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A tail of two worlds: Stock crashes, market contexts,

and expected returns*

S. Ghon Rhee

Shidler College of Business, University of Hawaii at Manoa

Room D-311b, 2404 Maile Way, Honolulu, HI 96822 USA

Tel: 1 808 956-8582

Email: rheesg@hawaii.edu

Feng Wu (Harry)**

School of Accounting and Finance, Hong Kong Polytechnic University

M742 Li Ka Shing Tower, Hung Hom, Kowloon, Hong Kong, China

Tel: 852 2766-7077

Email: fengwu@polyu.edu.hk

March 21, 2018

Abstract: Stocks with the potential for crashes in better market conditions are compensated by

higher expected returns than stocks with the potential for equally severe crashes in worse market

contexts. The impact of market context on returns is more pronounced among stocks with less

institutional holdings and when the potential crash is rarer and bigger. The effect also becomes

stronger in recent decades. Beta, size, book-to-market ratio, momentum, liquidity, investment

growth, and profitability cannot explain this phenomenon, which is not driven by micro-sized or

penny stocks either. These results are consistent with the implication of the salience-based asset

pricing model.

Key Words: Salience theory, Asset pricing, Behavioral finance

JEL Codes: A10, G10, G12, G40, G41

* We are indebted to Mauboussin (2006) for this title, whose Chapter 25 is designated: “A tail of two worlds: Fat tails

and investing.”

** Feng Wu (Harry) gratefully acknowledges financial support from the General Research Fund (No. B-Q50N) of the

University Grants Committee of Hong Kong.

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Economists have long recognized that agents do not evaluate an asset’s payoffs in isolation;

rather, they assess them within a payoff context.1 However, it is unclear how the context affects

the value of the asset. This paper examines the effect of the context on asset pricing in equity

markets. We propose that a severe price plunge of an individual stock will be more painful if the

overall market – taken as the context – performs better than the stock, whereas a plunge will be

less painful if the market also crashes. When the potential collapses or tail risks of individual stocks

are exactly the same, the effects on investor utility, investment decisions, and asset prices differ,

depending on what occurs in the market as a whole. In other words, not all tails are created equal.

For our theoretical explanation of the asset pricing effects of payoff contexts, we turn to

the salience theory of Bordalo, Gennaioli, and Shleifer (BGS) (2012, 2013). This theory suggests

that, holding constant the prospects of an asset’s crash, improvements in market conditions make

the crash more salient, rendering the asset more underpriced relative to other assets in the market,

and inducing a lower price level and a higher expected return in the cross-section. The salience

effect comes from the comparison of the asset’s payoff in a certain state with the average payoff

delivered by all available assets (i.e., market performance) in the same state. It is especially strong

in a low-probability crash state, in which the asset’s payoff is disastrous and its salience increases

the pain.

To empirically verify the proposed market context effect, we focus on the lower tail states

of individual stocks and investigate how the expected market context performance for a given level

of a stock crash affects the expected stock returns in the U.S. market for a sample period from July

1962 to December 2014. We find that a stock crash expected to occur in a better-performing market

commands a premium relative to a crash occurring in a worse-performing market, even though

both crashes are of the same probability and magnitude. This market context effect is more evident

when the potential stock crash is more severe; it becomes weaker as the magnitude of the stock’s

expected tail loss declines or as the crash becomes less rare. These findings support the asset

1 The economic rationale can be traced to the 18th century when Smith (1776) acknowledged the motive of questing

for social status.

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pricing implication of the salience theory, as well as its prediction that rarer, more severe crashes

lead to more salience-based overweighting of crash-state payoffs and underpricing of an asset.

According to BGS, salience arises from narrow framing such that it is shaped by the payoffs

of individual assets, and the salience effect comes from people’s limited cognitive resources such

that they focus only on sensory/salient prospects. We provide evidence consistent with these

premises. The market context effect is greater for stocks that have fewer institutional holders,

because institutional investors are less subject to narrow framing and to salience distortion. We

find also that the market context effect has become stronger in recent decades, suggesting that

investors’ cognitive abilities to make investment decisions have declined as the increasing asset

population of the market forces investors to concentrate on the salient portions of opportunity sets.

To our knowledge, this article offers the first empirical examination of the asset pricing

implications of the salience theory. It complements existing studies that investigate the effect of

salience on equity investments. For example, Barber and Odean (2008), using a similar cognitive

limitation argument, confirm that investors, especially individual investors, tend to buy salient,

attention-grabbing stocks. Hartzmark (2015) finds that individual investors are more likely to trade

extreme winners or extreme losers in their portfolios; this finding can be attributed to the increased

psychological salience of extreme-ranked stocks. Chen, Chou, Ko, and Rhee (2018) report that

rank- or sign-based momentum strategies outperform the traditional momentum, and explain this

outperformance using a salience-based hypothesis. Whereas these studies focus on how salience

influences investors’ trading behaviors, we examine how investors’ behaviors (i.e., reactions to

salience) affect asset prices.

Our work contributes to studies of the asset pricing effects of context- or reference-

dependent preferences, especially those based on the contemporaneous evolution of a payoff and

its reference. By taking as a reference the per capita consumption level (the Joneses) in a country

(or community) with undiversifiable wealth, Gomez, Priestley, and Zapatero (2009) demonstrate

that the “keeping up with the Joneses” (KUJ) motive drives down the prices (and drives up the

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expected returns) of assets that do not co-move with the non-hedgeable components of the local

Joneses. In a similar international setting, Solnik and Zuo (2012) propose that investors are willing

to accept higher prices and lower expected returns for home assets, because they are concerned

about potential regret if they invest in foreign assets (rather than domestic assets, which are taken

as the reference) but foreign assets underperform domestic ones.2 Our paper differs from these

studies in three dimensions. First, in accordance with the salience theory, we take the overall

market, rather than a fraction of the market, as the reference (we consider local Joneses or domestic

assets as part of the entire global portfolio). In contrast, the KUJ and regret models of Gomez et

al. (2009) and Solnik and Zuo (2012) examine the bias toward local or home assets. Investors take

their own local or home assets as the reference and have heterogeneous holdings in equilibrium.3

Second, we focus on the crash states of individual assets to show that the market context effect is

stronger in such cases. The existing KUJ- and regret-based asset pricing models do not differentiate

crash states from non-crash states. Our findings suggest that future research could examine the

potentially different impacts of KUJ and regret behaviors in different conditions (for example, the

very poor or extreme losers may have different sensitivities to the Joneses or regret). Third, by

supplementing existing studies that use covariance to measure the relation between an asset and

its reference, we directly gauge the expected reference payoff conditional on the asset’s crash states,

thereby making it easier to compare the payoffs and more intuitive to explain the context effect.

The market context effect that we document also facilitates a better understanding of some

existing (and sometimes puzzling) asset pricing phenomena, including the beta anomaly (high-

beta/low-return) (Black, Jensen, and Scholes, 1972; Fama and French, 1992; Bali, Brown, Murray,

and Tang, 2017) and idiosyncratic extreme-risk premium (Huang, Liu, Rhee, and Wu, 2012). A

2 Regret results from a comparison of the outcome from the chosen option and the counterfactual outcome from a

forgone alternative if the alternative turns out to be better (Bell, 1982; Loomes and Sugden, 1982).

3 This suggests that there are multiple references in Gomez et al.’s (2009) and Solnik and Zuo’s (2012) models. Gali

(1994) shows that if all agents hold the same global portfolio and take it as the reference, the KUJ motive translates

into a lower price of the single systematic factor, and equilibrium prices are identical to those in a KUJ-free economy

after adjusting for the degree of risk aversion.

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crash payoff context hinges on the connection between the individual stock returns and the market

returns. It may relate to other measures that, in various ways, also connect stock returns with

market returns. Specifically, we find that among stocks with high risks of crash, an improved

market context (i.e., higher conditional market return) is associated with smaller beta, larger

idiosyncratic volatility (IV), and higher idiosyncratic tail risk. The market environment for

substantial price drops may thus constitute a significant part of individual stock risk, including

systematic and idiosyncratic risks. Consistent with this conjecture, we find that the premia to beta

(which is negative, in line with the beta anomaly) and idiosyncratic tail risk can be explained

largely by the market context effect, but not vice versa. We document a similar but weaker result

for the mutual influences between the effects of market context and the non-extreme IV measure.

These findings shed new light on the sources of the beta anomaly and equity premium to

idiosyncratic risk, especially when stocks are exposed to crash risks.

Our research also supplements studies of the impact of crashes on stock prices (Bali,

Demirtas, and Levy, 2009). We investigate whether the expected context of a potential crash, rather

than the crash itself, matters with regard to asset pricing. Our work, combined with existing

findings about crash risk premia, provides a more complete description of extreme risk and its

pricing.

The remainder of this article is organized as follows: In Section I, we introduce an

illustrative model of the market context effect on the basis of the salience theory of BGS (2012,

2013), and develop testable hypotheses. In Section II, we describe the market context measurement

and report the main results for the market context effect on stock returns. We examine the

interactions between the asset pricing effects of market context and beta, IV, and idiosyncratic tail

risk in Section III, followed by the robust test results in Section IV. Section V concludes.

I. An Illustrative Model for the Market Context Effect on Asset Prices

We use a simple, relatively stylized model to develop testable hypotheses with regard to

the asset pricing effect of the market context for a stock crash. The model is built on the general

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framework of BGS’ (2012, 2013) salience theory, in which agents overweight more salient payoffs

– that is, those that differ most from the average payoff of all assets – and that such distortions are

strongest “in the presence of extreme payoffs, particularly when these occur with a low probability”

(BGS, 2012, p. 1245). Further, BGS (2013, p. 625) state that “an investor’s willingness to pay for

an asset is context dependent,” and “(c)hanges in background context affect the salience of an

asset’s payoffs and thus, its price.”4

We explore the salience-based context effect in a low-probability extreme-loss situation

(i.e., crash) for an individual asset. We start with BGS’s (2013, p. 626) parsimonious model setting,

in which there are only two states of nature, s = 1, 2, and the market has only two assets: a risk-

free asset F with constant payoff 0f and a risky asset which delivers a low payoff 2 0x in State

2 that occurs with probability 2 and a high payoff 1 2 , 0x x G G in State 1 that occurs with

probability 1 21 . Assume 2 2x f x G , such that the sure payoff lies between the high and

low risky payoffs. State 2 with the low payoff represents bad times and, if 2x (and 2 ) is small

enough, the time a crash occurs for the risky asset. To highlight the role of salience and simplify

the illustration, we follow BGS (2012, 2013) and assume a linear utility function and no time value

of money (i.e., risk neutral without time discounting).

In a salience-free world, the price of the risky asset is as follows:

1 1 2 2s ssp x x x . (1)

With the salience effect, the risky asset’s payoff in each state would be overweighted or

underweighted, depending on the percentage difference between the asset’s payoff and the average

payoff in the market. The market average payoff is 21

2

x G fm

in State1 and 2

22

x fm

in

State 2. Because 2 2x f x G , the risky asset’s payoff is lower than the market payoff in bad

4 However, the model we introduce subsequently is highly simplified and for illustration purpose only. The mechanism

is not a complete description of all components of salience theory; for a complete description, see BGS (2012, 2013).

6

times (State 2) and higher than the market payoff in good times (State 1), such that

22 2

2

x fx m

and 2

2 12

x G fx G m

. If the percentage payoff difference between the

risky asset and the market is larger in the downside State 2 than in the upside State 1 (i.e.,

2 22 2/ ( ) / ( )

2 2

x f x G fx x G

), investors who adopt salience-related thinking perceive the

risky asset’s downside (State 2) payoff to be salient and assign it a larger weight of 2 1 . The

weight given to the upside (State 1) payoff 1 is correspondingly reduced such that 1 1 2 2 1 .

In other words, the expected salience distortion is zero, and the salient payoff is overweighted at

the expense of the relatively non-salient one.

With payoffs adjusted by the salience weights, the risky asset’s price p is given by the

following (see Appendix A for proof):

1 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 2

1 1 2 2

( ) (1 )

[ (1 ) ]

p x x

x x G

x x G

x x

, (2)

where 2

2

1 (1 )G

x .

The first row of Equation (2) shows that the expected value of salience-weighted payoffs

determines the risky asset’s price, which is the standard formula for the asset value in the salience

theory. The second row indicates that the price can be expressed as the expected value of

unweighted payoffs (i.e., price in the salience-free world), plus an adjustment term from the bad

state. Therefore, the difference between asset pricing with and without salience-related thinking is

due to the downside (crash) state. The third and fourth rows of Equation (2) suggest that the asset’s

price is also equal to the expected value of the unweighted payoff in the good state and the salience-

adjusted payoff in the bad state. Equivalently, the salience weights can be rescaled such that the

weight assigned to the good state is 1 (as in the salience-free world), and only the weight of the

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bad state is adjusted. The implication of these expressions is that the salience effect on asset prices

can be derived from the downside, and especially the crash state.

For a given payoff in the downside (State 2), the salience weight 2 is influenced by the

average market payoff2m in the same state that acts as the context (or reference) for the risky asset’s

payoff. A better-performing market makes the asset’s payoff more salient in the bad state and

causes it to acquire a larger salience weight. Therefore, 2

2

0m

. Another key implication of the

salience theory is that the salience weight is more distorted in states with extreme payoffs. For the

downside state, the smaller the payoff, the larger the salience-related market context effect, or

2

2

2 2

0m x

. In summary, the market context drives the salience effect, which is more sensitive to

an enlarged crash.

When these conditions are satisfied, and keeping the risky asset’s payoff 2x in the downside

state unchanged, the asset’s price in Equation (2) is influenced by the market context (i.e., the

average market payoff 2m ) as follows:

22

2 2

0p

Gm m

. (3)

Ceteris paribus, a higher market context return in the given downside state should depress

the risky asset’s price and require a higher expected return for the asset; this prediction constitutes

the first hypothesis that we examine.

By also considering the pricing effect of the market context in the downside states with

different levels of asset payoff (crash severity) 2x , we obtain the following:

2 2

22

2 2 2 2

0p

Gm x m x

. (4)

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Thus, the market context effect on asset prices tends to be stronger for more severe crash

states with smaller 2x values. We compare the market context effects at different crash levels; at

each crash level, 2x is constant. We are interested in how the market context works, given a certain

level of crash (high or low). For a lower asset payoff2x ,

2

p

m

is also lower, such that it becomes

more negative, suggesting that the market context has a stronger impact on the risky asset’s price.

This is the second hypothesis that we test empirically.

To facilitate the illustration of our predictions, we present a simple example in which we

express the salience-adjustment term 2(1 )G from the second row of Equation (2) as an explicit

function of the difference between the risky asset’s payoff and the conditional market payoff in

the downside (crash) state. Specifically, we write the salience weight 2 as a function of 2 2( )x m ,

such that

2 2 2

2

(1 ) ( ), 0G x mx

, (5)

implying that

2 2 2

2

1 ( )x mx G

. (6)

We can verify that 2

2 2

0m x G

and

2

2

2

2 2 2

0m x x G

, consistent with the implications

of salience weight. Therefore,

21 1 2 2 2 2 2 1 1 2 2 2

2 2

( ) ( )( ) ( ) (1 )m

p x x x m x xx x

. (7)

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Given a certain level of 2x in the asset’s downside state, a better-performing market (i.e.,

higher2m ) leads to a lower asset price, because 2

2 2

0p

m x

, and

2

p

m

is more negative for a

smaller 2x (

2

2

2

2 2 2

0p

m x x

), so the effect of the market context 2m on asset price p is more

prominent in more severe crash states with lower levels of payoff 2x .

