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. Our main result
shows that crash perception is strongly priced, with average annual Sharpe ratios of 0.9 and 1.1 on
long-short hedge portfolios formed on perceived co-crash and idiosyncratic crash risks, respectively.
28
References
Albagli, E., C. Hellwig, and A. Tsyvinski, 2014, Dynamic dispersed information and the credit
spread puzzle. NBER Working Paper No. 19788.
Alti, Aydogan and Paul C. Tetlock, 2014, Biased Beliefs, Asset Prices, and Investment: A Structural
Approach, Journal of Finance 69, 325-361.
Amihud, Y., 2002, Illiquidity and stock returns: cross-section and time-series e¤ects. Journal of
Financial Markets 5, 31-56.
An, B.-J., A. Ang, T. G. Bali, and N. Cakici, 2014, The joint cross section of stocks and options.
Journal of Finance 69, 2279-2337.
Ang, A., J. Chen, and Y. Xing, 2006, Downside risk, Review of Financial Studies 19, 1191-1239.
Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang, 2006, The cross-section of volatility and expected
returns, Journal of Finance 61, 259-299.
Bakshi, G., N. Kapadia and D. Madan, 2003, Stock return characteristics, skew laws, and the
di¤erential pricing of individual equity options, Review of Financial Studies 16, 101-143.
Bali, T.G., N. Cakici, and R. F. Whitelaw, 2011, Maxing out: Stocks as lotteries and the crosssection of expected returns, Journal of Financial Economics 99, 427-446.
Bali, T.G., N. Cakici, and R. F. Whitelaw, 2014, Hybrid tail risk and expected stock returns: When
does the tail wag the dog? Review of Asset Pricing Studies 4.2: 206-246.
Barberis, N., 2013, The psychology of tail events: Progress and challenges, American Economic
Review 103, 611-616.
Barberis, Nicholas, Robin Greenwood, Lawrence Jin, and Andrei Shleifer, 2014, X-CAPM: An
extrapolative capital asset pricing model, Journal of Financial Economics forthcoming.
Barberis, N, and M. Huang, 2008, Stocks as lotteries: The implications of probability weighting for
security prices, American Economic Review 98, 2066-2100.
Barberis, Nicholas, Andrei Shleifer, and Robert W. Vishny, 1998, A model of investor sentiment,
Journal of Financial Economics 49, 307-343.
Baron M., and W. Xiong, 2014, Credit expansion and neglected crash risk, working paper, Princeton
University.
Barro, R. J., 2006, Rare disasters and asset markets in the twentieth century, Quarterly Journal of
Economics 121, 823-866.
29
Black, F., and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political
Economy 81, 637-654.
Bleck, A., and X. Liu, 2007, Market transparency and the accounting regime, Journal of Accounting
Research 45, 229-256.
Bollen, N. P. B., and R. E. Whaley, 2004, Does net buying pressure a¤ect the shape of implied
volatility functions? Journal of Finance 59, 711-753.
Bollerslev, T., V. Todorov, and S. Z. Li, 2013, Roughing up Beta: Continuous vs. Discontinuous
Betas, and the Cross-Section of Expected Stock Returns, working paper.
Boyer, B., T. Mitton, and K. Vorkink, 2010, Expected idiosyncratic skewness, Review of Financial
Studies 23, 169-202.
Britten-Jones, M., and A. Neuberger, 2000, Option prices, implied price processes, and stochastic
volatility, Journal of Finance 55, 839-866.
Brunnermeier, M. K., C. Gollier, and J. Parker, 2007, Optimal beliefs, asset prices and the preference for skewed returns, American Economic Review (P&P) 97, 159-165.
Carr, P., and L. Wu, 2009, Variance risk premia, Review of Financial Studies 22, 1311-1341.
Carhart, M., 1997, On persistence in mutual fund performance, Journal of Finance 52, 57-82.
Cecchetti, Stephen G., Pok-sang Lam, and Nelson C. Mark, 2000, Asset Pricing with Distorted
Beliefs: Are Equity Returns Too Good to Be True, American Economic Review 90, 787-805.
Chabi-Yo, F., Ruenzi, S., and W. Florian, 2014, Crash sensitivity and the cross-section of expected
stock returns, working paper.
