven overpricing.

JEL classification: G11; G12; G14

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Keywords: Price crashes; Overpricing; Anomalies; Institutional investors; Rational speculative bubbles

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We thank Jens Hilscher (the referee) and G. William Schwert (the editor) for valuable comments that significantly improved the paper. All remaining errors are ours. This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2017S1A5A8021250). †

Corresponding author: School of Business, Ajou University, 206 Worldcup-Ro, Yeongtong-Gu, Suwon 16499, Republic of Korea; Phone: +82-31-219-2716; E-mail: [email protected] ‡

College of Business, Korea Advanced Institute of Science and Technology (KAIST), 85 Hoegiro, DongdaemunGu, Seoul 02455, Republic of Korea; Phone: +82-2-958-3521; E-mail: [email protected]

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1. Introduction

It is well established in the finance literature that various firm characteristics are able to predict stock returns in the cross-section. Since numerous cross-sectional relations, classified as anomalies, remain

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unexplained by standard asset pricing theories stating that riskier assets should command higher expected returns, recent finance research attributes cross-sectional anomalies largely to mispricing rather than to rational pricing.1 Moreover, several empirical studies recently document that abnormal profits earned by the long–short portfolio strategies based on anomaly characteristics mainly come from taking short

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positions in overpriced stocks rather than long positions in underpriced stocks (Nagel, 2005; Stambaugh, Yu, and Yuan, 2012; Avramov, Chordia, Jostova, and Philipov, 2013). Considering that overpricing is more prevalent than underpricing in the stock market due to short-sale constraints, this implies that firm characteristics shown to predict future returns are likely to be related to the extent of relative overpricing

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in the cross-section.2

To explain a variety of anomalies in association with overpricing, two central questions arise of what

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drives stocks in the short-leg portfolios to be overpriced and why their abnormally low returns are not arbitraged away. To answer the first question, the classical argument in the limits-to-arbitrage literature is

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that sophisticated traders are always willing to trade against mispricing unless arbitrage constraints are binding, and thus any price deviations from fundamentals result from speculative demand of behavioral

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traders.3 Consistent with this argument, Stambaugh et al. (2012) find empirical evidence that 11 cross-

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sectional anomalies are stronger following high levels of investor sentiment since overpricing of the 1 A recent work by McLean and Pontiff (2016) documents that, using 97 different variables shown to predict cross-sectional stock returns, the average long-short portfolio return based on each predictor shrinks 58% postpublication, consistent with mispricing accounting for the cross-sectional anomalies and investors learning about and trading against the mispricing. 2

The role of short-sale constraints in limiting overpricing being arbitraged away is first addressed by Miller (1977) and it is recently termed by Stambaugh, Yu, and Yuan (2015) as arbitrage asymmetry, meaning that investors willing to buy an underpriced stock are reluctant or unable to sell short an overpriced stock due to short-sale constraints. 3

Theoretical works on limits to arbitrage with this classical argument include Miller (1977), Harrison and Kreps (1978), Delong et al. (1990a), Dow and Gorton (1994), and Shleifer and Vishny (1997). -1-

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short-leg portfolios is more prevalent when investor sentiment is high.4 Their findings indicate that, not only does the broad set of anomalies arise from overpricing, but they are driven by market-wide investor sentiment. Regarding the second question, the classical argument predicts that overpricing cannot be fully exploited by rational traders in the presence of short-sale constraints, and consequently, the cross-

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sectional anomalies, to the extent that they reflect overpricing, should be more pronounced among stocks with greater short-sale constraints. This prediction is evidenced by several empirical studies documenting that various anomalies are stronger among stocks with a lower fraction of shares owned by institutions (Nagel, 2005; Campbell, Hilscher, and Szilagyi, 2008; Conrad, Kapadia, and Xing, 2014; Stambaugh et

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al., 2015). 5 This line of empirical studies consistently supports that the cross-sectional return predictability may be the result of overpricing, driven by behavioral speculators and not exploited by rational arbitrageurs.

Inspired by these recent studies on cross-sectional anomalies, our paper constructs an ex ante measure

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of potential overpricing for individual stocks and investigates the cross-sectional relation between our measure of overpricing and future stock returns. In addition, we explore the sources behind the cross-

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sectional relation that differentiate it from other cross-sectional anomalies documented previously. Overpricing cannot be determined by the currently observed price but can be verified only after a

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substantially low return is realized as the overpricing is corrected. This means that currently overpriced stocks are very likely to have extremely low returns realized in the future, and moreover, they are more

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likely to do so as the stock prices deviate more widely from fundamentals and become closer to the peak

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of overvaluation. For these reasons, we assume that stocks with a high probability of extremely low future

4 According to Baker and Wurgler (2006), one possible definition of investor sentiment is the propensity to speculate. 5

Nagel (2005) documents that the underperformance of stocks with a high market-to-book, analyst forecast dispersion, turnover, or volatility is most pronounced among stocks with low institutional ownership. Campbell et al. (2008) find that puzzling low returns on financially distressed firms are more pronounced for stocks with informational or arbitrage-related frictions. Conrad et al. (2014) find that stocks with a high predicted probability of extreme positive returns earn abnormally low returns and the cross-sectional relation is much stronger in stocks with higher arbitrage costs. Stambaugh et al. (2015) document that the idiosyncratic volatility puzzle is more pronounced among overpriced stocks, particularly for stocks that are less easily shorted. -2-

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returns may be currently overpriced, and estimate the ex ante probability of large negative returns, or price crashes, as a measure of potential overpricing for individual stocks. To predict the ex ante probability of price crashes for individual stocks, we employ a generalized logit model, extending the binary logit framework adopted by Campbell et al. (2008) and Conrad et al. (2014).

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Conrad et al. (2014) use a binary logit model to predict the probability of extreme positive returns, or jackpots. Since stocks with a high probability of extreme positive returns (jackpots) are very likely to have extreme negative returns (crashes) as well, however, the probability of jackpots predicted from the binary logit model may be entangled with the probability of crashes, although jackpots and crashes are

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mutually exclusive events, and thus it may actually proxy for volatility. To overcome the same problem likely to arise if the probability of crashes were estimated from a binary logit model, we assume individual stocks to have ternary future payoffs of extreme negative returns, extreme positive returns, or non-extreme returns. Then, we build a logit model jointly predicting the two mutually exclusive events,

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crashes and jackpots, and by evaluating the out-of-sample predictive accuracy of our model, we confirm that crashes are indeed disentangled from jackpots and thus the crash probability predicted from our logit

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model measures left-tail risk rather than volatility. We further investigate the response of stock prices around the peak of the crash probability and find that the ex ante probability of crashes predicted from our

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logit model can time the peaks of price appreciation and crashes, implying that it effectively measures the degree of overpricing. The estimation results of the generalized logit model show that high past market

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returns, low past individual returns relative to the market return, a firm’s high volatility, high skewness, low age, few tangible assets, and high sales growth predict a high probability of the firm’s future extreme

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negative returns, indicating that these features are related to overpricing. By constructing decile portfolios sorted on the out-of-sample predicted probability of price crashes,

we find that stocks with higher crash probability subsequently earn lower average returns. The long–short strategy of going long the lowest and short the highest crash probability decile yields a Fama–French (1993) three-factor alpha of 0.96% per month, with a t-statistic of 4.76, which is largely due to the substantial underperformance of the highest decile. The highest crash probability portfolio consists of -3-

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small and growth stocks with low prior returns in recent months. Although the crash probability is highly associated with several variables related to volatility or skewness as well as other firm characteristics known to predict cross-sectional stock returns, we find that the negative relation between crash probability and future returns remains strong, even if a large set of predictors are controlled for, in both

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portfolio- and firm-level analyses. The control variables include the jackpot probability of Conrad et al. (2014), failure probability of Campbell et al. (2008), idiosyncratic volatility of Ang, Hodrick, Xing, and Zhang (2006), the maximum daily return of Bali, Cakici, and Whitelaw (2011), analyst forecast dispersion of Diether, Malloy, and Scherbina (2002), and firm characteristics such as size, the book-to-market ratio,

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past returns, liquidity, and turnover, all of which are closely related to both crash probability and stock returns in the cross-section. Moreover, the crash probability effect we find is clearly distinct from the 11 cross-sectional anomalies examined by Stambaugh et al. (2012, 2015), which indicates that our crash probability is a different measure of overpricing from the mispricing measure based on the 11 anomalies

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constructed by Stambaugh et al. (2015).

To look into the source of overpricing, we investigate whether the underperformance of stocks with

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high crash probability becomes stronger when limits to arbitrage become higher. Consistent with previous studies on anomalies, the crash probability effect is particularly pronounced among small and low-priced

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stocks. On the other hand, we find no clear evidence that the negative relation between crash probability and stock returns is pronounced in illiquid and low analyst coverage stocks. The most interesting is that

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stocks with high crash probability earn abnormally low returns regardless of the firms’ institutional ownership, which means that overpricing of high crash probability stocks is not arbitraged away even

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among stocks largely owned by institutions. In addition, we examine the effect of market-wide sentiment on overpricing and find that overpricing of high crash probability stocks is prevalent irrespective of market-wide sentiment, clearly different from the findings of Stambaugh et al. (2012). This indicates that the crash probability effect may not be entirely driven by investor sentiment. Our findings that overpricing of high crash probability stocks is neither related to institutional ownership nor to investor sentiment contradict the classical argument in the limits-to-arbitrage literature. -4-

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The classical theories on limits to arbitrage predict that overpricing is inversely related to institutional ownership, because institutional investors, as long as they are sophisticated, always trade against mispricing unless arbitrage is limited by frictions such as noise trader risk or short-sale constraints. Alternatively, reasoning along the line of rational speculation theories can help to understand our

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empirical findings. For example, DeLong et al. (1990b) document that rational traders could bid up the prices of securities above fundamental values with anticipation for positive feedback traders buying the securities at even higher prices in the next period. Abreu and Brunnermeier (2002, 2003) document that rational investors optimally choose to ride a price bubble even if they are aware of the overvaluation,

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when coordinated selling pressure is necessary to burst the bubble but other arbitrageurs are not yet likely to attack the bubble. In their models, rational traders can make the highest profit by riding the bubble as it continues to grow and exiting the market just prior to the crash. These theories consistently imply that rational investors may not actively trade against mispricing although short-sale constraints are not binding,

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and even these investors can contribute to price deviations from fundamentals by riding bubbles, to exploit capital gains in the short run.6 The theories provide an explanation of why overpricing is not

with investor sentiment.

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arbitraged away even among stocks mainly held by sophisticated institutions and not clearly associated

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We provide further evidence that supports the rational speculation argument but refutes the classical argument, by examining changes in aggregate institutional holdings prior to forming portfolios sorted by

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the crash probability. We find that stocks with higher crash probability are bought more heavily by institutions during the six quarters prior to portfolio formation. Moreover, institutional holdings for a firm

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increase continuously up until the firm enters into the highest decile sorted by crash probability, while they slow down afterward. These results indicate that institutional investors as a whole have a tendency to

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Empirical documentation also supports this line of theories. Brunnermeier and Nagel (2004) document that hedge funds invested heavily in technology stocks during the technology bubble, not exerting any correcting force on their prices, although this does not seem to be the result of unawareness of the bubble. Griffin et al. (2011) provide evidence that institutions bought technology stocks more aggressively than individuals did during the technology bubble, which shows that rational investors sometimes fail to trade against mispricing. -5-

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buy an overvalued security until its price reaches the peak of the bubble. Our findings are consistent with the prediction of the rational speculation theories that sophisticated institutions may not correct overpricing but ride price bubbles, destabilizing stock prices. Finally, to rule out a behavioral explanation and confirm that institutional investors are sophisticated

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and make profits from speculative trading, as predicted by the rational speculation argument, we evaluate the performance of institutions grouped into decile by the holding-weighted average of crash probability. We construct portfolios replicating the quarterly holdings of each group of institutions, and find that a tenth of institutions holding portfolios tilted the most toward high crash probability stocks earn a quarterly

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four-factor alpha of 0.58% (t-statistic = 3.11), which is 0.39% (t-statistic = 2.21) higher than the other institutions do. This finding indicates that institutional investors who actively speculate on overpriced stocks outperform the others, even though they overweight stocks with a negative alpha. This surprising result cannot be true unless institutional speculators have skill in timing the peak and burst of price

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bubbles of individual stocks, which implies that they may ride bubbles and successfully get out of them prior to price crashes. Consequently, our evidence excludes a behavioral explanation and strongly

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supports rational speculation as a source of overpricing of high crash probability stocks. Our study is related to an extensive literature on the relation between firm characteristics and the

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cross-section of stock returns. Our finding of the low average returns of stocks with high crash probability is very strong and robust to controls for numerous variables related to volatility and skewness as well as

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firm characteristics, which confirms that the crash probability effect we find is clearly distinguished from other cross-sectional anomalies documented in the literature. Our evidence also suggests that the strong

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predictive power of crash probability could be attributed to its ability to time the peaks of overpricing, as well as the ability to pick relatively overpriced stocks. Moreover, our empirical findings clearly differ from the recent findings of Nagel (2005) and Stambaugh et al. (2012) in that overpricing is also present among stocks largely owned by institutions and not associated with variations in investor sentiment. These results suggest that the low average returns of stocks with high crash probability may be the result of overpricing, driven by both rational and behavioral speculators and not exploited even if short selling is -6-

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not impeded. Our evidence casts doubt on the common presumption underlying both the efficient market hypothesis and the classical argument in the limits-to-arbitrage literature that sophisticated traders always stabilize stock prices as arbitrageurs by trading against mispricing whenever possible. Our study suggests an alternative explanation for overpricing of stocks with high crash probability

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based on the rational speculation theories of DeLong et al. (1990b) and Abreu and Brunnermeier (2002, 2003), and provides evidence consistent with their predictions. Our work is closely related to Edelen, Ince, and Kadlec (2016), who recently document that, based on seven well-known stock return anomalies, institutional investors have a strong propensity to buy stocks classified as overvalued, which means that

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they trade contrary to anomaly prescriptions. Our empirical findings on the crash probability effect are very similar to their results, but our explanation on speculative trading of institutions is different from theirs. While Edelen et al. (2016) suggest a behavioral explanation that agency-induced preferences drive portfolio managers to seek stock characteristics associated with poor long-run performance, we provide

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evidence that institutional speculators are sophisticated and can rationally time bubbles and crashes, which can rule out the behavioral explanation and strongly support the rational speculation argument. All

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the results of this study consistently imply that the abnormally low returns on stocks with high crash probability may be at least partially due to rational speculative bubbles driven by sophisticated traders,

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not entirely to irrational speculative bubbles driven by behavioral traders. The remainder of this paper proceeds as follows. Section 2 describes the data, defines the event of

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price crashes, and presents the empirical framework for predicting the ex ante probability of price crashes. Section 3 shows empirical results on the relation between the predicted probability of price crashes and

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the cross-section of stock returns. Section 4 explores the sources behind the cross-sectional relation and provides empirical evidence supporting the rational speculation argument. Section 5 concludes the paper.

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2. Generalized logit model of price crashes

2.1. Data We obtain the monthly and daily stock files from the Center for Research in Security Prices (CRSP)

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database over the period January 1926 through December 2015. Our sample includes NYSE, Amex, and Nasdaq ordinary common stocks with CRSP share codes 10 and 11. Stocks with an end-of-month price below $5 per share are excluded from the sample. The monthly and daily stock returns are adjusted for any delisting returns provided in the CRSP database. Accounting variables to calculate various firm

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characteristics are obtained from annual Compustat data. We match the annual accounting data for the end of year t − 1 with the monthly CRSP data for June of year t to May of year t + 1. Since the variables calculated using the Compustat data start in 1951, our analysis is restricted to the period June 1952 to December 2015.

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We obtain data on institutional ownership from the Thomson Reuters Institutional (13F) Holdings database. Since data on the quarterly institutional holdings start in the first quarter of 1980, the analysis

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based on institutional holdings data is confined to the period after 1980. We assume that stocks that appear in CRSP but have no reported holdings in 13F are not held by institutions. We also use the

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unadjusted file from the Institutional Brokers’ Estimate System (I/B/E/S) database, which provides data

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on analyst earnings forecasts starting in January 1976.

