Subjective Belief, Crash Perception, and Cross-Sectional Stock Return George P. Gaoy Cornell University Zhaogang Songz Federal Reserve Board Liyan Yangx University of Toronto This Draft: June 2015 Abstract Recent studies show investors form incorrect beliefs about stock market crash. We investigate the pricing implications of such biased beliefs for cross-sectional expected stock returns. We construct a stylized model that characterizes stock returns in equilibrium being proportional to investors’ subjective beliefs on stock crash (ex-ante crash perceptions). Such crash perceptions contain a market-driven systematic component (co-crash perception) and a stockspeci…c component (idiosyncratic crash perception). We suggest option-based measures of crash perceptions and …nd they are strongly connected to pessimistic beliefs re‡ected in the survey forecasts of aggregate economy and …rm prospects. Crash perceptions are strongly priced, with average annual Sharpe ratios of 0.9 and 1.1 on long-short hedge portfolios formed on co-crash and idiosyncratic crash perceptions, respectively. Keywords: Belief, crash perception, co-crash, idiosyncratic crash JEL classi…cations: G11; G12; G23 The analysis and conclusions set forth are those of the authors and do not indicate concurrence by the Federal Reserve System. y Samuel Curtis Johnson Graduate School of Management, Cornell University. Email: pg297@cornell.edu; Tel: (607) 255-8729. z Board of Governors of the Federal Reserve System, Mail Stop 165, 20th Street and Constitution Avenue, Washington, DC, 20551. E-mail: Zhaogang.Song@frb.gov. x Rotman School of Management, University of Toronto, Canada. Email: liyan.yang@rotman.utoronto.ca; Tel: (416) 978-3930. 1 Introduction Investors’ subjective beliefs a¤ect asset prices and returns. Prior theoretical studies have shown investors’biased subjective beliefs about mean growth rates of endowment, earnings, or investment can account for many empirical regularities of stock returns and volatilities (Barberis, Shleifer, and Vishny (1998); Daniel, Hirshleifer, and Subrahmanyam (1998); Cecchetti, Lam, and Mark (2000); Scheinkman and Xiong (2003); Brunnermeier, Gollier, and Parker (2007); Alti and Tetlock (2014); Barberis, Greenwood, Jin, and Shleifer (2014)). Intuitively, however, the subjective belief is most likely to be biased about tail (or crash) risks of the economy, as the very infrequent nature and hence the limited sample of crash events make it impossible for investors to learn the objective crash risk (Weitzman (2007)). Such a biased belief on crash events can have important pricing implications given the extreme impact of potential tail events. Several recent studies including Gennaioli, Shleifer, and Vishny (2012, 2013), Baron and Xiong (2014), and Jin (2014) have investigated how incorrect crash risk beliefs a¤ect the aggregate stock market return and …nancial instability. However, little is known about the pricing implications of biased subjective beliefs on crash risk for cross-sectional stock returns. Our paper attempts to …ll in this gap. There are (at least) two potential aspects of subjective beliefs about stock crash (crash perception) for the cross section of expected stock returns. First, incorrect beliefs of crash risk can occur on the aggregate market level. Such biased beliefs will impact stock returns through the e¤ect on the stochastic discount factor. Second, and importantly, incorrect beliefs of crash risk can also happen on individual …rm level, independent of the crash risk belief on the aggregate market. For example, outside investors tend to believe (and overestimate) that …rm managers will withhold or delay bad news for extended periods and stock price will crash when the bad news are …nally released (Jin and Myers (2006); Bleck and Liu (2007); Hutton, Marcus, and Tehranian (2009); Kothari, Shu, and Wysocki (2009); Kim and Zhang (2013)). This biased belief about stock crash, resulting from …rm-level information asymmetry, is likely to be independent of the market crash perception. Moreover, it can have substantial heterogeneity across individual …rms because of their heterogeneous information disclosure and opaqueness. The …rst aspect above — the market-level crash perception — seems to have pricing implication naturally. For the second aspect, however, it remains unclear how, if at all, the …rm-speci…c crash perception is priced in the cross-sectional 1 stock returns. In this paper, we conduct a comprehensive analysis of the pricing implications of biased beliefs about crash risk, at both the aggregate market and individual stock level, for the cross-sectional stock returns. We …rst motivate our study by constructing a stylized model in which a representative agent’s subjective beliefs on both the aggregate market crash risk and individual …rm crash risk are di¤erent from their corresponding objective crash probabilities. In particular, the agent assigns larger probabilities to the disaster events of consumptions and dividends that have almost zero occurring probabilities. The larger probability she puts on the crash event captures the perception of crash risk. Since the representative agent maximizes the utility under her subjective belief, the model shows that in equilibrium individual stock returns are functions of not only the traditional objective aggregate crash risk, but also the following two components of crash perception: (1) a systematic component, dubbed co-crash perception, which captures the covariation between a stock’s crash conception and the market’s crash perception; and (2) a stock-speci…c idiosyncratic component, dubbed idiosyncratic crash perception, which captures the part of stock crash perception that is unexplained by the market. Our main task is then to empirically test how systematic and idiosyncratic components of crash perception a¤ect cross-sectional expected stock returns. However, measuring investors’subjective belief empirically about stock crash is generally hard, if not impossible. A recent work, Greenwood and Shleifer (2014), uses various surveys to extract investor expectations of future stock market returns. An ideal measure of stock crash belief can be constructed should similar surveys on tail probabilities of individual stocks are available. Unfortunately, this type of crash-event surveys is not readily available. Instead, in our empirical exercise, we employ both the individual options on the U.S. common stocks and the S&P 500 index options from January 1996 through December 2012 to construct measures of crash perception at both the individual stock level and the aggregate market level. Our measures of crash perception rely on out-of-the-money (OTM) put options that are most informative about investors’perception of stock crash.1 Building on the model-free implied volatility studies in Britten-Jones and Neuberger (2000), Carr and Wu (2009), and Du and Kapadia (2012), such crash perception measures have volatility risk purged and depends exclusively on the 1 We also construct our measure of perceived crash risk by combining both OTM calls and puts together. Results with this alternative measure are similar. 2 left tail of the jump measure embedded in OTM puts. Speci…cally, this measure is a weighted average of OTM put option prices across moneyness, with higher weights on deeper OTM puts. As a result, this measure is expected to mainly capture the perception of crash risk rather than the objective crash risk given the rare nature of crash events associated with deep OTM puts.2 Admittedly, our crash perception measures based on option prices do not exclude preference (risk premium) components by construction and may be polluted by them. To corroborate these crash perception measures mainly capturing subjective belief on crash risk, we show they are strongly connected to pessimistic beliefs re‡ected in the survey forecasts of future aggregate and …rm-level economic conditions. In particular, we use the economist forecasts on real GDP growth from the Blue Chip Financial Forecasts (BCFF) data and the …nancial analyst forecasts on …rm earnings from the I/B/E/S earnings forecast data. We measure the extent of pessimistic beliefs across forecasters using the forecast skewness as well as lower percentile. At the aggregate market level, we …nd the market crash perception is higher as the economists form more pessimistic (and less optimistic) beliefs on future real output. At the …rm-level, we …nd stocks with higher crash perception are those about which the analysts formed more pessimistic beliefs in terms of the …rms’ future earnings and growth. Moreover, at both market and individual stock levels, the negative relation between crash perception and forecast skewness becomes stronger as the average belief is tilted more to the left tail. Overall, these results uphold our crash perception measures as capturing the subjective belief on stock crash risk strongly. We then conduct our main empirical analysis regarding the pricing implications of crash perception. We show that co-crash perception is strongly priced in the cross section of stock returns. When portfolios are formed monthly, the di¤erence in equal-weighted returns between the portfolios with the highest and lowest co-crash is 1.08% per month (with a Newey-West t-statistic of 2.7). The return spread based on value-weighted portfolios is of similar magnitude, albeit less statistically signi…cant at 1.06% per month (with a t-statistic of 1.7). The return pattern associated with perceived co-crash is strongest at monthly portfolio formation: the signi…cance of return spreads between high and low perceived co-crash stocks becomes dissipated at quarterly and semi-annual formation frequencies. 2 Using option prices to construct skewness measures, Conrad, Dittmar, and Ghysels (2013, p. 86) argue that “options re‡ect a true ex ante measure of expectations.” 3 Idiosyncratic crash perception has an even stronger explanatory power on cross-sectional stock returns. When portfolios are monthly formed, return spreads of high-minus-low perceived idiosyncratic crash portfolios are 0.91% and 1.08%, respectively, when stock returns are equal and value weighted. These spreads are at least three standard errors from zero. Furthermore, in contrast to co-crash, the e¤ect of perceived idiosyncratic crash on future stock returns is not short-lived. In fact, the return spreads between high and low idiosyncratic crash perception stocks at quarterly, semi-annual, and annual portfolio formation frequencies are not only statistically signi…cant but also economically large, ranging from 0.83% (with a t-statistic of 2.3) to 1.45% per month (with a t-statistic of 3.7). Are return variations driven by co-crash and idiosyncratic crash perceptions distinct from each other, as predicted by our stylized model? We perform an analysis of two-way sequentially sorted portfolios. Controlling the perceived co-crash, the return spread of high-minus-low perceived idiosyncratic crash portfolio remains strong and signi…cant, and vice versa. Hence, both co-crash and idiosyncratic crash perceptions are important determinants of cross-sectional stock returns and neither one dominates. Moreover, we show that excess returns of long-short hedge portfolios formed on perceived co-crash and idiosyncratic crash are robust to risk adjustment using a set of traditional risk factors, including the market, size and book-to-market factors (Fama and French (1993)), the momentum factor (Carhart (1997)), the short-term return reversal factor (Jegadeesh (1990)), and the market liquidity risk factor (Pastor and Stambaugh (2003)). For example, the standard Fama-French-Carhart four-factor alphas of these hedge portfolios vary from 0.80% to 0.99% per month. To examine whether the excess return earned on crash perception is merely compensation that investors require for bearing their losses when crash shocks are realized (i.e., risk premium for the objective crash risk implied from ex-post historical crash shocks), we conduct two detailed checks. First, we study whether the returns on crash perception are simply premiums for crash risks associated with macroeconomic downturns and liquidity crunches. We collect a set of market, liquidity, macroeconomic, and disaster risk factors, including the market excess return, PastorStambaugh (2003) and Hu-Pan-Wang (2013) liquidity factors, market volatility, default risk, and term spread. We …nd that these liquidity and macroeconomic risk factors generally cannot explain the return spreads between low and high crash perception portfolios. Instead, the perceived co-crash 4 and idiosyncratic crash long-short portfolios provide hedges against market downturns. Second, we collect a set of variables of (or potentially related to) crash risk based on historically realized crash events of stocks, including the downside market beta (Ang, Chen, and Xing (2006)), tail risk beta (Kelly and Jiang (2014)), co-skewness (Harvey and Siddique (2000)), co-kurtosis (Dittmar (2002)), hybrid tail risk (Bali, Cakici, and Whitelaw (2014)), maximum daily return (Bali, Cakici, and Whitelaw (2011)), idiosyncratic volatility (Ang et al. (2006)), and idiosyncratic skewness (Boyer, Mitton, and Vorkink (2010)).3 Results from Fama-MacBeth (1973) cross-sectional regressions show that none of these variables drives the return variations associated with crash perception. Overall, ex-ante crash perception is distinct from crash risk implied from ex-post crash shocks in a¤ecting cross-sectional stock returns.4 Because our measures of crash perception rely on stock options, we also investigate whether the existing option-based variables in the literature could drive our results. These variables include implied volatility slope (Yan (2011)), volatility spread (Cremers and Weinbaum (2010)), implied volatility smirk (Xing, Zhang, and Zhao (2010)), option-to-stock volume ratio (Johnson and So (2012)), implied volatility innovation (An et al. (2014)), and risk-neutral moments of volatility, skewness, and kurtosis (Conrad, Dittmar, and Ghysels (2013)). Our results are robust to these option-based …rm characteristics. Furthermore, we examine a set of option-based tail risk factors (such as implied volatility skewness and high-order moment risk) and …nd these factors cannot explain our results, reinforcing the unique role of our measure in investigating the pricing of subjective belief about crash risk.5 As discussed above, our option-based crash perception measures are certainly imperfect in capturing subjective beliefs because option price contains both preference and belief elements. Although we cannot completely preclude the preference element, in our regression analysis, cocrash and idiosyncratic crash perception remain both economically and statistically signi…cant in presence of control variables of co-skewness, co-kurtosis, idiosyncratic skewness, and return 3 A recent study by Bollerslev, Todorov, and Li (2013) decompses the market beta into the jump and continuous components, and shows that the former is priced in the cross section of stock returns while the latter is not. 4 One may argue that due to the peso problem, our results only show the pricing of objective crash risk and do not imply the distinction of subjective crash risk. Though plausible, this interpretation is inconsistent with the observation that the return spread of high-minus-low crash perception portfolios is signi…cantly positive during most crisis periods, including the recent …nancial crisis that is fairly extreme in the history of the U.S. economy. 5 The robustness of our results to the funding liquidity variables and implied volatility slope also implies that returns of crash perception portfolios are not merely driven by …nancial intermediary constraints (Bollen and Whaley (2004); Garleanu, Pedersen, and Poteshman (2009); Chen, Joslin, and Ni (2013)). 