This basic framework of the salience theory builds on the fundamental behavioral

foundation of limited cognitive resources and narrow framing. Therefore, factors that contribute

to the agents’ cognitive limitations and narrow-framed behavior also influence salience and its

impact on asset prices. For example, individual investors may have more limited resources for

comprehending the complete payoff structure and may be more subject to a narrow framing bias,

especially when facing overwhelming opportunity sets. For these reasons, we predict that prices

of stocks that tend to be held by individual investors are more sensitive to the market context effect,

and this effect tends to be more pronounced when investors must choose from a larger asset pool.

We also test these predictions.

II. Relationship Between the Market Context of a Stock Crash and Expected Returns

A. Measurement

We test the market context effect by focusing on the crash states of an individual stock in

the left-tail of its return distribution. Specifically, we examine the case in which investors envision

an extreme loss of a stock as well as its context, the contemporaneous market performance, to

determine how the expectation of the market context affects the stock’s price and expected returns,

given the same level of stock crash. Empirically, we need to estimate the magnitude of the crash

loss for each individual stock, as well as the associated market return conditional on the crash

states of the stock.

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Our measurement scheme closely follows the preceding intuition. To measure the expected

level of an individual stock loss in its crash states, we use expected tail loss (ETL), which is the

average loss below the Value-at-Risk (VaR) of a low probability (e.g., 1%). Conditional on a

stock’s tail states, we estimate expected market return and use it as a proxy for the market context

(MKTCON). Economically, MKTCON indicates the average performance of all stocks when an

individual stock suffers a disastrous price drop.

We estimate ETL and MKTCON both nonparametrically and parametrically. In the

nonparametric method, we select the lowest return data for each stock within a bottom small-

probability quantile of its empirical return distribution and use their average to measure the stock’s

ETL. We use the average of the market returns that are associated with the lowest stock returns to

measure MKTCON conditional on the downside tail states of each stock. In the parametric method,

we estimate ETL and MKTCON based on the extreme value theory (EVT), the details of which

are explained in Appendix B. The particular EVT model we adopt is able to describe the behavior

of one random variable (market return) given that another random variable (stock return) is in its

extreme region (lower tail states), and ensures that the estimates are robust to the parent

distributions of the stock and market returns. This attribute makes the parametric extreme value

approach suitable for our study, which enables us to predict a stock crash level and its associated

expected market context in an easy and technically rigorous way. For this reason, we report the

results under the EVT-based measurement scheme in the main text, and replicate the key tests

using ETL and MKTCON estimated from the nonparametric empirical distributions in the Internet

Appendix. These two methods generate remarkably consistent findings with regard to the asset

pricing effect of the market context.

Our measurement is guided by the economic intuition of the payoff context in the salience-

based theory. The core idea is that the market context return is conditional on the low-probability

crash returns of an individual stock, which requires that the measurement is directional in a non-

causal sense. This is in contrast to the existing measures, such as systematic risk proxies (e.g.,

market beta), which switch the conditioning and gauge an individual stock return in different

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market states. The reversed conditioning radically changes the interpretation of the measures. In

this paper, we primarily consider the direction from stock to market, which quantifies the market

performance when a stock is in distress. In Section III, we discuss the difference and connection

between the asset pricing effects of market context and systematic risk in more detail.

Our methods described above are not the only ways to measure the market context of a

stock’s tail loss; any proxy that indicates the level of market return conditional on the crash states

of the stock can serve the purpose. For example, the co-movement between a stock and the market

across the left-tail states of the individual stock return distribution also reflects the market context

of the crash states of the stock. In the Internet Appendix, we demonstrate that our main results are

qualitatively unchanged within this co-movement-based measurement framework.

B. Summary Statistics of Stock Crash and Associated Conditional Market Return

We use daily returns retrieved from the Center for Research in Security Prices (CRSP)

database to estimate ETLs and MKTCONs for all common stocks traded on the New York Stock

Exchange (NYSE), the NYSE MKT [formerly the American Stock Exchange (AMEX)], and the

Nasdaq from July 1962 to December 2014. To ensure that we can sufficiently detect the tail risk

of stocks, we adopt a sampling period of five years (with at least 1,000 non-missing daily return

observations) on a rolling window basis to conduct the estimation. Therefore, the first set of valid

estimates appear at the end of June 1967. To reduce potential microstructure biases in the daily

data, especially those of small and low-priced stocks, we follow the standard procedures in the

literature and require each sample stock’s price to be no lower than $5 and market capitalization

to be no lower than the 10th percentile of NYSE stocks at the end of each month of each estimation

period. In Section IV, we show that our results are robust to alternative data screening schemes.

We use the CRSP value-weighted index return as the proxy for market return in the main tests and

show that using other indexes delivers similar findings in robustness tests.

Table I reports basic statistics for the estimates of key variables under the scheme of ETL

with a 1% probability. Because the main purpose of this study is to examine the effect of

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MKTCON among stocks with the same or similar ETL levels, we form five subsamples each

month according to ETL and report the statistics of MKTCON for each subsample.5 Panel A shows

that MKTCON does not differ much across subsamples with different ETLs. Stocks with the

largest crash losses (in Subsample 1) are not associated with the biggest market drops: The mean

and median of MKTCON both have magnitudes similar to other ETL subsamples. This finding is

important because it suggests that not all individual stocks crash in a diving market; some plunge

in a calm market. The prospect of a steeper plummet of a stock does not always mean the prospect

of a worse market condition; otherwise, a more negative ETL would be associated with a smaller

MKTCON, which is not the case. This evidence is especially relevant to our study, because it

implies that a potential crash of a stock can occur either when the market as a whole is also doing

badly or when the market is not doing so badly or even pretty well. Hence, a stock can crash in

differing contexts. Our goal is to examine whether (and how) the market context (rather than the

crash itself) affects investors’ behavior and thus stock prices.

Although the relation between ETL and MKTCON does not exhibit any obvious pattern,

given a crash of a certain level, its associated market condition may be related to different firm

characteristics and risks. For example, crashes of small stocks may have higher MKTCONs

because they can drop drastically while the market remains stable, which naturally invokes higher

idiosyncratic risk, especially idiosyncratic tail risk. We examine this issue in Panel B of Table I.

We consider the levels of a menu of firm characteristics and risk variables in various MKTCON

groups (quintiles), conditional on a certain level of ETL. As our focus is the market context of the

crash states of stocks, we report only the relevant results for the subsample with large stock drops

(i.e., ETL Subsample 1 of Panel A).

The first column of Panel B confirms that the market context can be substantially different

for a similar stock crash level: Whereas MKTCON exhibits substantial variation across the

quintiles, ETL in the second column remains stable because these stocks belong to the same ETL

5 We winsorize ETL estimates from above the top 1% and below the bottom 1% of the full sample to eliminate possible

outlier effects caused by unrealistically small or large values under the EVT-based measurement scheme.

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subsample. The bottom quintile has a low market context return of -1.9255% per day, and the top

quintile has a relatively higher market context return of -0.4546%, inducing a difference of 1.4709%

per day, which is 1.5 times the standard deviation of daily CRSP market returns in our sample

period (0.9974%). This evidence suggests that the variation of MKTCON is not only economically

significant (i.e., reflects different market contexts for a given level of stock crash) but also

statistically significant (i.e., delivers reliable references of the market context’s impact on stock

prices in the cross-section).

The other columns of Panel B show the firm characteristics and risk variables (with

definitions detailed in Appendix C) in MKTCON quintiles. Higher market context returns are

associated with smaller firm sizes, higher book-to-market ratios (B/M), lower momentum returns,

and more illiquidity.6 Taken together, this evidence suggests that if a stock crashes in a relatively

better market context, it is more likely to be a small stock, a value stock, a past loser, or a stock

that lacks liquidity. We will show that the market context effect on asset prices is different from

the effects of these characteristic variables. The last three columns further show that such a stock

tends to have lower systematic risk (beta) and idiosyncratic risk, including IV and idiosyncratic

tail risk (proxied by idiosyncratic ETL). The risk connotation of the market context appears to be

stronger with regard to systematic risk and idiosyncratic tail risk, given that beta and idiosyncratic

ETL’s variations across MKTCON groups are larger than that of IV. Consistent with this

observation, in Section III, we show that MKTCON exhibits a bigger influence on the asset pricing

effects of beta and idiosyncratic ETL than on the IV effect.

C. Market Context of Stock Crash and the Cross-Section of Expected Returns

We use standard portfolio and Fama and MacBeth (1973) regression analyses to detect the

market context’s influence on expected stock returns in the cross-section. To deliver unequivocal

references with regard to the impact of MKTCON, we carefully control for ETL in all tests to

6 We winsorize estimates of these variables, together with the systematic and idiosyncratic risk measures subsequently

introduced, from above the top 1% and below the bottom 1% of their full samples.

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ensure that the market context effect is examined at the same levels of stock crash. Our paper aims

to supplement existing studies on crash risk by exclusively considering the market context of the

crash, rather than the crash itself. Panel A of Table II reports the time-series means of value-

weighted excess returns (monthly returns minus one-month T-bill rate) for quintile portfolios

formed by MKTCON of the previous month-end, in each of the five ETL subsamples.7 In other

words, we consider the market context’s relationship with expected returns among stocks with

similar ETLs. Two main findings emerge: First, high-MKTCON portfolios have high expected

return in each of the ETL subsamples. Second, the return differences between high- and low-

MKTCON portfolios are larger and statistically more significant in the low-ETL (i.e., more crash)

subsamples than in the high-ETL (i.e., less crash) subsamples. For example, in ETL Subsample 1

with a large scale of cash losses, the top MKTCON quintile is associated with a mean expected

return of 81.55 basis points (bps) per month, which is much larger than the mean expected return

of only 8.37 bps in the bottom MKTCON quintile. The difference (73.18 bps per month, translating

into a 10.91% annual return spread in a monthly compounding scheme) is highly significant in

both economic and statistical terms (t-statistic = 3.24). Both the magnitude and significance of this

return spread become monotonically lower as the average scale of the crash loss becomes less

severe. In ETL Subsample 2, the spread decreases to 50.63 bps with a t-statistic of 2.39, and in

ETL Subsample 5, which includes stocks with the smallest crash scales, the spread is reduced to

an insignificant 5.34 bps. These findings suggest that when investors face the prospect of severe

crash states of an individual stock, they tend to require a higher expected return for the stock if its

crash occurs in a better market environment than in a deteriorated market environment. According

to the salience theory, a better market context makes the crash more salient (or painfully salient).

Therefore, the stock must promise a higher expected return to attract investors to hold it. Our

evidence also confirms that the context effect is pronounced in the presence of more severe crashes

because the salience-based distortion becomes larger in such situations.

7 One-month T-bill rate data are downloaded from Kenneth French’s data library.

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Our test for the market context effect on stock returns in Panel A applies to stocks with

roughly similar levels of ETL, because a five-subsample scheme can ensure only that there is no

big difference in ETL among stocks in each subsample. However, the within-subsample variations

of ETL still exist, which may contaminate our inferences about the market context effect due to

the possible correlation between MKTCON and ETL, even in the same ETL subsample.8 To factor

out the influence from ETL, we provide an additional control for ETL in the portfolio analysis.

Specifically, in each of the ETL subsamples, before sorting stocks by MKTCON, we first sort

them by ETL and create five ETL subportfolios. Then, within each ETL subportfolio, we form

MKTCON quintiles. We compute each MKTCON quintile’s value-weighted mean excess return

across all ETL subportfolios, and report the results in Panel B. After further controlling for ETL,

the market context exhibits a similar impact on expected returns, especially in the ETL subsample

with the most severe crash, in which the high-minus-low return spread between the top and bottom

MTKCON quintiles is 72.75 bps per month. Although its magnitude is slightly lower than the

corresponding spread of 73.18 bps in Panel A, it enjoys a higher significance level, with a t-statistic

of 4.91. This additional controlling scheme for ETL delivers a more accurate reference of the

market context effect. Therefore, we adopt this approach in the following portfolio analyses.

D. Market Context Effect After Controlling for Firm Characteristics and Traditional Asset Pricing

Factors

In Table III, we examine the relation between the market context and expected returns

beyond the effects of beta, size, book-to-market ratio, momentum, and illiquidity. These are

commonly accepted variables that can influence asset prices. These variables also show non-trivial

associations with MKTCON (as in Panel B of Table I for stocks with large crash risk). These

associations (except for momentum) point to the same direction of relation with expected returns

as MKTCON. Panel A of Table III reports return spreads between the top and bottom MKTCON

quintiles after controlling for beta, size, B/M, momentum, and Amihud (2002) illiquidity measures

8 Table I, Panel B, shows that MKTCON and ETL tend to be negatively associated within the ETL subsample of large

stock drops.

16

for each ETL subsample. Specifically, to control for beta, we sort stocks into quintiles in sequence

by ETL, beta, and MKTCON, then compute the value-weighted average return of each MKTCON

quintile across all ETL-beta portfolios. We apply controls for other characteristic variables in a

similar manner. In the subsample with severe stock crashes, none of the control variables can

subsume the market context’s effect on expected returns, though controlling for them generally

reduces the magnitude of the cross-MKTCON return spread by various degrees. The market

context effect is not as robust in other subsamples with less severe crash prospects, especially after

we control for size, B/M, and illiquidity. This evidence highlights the importance of focusing on

the stocks’ crash states when examining the potential context effect, as suggested by the salience

theory.

Panel B of Table III reports the spreads of alpha between high-MKTCON and low-

MKTCON portfolios in different ETL subsamples. We estimate alpha spreads as the intercepts of

time-series regressions of the cross-MKTCON return spreads obtained from Table II, Panel B, on

the capital asset pricing model (CAPM) market factor, Fama and French’s (1993) three factors,

and Carhart’s (1997) four factors, with monthly factors data obtained from Kenneth French’s data

library. The top MKTCON portfolio has significantly higher CAPM alpha, three-factor alpha, and

four-factor alpha values than the bottom MKTCON portfolio in the subsample of stocks with the

largest potential crashes. Therefore, among these stocks, the impact of the market context on

expected returns still holds after beta, size, book-to-market, and momentum effects are

simultaneously controlled. Consistent with the results in Panel A, the MKTCON effect is weaker

in other ETL subsamples as the alpha spreads decrease and, in some cases, become insignificant.

We reach similar and consistent conclusions in Panel C, in which we report value-weighted

Fama-MacBeth regression coefficients of MKTCON after controlling for ETL and other variables,

including beta, logarithms of size and B/M, momentum, and illiquidity. In the subsample of the

largest stock crashes, we obtain a MKTCON coefficient of 0.7009 when we control for ETL only,

suggesting that for a one-percent increase (roughly equal to the standard deviation of MKTCON,

as shown in Panel A of Table I) in market-context returns, the expected return of the stock would

17

be 0.7009 percent higher in the following month. Although controlling for other variables

generally reduces this coefficient, it stays at a high level of no less than 0.5359. All the MKTCON

coefficients in different regression models are statistically significant with t-statistics of no less

than 2.89. This evidence supplements the findings in Panels A and B and confirms that the market

context effect on stock returns is not driven by traditional asset pricing variables.