2.2. Definition of price crashes

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We call the event that a stock price bubble bursts a price crash. In other words, we refer to the low-

probability event of large negative returns as a price crash throughout the paper. We specifically define the event of price crashes in an analogous way to Conrad et al. (2014) defining jackpot returns. A price crash is defined as the event of log returns less than −70% over the next 12 months. The cutoff of −70% approximately corresponds to a capital loss of 50% if the dividend yield is negligible. Symmetrically, we

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define a jackpot as achieving log returns greater than 70% over the next 12 months, which approximately corresponds to a capital gain of 100%.7 The difference between our study and that of Conrad et al. (2014) is that we mainly consider the probability of the left tail, not the right tail, of a return distribution, even though both studies investigate

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the effect of expected extreme returns on subsequent stock returns. We focus on the possibility of the event of extremely low returns for the following reasons. First, it has been documented that various crosssectional anomalies arise largely from the underperformance of stocks classified as relatively overvalued, rather than from the outperformance of stocks classified as relatively undervalued (Nagel, 2005;

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Stambaugh et al., 2012; Avramov et al., 2013), which is consistent with the prevalence of overpricing in the stock market. This implies that firm characteristics related to the degree of overpricing of individual stocks are likely to predict future stock returns in the cross-section. On the other hand, overpricing is not directly observed, and thus it can be determined only after a large negative return is observed in the future.

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This means that a stock is highly likely to be currently overpriced if it has a high ex ante probability of extremely low returns. We therefore argue that the ex ante probability of price crashes can measure the

section of future returns.

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degree of potential overpricing of individual stocks, and moreover, it should be related to the cross-

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Second, a stylized fact is that the largest movements in the stock market are often declines rather than rises. For example, during the period June 1952 to December 2015, seven out of the ten largest daily

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returns in the CRSP value-weighted index were negative.8 Moreover, an extensive literature documents that aggregate stock market returns exhibit negative skewness, or asymmetric volatility, that is, a higher

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volatility associated with negative returns (French, Schwert, and Stambaugh, 1987; Nelson, 1991; Campbell and Hentschel, 1992; Engle and Ng, 1993; Glosten, Jagannathan, and Runkle, 1993; Bekaert

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Although Conrad et al. (2014) define jackpot returns as annual log returns greater than 100%, they show that the jackpot effect they find is robust to different cutoffs. 8 Of the ten largest daily movements, three occurred in October 1987 and the remaining seven occurred in October 2008, both of which correspond to notorious periods of stock market declines. Moreover, the three positive returns were due to the recovery following large negative shocks, rather than independent price increases.

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and Wu, 2000). This asymmetry suggests that the event of extremely negative returns, rare but more frequent than jackpots, could affect stock prices significantly but differently, because investors could be more concerned about negative shocks than positive shocks.9 For these reasons, we investigate the predictive ability of the probability of price crashes for the cross-section of stock returns in this paper and

anomalies documented in previous studies.

2.3. Prediction of price crashes with a generalized logit model

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whether the relation between the crash probability and future returns is distinguished from various

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We employ a generalized logit model to estimate the ex ante probability of extreme negative returns, or crash probability, extending the binary logit model of Campbell et al. (2008) and Conrad et al. (2014). We basically follow the method of Conrad et al. (2014) to predict price crashes but regard future returns of individual firms as a ternary event of extreme negative returns, extreme positive returns, or non-

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extreme returns, rather than considering each extreme payoff as a binary event. Since highly volatile stocks are likely to have high probabilities of both crashes and jackpots, the two probabilities may be

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positively correlated, which makes it difficult to identify the effects of crashes and jackpots separately. A real problem could arise if we applied binary logit models to each extreme event and estimated each of

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the probabilities of crashes and jackpots independently, since the sum of the predicted probabilities of crashes and jackpots could exceed one for highly volatile stocks despite mutual exclusiveness of the two

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events. To impose a restriction that the sum of the two probabilities cannot exceed one, and moreover, to distinguish the effect of crash probability on stock returns from both the jackpot and volatility effects, we

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define price crashes and jackpots as exclusive but mutually dependent, and jointly predict the probabilities of the two events through a generalized logit model.

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Although individual stock returns are known to be positively skewed, the crash event occurs more frequently than the jackpot event does in our sample. Among the total 1,443,258 firm-month observations with non-missing data for all variables during the period June 1952 to December 2014, the number of firm-months that realize a crash (jackpot) over the next 12 months is 90,568 (64,094). - 10 -

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More specifically, we model the probabilities of crashes and jackpots over the next 12 months with the following distribution:

Prt (Yi ,t ,t 12  1) 

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1  exp( 1   1 X i ,t )  exp(1  1 X i ,t ) exp(1  1 X i ,t )

,

(1)

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Prt (Yi ,t ,t 12  1) 

exp( 1   1 X i ,t )

1  exp( 1   1 X i ,t )  exp(1  1 X i ,t )

where Yi,t,t+12 is a ternary variable that equals −1 if firm i’s log return during months t + 1 to t + 12 is less than −70%, one if the same return is greater than 70%, and zero otherwise. Xi,t denotes a vector of

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explanatory variables known at the end of month t. Eq. (1) indicates that an increase in the value of α−1 + β−1Xi,t or α1 + β1Xi,t predicts a higher probability of a crash or jackpot, respectively, over the next 12 months.

We choose explanatory variables to predict extreme negative and extreme positive returns, following

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previous empirical studies on skewness. They include the past market return (RM12), past individual

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return in excess of the market return (EXRET12), total volatility (TVOL), total skewness (TSKEW), firm size (SIZE), detrended turnover (DTURN), firm age (AGE), tangible assets (TANG), and sales growth

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(SALESG). The first six variables are employed following Chen, Hong, and Stein (2001) and Boyer, Mitton, and Vorkink (2010). Chen et al. (2001) forecast skewness based on cross-sectional regression

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specifications and find that an increase in trading volume and positive past returns predict negative skewness. We decompose individual stock returns over the past 12 months into the market return, defined

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as the CRSP value-weighted return, and the excess return relative to the market return, since the marketwide and idiosyncratic variations in past returns may differently affect future extreme returns. Boyer et al. (2010) document that firm characteristics such as firm size and idiosyncratic volatility are also important predictors of future idiosyncratic skewness. In addition to these six variables, Conrad et al. (2014) employ three new variables to predict future jackpot returns and find that young and rapidly growing firms with fewer tangible assets are likely to experience extremely high returns. Based on their findings, we use the - 11 -

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same variables to predict crashes and jackpots, since we assume that the variables related to skewness and extreme outcomes could forecast extreme returns in both directions with either the same or opposite signs. Annual accounting data such as tangible assets and sales growth at year-end are matched with monthly observations from June of the next year to ensure all information is observable at the beginning of the 12-

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month period over which a crash or jackpot is measured. Further details on these variables are provided in the Appendix.

Table 1 shows the in-sample estimation results of our generalized logit model. To account for the correlation between observations across both firms and time, we cluster standard errors by firm and

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month following Petersen (2009) and Thompson (2011). The coefficients on all the predictors are statistically significant at the 5% significance level, with the exception of DTURN as a predictor of crashes. For the past market return (RM12), past excess return (EXRET12), and firm size (SIZE), the estimated coefficients for crashes and jackpots have signs opposite to each other. Price crashes are more

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likely to occur as the market return increases but the excess return decreases and firm size increases, whereas jackpots become more probable as the market return declines but the excess return increases and

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firm size decreases. For the other predictors, the signs of the estimated parameters for crashes and jackpots are identical, which implies that stocks with high crash probability also tend to have high jackpot

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probability. The result indicates that stocks with higher volatility, higher skewness, lower detrended turnover, lower age, fewer tangible assets, and higher sales growth tend to have higher probabilities of

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future extreme returns in both directions. We also report the changes in odds ratios for a one standard deviation change in each variable, where the odds ratio of each extreme event (crash or jackpot) is defined

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as the probability of the event divided by the probability of a non-extreme event. The result shows that total volatility (TVOL) has the largest impact on the odds ratios of both crashes and jackpots: A one standard deviation increase in TVOL increases the odds ratio of crashes by 90.98% and that of jackpots by 56.61%. For crashes, the next most influential variables are AGE, RM12, TANG, and SALESG. For jackpots, SIZE and AGE are relatively important predictors, which is consistent with the results of Conrad

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et al. (2014). The pseudo-R2 value of our generalized logit model, proposed by Nagelkerke (1991), is 13.4%.

2.4. Evaluating predictive accuracy

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The results that many of the explanatory variables predict both crashes and jackpots with the same direction in-sample can raise the concern that our logit framework may fail to disentangle the crash probability from the jackpot probability and that the two predicted probabilities may be proxies for future volatility at best. To test whether our generalized logit model predicts future price crashes reliably out-of-

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sample, we re-estimate the model using only historically available data for every month recursively, over expanding estimation windows, starting from June 1952. We then calculate the out-of-sample predicted probability of price crashes in month t with a set of estimated parameters from the window ending in month t − 12 to ensure that the out-of-sample predictors are not observed before the period over which

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future extreme returns are measured. To ensure that the crash probabilities are predicted using a sufficient number of observations, the predicted probabilities before November 1971 are discarded.

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Based on the out-of-sample forecasts for crash probability over the period November 1971 to December 2014, we compute the accuracy ratio proposed by Moody’s, following the definition used by

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Vassalou and Xing (2004), to evaluate the ability of the model to predict actual crashes over the next 12 months. 10 The out-of-sample predicted crash probability from our generalized logit model has an

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accuracy ratio of 0.518, which is slightly lower than but comparable to the number obtained by Vassalou and Xing (2004). Since the recursive estimation of our generalized logit model also yields the out-of-

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sample predicted probability of jackpot returns, we can calculate the accuracy ratio for the jackpot or nonextreme event as well. The accuracy ratios for the jackpot and non-extreme event are 0.371 and 0.402, respectively. On the other hand, if our logit model failed to disentangle crash risk from jackpot risk and the predicted crash probability measured future volatility, the crash probability would predict actual 10

Under the definition of Vassalou and Xing (2004), a perfect model has an accuracy ratio of one and a completely uninformative model has an accuracy ratio of zero. - 13 -

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crashes and jackpots with equal accuracy. To demonstrate whether this is indeed the case, we also examine whether the crash probability predicts actual jackpots over the next 12 months and find that the corresponding accuracy ratio is only 0.273, which is much lower than the accuracy ratio for actual crashes of 0.518. The numbers we obtain indicate that the crash probability predicted from our generalized logit

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model contains considerable information for actual crashes but not for actual jackpots. Therefore, we confirm that our crash probability effectively measures left-tail risk rather than volatility.

Another way to determine whether our logit model isolates left-tail events from right-tail events is to compare the cross-sectional relations to stock returns of the crash and jackpot probabilities predicted from

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the same model. If our logit model failed to isolate crash probability from jackpot probability or volatility, the cross-sectional effect of the crash probability on stock returns could not be distinguished from that of the jackpot probability, and eventually, these cross-sectional effects would indicate the relation between volatility and stock returns. To address this issue, we compare the cross-sectional relation of each of the

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two probabilities and stock returns, and provide evidence showing the differences between them in

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Section 3.3.

2.5. Event study

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This section examines whether our generalized logit model indeed predicts future price crashes in a different way. If our model accurately forecasts price crashes, then the predicted probability of crashes

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would be elevated as a price bubble grows and it would peak when the bubble is on the verge of bursting. We therefore investigate the response of stock prices around the peak of the predicted crash probability by

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employing an event study. Specifically, we define the event as the new entry of a stock into the top 10% of all sample stocks when ranked by the ex ante probability of crashes at the end of each month. To avoid look-ahead bias, we use the out-of-sample predicted probability of price crashes, calculated using only historically available data for every month as described in Section 2.4, and call it the crash probability (CRASHP) hereafter. The end of the month in which a stock enters into the top 10% is defined as the event day. Then, daily abnormal returns (ARs) and cumulative abnormal returns (CARs) are calculated - 14 -

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over the event window, comprised of 126 pre-event trading days, the event day, and 126 post-event trading days, based on the market model estimated from the window of 126 trading days prior to the event window. Fig. 1 presents the average ARs and CARs over the event window during the period January 1972 to December 2015.

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The result in Fig. 1 shows that the response of stock prices before and after the crash probability peaks accords closely with our prediction. The CAR increases slowly but obviously during a horizon from d − 126 to d − 21, shoots up from d − 21 to the event day d, and then steadily declines during the post-event days. The sharply increasing returns before the crash probability peaks and the sudden reversal after the

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peak could be interpreted as growing price bubbles followed by bursting of them, if we assume that the price rises do not entirely arise from changes in fundamentals but they can originate in part from price bubbles. Although we cannot tell a source behind the price response at this stage, the analysis in this section suggests that the ex ante probability of crashes predicted from our logit model can time the peaks

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and bursts of price bubbles, which implies that it could effectively measure the degree of overpricing of individual stocks. We formally test the predictive ability of the crash probability for the cross-section of

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stock returns and investigate its source in Sections 3 and 4.

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3. Probability of price crashes and the cross-section of stock returns

3.1. Portfolios sorted by the predicted probability of crashes

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In this section, we examine the cross-sectional relation between the probability of price crashes and

subsequent returns. In particular, we construct decile portfolios sorted on the crash probability and observe the returns on each portfolio realized in the subsequent month. We use the out-of-sample predicted crash probability (CRASHP), to avoid look-ahead bias. Based on CRASHP, we form decile portfolios at the end of each month t and calculate the monthly returns on each decile realized in month t + 2. We allow one month of waiting between the portfolio ranking and holding periods to eliminate the - 15 -

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effect of short-term reversals. Since the out-of-sample predicted crash probability starts from November 1971, the portfolio returns begin in January 1972. Table 2 shows the mean monthly excess returns and risk-adjusted returns on decile portfolios sorted by crash probability. Panel A reports the value-weighted portfolio returns and Panel B reports the equal-

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weighted portfolio returns. In Panel A, the mean excess return from decile 1 to decile 8 does not show a monotonic pattern. However, it decreases sharply from decile 8 to decile 10, with the lowest excess return of −0.21% per month in decile 10. This cross-sectional pattern is commonly observed in previous studies (Diether et al., 2002; Ang et al., 2006; Bali et al., 2011; Stambaugh et al., 2012; Conrad et al., 2014),

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indicating that stocks in the highest decile are likely to be considerably overpriced. The return difference between decile 10 and decile 1 is −0.80% per month, with a t-statistic of −2.24. To control for conventional risk factors, we also report the capital asset pricing model (CAPM) alphas, Fama–French (1993) three-factor alphas, and Carhart (1997) four-factor alphas.11 The alphas on the zero-cost portfolio

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buying the highest and selling the lowest CRASHP decile are −1.32% (t-statistic = −4.24) for the CAPM, −0.96% (t-statistic = −4.76) for the three-factor model, and −0.60% (t-statistic = −3.20) for the four-factor

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model, all statistically significant. It is worth highlighting that these t-statistics in magnitude exceed a cutoff of 3.0 recommended by Harvey, Liu, and Zhu (2016), indicating that the cross-sectional difference

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in alphas shown in Table 2 is significant enough to reject the null hypothesis of no difference under a multiple testing framework. These results show that the cross-sectional difference in returns between the

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highest and lowest CRASHP portfolios cannot be explained by risk factors, largely due to abnormally low returns on the highest CRASHP decile portfolio. The equal-weighted portfolio returns shown in Panel B

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exhibit very similar patterns, confirming the finding that stocks with high crash probability subsequently earn lower returns. In Panel C of Table 2, we report the factor loadings of the value-weighted excess returns in the

Carhart (1997) four-factor regression for the CRASHP-sorted portfolios. The loadings on all the four 11

We obtain data on the monthly factor portfolio returns for the Fama–French three-factor model and the Carhart four-factor model from Kenneth French’s website, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/. - 16 -

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factors display substantial variations across deciles. The loading on MKT increases almost monotonically from the lowest to the highest CRASHP decile. The loading on SMB increases very clearly from −0.27 in the lowest to 0.83 in the highest CRASHP decile, while the loading on HML shows an apparent pattern of decreasing from 0.18 to −0.91 as the crash probability increases. In particular, the highest CRASHP decile

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loads positively on SMB but negatively on HML, indicating that it largely consists of small and growth stocks. On the other hand, the loading on WML does not change monotonically but tends to decrease from the lowest to the highest CRASHP decile. The highest CRASHP decile has a negative and significant loading of −0.30 on WML, which implies that it is likely to include loser stocks on average.12

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To investigate the economic significance of the cross-sectional relation between CRASHP and future returns, we examine the cumulative profits of buy-and-hold trading strategies for each decile sorted on CRASHP during the period January 1972 to December 2015. Specifically, we calculate the cumulative values in log scale of the value-weighted decile portfolios when a dollar is invested in each portfolio at

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the beginning of the period and held through the period with monthly rebalancing based on one-monthlagged CRASHP. The result is shown in Fig. 2. At the end of December 2015, the cumulative log return of

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the investment in the lowest CRASHP decile (P1) is 486.6%, higher than that of the CRSP value-weighted index (RM), 423.5%. In contrast, the same strategy investing in the highest CRASHP decile (P10) yields a

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cumulative log return of −143.4%, which shows that the highest CRASHP decile severely underperforms during the sample period. These results imply that stocks in the highest CRASHP are significantly

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overpriced.