5 maximum, which are shown in prior studies to capture investor preferences (Harvey and Siddique (2000); Dittmar (2002); Mitton and Vorkink (2007); Barberis and Huang (2008); Boyer, Mitton, and Vorkink (2010); Bali, Cakici, and Whitelaw (2011)). Furthermore, among the cross-section of our sample stocks, we …nd high crash perception forecasts less negative return skewness or downside volatility in the future, using measures of negative coe¢ cient of skewness and down-to-up volatility in Cheng, Hong, and Stein (2001). These evidence, in addition to the signi…cant relation between crash perception and pessimistic survey forecast, corroborate our option-based measures re‡ecting investors’subjective beliefs about stock crash risk to a large extent. Our paper contributes to the recent literature on the biased/distorted belief of crash risk that mainly focuses on the aggregate stock market as discussed above, as well as the disaster risk literature including Barro (2006), Gabaix (2012), Wachter (2013) among others. Moreover, di¤erent from recent studies such as Spitzer (2006), Chollete and Lu (2011), Bali, Cakici, and Whitelaw (2014), Chapman and Gallmeyer (2014), Kelly and Jiang (2014), and Chabi-Yo, Ruenzi, and Weigert (2014) that examine the pricing of objective tail risk for cross-sectional stock returns, we focus on the pricing of the subjective belief about crash risk. Our empirical evidence strongly suggests the importance of incorrect crash risk belief on explaining cross-sectional stock return variation, which is consistent with theoretical predictions in the dynamic general equilibrium model of Jin (2014). The remainder of the paper is organized as follows. Section 2 illustrates the pricing of biased subjective beliefs about crash risk and constructs the measures of crash perception. Section 3 examines relations between crash perception and subjective belief. Section 4 provides our key …ndings that both co-crash and idiosyncratic crash perceptions are priced in the cross section of U.S. stock returns. Section 5 provides robustness checks. Section 6 concludes. 2 Ex-ante Crash Perception: Motivation and Measure In this section, we …rst present a parsimonious theoretical approach to formalize the idea of subjective beliefs about crash risk (crash perception). Then we discuss how to use options to measure crash perception. 6 2.1 A theoretical perspective of crash perception The main contribution of our paper is empirical. Hence, the model we present in the following is very stylized and is used only as a guidance to construct variables and design empirical strategies later. The essence of the model is to demonstrate how both systematic and idiosyncratic components of crash perception can be priced in equilibrium. Environment We consider a pure exchange economy with two dates (t = 0; 1). There is one consumption good and one representative agent. At date 0, the representative agent chooses consumption plans and asset allocations, subject to the standard budget constraint, to maximize ! 1 ~1 C C 0 b ; E + 1 1 1 where > 0 is the relative risk aversion coe¢ cient, and C0 > 0 and C~1 > 0 are the agent’s b ( ) refers to an expectation operator consumption at dates 0 and 1, respectively. The operator E with respect to the agent’s subjective beliefs (perceptions), which will be speci…ed shortly. The tradable assets include a risk-free asset, many risky assets (stocks), and derivatives either on portfolios or on individual risky assets. The risk-free asset is in zero net supply, and we use Rf to denote its gross return, which is an endogenous constant in equilibrium. A typical risky asset n o ~ i;1 2 R2 . We i, which has a net supply of one share, is a claim to a dividend stream Di;0 ; D ++ ~i = denote its price by Pi , and so the gross return on stock i is R to denote logarithmic quantities; that is, c0 log C0 , di;0 ~ i;1 D Pi . log Di;0 , rf We use lower case letters log Rf , and r~i ~i, log R and so on. Technology The data generating processes for the aggregate endowments (consumptions) and for dividends on stock i admit a structure with disaster risks as formalized by Barro (2006). Speci…cally, the consumption growth rate c~ c~1 c0 and the dividend growth rate d~i d~i;1 di;0 are generated as follows: c~ = gC + d~i = gi + C~ "i i~ 7 + J~C ; + ~ + J~i ; i;J JC (1) (2) where gC > 0 and gi > 0 are constants, ~ and ~"i follow standard normal distributions with a correlation coe¢ cient of ! i = Corr (~; ~"i ), and J~C and J~i represent downside jumps: 8 8 > > < 0; < 0; with prob. 1 qC ; with prob. 1 qi ; ~ ~ JC = and Ji = > > : LC ; : Li ; with prob. qC ; with prob. qi ; with LC > 0, Li > 0, i;J > 0, qC 0 and qi (3) 0. As in Barro (2006), the random variables ~ and ~"i represent ‡uctuations in the normal periods, while the random variables J~C and J~i pick up low probability disasters (J~C and J~i are mutually independent and they are independent of ~ and ~"i ). In these disaster events, consumptions or dividends jump down sharply. The constants LC > 0 and Li > 0 capture the sizes of the downside jumps, and qC 0 and qi 0 are their occurring probabilities. The parameter i;J > 0 controls how dividends co-move with consumptions in events of large downside consumption jumps. As we will explain later, i;J corresponds to the systematic component of crash perception in our empirical analysis. Parameters gC > 0 and gi > 0 represent the average growth rates of consumption and dividends in normal periods, respectively. Beliefs We assume that the representative agent’s subjective belief assigns larger probabilities to the disaster events of consumption and dividends. We make this assumption for several reasons. First, the recent psychology literature suggests that people often overestimate the likelihood of rare, extreme events.6 Disaster events exactly share these two features of low probability and extreme outcomes. Second, the di¤erence between subjective and objective probabilities could be due to the fact that investors have extremely limited information about the objective crash risk (Weitzman (2007)). This view is particularly true from an ambiguity-aversion perspective. The literature suggests that ambiguity is appropriate when decision makers lack enough information to assess the relevant distribution over payo¤s (Heath and Tversky (1991); Epstein and Schneider (2008)). Since crashes are rare events, it is arguable that traders do not have enough information to form a prior. Under Gilboa and Schmeidler’s (1989) max-min ambiguity aversion preferences, traders will choose the most pessimistic view for crash events, which e¤ectively assigns larger probabilities to these 6 For example, Barberis (2013, ps. 611-612) summarizes: “A very rough, …rst-pass summary of the psychology literature, then, is that a person’s thinking about a tail event is subject to two forces: an event whose true probability, unknown to the individual, is 0:001, say, will …rst be judged more probable than it actually is— to have probability 0:002, say— and will then be weighted by even more than 0:002 in the individual’s decision making: by (0:002), say, where (0:002) > 0:002, and where the exact value of (0:002) can be determined from the probability weighting function.” 8 events. Finally, Albagli, Hellwig, and Tsyvinski (2014) construct a noisy rational expectations equilibrium model and show that when traders learn information from prices, the tail risk is larger under the marginal investor’s subjective belief than under the objective probability, which therefore endogenously generates our assumption. Formally, in the representative agent’s mind, he understands that the consumption growth rate d~i are generated by (1)-(3), but he thinks that J~C = c~ and the dividend growth rate with a probability q^C > qC and that J~i = Li with a probability q^i > qi . Two explanations are qC that he puts on the event of J~C = in order. First, the extra weight of q^C LC LC captures how much the representative agent overestimates the event that the aggregate consumption jumps down signi…cantly, which is his subjective belief about market-wide crash. Given that parameter i;J in (2) controls the intensity that stock i’s dividends are subject to this overestimation of downside consumption jumps, we use i;J to theoretically capture the systematic component of crash perception, dubbed “co-crash perception”. Second, the extra weight of q^i on the event of J~i = qi that he puts Li describes how much the representative agent’s subject belief about stock i’s crash that is not driven by the aggregate-level market crash, which we use to theoretically capture the …rm-level idiosyncratic component of crash perception, dubbed “idiosyncratic crash perception”. Asset prices According to Cochrane (2005, p. 17), up to a Taylor approximation, the following pricing equation holds for any asset under the representative agent’s subjective belief: b R ~i E where Rf = ^ i; ^ c c; (4) d R ~ i ; c~ Cov and ^ c = Vd ar ( c~) ; (5) Vd ar ( c~) d ( ; ) and Vd where Cov ar ( ) are the covariance and variance operators under the agent’s belief. ^ i; c = Historical data are generated under the objective probability speci…ed by (1)-(3). So, if we de…ne the consumption beta i; c and risk premium c according to the objective probability, ~ i ; c~ Cov R i; c = and V ar ( c~) c = V ar ( c~) ; (6) by (4), we then have: ~ i = Rf + E R i; c ^ c+ i; c ^c 9 i; c c h ~i + E R b R ~i E i : (7) That is, the average return on stock i is determined by the traditional risk factors (Rf + i; c c ), adjusted by terms re‡ecting the misperceptions of disaster risks. i We de…ne the price-dividend ratio as fi DPi;0 . Direct computation shows: h i i;J d ^ ^c ~ ~ = V ar J V ar J ; c C C i; c i; c fi i h i 1 h ~i b R ~i b ( di ) = 1 E i;J J~C + J~i b i;J J~C + J~i : E R E E ( di ) E E fi fi b J~i = Li (^ Plugging the above two expressions into (7) and noting that E J~i E qi qi ) yield: ~ i = Rf + E R i; c c + i;J fi h Vd ar J~C V ar J~C + E J~C b J~C E i + 1 Li (^ qi fi qi ) : (8) The second and third terms in (8) show that, for a given price-dividend ratio fi , the ex~ i increases with the co-crash perception pected stock return E R crash perception (^ qi ^ ^ i; c c ~i term E R i; c c i;J and with the idiosyncratic qi ). Alternatively, in view of equation (7), a high b R ~ i , while a high (^ E qi ~i and a high E R b R ~ i in (7). E In addition, increasing i;J and (^ qi i;J leads to both a high qi ) results in a high adjustment qi ) will lower price-dividend ratio fi , which further in- ~i creases the expected stock return E R in (8). To see this, by the pricing formula (4), we can compute: fi = E egi which decreases with i;J gC + and (^ qi "i;t+1 i~ C ~t+1 h b e( E i;J i )J~C E b eJ~i ; qi ). This is because both co-crash and idiosyncratic crash perceptions lower the representative agent’s perception about the stock’s future dividends. To sum up, we have the following proposition. ~ i on stock i is determined by Proposition 1 In the model economy above, the expected return E R the risk-free rate Rf , compensations for the traditional consumption risk i; c c , and compensations for the subjective belief about crash risks. In addition, the misperception compensation increases with both the co-crash perception 2.2 i;J and with the idiosyncratic crash perception (^ qi qi ). Measures of crash perception Options re‡ect traders’subjective beliefs through equilibrium. We use individual equity OTM put options and the S&P 500 index OTM put options to quantify each stock’s crash perception and the 10 market’s crash perception, respectively. We follow the methodology used in Gao, Gao, and Song (2013), which develops a rare disaster concern index to measure hedge fund performance. In this section, we describe intuitions behind the measure of a stock’s crash perception, and decompose it into a systematic component (co-crash perception) and a stock-speci…c component (idiosyncratic crash perception). Our discussions focus on empirical estimates, option data, and sample descriptive statistics. 2.2.1 Option-based measures To set the stage, we …rst discuss how to estimate the total crash perception for each individual …rms and the market. In the next section, we discuss how to decompose a …rm-level total crash perception into systematic co-crash and idiosyncratic crash perceptions. Essentially, the crash perception measure equals the price di¤erence between two option-based replication portfolios of variance swap contracts that deliver payments equal to the extent of stock/market price variations over a period [t; T ].7 The …rst accounts for price variations induced by jumps associated with mild price movement and the second incorporates price variations induced by jumps associated with extreme price movement (i.e., stock crash). In principle, both upside and downside jumps can contribute to stock price variations over the period [t; T ] (e.g., the extreme deviation of time-T stock price ST from time-t price St ). We focus on downside crash events associated with unlikely but extreme negative stock price jumps, motivated by studies showing that investors are more concerned about downside price swings (Liu, Pan, and Wang (2005); Ang, Chen, and Xing (2006); Barro (2006); Gabaix (2012); Wachter (2013)). In particular, the price of the …rst replication portfolio is Z 1 2er P (St ; K; T )dK; IV 2 K<St K the price of the second replication portfolio is Z 2er 1 ln (K=St ) P (St ; K; T )dK; V K2 K<St and the measure of total crash perception (T CR) is de…ned as Z 2er ln (St =K) T CR V IV = P (St ; K; T )dK; K2 K<St where r is the constant risk-free rate, 7 T (9) t is the time-to-maturity, St is the stock price at See Carr and Wu (2009) for detailed discussions of variance swap contracts and replication portfolios. 11 time t, and P (St ; K; T ) is the time-t price of the OTM put with strike price K and maturity date T . Under general assumptions on the stock price process (e.g., Merton (1976)), T CR captures all of the high-order ( 3) moments of the price jump distribution with negative sizes. A close inspection of (9) shows that T CR is a weighted average of OTM put prices across moneyness with higher weights on deeper OTM puts. As a result, this measure is expected to mainly capture the perception of crash risk rather than the objective crash risk given the rare nature of the crash events associated with deep OTM puts. We note that T CR is equal to a crash insurance price under no-arbitrage conditions only. The methodology above can be equally applied to an index. Therefore, we can follow the same methodology to empirically estimate the aggregate market’s crash perception. In the end, for each month t, we can compute a measure T CRi;t for stock i’s total crash perception, and a measure T CRM KT;t for the market’s total crash perception. 2.2.2 Co-crash and idiosyncratic crash perceptions We use a regression approach to decompose each stock’s total crash perception into a systematic component (co-crash) and a stock-speci…c component (idiosyncratic crash). The former captures the sensitivity of a stock’s perceived crash to the market’s perceived crash, whereas the latter captures a stock’s perceived crash that is unexplained by its covariation with the market. In particular, we perform the following rolling-window time series regression for each stock at the end of month m: T CRi;t = ai + i T CRM KT;t + "i;t ; for t = m; m 1; :::; m 23; (10) where T CRi;t is the estimated total crash perception of stock i in month t, and T CRM KT;t is the estimated market’s total crash perception in month t, both through equation (9). The co-crash perception of stock i is the estimated regression coe¢ cient i and the idiosyncratic crash perception of stock i is the estimated regression intercept ai . In relation to our theoretical perspective in Section 2.1, the market-level total crash perception T CRM KT measured in equation (10) re‡ects the subjective belief about the aggregate consumption crash q^C qC in the model. Similarly, the stock-level total crash perception T CRi measured in equation (10) re‡ects the subjective belief about individual stock crash, which is, in the model 12 economy, jointly determined by i;J (^ qC qC ) and q^i qi ; and as a …rst-order approximation in empirical setting, these two components are estimated as regression coe¢ cient and intercept in equation (10). 