In an influential paper, Harvey, Liu, and Zhu (2016) note that in the cross-sectional studies

of a proposed factor/variable and expected stock returns, a more stringent requirement for the t-

statistic of higher than 3.00 is needed to reliably show that the asset pricing effect of the new

factor/variable is not the result of data-mining. We note from Table III that in both the portfolio

analysis (even when we use quintile portfolios rather than decile portfolios as in many previous

studies) and the regression analysis, for stocks with high crash risk, the market context measure

passes this hurdle with t-statistics higher than 3.00 in most cases. The only exceptions are the

regressions reported in the last two columns of Panel C, in which the t-statistics are approximately

2.90. The market context effect has sound theoretical support, and a “factor derived from a theory

should have a lower hurdle…” (Harvey et al., 2016, p. 7), suggesting that our results are unlikely

to be subject to data-mining concerns.

To show the robustness of our findings, we expand the analysis by adding more controls,

including the recently introduced investment and profitability factors of Fama and French (2015)

and Hou, Xue, and Zhang (2015) in Section IV. The new results are consistent with our findings

in Table III. We also generate a factor that captures the returns associated with the market context

among stocks with large crash exposures; existing factors are unable to completely explain the

market context factor, and vice versa. This evidence suggests that the market context reflects asset

pricing information that is distinct from that of other factors.

E. Crashes with Different Probabilities, Institutional Holdings, and Time Trends

In this subsection, we explore the market context’s effect on expected returns conditional

on stock crashes measured at different probabilities, which complements our preceding analysis

18

of different crash levels at the same probability. In our setting, a larger probability indicates a less-

rare downside state and is more distant from being considered a crash. We propose a weaker

market context effect based on left-tail stock losses estimated with a higher probability. We also

examine the implications of cognitive limitations and narrow framing, which underline the salience

theory’s prediction about asset prices. To achieve this goal, we check the market context effect

among stocks with different levels of institutional holding as well as the time-varying trend of the

effect. We expect the market context effect to be more pronounced among stocks with less

institutional ownership because non-institutional/individual investors are more influenced by

narrow-framed thinking. We also expect that the effect becomes stronger in recent decades because

the increasing equity listings over time further limit the attention granted to each stock. Due to

space limitations, we present only the results for the ETL subsample with the largest crash losses.

In Panel A of Table IV, we report mean excess returns and Carhart (1997) four-factor

alphas for the top and bottom MKTCON quintiles, as well as the spreads between them, according

to firm ETL estimates associated with probabilities of 1%, 2%, and 4%. The last column reports

Fama-MacBeth regression coefficient of MKTCON after controlling for ETL. Because MKTCON

measures tend to be clustered in higher probability cases which may induce a upward bias in the

regression coefficient, we use their percentile rank values in the regressions to help reduce the bias

while maintaining the directional relationship between MKTCON and returns.9 Both the portfolio

and regression analyses reveal a clear pattern: The return spread, alpha spread, and regression

coefficient all decline monotonically from the 1%- to the 4%-probability case, though they remain

qualitatively consistent. These results show that stock prices (and thus expected returns) are less

affected by the market context of the payoff in a less rare state.

We check the market context effect among stocks with different levels of institutional

ownership in Panel B. We use MKTCON conditional on 1%-ETL and measure the percentage of

institutional holding using the Thomson Reuters Ownership (13f) database. We assign stocks into

9 The standard deviation of MKTCON conditional on 4%-ETL is 0.7216, smaller than the corresponding values of

0.8892 for the 2%-ETL case and 1.0730 for the 1%-ETL case.

19

three groups on a monthly basis: high institutional holding (stocks with institutional ownership

percentages higher than the median level), low institutional holding (stocks with institutional

ownership percentages lower than or equal to the median level), and no institutional holding

(MKTOCN sample stocks without institutional ownership information in 13f). 10 Because

institutional ownership data are available only from 1980, we report our results accordingly. Panel

B shows that the return and alpha spreads between the top and bottom MKTCON quintiles, as well

as the Fama-MacBeth coefficients of ranked MKTCON, monotonically decline from the less-

institutional-holding to more-institutional-holding groups, and their values more than double in

the no-institutional-holding group compared with in the high-institutional-holding group.11 This

evidence supports the prediction that stocks favored by unsophisticated individual investors are

more subject to salience distortion because these investors are more liable to use narrow-framed

thinking. Moreover, institutional investors normally operate according to a benchmarking

performance evaluation scheme and therefore may be sensitive and averse to underperforming the

benchmark (Roll, 1992); for this reason, the context effect could be stronger among institutional

investors. Our evidence of the cross-sectional market context effect suggests that the

benchmarking concerns of institutional investors, if any, are dominated by the salience-related

thinking of individual investors.

In Panel C, we report the market context effect according to MKTCON conditional on 1%-

ETL in different sample periods. We consider three periods: 1967 (July) to 1980, 1981 to 1997,

and 1998 to 2014. The return spread, alpha spread, and MKTCON coefficient values show a

generally increasing trend over time, and the market context effect is much more prominent after

1980. Meanwhile, the month-end average numbers of listed stocks with valid MKTCON estimates

in the three periods are approximately 1,010, 1,406, and 1,424, respectively, also exhibiting an

10 Although “no 13f ownership” does not always mean no institutional holding and our definition of “no institutional

holding” is subject to a potential missing data problem, we believe this problem is not severe enough to reject the no-

institutional-holding group as an approximation for stocks held more by non-institutional or individual investors.

11 Similar to Panel A, MKTCON estimates are clustered more in the more-institutional-holding groups than in the

less-institutional-holding groups. We use percentile rank values to abate potential bias in the regression coefficient.

20

increasing trend. One potential explanation that reconciles these two increasing patterns is that the

expansion of asset pools over time makes it more difficult for investors to allocate their cognitive

resources to all assets equally. The limited attention is attracted more to salient stocks, so there is

a more pronounced market context effect in recent decades. Of course, other factors may contribute.

For example, the rapid development of index funds and index exchange-traded funds (ETFs) after

the late 1970s makes information about the average market performance readily available and

helps investors quickly obtain comparisons of stock performances and the market average, thereby

magnifying the salience effect.12 This evolution of investment-opportunity sets and informational

environments demands a reconstruction of attention allocation, which overweights more salient

payoff situations. Our evidence is consistent with this logic.

III. Asset Pricing Effects of the Market Context and Systematic and Idiosyncratic Risk

Measures

In this section, we investigate how the market context effect interacts with existing

systematic and idiosyncratic risk measures (beta, IV, and idiosyncratic ETL), in particular, how

much of the market context effect can be subsumed by existing risk variables and vice versa.13 If

the crash state is important enough, we expect to see a non-trivial influence of MKTCON on the

effects of existing risk measures.

We first revisit the systematic risk measure beta; in the first four columns of Table V, we

report the results for the ETL subsample with the largest crash scales. Panel A shows a negative

relationship between beta and expected returns. The return spread between the highest and lowest

12 We note that the First Index Investment Trust (predecessor of the Vanguard 500 Index Fund), one of the earliest

index funds, started in 1975, and one of the earliest index ETFs, the S&P 500 Depository Receipt (SPDR), was created

in 1993. Given our five-year estimation window, the potential influences of these two funds on salience and the related

market context effect would appear from the early 1980s and 1998, respectively, which coincide with the beginning

years of our second and third sample periods.

13 Because we focus on the connection between stock and market returns, we use the single-factor market model when

measuring idiosyncratic risks (IV and idiosyncratic ETL).

21

beta quintiles is -23.86 bps per month, and the Fama-MacBeth coefficient of beta is -0.3820.14

These values, though not significantly different from zero, are qualitatively consistent with

documented beta anomalies that suggest a lower expected return for higher beta.15 In Panel B, this

beta effect virtually disappears after we control for MKTCON; the return spread reduces to almost

zero (0.08 bps, t-statistic = 0.01), and the regression coefficient drops to -0.0326 (t-statistic = -

0.12). In contrast, Panel C shows that the market context effect is still significant after we control

for beta, especially in the regression case in which the MKTCON coefficient is only slightly

reduced (from 0.7009 in Table III, Panel C to 0.6548). Taken together, these results indicate that

the market context has a non-negligible role in beta’s pricing effect among stocks with high crash

risk, but beta cannot subsume the market context effect. Given the crash level, a higher conditional

market return is associated with a lower beta (Table I, Panel B), the association between high

MKTCON and high return, thus, is consistent with the association between low beta and high

return (i.e., beta anomaly). Our evidence suggests that the former can explain the latter (but not

vice versa). This provides another clue, along with existing explanations, to the puzzling

phenomenon of the negative beta premium.16

In Columns 5 through 8 of Table V, we report the mutual influences between the effects of

MKTOCN and IV. IV is not significantly related to expected stock returns (Panel A), which is not

surprising given the lack of consistent evidence for the pricing of idiosyncratic risk (Bali, Cakici,

Yan, and Zhang, 2005; Han, Hu, and Lesmond, 2015), and IV cannot subsume the market context

effect either (Panel C).17 The impact of MKTCON on IV is weak (Panel B). Although we find that

14 To compare with the MKTCON effect, we control for stock ETL before conducting the portfolio and regression

analyses; we use this method throughout this section.

15 Note that in many beta anomaly studies (Fama and French, 1992; Bali et al., 2017), the negative premium to beta is

not statistically significant, consistent with our finding.

16 In the existing literature, Black et al. (1972) and Black (1993) propose that the beta anomaly is driven by the

divergence between risk-free borrowing and lending rates. Frazzini and Pedersen (2014) explain the anomaly via

market demand pressure on high-beta stocks exerted by leverage-constrained investors. Bali et al. (2017) attribute the

phenomenon to the investors’ chase for lottery-like stocks.

17 The extensively studied IV anomaly (high-IV/low-return) documented by Ang, Hodrick, Xing, and Zhang (2006)

is based on IV estimated over a one-month period using daily return data, which is different from the 60-month daily

22

controlling for MKTCON invokes large changes in the IV-based return spread and the regression

coefficient of IV, the directions of these changes are not consistent. The return spread is reduced

and the regression coefficient is increased. This outcome may be due to the relatively weak

relationship between MKTCON and IV (Table I, Panel B). The bottom-line conclusion is that the

market context effect on stock returns is not driven by the idiosyncratic volatility of the stock.

We find that MKTCON exhibits much stronger influences on the asset pricing effect of

idiosyncratic tail risk, as shown in the last four columns of Table V. Panel A shows that

idiosyncratic ETL is significantly and positively priced in the cross-section, with a return spread

of -51.79 bps per month and a Fama-MacBeth regression coefficient of -0.1994 (a more negative

idiosyncratic ETL indicates a larger idiosyncratic tail risk), consistent with the prior literature

(Huang et al. 2012). However, Panel B shows that the effect vanishes when MKTCON is

controlled. Both the return spread and regression coefficient of idiosyncratic ETL become much

smaller in magnitude and statistically insignificant. Therefore, in the stocks’ crash states,

idiosyncratic tail risk is largely affected by the market context in its asset pricing effect, suggesting

that a large-stock-down/small-market-change scenario constitutes an important component of

idiosyncratic tail risk. In contrast, as shown in Panel C, idiosyncratic ETL cannot completely

explain the MKTCON effect, highlighting the uniqueness of our market context measure that is

related but not confined to tail risk.

IV. Robustness Tests

A. Market Context Effects in Different Scenarios

We first examine potential microstructure noise in the daily returns of micro-cap or penny

stocks. We partially account for this issue in the main tests by screening stocks according to their

sizes and prices at the end of each month (i.e., requiring the size to be larger than those in the

return estimation window that we use (refer to Appendix C for details). Our purpose is to make a fair comparison

between MKTCON and IV according to the same and long enough estimation window, rather than trying to explain

the IV anomaly according to the short-term approach of Ang et al. (2006).

23

bottom NYSE decile and the price to be no lower than $5). Nevertheless, the MKTCON measure

is based on daily return data, and we make use of the region of the distribution with the largest

daily price drops of each individual stock. Setting minimum size and price thresholds for only the

last day of each month cannot guarantee that all the return data that can possibly enter into our

MKTCON measurement are immune from potential microstructure noises of very small or very

cheap stocks (though generally, it can do so if the month-end is a good representation of the entire

month). To ease this concern, we adopt the rather aggressive data filtering approach of deleting a

stock from the sample if its market capitalization assigns it to the lowest NYSE size decile or its

price is lower than $5 in any day of each five-year estimation window. The surviving stocks are

more likely to be free of microstructure (including illiquidity) noise. Based on this alternative

screening scheme, we report in Panel A of Table VI excess returns and four-factor alphas of the

top and bottom MKTCON quintiles, the spreads between them, and Fama-MacBeth coefficient of

MKTCON, of stocks with large crash drops (i.e., ETL Subsample 1, for all analyses in Table VI).

The spread and coefficient values are only slightly smaller than the corresponding values of the

main tests, suggesting that the market context effect is not unique to micro-cap or penny-stocks.

In Panel B of Table VI, we measure MKTCON using the S&P 500 returns rather than

CRSP market index returns. Although the more comprehensive market index is more consistent

with the requirements of the salience theory, the examination of the S&P 500 has its own merits:

It is widely disseminated as a proxy for the market and is therefore a much more salient indicator.

Whether the market context effect according to the S&P 500 is stronger is an open empirical issue.

Our results offer mixed evidence. In the S&P 500 case, the MKTCON-based return spread is lower

than in the CRSP index case by a small margin, but the alpha spread and the regression coefficient

are higher in the S&P 500 case, also by small margins. Despite these findings, we conclude that

the market context effect is not very sensitive to various market proxies.

In Panel C, we report the equal-weighted (EW) portfolio return spread, alpha spread, and

regression coefficient for the CRSP index-based MKTCON measure. The values are generally

smaller than in the value-weighted (VW) case, but all key results hold because the spreads and

24

coefficient maintain their consistent signs and are statistically significant. By comparing the top

and bottom MKTCON mean excess returns with those in the VW case (Table II, Panel B), we find

that the relatively smaller return spread in the EW case is the result of the higher mean return of

the low-MKTCON quintile (37.82 bps for EW vs. 12.09 bps for VW). The mean returns in both

weighting schemes for the high-MKTCON quintile are quite similar (86.34 bps for EW vs. 84.83

bps for VW). Small sizes have some influence only among stocks that tend to crash in a worse

market, i.e., those with less-salient payoffs in the crash states. The overall evidence confirms that

in general, small stocks do not drive the market context effect.

In Panel D, we examine the sensitivity of our results to stocks’ left-tail return data involved

in the market context measurement. As explained in Appendix B, to distinguish the tail parts of

each stock’s return distribution, we adopt the top and bottom 5% quantiles as the thresholds in the

main tests. As a robustness check, we alternatively use the thresholds defined by the top and bottom

10% quantiles. Relative to the 5%-quantile-threshold scheme, the 10% tails produce less extreme

data for ETL and MKTCON estimations. We expect these data will not capture large-scale crash

risk as accurately as in the main tests; therefore, we anticipate that the market context effect would

be weaker. Consistent with this prediction, we find that the return and four-factor alpha spreads

and the Fama-MacBeth coefficient of MKTCON all have reduced magnitudes, though they remain

positive and statistically significant.

B. Additional Analyses of Market Context Effects After Controlling for More Risk Factors

We investigate the market context effect by controlling for more risk variables in addition

to the conventional variables examined in the main tests. We focus on the investment (I/A) and

profitability (OP or ROE) variables of Fama and French (2015) and Hou et al. (2015) because they

are backed by fundamental economic theories and are becoming widely accepted by the

literature.18 The I/A variable is negatively associated with expected stock returns and OP or ROE

18 Appendix C details the estimations of these variables. We winsorize the estimates from above the top 1% and below

the bottom 1% of their full samples.

25

is positively priced. We find that, among stocks with high crash risk (i.e., those in ETL Subsample

1), larger MKTCON is associated with lower I/A, but the variations of OP and ROE across

MKTCON groups do not exhibit obvious trends.