Meanwhile, the zero-cost portfolio strategy going long the lowest and short the highest CRASHP

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decile yields an annualized mean return of 9.64% and standard deviation of 26.81%. During the same 12

The negative WML loading of the highest CRASHP decile may seem contrary to the finding of Section 2.5 that stocks with high crash probability experience price rises until the prices arrive at the peak of overvaluation, shown in Fig. 1. These results can be attributed to the persistent cross-sectional ranking based on CRASHP. By examining the transition across decile portfolios from month to month, we find that stocks in the highest CRASHP decile have an 86.4% chance of being in the same decile next month. This finding indicates that, although the stock prices inflate prior to the new entry into the highest decile, as shown in Fig. 1, most of the stocks in the highest decile may have entered into the decile in the previous month or earlier and thus have already experienced price declines. Therefore, the highest CRASHP decile may be primarily comprised of past loser stocks, which leads to a negative WML loading. - 17 -

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period, the CRSP value-weighted portfolio has an annualized mean excess return and standard deviation of 6.31% and 15.78%, respectively. The Sharpe ratio of the CRASHP-based strategy is 0.36, slightly lower than but comparable to that of the market, 0.40. Consistent with the nonzero alpha of the CRASHPbased strategy, an ex post optimal combination of the market and CRASHP-based strategy has a Sharpe

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ratio of 0.81, more than double that of the market. For comparison, HML and WML have an ex post optimal Sharpe ratio of 0.67 and 0.73, respectively, when combined with the market.

3.2. Crash probability and firm characteristics

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In the previous section, we find that stocks with a higher predicted probability of crashes earn significantly lower returns, on average, which is not accounted for by risk factors. Since the literature documents that various firm characteristics are related to the cross-section of stock returns, our finding of the cross-sectional relation between crash probability and stock returns could be due to other variables

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closely linked to crash probability. In this section, we examine various characteristics of portfolios sorted on crash probability and investigate whether controlling for these characteristics affects the cross-

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sectional relation between crash probability and stock returns. Table 3 presents firm characteristics and other related variables for decile portfolios sorted by

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CRASHP. These include the out-of-sample predicted jackpot probability (JACKPOTP), failure probability (FAILP), idiosyncratic volatility (IVOL), idiosyncratic skewness (ISKEW), maximum (MAX) and

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minimum (MIN) daily returns in month t, the price per share (PRC), institutional ownership (IO), analyst coverage (COVER), analyst forecast dispersion (DISP), the market beta (BETA), the log of market

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capitalization (SIZE), the book-to-market ratio (BM), the return in month t (REV), the return over month t − 6 to month t − 1 (MOM), illiquidity (ILLIQ), and share turnover (TURN). The definitions of these variables are given in the Appendix. We report the cross-sectional mean of each variable winsorized at the 25th and 75th percentiles for each decile, averaged across months over the period January 1972 to December 2015.

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The first set of characteristics includes variables related to volatility, skewness, or extreme payoffs, all of which are expected to be highly associated with CRASHP. Both JAKCPOTP and FAILP are analogous to CRASHP in that they are the ex ante probability of a rare event predicted by a logit model. As expected, both JACKPOTP and FAILP increase monotonically from the lowest to the highest CRASHP decile. The

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cross-sectional correlations of these variables with CRASHP are 0.73 and 0.33, respectively. The high correlations, combined with the negative relations between each of the two variables and the cross-section of returns documented by Conrad et al. (2014) and Campbell et al. (2008), raise the concern that the cross-sectional relation between CRASHP and future stock returns we document may disappear after

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controlling for the jackpot or failure probability. In addition, idiosyncratic volatility (IVOL) and skewness (ISKEW) increase with CRASHP, which indicates that stocks with a high CRASHP experience highly volatile and skewed returns prior to portfolio formation. The patterns in MAX and MIN across deciles are also clear and consistent with IVOL and ISKEW.

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The next set of variables is considered as measures of limits to arbitrage. It is known in the literature that the price per share (PRC) is inversely related to trading costs, institutional ownership (IO) indicates

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investor sophistication or ease of short selling, and analyst coverage (COVER) and analyst forecast dispersion (DISP) measure information uncertainty (Bhardwaj and Brooks, 1992; Hong, Lim, and Stein,

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2000; Nagel, 2005; Kumar and Lee, 2006; Zhang, 2006). Table 3 shows that stocks with a higher CRASHP tend to have lower prices, lower institutional ownership, lower analyst coverage, and higher

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analyst forecast dispersion. Both PRC and COVER monotonically decrease and DISP monotonically increases from the lowest to the highest CRASHP decile, which indicates strong associations of CRASHP

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with these variables. On the other hand, institutional ownership displays a weaker association with CRASHP than the other variables do, since the decreasing pattern of IO is not monotonic. Nonetheless, the highest CRASHP decile has the lowest institutional ownership. The relations of CRASHP with these variables consistently indicate that stocks with high crash probability have high arbitrage costs and mispricing is thus not likely to be eliminated. The effect of limits to arbitrage on the cross-sectional relation between CRASHP and stock returns is discussed in Section 4.1. - 19 -

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The final set of variables comprises firm characteristics known as determinants of stock returns in the cross-section. The market beta (BETA) increases from the lowest to the highest CRASHP decile. On the other hand, firm size (SIZE) and the book-to-market ratio (BM) clearly decrease, indicating that the highest CRASHP decile consists of small and growth stocks, on average. This result is consistent with that

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of Fama and French (1996), who find a large negative unexplained return on the smallest size and lowest book-to-market portfolio. In particular, the decreasing pattern in BM, the most common measure of valuation, across deciles confirms that CRASHP measures overvaluation of individual stocks. The return over month t (REV) and the past return over months t − 6 to t − 1 (MOM) do not monotonically vary

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across deciles but show a humped shape approximately. These results are consistent with our finding in Section 2.5 that stocks with a higher CRASHP are more likely to experience price rises recently until the crash probability peaks but price declines after the peak.

Meanwhile, Amihud’s (2002) illiquidity

(ILLIQ) shows that stocks become less liquid as the predicted probability of crashes increases. Share

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turnover (TURN) also displays an increasing pattern across deciles. This result is consistent with the idea that a large trading volume causes a price increase since it affects a stock’s visibility and subsequent

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demand (Gervais, Kaniel, and Mingelgrin, 2001).

In sum, the results reported in Table 3 show clear and strong relations of CRASHP with various firm

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characteristics, which indicates that the negative relation of CRASHP with subsequent stock returns could depend largely on such characteristics that determine the cross-section of stock returns. To investigate

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whether the crash probability can predict the cross-section of stock returns even after controlling for the various characteristics, we orthogonalize the crash probability to each of the firm characteristics and

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construct portfolios sorted by the orthogonalized variable. If the crash probability, compared to the various firm characteristics, contains additional information for future returns, the orthogonalized variable should also predict future returns. Specifically, we regress CRASHP on a control variable each month in the cross-section and obtain the residuals. The residual crash probability is highly correlated with CRASHP but not correlated with the control variable. Then, we construct decile portfolios sorted on the residual crash probability at the end of each month t and calculate the monthly value-weighted returns on - 20 -

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each portfolio realized in month t + 2. In Table 4, we report the Fama–French three-factor alphas on decile portfolios sorted by the residual crash probability orthogonal to each of the characteristics. The control variables are the same as those reported in Table 3. Table 4 shows that controlling for various characteristics does not alter the previous finding of the

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crash probability effect. In particular, the risk-adjusted returns tend to decrease from decile 1 to decile 10, with a sharp drop to −0.68% per month in decile 10, after controlling for JACKPOTP. The zero-cost portfolio buying decile 10 and selling decile 1 yields −1.10% (t-statistic = −4.87) per month. In spite of the high correlation between the two variables and the jackpot effect documented by Conrad et al. (2014),

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this evidence shows that the negative relation between CRASHP and future returns in the cross-section is distinguished from the jackpot effect. When controlling for FAILP, we also find a non-monotonic but decreasing pattern across deciles, which leads to a significant difference in alphas of −0.77% (t-statistic = −3.29) between decile 10 and decile 1. This result confirms that the crash probability has distinct

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information, not contained in failure probability, to predict future returns, and therefore the crash effect we find differs from the failure probability puzzle documented by Campbell et al. (2008).

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For the other 15 characteristics, the risk-adjusted returns on the zero-cost portfolios controlling for a characteristic range from −0.37% to −1.08% per month, on average, which is somewhat reduced in

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magnitude for several characteristics in comparison with the results in Table 2 but still statistically significant. For example, the zero-cost portfolio buying decile 10 and selling decile 1 earns −0.37% (t-

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statistic = −2.33) when controlling for IVOL, −0.63% (t-statistic = −3.28) when controlling for MAX, −0.91% (t-statistic = −4.01) when controlling for DISP, −0.65% (t-statistic = −3.08) when controlling for

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SIZE, −0.86% (t-statistic = −4.25) when controlling for BM, and −0.82% (t-statistic = −4.02) when controlling for MOM. When controlling for PRC, the zero-cost portfolio earns −0.45% (t-statistic = −1.71) per month, only marginally significant. The results in Table 4 provide evidence that the crash probability, even orthogonalized to various characteristics, can predict the cross-section of stock returns. In particular, controlling for the variables negatively related to stock returns and positively related to CRASHP does not attenuate the cross-sectional pattern of the CRASHP-sorted portfolios. - 21 -

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3.3. Firm-level cross-sectional regressions In the previous section, we show that the relation between crash probability and stock returns is robust to controlling for each of the various firm characteristics closely associated with the cross-section of both

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crash probability and stock returns. In this section, we use firm-level regression analyses to compare the predictive power of the variables related to volatility or skewness, when simultaneously controlling for the firm characteristics. Table 5 reports the Fama and MacBeth (1973) estimates for the cross-sectional regression coefficients of stock returns in month t on subsets of the firm characteristics in month t − 1.

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Each column of Table 5 represents a different specification of the cross-sectional regression. The control variables common to all the specifications include BETA, SIZE, BM, REV, MOM, ILLIQ, and TURN. In Table 5, Column 1 shows a benchmark model without any volatility-related predictors.13 Column 2 presents the result of the cross-sectional regression on CRASHP, which confirms the negative and

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significant relation between CRASHP and future stock returns, not diminished by controlling simultaneously for the characteristics. The average slope coefficient on CRASHP in Column 2 is −6.72

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with a t-statistic of −6.61. Multiplying the spread in the mean CRASHP between decile 1 and decile 10, reported in Table 3, by the average slope yields a monthly premium of 1.17%, larger in magnitude than

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the return difference reported in Table 2. In addition, it is noteworthy that the t-statistic for the coefficient on CRASHP is far greater than a cutoff of 3.0 suggested by Harvey et al. (2016), which confirms that the

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crash effect we find is highly significant even using a multiple testing framework. To compare the predictive ability of CRASHP to that of other predictors related to volatility or

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skewness, given the common control characteristics, JACKPOTP, FAILP, IVOL, ISKEW, MAX, MIN, and DISP are each included in turn in the cross-sectional regressions. For JACKPOTP, Column 3 shows a negative relation between JACKPOTP and stock returns, as documented in Conrad et al. (2014), but the

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Although it has been well documented in the literature that the illiquidity premium exists in stock markets, ILLIQ is negatively and significantly related to stock returns in our cross-sectional regressions, because we include only stocks with a price of $5 and above in our sample. When stocks with a price below $5 are added to the sample, the coefficient on ILLIQ becomes positive and significant. - 22 -

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coefficient is not statistically significant.14 When included together with CRASHP in Column 4, however, the coefficient on JACKPOTP turns out to be positive, while the negative coefficient on CRASHP is unaffected. This finding indicates that, although the two probabilities share very similar features, only CRASHP has a strong and negative relation to stock returns and the jackpot effect of Conrad et al. (2014)

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is neither strong nor robust to controls for other predictors. Columns 5 and 6 compare CRASHP and FAILP as predictors of returns in the cross-section. The negative relation between CRASHP and returns does not disappear when FAILP is added in Column 6, though the coefficient is slightly reduced compared to Column 2. This result confirms that CRASHP

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contains information on future stock returns differentiated from the information contained in FAILP, despite the high correlation between the two variables. Turning to the results on IVOL shown in Columns 7 and 8, both CRASHP and IVOL retain their predictive power for future returns after controlling for each other, but the coefficients on both variables become smaller in magnitude.

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In Columns 9 and 10, we present the estimation results of the cross-sectional regressions on ISKEW as a variable related to extreme payoffs of stocks. Although stocks with a high CRASHP tend to have a high

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ISKEW, the negative relation between CRASHP and future returns is not affected by ISKEW in our analysis. In addition, we compare the effect of CRASHP with the effects of MAX and MIN, following Bali

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et al. (2011), and the results are shown in Columns 11 and 12. When the common firm characteristics are controlled for, both MAX and MIN negatively predict stock returns without CRASHP in Column 11.

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However, the coefficient on MAX becomes insignificant by adding CRASHP in Column 12. In contrast,

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the negative coefficient on CRASHP is sustained and not influenced by MAX and MIN. Our last volatility-

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Contrary to the findings of Conrad et al. (2014), we find that the jackpot probability predicted from our generalized logit model cannot significantly predict the cross-section of stock returns in both the portfolio- and firmlevel analyses. These different results can be attributed to our generalized logit model that successfully disentangles crashes and jackpots. Since Conrad et al. (2014) predict their jackpot probability by employing the binary logit model, their jackpot probability can be a proxy for the crash probability or volatility, which leads to the negative relation between the jackpot probability and stock returns they document. When we predict jackpot probabilities using the same binary logit model of Conrad et al. (2014), the jackpot probability can significantly predict stock returns with a negative coefficient. However, their jackpot probability becomes insignificant after controlling for our crash probability, while the negative and significant coefficient on our crash probability is not altered by controlling for their jackpot probability. - 23 -

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related variable, DISP, is included in the cross-sectional regressions shown in Columns 13 and 14. The negative CRASHP–return relation is unchanged, though the size of the coefficient shrinks, by controlling for DISP. Finally, Columns 15 to 17 in Table 5 are to emphasize the importance of left-tail risk as a return

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predictor compared to right-tail risk or volatility. In Section 2, we provide evidence that our crash probability measure indicates left-tail risk rather than volatility by evaluating the predictive accuracy of our logit model. To confirm that CRASHP indeed measures left-tail risk, and moreover, to demonstrate that left-tail risk predicts stock returns more strongly than right-tail risk or volatility does, we consider the

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cross-sectional regression models including CRASHP, JACKPOTP, and IVOL simultaneously, shown in Column 15.15 The result shows that JACKPOTP has a positive coefficient, while both CRASHP and IVOL are negatively related to stock returns, which confirms that a negative relation between right-tail risk and returns is not retained after controlling for left-tail risk or volatility. In Column 16, we augment

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the regression with MAX and MIN, the variables related to left- and right-tail events realized recently, respectively, although they are not ex ante estimates for future extreme events.16 The result indicates that,

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not only do the two right-tail variables have positive coefficients, IVOL also loses its predictive power. In

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contrast, both CRASHP and MIN are still negatively and significantly related to future returns. Moreover,

15

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Since idiosyncratic volatility (IVOL) is not an ex ante estimate for future volatility, it could contain clearly different information from crash probability even if crash probability were not isolated from jackpot probability. Nevertheless, we do not construct an ex ante volatility measure and use IVOL of Ang et al. (2006), since we think that the firm-level volatility prediction is beyond the scope of this study. Although Conrad, Dittmar, and Ghysels (2013) document a negative cross-sectional relation between ex ante risk-neutral volatility and subsequent returns, we do not borrow their measure because the relation between physical and risk-neutral volatility is not clear. One justification for using IVOL is that the cross-sectional ranking based on IVOL is quite persistent, which implies that IVOL may contain substantial information for future volatility. When examining the transition across decile portfolios sorted by IVOL from month to month, for each decile portfolio the average probability that a stock in a decile portfolio in one month will be in the same decile again in the subsequent month exceeds 10%, the number it should be if IVOL in one month had no information about IVOL in the next month. In particular, stocks in the highest (lowest) IVOL decile have a 36% (48%) chance of being in the same decile and a 73% (81%) chance of being in deciles 8 to 10 (1 to 3) next month. 16

We find that MAX and MIN provide persistent cross-sectional rankings from month to month, as documented by Bali et al. (2011), which implies that these variables contain information for future extreme events. For MAX, stocks in the highest (lowest) MAX decile have a 25% (37%) chance of being in the same decile and a 58% (67%) chance of being in deciles 8 to 10 (1 to 3) next month. For MIN, stocks in the highest (lowest) MIN decile have a 27% (40%) chance of being in the same decile and a 61% (71%) chance of being in deciles 8 to 10 (1 to 3) next month. - 24 -

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even when we omit the two right-tail variables in Column 17, IVOL does not recover its predictive power but the left-tail variables, CRASHP and MIN, remain highly significant. These results consistently show that the negative relation between CRASHP and stock returns clearly differs from the negative relation between volatility and stock returns, and that crash probability, or left-tail risk, is a more crucial

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determinant of the cross-section of stock returns than volatility or right-tail risk. To sum up, the cross-sectional regressions at the firm level provide strong evidence of a negative cross-sectional relation between crash probability and future stock returns. The predictive ability of CRASHP is maintained even after controlling simultaneously for various related characteristics that also

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predict the cross-section of stock returns. From both the portfolio-level and firm-level evidence, we confirm that the cross-sectional effect of crash probability is accounted for by neither various firm characteristics nor risk factors in the literature.