2.2.3 Descriptive statistics Our sample consists of CRSP common stocks (with share codes 10 or 11) that have available OptionMetrics OTM puts from January 1996 through December 2012. Table 1 presents summary statistics. Overall, we have 4545 optionable stocks in history, ranging from 1564 stocks in 1996 to 1658 stocks in 2012. These stocks on average are mid- and big-cap stocks: the pooled average of monthly market equity is 8.34 billion and that of size decile ranking is 6.2 (decile breakpoints are determined using only NYSE stocks). In addition, these stocks have active option and stock trading activities.8 For example, the average stock turnover rate is 25:2% per month and the average trading volume of all types of option contracts is 47; 720 per month (each equity option contract corresponds to 100 shares). To estimate stocks’crash perception, we use daily data of U.S. equity options and clean them by a few …lters, such as removing observations where option prices violate no-arbitrage bounds. We then select only OTM put options with maturities longer than 7 days and shorter than 60 days and generate a 30-day implied volatility curve. Equipped with the implied volatility curve to compute the option prices, we then estimate the total crash perception according to a discretization of equation (9) on each day. After obtaining these daily estimates, we take the daily average over a month to obtain T CRi;t , which we further decompose via equation (10) into perceived co-crash i and idiosyncratic crash ai in each month. As the crash perception measures are based on variance swap contracts, it is conceivable that they may di¤er by their scales of volatility levels across stocks. To obtain measures that are robust to the scale di¤erence in volatility levels, we divide T CRi;t by the daily standard deviation over the month t, and ai by the standard deviation of regression residuals "i;t .9 We use these normalized measures in all our empirical analysis that are 8 In Appendix 2, we report summary statistics of daily open interest of individual equity options. OTM options are not illiquid. For example, the median of daily open interest for OTM puts that protect a 5% price drop in future 14-60 days is 219 contracts over our sample period 1996-2012. A recent paper, Muravyev and Pearson (2014), actually show that transaction costs of equity options are lower than those meausred by conventional bid-ask spreads. 9 Within a month, we require at least 15 non-missing daily estimates available in order to calculate the daily average and standard deviation. 13 cross-sectionally comparable. We note that co-crash perception i, as a correlation measure, does not depend on volatility levels of stocks and hence no further normalization is needed. Last, to ensure that we have a reasonable number of observations in estimating regressions (10), we require stocks to have at least 18 months of total crash perception measures available. Table 2 reports summary statistics of stock-level total crash, co-crash, and idiosyncratic crash perceptions. We observe large cross-sectional variations of a stock’s crash perceptions. For example, the idiosyncratic crash perception varies from 0.1 (the bottom one percentile) to 4.9 (the top one percentile) over the full sample, from -0.01 to 5.2 over the …rst half sample, and from 0.2 to 4.5 over the second half sample. 3 Crash Perception and Subjective Belief In this section, we investigate how well our option-based measures capture biased subjective beliefs on stock crash risk. We document strong connection between crash perception measures and pessimistic beliefs of economic agents in survey forecasts of both aggregate economic conditions (such as real GDP growth) and individual …rm prospects (such as earnings). These forecasts are ex-ante and re‡ect beliefs of important market participants. Hence, the strong connection points to our crash perception measures as capturing the subjective belief about stock crash risk reasonably well. Moreover, we also study whether a …rm’s …nancial disclosure (such as readability and tone) is related to stock’s crash perception, from an investor’s perspective. Finally, we ask whether crash perception can correctly “forecast future crashes” in terms of conditional skewness of stock return distribution. 3.1 3.1.1 Subjective Belief in Survey Forecasts Market crash perception We obtain the U.S. real GDP forecasts from the Blue Chip Financial Forecasts (BCFF) survey published by Aspen Publishers. The GDP forecasts are for quarter-on-quarter growth expressed at annualized rate (in percentage). Each month, the BCFF surveys a large number of economists from banks, broker-dealers, consulting …rms, etc. During our sample period from January 1996 through December 2012, on average there are 56 forecasters providing one-quarter-ahead GDP 14 forecasts per month (the minimum number of forecasters is 40 in March 2001 and the maximum is 91 in October 2012). At the end of each month, we calculate the skewness of these forecasts (across individual forecasters) that captures how pessimistic the forecasters are in their beliefs on the future aggregate economy relative to the average belief.10 We also use the 10th percentile of the forecasts that captures the subjective belief of the excessively pessimistic forecaster as an additional measure. Finally, we compute the regular consensus (median) and the dispersion (standard deviation) measures as well as the interquartile range (the di¤erence between 75th and 25th percentile) of these forecasts. Figure 1 presents monthly time series of the market’s total crash perception and three forecast summary statistics, skewness, 10th percentile, and dispersion. The market-level crash perception is estimated via equation (9) using the S&P 500 index OTM put options. We observe that when the market perceives higher level of crash risk for next month, the professional forecasts of the future real GDP growth display more negative skewness and lower 10th percentile. That is, the market crash perception is strongly connected with the pessimistic belief on the aggregate economy. We also …nd positive correlation between the market crash perception and forecast dispersion, implying that forecasters tend to disagree more when they are more pessimistic. We then perform a time series regression analysis to formally describe the contemporaneous relation between the market’s crash perception and the professional forecasters’ beliefs on real output. Table 3 presents regression results in detail. In model speci…cation (1), we run the following regression T CRM KT;t = Intercept + The coe¢ cient 1 1 Skewnesst + 2 M axf0; Skewnesst g + "t : (11) captures the average relation between forecast skewness and market crash per- ception, whereas the coe¢ cient negative estimate -0.20 of 1 2 captures this relation conditional on a negative skewness. The (with a Newey-West t-statistic of -2.7) implies that the market crash perception is signi…cantly consistent with an excessively pessimistic subjective belief on the aggregate economy. Moreover, this negative relation between market crash perception and forecast skewness is stronger conditional on a negative skewness: the coe¢ cient estimate -0.27 of 10 2 (that We note that the BCFF provides forecasts of the mean future real GDP growth across various professional forecasters. Hence, they are not forecasts on crash risk. Nonetheless, the forecast skewness across individual forecasters captures the extent of the pessimistic subjective beliefs relative to the concensus belief, which can re‡ect the belief on crash risk of the "tail forecaster". 15 remains in the regression only when Skewnesst < 0 ) dominates the 1 estimate for the average rela- tion. In addition, the model speci…cations (5) and (7) also show the signi…cant consistency between the market crash perception measure and subjective belief of the strongly pessimistic forecaster. Other model speci…cations con…rm the expected relation between crash perception and consensus forecast and forecast dispersion. Overall, our option-based market crash perception measure is strongly connected to the subjective belief regrading the future aggregate economy. 3.1.2 Stock crash perception We further investigate whether our crash perception measures of individual stocks capture investors’ subjective beliefs on stock crash risk. We extract …nancial analyst one-year-ahead earnings (FYR1) forecasts and long-term-growth (LTG) forecasts from I/B/E/S. These non-stock-price-based variables are less likely to contain agents’ preferences and are “cleaner” to tease out the belief element of stock crash risk. At the end of each month, we collect each analyst’s most recent (and valid) forecasts and then calculate the cross-sectional skewness among all analysts’ forecasts (we require at least …ve available forecasts in estimating skewness). Table 4 presents the results from cross-sectional regressions of stocks’ total crash perception on analyst forecast skewness. These cross-sectional regressions are similar to the regression (11) in including both the Skewness and M axf0; Skewnessg to gauge the relation between crash perception measures and subjective beliefs in di¤erent scenarios. The stock-level crash perceptions are estimated via equation (9) using individual stock’s OTM put options. Similar to Fama and MacBeth (1973), we run regressions at each point of time and report time-series averages of regression coe¢ cients (and Newey-West t-statistics). The …rst two columns of Table 4 show the negative relation between stock crash perception and analyst forecast skewness – a high crash perception measure is signi…cantly consistent a strongly pessimistic subjective belief on …rms’future earnings and growth. Moreover, this negative relation is strengthened when the analysts’beliefs are pessimistic on average. For example, the regression coe¢ cient capturing the relation between crash perception measures and FYR1 forecast skewness (conditional on being negative ) is -1.25, dominating the regression coe¢ cient -0.54 for the average relation. Overall, these cross-sectional results on stock crash perception are similar to those from the previous section on market crash perception, corroborating the signi…cant consistency between 16 our crash perception measures and subjective beliefs about stock crash risk. 3.2 Accounting disclosure As discussed in the Introduction, when a …rm discloses its …nancial information in a non-transparent way (for example, little readability in its annual report), investors hold strong (distorted) subjective belief that managers withhold bad news that can lead to future stock crash even if the managers did not do so. Moreover, when the tone of a …rm’s …nancial disclosures is less favorable (for example, the contextual information conveyed in the narrative R&D disclosure is negative), investors tend to form beliefs on the lack of …rm’s growth opportunities and its poor performance in the future. In a nutshell, the increased information asymmetry between managers and investors, particularly when investors discern opaque and/or negative textual information from …rms’ disclosures, can induce investors to form biased subjective beliefs on stock crash risk. Therefore, measures of the readability of …rm …nancial disclosures and the tone in narrative R&D disclosures can proxy for subjective beliefs on stock crash risk to some extent. In this section, we document close relations between our option-based crash perception measures and these readability and tone measures, providing further evidence that the our crash perception measures capture the biased subjective belief on crash risk well. The results in columns (3) – (9) of Table 4 show the signi…cant positive relation between stock crash perception and various readability measures of …nancial disclosures, including the Fog index, the length of …lings, the document …le size, and the number of unique words (see variable details in Li (2008), Merkley (2013), and Loughran and McDonald (2014)). In addition, …rms with less positive tone in narrative R&D disclosures have signi…cantly higher stock crash perception; and interestingly such a relation doesn’t show up if we check the tone of non-R&D disclosures.11 With respect to the textual measures of positive, negative, and uncertain tones based on the summary data for all 10-K variants (see details in Loughran and McDonald (2011)), we also …nd signi…cant positive associations between crash perception and negative/uncertain tone, and negative association between crash perception and positive tone. 11 We would like to thank Feng Li, Ken Merkley, and Bill McDonald to make their data available to us. 17 3.3 Forecast future return skewness Can stocks’total crash perception indeed predict future crashes in the cross section? In this section, we study how, if at all, our crash perception measures relate to future return skewness. Such an exercise can shed light on whether the crash perception measures capture biased subjective belief on crash risk or objective crash risk. Following Chen, Hong, and Stein (2001), we measure a stock’s crash risk ex post by the negative coe¢ cient of skewness (NCSKEW) and down-to-up volatility (DUVOL). These measures are estimated using the stock’s daily (log) returns in excess of valueweighted CRSP market returns. An increase in NCSKEW or DUVOL indicates more “crash prone” or left skewed return distribution. Table 5 presents results of cross-sectional regressions of stocks’ NCSKEW or DUVOL measured over a future k-month horizon on their total crash perception measured as of month t. Results are striking. Crash perception measures signi…cantly predict future return skewness but in an opposite way to what the correct beliefs of stock crash will suggest. In the case of 6-month forecast horizon, for example, stocks with high crash perception today in fact have realized less negative return skewness or downside volatility in the future. The regression coe¢ cients to predict NCSKEW and DUVOL are -0.136 and -0.030, respectively, both are at least three standard errors from zero. These results provide supporting evidence on the incorrect subjective beliefs of crash risk on individual stocks, which is an essential feature in our stylized model in Section 2 to characterize how crash perception is priced in equilibrium.12 4 Crash Perception and Cross-Sectional Stock Returns In this section, we examine the pricing of crash perception in the cross-sectional equity returns. We …rst present a set of portfolio analysis to illustrate the economic signi…cance of crash perceptions in expected returns. We then perform a regression-based multivariate analysis that allows us to control a large set of …rm characteristics and factors that are likely correlated with our crash perception measures. Our empirical tests focus on the systematic component (co-crash) and the stock-speci…c com12 We also conduct a similar exercise of market crash perception predicting future market return skewness. Although we also obtain negative coe¢ cients, none of them is statistically signi…cant, which suggests the limited statistical power in performing market-wide time series predictive regression. 18 ponent (idiosyncratic crash) of stock crash perception that are shown to to directly a¤ect expected stock return (Proposition 1). Nevertheless, as an initial evidence on stock crash perception being priced in the cross section, we perform the analysis on total crash perception (TCR). We monthly form decile portfolios from January 1996 through November 2012, hold them for one month, and calculate both equal-weighted (EW) and value-weighted (VW) returns.13 Decile 1 (10) consists of stocks with the lowest (highest) total crash perception, and the “High-Low”long-short hedge portfolio consists of going long on stocks in decile 10 and going short on stocks in decile 1. Portfolios are well diversi…ed since on average each decile has 132-133 stocks. Appendix 3 presents detailed results. We observe an increasing return pattern from low-TCR stocks to high-TCR stocks. On a value-weighted basis, for example, the monthly return spreads between high and low perceived total crash deciles are 0.56% (with a Newey-West t-statistic of 2.4). 