Table VII reports the results from portfolio and regression analyses that resemble Table III

in format. In Panel A, the high-minus-low return differences across MKTCON quintiles remain

significantly positive after controlling for I/A, OP, and ROE in ETL Subsample 1, and become

weaker in other subsamples. Panel B shows that alpha spreads between the top and bottom

MKTCON quintiles from various expanded factor models are all significantly positive in ETL

Subsample 1, but not in other subsamples.19 In Panel C, Fama-MacBeth regression coefficients of

MKTCON after controlling for ETL, together with additional variables, remain positive and highly

significant in ETL Subsample 1, and decrease in magnitude and weaker in significance as a

potential stock crash becomes less severe in the other subsamples. Overall, the evidence in Table

VII is consistent with our main findings, suggesting that the market context effect is robust to more

risk factors.

To further explore the role of the market context in predicting the cross-sectional variation

of stock returns and its association with existing risk factors, we conduct a factor-mimicking

exercise. We generate a market context factor RCON in ETL Subsample 1 that captures the returns

associated with MKTCON, and report the average factor returns of RCON and its alphas with

19 The traditional four factors are excess market return (EM), small-minus-big (SMB), high-minus-low (HML), and

winner (up)-minus-loser (down) (UMD). The expanded models contain additional factors including the liquidity factor

LIQ of Pastor and Stambaugh (2003), the conservative-minus-aggressive (CMA) factor for investment and robust-

minus-weak (RMW) factor for profitability from Fama and French (2015), and the investment factor RIA and

profitability factor RROE from Hou et al. (2015). We denote the size factor in Hou et al. (2015) by RME, and

download the LIQ data from Lubos Pastor’s website, available from 1968. We compute the factors in Hou et al.

(2015) using their procedures starting from January 1972. Data for all other factors start from July 1967; we download

them from Kenneth French’s data library. The factor data are at monthly frequency.

26

respect to different existing-factor models in Panel A of Table VIII.20, 21 Column 1 shows that the

average RCON returns, though slightly different because of different reporting periods, are all

above 50 bps per month and highly significant. Alphas (α) from the various models reported in

Column 2 are also significantly positive and are in the range of 28.91 to 66.78 bps per month.

Loadings (β) of the various models reported in the rest of the panel also reveal that the market

context has its own asset pricing effect that cannot be fully explained by existing risk factors.

Panel B of Table VIII shows the explanatory power of RCON, together with EM and our

size factor RME, on the existing factors. The existing factors exhibit significantly positive alphas,

and factor loadings on RCON are significantly positive in all models, except LIQ and UMD. Thus,

the market context factor is associated with various other factors but cannot completely explain

them. Together with evidence from Panel A, these findings imply that, among stocks with a high

20 Following the common technique in the literature, at the beginning of each month, we form two size portfolios

according to market capitalization of the most recent June, using NYSE median breakpoint, and then independently

form three MKTCON portfolios according to the 30% and 70% NYSE quantiles of MKTCON estimates, after

controlling for stock ETL. To control for ETL, we follow the usual practice and first create five ETL groups, using

NYSE quintile breakpoints; then, within each group, we form three MKTCON portfolios as previously described. We

compute value-weighted return of the month for each of these 30 (= 2 × 5 × 3) intersection groups and then calculate

the average returns for the three MKTCON portfolios across the ETL groups. We take the MKTCON factor RCON

as the average return of the two top 30% MKTCON portfolios minus the average return of the two bottom 30%

MKTCON portfolios. We obtain the size factor in this setting, which we denote by RME following Hou et al. (2015),

as the difference between the average return of the three large-size portfolios and the average return of the three small-

size portfolios. In contrast with the simple high-minus-low quintile return difference, RCON uses NYSE breakpoints

in all sortings, reflects return difference between the top and bottom 30% (rather than the top and bottom quintiles) of

MKTCON portfolios, and is neutral to both ETL and firm size. The approach of controlling size and using NYSE

breakpoints also alleviates the micro-cap bias that drives many asset pricing anomalies (Fama and French 2008; Hou,

Xue, and Zhang 2017).

21 Because of the availability of factor data, we report results for models involving Hou et al.’s (2015) factors for the

period of 1972-2014. In the other models, the reporting period is 1968-2014 when LIQ is included, and 1967 (July)-

2014 otherwise.

27

potential crash risk, the market context delivers information that is different from well-known

existing factors; therefore, its impact on asset prices is also distinct from those of existing factors.22

V. Conclusion

We show that the impact of a stock’s crash/tail risk on its price differs according to the

market conditions/contexts in which the crash potentially occurs. By using ETL to measure

expected loss in an individual stock’s downside tail states, we find that the associated market return,

which is conditional on the stock’s tail loss, is positively associated with expected stock returns,

even after we control for ETL. The evidence suggests that investors care about the market context

of a potential crash and require higher compensation to hold a stock that may collapse in a better-

performing market.

Our evidence is consistent with the implications of BGS’ (2012, 2013) salience theory

which suggests that agents with limited cognitive resources and narrow framing behavior

overweight salient payoffs that differ more from the average payoff level in the market, and the

strongest salience distortions occur in extreme payoffs with low probabilities. We illustrate this

economic intuition for downside crash losses, using a simple asset pricing model. We demonstrate

that the market context has a greater impact on stock prices in situations with more severe and

rarer crashes. The market context effect is also more prominent among stocks more likely to be

held by individual investors who are more subject to cognitive limitations and narrow-framed

thinking. In addition, the effect becomes stronger in the post-1980 era when more public firms

emerged. It explains a major part of the asset pricing effects of beta and idiosyncratic tail risk,

especially among stocks with substantial crash exposure.

Finally, though the foundation for this study is based mainly on the salience theory, it does

not rule out other possible models that may also predict the market context’s effect on asset prices.

22 The RCON factor is constructed only among high-crash-risk stocks. In this sense, it is not a usual (or genuine) factor

constructed over all sample stocks. Our goal in conducting this analysis is to confirm the robustness of the market

context effect to various existing factors, rather than to propose a new factor that co-exists with others.

28

The economic implications of salience through a comparison with a reference potentially relate to

other context-dependent concerns, depending on different market settings. However, a

comprehensive examination of various candidate theories is beyond the scope of this study.

Appendix A: Proof of Equation (2)

The risky asset’s price in a salience-free world is expressed as:

1 1 2 2 1 2 2 2 2 1( ) ( )p E x x x x G x x G , (A1)

where we know 1 2x x G and 1 2 1 .

Because 1 1 2 2 1 , the price of the risky asset with salience effect is:

1 1 1 2 2 2 1 1 2 2 2 2

2 1 1 2 1 1 1 1

2 1 2 2 2

2 1 2 2

( )

( )

( ) (1 ) (1 )

( ) (1 )

p x x x G x

x G x G G G

x G G G

x G G

. (A2)

Using the result from (A1), we have:

2 2 1 1 2 2 2 2( ) (1 ) ( ) (1 )p E x G x x G . (A3)

This proves the second row of Equation (2). Proofs of the third and fourth rows are

straightforward.

Appendix B: Extreme Value Approach for ETL and MKTCON Measurement

In this parametric approach, we adopt the generalized Pareto distribution (GPD), a classic

EVT model, to describe distribution tails and estimate ETL, and estimate MKTCON with a novel

multivariate extreme value algorithm introduced by Heffernan and Tawn (2004).

As a first step, following the standard treatments of Coles and Tawn (1991) and Hildal,

Poon, and Tawn (2011), we negate the natural logarithm of individual stock i’s return Ri to

29

concentrate on the upper-tail region. That is, we create a new variable log(1 )i iY R . We then

apply GPD to the tail parts of the distribution of iY . The cumulative distribution function (CDF)

for iY is as follows:

1/

1/

( )[1 ( )]for

( ) ( ) for

for 1 [1 ( )][1 ( )]

i i

i i

i

i i i i

i ii

i i

Y

Y Y iYi

Y Y Y Y

Y Y

Y Y i

i

y dF d

y d

F y F y d y u

y u y uF u

, (B1)

whereiYd and

iYu are sufficiently low and high tail thresholds, respectively. The left (belowiYd ) and

right (aboveiYu ) tails are described by GPD with a shape parameter ξi and a scale parameter δi > 0,

as in the first and third rows, respectively. The nonparametric empirical distributioniYF in the

second row describes the return characteristics of the non-tail part. We can obtain the CDF

analogously for the negated returns of the market mY .

In our analysis, GPD tails facilitate an easy estimation of ETL of iY for a certain low-

probability level, which is the average beyond VaR. For example, for stock i, the 1%-ETL of iY

in the right-tail (using of the upper tail of the negated return series) is:

1%

1% 1%, where {[100(1 ( ))] 1}1 1

i i i

i i i i i

Y i i Y iY Y Y Y Y

i i i

VaR uETL VaR u F u

. (B2)

The ETL of Ri in the original unlogged scale, which indicates the expected return level of

the left-tail states, is given by:

1% 1%exp( ) 1

i iR YETL ETL . (B3)

Another key variable, expected conditional market return MKTCONi, is obtained from the

extremal dependence between stock i and market m, following Heffernan and Tawn (2004) and

30

Hilal et al. (2011). We first remove the effect of margins by transforming (Ym, Yi) into another set

of random variables (Zm, Zi) that have common Laplace margins, as follows (and market margin

can be transformed in a similar manner):

log[2 ( )] if ( ) 0.5

log[2(1 ( ))] if ( ) 0.5

i i

i i

Y i Y i

i

Y i Y i

F Y F YZ

F Y F Y

. (B4)

Extremal dependence of the transformed random variables (Zm, Zi) is determined by the

asymptotic structure of the conditional distribution of Zm | Zi = z as z→∞. Heffernan and Tawn

(2004) and Hilal et al. (2011) show that, for real normalizing functions| |( )m i m ia z z and

|

| ( ) mi

m ib z z

, withii Zz Z u for a high threshold

iZu ,

|

| | ||

| 2

| | |, with , ( , )mi

mi mi mimi

i m i

m m i m i m i

Z zZ z z S S S N

z

. (B5)

Therefore, the mean of the random variable |im i ZZ Z z u can be easily obtained as

|

| |( | ) mi

im i Z m i m iE Z Z z u z z

. (B6)

To estimate the expected market return MKTCONi associated with stock i’s 1%-ETL, we

first obtain the quantile probability ( )iY iF Y corresponding to the value of 1%-ETL in the upper tail

of iY by1

1 (1 ) 0.01 , transform it into the corresponding value Zi using Equation (B4), and

apply Equations (B5) and (B6) to estimate E(Zm | Zi = z) using the maximum likelihood method.

We transform E(Zm) back into E(Ym), and then into the return form of the original scale, which is

the proxy for MKTCONi, indicating the conditional market return when return of stock i is

expected to be at its 1%-ETL level.

Estimation of ETL and MKTCON according to the extreme value method as described

above involves the selection of tail thresholds beyond which observations can be considered

31

extreme. We choose the top 5% and bottom 5% quantiles as the thresholds in the main tests, and

check the robustness of the results to alternative quantile thresholds in Section IV.

Appendix C: Estimations of Existing Firm Characteristics and Risk Variables

Firm size: Market capitalization (product of price per share and number of shares outstanding) of

the end of the most recent June, reported in millions of dollars.

Book-to-market ratio (B/M): The ratio of book value of equity of the previous fiscal year ending

in the preceding calendar year over market value of equity at the end of the preceding calendar

year, measured at the end of each June using data obtained from COMPUSTAT’s annual file.

Book value of equity is shareholders’ book equity plus deferred taxes and investment tax credit,

minus the book value of preferred stock. Shareholders’ book equity is obtained from

COMPUSTAT annual item SEQ, if available. Otherwise, it is measured as the book value of

common equity (item CEQ) plus the par value of preferred stock (item PSTK) or the difference

between book values of total assets (item AT) and total liabilities (item LT). Deferred taxes and

investment tax credit correspond to item TXDITC (zero if missing). Depending on availability, we

take the preferred stock’s book value as its redemption value (item PSTKRV), liquidating value

(item PSTKL), or par value (item PSTK), in that order. This method for book value computation

follows Davis, Fama, and French (2000). We assign B/M computed at the end of June of each year

to July to December of the same year and January to June of the following year.

Momentum: Past 11-month return, skipping the most recent month.

Illiquidity: The average ratio of absolute daily return over dollar volume as in Amihud (2002),

estimated in a 12-month period with at least 200 non-missing daily return data and is scaled by

106. We follow Gao and Ritter (2010) to adjust for institutional differences in Nasdaq and

NYSE/AMEX volumes by applying divisors of 2.0, 1.8, 1.6, and 1.0 to the volumes of Nasdaq

stocks for the periods before February 2001, February 2001-December 2001, January 2002-

December 2003, and after 2003, respectively.

32

Beta: The covariance between stock and market returns divided by the variance of market returns,

estimated based on the past 60 months’ daily return data with a minimum of 1,000 observations.

Idiosyncratic volatility (IV): The standard deviation of regression residuals of the market model,

estimated based on the past 60 months’ daily return data with a minimum of 1,000 observations.

Idiosyncratic ETL: The nonparametric 1%-ETL based on regression residuals of the market model,

i.e., the average of the 1% lowest residual values, estimated based on the past 60 months’ daily

return data with a minimum of 1,000 observations.

I/A: The annual change in total assets (COMPUSTAT annual item AT) divided by one-year-lagged

total assets, following Fama and French (2015) and Hou et al. (2015). At the end of June of each

year t, we compute I/A for the previous fiscal year ending in the preceding calendar year and then

assign the value to July-December of year t and January-June of year t + 1.

OP: Revenues (COMPUSTAT annual item REVT) minus cost of goods sold (item COGS), minus

selling, general, and administrative expenses (item XSGA, zero if missing), minus interest expense

(item XINT, zero if missing), divided by book equity as detailed in the estimation of B/M. This

definition follows Fama and French (2015). At the end of June of each year t, we compute OP for

the previous fiscal year ending in the preceding calendar year and then assign the value to July-

December of year t and January-June of year t + 1.

ROE: Return on equity, estimated for each fiscal quarter using COMPUSTAT’s quarterly file as

income before extraordinary items (COMPUSTAT quarterly item IBQ) divided by one-quarter-

lagged book equity. We compute quarterly book equity as shareholders’ equity plus balance sheet

deferred taxes and investment tax credit (item TXDITCQ, zero if missing), minus the book value

of preferred stock. We obtain shareholders’ equity from item SEQQ, or common equity (item

CEQQ) plus the carrying value of preferred stock (item PSTKQ), or the difference between total

assets (item ATQ) and total liabilities (item LTQ), in that order, depending on availability. The

book value of preferred stock corresponds to the redemption value of preferred stock (item

PSTKRQ), if available, or its carrying value (item PSTKQ). We use the quarterly variable ROE in

33

the months immediately after the most recent public quarterly earnings announcement dates

(COMPUSTAT quarterly item RDQ) and allow a maximum six-month lag between the end of

fiscal quarter that corresponds to its announced earnings and each month with matched ROE. We

estimate ROE for the months after 1971 because data for public quarterly earnings announcement

dates largely are not available before 1972. This definition follows Hou et al. (2015).