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3.4. Crash probability effects and other cross-sectional anomalies

In this section, we ask whether the crash probability effect is linked to the cross-sectional anomalies

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examined by Stambaugh et al. (2012, 2015). Stambaugh et al. (2012) examine 11 cross-sectional anomalies known to survive adjustments for risk exposure and provide evidence that these 11 anomalies

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are at least partially caused by sentiment-driven overpricing. Since our finding of the negative relation between crash probability and stock returns is also primarily due to the overpricing of stocks with high

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crash probability, there could be concerns that our finding is only another form of one of numerous anomalies confirmed by the literature. Therefore, we investigate the role of the 11 anomalies on the cross-

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sectional relation between crash probability and future returns, and provide evidence that our finding is an independent phenomenon of these anomalies. On the other hand, Stambaugh et al. (2015) propose a mispricing measure of individual stocks by combining cross-sectional ranks based on each of the 11 anomaly characteristics. We also investigate whether our crash probability, as an ex ante measure of potential overpricing, can still predict future returns after controlling for their mispricing measure.

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Since two of the 11 anomaly variables—the failure probability of Campbell et al. (2008) and the momentum of Jegadeesh and Titman (1993)—are examined in Sections 3.2 and 3.3, we investigate only the remaining nine anomaly variables in this section. These include Ohlson’s (1980) O-score (OSCR); the net stock issues (NSI) of Ritter (1991); the composite equity issues (CEI) of Daniel and Titman (2006);

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the total accruals (TA) of Sloan (1996); the net operating assets (NOA) of Hirshleifer et al. (2004); the gross profitability (GP) of Novy-Marx (2013); the asset growth (AG) of Cooper, Gulen, and Schill (2008); the return on assets (ROA) of Fama and French (2006); and the abnormal capital investment (ACI) of Titman, Wei, and Xie (2004). In addition, we use the mispricing measure (MISP) for individual stocks of

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Stambaugh et al. (2015), obtained from Yu Yuan’s website.17 The definitions of these variables are given in the Appendix.

Table 6 presents the results of firm-level cross-sectional regressions including together the crash probability and each of the nine anomaly variables, when the same set of control characteristics as in

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Table 5— BETA, SIZE, BM, REV, MOM, ILLIQ, and TURN—are included. Excluding OSCR and AG, the anomaly variables have statistically significant predictive power, with signs consistent with those

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documented by previous studies, that is, positive for GP and ROA and negative otherwise. These results imply that the predictive power of these anomaly variables, previously documented in the literature, is

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hardly affected by the crash probability. More importantly, our measure of crash probability has a highly significant and negative coefficient in all the specifications through Columns 1 to 9. Compared to the

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results shown in Table 5, the magnitudes of the coefficients on CRASHP are not affected by the anomaly variables. When controlling for the mispricing measure of Stambaugh et al. (2015) in Column 10, the

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coefficient on the crash probability remains negative and significant, indicating that our measure of overpricing clearly differs from theirs. The results in Table 6 confirm that the negative cross-sectional relation between crash probability and future returns is not altered by controlling for the broad set of

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See http://www.saif.sjtu.edu.cn/facultylist/yyuan/. - 26 -

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anomaly variables examined by Stambaugh et al. (2012, 2015), strengthening the robustness of our findings.

4.1. Crash probability effects and limits to arbitrage

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4. Probability of price crashes and sources of overpricing

We find in Section 3.2 that the crash probability is closely associated with proxies for arbitrage costs

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in the cross-section. In this section, we investigate whether the overpricing of stocks with high crash probability is stronger as limits to arbitrage are higher. We use five variables as measures of limits to arbitrage: firm size (SIZE), the price per share (PRC), illiquidity (ILLIQ), analyst coverage (COVER), and institutional ownership (IO). The first three variables are known to be related to trading costs. Lower

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analyst coverage indicates higher information uncertainty and lower institutional ownership indicates lower investor sophistication and greater short-sale constraints (Bhardwaj and Brooks, 1992; Hong, Lim,

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and Stein, 2000; Nagel, 2005; Kumar and Lee, 2006; Zhang, 2006; Fama and French, 2008). Based on each of the five variables, we classify sample stocks as high and low limits-to-arbitrage groups each

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month and then, within each group, we construct quintile portfolios sorted by CRASHP. The monthly value-weighted portfolio returns on each quintile for each group are measured two months later after

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portfolio formation.

When we classify stocks based on the limits-to-arbitrage variables other than SIZE, we control for

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firm size to disentangle the separate effect of those variables from that of firm size. Specifically, one of the log of PRC, the log of ILLIQ, COVER, and the logit of IO is regressed on SIZE and squared SIZE in the cross-section, and the residuals are denoted by the residual price (RPRC), residual illiquidity (RILLIQ), residual analyst coverage (RCOVER), and residual institutional ownership (RIO), respectively. Then, sample stocks are sorted into terciles based on each of the residual variables each month. We classify stocks included in the highest RILLIQ or the lowest SIZE, RPRC, RCOVER, or RIO tercile as the illiquid, - 27 -

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small, low price, low analyst coverage, or low institutional ownership group, respectively. These groups represent high limits-to-arbitrage groups. Likewise, we classify stocks included in the lowest RILLIQ or the highest SIZE, RPRC, RCOVER, or RIO tercile as the liquid, big, high price, high analyst coverage, or high institutional ownership group, respectively. These represent low limits-to-arbitrage groups. The

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definitions of these residual variables are given in the Appendix. Table 7 presents returns on the zero-cost portfolios buying the highest and selling the lowest CRASHP quintiles for different groups classified by the limits-to-arbitrage variables. For firm size, although the average return on the zero-cost portfolio is negative for both the small and big groups, it is statistically

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significant only for small stocks. For example, the mean returns and four-factor alphas are −0.89% (tstatistic = −3.34) and −1.00% (t-statistic = −5.57) per month for small firms and −0.22% (t-statistic = −0.82) and −0.06% (t-statistic = −0.44) per month for large firms, respectively. These results indicate that the effect of crash probability is clearly observed in small firms but weak in large firms. When we classify

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stocks by price per share, we find similar results that the crash probability effect is strong in low-priced stocks but not present in high-priced stocks.

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For the next three limits-to-arbitrage variables, the difference in the zero-cost portfolio returns between the high and low limits-to-arbitrage groups is less clear. When stocks are classified by liquidity,

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the three-factor alpha on the zero-cost portfolio return is statistically significant for both the illiquid and liquid groups and even it is lower in liquid firms than in illiquid firms, contrary to the prediction of the

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classical theories on limits to arbitrage. The four-factor alpha becomes insignificant in both groups. The difference in alphas between the two liquidity groups is not different from zero. For analyst coverage, the

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zero-cost portfolio return, adjusted by the Fama–French three factors, is significantly negative and very similar for both the low- and high-coverage groups, but for high-coverage firms, it becomes insignificant after additionally controlling for the momentum factor. The zero-cost portfolio based on CRASHP earns a more negative four-factor alpha in low-coverage firms than in high-coverage firms, but the difference between the two groups is not statistically significant.

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On the other hand, an interesting result is observed for groups classified by institutional ownership. The risk-adjusted returns on the zero-cost portfolio are negative and significant for both the low- and high-institutional ownership groups. For high-institutional-ownership stocks, the four-factor alpha is −0.40% (t-statistic = −1.81) per month, smaller in magnitude than for low-institutional-ownership stocks

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but still marginally significant. Moreover, the difference in alphas between the low- and high-institutional ownership groups is negative but not statistically different from zero in any case. These are surprising results in comparison to the previous studies on anomalies (Nagel, 2005; Campbell et al., 2008; Conrad et al., 2014; Stambaugh et al., 2015). When focusing on the highest CRASHP quintile, we find clear

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evidence that the overpricing of high CRASHP stocks is also prevalent among stocks with high institutional ownership.18 Our findings are evidently not consistent with the literature that argues that overpricing is persistent among stocks with low institutional ownership but not present among stocks with high institutional ownership.

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The literature predicts that overpricing is inversely related to institutional ownership for two reasons. First, it is more difficult to arbitrage away mispricing as a lower proportion of shares is owned by

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institutions, since individuals are likely to be unsophisticated noise traders and cause prices to deviate from fundamental values, and this noise trader risk deters rational arbitrageurs from exploiting mispricing

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(DeLong et al., 1990a). Second, since the main suppliers of stock loans to short sellers are institutional investors, stocks with lower institutional ownership are more costly to sell short (D’Avolio, 2002; Nagel,

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2005). This prevents the overpricing of stocks owned mainly by individuals from being readily eliminated. The common premise of these two reasons is that sophisticated, professional investors unexceptionally act

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as rational arbitrageurs, which means that they always trade against mispricing unless arbitrage is restricted by frictions in financial markets. This premise leads to the conclusion that overpricing should be less persistent among stocks held mainly by institutions because institutions must take a short position in

18

For the high-institutional-ownership group, we find that the three- and four-factor alphas on the highest CRASHP quintile are −0.59% (t-statistic = −3.20) and −0.41% (t-statistic = −2.44) per month, respectively. For the low-institutional-ownership group, the three- and four-factor alphas on the highest CRASHP quintile are −0.64% (tstatistic = −2.61) and −0.45% (t-statistic = −1.86) per month, respectively. - 29 -

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overpriced stocks whenever they become aware of the overvaluation. However, our findings in Table 7 cannot be explained by this line of literature. An alternative explanation for our empirical results can be found in theories that view sophisticated investors as rational speculators. According to DeLong et al. (1990b), rational traders can buy and bid up

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prices above fundamental values because they anticipate that positive feedback traders will buy the securities at even higher prices in the next period. In Abreu and Brunnermeier (2002, 2003), since a price bubble bursts only if arbitrageurs are able to coordinate their selling strategies due to capital constraints, rational investors, who know that the asset is overvalued but that other arbitrageurs are not likely to trade

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against the bubble yet, maximize their profits by riding the bubble as it grows and leaving the market just prior to the crash. These theories consistently imply that trading against mispricing is not always optimal for rational investors, and moreover, they even contribute to price movements away from fundamentals. We argue that these theories predict that overpricing, or a price bubble, is not necessarily related to

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institutional ownership, which is differentiated from the prediction of the classical argument in the limitsto-arbitrage literature. This is because sophisticated traders may not sell short overpriced stocks even

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though they can, but they may even ride the bubble; hence the overpricing of stocks owned mainly by institutions may not be arbitraged away for a while. Therefore, our findings in this section are more

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consistent with the rational speculation argument than with the classical argument.

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4.2. Crash probability effects across different sub-periods In the previous section, we find evidence that the overpricing of stocks with high crash probability

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still persists even if the stocks are largely owned by institutions. This evidence is definitely inconsistent with the classical argument in the literature, and alternatively, it appears to be consistent with the literature on rational speculation. The rational speculation argument further suggests that the overpricing of high crash probability stocks could be partly driven by rational speculators, not entirely by behavioral, sentiment-driven investors, whereas the classical argument predicts that price deviations from fundamentals are driven solely by noise traders. - 30 -

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In this section, we follow Stambaugh et al. (2012) for further investigation. Stambaugh et al. (2012) document that various cross-sectional anomalies are stronger following periods of high investor sentiment, since the stocks in the short-leg of each anomaly-based strategy, relatively overpriced compared to the stocks in the long-leg, are more overpriced when sentiment is high.19 Their findings indicate that the

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broad set of anomalies examined reflects sentiment-driven overpricing, at least partially. To investigate whether the overpricing of high crash probability stocks is associated with variations in market-wide sentiment, we use the monthly market-wide sentiment index constructed by Baker and Wurgler (2006), obtained from Jeffrey Wurgler’s website.20 Specifically, we classify each month from January 1972 to

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October 2015 as a high-sentiment period if the sentiment index in the previous month is above the median value over the sample period and as a low-sentiment period otherwise.21 We then separately calculate the average value-weighted returns on decile portfolios sorted by CRASHP for the high- and low-sentiment periods. We report the average excess returns adjusted by the Fama–French three-factor model following

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each period in Panel A of Table 8.

During the months following high investor sentiment, the long–short strategy that goes long the

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lowest CRASHP decile and short the highest CRASHP decile yields 1.19% (t-statistic = 4.07) per month, on average, when adjusted for the three risk factors. Moreover, the profit comes from considerably low

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returns on the short-leg, the highest CRASHP decile, of −1.01% (t-statistic = −3.79). On the other hand, a similar pattern is also observed during the months following low investor sentiment, although the slope of

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decreasing returns across deciles is substantially reduced and non-monotonic. During low-sentiment periods, the average return on the long–short strategy is 0.74% (t-statistic = 2.87) per month, less than the

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average return during high-sentiment periods but still statistically and economically significant. As a result, the difference in the long–short strategy returns between high- and low-sentiment periods does not differ statistically from zero. This result is not consistent with that of Stambaugh et al. (2012). In addition, 19

In their work, the highest-performing decile of each anomaly is called the long-leg, and the lowest-performing decile is called the short-leg. 20

See http://people.stern.nyu.edu/jwurgler/. We use the data updated as of March 31, 2016.

21

Since the monthly series of the sentiment index ends September 2015, the sample period is restricted. - 31 -

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the insignificant difference in the returns on the highest CRASHP decile between the two periods shows why our result differs. The stocks in the highest CRASHP decile are more overpriced during highsentiment periods, but they are also overpriced during low-sentiment periods. This finding suggests that, even though variations in market-wide sentiment could partially drive the cross-sectional relation between

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CRASHP and stock returns, the overpricing of high CRASHP stocks could also be substantially driven by other factors as well.

To examine whether the overpricing of high CRASHP stocks is influenced by variations in economic states but not by market-wide sentiment, we further classify each month into two sub-periods based on

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each of the alternative state indicators. We choose three state variables: the National Bureau of Economic Research (NBER) recession indicator; the market excess return, defined as the CRSP value-weighted return in excess of the one-month T-bill rate; and the liquidity innovation of Pastor and Stambaugh (2003). The average risk-adjusted returns on the CRASHP-sorted decile portfolios following expansion and

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recession periods, up- and down-market periods, and high- and low-liquidity periods are reported in Panels B to D of Table 8, respectively. In short, the long–short strategy based on CRASHP earns highly

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significant abnormal returns for any sub-period, and consequently, there is no significant difference in the long–short portfolio returns across states, no matter what state variables are used. When focusing on the

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highest CRASHP decile, we find no evidence that the overpricing of the highest CRASHP decile is pronounced in one particular state. On the one hand, these findings show that the negative cross-sectional

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relation between CRASHP and stock returns is driven neither by hedging demand against changes in economic states nor by investor sentiment. On the other hand, the results in Table 8 show that the

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overpricing of high CRASHP stocks we find is a very strong and robust phenomenon across different subperiods.

Our finding that stocks with high crash probability are considerably overpriced during both high- and

low-sentiment periods is inconsistent with both the predictions based on classical theories on limits to arbitrage and the empirical findings of Stambaugh et al. (2012). Moreover, the results in Table 8, integrated with those in Table 7, suggest that the low average returns of stocks with high crash probability - 32 -

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may not be completely due to sentiment-driven overpricing but partly due to rational speculative bubbles. Meanwhile, our analyses in Sections 4.1 and 4.2 do not provide a direct test of the rational speculation theories in the bubble literature. Although our evidence does not rule out other alternative explanations, the crash probability effect appears to be more consistent with and better explained by the rational

the other various anomalies examined previously.