4.1 Returns on co-crash and idiosyncratic crash perception portfolios To set the stage, we …rst examine the characteristics of stocks with high levels of crash perceptions. Table 6 presents …rm characteristics of decile portfolios formed on crash perception (Appendix 1 provides a brief description of characteristic variables). We monthly rank stocks into ten groups according to their co-crash (Panel A) and idiosyncratic crash perceptions (Panel B), and then calculate the equal-weighted average of …rm characteristics within each group. Compared with low co-crash stocks, high co-crash stocks have lower book-to-market equity, lower idiosyncratic return volatility, higher return kurtosis, lower negative earnings surprise, and lower level and uncertainty of analyst long-term growth forecasts. Similar to high co-crash stocks, high idiosyncratic crash stocks also have lower book-to-market equity, idiosyncratic volatility, and negative earnings surprise than low idiosyncratic crash stocks. Interestingly, stocks with high idiosyncratic crash perceptions are return winners and have much lighter tails (sample kurtosis based on realized historical returns) within the past one year. In addition, these stocks have slow sales growth over the past three years, and among these stocks, we see optimism but also high disagreement in analysts’long-term growth forecasts. Overall, our sample includes economically important stocks on the U.S. equity market. Portfo13 CRSP monthly delisting returns are used whenever available; if not, we use the historical industry average of delisting returns. 19 lios formed on investors’subjective beliefs of stock crash risk display strong heterogeneity in various …rm characteristics, which provides an interesting task in studying stock returns and performing asset pricing tests. We now turn our main focus to portfolios formed using co-crash and idiosyncratic crash perception. Similar to the portfolio analysis above, we rank stocks into ten deciles according to their perceived co-crash and idiosyncratic crash, and also construct long-short hedge portfolios. To get a complete picture of our results, we consider portfolio formation at monthly, quarterly, semi-annual, and annual frequencies. At a monthly formation, for example, we form deciles at the end of each month from December 1997 through November 2012, and hold portfolios for one month; at a quarterly formation, we form deciles at the end of each quarter from 1997Q4 through 2012Q3, and hold portfolios for three months; and so on. The …rst date of constructing portfolios is at the end of December 1997 because OptionMetrics data start in January 1996 and we require a 24-month horizon to run regression (10) and estimate the co-crash and idiosyncratic crash perceptions. Each decile on average contains 65-66 stocks. We report both equal-weighted (EW) and value-weighted (VW) portfolio returns. To estimate risk-adjusted abnormal returns (alphas), we use the Fama-French (1993) three factors augmented with the Carhart (1997) momentum factor, the Fama-French-Carhart factors augmented with the Jegadeesh (1990) short-term reversal factor, the Pastor-Stambaugh (2003) market liquidity risk factor, and the Fung-Hsieh (2001) seven factors including option-based lookback straddles and macro-based default risk and term risk factors. 4.1.1 Univariate sorts Table 7 presents returns on perceived co-crash portfolios. When portfolios are monthly formed, equal-weighted low (high) perceived co-crash stocks earn -0.11% (0.96%) per month and the return di¤erence is 1.08% per month (with a Newey-West t-statistic of 2.7). The return spread based on value-weighted portfolios is of similar magnitude, albeit less statistically signi…cant, at 1.06% per month (with a t-statistic of 1.7). Risk-adjusted abnormal returns of the high-minus-low perceived co-crash portfolios are economically large, varying from 0.96% per month (benchmarked on the Fama-French-Carhart four-factor model) to 1.6% (benchmarked on the Fung-Hsieh seven-factor model). This return pattern associated with co-crash perception is strongest at monthly portfolio 20 formation: the signi…cance of return spreads becomes dissipated at quarterly and semi-annual formation frequencies. Table 8 presents returns on perceived idiosyncratic crash portfolios. Three main results arise. First, stocks with high idiosyncratic crash perception earn signi…cantly higher excess returns than stocks with low idiosyncratic crash perception. For example, when portfolio formation is at a monthly frequency, return spreads of high-minus-low perceived idiosyncratic crash portfolios are 0.91% and 1.08%, respectively, when stock returns are equal and value weighted in portfolios. These spreads are at least three standard errors from zero. Second, these return spreads are largely driven by going long in stocks with high perceived idiosyncratic crash (decile 10): equal- and value-weighted monthly excess returns are 1.20% (with a t-statistic of 2.2) and 1.0% (with a tstatistic of 2.4), respectively. Lastly, the e¤ect of perceived idiosyncratic crash on predicting future stock returns is not short-lived. In fact, the return di¤erences between high and low perceived idiosyncratic crash stocks at quarterly, semi-annual, and annual portfolio formation frequencies are not only statistically signi…cant but also economically large, ranging from 0.83% (with a t-statistic of 2.3) to 1.45% per month (with a t-statistic of 3.7). Results of abnormal returns based on di¤erent benchmark factors do not change our conclusions. To examine whether our return results come from a particular sub-sample period, we calculate year-by-year annual returns and Sharpe ratios of long-short portfolios formed on co-crash and idiosyncratic crash perceptions. Figure 2 presents these results (see details of long-short portfolio construction therein). The outperformance of high perceived co-crash and idiosyncratic crash stocks is not restricted to a particular year. Interestingly, during years of negative returns from investing in co-crash perception portfolios, returns from investing in idiosyncratic crash perception portfolios are positive, and vice versa. Overall, the averages of annual returns of long-short perceived co-crash and idiosyncratic crash portfolios are 12.9% and 10.9% over the 15-year sample from 1998 through 2012, and the averages of annual Sharpe ratios are 0.89 and 1.07. 4.1.2 Double sorts To investigate the respective power of perceived co-crash and idiosyncratic crash in explaining cross-sectional expected returns, we perform an analysis on two-way sequentially sorted portfolios. Table 9 reports equal-weighted return results of these double-sorted 5 21 5 portfolios (each portfolio on average contains 26-27 stocks). Value-weighted returns are qualitatively similar (results are available upon request). In Panel A, we rank stocks into 25 portfolios …rst on perceived co-crash and then on perceived idiosyncratic crash. After we control the e¤ect of co-crash perception, the average returns of perceived idiosyncratic crash portfolios monotonically increase from the bottom quintile (0.23% with a t-statistic of 0.4) to the top quintile (1.10% with a t-statistic of 2.1), and the return di¤erence between these two quintiles is 0.87% per month (with a t-statistic of 3.2). In addition, the e¤ect of idiosyncratic crash perception is stronger among stocks with low co-crash: the return spread of high-minus-low perceived idiosyncratic crash portfolios is signi…cant at 1.1% per month among the bottom quintile of perceived co-crash, whereas it is insigni…cant at 0.55% among the top quintile of perceived co-crash. Results of abnormal returns benchmarked on the Fama-French-Carhart four factors are qualitatively similar. In Panel B, we rank stocks into 25 portfolios …rst on perceived idiosyncratic crash and then on perceived co-crash. In the presence of idiosyncratic crash perception, the average returns of perceived co-crash portfolios also monotonically increase from the bottom quintile (0.22% with a t-statistic of 0.3) to the top quintile (1.19% with a t-statistic of 2.3), and the return di¤erence between these two quintiles is 0.97% per month (with a t-statistic of 2.2). Turning to the e¤ect of co-crash perception within each idiosyncratic crash quintile, we …nd no signi…cant return spreads among stocks with high idiosyncratic crash perception (0.13% with a t-statistic of 0.2), whereas we …nd much larger return spreads among stocks with low idiosyncratic crash perception (1.08% with a t-statistic of 1.8). These results are consistent with implications from our empirical procedures of decomposing a stock’s perceived total crash into a systematic component (co-crash perception) and a stock-speci…c component (idiosyncratic crash perception). In summary, our analysis on double-sorted portfolios suggests that both co-crash and idiosyncratic crash perceptions are important determinants of cross-sectional stock returns and neither one dominates the other. 4.2 Crash perception and realized crash shocks In this subsection, we investigate how subjective belief about crash risk (ex-ante crash perception) is di¤erent from objective crash risk implied from ex-post historical crash shocks in a¤ecting cross22 sectional stock returns. First, we study whether stocks with high crash perception earn higher expected returns simply by being more exposed to liquidity and macroeconomic risk factors. We use two market-wide liquidity risk factors, the Pastor-Stambaugh (2003) market liquidity innovation and the monthly change of Hu-Pan-Wang (2013) “noise” measure. Macroeconomic risk factors include the market volatility risk (measured as the monthly change in CBOE Volatility Index (VIX)), the term risk (measured as the monthly change in the term spread between the US 10-year bond yield and 3month T-bill rate), and the default risk (measured as the monthly change in the default spread between the Moody’s Aaa and Baa corporate bond yield). We also use the CRSP value-weighted market excess return as the market factor. The …rst six columns of Table 10 present the factor exposure of perceived co-crash and idiosyncratic crash portfolios with respect to the liquidity and macroeconomic risk factors.14 Stocks with high co-crash and idiosyncratic crash perceptions are generally not exposed to liquidity and macroeconomic shocks (decile 10 portfolios have economically small and statistically insigni…cant exposure to the majority of these factors). Moreover, we observe that liquidity and macroeconomic factors are usually not signi…cant in explaining the return spreads between low and high perceived crash risk portfolios: factor loadings are less than one standard error from zero most of the time. One exception is the negative loading of equal-weighted perceived co-crash hedge portfolio (i.e., High-Low portfolio) on the Hu-Pan-Wang liquidity risk factor (-0.008 with a t-statistic of -2.5). However, the loading of the perceived idiosyncratic crash hedge portfolio on this factor is positive (0.007 with a t-statistic of 1.8), which goes in a wrong direction to explaining return spreads earned on idiosyncratic crash perception. Regarding exposure to market risk, perceived co-crash and idiosyncratic crash hedge portfolios in fact provide hedges against market downturns: their loadings on the market factor are all negative (from -0.47 to -0.11) and most are signi…cant. These results deliver evidence that portfolio returns on crash perceptions are not merely compensations for bearing losses during macroeconomic downturns and liquidity crunches. Second, to examine whether return patterns associated with crash perception are solely the manifestation of realized crash shocks on stock markets, we perform a set of Fama-MacBeth (1973) 14 For brevity, we only report results of deciles 1, 5, and 10, and the long-short hedge portfolios. The results of all deciles are available upon request. 23 cross-sectional regressions by including variables related to stocks’tail distributions based on historical observations. Panel A of Table 11 provides results of regression coe¢ cients and Newey-West (1987) t-statistics when we regress stocks’realized excess returns in month t + 1 on crash perception measures and various subsets of the explanatory variables as of month t. To reduce measurement errors in stocks’ crash perception (e.g., co-crash) and make regression coe¢ cients comparable across di¤erent model speci…cations, we use each stock’s crash perception decile rankings as regressors when performing cross-sectional regressions at each point of time (see Table 6 for details of decile portfolio construction). The existing literature documents that the cross-sectional return variation is associated with downside market risk (downside market beta) (Ang, Chen, and Xing (2006)), idiosyncratic volatility (Ang et al. (2006)), systematic and idiosyncratic skewness (Harvey and Siddique (2000)), systematic kurtosis (Dittmar (2002)), a stock’s lottery characteristic measured by maximum daily return (Bali, Cakici, and Whitelaw (2011)), hybrid tail covariance risk (Bali, Cakici, and Whitelaw (2014)), and tail risk beta (Kelly and Jiang (2014)). In all regression speci…cations, the coe¢ cients on total crash, co-crash, and idiosyncratic crash perceptions are positive and statistically signi…cant. Moreover, the magnitudes of these coe¢ cients do not vary much across di¤erent speci…cations, which implies that return spreads between the top and bottom deciles of crash perception are similar in the presence of various tail measures based on historical observations. Additionally, the seventh column of Table 10 shows no signi…cant exposure of crash perception portfolios with respect to the Kelly and Jiang (2014) stock market tail risk factor. These results provide corroborating evidence that the e¤ect of subjective belief about crash risk on expected stock return is not simply captured by stock crash risk implied from realized historical crash events. 4.3 Crash perception and option information As our measure of subjective belief is based on out-of-the-money put options, it is imperative to check whether the ex-ante crash perception has been fully captured by well-known option-based variables that describe risk-neutral moments and account for volatility risk, jump risk, informed trading in option markets, and slow information incorporation in equity markets. These option characteristics are shown to have return predictability in various studies (Cremers and Weinbaum 24 (2010); Xing, Zhang, and Zhao (2010); Yan (2011); Johnson and So (2012); An et al. (2014); Conrad, Dittmar, and Ghysels (2013)). In Panel B of Table 11, we control for a set of option-based variables, including option-tostock volume ratio, at-the-money implied volatility, slope of implied volatility, among others. In all speci…cations, the coe¢ cients of co-crash and idiosyncratic crash perceptions remain positive and signi…cant, and are quantitatively similar across di¤erent controls of option-based variables. For example, returns spreads between the top and the bottom deciles of perceived co-crash range from 0.79% (0.00079 10 in speci…cation (4)) to 0.92% (0.00092 10 in speci…cation (6)) per month; and those of perceived idiosyncratic crash range from 0.74% to 0.88%. These results strongly support that our measures of systematic and stock-speci…c components of subjective belief about crash risk are not simply manifestations of option-based variables on implied volatility and informed trading. Furthermore, the last four columns of Table 10 also report factor exposure of crash perception portfolios with respect to a set of S&P 500 index-option-based factors, including the implied volatility skewness (Xing, Zhang, and Zhao (2010)) and the high-order moment risk of variance, skewness, and kurtosis (Bakshi, Kapadia, and Madan (2003)).15 We observe that these option-based factors are rarely signi…cant in explaining the return spreads between low and high crash perception portfolios. One exception is the negative loading of the equal-weighted perceived co-crash hedge portfolio on the risk-neutral variance (-0.947 with a t-statistic of -2.0). However, the positive loading of the perceived idiosyncratic crash hedge portfolio on this factor (0.565 with a t-statistic of 1.9) goes in a wrong direction to explaining the return spread of perceived crash risk portfolios. Overall, these results show that a stock’s crash perception is priced in the cross-section, and this e¤ect cannot be explained by option-based variables that have been proposed in the literature. 4.4 Crash perception and other …rm characteristics We complete our regression analysis by controlling for explanatory variables that account for commonly used …rm characteristics. These variables include size and book-to-market equity (Fama and French (1992)), past short- and intermediate-term returns (Jegadeesh (1990); Jegadeesh and Tit15 In an (unreported) analysis, we also use other option-based disaster risk factors such as the slope of implied volatility (Yan (2011)), the implied volatility spread (Cremers and Weinbaum (2010)), ATM implied volatility, and the delta-hedged option return spreads between S&P 500 index OTM and ATM puts. Our results are robust to all these factors. 