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35

Table I

Summary Statistics

MKTCON is a proxy for the market context of a potential stock crash, measured as expected market return (in percentage) conditional on the left-tail states of an individual

stock. Expected magnitude of the individual crash is measured by the stock’s predicted ETL (in percentage) with a 1% probability. ETL is estimated according to the GPD

modeling of the downside tail of the stock’s return distribution, and MKTCON is estimated based on the conditional extreme value approach of Heffernan and Tawn

(2004). The estimation is conducted on a five-year rolling window basis using daily return data (with at least 1,000 non-missing observations) of all common stocks listed

on NYSE, NYSE MKT (former AMEX), and Nasdaq from July 1962 to December 2014. In each estimation window, a stock is deleted if its price is below $5 or its market

capitalization is below the 10th percentile of NYSE stocks at the end of any given month. Market returns are approximated by CRSP value-weighted index returns. The

top and bottom 5% quantiles are used as thresholds to identify tail observations for GPD and extremal dependence estimations. Panel A reports summary statistics of

MKTCON in each of the five ETL subsamples formed each month with Subsample 1 representing stocks with the largest potential crash losses (i.e., the lowest or most

negative ETLs). ETL estimates are winsorized from above the top 1% and below the bottom 1% of the full sample before the subsamples are formed. Panel B reports

average values of MKTCON, ETL, size (in millions of dollars), book-to-market ratio (B/M), momentum (in percentage), the Amihud (2002) illiquidity measure, beta,

idiosyncratic volatility (Idio Vol), and idiosyncratic ETL (Idio ETL, in percentage) for five MKTCON groups formed among stocks in ETL Subsample 1. Stock

characteristics and risk variables are winsorized from above the top 1% and below the bottom 1% of their full samples.

Panel A: Summary Statistics of MKTCON in Different ETL Subsamples

Mean StdDev Skewness Kurtosis Minimum 25% Median 75% Maximum

ETL subsample 1 (mean ETL = -11.7610) -1.2333 1.0730 -1.9708 4.3594 -6.5216 -1.4752 -1.0094 -0.5966 1.5811

ETL subsample 2 (mean ETL = -8.6813) -1.2691 1.0826 -1.8525 3.7649 -6.4813 -1.5152 -1.0204 -0.6176 1.1479

ETL subsample 3 (mean ETL = -7.2226) -1.2958 1.0676 -1.7388 3.1501 -6.0699 -1.5440 -1.0205 -0.6487 1.1665

ETL subsample 4 (mean ETL = -6.0636) -1.3157 1.0734 -1.7396 3.1242 -6.3914 -1.5784 -1.0293 -0.6588 1.1143

ETL subsample 5 (mean ETL = -4.6413) -1.2766 1.0952 -1.7679 3.1575 -6.0226 -1.5565 -0.9752 -0.5955 1.2748

Panel B: Mean Stock Characteristic and Risk Variable Values in Different MKTCON Groups in ETL Subsample 1 (with Large Stock Drops)

MKTCON ETL Size B/M Momentum Illiquidity Beta Idio Vol Idio ETL

MKTCON group 1 (low context return) -1.9255 -11.4941 2748.1400 0.6457 13.8473 0.0670 1.6084 2.7552 -9.3701

MKTCON group 2 -1.5436 -11.6403 1815.9788 0.6729 13.4304 0.1116 1.4460 2.8663 -10.2021

MKTCON group 3 -1.2645 -11.7705 1427.2239 0.6972 12.8344 0.1576 1.3081 2.9058 -10.7140

MKTCON group 4 -0.9586 -11.9109 1049.5069 0.7119 12.5109 0.2222 1.1311 2.9019 -11.1461

MKTCON group 5 (high context return) -0.4546 -11.9963 824.5531 0.7571 10.8666 0.6478 0.8549 2.9282 -11.5285

36

Table II

Excess Returns of Market Context Portfolios

Stocks are grouped into quintile portfolios each month based on their MKTCON estimates within each of the five

ETL subsamples created according to ETL estimates of the month. Panel A reports value-weighted average monthly

percentage returns in excess of the risk-free rate (proxied by one-month T-bill rate) of the following month, as well

as the differences between the top and bottom MKTCON quintiles (High - Low). Panel B reports corresponding

results after further controlling for ETL. Specifically, within each ETL subsample, stocks are first sorted by ETL into

five ETL groups, and within each group, further sorted into MKTCON quintiles. Each MKTCON quintile’s value-

weighted mean excess return across all ETL portfolios is then computed. Newey and West (1987) robust t-statistics

are reported in brackets.

Panel A: Excess Returns of MKTCON Portfolios

MKTCON Portfolios

1 Low 2 3 4 5 High High - Low

ETL subsample 1 (with large stock drops) 0.0837 0.5883 0.6579 0.7825 0.8155 0.7318

[3.24]

ETL subsample 2 0.2116 0.5760 0.6593 0.7087 0.7179 0.5063

[2.39]

ETL subsample 3 0.4259 0.4751 0.5935 0.7623 0.7790 0.3532

[1.94]

ETL subsample 4 0.5027 0.4925 0.6133 0.6699 0.6181 0.1153

[0.75]

ETL subsample 5 (with small stock drops) 0.4722 0.5704 0.5783 0.5329 0.5256 0.0534

[0.40]

Panel B: Excess Returns of MKTCON Portfolios After Controlling ETL

MKTCON Portfolios

1 Low 2 3 4 5 High High - Low

ETL subsample 1 (with large stock drops) 0.1209 0.5349 0.8079 0.6819 0.8483 0.7275

[4.91]

ETL subsample 2 0.3093 0.4354 0.7119 0.6699 0.6845 0.3752

[2.91]

ETL subsample 3 0.4483 0.4962 0.6209 0.8202 0.6441 0.1958

[1.71]

ETL subsample 4 0.5270 0.6126 0.7086 0.6977 0.6408 0.1137

[1.18]

ETL subsample 5 (with small stock drops) 0.5565 0.5304 0.6563 0.6133 0.5522 -0.0043

[-0.05]

37

Table III

Market Context Effects After Controlling for Traditional Asset Pricing Variables

Panel A reports value-weighted mean return spreads (in percentage) between the top and bottom MKTCON quintile

portfolios in each ETL subsample, after controlling for ETL and one of the traditional asset pricing variables (beta,

size, B/M, momentum, illiquidity). To control for beta, within each ETL subsample, stocks are first sorted by ETL

into five ETL subgroups, and within each ETL subgroup, further sorted into five beta groups, and then into MKTCON

quintiles within each ETL-beta group. Other variables are controlled in a similar manner. Panel B reports the spreads

of alpha (in percentage) between the top and bottom MKTCON quintiles in each ETL subsample. Alpha spreads are

estimated as the intercepts of time-series regressions of the cross-MTKCON return spreads obtained from Table II,

Panel B, on the market factor (CAPM alpha spread), the Fama and French (1993) three factors (3-factor alpha spread),

or the Carhart (1997) four factors (4-factor alpha spread). Panel C reports value-weighted Fama and MacBeth (1973)

regression coefficients of MKTCON after controlling for ETL and traditional asset pricing variables in each ETL

subsample. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Return Spreads between Top and Bottom MKTCON Quintile Portfolios After Controlling Traditional Asset Pricing Variables

Controlling

Beta

Controlling

Size

Controlling

B/M

Controlling

Momentum

Controlling

Illiquidity

ETL subsample 1 (with large stock drops) 0.3804 0.3806 0.4818 0.5101 0.3437

[3.57] [3.66] [4.41] [4.73] [3.20]

ETL subsample 2 0.2226 0.1556 0.0706 0.3728 -0.0160

[2.52] [1.89] [0.79] [4.20] [-0.19]

ETL subsample 3 0.0643 0.0246 0.1071 0.2460 0.0810

[0.85] [0.35] [1.38] [3.19] [1.07]

ETL subsample 4 -0.0149 -0.0280 0.0838 0.0747 -0.0154

[-0.22] [-0.45] [1.24] [1.10] [-0.24]

ETL subsample 5 (with small stock drops) 0.0259 -0.0840 0.1117 0.0148 -0.1458

[0.48] [-1.68] [2.00] [0.27] [-2.78]

Panel B: Alpha Spreads between Top and Bottom MKTCON Quintile Portfolios

CAPM Alpha

Spread

3-Factor Alpha

Spread

4-Factor Alpha

Spread

ETL subsample 1 (with large stock drops) 0.9521 0.6781 0.5944

[6.59] [4.61] [3.92]

ETL subsample 2 0.5660 0.3369 0.1636

[4.47] [2.72] [1.23]

ETL subsample 3 0.3243 0.1547 0.0603

[2.84] [1.33] [0.49]

ETL subsample 4 0.2321 0.0680 0.0302

[2.42] [0.72] [0.31]

ETL subsample 5 (with small stock drops) 0.0924 -0.0936 -0.0583

[1.09] [-1.14] [-0.69]

(Continued)

38

Table III – Continued

Panel C: Coefficients of MKTCON from Fama-MacBeth Regressions After Controlling ETL and Different Explanatory Variables

Controlling

ETL

Controlling

ETL,

Controlling

ETL,

Controlling

ETL,

Controlling

ETL,

Beta Beta, ln(Size),

ln(B/M)

Beta, ln(Size),

ln(B/M),

Beta, ln(Size),

ln(B/M),

Momentum Momentum,

Illiquidity

ETL subsample 1 (with large stock drops) 0.7009 0.6548 0.6228 0.5359 0.5374

[3.99] [3.56] [3.25] [2.91] [2.89]

ETL subsample 2 0.4725 0.4316 0.3538 0.2987 0.3004

[3.03] [2.73] [2.39] [2.06] [2.07]

ETL subsample 3 0.2935 0.3965 0.2498 0.2531 0.2514

[2.08] [2.41] [1.65] [1.69] [1.66]

ETL subsample 4 0.1834 0.0236 0.0117 0.0071 0.0057

[1.38] [0.16] [0.08] [0.06] [0.05]

ETL subsample 5 (with small stock drops) 0.0172 -0.1396 -0.0879 -0.1257 -0.1343

[0.13] [-1.17] [-0.76] [-1.16] [-1.23]

39

Table IV

Market Context Effects Conditional on Crashes with Different Probabilities, among Stocks with Different

Institutional Holdings, and within Different Time Periods

Panel A reports mean excess returns (in percentage, after controlling for ETL) and Carhart (1997) four-factor alphas

(in percentage, after controlling for ETL) for the top and bottom MKTCON quintiles and the spreads between them,

as well as Fama-MacBeth regression coefficients of the percentile rank values of MKTCON (after controlling for

ETL), based on ETL estimates associated with probabilities of 1%, 2%, and 4%, in the subsample with the largest

potential crash losses (i.e., ETL Subsample 1). Panel B reports the return, spread, and coefficient values for the 1%-

ETL case, among stocks with different levels of institutional ownership. Stocks in the high- and low-institutional-

holding level groups have percentage institutional holdings above and below the median level in the Thomson

Reuters Ownership (13f) database, respectively. Stocks in the no-institutional-holding group are those without

ownership information in the 13f database. Panel C reports similar return and spread values and coefficients of

unranked MKTCON for the 1%-ETL case in three different time periods of 1967 (July)-1980, 1981-1997, and 1998-

2014. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: MKTCON Effects Based on ETLs with Different Probabilities

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low Ranked MKTCON

MKTCON based on 1%-ETL 0.1209 0.8483 0.7275 -0.3746 0.2197 0.5944 1.0570

[4.91] [3.92] [3.76]

MKTCON based on 2%-ETL 0.1849 0.7538 0.5689 -0.2772 0.0706 0.3478 0.9398

[3.66] [2.17] [3.22]

MKTCON based on 4%-ETL 0.3171 0.7434 0.4262 -0.1565 0.0393 0.1958 0.6901

[2.70] [1.19] [2.30]

Panel B: MKTCON Effects among Stocks with Different Institutional Holding Levels

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low Ranked MKTCON

No institutional holding -0.0864 0.7646 0.8509 -0.7376 0.1103 0.8480 1.9606

[2.62] [2.38] [3.11]

Low institutional holding

level 0.1913 0.6889 0.4975 -0.4600 0.0659 0.5259 0.8454

[1.98] [1.94] [1.80]

High institutional holding

level 0.7028 1.1042 0.4014 0.0359 0.3652 0.3293 0.8034

[1.86] [1.46] [2.10]

Panel C: MKTCON Effects in Different Sample Periods

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low MKTCON

1967 (July) - 1980 0.2109 0.4656 0.2547 -0.0664 -0.1089 -0.0425 0.0300

[1.01] [-0.18] [0.10]

1981 - 1997 0.1342 1.0730 0.9388 -0.2640 0.4555 0.7195 0.9191

[3.82] [2.91] [3.39]

1998 - 2014 0.0360 0.9276 0.8916 -0.6571 0.3270 0.9840 1.0155

[2.77] [3.52] [3.11]

40

Table V

Effects of Market Context and Systematic and Idiosyncratic Risk Measures

Panel A reports mean excess returns (in percentage, after controlling for ETL) for the top and bottom quintiles (and the spread between them) formed by the risk measures

of beta, idiosyncratic volatility, and idiosyncratic ETL (in percentage), as well as Fama-MacBeth regression coefficients of the risk measures (after controlling for ETL),

in the subsample with the largest potential crash losses (i.e., ETL Subsample 1). To control for ETL in the portfolio analysis, stocks are first sorted by ETL into five groups,

and within each group, further sorted into quintiles based on the risk measures. Panel B reports corresponding results after further controlling for MKTCON. In the

portfolio analysis, stocks are first sorted by ETL into five groups, and within each ETL group, further sorted into five MKTCON groups, and then into quintiles by the

risk measures within each ETL-MKTCON group. Panel C reports mean excess returns, return spreads, and Fama-MacBeth coefficient values for MKTCON after

controlling for ETL and each of the risk measures, with the controlling conducted in a similar manner. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Excess Returns of Portfolios Formed by Beta, Idiosyncratic Volatility, or Idiosyncratic ETL

Beta Portfolios Coefficient of Idio Vol Portfolios Coefficient of Idio ETL Portfolios Coefficient of

1 Low 5 High High - Low Beta

1 Low 5 High High - Low Idio Vol

1 Low 5 High High - Low Idio ETL

0.7709 0.5323 -0.2386 -0.3820 0.4727 0.6172 0.1444 0.0241 0.8545 0.3366 -0.5179 -0.1994

[-1.39] [-1.60] [0.88] [0.14] [-3.76] [-2.86]

Panel B: Excess Returns of Portfolios Formed by Beta, Idiosyncratic Volatility, or Idiosyncratic ETL After Controlling MKTCON

Beta Portfolios Coefficient of Idio Vol Portfolios Coefficient of Idio ETL Portfolios Coefficient of

1 Low 5 High High - Low Beta

1 Low 5 High High - Low Idio Vol

1 Low 5 High High - Low Idio ETL

0.7554 0.7562 0.0008 -0.0326 0.6498 0.5982 -0.0516 0.0600 0.6739 0.5910 -0.0829 -0.0868

[0.01] [-0.12] [-0.45] [0.33] [-0.77] [-0.89]

Panel C: Excess Returns of MKTCON Portfolios After Controlling Beta, Idiosyncratic Volatility, or Idiosyncratic ETL

Controlling Beta Coefficient of Controlling Idio Vol Coefficient of Controlling Idio ETL Coefficient of

1 Low 5 High High - Low MKTCON

1 Low 5 High High - Low MKTCON

1 Low 5 High High - Low MKTCON

0.4135 0.7939 0.3804 0.6548 0.4064 0.7747 0.3683 0.5999 0.4408 0.7881 0.3473 0.6021

[3.57] [3.56] [3.31] [3.32] [3.19] [2.72]

41

Table VI

Robustness of Market Context Effect

In Panel A, MKTCON (and ETL) are estimated for stocks whose prices are at least $5 and whose market

capitalizations are larger than those in the bottom NYSE decile in all days of each five-year estimation window. The

panel reports mean excess returns and Carhart (1997) four-factor alphas (all in percentage, after controlling for ETL)

for the top and bottom MKTCON quintiles and the spreads between them, as well as Fama-MacBeth regression

coefficient of MKTCON (after controlling for ETL), in the subsample with the largest potential crash losses (i.e.,

ETL Subsample 1). Panel B reports the return, spread, and coefficient values for MKTCON estimated using the S&P

500 index return as a proxy for the market return. Panel C reports corresponding equal-weighted portfolio and

regression values for MKTCON estimated using CRSP market returns. Panel D reports value-weighted portfolio and

regression values for MKTCON based on CRSP market returns, where MKTCON (and ETL) are estimated by

applying a 10% quantile threshold scheme for tail observation identification and extremal dependence specification.

Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Alternative Data Screening Scheme

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low MKTCON

Exclude micro and 0.1565 0.8150 0.6585 -0.3428 0.2174 0.5602 0.6663

penny stocks each day [4.45] [3.68] [3.81]

Panel B: Alternative Market Return

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low MKTCON

S&P 500 return 0.1032 0.7998 0.6966 -0.3978 0.2144 0.6121 0.7263

[4.68] [4.00] [4.21]

Panel C: Alternative Weighting Scheme

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low MKTCON

Equal-weighted 0.3782 0.8634 0.4852 -0.1608 0.1897 0.3505 0.3432

[4.25] [3.02] [2.61]

Panel D: Alternative Tail Threshold in ETL and MKTCON Estimations

Excess Returns of MKTCON Portfolios 4-Factor Alphas of MKTCON Portfolios Coefficient of

1 Low 5 High High - Low 1 Low 5 High High - Low MKTCON

10%-ptl tail 0.1481 0.7596 0.6115 -0.3219 0.0723 0.3942 0.5587

[4.01] [2.54] [2.79]

42

Table VII

Market Context Effects After Controlling for More Asset Pricing Variables

Panel A reports value-weighted mean return spreads (in percentage) between the top and bottom MKTCON quintile portfolios in each ETL subsample, after

controlling for ETL and the investment variable (I/A) or the profitability variable (OP or ROE), using the same sorting techniques as in Table III. Panel B

reports the spreads of alpha (in percentage, after controlling for ETL) between the top and bottom MKTCON quintiles in each ETL subsample. Alphas are

estimated from a five-factor model including the Fama and French (1993) three factors and the momentum and liquidity factors (FF3 + UMD + LIQ in Column

one), a five-factor model from Fama and French (2015) (FF5 in Column two), a seven-factor model including the Fama and French (2015) five factors and

the momentum and liquidity factors (FF5 + UMD + LIQ in Column three), a four-factor model from Hou et al. (2015) (HXZ4 in Column four), and a six-

factor model including the Hou et al. (2015) four factors and the momentum and liquidity factors (HXZ4 + UMD + LIQ in Column five). Panel C reports, in

each ETL subsample, value-weighted Fama and MacBeth (1973) regression coefficients of MKTCON after controlling for ETL together with the five risk

variables from Fama and French (2015) (Column one), with seven variables expanding Fama and French (2015) by momentum and illiquidity (Column two),

with the four variables from Hou et al. (2015) (Column three), with Hou et al.’s (2015) variables plus book-to-market ratio (Column four), and with Hou et

al.’s (2015) variables augmented by book-to-market ratio, momentum, and illiquidity (Column five). Similar to the asset pricing variables used in previous

tables, I/A, OP, and ROE are winsorized from above the top 1% and below the bottom 1% of their full samples. Results for models involving ROE or Hou et

al.’s (2015) factors are reported for the period of 1972-2014. In other models, the reporting period is 1968-2014 when LIQ is included and 1967 (July)-2014

otherwise. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Return Spreads between Top and Bottom MKTCON Quintile Portfolios After Controlling Investment and Profitability Variables

Controlling I/A Controlling OP Controlling ROE

ETL subsample 1 (with large stock drops) 0.3254 0.5616 0.5829

[2.95] [5.04] [4.91]

ETL subsample 2 0.2560 0.3835 0.2623

[2.83] [4.20] [2.69]

ETL subsample 3 0.1073 0.0842 0.1515

[1.37] [1.07] [1.81]

ETL subsample 4 0.0607 0.1295 0.1455

[0.89] [1.91] [2.00]

ETL subsample 5 (with small stock drops) 0.0981 0.1162 0.1586

[1.73] [2.05] [2.58]

(Continued)

43

Table VII – Continued

Panel B: Alpha Spreads between Top and Bottom MKTCON Quintile Portfolios Based on Various Factor Models

5-Factor Alpha Spread 5-Factor Alpha Spread 7-Factor Alpha Spread 4-Factor Alpha Spread 6-Factor Alpha Spread

(FF3 + UMD + LIQ) (FF5) (FF5 + UMD + LIQ) (HXZ4) (HXZ4 + UMD + LIQ)

ETL subsample 1 (with large stock drops) 0.5957 0.4191 0.3857 0.4341 0.4362

[3.86] [2.71] [2.41] [2.51] [2.51]

ETL subsample 2 0.1958 0.0383 -0.0287 -0.1518 -0.1184

[1.46] [0.31] [-0.22] [-1.07] [-0.82]

ETL subsample 3 0.0579 0.0229 -0.0329 0.0413 0.0283

[0.48] [0.19] [-0.27] [0.30] [0.21]

ETL subsample 4 0.0429 0.0289 0.0210 -0.0015 0.0015

[0.43] [0.29] [0.20] [-0.01] [0.01]

ETL subsample 5 (with small stock drops) -0.0551 -0.0177 0.0061 0.0559 0.0562

[-0.65] [-0.21] [0.07] [0.61] [0.61]

Panel C: Coefficients of MKTCON from Fama-MacBeth Regressions After Controlling ETL and Different Explanatory (Including Investment and Profitability) Variables

Controlling ETL, Controlling ETL, Controlling ETL, Controlling ETL, Controlling ETL,

Beta, ln(Size),

ln(B/M), Beta, ln(Size), ln(B/M), Beta, ln(Size), Beta, ln(Size), ln(B/M), Beta, ln(Size), ln(B/M),

I/A, OP I/A, OP, Mom, Illiq I/A, ROE I/A, ROE I/A, ROE, Mom, Illiq

ETL subsample 1 (with large stock drops) 0.6696 0.5848 0.8671 0.7872 0.7566

[3.51] [3.21] [4.19] [3.83] [3.78]

ETL subsample 2 0.2709 0.2485 0.3471 0.3697 0.3095

[1.87] [1.74] [2.13] [2.38] [2.04]

ETL subsample 3 0.2784 0.2889 0.3967 0.3062 0.3341

[1.87] [1.93] [2.40] [1.93] [2.11]

ETL subsample 4 0.0283 0.0209 -0.0047 0.0136 0.0036

[0.21] [0.17] [-0.03] [0.09] [0.03]

ETL subsample 5 (with small stock drops) -0.0785 -0.1221 -0.1118 -0.0535 -0.0961

[-0.69] [-1.14] [-0.92] [-0.45] [-0.83]

44

Table VIII

Market Context Factor and Existing Factors

In ETL Subsample 1, at the beginning of each month, two size portfolios are formed according to market capitalization of the most recent June using NYSE

median breakpoint. Independently, three MKTCON portfolios are formed according to the 30% and 70% NYSE quantiles of MKTCON estimates, after

controlling for stock ETL. The market context factor RCON is taken as the average return of the two top 30% MKTCON portfolios minus the average return

of the two bottom 30% MKTCON portfolios. The size factor RME in this setting is obtained as the difference between the average return of the three big size

portfolios and the average return of the three small size portfolios. Panel A reports average returns (in percentage) of RCON and its alphas (α, in percentage)

from various factor models, including the CAPM (Model 1), the Fama and French (1993) three-factor model (Model 2), the Carhart (1997) four-factor model

(Model 3), four-factor plus liquidity model (Model 4), the Fama and French (2015) five-factor model with (Model 6) and without (Model 5) being augmented

by momentum and liquidity factors, and the Hou et al. (2015) four-factor model with (Model 8) and without (Model 7) being augmented by momentum and

liquidity factors [the size factor in Hou et al. (2015) is denoted by RME]. Loadings (β) on existing factors are also reported. Panel B reports average returns

(in percentage) of existing factors including HML [constructed as in Fama and French (1993)], UMD, LIQ, CMA, RMW, RIA, and RROE, as well as their

alphas (in percentage) in the models involving the market, size (RME as constructed in this paper), and market context factors. Factor loadings on EM, RME,

and RCON are also reported. Results for models involving RME, RIA, and RROE are reported for the period of 1972-2014. In other models, the reporting

period is 1968-2014 when LIQ is included and 1967 (July)-2014 otherwise. Newey and West (1987) robust t-statistics are reported in brackets.

(Continued)

45

Table VIII – Continued

Panel A: Average Monthly Returns and Alphas of the Market Context Factor and Loadings on Existing Factors in Different Models

Model Ave. Return α β

EM SMB/RME HML UMD LIQ CMA RMW RIA RROE

1 0.5065 0.6678 -0.3241

[3.30] [4.71] [-8.67]

2 0.5065 0.5152 -0.2337 -0.0983 0.3466

[3.30] [3.56] [-4.98] [-1.35] [4.66]

3 0.5065 0.4621 -0.2225 -0.0969 0.3662 0.0600

[3.30] [3.09] [-4.84] [-1.26] [4.88] [1.37]

4 0.5151 0.4604 -0.2239 -0.0974 0.3615 0.0606 0.0122

[3.33] [3.11] [-4.90] [-1.25] [4.82] [1.38] [0.23]

5 0.5065 0.3076 -0.1958 0.0427 0.2621 0.1730 0.4795

[3.30] [2.20] [-5.01] [0.92] [2.82] [1.37] [6.52]

6 0.5151 0.2891 -0.1942 0.0411 0.2779 0.0325 0.0076 0.1518 0.4716

[3.33] [2.04] [-4.98] [0.88] [2.89] [0.79] [0.14] [1.16] [6.13]

7 0.5233 0.3456 -0.2497 0.0126 0.4253 0.2578

[3.19] [2.13] [-5.56] [0.14] [4.04] [3.26]

8 0.5233 0.3499 -0.2576 0.0405 -0.0937 0.0266 0.4217 0.3456

[3.19] [2.20] [-6.00] [0.53] [-1.66] [0.56] [4.11] [3.58]

(Continued)

46

Table VIII – Continued

Panel B: Average Monthly Returns and Alphas of Existing Factors and Loadings on the Market, Size, and Market Context Factors

Factor Ave. Return α β

EM RME RCON

HML 0.3653 0.2368 -0.1035 0.2275 0.2638

[2.75] [1.92] [-2.85] [4.49] [4.55]

UMD 0.6674 0.7361 -0.1255 -0.0674 0.0148

[3.56] [3.55] [-1.94] [-0.69] [0.15]

LIQ 0.4203 0.4463 -0.0461 -0.0537 0.0137

[2.73] [2.97] [-0.95] [-0.98] [0.27]

CMA 0.3663 0.3307 -0.1282 0.1927 0.1187

[4.03] [4.13] [-6.73] [6.77] [3.69]

RMW 0.2615 0.1969 -0.0580 -0.0682 0.2120

[2.52]] [1.88] [-2.16] [-1.28] [3.07]

RIA 0.4023 0.3682 -0.1058 0.1523 0.1192

[4.72] [4.69] [-5.69] [5.16] [3.70]

RROE 0.5297 0.5141 -0.0593 -0.1702 0.1512

[4.40] [4.44] [-1.55] [-3.41] [3.02]

47

A tail of two worlds: Stock crashes, market contexts,

and expected returns

Internet Appendix

Section A reports key results about the asset pricing effect of the market context estimated using the

nonparametric method based on empirical distributions. Section B examines an alternative market

context measure estimated by the co-movement of stock and market returns across the tail states of the

individual stock return distribution.

48

A. Effect of Nonparametric Market Context Estimator

In the main text, we adopt the extreme value approach as the primary measurement

framework, because it delivers a technically rigorous and economically intuitive estimation for the

expected market context payoff conditional on the extreme crash states of a stock. Although

statistically superior, this parametric method is based on asymptotic arguments and the estimate

can only be taken as an approximation in a finite sample. To show that the asset pricing effect of

the market context is not specific to this particular estimation scheme, we adopt an alternative set

of measures for ETL and MKTCON based on non-parametric empirical distributions. Specifically,

from each five-year estimation window, we select the bottom 1% lowest daily return data (with at

least 10 observations) for each stock, and compute their average to measure the 1%-ETL.

Corresponding to each of these lowest stock returns, there is an associated market-index return;

we use the average of these market returns to measure MKTCON. Table AI reports the results.

The first column in Panel A shows that the value-weighted return difference between the

top and bottom quintiles of the market context estimated from empirical distributions (denoted by

MKTCON_emp) is 65.72 bps per month in ETL Subsample 1, which is statistically significant.

The return spread after stock ETL is controlled is reported in the second column, and other columns

report the spreads when beta, firm size, book-to-market ratio (B/M), momentum, illiquidity,

investment (I/A), and profitability (OP or ROE) are additionally controlled. In all these cases, the

return differences range from 33.74 to 71.64 bps and the t-statistics are between 3.26 and 5.29. In

other subsamples with less severe individual stock price crashes, the effect becomes weaker.

Panels B and C report alpha spreads between the top and bottom MTKCON_emp quintiles

and Fama-MacBeth regression coefficients of MKTCOM_emp in different ETL subsamples. For

both alpha and regression analyses, we consider all the models (and the associated factors) in the

main tests, including the CAPM, the Fama and French (1993) three-factor model, the Carhart

(1997) four-factor model, the Hou, Xue, and Zhang (2015) four-factor model, and the Fama and

French (2015) five-factor model, as well as their extensions. In all these models, the results are

similar to those in the main tests. Overall, Table AI confirms the robustness of our empirical results

on the asset pricing effect of the market context, which mitigates concerns about potential

49

estimation errors in the GPD and extremal dependence parameters used for our primary

measurement in the main text.

B. Effect of Conditional Co-Movement Measure

The salience of an asset’s payoff from a comparison with the concurrent market payoff as

suggested by Bordalo, Gennaioli, and Shleifer (2012, 2013) is a highly appealing concept. Both

the parametric and nonparametric measurement schemes we have discussed so far closely follow

the economic intuition of the salience theory by gauging the expected levels of stock crash and the

conditional market return. These schemes, however, are by no means the only ways to detect the

market context of a stock crash. In this section, we examine a measure that is built on the co-

movement between a stock and the market across the tail states of the stock’s return distribution,

and explore its impact on asset prices.

We estimate the conditional co-movement measure, denoted by C-COMVT, in the

following way:

- ( )( )i i

i i

i i i m m R h

R h

C COMVT R h R h n

, (B1)

where hi and hm are return thresholds corresponding to a low quantile of the return distributions of

stock i and market m, respectively, andi iR hn is the number of stock returns that fall below its

threshold. Obviously, C-COMVT considers only the left-tail states of the individual stock, and

with this condition satisfied, it indicates how far the market return is from its own threshold. If the

market is above its threshold, C-COMVT will be negative; otherwise, it will be positive. A more

negative C-COMVT means the market performs better when the stock is in its downside tail states.

From the standpoint of the salience theory, it makes the stock’s loss more salient. Therefore, we

take C-COMVT as an alternative measure for the market context.