4.3. Crash probability effects and changes in institutional holdings

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speculation argument than by the classical argument. This differentiates the crash probability effect from

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In the previous section, we suggest an explanation that the overpricing of stocks with high crash probability may be, at least partially, driven by the rational speculation of institutional investors. In this section, to find evidence supporting the rational speculation argument, we examine changes in institutional holdings prior to the formation of portfolios sorted by crash probability. Following Edelen et

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al. (2016), we focus on the number of institutions holding a stock, as well as the percent of shares held by institutions, since the change in the number of institutions is more likely to track new and closed positions,

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whereas the change in institutional ownership includes adjustments to ongoing positions. We also choose the six-quarter period prior to portfolio formation to measure changes in institutional holdings.

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Specifically, at the end of each quarter, the change in the number of institutions (∆INUM) during the prior six quarters is scaled by the average number in the same market capitalization decile as of the

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beginning of the six-quarter period. Stocks are sorted into decile portfolios based on crash probability at the end of each quarter. For each decile, we calculate the time-series and cross-sectional averages of both

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the change in the number of institutions (∆INUM) and the change in institutional ownership (∆IO), plotted in Panel A of Fig. 3. The two measures for changes in institutional holdings mirror each other very closely. The changes in institutional holdings during the prior six quarters monotonically increase as the crash probability increases from decile 1 to decile 9, and they drop from decile 9 to decile 10. This pattern may be related to the persistent cross-sectional ranking based on crash probability. The highest CRASHP decile at each quarter includes many stocks that have been in the same decile from the previous quarter or - 33 -

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earlier. If high crash probability stocks experience an increase in institutional holdings until they enter into the top decile but a decrease in institutional holdings thereafter, the change in institutional holdings can decrease from decile 9 to decile 10 while it increases from decile 1 to decile 9. Although the increasing pattern is not extended to the highest CRASHP decile, the change in institutional holdings is

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undeniably greater for the highest CRASHP decile than for the six lowest CRASHP deciles. Interestingly, we find in Table 3 that stocks with higher crash probability tend to have lower institutional ownership, although the relation is not monotonic. These findings imply that, in spite of the relatively low level of institutional holdings, stocks with higher crash probability are bought more heavily by institutions during

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the recent six quarters.

In Panel B of Fig. 3, we investigate how the level of institutional holdings for a firm changes around the quarter when the firm enters into the highest CRASHP decile. The number of institutions (INUM) is scaled by the average number in the same market capitalization decile each quarter. For firms entering

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into the highest CRASHP decile each quarter, we present the average number of institutions (INUM) and the average institutional ownership (IO) of the firms, in excess of the mean of each measure for all firms

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that have non-missing crash probability that quarter, for the six quarters before and after the entry into the highest decile. The number of institutions (INUM) clearly increases during the six quarters prior to the

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entry into the highest decile, and particularly, it increases more sharply during the latest two quarters. In contrast, the number of institutions decreases steadily during the next four quarters after the firms enter

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into the highest CRASHP decile. Institutional ownership (IO) also displays an increasing pattern during the pre-entry period but peaks two quarters earlier than the entry into the highest decile, remains flat

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during a quarter, drops in the event quarter, and then slows down afterward. Even though the peaks of the two measures do not coincide with each other, both measures of institutional holdings indicate that institutional investors show a strong propensity to buy stocks with high crash probability, which are relatively overpriced, during the six quarters before the stock prices reach the peaks of the bubbles. Since the institutional ownership (IO) represents institutions’ aggregate holdings, our results do not mean that all institutional investors buy overvalued stocks prior to the entry into the highest CRASHP decile. - 34 -

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Moreover, some institutional speculators may sell overvalued stocks in advance when the stocks continue to appreciate, while others may sell them just prior to the crash. Nevertheless, the finding that the number of institutions (INUM) peaks at the quarter when the crash probability reaches the top decile implies that at least some institutions continue to enter into the market and buy overpriced stocks until their prices

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reach a peak. The results in this section support the rational speculation hypothesis as an explanation for the crash probability effect. Our findings indicate that stocks with higher crash probability exhibit greater institutional demand in the cross-section and institutions are net buyers for stocks close to but not yet at

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the peak of overvaluation. These findings imply that sophisticated institutions may drive the prices of currently overpriced stocks even farther away from fundamentals through speculative trading, as predicted by DeLong et al. (1990b) and Abreu and Brunnermeier (2002, 2003). The results throughout Sections 4.1 to 4.3 provide evidence that the underperformance of high crash probability stocks may arise,

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at least partially, from speculative bubbles driven by institutional investors.

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4.4. Rational speculation and institutional performance In the previous section, we find evidence that institutions in aggregate may destabilize asset prices by

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buying overpriced stocks with high crash probability. Our findings support the rational speculation argument against the classical argument, in that institutions do not actively correct mispricing and they

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even speculate on overvalued stocks. However, we cannot yet leave out other explanations. For example, Edelen et al. (2016) find that, based on seven well-known anomalies, institutions trade contrary to

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anomaly prescriptions by buying overvalued stocks, and provide some evidence on a behavioral interpretation that agency conflicts or other behavioral biases may drive institutions to track stock characteristics related to overvaluation. Our goal in this section is to find evidence that institutions involved in speculative trading on high crash probability stocks are sophisticated and thus outperform the others, which strongly supports the rational speculation argument but rejects the behavioral interpretation. If at least some institutions optimally choose to buy overpriced stocks, as argued by Abreu and - 35 -

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Brunnermeier (2002, 2003), they can make the highest profit by selling them just prior to the crash, and therefore, they will outperform the other investors who forgo much of the capital gains by selling them in advance or not riding the bubble. To evaluate institutional performance, following Lewellen’s (2011) approach, we construct portfolios

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mimicking quarterly institutional holdings reported in the 13F database by group. As documented by Lewellen (2011), institutions as a whole do little more than hold the market portfolio and thus their aggregate returns have a nearly perfect correlation with the value-weighted index. Therefore, we classify institutions by the holding-weighted average of crash probability and examine the performance of

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institutions that tilt the most toward high crash probability stocks compared to that of the others. It is worth noting that, if institutional performance depended only on stock-picking ability, institutions buying high crash probability stocks should earn a negative alpha, as expected from the negative relation between crash probability and risk-adjusted returns documented in Section 3. In other words, if institutions earn a

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positive alpha in spite of their portfolios tilted toward high crash probability stocks with negative alphas, it is likely to result from the ability to time the bubbles and crashes of individual stocks.

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Specifically, at the end of each quarter, all institutions are sorted into deciles by the average CRASHP of stocks they hold weighted by the value of each institution’s holdings. Then, institutional holdings are

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aggregated within each group and each group of institutions is considered as one big investor. By construction, institutions in the highest 10% group most likely hold the largest fraction of their portfolios

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invested in stocks with high crash probability. To directly compare the portfolios held by each group, we sort stocks into decile based on CRASHP each quarter and calculate the fraction of the institutional

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portfolio invested in the CRASHP-sorted decile for each institutional group. Table 9 reports the portfolio weight invested by each institutional group in the decile portfolios sorted by CRASHP, averaged across quarters over the period 1980 to 2015. We also report the weight of each CRASHP decile in the market portfolio and in the institutions’ aggregate portfolio for comparison. As expected from the negative relation between firm size and crash probability documented in Section 3.2, the market value weight clearly decreases from the lowest to the highest CRASHP decile. The - 36 -

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aggregate portfolio held by all institutions is very similar to the market portfolio, which indicates that institutions as a whole do not tilt their holdings toward or away from high crash probability stocks, on average. Focusing on institutional groups, institutional investors are successfully classified according to their biases toward high crash probability stocks. While institutions with lower CRASHP of holdings

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underweight higher CRASHP deciles compared to the market portfolio, institutions with higher CRASHP of holdings clearly overweight higher CRASHP deciles. In particular, the portfolio of the highest 10% group appears to deviate most from the market portfolio, with an increasing weight across deciles. For example, the highest CRASHP decile (decile 10), only 2.3% of the market portfolio, accounts for 14.7%

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of the portfolio held by the highest 10% group, and the three highest CRASHP deciles (decile 8 to decile 10) account for 43.7% of the highest 10% group’s portfolio and 10.3% of the market portfolio, respectively. These results indicate that the highest 10% institutions based on CRASHP of holdings are likely to represent a group of institutional investors taking a speculative trading strategy most actively.

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For comparison, when the lowest 90% institutions are also aggregated into one group, their portfolio does not differ much from the market portfolio. We also classify institutions in a more extreme way into the

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highest 1% group and the others based on CRASHP of their holdings. The highest 1% institutions have 49.4% of their portfolio invested in the highest CRASHP decile and 81.0% in the three highest CRASHP

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deciles, indicating that their portfolio is extremely biased toward high crash probability stocks. Turning to an evaluation of performance by each institutional group, we calculate returns of portfolios

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replicating the quarterly holdings of each group of institutions, by weighting returns on each stock in proportion to the value held by each group at the end of the previous quarter. Table 10 reports the average

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quarterly excess returns and CAPM, Fama–French (1993) three-factor, and Carhart (1997) four-factor alphas for institutional groups.22 Focusing on the ten institutional groups classified by CRASHP of holdings, the mean excess return tends to increase from the lowest 10% (group 1) to the highest 10%

22 To get quarterly factor portfolio returns, quarterly returns are compounded from monthly returns for each of the long and short sides of MKT, SMB, HML, and WML, respectively. Then, the long-short portfolio returns are obtained from the difference in quarterly returns between the long and short sides.

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group (group 10). The highest 10% institutions earn an average excess return of 2.80% per quarter, higher than all the other nine groups. When institutions in the nine groups are aggregated into one big group, the lowest 90% group has an average excess return of 2.16% per quarter, 0.64% lower than the highest 10% group, although the difference is not statistically significant.

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Once adjusted for risk factors, the highest 10% institutions earn a three-factor alpha of 0.53% (tstatistic = 1.38) per quarter, higher than that of the other nine groups but neither significant itself nor statistically different from the alpha of the lowest 90% institutions. Controlling for Carhart’s (1997) four factors, the highest 10% institutions have the best performance with a four-factor alpha of 0.58% (t-

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statistic = 3.11) per quarter, compared to the other nine groups with alphas of 0.16% to 0.27%.23 The difference in quarterly four-factor alphas between the highest 10% and the lowest 90% group is 0.39% (tstatistic = 2.21), which shows the outperformance of institutions that tilt the most toward high crash probability stocks. On the other hand, the highest 1% institutions based on CRASHP of holdings earn a

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four-factor alpha of 1.42% (t-statistic = 2.22), and the difference in four-factor alphas between the highest 1% institutions and the others is 1.23% (t-statistic = 1.95) per quarter. Although these institutions are only

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a tiny minority, the results confirm that institutions whose investments are more concentrated on high crash probability stocks perform much better than the other institutions.24

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In sum, we find that institutional investors who actively bet on stocks with high crash probability outperform the other institutions. This finding could be quite surprising if stock-picking skill of

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institutional investors determined their performance, because institutions should have sold overpriced

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stocks to earn a positive alpha. Our finding consequently implies that at least some of the institutional

23 While Lewellen (2011) documents that there is little evidence that any institutional group has stock-picking ability measured by the four-factor alpha, our results in Table 10 show that seven out of the ten groups have a significant and positive four-factor alpha, largely due to the omission of stocks with a price per share below $5 from the sample. 24

The finding that institutions excessively holding high crash probability stocks earn a higher four-factor alpha suggests that stocks that have recently entered into the highest CRASHP decile may have different returns from stocks that have already been in the highest decile for a few periods. We find that stocks that have been in the top decile during the previous quarter have a monthly four-factor alpha of −0.72% (t-statistic = −3.46) while stocks that have entered into the top decile during the recent quarter have a four-factor alpha of −0.20% (t-statistic = −0.90). - 38 -

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speculators may ride price bubbles and successfully avoid price crashes, as predicted by the rational speculation theories. We conclude that the institutional speculators are sophisticated and have skill in rationally timing the correction of overpricing. Our evidence rules out the behavioral explanation but clearly supports the rational speculation argument as a source of overpricing of high crash probability

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stocks.

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5. Conclusion

This study investigates the cross-sectional relation between the predicted probability of price crashes and future stock returns and finds strong evidence that differentiates our results from other cross-sectional anomalies examined extensively in the literature. First, we find that stocks with a high probability of price

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crashes earn substantially low returns on average, which is not accounted for by various firm characteristics known as determinants of the cross-section of stock returns. Second, we find evidence that

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the overpricing of stocks with high crash probability is not arbitraged away even among stocks owned largely by institutions. Moreover, the negative cross-sectional relation between crash probability and

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stock returns is not associated with variations in market-wide sentiment. These findings cast doubt on the role of institutional investors as rational arbitrageurs as assumed in the classical argument in the literature,

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which predicts that overpricing is inversely related to institutional ownership and that mispricing is primarily driven by sentiment of behavioral traders. Third, we find that stocks with higher crash

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probability exhibit greater institutional demand and institutional investors have a tendency to buy stocks close to but not yet at the peak of overvaluation. This evidence suggests that the overpricing of high crash probability stocks could also be driven by institutions, not solely by individual traders. Finally, we find evidence that institutions who most actively speculate on stocks with high crash probability outperform the other institutions, despite a negative alpha of those stocks. This finding implies that institutions are sophisticated and have skill in timing the peak and burst of price bubbles. - 39 -

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Our distinctive findings can be well understood along the line of theories on rational speculation that presumes that sophisticated traders do not always stabilize asset prices but destabilize them by riding a price bubble to maximize profits. We conclude that the underperformance of stocks with high crash probability could result partially from rational speculative bubbles driven by sophisticated traders, not

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entirely from sentiment-driven overpricing by noise traders. The different sources of overpricing seem to enable the crash probability effect to survive after controlling for other related anomaly variables. Our evidence also suggests that the superior predictive power of crash probability could stem from both the

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ability of timing the peak of price bubbles and the ability of picking overpriced stocks.

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Appendix. Variable definitions The definitions of all the variables in this study are provided in alphabetical order. ACI: Abnormal capital investment at the end of June in year t to May in year t + 1 is defined as capital expenditure for year t − 1 divided by the previous three-years’ average of capital expenditure, minus

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one. The capital expenditure (Compustat annual item CAPX) is scaled by sales (item SALE) for the same year.

AG: Asset growth at the end of June in year t to May in year t + 1 is the annual change in total assets (Compustat annual item AT) for year t − 1 divided by total assets for year t − 2.

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AGE: Firm age is the number of years since the firm’s first appearance on the CRSP monthly stock file. BETA: Market beta is estimated from the time-series regression of daily excess returns on market excess returns, defined as the CRSP value-weighted returns in excess of the one-month T-bill rate, in each month. To avoid issues related to nonsynchronous trading, we use the lag and lead of the market

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excess return as well as the current market return. Then, the market beta is defined as the sum of the three slope coefficients on the lag, lead, and current market excess returns. We exclude stocks with

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fewer than 15 daily excess returns a month.

BM: Book-to-market ratio at the end of June in year t to May in year t + 1 is the ratio of book equity for

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the end of year t − 1 to market equity at the end of December in year t − 1. The book equity is defined as stockholders’ equity (Compustat annual item SEQ), plus deferred taxes and investment

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tax credit (item TXDITC) if available, minus the book value of preferred stocks. The book value of preferred stocks is the preferred stock redemption value (item PSTKRV) if available, the liquidating

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value (item PSTKL) if available, or the par value (item PSTK).

CEI: Composite equity issues at the end of June in year t to May in year t + 1 is measured as the log of the ratio of market equity at the end of June in year t to market equity at the end of June in year t − 5, minus the cumulative log return from the end of June in year t − 5 to the end of June in year t. COVER: Analyst coverage is the log of one plus the number of estimates for earnings forecasts (I/B/E/S unadjusted file item NUMEST) for the current fiscal year (item FPI = 1). - 41 -

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CRASHP: Crash probability at the end of month t is the probability of log returns less than −70% from month t + 1 to month t + 12, predicted out-of-sample from the generalized logit model specified in Eq. (1). DISP: Analyst forecast dispersion of Diether, Malloy, and Scherbina (2002) is the standard deviation of

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earnings forecasts (I/B/E/S unadjusted file item STDEV) divided by the absolute value of the mean of earnings forecasts (item MEANEST) for the current fiscal year (item FPI = 1).

DTURN: Detrended turnover is the six-month average of monthly share turnover subtracted by the prior 18-month average.