25 man (1993)), net stock issuance (Ponti¤ and Woodgate (2008)), operating accruals (Sloan (1996)), and illiquidity (Amihud (2002)). We report regression estimates in Panel C of Table 11. Results of the …rst four regression speci…cations con…rm our above-mentioned baseline portfolio results. In the remaining speci…cations (5) and (6), the coe¢ cients on total crash, co-crash, and idiosyncratic crash perceptions are all positive and statistically signi…cant, showing that the explanatory power of subjective belief about crash risk for cross-sectional stock returns is not subsumed by commonly used …rm characteristics. 5 Robustness Checks In this section, we investigate industry e¤ects behind …rm-level crash perception and check alternative measures of crash perception. 5.1 Industry e¤ects Are there any industry e¤ects driving stock return patterns associated with crash perception? In this subsection, we …rst examine industry compositions among stocks with the highest perceived co-crash and idiosyncratic crash. Figure 3 presents a time series of industry composition within the two top deciles of co-crash and idiosyncratic crash perception. Particularly, we look for industries with the highest likelihood of being ranked into these top deciles at the end of each month of portfolio formation (see Figure 3 for details).16 We group stocks into the Fama-French 12-industry classi…cation according to their Standard Industry Classi…cation (SIC) codes. The …gure suggests that to some extent the investors’ belief about stock crash is industry wide. There are some industries that are more likely than others to enter into portfolios with the highest crash perception, and importantly, such industry compositions are also time-varying in response to changing macroeconomic conditions. For example, industry 7 (telephone and television transmission) accounts for the largest fraction among stocks with the highest perceived co-crash during the earlier sample period 02/1999-03/2000, whereas industries 10 (healthcare and medical equipment) and 11 (…nance) dominate the largest fractions during the later sample periods 07/2006-01/2009 and 03/2009-02/2011, respectively. 16 In Appendix 4, we report summary statistics regarding the likelihood of the Fama-French 12 industries that are ranked within each perceived co-crash decile and each perceived idiosyncratic crash decile. 26 Because of industry common components behind the crash perception of individual stocks, we further investigate whether our …ndings are completely explained by industry e¤ects. We perform two additional analyses: (1) industry-neutral portfolios formed on crash perception; and (2) standard portfolios formed on industry-adjusted crash perception. For brevity, we discuss the main results below and the full set of tables is reported in the Appendix. To construct industry-neutral portfolios, we …rst sort stocks into ten groups based on their crash perception within each of the Fama-French 12 industries, and then form ten decile portfolios by combining stocks across industries. These portfolios are industry neutral because decile 1 (10) contains stocks with the lowest (highest) perceived crash from each industry. Compared with the main results in Tables 7 and 8, we …nd that return patterns associated with crash perception are not completely attributed to industry e¤ects, particularly at monthly frequency of portfolio formation. Return spreads of high-minus-low perceived co-crash and idiosyncratic crash portfolios, for example, range from 0.52% to 0.82% per month (with t-statistics varying from 1.7 to 2.6). Detailed results are shown in Appendix 5. To construct portfolios based on industry-adjusted perceived crash risk, we …rst demean each stock’s crash perception by industry, and then rank stocks into ten decile portfolios according to their demeaned measures. Results are qualitatively similar to those of industry-neutral portfolio analysis. For example, at monthly formation, return spreads of high-minus-low perceived co-crash and idiosyncratic crash portfolios range from 0.49% to 0.93% per month (with t-statistics varying from 1.6 to 2.4). Detailed results are shown in Appendix 6. 5.2 Alternative measures of crash perception In the main analysis, we use the OTM put prices to capture the subjective belief of crash risk (rather than the objective crash risk). The …rst alternative approach is to combine both OTM calls and puts together in constructing a measure of crash perception, similar to that described in equation (9). This measure concerns the protection against the unlikely but extreme downside net of upside price movement. Panel A of Appendix 7 shows that excess returns of high total crash/co-crash/idiosyncratic crash are signi…cantly higher than those of low crash stocks under this alternative measure of crash perception. For example, equal-weighted return spreads of high-minuslow perceived total crash, co-crash, and idiosyncratic crash portfolios are 0.40%, 0.73%, and 0.70% 27 per month, respectively, all statistically signi…cant. Value-weighted return spreads are of similar magnitudes. Throughout the paper so far, we measure crash perception over next month. As a robustness check on horizon, we also measure crash perception over the next two months, using options with maturities longer than 30 days and shorter than 90 days to generate a 60-day implied volatility curve. Panel B of Appendix 7 presents excess returns of crash perception portfolios. Results are similar to those in our baseline analysis. Equal-weighted return spreads of high-minus-low perceived total crash, co-crash, and idiosyncratic crash portfolios are 0.39%, 0.91%, and 0.85% per month, respectively, all statistically signi…cant. Value-weighted return spreads are of similar magnitudes. 6 Conclusion In many scenarios, economic agents do not know the true probability of future events and make decisions based on the biased subjective beliefs. In this paper we explore the pricing implications of the di¤erence between perception (the subjective belief) and truth (the objective probability) in the context of stock crash risk. We believe stock crash risk is an ideal setting for testing the pricing implications of the perception for two reasons. First, crash events are rare and traders may have a di¢ cult time forming a correct belief. Second, option price data are readily available on individual stocks, which re‡ect traders’subjective beliefs through equilibrium. We use out-of-the-money put options on both the S&P 500 Index and individual equities to measure two components of crash perception: perceived co-crash, a systematic component capturing the covariation between a stock’s crash perception and the market’s crash perception; and perceived idiosyncratic crash, a stock-speci…c component capturing the part of stock-crash perception that is unexplained by the market. Although option prices contain both preference and belief elements, we show the crash perception measures are signi…cantly connected to pessimistic beliefs re‡ected in the survey forecasts of future aggregate economy and …rm-level conditions, which corroborates our crash perception measures capturing subjective beliefs on crash risk well. 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Yan, S., 2011, Jump risk, stock returns, and slope of implied volatility smile, Journal of Financial Economics 99, 216-233. 34 Table 1: CRSP/OptionMetrics stock sample descriptive statistics The sample consists of CRSP common stocks (share code 10 or 11) with available out-of-themoney (OTM) put options in OptionMetrics database from January 1996 through December 2012. In addition to the number of optionable stocks by year, we also report the pooled average of each of the following monthly variables: (1) average daily open interest of all option contracts within a month; (2) monthly trading volume of all option contracts (each contract is in 100 shares); (3) monthly trading volume of stock shares (in hundred); (4) monthly turnover of stock shares (in percent); (5) market equity (in $ million); and (6) book-to-market equity. Appendix 1 provides a description of all variables used in our empirical analysis. 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 All Number of Optionable Stocks 1564 1844 1972 2036 1951 1792 1709 1605 1740 1709 1785 1881 1825 1747 1777 1796 1658 4545 Option Open Interest Option Volume Stock Volume Stock Turnover 10207 11158 13969 17733 24181 32771 36401 44642 53835 63843 72736 83075 77102 64130 71181 76558 62965 49378 11987 13219 15239 18844 25045 29545 29145 32544 40736 50691 60382 71476 76337 77822 73867 82878 78359 47720 118656 129125 158760 193184 305050 368457 377117 357504 357235 371777 409723 482064 695834 727926 631290 606954 542373 412188 18.9 18.5 18.4 20.7 25.4 23.9 23.7 23.7 24.1 23.8 25.8 28.9 35.1 31.5 27.6 27.6 25.8 25.2 Market Equity ($MM) 5234.2 5748.0 7209.2 8564.1 9897.0 8697.2 8030.5 7962.6 8815.2 9463.5 9575.5 9959.0 8233.6 6645.0 7776.0 8940.7 10187.5 8343.4 B/M 0.46 0.45 0.42 0.43 0.43 0.48 0.51 0.58 0.55 0.46 0.44 0.45 0.48 0.71 0.76 0.61 0.59 0.52 Table 2: Summary statistics of stock's crash perception We report summary statistics of perceived total crash, co-crash, and idiosyncratic crash over the cross section of stocks in different sample periods. For perceived total crash, the full sample is from 1996 through 2012, the first half sample is from 1996 through 2004, and the second half sample is from 2005 through 2012. For perceived co-crash and idiosyncratic crash, the full sample is from 1998 through 2012, the first half sample is from 1998 through 2005, and the second half sample is from 2006 through 2012. The statistics of perceived co-crash are multiplied by 100 in the table. Mean Std. Dev. Percentiles 1% 5% 10% 25% 50% 75% 90% 95% 99% Total Crash Perception Full 1st Half 2nd Half Sample 9.30 10.42 8.16 134.96 116.27 151.70 1.07 1.76 2.26 3.36 5.27 8.75 14.95 20.90 44.89 1.05 1.73 2.25 3.46 5.70 10.18 17.48 24.38 51.89 1.09 1.79 2.26 3.29 4.95 7.60 12.13 16.73 35.84 Co-Crash Perception Full 1st Half 2nd Half Sample -0.07 -0.12 -0.04 0.52 0.50 0.53 -2.01 -0.61 -0.28 -0.08 -0.01 0.02 0.14 0.27 0.83 -2.22 -0.91 -0.46 -0.12 -0.01 0.03 0.13 0.25 0.75 -1.66 -0.39 -0.19 -0.06 -0.01 0.02 0.14 0.30 0.91 Idiosyn. Crash Perception Full 1st Half 2nd Half Sample 1.88 2.15 1.70 1.03 1.11 0.92 0.10 0.49 0.71 1.12 1.76 2.45 3.25 3.80 4.87 -0.01 0.55 0.84 1.37 2.05 2.79 3.61 4.14 5.22 0.16 0.46 0.65 1.00 1.59 2.20 2.90 3.41 4.49 Table 3: Market crash perception and one-quarter-ahead real GDP forecast distribution This table presents results from (contemporaenous) time series regressions of market crash perception on the statistics of real GDP forecast distribution. The market-level total crash perception (TCR) is measured as a weighted average of OTM put prices across moneyness of S&P 500 index options with 30-day maturities (see equation (9) in the paper for details). The U.S. real GDP forecasts for quarter-on-quarter growth, expressed in annualized percentage points, are obtained from the Blue Chip Financial Forecasts (BCFF) surveys. At the end of each month from January 1996 through December 2012, we estimate forecast distribution across all nonmissing values provided by professional forecasters, including the lower 10th percentile, the standard deviation (dispersion), the median (consensus), the skewness, and the interquartile range of one-quarter-ahead real GDP growth forecasts. We report Newey-West robust t -statistics in parentheses. Real GDP Forecast Skewness Max{0, –Skewness} Lower 10th Percentile (1) -0.1990 (-2.74) -0.2710 (-2.06) Consensus (2) -0.4600 (-2.64) Dispersion Interquartile Range Intercept Adjusted R-square 0.4704 (5.51) 0.01 1.6145 (3.17) 0.33 (3) (4) 1.3745 (2.19) -0.3160 (-2.77) 0.9555 (2.14) -0.6580 (-1.55) 0.33 0.4998 (1.50) 0.45 (5) (6) -0.3620 (-2.69) 1.0089 (3.62) 0.35 (7) -0.2120 (-3.27) 0.8211 (2.13) -0.3770 (-1.20) 0.36 0.4984 (1.58) 0.2850 (0.81) 0.43 Table 4: Firm-level total crash perception, analyst forecast skewness, and financial disclosures This table presents results from cross-sectional regressions of stocks' total crash perception on analyst forecast skewness (AFSKEW), annual report readability, R&D disclosure tone, and tones of various financial disclosures. We perform regressions at the end of each period and report time-series averages of regression coefficients (and Newey-West t -statistics in parentheses). In Panel A, the regressors include the sample skewness of analysts' one-year-ahead earnings forecasts (FYR1) and long-term-growth forecasts (LTG). We require at least five available forecasts in estimating these forecast skewness. In Panel B, we use readability measure of 10-K filings, including the Fog Index (FOG), the log of number of words (LENGTH). We follow Li (2008) and require at least 3,000 words in the annual report. We also use narrative R&D (and non R&D) disclosure tones in Merkley (2013), which is estimated as the number of positive sentences less negative sentences divided by total sentences. In Panel C, we use readability and tone measures of Loughran and McDonald (2011, 2014) based on their summary data for all 10-K variants, including document file size, number of unique words, number (and percent) of positive/negative/uncertaint words. All regressions contain intercepts but their estimates are not presented. (1) (2) (3) Panel A: Analyst forecast skewness (AFSKEW) AFSKEW(FYR1) -0.5408 (-2.62) Max{0, –AFSKEW(FYR1)} -1.2506 (-3.45) AFSKEW(LTG) -0.5184 (-1.87) Max{0, –AFSKEW(LTG)} -0.4766 (-0.53) Panel B: FOG (Li, 2008) and R&D disclosures (Merkley, 2013) FOG 0.1159 (2.37) LENGTH R&D Disclousre Tone (4) 0.1579 (2.01) NonR&D Disclousre Tone Panel C: 10-X file summaries (Loughran and McDonald, 2011, 2014) Net File Size (5) (6) (7) (8) -1.7210 (-6.21) 3.2266 (0.56) 0.2701 (4.65) No. Unique Words No. Words Negative 0.6180 (5.79) No. Words Positive No. Words Uncertain Pct. Words Negative 0.4354 (5.18) -0.3250 (-3.52) 0.2497 (1.68) Pct. Words Positive Pct. Words Uncertain Avg. # of Stocks Avg. Adjusted R-square # of Time-Series Obs. 981 0.00 203 540 0.00 203 125 0.00 136 125 0.00 136 (9) 537 0.01 13 451 0.01 203 451 0.01 203 451 0.02 203 0.3751 (3.74) 0.5104 (7.73) 0.2447 (1.37) 0.5751 (3.45) 451 0.04 203 Table 5: Firm-level total crash perception and forecasting crashes This table presents results from cross-sectional regressions of stocks' future "crashes" (conditional skewness of return distribution) on their total crash perception as of month t . We perform regressions at the end of each period and report time-series averages of regression coefficients (Fama and MacBeth, 1973). We follow Chen, Hong, and Stein (2001) in constructing crash measures of "negative coefficient of skewness" (NCSKEW) and "down-to-up volatility" (DUVOL). Forecasting horizons consist of 1-6 months in the future. We use log changes in price and daily observations over the forecasting period to estimate NCSKEW and DUVOL. An increase in NCSKEW indicates more "crash prone" (or left skewed in the return distribution). Intercepts are included in regressions but not reported. Newey-West (1987) t -statistics are reported in parentheses. Panel A: Predict negative coefficient of skewness (NCSKEW) Forecast Horizons (1) (2) (3) 1 month -0.0124 (-0.53) 3 months -0.1043 (-3.72) 6 months -0.1361 (-4.73) Avg. # of Stocks 1322 1316 1301 Avg. Adjusted R-square 0.00 0.00 0.00 # of Time-Series Obs. 203 203 203 Panel B: Predict down-to-up volatility (DUVOL) Forecast Horizons (1) (2) 1 month 0.0135 (0.78) 3 months -0.029 (-2.64) 6 months Avg. # of Stocks Avg. Adjusted R-square # of Time-Series Obs. 1321 0.00 203 1316 0.00 203 (3) -0.030 (-3.29) 1301 0.00 203 Table 6: Firm characteristics of decile portfolios formed on crash perception At the end of each month we rank stocks into ten deciles according to their co-crash perception (Panel A) and idiosyncratic crash perception (Panel B). Decile 1 (10) consists of stocks with the lowest (highest) crash perception. Firm characteristics are as follows: book-to-market equity, past 11-month cumulative return (we skip the most recent month following the momentum literature), idiosyncratic return volatility (the standard deviation of residuals from Fama-French three-factor daily return regressions within the past one year), daily excess return skewness and kurtosis within the