We estimate C-COMVT using a 5%-quantile daily return threshold on a five-year rolling

window basis, to be consistent with the primary EVT-based market context measurement (as

explained in Appendix B of the main text), and report the key results in Table BI. There is a clearly

negative relation between C-COMVT and expected returns pertaining to stocks with large crash

losses (i.e., in the subsample with most negative ETLs estimated via the extreme value approach):

Return spreads, alpha spreads, and regression coefficients of C-COMVT are negative in all cases.

50

Since a lower C-COMVT implies a better market performance when a stock is in its left-tail states,

these findings are largely consistent with our conclusion about the asset pricing effect of the market

context.

Unlike our primary measurement that directly translates the economic intuition of stock

crash and its market context, the structure of C-COMVT is based on the co-movement between a

stock and the market, which potentially has dual implications with regard to its impact on asset

prices. On the one hand, if we strictly control for the tail-return variations of individual stocks,

then C-COMVT mainly reflects the associated market performance, and thus can serve as a proxy

for the market context. Consistent with this idea, the analyses in Table BI are conducted within

subsamples with similar levels of stock ETL, and further control for ETL in each subsample by

averaging across five ETL subportfolios. This approach significantly limits cross-sectional

variation of individual stocks’ tail returns, which enables C-COMVT to become a reflection of the

market context.1 We therefore detect an asset pricing effect consistent with our primary MKTCON

measure. This is especially evident when individual stock returns are at substantially low levels

(e.g., within ETL Subsample 1) so that salience exerts a more pronounced influence on stock prices,

which is also implied by the salience theory.

On the other hand, if we allow sufficient variations in individual stocks’ downside returns,

a larger C-COMVT can also indicate that the market is declining along with a stock’s decline,

which constitutes an overall downside risk to an investor’s total portfolio. Such risk requires higher

expected payoffs as a compensation, resulting in a positive relation between C-COMVT and stock

returns. In fact, Bali, Cakici, and Whitelaw (2014) have documented a positive pricing effect of a

conditional co-movement measure similarly constructed as C-COMVT.2 In Bali et al.’s (2014)

setting, however, individual stocks’ tail returns are not controlled because their cross-sectional

analyses are conducted in the full sample (rather than in each ETL subsample). Moreover, Bali et

al. (2014) estimate their measure from a sampling period of no more than one year, because they

1 We conduct the analyses in ETL subsamples (and control for ETL in each of them) to make the results in Table BI

comparable with those in the main tests. Relative to the full sample, the cross-sectional variation of stock tail returns

in each ETL subsample is considerably smaller, and further declines after ETL is additionally controlled in each

subsample. 2 Bali et al.’s (2014) measure is called the hybrid tail covariance risk, which is estimated as ( )( )

i i

i i m m

R h

R h R h

with

notations the same as in Equation (B1). C-COMVT takes the average of this value since there may be different

numbers of observations in the tail states (i.e., returns below a certain quantile) of different stocks.

51

mainly consider “the more frequent but less extreme tail events that occur on a regular basis” (Bali

et al., 2014, p. 208). A relatively short period is less likely to detect crash risk (i.e., infrequent

event of extreme magnitude) and thus the market context effect is less salient. This is in contrast

to a much longer estimation window (e.g., five years) which adequately captures a rare, extreme

risk and hosts a stronger market context effect. Therefore, the asset pricing effect of conditional

co-movement-based measures hinges on whether there is sufficient extremeness in individual tail

losses (determined by the length of sampling period) and whether the variation of such losses is

adequately controlled so that the market performance has more context connotation.

To highlight the influences of sampling period and stock loss variation on the pricing effect

of C-COMVT, we estimate C-COMVT using rolling windows with lengths of one, three, and five

years, while keeping all other aspects (e.g., quantile threshold for tail-return identification) of the

estimation unchanged. Based on these C-COMVT estimates, we conduct the portfolio, alpha, and

regression analyses in the full sample and do not control for the level of ETL. Table BII shows

that, as we shorten the estimation window length, the relation between C-COMVT and stock

returns becomes more positive in the cross-section of the full sample. In particular, in the one-

year-window scenario, the return and alpha spreads and Fama-MacBeth coefficients of C-COMVT

are significantly positive in general, which is consistent with the findings of Bali et al. (2014).3

The results are largely reversed when the estimation window becomes longer and covers more

extreme crash losses, although the C-COMVT effect is not statistically significant for many cases

(especially in the regressions). This evidence suggests that, as the estimation period expands, for

a given quantile threshold, tail observations below the threshold invoke more crash connotations,

so the implication of the C-COMVT measure tilts away from portfolio risk at one end of the

spectrum and toward the market context at the other end. Larger C-COMVT can indicate both a

higher portfolio risk and a worse market context, with opposite asset pricing effects. Determination

of the dominating effect depends on whether C-COMVT has more portfolio risk or market context

implications – that is, whether C-COMVT is estimated from a short or long sampling period.4

3 These are equal-weighted results, to be consistent with Bali et al. (2014) who report only equal-weighted results for

the one-year-window scenario. 4 Bali et al. (2014) adopt a model setting in which investors have concentrated holdings in a stock together with an

investment in a well-diversified market portfolio. Our simple salience-based model in Section I of the main text can

fit such a setting too if we take the market portfolio as a reference/context for the stock in its crash states. In this case,

C-COMVT has the suggested dual implications.

52

The positive C-COMVT-return relation in the one-year-window case is also partially

contributed by the variations of tail losses of individual stocks because the tests in Table BII (and

in Bali et al., 2014) do not control for the change of stock crash. To investigate to what extent the

C-COMVT effect is driven by tail risk, we report results from return spread, alpha spread, and

regression analyses in the one-year-window case after controlling for stock ETL in Table BIII.

These results reflect the residual asset pricing effect of C-COMVT caused by the downside

movement risk of the market. A comparison between the last rows of Table BII’s panels and the

corresponding values in Table BIII reveals that the magnitudes (and the associated t-statistics) of

C-COMVT premia in the forms of return spread, alpha spread, and regression coefficient are

substantially reduced after individual stocks’ ETLs are controlled in the full-sample analyses. In

several cases the premia become insignificant or even negative. The evidence suggests a nontrivial

influence from stock tail risk in the pricing of the conditional co-movement measure.

In summary, findings in this section lend support to our conjecture about the distinction

between the Bali et al. (2014) model and the salience theory, which enriches potential economic

implications of the conditional co-movement measure.5

REFERENCES

Adrian, T., and M. Brunnermeier. 2016. CoVaR. American Economic Review 106, 1705-1741.

Bali, T., N. Cakici, and R. Whitelaw. 2014. Hybrid tail risk and expected stock returns: When does the tail

wag the dog? Journal of Asset Pricing Studies 4, 206-246.

Bordalo, P., N. Gennaioli, and A. Shleifer. 2012. Salience theory of choice under risk. Quarterly Journal of

Economics 127, 1243-1285.

5 Another study that involves a risk measure conditional on an individual firm’s particular state is CoVaR from Adrian

and Brunnermeier (2016) which is defined as the VaR of the financial system conditional on an institution being under

distress. CoVaR is a conditional tail-dependency measure, which mainly captures the co-crash risk of both the

institution and the financial system. If the institution and the system collapse together, CoVaR has some overlapping

implications with other market context measures. However, for the situations in which the system does not fall along

with the institution, CoVaR is not effective as a measure for market context because it cannot differentiate between,

say, an increased system and a slightly decreased system because in both cases there is no co-crash (and CoVaR values

should be similar). In other words, CoVaR is a noisier indicator for the market context when the institution losses are

more salient relative to the system performance. After all, CoVaR is designed to reflect systemic risk in the financial

industry rather than the context of a firm’s particular payoff. Moreover, unlike Bali et al. (2014) who introduce the

conditional co-movement measure to examine its asset pricing effect, Adrian and Brunnermeier (2016) introduce

CoVaR without asset pricing concerns either theoretically and empirically. Therefore, as our paper is to explore the

asset pricing implication of salience theory, it is important to investigate how the conditional co-movement measure,

but not necessarily CoVaR, is related to market context in the cross-section of stock returns. Our paper aims at testing

the salience theory, not introducing a new asset pricing measure that is different from existing measures. We believe

the consistent results from the EVT-based, empirical-distribution-based, and conditional co-movement-based market

context measures have provided sufficient support to the salience theory even without checking the stock return effect

of CoVaR that neither provides a complete description of market context nor exists as an asset pricing measure.

53

Bordalo, P., N. Gennaioli, and A. Shleifer. 2013. Salience and asset prices. American Economic Review:

Papers & Proceedings 103, 623-628.

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

Fama, E., and K. French. 1993. Common risk factors in the returns on stocks and bonds. Journal of

Financial Economics 33, 3-56.

Fama, E., and K. French. 2015. A five-factor asset pricing model. Journal of Financial Economics 116, 1-

22.

Fama, E., and J. MacBeth. 1973. Risk, return, and equilibrium: Empirical tests. Journal of Political

Economy 81, 607-36.

Hou, K., C. Xue, and L. Zhang. 2015. Digesting anomalies: An investment approach. Review of Financial

Studies 28, 650-705.

Newey, W., and K. West. 1987. A simple, positive semi-definite, heteroskedasticity and autocorrelation

consistent covariance matrix. Econometrica 55, 703-708.

54

Table AI. Effect of Market Context Estimated from Nonparametric Empirical Distributions

This table presents the asset pricing effect of the market context estimated from the empirical distributions of stock and market returns, denoted by

MKTCON_emp, in five ETL (estimated from empirical distributions) subsamples. Panel A reports value-weighted mean return spread (in percentage) between

the top and bottom MKTCON_emp quintile portfolios in each ETL subsample, and the spreads after controlling for ETL as well as other variables including

beta, size, book-to-market ratio (B/M), momentum, illiquidity, the investment variable (I/A), and the profitability variable (OP or ROE), using the same sorting

techniques as in Tables II, III, and VII of the main text. Panel B reports the spreads of alpha (in percentage, after controlling for ETL) between the top and

bottom MKTCON_emp quintiles in each ETL subsample. Alphas are estimated from the CAPM, the Fama and French (1993) three-factor model (FF3), the

Carhart (1997) four-factor model, the Hou et al. (2015) four-factor model (HXZ4), and the Fama and French (2015) five-factor model (FF5), as well as their

extensions. Panel C reports, in each ETL subsample, value-weighted Fama and MacBeth (1973) regression coefficients of MKTCON_emp after controlling

for ETL and various asset pricing variables reflected in the factor models in Panel B. Similar winsorization is applied to asset pricing variables as in the

relevant tests in the main text. Results for models involving ROE or Hou et al.’s (2015) factors are reported for the period of 1972-2014. In other models, the

reporting period is 1968-2014 when the liquidity factor LIQ is included and 1967 (July)-2014 otherwise. Newey and West (1987) robust t-statistics are reported

in brackets.

Panel A: Return Spreads between Top and Bottom MKTCON_emp Quintile Portfolios Before and After Controlling ETL and Existing Asset Pricing Variables

High - Low Controlling

ETL

Controlling

Beta

Controlling

Size

Controlling

B/M

Controlling

Momentum

Controlling

Illiquidity

Controlling

I/A

Controlling

OP

Controlling

ROE

ETL subsample 1 (with large stock drops) 0.6572 0.7164 0.3950 0.3374 0.4272 0.5776 0.3927 0.4732 0.5268 0.5604

[2.98] [4.87] [3.74] [3.26] [3.87] [5.29] [3.64] [4.23] [4.71] [4.66]

ETL subsample 2 0.4915 0.4757 0.0957 0.0701 0.1219 0.4444 -0.0130 0.2063 0.3453 0.3504

[2.57] [3.74] [1.13] [0.88] [1.40] [5.00] [-0.15] [2.34] [3.98] [3.62]

ETL subsample 3 0.1678 0.1932 -0.0532 0.0479 0.1021 0.1613 -0.0679 0.0696 0.0499 0.1616

[0.99] [1.76] [-0.73] [0.67] [1.33] [2.10] [-0.94] [0.90] [0.65] [1.93]

ETL subsample 4 0.1381 0.1740 -0.0286 -0.0005 0.0948 0.1458 -0.0325 0.1093 0.1546 0.1300

[0.94] [1.79] [-0.44] [-0.01] [1.40] [2.15] [-0.51] [1.58] [2.25] [1.78]

ETL subsample 5 (with small stock drops) -0.0332 0.0291 0.0050 -0.1182 0.0779 0.0670 -0.1056 0.0673 0.0122 0.0890

[-0.26] [0.36] [0.10] [-2.39] [1.44] [1.19] [-2.07] [1.20] [0.22] [1.46]

(Continued)

55

Table AI – Continued

Panel B: Alpha Spreads between Top and Bottom MKTCON_emp Quintile Portfolios

CAPM Alpha

Spread

3-Factor Alpha

Spread

4-Factor Alpha

Spread

5-Factor Alpha

Spread

5-Factor Alpha

Spread

7-Factor Alpha

Spread

4-Factor

Alpha Spread

6-Factor Alpha

Spread

(FF3 + UMD) (FF3 + UMD +

LIQ)

(FF5) (FF5 + UMD +

LIQ)

(HXZ4) (HXZ4 + UMD

+ LIQ)

ETL subsample 1 (with large stock drops) 0.9261 0.6430 0.5564 0.5506 0.3728 0.3303 0.4447 0.4284

[6.46] [4.35] [3.62] [3.51] [2.43] [2.07] [2.62] [2.50]

ETL subsample 2 0.6365 0.4269 0.3070 0.2937 0.1199 0.0503 0.0253 0.0198

[5.06] [3.35] [2.25] [2.17] [0.94] [0.38] [0.17] [0.14]

ETL subsample 3 0.2970 0.1419 0.0974 0.0886 0.0056 -0.0159 0.0223 0.0188

[2.68] [1.26] [0.83] [0.75] [0.05] [-0.13] [0.17] [0.15]

ETL subsample 4 0.2788 0.0979 0.0633 0.0921 0.0373 0.0505 0.0242 0.0497

[2.86] [1.00] [0.63] [0.90] [0.36] [0.48] [0.21] [0.44]

ETL subsample 5 (with small stock drops) 0.1016 -0.0542 -0.0139 -0.0084 0.0293 0.0582 0.0513 0.0513

[1.25] [-0.68] [-0.17] [-0.10] [0.36] [0.70] [0.58] [0.58]

(Continued)

56

Table AI – Continued

Panel C: Coefficients of MKTCON_emp from Fama-MacBeth Regressions After Controlling ETL and Different Explanatory Variables

Controlling

ETL

Controlling

ETL, Beta

Controlling

ETL, Beta,

ln(Size),

ln(B/M)

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

Mom

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

Mom, Illiq

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

I/A, OP

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

OP, Mom,

Illiq

Controlling

ETL, Beta,

ln(Size),

I/A, ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

I/A, ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

ROE, Mom,

Illiq

ETL subsample 1 (with large stock drops) 0.6135 0.5925 0.5199 0.4508 0.4617 0.5480 0.4979 0.7517 0.6684 0.6531

[3.84] [3.69] [3.34] [2.96] [3.02] [3.57] [3.32] [4.44] [4.02] [3.99]

ETL subsample 2 0.4283 0.3512 0.2609 0.2388 0.2510 0.1812 0.1930 0.3354 0.3190 0.2812

[3.38] [2.46] [1.91] [1.82] [1.91] [1.37] [1.50] [2.36] [2.33] [2.11]

ETL subsample 3 0.1694 0.2009 0.1071 0.1078 0.1021 0.1342 0.1365 0.2601 0.1782 0.1892