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EXRET12: Measured for each stock at the end of month t as the log return on the stock from month t − 11 to month t, in excess of the same return on the CRSP value-weighted index (RM12). FAILP: Following Campbell, Hilscher, and Szilagyi (2008), failure probability is defined by FAILPi ,t 

M

where

1 , 1  exp   DISTi ,t 

DISTi ,t  9.164  20.264  NIMTAAVGi ,t  1.416  TLMTAi ,t  7.129  EXRETAVGi ,t

1  NIMTAi,t   NIMTAi,t 1  1   12

  11 NIMTAi ,t 11  ,

1  EXRETi,t   EXRETi,t 1  1   12

  11EXRETi ,t 11  ,

PT

NIMTAAVGi ,t 

ED

1.411 SIGMAi ,t  0.045  RSIZEi ,t  2.132  CASHMTAi ,t  0.075  MBi ,t  0.058  PRICEi ,t ,

CE

EXRETAVGi ,t 

and ϕ = 2−1/3. NIMTA is net income (Compustat quarterly item NIQ) divided by the sum of market

AC

equity (price per share times the number of outstanding shares) and total liabilities (item LTQ), EXRET is the firm’s log return in excess of the log return on Standard & Poor’s (S&P) 500 index, TLMTA is total liabilities divided by the sum of market equity and total liabilities, SIGMA is the annualized three-month rolling sample standard deviation of daily returns, RSIZE is the log ratio of market equity to the total market value of the S&P 500 index, CASHMTA is cash and short-term investments (item CHEQ) divided by the sum of market equity and total liabilities, MB is the market- 42 -

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to-book equity ratio, and PRICE is the log of the price per share. We winsorize each of the explanatory variables at their fifth and 95th percentiles of the pooled distribution of all firm-month observations. GP: Gross profitability at the end of June in year t to May in year t + 1 is measured as total revenue

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(Compustat annual item REVT) minus the cost of goods sold (item COGS) for year t − 1, scaled by total assets (item AT).

ILLIQ: Illiquidity at the end of month t denotes the price impact measure proposed by Amihud (2002), defined as the average of the ratio of the absolute daily return to the dollar trading volume in month t.

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IO: Institutional ownership is the fraction of shares owned by institutions, obtained from the Thomson Reuters Institutional (13F) Holdings database.

ISKEW: Idiosyncratic skewness at the end of month t is defined as the skewness of the residuals from the time-series regression of daily excess returns from month t − 5 to month t on the Fama–French three

M

factors. We exclude stocks with fewer than 50 daily excess returns during the six-month period. IVOL: Idiosyncratic volatility at the end of month t is defined as the standard deviation of the residuals

ED

from the time-series regression of daily excess returns in month t on the Fama–French three factors. We exclude stocks with fewer than 15 daily excess returns each month.

PT

JACKPOTP: Jackpot probability at the end of month t is the probability of log returns greater than 70% from month t + 1 to month t + 12, predicted out-of-sample from the generalized logit model

CE

specified in Eq. (1).

MAX: Maximum daily return within month t. We exclude stocks with fewer than 15 daily returns each

AC

month.

MIN: Negative of the minimum daily return within month t. We exclude stocks with fewer than 15 daily returns each month.

MISP: Mispricing measure for a stock, ranging between zero and 100, is defined as the arithmetic average of its cross-sectional ranking percentile for each of the 11 anomaly variables examined by Stambaugh et al. (2012) at the end of each month. - 43 -

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MOM: Momentum, measured at the end of month t, is defined as the cumulative return from month t − 6 to month t − 1. NOA: Net operating assets at the end of June in year t to May in year t + 1 is defined as operating assets minus operating liabilities for year t − 1, scaled by total assets (Compustat annual item AT) for year t

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− 2. Operating assets is total assets minus cash and short-term investments (item CHE). Operating liabilities is total assets minus debt in current liabilities (item DLC) minus long-term debt (item DLTT) minus minority interest (item MIB) minus preferred stock (item PSTK) minus common equity (item CEQ).

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NSI: Net stock issues at the end of June in year t to May in year t + 1 is measured as the log of the ratio of split-adjusted shares outstanding at the end of year t − 1 to the split-adjusted shares outstanding at the end of year t − 2, where split-adjusted shares outstanding is common shares outstanding (Compustat annual item CSHO) times the adjustment factor (item AJEX). OSCR: Following Ohlson (1980), the O-score is defined by

ED

M

OSCR  1.32  0.407  ASIZE  6.03  TLTA  1.43  WCTA  0.0757  CLCA 1.72  OENEG  2.37  NITA  1.83  FUTL  0.285  INTWO  0.521 CHIN ,

where ASIZE is the log of total assets (Compustat annual item AT) divided by the gross domestic

PT

product (GDP) deflator, TLTA is the ratio of total liabilities (item LT) to total assets, WCTA is the ratio of working capital (item WCAP) to total assets, CLCA is the ratio of current liabilities (item

CE

LCT) to current assets (item ACT), OENEG is one if total liabilities exceed total assets and zero otherwise, NITA is the ratio of net income (item NI) to total assets, FUTL is funds from operations

AC

(item FOPT) divided by total liabilities, INTWO is one if net income has been negative for the last two years and zero otherwise, and CHIN is the difference in net income between the current and previous years divided by the sum of the absolute values of the two values.

RCOVER: Residual analyst coverage is defined as the residuals from the cross-sectional regression of analyst coverage (COVER) on firm size (SIZE) and squared firm size (SIZE2) for each month. REV: Short-term reversal represents the return on a stock over month t. - 44 -

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RM12: Measured at the end of month t as the log return of the CRSP value-weighted index from month t − 11 to month t. RILLIQ: Residual illiquidity is defined as the residuals from the cross-sectional regression of the log of illiquidity (ILLIQ) on firm size (SIZE) and squared firm size (SIZE2) for each month. If the values of

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ILLIQ are below 0.0001, they are replaced with 0.0001. RIO: Residual institutional ownership is defined as the residuals from the cross-sectional regression of the logit of institutional ownership (IO) on firm size (SIZE) and squared firm size (SIZE2) for each month, following Nagel (2005). If the values of IO are below 0.0001 or above 0.9999, they are

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replaced with 0.0001 or 0.9999, respectively.

ROA: Return on assets at the end of month t is defined as income before extraordinary items (Compustat item IBQ) for the most recently announced quarter, divided by one-quarter-lagged total assets (item ATQ).

M

RPRC: Residual price is defined as the residuals from the cross-sectional regression of the log of the per share price (PRC) on firm size (SIZE) and squared firm size (SIZE2) for each month.

ED

SALESG: Sales growth at the end of June in year t to May in year t + 1 is the log difference between sales (Compustat annual item SALE) at the end of year t − 1 and sales at the end of year t − 2.

PT

SIZE: Firm size is defined as the log of the price per share times the number of shares outstanding. TA: Total accruals at the end of June in year t to May in year t + 1 is defined as the annual change for year

CE

t − 1 in non-cash working capital minus depreciation (Compustat annual item DP), scaled by the last two-years’ average of total assets (item AT). Non-cash working capital is non-cash current assets,

AC

measured as current assets (item ACT) minus cash and short-term investments (item CHE), minus current liabilities (item LCT) less debt in current liabilities (item DLC) and income taxes payable (item TXP).

TANG: Tangible assets at the end of June in year t to May in year t + 1 are property, plant, and equipment (Compustat annual item PPEGT) divided by total assets (item AT) for the end of year t − 1. TSKEW: Total skewness at the end of month t is calculated using daily log returns from month t − 5 to - 45 -

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month t. We exclude stocks with fewer than 50 daily log returns during the six-month period. TURN: Turnover is the ratio of the monthly trading volume to the number of shares outstanding. TVOL: Total volatility at the end of month t is the standard deviation of daily log returns from month t − 5

AC

CE

PT

ED

M

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CR IP T

to month t. We exclude stocks with fewer than 50 daily log returns during the six-month period.

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Table 1 Parameter estimates of the prediction model for extreme stock returns.

Crash z-Statistic

Intercept

-4.350

-17.309

RM12

1.084

3.649

EXRET12

-0.089

TVOL

% Change in odds ratio for a σ change

Coefficient

z-Statistic

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Coefficient

Jackpot

% Change in odds ratio for a σ change

-7.627

18.93%

-1.036

-5.054

-15.26%

-2.081

-3.49%

0.199

4.643

8.29%

46.206

25.512

90.98%

32.034

19.021

56.61%

TSKEW

0.047

5.624

5.17%

0.025

2.892

2.73%

SIZE

0.058

3.327

11.99%

-0.144

-9.021

-24.30%

DTURN

-0.004

-0.432

-0.42%

-0.047

-5.013

-5.40%

AGE

-0.027

-14.580

-35.06%

-0.015

-11.577

-21.83%

TANG

-0.468

-9.490

-16.57%

-0.216

-5.095

-8.01%

SALESG

0.339

14.050

12.37%

0.196

7.321

6.97%

AC

CE

PT

ED

-1.781

M

Variable

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This table reports the estimated parameters of the generalized logit model specified in Eq. (1) for predicting extreme stock returns. A crash is defined as log returns less than −70% over the next 12 months and a jackpot is defined as log returns greater than 70% over the next 12 months. The variable RM12 is the log return of the CRSP value-weighted index over the past 12 months; EXRET12 is the log return over the past 12 months in excess of RM12; TVOL and TSKEW are the standard deviation and skewness, respectively, of daily log returns over the past six months; SIZE is the log of market capitalization; DTURN is the detrended turnover, defined by the six-month average of monthly share turnover minus the prior 18-month average; AGE is the number of years since the firm’s first appearance on the CRSP monthly stock file; TANG is tangible assets divided by total assets; and SALESG is the sales growth over the prior year. All explanatory variables are constructed to be observable at the beginning of the 12-month period over which a crash or jackpot is measured. The z-statistics are calculated based on standard errors clustered by firm and month. The sample includes all available firm-month observations during the period June 1952 to December 2015.

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R2 0.134

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Table 2 Returns on portfolios sorted by the out-of-sample predicted probability of price crashes.

1

2

3

4

5

8

9

10

10−1

0.45 (1.58) -0.22 (-2.08) -0.06 (-0.64) -0.07 (-0.72)

0.50 (1.66) -0.23 (-1.55) -0.07 (-0.61) 0.07 (0.62)

0.36 (1.01) -0.43 (-2.25) -0.21 (-1.49) -0.11 (-0.86)

-0.21 (-0.47) -1.12 (-4.17) -0.81 (-4.55) -0.52 (-3.17)

-0.80 (-2.24) -1.32 (-4.24) -0.96 (-4.76) -0.60 (-3.20)

0.90 (3.43) 0.27 (1.73) 0.14 (2.09) 0.19 (2.64)

0.76 (2.59) 0.07 (0.45) -0.01 (-0.16) 0.09 (1.35)

0.52 (1.55) -0.23 (-1.30) -0.26 (-3.22) -0.12 (-1.48)

-0.09 (-0.23) -0.94 (-4.20) -0.87 (-7.51) -0.56 (-4.99)

-0.81 (-2.72) -1.27 (-4.85) -1.01 (-6.98) -0.67 (-4.62)

6

7

0.65 (2.61) 0.03 (0.38) 0.15 (1.93) 0.21 (2.30) 0.88 (3.61) 0.30 (2.12) 0.13 (1.88) 0.17 (2.58)

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Panel A: Value-weighted portfolios sorted by CRASHP 0.59 0.47 0.54 0.61 0.67 Excess return (3.58) (2.48) (2.65) (2.64) (2.75) CAPM α 0.20 0.00 0.06 0.06 0.09 (3.35) (-0.02) (0.74) (0.89) (1.27) 0.15 -0.05 0.03 0.07 0.15 Three-factor α (3.07) (-0.89) (0.48) (1.09) (2.14) 0.08 -0.07 -0.01 0.01 0.11 Four-factor α (1.61) (-1.23) (-0.21) (0.12) (1.67)

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This table reports the returns on decile portfolios sorted by the predicted probability of price crashes. At the end of each month t, decile portfolios are constructed by sorting stocks based on crash probability (CRASHP). The monthly portfolio returns on each decile realized in month t + 2 are calculated and the mean excess returns, CAPM alphas, Fama–French three-factor alphas, and Carhart four-factor alphas are reported. Column 1 shows the results for the lowest CRASHP decile, Column 10 is for the highest CRASHP decile, and Column 10−1 is for the zero-cost portfolio buying the highest and selling the lowest CRASHP decile. Panels A and B report the value- and equalweighted returns, respectively, and Panel C reports the factor loadings of the value-weighted excess returns in the Carhart four-factor regression. The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to December 2015.

Panel B: Equal-weighted portfolios sorted by CRASHP

Three-factor α Four-factor α

0.78 (4.14) 0.34 (2.84) 0.11 (1.73) 0.14 (2.44)

0.85 (4.17) 0.39 (2.97) 0.17 (2.41) 0.19 (3.04)

0.91 (4.24) 0.40 (3.04) 0.19 (3.05) 0.21 (4.25)

0.95 (4.07) 0.41 (2.87) 0.22 (3.68) 0.23 (3.95)

M

CAPM α

0.73 (4.33) 0.33 (3.39) 0.13 (1.94) 0.11 (1.77)

ED

Excess return

AC

CE

PT

Panel C: Loadings on the four factors of value-weighted portfolios sorted by CRASHP 0.84 0.95 0.97 1.06 1.05 1.07 1.15 1.19 1.23 1.34 0.50 MKT (39.81) (66.52) (50.08) (45.07) (66.55) (43.76) (30.12) (48.35) (24.57) (20.31) (6.28) SMB -0.27 -0.14 -0.09 0.02 0.12 0.20 0.27 0.45 0.70 0.83 1.09 (-14.66) (-4.38) (-2.35) (0.50) (5.19) (6.11) (6.18) (10.01) (9.57) (10.63) (13.13) 0.18 0.13 0.09 -0.02 -0.14 -0.31 -0.39 -0.48 -0.63 -0.91 -1.09 HML (4.74) (3.44) (1.24) (-0.42) (-4.01) (-10.17) (-8.17) (-8.94) (-8.68) (-7.28) (-7.27) 0.07 0.02 0.05 0.07 0.04 -0.06 0.01 -0.14 -0.11 -0.30 -0.37 WML (3.27) (0.99) (1.35) (4.05) (1.40) (-1.73) (0.19) (-3.32) (-2.14) (-3.90) (-4.41)

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Table 3 Characteristics of portfolios sorted by crash probability.

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This table reports various firm characteristics for each decile portfolio sorted by the predicted probability of price crashes. At the end of each month t, decile portfolios are constructed by sorting stocks based on crash probability (CRASHP). Then, we calculate the 25% winsorized means of various characteristics for each decile in each month and report their time-series averages. The firm characteristic variables are the crash probability (CRASHP), jackpot probability (JACKPOTP), failure probability (FAILP), idiosyncratic volatility (IVOL), idiosyncratic skewness (ISKEW), the maximum daily return in month t (MAX), the minimum daily return in month t (MIN), the price per share (PRC), institutional ownership (IO), analyst coverage (COVER), analyst forecast dispersion (DISP), the market beta (BETA), the log of market capitalization (SIZE), the book-to-market ratio (BM), the return in month t (REV), the return over months t − 6 to t − 1 (MOM), illiquidity (ILLIQ), and share turnover (TURN). The sample period is January 1972 to December 2015 for all the variables but FAILP, which begins August 1972; IO, which begins March 1980; and COVER and DISP, which begin January 1976. Ranking on CRASHP 2

3

4

5

6

0.014 0.019 0.031 1.327 0.259 3.472 3.079 27.464 0.487 2.123 0.047 0.692 6.256 0.851 0.963 6.510 0.079 0.058

0.021 0.024 0.032 1.494 0.305 3.864 3.398 24.564 0.489 1.984 0.050 0.714 5.872 0.820 1.030 7.218 0.143 0.061

0.028 0.029 0.033 1.637 0.328 4.228 3.676 22.959 0.495 1.914 0.050 0.761 5.660 0.799 1.105 7.895 0.211 0.067

0.036 0.034 0.034 1.785 0.356 4.617 3.970 21.706 0.496 1.876 0.050 0.821 5.538 0.752 1.225 8.545 0.257 0.075

0.046 0.039 0.037 1.953 0.382 5.042 4.315 19.820 0.483 1.830 0.053 0.884 5.392 0.724 1.224 8.914 0.288 0.083

M

ED

PT

7

8

9

10

0.058 0.045 0.040 2.164 0.401 5.601 4.742 17.853 0.466 1.789 0.059 0.975 5.257 0.693 1.275 8.909 0.315 0.093

0.075 0.052 0.045 2.418 0.431 6.265 5.250 16.241 0.447 1.739 0.066 1.068 5.120 0.651 1.235 8.572 0.369 0.107

0.102 0.060 0.054 2.767 0.463 7.183 5.956 14.224 0.416 1.707 0.081 1.191 4.960 0.601 1.076 6.781 0.422 0.123

0.180 0.071 0.090 3.488 0.575 9.112 7.290 11.470 0.353 1.647 0.119 1.336 4.766 0.534 1.119 2.885 0.541 0.163

AN US

1 0.006 0.011 0.030 1.130 0.191 2.973 2.684 33.217 0.498 2.464 0.043 0.666 7.184 0.901 1.008 6.841 0.035 0.061

AC

CE

CRASHP JACKPOTP FAILP (%) IVOL ISKEW MAX MIN PRC IO COVER DISP BETA SIZE BM REV MOM ILLIQ TURN

- 54 -

ACCEPTED MANUSCRIPT

Table 4 Returns on portfolios sorted by crash probability orthogonalized to other characteristics.