past one year, the average of standardized unexpected earnings (SUE) over the past four quarters, the average of sales growth (SG) over the past three years (seasonal sales growth is based on quarter-to-quarter change in sales), and the median and standard deviation of analyst long-term growth (LTG) forecast. Portfolio characteristics are calculated as simple equal-weighted average of firm characteristics. The table reports time-series average of portfolio characteristics over the sample period (1998-2012), with all firm characteristics calculated as of portfolio formation. The last row report t -statistics of the difference in characteristics between high and low crash portfolios. Panel A: Co-crash portfolios Portfolio B/M Return Ranking Equity (t-2, t-12) 1 - Low 0.495 0.227 2 0.426 0.163 3 0.421 0.152 4 0.435 0.139 5 0.441 0.130 6 0.443 0.142 7 0.444 0.145 8 0.449 0.166 9 0.462 0.209 10 - High 0.452 0.198 High - Low -0.043 -0.029 (-4.4) (-0.8) Idiosyn. Volatility 0.034 0.027 0.023 0.021 0.019 0.019 0.019 0.020 0.023 0.029 -0.005 (-7.9) Panel B: Idiosyncratic crash portfolios Portfolio B/M Return Idiosyn. Ranking Equity (t-2, t-12) Volatility 1 - Low 0.512 0.095 0.025 2 0.473 0.141 0.024 3 0.454 0.158 0.023 4 0.442 0.170 0.023 5 0.437 0.190 0.022 6 0.426 0.165 0.023 7 0.420 0.175 0.023 8 0.428 0.179 0.023 9 0.437 0.189 0.024 10 - High 0.443 0.205 0.024 High - Low -0.069 0.110 -0.001 (-9.5) (9.7) (-2.2) Skewness Kurtosis 0.162 0.127 0.119 0.107 0.093 0.106 0.122 0.136 0.153 0.182 0.020 (0.7) 7.270 5.247 4.982 5.016 4.910 4.734 4.980 5.113 5.506 7.700 0.430 (2.0) Skewness Kurtosis 0.167 0.162 0.146 0.137 0.115 0.098 0.098 0.103 0.127 0.154 -0.013 (-0.7) 8.152 5.844 5.450 5.269 5.042 5.097 4.906 5.055 5.109 5.559 -2.592 (-9.8) Avg. SUE (q-1, q-4) -0.43 -0.10 -0.09 -0.12 -0.15 -0.12 -0.11 -0.26 -0.16 -0.19 0.25 (4.4) Avg. SG (y-1, y-3) 0.74 0.32 0.23 0.19 0.19 0.17 0.17 0.18 0.22 0.64 -0.11 (-1.0) Median LTG 21.49 18.77 16.72 15.12 14.27 14.02 14.34 14.82 16.07 19.15 -2.34 (-5.2) Avg. SUE (q-1, q-4) -0.21 -0.24 -0.23 -0.24 -0.15 -0.16 -0.18 -0.15 -0.08 -0.08 0.13 (3.1) Avg. SG (y-1, y-3) 0.66 0.39 0.30 0.24 0.26 0.24 0.24 0.24 0.23 0.21 -0.45 (-5.0) Median LTG 15.69 15.97 16.22 16.28 16.38 16.53 16.75 16.54 16.59 16.91 1.23 (4.6) Std. LTG 7.01 5.92 4.86 4.34 3.96 3.81 4.06 4.17 4.68 6.08 -0.93 (-4.9) Std. LTG 4.75 4.91 4.64 4.75 4.75 4.69 4.82 4.88 4.82 5.16 0.41 (3.8) Table 7: Returns of co-crash perception portfolios This table presents monthly mean excess returns, abnormal returns (alphas), and Newey-West t -statistics of portfolios formed on stock's co-crash perception. In Panel A, we consider portfolio formation at monthly, quarterly, semi-annual and annual frequencies, and report both equal-weighted (EW) and value-weighted (VW) portfolio excess returns. In Panel B, we report alphas of monthly formed perceived co-crash portfolios using the following set of factors: (1) Fama-French-Carhart (FFC) market, size, value, and momentum factors; (2) FFC 4 factors augmented with short-term reversal factor; (3) FFC 4 factors plus Pastor-Stambaugh market liquidity risk factor; and (4) Fung-Hsieh primitive trend-following factors. On average, there are 65-66 stocks within each decile. Panel A: Monthly excess returns at different portfolio formation frequencies Monthly Formation Quarterly Formation Semi-annual Formation Portfolio Ranking EW VW EW VW EW VW 1 - Low -0.112 -0.481 0.226 -0.354 0.677 0.193 (-0.14) (-0.55) (0.27) (-0.38) (0.84) (0.20) 2 0.442 0.008 0.381 -0.106 0.680 0.391 (0.66) (0.01) (0.57) (-0.17) (1.03) (0.58) 3 0.441 -0.235 0.434 -0.288 0.572 -0.192 (0.76) (-0.41) (0.74) (-0.47) (0.95) (-0.33) 4 0.381 0.198 0.379 0.151 0.739 0.514 (0.79) (0.45) (0.78) (0.33) (1.59) (1.27) 5 0.435 0.239 0.345 0.212 0.374 0.311 (1.03) (0.60) (0.78) (0.53) (0.83) (0.76) 0.621 0.432 0.571 6 0.439 0.527 0.303 (1.52) (1.19) (1.44) (1.22) (1.28) (0.77) 7 0.955 0.761 0.756 0.558 0.572 0.258 (2.17) (2.02) (1.85) (1.61) (1.34) (0.65) 8 0.775 0.617 0.946 0.750 0.800 0.518 (1.79) (1.68) (2.08) (1.96) (1.70) (1.24) 9 1.187 0.855 1.182 0.903 0.855 0.426 (2.27) (1.71) (2.38) (1.74) (1.65) (0.85) 10 - High 0.964 0.577 0.914 0.552 0.752 0.274 (1.49) (0.79) (1.41) (0.77) (1.14) (0.40) High - Low 1.076 1.058 0.688 0.907 0.075 0.082 (2.66) (1.70) (1.74) (1.56) (0.21) (0.14) Annual Formation EW VW 1.112 0.666 (1.35) (0.73) 0.917 0.498 (1.39) (0.82) 0.968 0.330 (1.63) (0.58) 0.790 0.563 (1.65) (1.39) 0.529 0.241 (1.20) (0.62) 0.389 0.577 (1.34) (1.04) 0.540 0.278 (1.30) (0.70) 0.453 -0.123 (0.95) (-0.25) 0.546 0.119 (1.11) (0.26) 0.627 -0.209 (0.94) (-0.29) -0.486 -0.875 (-1.17) (-1.41) Panel B: Monthly abnormal returns based on different benchmark factors FFC 4 Factors FFC 4 Factors + ST Rev FFC 4 Factors + PS LIQ Portfolio Ranking EW VW EW VW EW VW 1 - Low -0.536 -0.788 -0.559 -0.804 -0.467 -0.718 (-1.74) (-1.81) (-1.87) (-1.90) (-1.48) (-1.78) 2 0.106 -0.316 0.062 -0.336 0.039 -0.268 (0.42) (-1.10) (0.27) (-1.19) (0.15) (-0.90) 3 0.116 -0.398 0.076 -0.433 0.028 -0.430 (0.57) (-1.75) (0.39) (-2.03) (0.13) (-1.95) 0.084 4 0.140 0.099 0.052 0.009 -0.020 (0.67) (0.41) (0.51) (0.28) (0.05) (-0.11) 5 0.125 0.046 0.099 0.009 0.046 0.002 (0.83) (0.29) (0.66) (0.06) (0.31) (0.01) 6 0.256 0.201 0.231 0.183 0.169 0.221 (1.64) (1.19) (1.47) (1.11) (1.07) (1.23) 7 0.543 0.462 0.564 0.479 0.485 0.487 (3.12) (3.20) (3.17) (3.38) (2.71) (3.29) 8 0.304 0.300 0.311 0.319 0.255 0.282 (1.79) (1.63) (1.82) (1.79) (1.56) (1.49) 9 0.682 0.439 0.729 0.505 0.683 0.525 (3.35) (1.62) (3.65) (1.92) (3.14) (1.80) 10 - High 0.452 0.167 0.463 0.246 0.505 0.287 (2.19) (0.45) (2.14) (0.62) (2.27) (0.71) High - Low 0.988 0.955 1.022 1.050 0.972 1.005 (2.54) (1.51) (2.67) (1.66) (2.33) (1.55) FH 7 Factors EW VW -0.698 -1.079 (-1.91) (-2.07) -0.109 -0.484 (-0.42) (-1.56) -0.086 -0.801 (-0.33) (-2.46) -0.255 -0.436 (-1.30) (-1.87) 0.034 0.066 (0.18) (0.34) 0.272 0.108 (1.46) (0.55) 0.566 0.482 (3.33) (2.76) 0.403 0.428 (1.95) (1.88) 0.731 0.611 (2.79) (1.65) 0.553 0.519 (1.89) (0.93) 1.251 1.597 (2.86) (2.06) Table 8: Returns of idiosyncratic crash perception portfolios This table presents monthly mean excess returns, abnormal returns (alphas), and Newey-West t -statistics of portfolios formed on stock's idiosyncratic crash perception. In Panel A, we consider portfolio formation at monthly, quarterly, semiannual and annual frequencies, and report both equal-weighted (EW) and value-weighted (VW) portfolio excess returns. In Panel B, we report alphas of monthly formed perceived idiosyncratic crash portfolios using the following set of factors: (1) Fama-French-Carhart (FFC) market, size, value, and momentum factors; (2) FFC 4 factors augmented with short-term reversal factor; (3) FFC 4 factors plus Pastor-Stambaugh liquidity factor; and (4) Fung-Hsieh primitive trend-following factors. On average, there are 65-66 stocks within each decile. Panel A: Monthly excess returns at different portfolio formation frequencies Monthly Formation Quarterly Formation Semi-annual Formation Portfolio Ranking EW VW EW VW EW VW 1 - Low 0.284 -0.078 0.149 -0.349 0.314 -0.552 (0.50) (-0.14) (0.26) (-0.58) (0.53) (-0.96) 2 0.568 0.480 0.730 0.517 0.564 0.485 (1.03) (1.11) (1.31) (1.12) (1.01) (1.09) 3 0.368 0.282 0.157 0.213 0.416 0.219 (0.69) (0.61) (0.28) (0.50) (0.77) (0.50) 4 0.483 0.052 0.658 0.349 0.627 0.269 (0.98) (0.13) (1.37) (0.83) (1.24) (0.67) 5 0.482 0.388 0.517 0.323 0.471 0.338 (0.92) (0.83) (1.02) (0.73) (0.97) (0.80) 0.524 0.403 0.470 6 0.494 0.607 0.503 (1.04) (0.95) (0.93) (1.19) (1.20) (1.16) 7 0.483 0.138 0.604 0.172 0.720 0.439 (1.00) (0.33) (1.21) (0.41) (1.41) (1.07) 8 0.726 0.373 0.705 0.326 0.821 0.373 (1.34) (0.89) (1.33) (0.80) (1.58) (0.85) 9 0.981 0.734 0.982 0.729 0.847 0.594 (1.86) (1.75) (1.85) (1.64) (1.57) (1.34) 10 - High 1.195 1.000 1.159 0.857 1.153 0.899 (2.19) (2.36) (2.08) (1.99) (2.01) (1.97) High - Low 0.911 1.078 1.010 1.206 0.839 1.451 (3.22) (3.00) (3.25) (2.93) (2.76) (3.72) Annual Formation EW VW 0.281 -0.429 (0.46) (-0.72) 0.479 0.169 (0.86) (0.39) 0.672 0.358 (1.35) (0.91) 0.602 0.214 (1.20) (0.51) 0.546 0.484 (1.15) (1.17) 0.332 0.759 (1.55) (0.76) 0.635 0.405 (1.18) (0.99) 0.977 0.568 (1.87) (1.25) 0.970 0.649 (1.77) (1.48) 1.106 0.834 (1.83) (1.74) 0.825 1.263 (2.27) (3.06) Panel B: Monthly abnormal returns based on different benchmark factors FFC 4 Factors FFC 4 Factors + ST Rev FFC 4 Factors + PS LIQ Portfolio Ranking EW VW EW VW EW VW 1 - Low -0.100 -0.299 -0.115 -0.301 -0.120 -0.217 (-0.53) (-1.40) (-0.59) (-1.36) (-0.65) (-1.06) 2 0.178 0.261 0.164 0.264 0.158 0.289 (1.19) (1.36) (1.07) (1.34) (1.07) (1.45) 3 0.030 0.053 -0.001 0.040 0.002 0.058 (0.22) (0.37) (-0.00) (0.28) (0.02) (0.41) -0.165 4 0.137 0.121 -0.181 0.054 -0.180 (0.87) (-1.21) (0.78) (-1.37) (0.36) (-1.34) 5 0.109 0.134 0.085 0.128 0.066 0.169 (0.71) (0.74) (0.58) (0.71) (0.39) (0.92) 6 0.132 0.046 0.116 0.054 0.042 0.032 (0.85) (0.26) (0.76) (0.31) (0.28) (0.17) 7 0.100 -0.140 0.093 -0.154 0.065 -0.209 (0.64) (-0.89) (0.59) (-0.98) (0.39) (-1.29) 8 0.333 0.055 0.328 0.051 0.268 -0.018 (1.68) (0.29) (1.63) (0.26) (1.35) (-0.10) 9 0.578 0.443 0.586 0.452 0.547 0.484 (2.69) (2.07) (2.69) (2.17) (2.46) (2.44) 10 - High 0.699 0.597 0.704 0.600 0.673 0.559 (3.37) (2.78) (3.33) (2.79) (3.25) (2.74) High - Low 0.799 0.897 0.819 0.901 0.792 0.777 (2.94) (2.84) (2.93) (2.77) (3.05) (2.48) FH 7 Factors EW VW -0.026 -0.205 (-0.08) (-0.62) 0.015 0.276 (0.06) (0.97) -0.132 -0.079 (-0.65) (-0.41) -0.095 -0.355 (-0.66) (-2.20) 0.015 -0.125 (0.08) (-0.62) 0.144 0.112 (0.83) (0.55) -0.079 -0.077 (-0.48) (-0.36) 0.309 -0.019 (1.44) (-0.10) 0.561 0.505 (2.48) (2.01) 0.714 0.855 (3.24) (3.26) 0.739 1.060 (1.99) (2.26) Table 9: Returns of double-sorted co-crash and idiosyncratic crash perception portfolios At the end of each month from December 1997 through November 2012, we form two-way sequentially sorted portfolios, hold portfolios for one month, and calculate equal-weighted returns. In Panel A, we rank stocks into 25 portfolios first on co-crash perception then on idiosyncratic crash perception; in Panel B, we rank stocks into 25 portfolios first on idiosyncratic crash perception then on co-crash perception. The last column of each panel reports the Fama-French-Carhart four-factor alphas (in percent) of high-minus-low quintile, and the last two rows of each panel reports average returns (in percent) and Newey-West t -statistics (in parentheses) of five quintiles. On average, there are 26-27 stocks in each of 25 portfolios. Panel A: 5×5 portfolios sorted first on perceived co-crash then on idiosyncratic crash Idiosy. Idiosy. Crash Crash Co-Crash 2 3 4 5-1 1 - Low 5 - High 1 - Low -0.215 -0.289 -0.088 0.475 0.928 1.143 (-0.25) (-0.38) (-0.12) (0.63) (1.22) (2.34) 2 0.011 0.426 0.308 0.444 0.853 0.842 (0.02) (0.79) (0.58) (0.83) (1.54) (2.35) 3 0.447 0.497 0.333 0.441 0.915 0.468 (0.97) (1.23) (0.84) (1.07) (2.11) (1.41) 4 0.198 0.642 0.857 1.027 1.569 1.371 (0.47) (1.59) (2.20) (2.42) (2.84) (2.76) 5 - High 0.704 0.840 1.267 1.289 1.249 0.545 (1.25) (1.58) (2.18) (2.40) (2.05) (1.39) Average 0.229 0.423 0.535 0.735 1.103 0.874 (0.43) (0.88) (1.12) (1.54) (2.13) (3.20) FFC 4-Factor Alpha 1.012 (2.15) 0.727 (2.05) 0.254 (0.80) 1.054 (2.47) 0.362 (0.97) 0.682 (2.77) Panel B: 5×5 portfolios sorted first on perceived idiosyncratic crash then on co-crash Idiosy. Crash Co-Crash 1 - Low 2 3 1 - Low -0.240 (-0.32) -0.081 (-0.10) 0.005 (0.01) 0.214 (0.28) 1.193 (1.55) 0.218 (0.30) 0.018 (0.04) -0.031 (-0.05) 0.177 (0.31) 0.315 (0.54) 0.762 (1.34) 0.248 (0.48) 0.568 (1.12) 0.524 (1.17) 0.223 (0.52) 0.415 (0.91) 1.101 (2.42) 0.566 (1.35) 2 3 4 5 - High Average 4 Co-Crash 5 - High 5-1 0.915 (1.83) 0.642 (1.50) 0.825 (2.11) 0.586 (1.52) 1.055 (2.80) 0.805 (2.11) 0.842 (1.15) 1.051 (1.79) 1.255 (2.63) 1.457 (2.80) 1.324 (2.51) 1.186 (2.29) 1.082 (1.82) 1.133 (2.04) 1.250 (2.18) 1.242 (2.15) 0.131 (0.24) 0.967 (2.24) FFC 4-Factor Alpha 0.781 (1.40) 0.826 (1.67) 1.055 (2.23) 1.160 (2.35) -0.012 (-0.03) 0.762 (2.15) Table 10: Crash perception portfolios' exposures to market, liquidity, macroeconomic, and tail risk We monthly form decile portfolios based on stock's perceived co-crash (Panel A) and idiosyncratic crash (Panel B). This table presents portfolios' exposures to the following set of factors: (1) CRSP value-weighted market excess returns; (2) Pastor-Stambaugh market liquidity innovation; (3) HuPan-Wang "noise" measure of liquidity risk; (4) the change in CBOE Volatility Index (VIX); (5) the change in term spread; (6) the change in default yield; (7) Kelly-Jiang market tail risk; (8) implied volatility skewness based on the S&P 500 index OTM put and ATM call options; and (9) risk-neutral highorder moment risk of variance, skewness, and kurtosis based on the S&P 500 index options. In specifications (2)-(9) when performing time series regression of monthly portfolio returns on factors, we always include the market return factor. Panel A: Perceived co-crash portfolios Portfolio Market Liquidity Ranking Return Risk (P-S) Equal-weighted returns 1 - Low 1.909 -0.048 (16.69) (-0.71) 5 1.037 -0.010 (20.82) (-0.38) 10 - High 1.444 -0.077 (25.85) (-1.63) High - Low -0.465 -0.028 (-3.85) (-0.41) Value-weighted returns 1 - Low 1.875 0.023 (12.47) (0.29) 5 0.916 -0.022 (18.82) (-1.01) 10 - High 1.434 -0.069 (11.04) (-0.91) High - Low -0.441 -0.092 (-2.12) (-0.79) Noise Liq. (H-P-W) Volatility Risk Term Risk Default Risk Market Tail Imp. Vola. Risk (K-J) Skew. 0.006 (1.55) -0.002 (-1.04) -0.002 (-0.67) -0.008 (-2.48) 0.001 (0.90) -0.001 (-2.98) -0.000 (-0.01) -0.001 (-0.83) 0.011 (0.68) -0.006 (-0.80) 0.003 (0.27) -0.008 (-0.51) 0.020 (0.85) -0.007 (-0.42) -0.005 (-0.27) -0.025 (-1.20) -0.090 (-0.82) -0.048 (-0.83) -0.015 (-0.11) 0.075 (0.54) 0.000 (0.07) -0.001 (-0.25) -0.001 (-0.24) -0.001 (-0.31) 0.003 (1.68) -0.001 (-1.86) 0.001 (0.70) -0.001 (-0.46) -0.004 (-0.20) 0.003 (0.45) 0.006 (0.49) 0.010 (0.50) 0.016 (0.51) -0.013 (-1.03) -0.001 (-0.01) -0.016 (-0.49) -0.056 (-0.44) -0.099 (-1.64) -0.094 (-0.45) -0.038 (-0.16) RN Mom. (Var.) RN Mom. (Skew.) RN Mom. (Kurt.) 0.086 (1.05) -0.049 (-1.06) 0.019 (0.27) -0.066 (-0.79) 0.633 (1.28) -0.318 (-1.40) -0.314 (-1.19) -0.947 (-1.97) -0.005 (-0.49) 0.001 (0.40) 0.000 (0.02) 0.006 (0.52) 0.000 (0.28) -0.000 (-0.40) 0.000 (0.11) -0.000 (-0.19) 0.133 (1.09) -0.020 (-0.42) 0.133 (1.08) -0.000 (-0.00) 1.218 (2.08) -0.200 (-0.99) 0.173 (0.40) -1.045 (-1.37) 0.000 (0.02) 0.001 (0.23) -0.000 (-0.00) -0.000 (-0.02) -0.000 (-0.29) 0.000 (0.15) -0.001 (-0.66) -0.000 (-0.27) Panel B: Perceived idiosyncratic crash portfolios Portfolio Market Liquidity Noise Liq. Ranking Return Risk (P-S) (H-P-W) Equal-weighted returns 1 - Low 1.329 0.016 -0.005 (19.92) (0.34) (-1.67) 5 1.285 -0.042 0.000 (27.86) (-1.15) (0.09) 10 - High 1.217 -0.058 0.002 (26.96) (-1.88) (1.09) High - Low -0.112 -0.074 0.007 (-1.30) (-1.81) (1.82) Value-weighted returns 1 - Low 1.284 -0.010 -0.006 (19.70) (-0.33) (-2.25) 5 1.058 -0.048 0.002 (24.24) (-1.61) (1.06) 10 - High 0.882 -0.054 -0.000 (14.57) (-1.58) (-0.07) High - Low -0.402 -0.044 0.006 (-3.74) (-0.97) (1.56) Volatility Risk Term Risk Default Risk Market Tail Imp. Vola. Risk (K-J) Skew. 0.000 (0.36) -0.001 (-1.05) -0.001 (-1.52) -0.001 (-1.40) 0.002 (0.15) 0.003 (0.48) 0.011 (1.17) 0.009 (0.59) -0.008 (-0.46) -0.006 (-0.55) 0.012 (0.88) 0.020 (0.88) -0.059 (-0.69) -0.031 (-0.53) -0.059 (-0.60) 0.000 (0.00) -0.000 (-0.36) 0.001 (1.28) -0.001 (-0.83) -0.000 (-0.28) 0.006 (0.47) -0.007 (-1.22) -0.002 (-0.25) -0.008 (-0.45) -0.009 (-0.55) -0.004 (-0.53) 0.013 (0.82) 0.022 (0.80) -0.104 (-1.30) -0.055 (-1.01) -0.015 (-0.19) 0.089 (0.71) RN Mom. (Var.) RN Mom. (Skew.) RN Mom. (Kurt.) 0.080 (1.43) -0.057 (-1.37) 0.005 (0.09) -0.076 (-1.18) -0.547 (-1.85) -0.088 (-0.46) 0.018 (0.09) 0.565 (1.87) -0.009 (-1.35) 0.002 (0.72) -0.001 (-0.29) 0.008 (1.03) 0.001 (1.64) -0.000 (-1.15) 0.000 (0.91) -0.001 (-0.83) 0.119 (1.79) -0.066 (-1.60) -0.067 (-1.15) -0.186 (-1.87) -0.113 (-0.50) 0.144 (0.88) 0.008 (0.04) 0.121 (0.37) -0.002 (-0.36) 0.005 (1.31) -0.000 (-0.01) 0.002 (0.23) 0.000 (0.59) -0.000 (-0.93) 0.000 (0.66) 0.000 (0.03) Table 11: Fama-MacBeth cross-sectional regressions This table presents results from Fama-MacBeth (1973) regressions of stocks' realized excess returns in month t +1 on their decile rankings of crash perception and other explanatory variables as of month t . Crash perception includes total crash, co-crash, and idiosyncratic