Chan, Louis K.C, N. Jegadeesh and J. Lakonishok, 1996, Momentum strategies, Journal of Finance,
51, 1681-1713.
Chapman, D. A., and M. F. Gallmeyer, 2014, Aggregate Tail Risk, Economic Disasters, and the
Cross-Section of Expected Returns, working paper, University of Virginia.
Chen H, S. Joslin, and S. Ni, 2013, Demand for crash insurance, intermediary constraints, and
stock return predictability, working paper, MIT.
Cholette, L., and C.-C. Lu, 2011, The market premium for dynamic tail risk, Working Paper.
Cochrane, J. H., 2005, Asset Pricing. Princeton University Press.
Conrad, J., R. F. Dittmar, and E. Ghysels, 2013, Ex ante skewness and expected stock returns,
Journal of Finance 68, 85-124.
30
Cremers, M., and D. Weinbaum, 2010, Deviations from put-call parity and stock return predictability, Journal of Financial Quantitative Analysis 45, 335-367.
Daniel, Kent D., David Hirshleifer and Avanidhar Subrahmanyam, 1998, Investor psychology and
security market under- and over-reactions, Journal of Finance 53, 1839-1885.
Dittmar, R. F., 2002, Nonlinear pricing kernels, kurtosis preference, and evidence from the cross
section of equity returns, Journal of Finance 57, 369-403.
Du, J., and N. Kapadia, 2012, Tail and volatility indices from option prices, Working paper,
University of Massachusetts.
Epstein, L., and M. Schneider, 2008, Ambiguity, information quality and asset pricing, Journal of
Finance 63, 197-228.
Fama, E. F., and K. R. French, 1992, The cross-section of expected stock returns, Journal of
Finance 47, 427-465.
Fama, E. F., and K. R. French, 1993, Common risk factors in the returns on stocks and bonds,
Journal of Financial Economics 33, 3-56.
Fama, E. F., and J. D. MacBeth, 1973, Risk, return and equilibrium: Empirical tests, Journal of
Political Economy 81, 607-636.
Fama, E. F., and K. R. French, 2008, Dissecting anomalies, Journal of Finance 63, 1653-1678.
Fung, William, and David A. Hsieh, 2001, The risk in hedge fund strategies: theory and evidence
from trend followers, Review of Financial Studies 14, 313–341.
Gabaix, X., 2012, Variable rare disasters: An exactly solved framework for ten puzzles in macro…nance, Quarterly Journal of Economics 127, 645-700.
Gao, G. P., Gao P., and Z. Song, 2013, Do hedge funds exploit rare disaster concerns, Working
paper, Cornell University.
Garleanu, N., L. H. Pedersen, and A. M. Poteshman, 2009, Demand-based option pricing, Review
of Financial Studies 22, 4259-4299.
Gennaioli, Nicola, Andrei Shleifer, and Robert W. Vishny, 2012, Neglected Risks, Financial Innovation, and Financial Fragility, Journal of Financial Economics 104(3): 452-468.
Gennaioli, Nicola, Andrei Shleifer, and Robert W. Vishny, 2013, A Model of Shadow Banking,
Journal of Finance 68(4): 1331-1363.
Gilboa, I., and D. Schmeidler, 1989, Maxmin expected utility with a non-unique prior, Journal of
Mathematical Economics 18, 141-153.
31
Greenwood, R., and A. Shleifer. 2014. Expectations of Returns and Expected Returns, Review of
Financial Studies 27 (3): 714-746.
Harvey, C. R., and A. Siddique, 2000, Conditional skewness in asset pricing tests, Journal of
Finance 55, 1263-1295.
Heath, C., and A. Tversky, 1991, Preference and belief: Ambiguity and competence in choice under
uncertainty, Journal of Risk and Uncertainty 4, 5-28.
Hill, B. 1975. A simple general approach to inference about the tail of a distribution." The Annals
of Statistics, 3: 1163-1174.
Hu, X., J. Pan, and J. Wang, 2013, Noise as information for illiquidity, Journal of Finance 68,
2223-2772.