FAILP IVOL

MAX

IO COVER

AC

CE

DISP

BM

REV

MOM

ILLIQ TURN

PT

PRC

SIZE

0.43 0.29 0.13 0.07 -0.03 0.07 (2.40) (2.65) (1.66) (0.94) (-0.42) (1.16) 0.07 0.07 0.01 0.04 0.15 0.15 (0.72) (0.94) (0.19) (0.61) (1.85) (1.45) -0.12 0.07 0.05 0.06 0.07 0.09 (-1.22) (0.81) (0.72) (1.18) (1.32) (1.37) 0.07 0.05 0.08 0.12 0.05 0.15 (1.20) (1.01) (1.50) (1.69) (0.76) (2.14) 0.01 0.07 0.04 0.14 -0.01 0.11 (0.13) (1.13) (0.70) (2.42) (-0.18) (1.34) -0.01 0.04 0.13 0.00 0.12 0.01 (-0.15) (0.82) (2.15) (-0.07) (1.79) (0.21) 0.07 -0.08 -0.03 0.02 0.02 0.07 (0.83) (-1.13) (-0.36) (0.24) (0.22) (1.23) 0.10 0.08 0.17 -0.04 0.03 0.26 (1.51) (1.51) (2.72) (-0.55) (0.46) (3.22) 0.06 0.01 -0.05 0.01 0.04 0.09 (0.74) (0.08) (-0.81) (0.09) (0.59) (0.96) 0.14 0.05 0.06 0.00 0.12 0.16 (2.71) (0.75) (0.89) (-0.04) (1.52) (1.72) 0.14 0.05 0.03 -0.06 0.07 0.14 (2.40) (0.74) (0.56) (-0.87) (1.16) (1.66) 0.20 0.12 -0.06 0.08 -0.09 -0.02 (2.73) (1.46) (-0.65) (0.97) (-1.25) (-0.37) 0.09 0.09 0.01 0.06 0.11 0.28 (1.69) (1.60) (0.19) (0.95) (1.63) (2.83) 0.18 0.05 0.02 0.05 0.05 0.15 (3.37) (0.87) (0.28) (0.73) (0.74) (1.92) 0.07 -0.02 0.06 -0.06 0.12 0.23 (0.98) (-0.33) (0.80) (-0.93) (1.90) (2.39) 0.16 -0.01 0.05 0.04 0.11 0.17 (3.28) (-0.23) (0.63) (0.62) (1.54) (2.14) 0.07 0.10 0.07 0.09 0.11 -0.09 (1.11) (1.96) (1.38) (1.26) (1.46) (-1.28)

ED

MIN

BETA

3

M

ISKEW

2

Ranking on residual CRASHP 4 5 6 7

8

9

10

10−1

0.01 (0.18) -0.02 (-0.13) 0.07 (0.93) -0.12 (-1.03) 0.00 (0.03) 0.04 (0.46) 0.15 (1.29) -0.07 (-0.60) 0.04 (0.29) -0.17 (-1.39) -0.11 (-1.01) 0.07 (0.88) -0.07 (-0.61) -0.13 (-1.28) -0.02 (-0.24) -0.09 (-0.87) -0.02 (-0.19)

-0.05 (-0.54) -0.18 (-1.21) -0.01 (-0.12) -0.18 (-1.26) -0.15 (-1.41) -0.06 (-0.52) 0.09 (0.55) -0.09 (-0.56) -0.04 (-0.30) -0.13 (-0.88) -0.30 (-2.21) 0.02 (0.16) -0.27 (-1.89) -0.25 (-2.00) -0.11 (-0.83) -0.16 (-1.08) -0.29 (-2.48)

-0.68 (-4.96) -0.70 (-3.87) -0.48 (-3.35) -0.88 (-5.28) -0.62 (-3.74) -0.55 (-3.17) -0.37 (-1.83) -0.82 (-3.86) -0.70 (-3.51) -0.77 (-3.89) -0.81 (-4.34) -0.45 (-2.63) -0.77 (-4.51) -0.89 (-5.39) -0.75 (-4.43) -0.79 (-4.53) -0.86 (-5.00)

-1.10 (-4.87) -0.77 (-3.29) -0.37 (-2.33) -0.95 (-4.79) -0.63 (-3.28) -0.53 (-2.66) -0.45 (-1.71) -0.92 (-4.06) -0.77 (-3.15) -0.91 (-4.01) -0.96 (-4.49) -0.65 (-3.08) -0.86 (-4.25) -1.08 (-5.62) -0.82 (-4.02) -0.96 (-4.80) -0.94 (-5.12)

AN US

1 Controlling for JACKPOTP

CR IP T

This table reports the risk-adjusted returns on decile portfolios sorted by the residual crash probability orthogonalized to each of various firm characteristics. At the end of each month t, the residual crash probability is obtained from the cross-sectional regression of the predicted probability of price crashes (CRASHP) on each of the firm characteristics, and then, decile portfolios are constructed by sorting stocks based on the residual crash probability. The monthly value-weighted portfolio returns on each decile realized in month t + 2 are calculated and the portfolio returns adjusted for the risk factors of the Fama–French three-factor model are reported. The control characteristics include jackpot probability (JACKPOTP), failure probability (FAILP), idiosyncratic volatility (IVOL), idiosyncratic skewness (ISKEW), the maximum daily return in month t (MAX), the minimum daily return in month t (MIN), the price per share (PRC), institutional ownership (IO), analyst coverage (COVER), analyst forecast dispersion (DISP), the market beta (BETA), the log of market capitalization (SIZE), the book-to-market ratio (BM), the return in month t (REV), the return over months t − 6 to t − 1 (MOM), illiquidity (ILLIQ), and share turnover (TURN). The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to December 2015 for all the control variables but FAILP, which begins October 1972; IO, which begins May 1980; and COVER and DISP, which begin March 1976.

- 55 -

0.01 (0.10) -0.01 (-0.07) 0.07 (0.89) 0.01 (0.11) 0.12 (1.52) -0.01 (-0.09) 0.03 (0.27) -0.13 (-1.85) -0.02 (-0.23) 0.00 (0.03) -0.09 (-0.89) -0.02 (-0.28) -0.14 (-1.46) -0.04 (-0.43) -0.07 (-0.79) -0.02 (-0.22) -0.22 (-2.15)

CR IP T

ACCEPTED MANUSCRIPT

Table 5 Firm-level cross-sectional regressions.

(1) CRASHP

(2)

(3)

(4)

-2.89 (-1.01)

-7.21 (-6.69) 6.67 (2.22)

-6.72 (-6.61)

JACKPOTP

(5)

FAILP

(6)

(7)

-5.64 (-5.73)

-2.68 (-2.74)

-0.24 (-6.51)

MAX

DISP

REV MOM ILLIQ

AC

TURN

0.00 (-0.07) -0.07 (-1.72) 0.15 (2.70) -0.03 (-5.90) 0.01 (4.17) -0.04 (-3.35) -0.75 (-1.79)

0.01 (0.49) -0.06 (-1.60) 0.07 (1.57) -0.03 (-6.42) 0.01 (2.79) -0.04 (-3.42) 0.64 (1.40)

-0.01 (-0.21) -0.08 (-1.90) 0.17 (2.61) -0.03 (-5.88) 0.01 (2.91) -0.04 (-3.72) 0.12 (0.11)

PT

BM

0.02 (0.57) -0.12 (-3.17) 0.07 (1.52) -0.03 (-5.99) 0.01 (4.27) -0.03 (-3.07) 0.95 (1.87)

CE

SIZE

-0.01 (-0.31) -0.05 (-1.40) 0.16 (2.60) -0.03 (-5.85) 0.01 (4.57) -0.04 (-3.80) -0.70 (-1.41)

ED

MIN

0.02 (0.52) -0.13 (-3.32) 0.10 (1.90) -0.03 (-5.99) 0.01 (2.89) -0.03 (-3.11) 0.80 (0.99)

(9)

0.01 (0.31) -0.11 (-3.17) 0.12 (2.19) -0.03 (-5.23) 0.01 (4.54) -0.02 (-2.10) 0.94 (1.92)

(10)

(11)

-6.76 (-6.69)

(12)

(13)

-5.67 (-5.99)

(14)

(15)

(16)

(17)

-5.26 (-4.67)

-6.25 (-6.15) 8.28 (2.74)

-6.25 (-6.13) 8.91 (2.96)

-5.62 (-6.02)

-0.15 (-5.22)

-0.05 (-0.83)

0.02 (0.64)

0.01 (1.14) -0.08 (-6.18)

-0.08 (-7.54)

0.03 (1.14) -0.07 (-1.82) 0.07 (1.50) -0.04 (-8.12) 0.00 (2.51) -0.03 (-2.62) 1.61 (3.01)

0.04 (1.33) -0.14 (-3.84) 0.06 (1.39) -0.04 (-7.34) 0.01 (4.24) -0.02 (-2.18) 1.82 (3.17)

-0.12 (-4.52)

M

ISKEW

BETA

(8)

-5.70 (-6.15)

-2.04 (-2.70)

IVOL

AN US

This table reports the time-series averages of the coefficients from the cross-sectional regressions of stock returns on the lagged values of various characteristics. The variables CRASHP and JACKPOTP are the probabilities of extreme negative and extreme positive returns over the next 12 months, respectively, predicted out-of-sample from the generalized logit model in Eq. (1). The other firm characteristics include failure probability (FAILP), idiosyncratic volatility (IVOL), idiosyncratic skewness (ISKEW), the maximum daily return in month t (MAX), the minimum daily return in month t (MIN), analyst forecast dispersion (DISP), the market beta (BETA), the log of market capitalization (SIZE), the book-to-market ratio (BM), the return in month t (REV), the return over months t − 6 to t − 1 (MOM), illiquidity (ILLIQ), and share turnover (TURN). The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to December 2015 for all control variables but FAILP, which begins September 1972, and DISP, which begins February 1976.

0.02 (0.80) -0.14 (-3.87) 0.07 (1.43) -0.03 (-5.64) 0.01 (4.28) -0.02 (-2.25) 1.56 (2.81)

- 56 -

0.00 (-0.17)

0.01 (0.39)

-0.02 (-2.07) -0.10 (-8.94)

-0.01 (-0.33) -0.06 (-1.41) 0.16 (2.59) -0.03 (-5.84) 0.01 (4.36) -0.04 (-3.80) -0.67 (-1.37)

0.02 (0.55) -0.12 (-3.13) 0.07 (1.51) -0.03 (-6.03) 0.01 (4.07) -0.03 (-3.06) 0.99 (1.95)

0.03 (1.02) -0.11 (-2.94) 0.12 (2.19) -0.04 (-7.22) 0.01 (4.53) -0.02 (-2.22) 1.11 (2.22)

0.00 (0.87) -0.08 (-8.05)

0.04 (1.29) -0.14 (-3.84) 0.07 (1.41) -0.04 (-7.53) 0.01 (4.26) -0.02 (-2.24) 1.77 (3.11)

-0.22 (-2.08) -0.02 (-0.39) -0.08 (-1.97) 0.07 (1.07) -0.02 (-4.89) 0.01 (4.29) -0.12 (-2.53) -0.76 (-1.52)

-0.21 (-1.99) 0.00 (0.02) -0.14 (-3.29) -0.02 (-0.37) -0.03 (-5.33) 0.01 (3.72) -0.09 (-1.92) 0.27 (0.66)

0.02 (0.74) -0.07 (-1.92) 0.07 (1.53) -0.03 (-6.06) 0.00 (2.56) -0.03 (-2.70) 1.34 (2.57)

ACCEPTED MANUSCRIPT

Table 6 Firm-level cross-sectional regressions with anomaly variables.

OSCR NSI

(2) -6.29 (-5.27)

(3) -5.98 (-5.14)

(4) -6.50 (-5.31)

-1.15 (-4.13)

CEI

-0.33 (-3.73)

TA

-0.65 (-2.25)

GP

ROA

PT

ACI MISP

AC

BM

0.02 (0.57) -0.11 (-2.74) 0.07 (1.18) -0.03 (-6.64) 0.01 (3.55) -0.03 (-2.55) 1.02 (1.31)

REV

MOM

ILLIQ

TURN

0.02 (0.54) -0.10 (-2.58) 0.08 (1.30) -0.03 (-6.60) 0.01 (3.57) -0.03 (-2.85) 1.05 (1.32)

CE

SIZE

(7) -6.43 (-5.23)

(8) -6.04 (-5.15)

(9) -6.77 (-5.53)

(10) -4.91 (-5.14)

0.47 (2.71) -0.15 (-1.49)

ED

AG

BETA

(6) -6.65 (-5.43)

-0.42 (-2.92)

M

NOA

(5) -6.73 (-5.52)

AN US

CRASHP

(1) -6.70 (-5.56) -0.02 (-1.14)

CR IP T

This table reports the time-series averages of the coefficients from the cross-sectional regressions of stock returns on the lagged values of various anomaly variables and firm characteristics. The variable CRASHP is the probability of extreme negative returns over the next 12 months, predicted out-of-sample from the generalized logit model in Eq. (1). The anomaly variables include Ohlson’s (1980) O-score (OSCR), the net stock issues (NSI) of Ritter (1991), the composite equity issues (CEI) of Daniel and Titman (2006), the total accruals (TA) of Sloan (1996), the net operating assets (NOA) of Hirshleifer et al. (2004), the gross profitability (GP) of Novy-Marx (2013), the asset growth (AG) of Cooper et al. (2008), the return on assets (ROA) of Fama and French (2006), the abnormal capital investment (ACI) of Titman et al. (2004), and the mispricing measure (MISP) of Stambaugh et al. (2015). The other firm characteristics include the market beta (BETA), the log of market capitalization (SIZE), the book-to-market ratio (BM), the return in month t (REV), the return over months t − 6 to t − 1 (MOM), illiquidity (ILLIQ), and share turnover (TURN). The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to December 2015.

0.02 (0.53) -0.10 (-2.53) 0.07 (1.15) -0.03 (-6.67) 0.01 (3.50) -0.03 (-2.97) 1.26 (1.58)

0.01 (0.44) -0.10 (-2.56) 0.08 (1.26) -0.03 (-6.59) 0.01 (3.54) -0.03 (-2.63) 1.02 (1.28)

9.41 (4.77) -0.08 (-3.65)

0.01 (0.44) -0.10 (-2.59) 0.08 (1.45) -0.03 (-6.66) 0.01 (3.32) -0.03 (-2.54) 1.13 (1.39)

- 57 -

0.01 (0.39) -0.09 (-2.26) 0.12 (1.91) -0.03 (-6.73) 0.01 (3.55) -0.03 (-2.64) 0.98 (1.19)

0.02 (0.53) -0.10 (-2.45) 0.08 (1.30) -0.03 (-6.63) 0.01 (3.60) -0.03 (-2.63) 1.20 (1.50)

0.01 (0.49) -0.11 (-2.72) 0.14 (2.27) -0.03 (-6.83) 0.01 (2.90) -0.03 (-2.72) 1.07 (1.28)

0.02 (0.53) -0.11 (-2.79) 0.07 (1.17) -0.03 (-6.59) 0.01 (3.58) -0.03 (-2.60) 1.17 (1.42)

-0.03 (-8.85) 0.02 (0.73) -0.13 (-3.74) 0.04 (0.67) -0.03 (-5.80) 0.00 (2.25) -0.04 (-3.27) 1.19 (2.12)

CR IP T

ACCEPTED MANUSCRIPT

Table 7 Crash probability effects and limits to arbitrage.

Firm size Diff. -0.67 (-3.61) -0.64 (-3.29) -0.89 (-4.86) -0.94 (-5.35)

Low

High

Diff.