crash. The explanatory variables of Panels A, B, and C, respectively, control for return-moment and tail-distribution-based characteristics, option-based stock characteristics, and conventional firm characteristics. Intercepts are included in regressions but not reported. Newey-West (1987) t -statistics are reported in parentheses. Appendix 1 provides a description of all explanatory variables in regressions. Panel A: Return-moment and tail-distribution-related characteristics (1) (2) (3) (4) Total Crash 0.00038 0.00046 (2.99) (2.39) Co-Crash 0.00070 0.00153 (2.59) (2.76) Idiosyncratic Crash 0.00092 0.00124 (3.16) (2.87) Downside Beta 0.00000 -0.00006 (0.00) (-0.02) Idiosyncratic Volatility -0.16383 0.14294 (-1.40) (0.74) Systematic Skewness 0.00011 0.00020 (0.93) (1.24) Idiosyncratic Skewness -0.00001 -0.00035 (-0.02) (-0.48) Systematic Kurtosis -0.00001 -0.00001 (-0.67) (-0.68) MAX -0.01986 -0.02490 (-1.54) (-1.12) Hybrid Tail Covariance Risk 1.37324 0.39428 (2.89) (0.88) Lower Partial Moment (LPM) LPM Beta (5) 0.00045 (2.38) -0.06222 (-1.60) (6) 0.00154 (2.80) 0.00127 (2.99) 0.02979 (0.72) Market Tail Risk Beta Avg. # of Stocks Avg. Adjusted R-square # of Time-Series Obs. 1305 0.08 203 655 0.10 180 1317 0.01 203 655 0.03 180 1317 0.01 203 655 0.04 180 (7) 0.00047 (2.39) (8) 0.00157 (2.96) 0.00127 (3.06) 0.00123 (1.35) 0.00101 (1.07) 1317 0.01 203 655 0.03 180 (9) 0.00042 (2.44) 0.00172 (0.85) 1204 0.02 203 (10) 0.00152 (2.99) 0.00119 (2.98) 0.00333 (1.36) 641 0.05 180 Panel B: Option-based information Total Crash Co-Crash (1) 0.00031 (2.04) Idiosyncratic Crash Option-to-Stock Volume Ratio ATM Implied Volatility Slope of Implied Volatility Implied Volatility Smirk -0.01135 (-1.43) -0.00656 (-1.07) -0.06798 (-8.57) (2) 0.00091 (2.39) 0.00086 (2.45) -0.01294 (-1.13) 0.00474 (0.59) -0.08127 (-5.25) Volatility Spread (3) 0.00016 (1.08) (4) -0.00847 (-1.08) -0.00626 (-1.02) 0.00079 (2.29) 0.00074 (2.25) -0.00763 (-0.83) 0.00544 (0.72) -0.04911 (-7.49) -0.06377 (-5.11) Innovation of Imp. Vol. Call (5) 0.00029 (1.90) (6) -0.00943 (-1.18) -0.00616 (-1.00) 0.00092 (2.41) 0.00086 (2.43) -0.01158 (-1.01) 0.00535 (0.66) 0.09433 (10.76) 0.09034 (5.73) Risk-Neutral Variance (7) 0.00035 (2.34) (8) -0.01244 (-1.63) -0.00768 (-1.23) 0.00082 (2.26) 0.00088 (2.59) -0.01091 (-0.99) 0.00362 (0.45) 0.01348 (2.12) 0.01578 (1.92) 1312 0.06 202 653 0.08 180 Risk-Neutral Skewness Risk-Neutral Kurtosis Avg. # of Stocks Avg. Adjusted R2 # of Time-Series Obs. 1315 0.06 203 653 0.08 180 1312 0.06 203 653 0.08 180 1313 0.06 203 653 0.08 180 (9) 0.00018 (1.15) (10) -0.00941 (-1.32) 0.00085 (2.20) 0.00087 (2.51) -0.00267 (-0.29) -0.09556 (-1.96) 0.00618 (3.62) 0.00039 (1.23) 1136 0.06 203 -0.01260 (-0.21) 0.00446 (1.29) -0.00050 (-0.72) 638 0.08 180 Panel C: Commonly used firm characteristics (1) (2) Total Crash 0.00046 (2.36) Co-Crash 0.00112 (2.44) Idiosyncratic Crash Market Beta (3) (4) 0.00078 (2.65) 0.00154 (2.75) 0.00123 (2.83) 655 0.01 180 655 0.03 180 Log(Market Equity) Log(B/M Equity) Return (t-1) Return (t-2, t-12) Net Stock Issuance Operating Accruals Amihud Illiquidity Avg. # of Stocks Avg. Adjusted R-square # of Time-Series Obs. 1322 0.00 203 655 0.02 180 (5) 0.00035 (2.83) 0.00057 (0.14) -0.00026 (-0.33) 0.00116 (1.25) -0.01281 (-1.86) 0.00048 (0.17) -0.01819 (-4.90) 0.00113 (0.22) -0.02983 (-0.67) 1092 0.09 203 (6) 0.00069 (2.22) 0.00087 (2.85) 0.00174 (0.38) -0.00214 (-2.34) -0.00056 (-0.51) -0.01690 (-2.32) 0.00135 (0.34) -0.01260 (-3.13) -0.00015 (-0.02) -1.05915 (-2.13) 562 0.11 180 Figure 1: Market crash perception and one-quarter-ahead real GDP forecast The market-level total crash perception (TCR) is measured as a weighted average of OTM put prices across moneyness of S&P 500 index options. The U.S. real GDP forecasts for quarter-on-quarter growth, expressed in annualized percentage points, are obtained from the Blue Chip Financial Forecasts (BCFF) surveys. At the end of each month from January 1996 through December 2012, we estimate forecast distribution across all non-missing values provided by professional forecasters, including the 10th percentile (Pctl_10), the standard deviation (dispersion), and the skewness of one-quarter-ahead real GDP growth forecasts. The values of market TCR are multiplied by 100 and presented on the left axis, and the values of forecast statistics are presented on the right axis. Market TCR (LHS) Dispersion (RHS) Pctl_10 (RHS) Skewness (RHS) 5 7 4 6 3 5 2 1 4 0 3 -1 -2 2 -3 1 -4 0 Sep-12 Apr-12 Nov-11 Jan-11 Jun-11 Aug-10 Mar-10 Oct-09 May-09 Dec-08 Jul-08 Feb-08 Apr-07 Sep-07 Nov-06 Jun-06 Jan-06 Aug-05 Mar-05 Oct-04 May-04 Dec-03 Jul-03 Feb-03 Sep-02 Apr-02 Jun-01 Nov-01 Jan-01 Aug-00 Mar-00 Oct-99 May-99 Dec-98 Jul-98 Feb-98 Apr-97 Sep-97 Nov-96 Jun-96 Jan-96 -5 Figure 2: Annual returns and Sharpe ratios of long-short portfolios formed on co-crash and idiosyncratic crash perception At the end of each month from December 1997 through November 2012, we rank CRSP/OptionMetrics common stocks into 10 decile portfolios according to their co-crash and idiosyncratic crash perception estimated from out-of-the-money put options. A long-short portfolio on co-crash is constructed by going long on stocks in decile 10 (the hightest co-crash) and going short on stocks in decile 1 (the lowest co-crash). A long-short portfolio on perceived idiosyncratic crash is constructed in a similar way. Portfolios are monthly rebalanced and portfolio returns are equal weighted. This figure presents year-by-year annual returns and Sharpe ratios of long-short portfolios. Co-Crash Annual Return of Long-Short Portfolio Idiosyn. Crash 0.6 0.5 0.4 0.3 0.2 0.1 0 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 -0.1 -0.2 -0.3 Co-Crash Annual Sharpe Ratio of Long-Short Portfolio Idiosyn. Crash 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Figure 3: Industry composition within the highest crash perception portfolio This figure shows the time series of Fama-French 12-industry composition within the decile of the highest perceived co-crash (red line) and the decile of the highest perceived idiosyncratic crash (green line). Decile portfolios on stock's crash perception are monthly formulated (see Table 5 in detail). To get the time series of industry composition within the decile of the highest co-crash (Decile 10), for example, we find the industry with the highest likelihood of being ranked into this decile at time of portfolio formation, and then plot industries over time. Given an industry, we estimate its likelihood into a decile as follows: We count total number of stocks at portfolio formation, divide by ten to get expected number of stocks in each decile, and estimate the ratio between realized and expected number. The following is a brief description of Fama-French 12 industries: 1, Consumer NonDurables (Food, Tobacco, Textiles, Apparel, Leather, Toys); 2, Consumer Durables (Cars, TV's, Furniture, Household Appliances); 3, Manufacturing (Machinery, Trucks, Planes, Off Furn, Paper, Com Printing); 4, Energy (Oil, Gas, and Coal Extraction and Products); 5, Chemicals and Allied Products; 6, Business Equipment (Computers, Software, and Electronic Equipment); 7, Telephone and Television Transmission; 8, Utilities; 9, Wholesale, Retail, and Some Services (Laundries, Repair Shops); 10, Healthcare, Medical Equipment, and Drugs; 11, Finance; 12, Other (Mines, Constr, BldMt, Trans, Hotels, Bus Serv, Entertainment). High Co-Crash (Decile 10) High Idiosyn. Crash (Decile 10) 12 11 10 9 8 7 6 5 4 3 2 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr-00 Aug-00 Dec-00 Apr-01 Aug-01 Dec-01 Apr-02 Aug-02 Dec-02 Apr-03 Aug-03 Dec-03 Apr-04 Aug-04 Dec-04 Apr-05 Aug-05 Dec-05 Apr-06 Aug-06 Dec-06 Apr-07 Aug-07 Dec-07 Apr-08 Aug-08 Dec-08 Apr-09 Aug-09 Dec-09 Apr-10 Aug-10 Dec-10 Apr-11 Aug-11 Dec-11 Apr-12 Aug-12 1 Appendix 1: Descriptions of all variables used in our empirical analysis We extract stock price, return, trading volume, and historical industry classifications (SIC codes) from CRSP, and we obtain firm accounting information from Compustat. We also extract analyst forecasts from I/B/E/S, and collect U.S. equity and S&P 500 index options from OptionMetrics. The following table briefly describes the main variables used in our empirical analyses. Market equity : stock price times shares outstanding. We take the natural logarithm of this number. Book-to-Market equity : annual book value of equity divided by market value of equity. We follow the convention in Fama and French (1992) to calculate book value of equity. We also take the natural logarithm of this number. Net stock issuance : The log level change in split-adjusted shares outstanding (Fama and French, 2008). Operating accruals : The change of current assets excluding cash and short-term investments, minus the change of current liabilities excluding short-term debt and taxes payable, minus the change of depreciation and amortization (Sloan, 1996). Return(t-1) : Monthly holding period return at end of previous month t-1. Return(t-2,t-12) : Past 11-month cumulative returns from t-12 to t-2. Amihud illiquidity : Average of daily absolute value of return divided by dollar volume over a one-month period, scaled by 106 (Amihud, 2002). Earnings surprise : Standardized unexpected earnings (SUE) based on seasonal random walk model with drift. Earnings definition follows the convention in Chan, Jegadeesh, and Lakonishok (1996). Downside beta : it is estimated from regressing daily stock excess returns on the market factor where we use only market-down trading days over the past one year (i.e., the market daily return is below its average over the oneyear sample period) (Ang, Chen, and Xing, 2006). Idiosyncratic volatility : it is calculated as residuals from daily Fama-French three-factor regression over the past one-year horzion (Ang, Hodrick, Xing, and Zhang, 2006). Systematic skewness and Idiosyncratic skewness : we regress a stock's daily excess return on the market excess return and its squared term using the past one-year sample period. The systematic skewness is the coefficient on the squared term, and the idiosyncratic skewness is the sample skewness of regression residuals (Harvey and Siddique, 2000; Bali, Cakici, and Whitelaw, 2011). Systematic kurtosis : we regress a stock's daily excess return on the market excess return, its squared term, and its cubic term, using the past one-year sample period. The regression coefficient on the cubic term is the estimated systematic kurtosis (Dittmar, 2002). Maximum daily return (MAX) : the maximum daily return within a month (Bali, Cakici, and Whitelaw, 2011). Option open interest : the average daily open interest of all option contracts within a month (calls and puts regardless of the moneyness). Option volume : the monthly total trading volume of all option contracts (calls and puts regardless of the moneyness). At-the-money (ATM) implied volatility : we separately calculate arithmetic average of the Black-Scholes (1973) implied volatility of all traded ATM calls and puts. Then we take the arithmetic average of these two numbers. Option-to-Stock volume ratio (O/S) : the ratio between monthly option trading volume and the underlying stock trading volume (Johnson and So, 2012). Slope of implied volatility : the difference between the fitted 30-day implied volatlity of put option (at delta = -0.5) and that of call option (at delta = 0.5) (Yan, 2011). Implied volatility smirk : It is the volatility skew measure defined in Xing, Zhang, and Zhao (2010). Moneyness (strike-spot ratio) 0.95 and 1.05 is used as thresholds to define ITM, ATM, and OTM options. Volatility spread : we first match calls and puts with the same strike and time-to-maturity. Then we calculate the weighted average of the difference in implied volatilities between matched call and put pairs (Ofek, Richardson, and Whitelaw, 2004; Cremers and Weinbaum, 2010; Jin, Livnat, and Zhang, 2012). Innovation of implied volatility : We use OptionMetrics 30-day implied volatility surface to get end-of-month ATM implied volatility of calls (delta = 0.5). The innovation measure is the difference in implied volatilities between this month and last month (An, Ang, Bali, and Cakici, 2014). Hybrid tail covariance risk, lower partial moments (LPM), and co-lower partial moments (LPM beta) : We use each stock and the market daily returns within the past six months to construct these tail risk measures. The threshold of return distribution is the lower 10th percentile. Detailed procedures are described in Bali, Cakici, and Whitelaw (2014). Tail risk beta : We follow Kelly and Jiang (2014) in constructing market-level tail risk factor. We fix the threshold at the 5th percentile of the daily stock returns in the cross section at the end of each month. Then we apply Hill's (1975) power law estimator to the set of daily return observations for all common stocks in a month. After obtaining the tail risk factor, we use 60-month rolling-window regressions to estiamte each stock's tail risk beta (we require at least 36 months of stock returns). Risk-neutral moments : We follow Conrad, Dittmar, and Ghysels (2013) in constructing individual stocks' riskneutral moments (volatility, skewness, and kurtosis). Employing the method in Bakshi, Kapadia, and Madan (2003), we use a set of option prices with one-month maturity and different strikes to construct second, third, and fourth powers of stock returns. We average daily estimates to get monthly risk-neutral moments (require at least 10 daily estimates available within a month). Negative coefficient of skewness and down-to-up volatility : We follow Chen, Hong, and Stein (2001) in constructing these measures of return distribution. We drop stocks with more than five missing daily return observations and use log price change in calculating daily returns. We also adjust each stock's daily return by subtracting it from the value-weighted CRSP market index return. Appendix 2: Daily open interest of individual equity options This table presents medians of daily open interest of stock options by year. Individual equity (comon stock) options are from OptionMetrics, and they must have non-zero open interest, standard expiration dates, non-missing implied volatility, valid bid and ask prices, and no-arbitrage bounds. We select equity options with 14-60 days of maturity and divide them into six moneyness groups (i.e., K/S, the ratio between option strike and underlying stock price). The first part of table presents results of put options and the second part presents results of call options. K/S ≤ 0.90 0.90 < K/S ≤ 0.95 0.95 < K/S ≤ 1.00 1.00 < K/S ≤ 1.05 1.05 < K/S ≤ 1.10 K/S > 1.10 Put options 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 All 111 85 85 97 86 130 175 257 282 350 334 348 265 365 302 230 218 212 100 70 67 73 80 112 135 187 245 292 339 341 288 399 333 276 281 219 80 63 60 66 74 102 117 151 215 262 318 343 283 352 308 260 270 200 62 55 54 60 68 92 102 113 152 185 227 250 243 248 207 183 167 150 52 50 50 51 61 77 87 84 110 130 156 169 197 170 147 143 122 112 53 46 48 48 50 66 83 65 94 112 125 142 195 115 117 153 122 95 Call options 1996 1997 1998 1999 2000 90 78 67 78 65 113 99 81 93 98 131 113 98 105 115 147 122 106 121 130 140 116 102 120 135 166 137 125 139 141 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 All 70 75 134 128 140 140 139 90 173 150 128 129 104 116 110 192 189 205 232 225 161 308 237 193 201 167 145 143 235 275 304 355 344 241 423 365 280 307 234 170 175 268 326 374 468 456 332 519 473 383 418 297 171 174 255 310 367 444 451 365 493 418 361 367 280 181 205 266 347 415 455 450 413 376 373 347 323 263 Appendix 3: Returns of total crash perception portfolios This table presents monthly mean excess returns, abnormal returns (alphas), and Newey-West t -statistics of portfolios formed on stock's total crash perception. In Panel A, we consider portfolio formation at monthly, quarterly, semi-annual and annual frequencies, and report both equal-weighted (EW) and value-weighted (VW) portfolio excess returns (in percent). In Panel B, we report alphas of monthly formed perceived total crash portfolios using the following set of factors: (1) FamaFrench-Carhart (FFC) market, size, value, and momentum factors; (2) FFC 4 factors augmented with short-term reversal factor; (3) FFC 4 factors plus Pastor-Stambaugh market liquidity risk factor; and (4) Fung-Hsieh primitive trend-following factors. On average, there