Hutton, A. P., A. J. Marcus, and H. Tehranian, 2009, Opaque …nancial reports, R2 , and crash risk,
Journal of Financial Economics 94, 67-86.
Jegadeesh, N. 1990. Evidence of predictable behavior of security returns. Journal of Finance, 45(3),
881-898.
Jegadeesh, Narasimhan, and Sheridan Titman, 1993, Returns to buying winners and selling losers:
Implications for stock market e¢ ciency, Journal of Finance 48, 65-91.
Jin, L., 2014, A Speculative Asset Pricing Model of Financial Instability, working paper, Yale
University.
Jin, W., J. Livnat, and Y. Zhang, 2012, Option prices leading equity prices: Do option traders
have an information advantage? Journal of Accounting Research 50, 401-432.
Jin, L., and Myers, S.C., 2006. R2 around the world: new theory and new tests. Journal of Financial
Economics 79, 257–292.
Johnson, T. L., So, E. C., 2012. The option to stock volume ratio and future returns. Journal of
Financial Economics 106, 262–286.
Kelly, B., and H. Jiang, 2014, Tail risk and asset prices. Review of Financial Studies, 27, 2841-2871.
Kim, J-B., and L. Zhang, 2013, Financial reporting opacity and expected crash risk: Evidence from
implied volatility smirks, Contemporary Accounting Research, forthcoming.
Kothari, S.P., S. Shu, and P. Wysocki, 2009, Do managers withhold bad news? Journal of Accounting Research 47, 241–276.
Li, F., 2008, Annual report readability, current earnings, and earnings persistence. Journal of
Accounting and Economics, 45(2), 221-247.
32
Liu, J., J. Pan, and T. Wang, 2005, An equilibrium model of rare-event premia and its implication
for option smirks, Review of Financial Studies 18, 131-164.
Loughran, T., and McDonald, B., 2011, When is a liability not a liability? Textual analysis,
dictionaries, and 10-Ks. Journal of Finance, 66(1), 35-65.
Loughran, T., and McDonald, B., 2014, Measuring readability in …nancial disclosures. Journal of
Finance, 69(4), 1643-1671.
Merkley, K. J., 2013, Narrative disclosure and earnings performance: Evidence from R&D disclosures. Accounting Review, 89(2), 725-757.
Merton, R. C., 1976, Option pricing when underlying stock returns are discontinuous, Journal of
Financial Economics 3, 125-144.
Mitton, T., and K. Vorkink, 2007, Equilibrium underdiversi…cation and the preference for skewness,
Review of Financial Studies 2007, 1255-1288.
Muravyev D., and N. Pearson, 2014, Option Trading Costs Are Lower Than You Think, Working
Paper.
Newey, W. K., and K. D. West, 1987, A simple, positive semi-de…nite, heteroskedasticity and
autocorrelation consistent covariance matrix, Econometrica 55, 703-708.
Ofek, E., M. Richardson, and R. F. Whitelaw, 2004, Limited arbitrage and short sales restrictions:
Evidence from the options markets, Journal of Financial Economics 74, 305-342.
Pastor, L., and R. F. Stambaugh, 2003, Liquidity risk and expected stock returns, Journal of
Political Economy 111, 642-685.
Ponti¤, J., and A. Woodgate, 2008, Share issuance and cross-sectional returns, Journal of Finance
63, 921-945.
Scheinkman, Jose and Wei Xiong, 2003, Overcon…dence and Speculative Bubbles, Journal of Political Economy 111, 1183-1219.
Sloan, R. G., 1996, Do stock prices fully re‡ect information in accruals and cash ‡ows about future
earnings? Accounting Review 71, 289-315.
Spitzer, J. F., 2006, Market crashes, market booms and the cross-section of expected returns,
Working Paper.
Wachter, J. A., 2013, Can time-varying risk of rare disasters explain aggregate stock market volatility? Journal of Finance 68, 987-1035.
Weitzman, M. L., 2007, Subjective expectations and asset-return puzzles, American Economic
Review 97, 1102-1130.
33
Xing, Y., X. Zhang, and R. Zhao, 2010, What does the individual option volatility smirk tell us
about future equity returns? Journal of Financial Quantitative Analysis 45, 641-662.
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)