-0.98 0.11 -1.09 (-2.81) (0.38) (-3.67) -1.44 -0.21 -1.24 (-4.74) (-0.76) (-4.49) -1.20 0.16 -1.36 (-5.49) (0.72) (-4.56) -0.77 0.19 -0.95 (-3.75) (1.01) (-3.92)

Liquid

Diff.

Low

-0.39 (-1.51) -0.66 (-2.67) -0.59 (-3.00) -0.27 (-1.26)

-0.72 (-1.97) -1.21 (-3.35) -0.80 (-3.63) -0.36 (-1.65)

0.33 (1.13) 0.56 (1.79) 0.21 (0.76) 0.09 (0.30)

-0.19 (-0.61) -0.70 (-2.54) -0.56 (-2.70) -0.44 (-2.15)

CE AC

Analyst coverage

Illiquid

M

Big -0.22 (-0.82) -0.61 (-2.62) -0.28 (-1.81) -0.06 (-0.44)

PT

-0.89 (-3.34) CAPM α -1.26 (-5.27) Three-factor α -1.17 (-6.35) Four-factor α -1.00 (-5.57)

Liquidity

ED

Small Excess return

Price

AN US

This table reports returns on the zero-cost portfolios sorted by the predicted probability of price crashes for subgroups classified by limits to arbitrage. For firm size (SIZE), at the end of each month t, stocks are sorted into terciles on SIZE, and the smallest and largest terciles are classified as the small and big groups, respectively. For other limits-to-arbitrage variables, the residual price (RPRC), residual illiquidity (RILLIQ), residual analyst coverage (RCOVER), and residual institutional ownership (RIO) are obtained from the cross-sectional regressions of each of the log price per share, the log of illiquidity, analyst coverage, and the logit of institutional ownership on SIZE and squared SIZE. Then, at the end of each month t, stocks are sorted into terciles on each of RPRC, RILLIQ, RCOVER, and RIO. Stocks contained in the lowest RPRC (RILLIQ, RCOVER, or RIO) tercile are classified as the low price (liquid, low analyst coverage, or low institutional ownership) group. Stocks contained in the highest RPRC (RILLIQ, RCOVER, or RIO) tercile are classified as the high price (illiquid, high analyst coverage, or high institutional ownership) group. Then, within each of the ten groups, quintile portfolios are constructed by sorting stocks based on crash probability (CRASHP) at the end of month t. The monthly value-weighted returns on each portfolio realized in month t + 2 are calculated, and the mean excess returns, CAPM alphas, Fama–French three-factor alphas, and Carhart four-factor alphas of the zero-cost portfolios buying the highest and selling the lowest CRASHP quintile for each group are reported. The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to December 2015 for all the limits-to-arbitrage variables but COVER, which begins March 1976; and IO, which begins May 1980.

- 58 -

High

Diff.

-0.46 0.26 (-1.17) (1.00) -1.01 0.31 (-2.67) (1.13) -0.58 0.02 (-2.47) (0.09) -0.21 -0.23 (-0.94) (-0.99)

Institutional ownership Low

High

Diff.

-0.60 (-1.42) -1.27 (-3.36) -0.89 (-3.30) -0.66 (-2.44)

-0.51 (-1.55) -0.96 (-3.15) -0.66 (-2.79) -0.40 (-1.81)

-0.09 (-0.27) -0.31 (-0.91) -0.23 (-0.71) -0.25 (-0.76)

ACCEPTED MANUSCRIPT

Table 8 Crash probability effects across different sub-periods. This table reports the average risk-adjusted returns on decile portfolios sorted by the predicted probability of price crashes following sub-periods classified by several binary state indicators. At the end of each month t, decile portfolios are constructed by sorting stocks based on crash probability (CRASHP) and the monthly value-weighted portfolio returns on each decile realized in month t + 2 are calculated. The average risk-adjusted returns following sub-periods are the estimates of α1 and α2 in the regression rt  rf  1D1,t 1   2 D2,t 1   MKTt   SMBt   HMLt   t ,

2

3

5

6

7

8

9

10

10−1

0.16 (1.69) 0.16 (1.64) 0.00 (-0.01)

0.12 (1.08) 0.18 (1.76) -0.06 (-0.39)

0.01 (0.14) -0.13 (-0.85) 0.14 (0.83)

-0.23 (-1.44) 0.11 (0.85) -0.34 (-1.66)

-0.32 (-1.73) -0.12 (-0.66) -0.19 (-0.80)

-1.01 (-3.79) -0.63 (-2.81) -0.38 (-1.12)

-1.19 (-4.07) -0.74 (-2.87) -0.45 (-1.17)

0.09 (1.22) 0.05 (0.26) 0.05 (0.23)

0.22 (3.13) -0.11 (-0.66) 0.33 (1.86)

0.14 (1.60) 0.26 (1.33) -0.13 (-0.60)

-0.05 (-0.47) 0.03 (0.11) -0.08 (-0.30)

-0.05 (-0.42) -0.21 (-0.60) 0.16 (0.45)

-0.22 (-1.41) -0.38 (-1.25) 0.16 (0.47)

-0.75 (-4.12) -1.07 (-2.00) 0.32 (0.59)

-0.89 (-4.28) -1.28 (-2.12) 0.38 (0.62)

0.16 (1.87) -0.04 (-0.34) 0.20 (1.33)

0.11 (1.29) 0.21 (1.99) -0.10 (-0.80)

0.16 (1.47) 0.14 (1.17) 0.02 (0.14)

-0.04 (-0.35) -0.09 (-0.62) 0.05 (0.30)

-0.16 (-1.26) 0.06 (0.37) -0.22 (-1.13)

-0.33 (-1.92) -0.05 (-0.28) -0.27 (-1.18)

-0.80 (-3.68) -0.82 (-3.29) 0.02 (0.07)

-0.94 (-3.73) -0.99 (-3.41) 0.06 (0.16)

0.11 (1.22) 0.03 (0.33) 0.07 (0.56)

0.11 (1.21) 0.20 (2.18) -0.08 (-0.69)

0.11 (1.15) 0.21 (1.62) -0.10 (-0.66)

-0.09 (-0.80) -0.03 (-0.17) -0.06 (-0.35)

-0.02 (-0.19) -0.12 (-0.71) 0.10 (0.50)

-0.27 (-1.75) -0.14 (-0.58) -0.13 (-0.46)

-0.88 (-4.45) -0.71 (-2.60) -0.17 (-0.54)

-1.03 (-5.01) -0.87 (-2.57) -0.16 (-0.44)

0.04 (0.41) 0.12 (1.34) -0.08 (-0.61)

AC

CE

PT

ED

Panel A: High- vs. low-sentiment periods 0.18 0.09 0.13 High sentiment (2.81) (1.18) (1.53) 0.12 -0.18 -0.06 Low sentiment (1.76) (-2.91) (-0.65) 0.06 0.26 0.19 Difference (0.72) (2.85) (1.62) Panel B: Expansion vs. recession periods Expansion 0.14 -0.04 0.01 (2.68) (-0.82) (0.11) 0.21 -0.06 0.21 Recession (1.51) (-0.30) (1.36) -0.07 0.01 -0.20 Difference (-0.44) (0.07) (-1.25) Panel C: Up- vs. down-market periods 0.14 -0.09 0.03 Up market (2.04) (-1.27) (0.31) Down market 0.17 0.01 0.04 (2.35) (0.17) (0.38) Difference -0.04 -0.10 -0.01 (-0.36) (-0.92) (-0.07) Panel D: High- vs. low-liquidity periods 0.15 -0.06 0.02 High liquidity (2.52) (-0.95) (0.21) 0.15 -0.03 0.05 Low liquidity (1.64) (-0.38) (0.40) Difference 0.00 -0.02 -0.03 (-0.04) (-0.22) (-0.21)

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where D1,t−1 and D2,t−1 are dummy variables indicating each of the two states in month t − 1 and rt is the portfolio return in month t. Panel A shows the results for high- and low-investor sentiment periods classified by the median value of the sentiment index of Baker and Wurgler (2006). Panel B shows the results for expansion and recession periods classified by the NBER recession indicator. Panel C shows the results for up- and down-market periods classified based on the sign of the CRSP value-weighted return in excess of the one-month T-bill rate. Panel D shows the results for high- and low-liquidity periods classified by the sign of the liquidity innovation of Pastor and Stambaugh (2003). The numbers in parentheses are t-statistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is January 1972 to October 2015 in Panel A, January 1972 to September 2014 in Panel B, and January 1972 to December 2015 in Panels C and D.

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Table 9 Portfolio invested by institutions grouped by crash probability of holdings.

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This table reports the portfolio weight invested by institutions grouped by the average crash probability of their holdings. At the end of each quarter, all institutions are sorted into decile or percentile by the average crash probability (CRASHP) of stocks they hold, weighted by the value of each institution’s holdings, and then, institutional holdings are aggregated within each group. The lowest 90% group includes institutions with the average CRASHP of holdings below the 90th percentile of all institutions. Likewise, the highest 1% group includes institutions with the average CRASHP of holdings above the 99th percentile of all institutions and the lowest 99% group includes the others. We sort stocks into decile based on CRASHP each quarter, and calculate the fraction of the institutional portfolio invested by each institutional group in each decile sorted on CRASHP. The weights of each CRASHP decile in the market value-weighted portfolio and institutions’ aggregate portfolio are also calculated. The portfolio weights for each quarter are averaged across quarters. The sample period is the first quarter of 1980 to the fourth quarter of 2015. Ranking on CRASHP 1

2

3

4

5

6

Market

0.374

0.147

0.102

0.078

0.075

Aggregate institutions

0.353

0.148

0.107

0.082

Institutions grouped by CRASHP of holdings:

7

8

9

10

0.066

0.055

0.045

0.035

0.023

0.080

0.069

0.057

0.048

0.035

0.021

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Portfolio weight

0.605

0.185

0.089

0.042

0.031

0.021

0.012

0.008

0.004

0.002

2

0.503

0.180

0.106

0.062

0.052

0.040

0.027

0.018

0.010

0.004

3

0.442

0.175

0.113

0.074

0.064

0.050

0.036

0.025

0.015

0.006

4

0.414

0.163

0.113

0.077

0.070

0.057

0.043

0.033

0.021

0.010

5

0.381

0.155

0.110

0.082

0.077

0.064

0.051

0.039

0.026

0.014

6

0.332

0.148

0.112

0.089

0.087

0.073

0.059

0.048

0.034

0.019

7

0.279

0.135

0.112

0.095

0.095

0.084

0.071

0.059

0.044

0.025

8

0.218

9

0.149

Lowest 90% Highest 1%

ED 0.107

0.100

0.105

0.096

0.086

0.075

0.058

0.035

0.091

0.094

0.096

0.109

0.110

0.109

0.103

0.083

0.057

0.053

0.060

0.071

0.088

0.101

0.125

0.143

0.148

0.147

0.360

0.150

0.108

0.082

0.080

0.068

0.056

0.045

0.032

0.018

0.017

0.011

0.010

0.016

0.025

0.040

0.070

0.104

0.211

0.494

0.354

0.148

0.107

0.082

0.080

0.069

0.057

0.048

0.035

0.021

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Lowest 99%

0.065

0.118

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10 (highest 10%)

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1 (lowest 10%)

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Table 10 Performance of institutions grouped by crash probability of holdings.

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This table reports the performance of institutional investors grouped by the average crash probability of their holdings. At the end of each quarter, all institutions are sorted into decile or percentile by the average crash probability (CRASHP) of stocks they hold, weighted by the value of each institution’s holdings, and then, institutional holdings are aggregated within each group. The lowest 90% group includes institutions with the average CRASHP of holdings below the 90th percentile of all institutions. Likewise, the highest 1% group includes institutions with the average CRASHP of holdings above the 99th percentile of all institutions and the lowest 99% group includes the others. We calculate quarterly portfolio returns replicating the holdings of each group of institutions by weighting returns on each stock in proportion to the value held by each group at the end of previous quarter. The average quarterly excess returns, CAPM alphas, Fama–French three-factor alphas, and Carhart four-factor alphas for institutional groups are shown in percent per quarter. The numbers in parentheses are tstatistics adjusted using Newey–West (1987) standard errors with 12 lags. The sample period is the second quarter of 1980 to the fourth quarter of 2015. Institutions grouped by CRASHP of holdings

CAPM α Three-factor α

3

4

5

6

2.05 (3.91) 0.45 (2.39) 0.34 (2.05) 0.27 (1.68)

2.02 (3.56) 0.31 (2.76) 0.23 (1.56) 0.22 (1.90)

2.16 (3.79) 0.33 (3.33) 0.28 (4.01) 0.23 (3.10)

2.16 (3.68) 0.28 (2.77) 0.28 (3.79) 0.26 (3.05)

2.11 (3.44) 0.15 (2.42) 0.19 (3.57) 0.17 (2.77)

2.14 (3.49) 0.11 (1.40) 0.17 (2.75) 0.16 (2.76)

7

8

9

2.28 (3.57) 0.16 (2.28) 0.29 (4.36) 0.22 (3.29)

2.44 (3.79) 0.16 (1.06) 0.36 (2.57) 0.24 (2.43)

2.36 (3.40) -0.15 (-0.72) 0.20 (1.12) 0.16 (1.18)

AC

CE

PT

ED

Four-factor α

2

M

Excess return

Lowest 10%

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Highest Lowest Highest 10% (A) 90% (B) 1% (C) 2.80 (3.50) -0.12 (-0.30) 0.53 (1.38) 0.58 (3.11)

2.16 (3.51) 0.15 (2.48) 0.23 (4.83) 0.20 (3.75)

2.78 (2.38) -0.95 (-0.88) 0.17 (0.23) 1.42 (2.22)

Lowest 99% (D) 2.16 (3.51) 0.13 (2.08) 0.23 (4.63) 0.19 (3.67)

A−B

C−D

0.64 0.61 (1.50) (0.62) -0.27 -1.08 (-0.72) (-1.03) 0.30 -0.06 (0.82) (-0.08) 0.39 1.23 (2.21) (1.95)

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Fig. 1. Cumulative abnormal returns around entry into the top decile portfolio sorted by crash probability.

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This figure shows the daily abnormal returns and cumulative abnormal returns around the entry into the top decile portfolio sorted based on the predicted probability of price crashes. At the end of each month, decile portfolios are constructed by sorting stocks based on the crash probability (CRASHP). The event is defined as the entry of a firm into the highest CRASHP decile. We present the abnormal returns (AR, left axis) and cumulative abnormal returns (CAR, right axis) during an event window of 253 trading days, comprised of 126 pre-event days, the event day, and 126 post-event days. We calculate ARs and CARs based on the market model estimated from the window of 126 trading days prior to the event window. The sample period is January 1972 to December 2015.

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Fig. 2. Cumulative profit of trading strategies based on crash probability. This figure shows the cumulative

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log returns of the trading strategies based on the predicted probability of price crashes. At the beginning of each month t from January 1972 to December 2015, decile portfolios are formed by sorting stocks based on crash probability (CRASHP) in month t − 2. Then, the cumulative value in log scale of the value-weighted buy-and-hold strategy on each decile is calculated, with an initial investment of $1 at the beginning of January 1972. P1 and P10 denote the lowest and highest CRASHP deciles, respectively, P1−P10 is the difference between the two portfolios, and RM denotes the CRSP value-weighted portfolio.

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Fig. 3. Crash probability effects and changes in institutional holdings. This figure shows the relation between

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changes in institutional holdings and the predicted probability of price crashes. At the end of each quarter, decile portfolios are constructed by sorting stocks based on crash probability (CRASHP). The number of institutions (INUM) is defined as the number of institutions holding a stock, scaled by the average number of institutions in the same market capitalization decile each quarter (or as of the beginning of the period over which changes in institutional holdings are measured), and institutional ownership (IO) is defined as the fraction of shares owned by institutions each quarter. In Panel A, we plot the time-series and cross-sectional average of changes in institutional holdings during the six quarters prior to portfolio formation for each decile portfolio, where changes in institutional holdings are measured by both changes in the number of institutions (∆INUM, left axis) and changes in institutional ownership (∆IO, right axis). In Panel B, for firms that enter into the highest CRASHP decile each quarter, we present the average number of institutions (INUM, left axis) and the average institutional ownership (IO, right axis) of the firms in excess of the mean of each measure for all firms that have institutional holdings data that quarter, for six quarters before and after the entry into the highest CRASHP decile. We only include firms that have non-missing data for the 13-quarter window around the event. The sample period is the first quarter of 1980 to the fourth quarter of 2015.

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