are 132-133 stocks within each decile. Panel A: Monthly excess returns at different portfolio formation frequencies Monthly Formation Quarterly Formation Semi-annual Formation Portfolio Ranking EW VW EW VW EW VW 1 - Low 0.424 0.277 0.384 0.206 0.473 0.546 (0.95) (0.69) (0.88) (0.51) (1.02) (1.38) 2 0.382 0.366 0.403 0.404 0.425 0.409 (0.80) (0.83) (0.83) (1.01) (0.88) (1.09) 3 0.266 0.246 0.164 0.145 0.459 0.301 (0.58) (0.68) (0.34) (0.38) (0.96) (0.78) 4 0.334 0.431 0.519 0.649 0.647 0.643 (0.68) (1.17) (1.01) (1.81) (1.28) (1.78) 5 0.484 0.433 0.437 0.504 0.402 0.384 (0.98) (1.14) (0.87) (1.29) (0.79) (0.89) 0.488 0.524 0.506 6 0.550 0.564 0.413 (1.01) (1.43) (1.03) (1.41) (1.11) (0.98) 7 0.636 0.521 0.463 0.422 0.481 0.242 (1.32) (1.44) (0.94) (1.04) (0.97) (0.55) 8 0.650 0.478 0.637 0.572 0.579 0.503 (1.33) (1.28) (1.30) (1.47) (1.18) (1.31) 9 0.579 0.695 0.587 0.609 0.602 0.622 (1.22) (1.94) (1.22) (1.63) (1.24) (1.45) 10 - High 0.799 0.833 0.755 0.804 0.752 0.679 (1.71) (2.29) (1.54) (2.02) (1.55) (1.78) High - Low 0.374 0.556 0.370 0.598 0.279 0.132 (1.98) (2.39) (2.12) (2.58) (1.82) (0.70) Annual Formation EW VW 0.594 0.287 (1.27) (0.63) 0.504 0.268 (1.00) (0.72) 0.440 0.307 (0.86) (0.80) 0.657 0.662 (1.31) (1.81) 0.494 0.534 (0.92) (1.31) 0.389 0.564 (1.07) (0.93) 0.461 0.236 (0.91) (0.53) 0.577 0.435 (1.14) (0.99) 0.557 0.536 (1.09) (1.20) 0.751 0.677 (1.47) (1.66) 0.157 0.390 (1.09) (1.82) Panel B: Monthly abnormal returns based on different benchmark factors FFC 4 Factors FFC 4 Factors + ST Rev FFC 4 Factors + PS LIQ Portfolio Ranking EW VW EW VW EW VW 1 - Low -0.254 -0.236 -0.252 -0.226 -0.240 -0.194 (-2.08) (-1.54) (-2.04) (-1.48) (-1.90) (-1.21) 2 -0.254 -0.031 -0.247 -0.011 -0.266 0.031 (-2.57) (-0.23) (-2.52) (-0.08) (-2.66) (0.23) 3 -0.363 -0.167 -0.370 -0.158 -0.365 -0.128 (-3.37) (-1.49) (-3.45) (-1.41) (-3.16) (-1.12) 0.087 4 -0.276 -0.292 0.068 -0.309 0.083 (-2.56) (0.87) (-2.68) (0.68) (-2.91) (0.81) 5 -0.118 -0.007 -0.122 -0.020 -0.160 -0.078 (-0.98) (-0.05) (-1.00) (-0.17) (-1.25) (-0.66) 6 -0.146 0.079 -0.148 0.067 -0.205 0.017 (-1.30) (0.61) (-1.30) (0.53) (-1.84) (0.14) 7 0.012 0.073 0.012 0.084 -0.026 0.050 (0.10) (0.50) (0.10) (0.57) (-0.21) (0.33) 8 -0.015 -0.042 -0.025 -0.051 -0.052 -0.090 (-0.12) (-0.30) (-0.20) (-0.38) (-0.43) (-0.64) 9 -0.095 0.177 -0.109 0.177 -0.131 0.111 (-0.72) (1.17) (-0.83) (1.15) (-0.99) (0.73) 10 - High 0.161 0.275 0.111 0.264 0.120 0.224 (1.27) (2.35) (0.96) (2.26) (0.97) (1.94) High - Low 0.414 0.511 0.363 0.490 0.360 0.418 (2.37) (2.40) (2.26) (2.33) (2.07) (1.95) FH 7 Factors EW VW -0.197 -0.228 (-1.19) (-1.39) -0.456 -0.201 (-4.89) (-1.04) -0.430 -0.227 (-3.51) (-1.63) -0.426 -0.035 (-3.50) (-0.22) -0.171 -0.037 (-1.30) (-0.30) -0.163 0.086 (-1.38) (0.66) 0.015 0.159 (0.11) (0.95) 0.063 0.051 (0.40) (0.32) -0.010 0.189 (-0.06) (1.00) 0.316 0.483 (1.68) (2.42) 0.513 0.711 (2.70) (3.00) Appendix 4: Industry composition within stock's crash perception portfolios This table reports likelihood of Fama-French 12 industries that are ranked within each crash perception decile (crash perception portfolios include deciles formed on co-crash and idiosyncratic crash). Given an industry, we estimate its likelihood into a decile as follows: We count total number of stocks at portfolio formation, divide by ten to get expected number of stocks in each decile, estimate the ratio between realized and expected number, and calculate time-series average of the ratios over all portfolio formation months. To present results in the table, we normalize likelihoods of all industries within a decile so that the sum of probability equals one. The following is a brief description of Fama-French 12 industries: 1, Consumer NonDurables (Food, Tobacco, Textiles, Apparel, Leather, Toys); 2, Consumer Durables (Cars, TV's, Furniture, Household Appliances); 3, Manufacturing (Machinery, Trucks, Planes, Off Furn, Paper, Com Printing); 4, Energy (Oil, Gas, and Coal Extraction and Products); 5, Chemicals and Allied Products; 6, Business Equipment (Computers, Software, and Electronic Equipment); 7, Telephone and Television Transmission; 8, Utilities; 9, Wholesale, Retail, and Some Services (Laundries, Repair Shops); 10, Healthcare, Medical Equipment, and Drugs; 11, Finance; 12, Other (Mines, Constr, BldMt, Trans, Hotels, Bus Serv, Entertainment). Portfolio Ranking 1 - Consu. NonDurables 2 - Consu. 34 - Energy 5 - Chemi. Durables Manufac. Deciles are formed on co-crash perception 1 - Low 5.21 11.60 5.87 2 7.26 10.55 7.15 3 7.32 10.13 8.36 4 8.08 11.38 7.96 5 9.03 8.86 8.45 6 7.90 10.03 8.46 7 7.43 9.88 8.38 8 9.46 9.33 7.70 9 9.69 9.35 7.68 10 - High 6.88 9.58 5.80 Deciles are formed on idiosyncratic crash perception 1 - Low 5.78 8.25 5.46 2 7.04 9.13 7.29 3 8.18 10.82 6.82 4 7.83 10.58 8.17 5 7.22 10.62 8.61 6 7.50 9.36 8.62 7 7.96 8.71 8.27 8 8.34 8.82 8.47 9 7.13 10.41 8.31 10 - High 7.96 10.37 8.24 6 - Bus. Equip. 7 - Tele. 8Utilities 9 - Retail 10 Wholesale Healthcare 11 Finance 12 - Others 16.93 15.78 10.90 8.10 8.45 7.53 8.00 7.87 9.33 8.70 6.61 8.40 8.45 10.57 11.80 12.24 11.67 10.51 9.35 7.35 9.89 9.53 8.23 6.67 6.65 6.61 6.44 6.42 6.46 8.61 6.57 5.95 8.78 8.52 9.59 9.41 9.08 9.85 8.43 9.71 5.68 8.18 9.25 9.37 8.72 7.36 7.46 8.38 12.21 8.75 4.45 6.58 8.15 8.36 8.40 8.67 8.24 8.83 8.06 6.68 9.69 6.58 6.54 6.88 6.43 7.32 8.39 7.91 7.22 11.38 10.45 7.76 7.82 8.26 7.66 8.06 8.14 7.12 5.92 7.40 7.04 6.27 6.07 5.83 5.95 6.41 6.90 6.61 6.30 9.16 6.28 7.67 7.81 9.68 9.55 11.17 11.58 11.19 9.23 9.93 9.99 8.91 11.95 9.65 10.73 9.88 8.74 9.70 9.30 9.59 4.22 6.13 6.55 7.06 7.50 8.01 8.35 8.74 10.37 11.40 11.91 12.10 10.64 9.22 8.50 7.88 8.32 7.62 7.06 5.97 13.39 8.52 7.66 7.80 8.69 9.00 8.84 7.42 6.89 5.50 5.05 7.36 7.76 8.29 8.36 8.99 8.91 8.61 7.80 7.96 9.95 7.73 6.53 7.17 6.78 7.55 8.04 7.78 8.81 10.40 13.78 11.36 8.65 7.46 6.67 5.41 5.80 6.50 7.09 4.99 5.95 6.75 6.62 7.09 6.77 6.62 6.48 6.82 7.60 7.69 Appendix 5: Mean excess returns of industry-neutral portfolios formed on crash perception Within each of Fama-French 12 industries, we first rank stocks into ten groups according to their co-crash (Panel A) and idiosyncratic crash (Panel B). Then we form ten decile portfolios by combining stocks across industries: Decile 1 (10) consists of stocks with the lowest (highest) perceived crash risk from each Fama-French industry. The "High-Low" is a long-short hedge portfolio constructed by going long on stocks in decile 10 and going short on stocks in decile 1. We consider portfolio formation at monthly, quarterly, semi-annual and annual frequencies. This table shows both equal-weighted (EW) and valueweighted (VW) portfolio monthly excess returns (in percent) and Newey-West t -statistics (in parentheses). On average, there are 60-71 stocks within each decile. Panel A: Deciles are formed using co-crash perception Portfolio Monthly Formation Quarterly Formation Ranking EW VW EW VW 1 - Low 0.072 -0.383 0.432 -0.154 (0.10) (-0.51) (0.59) (-0.21) 2 0.383 0.215 0.339 -0.173 (0.63) (0.38) (0.54) (-0.29) 3 0.574 0.269 0.500 0.396 (1.07) (0.59) (0.89) (0.82) 4 0.413 0.190 0.576 0.373 (0.85) (0.47) (1.17) (0.91) 5 0.316 -0.047 0.308 -0.193 (0.69) (-0.11) (0.68) (-0.47) 0.792 0.124 0.490 6 0.057 (0.32) (1.75) (1.09) (0.14) 7 0.802 0.439 0.712 0.396 (1.83) (1.22) (1.70) (1.12) 8 0.828 0.781 0.891 0.900 (1.84) (1.91) (1.95) (2.13) 9 0.990 0.831 1.134 0.731 (1.95) (1.92) (2.23) (1.77) 10 - High 0.888 0.438 0.754 0.413 (1.44) (0.73) (1.27) (0.72) High - Low 0.815 0.821 0.322 0.567 (2.38) (1.71) (1.12) (1.25) Semi-annual Formation EW VW 0.636 0.168 (0.87) (0.23) 0.633 0.260 (1.03) (0.50) 0.669 0.481 (1.22) (1.02) 0.695 0.332 (1.47) (0.76) 0.395 0.121 (0.85) (0.30) 0.461 0.083 (1.05) (0.21) 0.812 0.560 (1.84) (1.46) 0.708 0.365 (1.50) (0.81) 0.954 0.449 (1.82) (1.04) 0.602 0.090 (0.98) (0.16) -0.033 -0.078 (-0.11) (-0.19) Annual Formation EW VW 0.949 0.303 (1.28) (0.42) 0.906 0.417 (1.50) (0.80) 0.788 0.526 (1.46) (1.11) 0.779 0.208 (1.65) (0.50) 0.624 0.209 (1.34) (0.55) 0.350 0.482 (1.12) (0.81) 0.642 0.280 (1.36) (0.64) 0.735 0.370 (1.61) (0.81) 0.630 0.111 (1.26) (0.26) 0.529 -0.326 (0.83) (-0.53) -0.421 -0.628 (-1.24) (-1.30) Panel B: Deciles are formed using idiosyncratic crash perception Portfolio Monthly Formation Quarterly Formation Ranking EW VW EW VW 1 - Low 0.431 -0.051 0.398 -0.096 (0.76) (-0.09) (0.68) (-0.15) 2 0.642 0.420 0.748 0.463 (1.15) (0.85) (1.35) (0.94) 3 0.560 0.545 0.509 0.236 (1.06) (1.17) (0.97) (0.49) 4 0.389 0.027 0.275 0.254 (0.77) (0.06) (0.50) (0.55) 5 0.596 0.454 0.641 0.375 (1.22) (1.09) (1.36) (0.92) 6 0.449 0.162 0.415 0.198 (0.86) (0.34) (0.82) (0.46) 7 0.579 0.280 0.568 0.267 (1.09) (0.68) (1.07) (0.65) 8 0.564 0.368 0.764 0.501 (1.15) (0.89) (1.51) (1.20) 9 0.883 0.241 0.888 0.129 (1.76) (0.59) (1.81) (0.33) 10 - High 0.955 0.672 0.879 0.527 (1.80) (1.81) (1.67) (1.42) High - Low 0.524 0.722 0.481 0.623 (2.63) (2.14) (2.23) (1.53) Semi-annual Formation EW VW 0.663 0.099 (1.12) (0.17) 0.610 0.292 (1.14) (0.62) 0.649 0.325 (1.15) (0.67) 0.341 0.273 (0.65) (0.61) 0.546 0.463 (1.11) (1.11) 0.554 0.260 (1.06) (0.57) 0.645 0.343 (1.25) (0.82) 0.801 0.528 (1.63) (1.30) 0.943 0.108 (1.85) (0.25) 0.793 0.530 (1.48) (1.26) 0.130 0.432 (0.61) (1.48) Annual Formation EW VW 0.757 0.528 (1.28) (1.12) 0.434 0.103 (0.80) (0.20) 0.674 0.107 (1.18) (0.22) 0.408 0.161 (0.90) (0.39) 0.624 0.344 (1.20) (0.74) 0.742 0.386 (1.51) (0.85) 0.779 0.437 (1.49) (1.15) 0.846 0.535 (1.68) (1.30) 0.969 0.155 (1.87) (0.35) 0.816 0.682 (1.48) (1.72) 0.059 0.154 (0.26) (0.54) Appendix 6: Mean excess returns of portfolios formed on industry-adjusted crash perception Stock's perceived co-crash and idiosyncratic crash are demeaned by industry (we use the Fama-French 12-industry classifciations). This table presents monthly mean excess returns (in percent) and Newey-West t -statistics (in parentheses) of portfolios formed on industry-adjusted co-crash (Panel A) and idiosyncratic crash (Panel B). Decile 1 (10) consists of stocks with the lowest (highest) crash perception, and "High-Low" is constructed by going long on stocks in decile 10 and going short on stocks in decile 1. We consider portfolio formation at monthly, quarterly, semi-annual and annual frequencies. The table shows both equal-weighted (EW) and value-weighted (VW) portfolio excess returns. On average, there are 65-66 stocks within each decile. Panel A: Deciles are formed using industry-adjusted co-crash perception Portfolio Monthly Formation Quarterly Formation Semi-annual Formation Ranking EW VW EW VW EW VW 1 - Low -0.037 -0.232 0.322 -0.199 0.587 0.062 (-0.05) (-0.27) (0.39) (-0.21) (0.73) (0.07) 2 0.550 0.411 0.383 0.316 0.689 0.730 (0.89) (0.69) (0.62) (0.50) (1.11) (1.21) 3 0.495 0.194 0.532 0.270 0.743 0.276 (0.96) (0.43) (0.98) (0.61) (1.38) (0.59) 4 0.524 0.307 0.630 0.323 0.682 0.338 (1.21) (0.81) (1.46) (0.88) (1.57) (0.88) 5 0.631 0.243 0.557 0.276 0.670 0.200 (1.55) (0.74) (1.35) (0.81) (1.60) (0.56) 0.585 0.109 0.442 6 -0.051 0.281 0.048 (0.30) (1.36) (1.05) (-0.14) (0.63) (0.12) 7 0.665 0.157 0.576 0.122 0.581 0.218 (1.53) (0.41) (1.34) (0.30) (1.31) (0.56) 8 0.915 0.654 0.873 0.594 0.754 0.613 (1.97) (1.38) (1.87) (1.28) (1.52) (1.36) 9 0.945 0.861 0.974 0.902 0.906 0.487 (1.67) (1.66) (1.75) (1.85) (1.69) (0.95) 10 - High 0.816 0.697 0.845 0.831 0.632 0.243 (1.23) (1.01) (1.29) (1.13) (0.96) (0.37) High - Low 0.852 0.929 0.524 1.030 0.046 0.181 (2.41) (1.63) (1.54) (1.85) (0.14) (0.32) Annual Formation EW VW 1.015 0.540 (1.25) (0.59) 0.823 0.619 (1.36) (1.05) 0.605 0.181 (1.21) (0.39) 0.701 0.272 (1.53) (0.67) 0.736 0.255 (1.70) (0.72) 0.168 0.577 (1.31) (0.46) 0.680 0.138 (1.54) (0.30) 0.539 0.543 (1.07) (1.31) 0.648 0.332 (1.20) (0.73) 0.712 0.352 (1.04) (0.51) -0.303 -0.189 (-0.84) (-0.33) Panel B: Deciles are formed using industry-adjusted idiosyncratic crash perception Portfolio Monthly Formation Quarterly Formation Semi-annual Formation Ranking EW VW EW VW EW VW 1 - Low 0.518 0.047 0.391 -0.143 0.573 -0.085 (0.92) (0.09) (0.70) (-0.25) (1.02) (-0.16) 2 0.756 0.650 0.860 0.701 0.633 0.352 (1.36) (1.36) (1.53) (1.46) (1.12) (0.80) 3 0.449 0.291 0.338 -0.035 0.469 0.352 (0.80) (0.58) (0.59) (-0.07) (0.84) (0.73) 4 0.394 0.168 0.391 0.383 0.526 0.248 (0.80) (0.39) (0.76) (0.87) (1.04) (0.57) 5 0.455 0.392 0.589 0.399 0.581 0.361 (0.93) (0.87) (1.24) (0.93) (1.20) (0.86) 6 0.488 0.112 0.368 0.240 0.551 0.267 (0.95) (0.25) (0.73) (0.55) (1.06) (0.59) 7 0.391 0.116 0.444 0.156 0.599 0.319 (0.73) (0.27) (0.83) (0.37) (1.11) (0.76) 8 0.667 0.370 0.891 0.496 0.831 0.489 (1.38) (0.95) (1.85) (1.23) (1.72) (1.16) 9 0.974 0.555 0.974 0.456 0.841 0.371 (1.86) (1.35) (1.85) (1.04) (1.58) (0.77) 10 - High 1.006 0.814 0.886 0.514 0.936 0.607 (1.91) (2.03) (1.70) (1.36) (1.75) (1.48) High - Low 0.488 0.767 0.496 0.658 0.363 0.692 (2.44) (2.34) (2.18) (1.66) (1.70) (1.93) Annual Formation EW VW 0.650 0.249 (1.17) (0.55) 0.426 0.061 (0.77) (0.13) 0.633 0.243 (1.12) (0.51) 0.486 0.206 (1.02) (0.50) 0.614 0.191 (1.24) (0.42) 0.710 0.577 (1.41) (1.43) 0.838 0.233 (1.64) (0.50) 0.805 0.413 (1.50) (0.95) 0.884 0.537 (1.66) (1.26) 1.001 0.808 (1.80) (1.95) 0.351 0.558 (1.50) (1.95) Appendix 7: Mean excess returns of portfolios formed on alternative measures of crash perception In Panel A, we measure stock's crash perception using both OTM put and call options, which takes into account both left and right tails of stock price. An increased value of this measure means that the market has increased ex-ante expecation on stock's future price going down (i.e., extreme downside movement along the left tail) and decreased ex-ante expectation on stock's future price rise up (i.e., extreme upside movement along the right tail). In Panel B, we use OTM puts expiring around 60 days to measure stock's crash perception in 60-day period. For each type of crash perception, we monthly construct decile portfolios and hold for one month. Decile 1 (10) consists of stocks with the lowest (highest) crash perception, and "High-Low" is constructed by going long on stocks in decile 10 and going short on stocks in decile 1. We then calculate both equal-weighted (EW) and value-weighted (VW) portfolio returns. This table presents monthly mean excess returns (in percent) and Newey-West t -statistics (in parentheses) of crash perception portfolios. Panel A: Crash perception measured by using both OTM calls and puts Portfolio Total Crash Perception Co-Crash Perception Ranking EW VW EW VW 1 - Low 0.403 0.341 0.086 -0.211 (0.94) (0.87) (0.11) (-0.27) 2 0.237 0.230 0.585 0.017 (0.51) (0.61) (0.92) (0.03) 3 0.363 0.347 0.451 -0.039 (0.80) (1.01) (0.84) (-0.08) 4 0.457 0.518 0.547 0.106 (0.98) (1.53) (1.18) (0.25) 5 0.418 0.361 0.555 0.237 (0.89) (1.00) (1.33) (0.62) 6 0.565 0.556 0.600 0.411 (1.22) (1.64) (1.59) (1.29) 7 0.576 0.493 0.677 0.447 (1.21) (1.32) (1.76) (1.34) 8 0.582 0.491 0.786 0.770 (1.27) (1.42) (1.88) (2.06) 9 0.636 0.763 0.986 0.674 (1.38) (2.15) (1.95) (1.44) 10 - High 0.798 0.784 0.819 0.675 (1.75) (2.18) (1.35) (1.25) 0.886 High - Low 0.395 0.443 0.733 (2.44) (1.90) (2.40) (1.91) Idiosyn. Crash Perception EW VW 0.447 0.116 (0.81) (0.22) 0.466 0.257 (0.83) (0.52) 0.475 0.312 (0.95) (0.71) 0.433 0.138 (0.83) (0.30) 0.577 0.249 (1.13) (0.57) 0.515 0.480 (1.00) (1.13) 0.624 0.293 (1.23) (0.72) 0.697 0.416 (1.41) (1.00) 0.715 0.572 (1.32) (1.37) 1.148 0.840 (2.10) (1.98) 0.701 0.724 (2.74) (2.14) Panel B: Crash perception measured over 60-day horizon Portfolio Total Crash Perception Co-Crash Perception Ranking EW VW EW VW 1 - Low 0.276 0.178 -0.030 -0.354 (0.67) (0.43) (-0.04) (-0.43) 2 0.447 -0.068 0.629 0.533 (0.99) (-0.19) (0.90) (0.81) 3 0.231 0.418 0.727 0.323 (0.51) (1.17) (1.26) (0.62) 4 0.424 0.384 0.379 0.249 (0.92) (1.12) (0.78) (0.56) 5 0.426 0.439 0.524 0.291 (0.92) (1.22) (1.21) (0.74) 6 0.416 0.556 0.527 0.354 (0.90) (1.59) (1.24) (0.95) 7 0.569 0.439 0.605 0.632 (1.26) (1.21) (1.45) (1.57) 8 0.446 0.690 0.865 0.694 (0.97) (1.94) (1.86) (1.68) 9 0.521 0.514 0.908 0.655 (1.12) (1.44) (1.79) (1.36) 10 - High 0.663 0.757 0.884 0.956 (1.46) (1.99) (1.26) (1.28) 1.310 High - Low 0.387 0.579 0.914 (2.26) (2.17) (2.00) (1.91) Idiosyn. Crash Perception EW VW 0.253 -0.127 (0.46) (-0.28) 0.324 0.224 (0.63) (0.52) 0.500 -0.003 (0.99) (-0.01) 0.560 0.382 (1.11) (0.94) 0.424 0.287 (0.87) (0.71) 0.646 0.361 (1.36) (0.91) 0.681 0.370 (1.44) (0.96) 0.764 0.615 (1.56) (1.63) 0.895 0.926 (1.78) (2.29) 1.097 0.623 (2.16) (1.57) 0.845 0.751 (2.82) (2.17)