[1.43] [1.57] [0.93] [0.97] [0.91] [1.17] [1.23] [2.04] [1.45] [1.59]

ETL subsample 4 0.1618 0.0219 0.0305 0.0356 0.0302 0.0451 0.0422 0.0338 0.0234 0.0320

[1.36] [0.17] [0.26] [0.34] [0.28] [0.39] [0.40] [0.27] [0.19] [0.29]

ETL subsample 5 (with small stock drops) 0.0162 -0.0546 -0.0193 -0.0396 -0.0402 -0.0079 -0.0282 -0.0735 -0.0399 -0.0322

[0.16] [-0.60] [-0.23] [-0.49] [-0.49] [-0.09] [-0.35] [-0.78] [-0.44] [-0.36]

57

Table BI. Effect of Co-Movement Measure Conditional on a Stock’s Tail States

This table presents the asset pricing effect of the conditional co-movement measure, denoted by C-COMVT, in five ETL (estimated via the extreme value

approach) subsamples. Panel A reports value-weighted mean return spread (in percentage) between the top and bottom C-COMVT quintile portfolios in each

ETL subsample, and the spreads after controlling for ETL as well as other variables including beta, size, book-to-market ratio (B/M), momentum, illiquidity,

the investment variable (I/A), and the profitability variable (OP or ROE), using the same sorting techniques as in Tables II, III, and VII of the main text. Panel

B reports the spreads of alpha (in percentage, after controlling for ETL) between the top and bottom C-COMVT quintiles in each ETL subsample. Alphas are

estimated from the CAPM, the Fama and French (1993) three-factor model (FF3), the Carhart (1997) four-factor model, the Hou et al. (2015) four-factor

model (HXZ4), and the Fama and French (2015) five-factor model (FF5), as well as their extensions. Panel C reports, in each ETL subsample, value-weighted

Fama and MacBeth (1973) regression coefficients of C-COMVT after controlling for ETL and various asset pricing variables reflected in the factor models in

Panel B. Similar winsorization is applied to asset pricing variables as in the relevant tests in the main text. Results for models involving ROE or Hou et al.’s

(2015) factors are reported for the period of 1972-2014. In other models, the reporting period is 1968-2014 when the liquidity factor LIQ is included and 1967

(July)-2014 otherwise. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Return Spreads between Top and Bottom C-COMVT Quintile Portfolios Before and After Controlling ETL and Existing Asset Pricing Variables

High - Low Controlling

ETL

Controlling

Beta

Controlling

Size

Controlling

B/M

Controlling

Momentum

Controlling

Illiquidity

Controlling

I/A

Controlling

OP

Controlling

ROE

ETL subsample 1 (with large stock drops) -0.6137 -0.6543 -0.2900 -0.5162 -0.4758 -0.4299 -0.2953 -0.4025 -0.5963 -0.6058

[-2.86] [-4.38] [-2.68] [-5.02] [-4.42] [-4.01] [-2.75] [-3.67] [-5.36] [-5.07]

ETL subsample 2 -0.3856 -0.4139 -0.0358 -0.0590 -0.1787 -0.2835 -0.0257 -0.1983 -0.3188 -0.1661

[-1.87] [-3.16] [-0.42] [-0.73] [-2.03] [-3.25] [-0.30] [-2.20] [-3.55] [-1.73]

ETL subsample 3 -0.4933 -0.3545 -0.1472 0.0028 -0.1179 -0.1882 -0.0605 -0.1713 -0.1490 -0.1595

[-2.75] [-3.23] [-2.00] [0.04] [-1.58] [-2.52] [-0.83] [-2.25] [-1.96] [-1.96]

ETL subsample 4 -0.1871 -0.1659 0.0958 -0.0042 -0.0421 -0.1119 0.0259 -0.1017 -0.1531 -0.1518

[-1.24] [-1.78] [1.49] [-0.07] [-0.63] [-1.69] [0.42] [-1.53] [-2.29] [-2.13]

ETL subsample 5 (with small stock drops) -0.1382 -0.0556 -0.0095 0.0665 -0.0696 -0.0708 0.1385 -0.0845 -0.0637 -0.0908

[-1.08] [-0.66] [-0.18] [1.34] [-1.29] [-1.29] [2.64] [-1.54] [-1.15] [-1.53]

(Continued)

58

Table BI – Continued

Panel B: Alpha Spreads between Top and Bottom C-COMVT Quintile Portfolios

CAPM Alpha

Spread

3-Factor Alpha

Spread

4-Factor Alpha

Spread

5-Factor Alpha

Spread

5-Factor Alpha

Spread

7-Factor Alpha

Spread

4-Factor Alpha

Spread

6-Factor Alpha

Spread

(FF3 + UMD) (FF3 + UMD +

LIQ)

(FF5) (FF5 + UMD +

LIQ)

(HXZ4) (HXZ4 + UMD

+ LIQ)

ETL subsample 1 (with large stock drops) -0.8924 -0.5928 -0.4848 -0.4846 -0.2645 -0.2185 -0.2105 -0.2192

[-6.13] [-4.00] [-3.16] [-3.12] [-1.71] [-1.36] [-1.21] [-1.26]

ETL subsample 2 -0.6001 -0.3672 -0.2366 -0.2421 -0.0285 0.0264 0.1574 0.1349

[-4.71] [-2.91] [-1.74] [-1.78] [-0.23] [0.20] [1.07] [0.91]

ETL subsample 3 -0.4886 -0.3462 -0.2747 -0.2707 -0.2316 -0.1912 -0.2224 -0.2063

[-4.48] [-3.10] [-2.35] [-2.34] [-2.02] [-1.64] [-1.73] [-1.63]

ETL subsample 4 -0.2804 -0.1095 -0.0732 -0.0720 -0.0572 -0.0374 -0.0188 -0.0113

[-3.00] [-1.18] [-0.76] [-0.73] [-0.59] [-0.37] [-0.18] [-0.10]

ETL subsample 5 (with small stock drops) -0.1422 0.0229 0.0089 0.0073 -0.0522 -0.0585 -0.0789 -0.0718

[-1.68] [0.28] [0.10] [0.09] [-0.63] [-0.68] [-0.88] [-0.79]

(Continued)

59

Table BI – Continued

Panel C: Coefficients of C-COMVT from Fama-MacBeth Regressions After Controlling ETL and Different Explanatory Variables

Controlling

ETL

Controlling

ETL, Beta

Controlling

ETL, Beta,

ln(Size),

ln(B/M)

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

Mom

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

Mom, Illiq

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

I/A, OP

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

OP, Mom,

Illiq

Controlling

ETL, Beta,

ln(Size),

I/A, ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M),

I/A, ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

ROE, Mom,

Illiq

ETL subsample 1 (with large stock drops) -40.6337 -38.4520 -35.3964 -33.6550 -34.7582 -37.4292 -36.1808 -49.1211 -45.7965 -48.4397

[-4.11] [-3.68] [-3.31] [-3.30] [-3.34] [-3.56] [-3.54] [-4.39] [-4.13] [-4.50]

ETL subsample 2 -34.5114 -24.6789 -16.8253 -15.6537 -16.0483 -11.9671 -13.1886 -17.2757 -19.8772 -16.5758

[-3.15] [-1.86] [-1.34] [-1.26] [-1.29] [-0.97] [-1.09] [-1.36] [-1.62] [-1.37]

ETL subsample 3 -27.1562 -33.4026 -20.4883 -20.6116 -21.7078 -23.2173 -24.2242 -34.3546 -25.5086 -29.2694

[-2.11] [-2.14] [-1.38] [-1.41] [-1.45] [-1.62] [-1.67] [-2.26] [-1.71] [-1.96]

ETL subsample 4 -27.7385 -13.5818 -12.7659 -9.2027 -8.9432 -14.8627 -11.4927 -4.3243 -8.3203 -5.7082

[-1.85] [-0.80] [-0.82] [-0.66] [-0.63] [-0.98] [-0.83] [-0.26] [-0.52] [-0.38]

ETL subsample 5 (with small stock drops) -8.7323 -1.1799 -0.9149 5.1354 6.4916 0.5251 6.3270 8.2562 0.2810 5.1929

[-0.46] [-0.06] [-0.06] [0.33] [0.41] [0.03] [0.40] [0.48] [0.02] [0.31]

60

Table BII. Effects of Conditional Co-Movement Measures Estimated from Different Rolling Windows

This table presents the full-sample asset pricing effects of C-COMVT estimated from rolling windows with the lengths of five years, three years, and one year.

For each estimation scheme, Panel A reports equal-weighted mean return spread (in percentage) between the top and bottom C-COMVT quintile portfolios in

the full sample, as well as the spreads after controlling for beta, size, book-to-market ratio (B/M), momentum, illiquidity, the investment variable (I/A), or the

profitability variable (OP or ROE), using similar sorting techniques as in Tables II, III, and VII of the main text. Panel B reports the spreads of alpha (in

percentage) between the top and bottom C-COMVT quintiles in the full sample. Alphas are estimated from the CAPM, the Fama and French (1993) three-

factor model (FF3), the Carhart (1997) four-factor model, the Hou et al. (2015) four-factor model (HXZ4), and the Fama and French (2015) five-factor model

(FF5), as well as their extensions. Panel C reports equal-weighted Fama and MacBeth (1973) regression coefficients of C-COMVT after controlling for various

asset pricing variables reflected in the factor models in Panel B. ETL is not controlled in the tests of this table. Similar winsorization is applied to asset pricing

variables as in the relevant tests in the main text. Results for models involving ROE or Hou et al.’s (2015) factors are reported for the period of 1972-2014. In

other models, the reporting period is 1968-2014 when the liquidity factor LIQ is included and 1967 (July)-2014 otherwise. Newey and West (1987) robust t-

statistics are reported in brackets.

Panel A: Return Spreads between Top and Bottom C-COMVT Quintile Portfolios Before and After Controlling Existing Asset Pricing Variables

High - Low Controlling

Beta

Controlling

Size

Controlling

B/M

Controlling

Momentum

Controlling

Illiquidity

Controlling

I/A

Controlling

OP

Controlling

ROE

5-year estimation window -0.1839 -0.1733 -0.0580 -0.1802 -0.2786 -0.0288 -0.1959 -0.2088 -0.2624

[-1.80] [-2.85] [-1.03] [-3.14] [-4.87] [-0.51] [-3.50] [-3.71] [-4.35]

3-year estimation window 0.0031 -0.0180 0.0784 0.0058 -0.1217 0.1355 -0.0293 -0.0304 -0.0789

[0.03] [-0.31] [1.46] [0.10] [-2.20] [2.44] [-0.54] [-0.54] [-1.32]

1-year estimation window 0.3529 0.1626 0.4093 0.2611 0.1139 0.4105 0.2326 0.2747 0.1834

[3.33] [2.94] [7.61] [4.70] [2.10] [7.43] [4.22] [4.87] [3.03]

(Continued)

61

Table BII – Continued

Panel B: Alpha Spreads (without Controlling ETL) between Top and Bottom C-COMVT Quintile Portfolios

CAPM Alpha

Spread

3-Factor Alpha

Spread

4-Factor Alpha

Spread

5-Factor Alpha

Spread

5-Factor Alpha

Spread

7-Factor Alpha

Spread

4-Factor Alpha

Spread

6-Factor Alpha

Spread

(FF3 + UMD) (FF3 + UMD +

LIQ)

(FF5) (FF5 + UMD +

LIQ)

(HXZ4) (HXZ4 + UMD

+ LIQ)

5-year estimation window -0.2309 -0.2063 -0.1875 -0.1556 -0.2595 -0.2083 -0.2899 -0.2320

[-2.22] [-2.31] [-1.74] [-1.43] [-2.90] [-1.99] [-2.48] [-1.93]

3-year estimation window -0.0259 0.0079 -0.0194 -0.0046 -0.0242 -0.0277 -0.0917 -0.0703

[-0.25] [0.09] [-0.19] [-0.04] [-0.28] [-0.27] [-0.84] [-0.59]

1-year estimation window 0.3504 0.4117 0.2779 0.2790 0.3809 0.2808 0.2329 0.2104

[3.31] [4.60] [3.07] [3.03] [4.25] [3.14] [2.26] [1.96]

Panel C: Coefficients of C-COMVT from Fama-MacBeth Regressions Before and After Controlling Different Explanatory Variables

No Controls Controlling

Beta

Controlling

Beta,

ln(Size),

ln(B/M)

Controlling

Beta,

ln(Size),

ln(B/M),

Mom

Controlling

Beta,

ln(Size),

ln(B/M),

Mom, Illiq

Controlling

Beta,

ln(Size),

ln(B/M),

I/A, OP

Controlling

Beta,

ln(Size),

ln(B/M),

I/A, OP,

Mom, Illiq

Controlling

Beta,

ln(Size),

I/A, ROE

Controlling

Beta,

ln(Size),

ln(B/M),

I/A, ROE

Controlling

Beta,

ln(Size),

ln(B/M), I/A,

ROE, Mom,

Illiq

5-year estimation window -8.3135 -3.8417 1.1324 -0.3830 0.6415 -0.2012 -0.4894 -2.2265 -6.0188 -6.4214

[-1.46] [-0.55] [0.23] [-0.08] [0.13] [-0.04] [-0.10] [-0.44] [-1.26] [-1.34]

3-year estimation window 3.9783 3.2499 7.8980 4.1318 4.9286 6.7608 3.9050 5.6254 2.9288 0.4800

[0.80] [0.61] [2.12] [1.15] [1.35] [1.83] [1.08] [1.49] [0.82] [0.13]

1-year estimation window 19.0954 11.9905 12.9795 7.0093 7.2411 12.1711 6.3415 8.9201 6.6772 2.4823

[5.53] [3.58] [4.90] [2.86] [2.88] [4.67] [2.56] [3.45] [2.64] [0.99]

62

Table BIII. Effect of Conditional Co-Movement Measure Estimated from One-Year Rolling Window After Controlling Stock Crash Risk

This table presents the full-sample asset pricing effect of C-COMVT estimated from a rolling window with a one-year length. The tests are corresponding to

those in the one-year-window scenario of Table BII, but with ETL controlled. Panel A reports equal-weighted mean return spreads (in percentage) between

the top and bottom C-COMVT quintile portfolios in the full sample, after controlling for ETL as well as other variables including beta, size, book-to-market

ratio (B/M), momentum, illiquidity, the investment variable (I/A), and the profitability variable (OP or ROE), using the same sorting techniques as in Tables

II, III, and VII of the main text. Panel B reports the spreads of alpha (in percentage, after controlling for ETL) between the top and bottom C-COMVT quintiles

in the full sample. Alphas are estimated from the CAPM, the Fama and French (1993) three-factor model (FF3), the Carhart (1997) four-factor model, the Hou

et al. (2015) four-factor model (HXZ4), and the Fama and French (2015) five-factor model (FF5), as well as their extensions. Panel C reports equal-weighted

Fama and MacBeth (1973) regression coefficients of C-COMVT after controlling for ETL and various asset pricing variables reflected in the factor models in

Panel B. Similar winsorization is applied to asset pricing variables as in the relevant tests in the main text. Results for models involving ROE or Hou et al.’s

(2015) factors are reported for the period of 1972-2014. In other models, the reporting period is 1968-2014 when the liquidity factor LIQ is included and 1967

(July)-2014 otherwise. Newey and West (1987) robust t-statistics are reported in brackets.

Panel A: Return Spreads between Top and Bottom C-COMVT Quintile Portfolios After Controlling ETL and Existing Asset Pricing Variables

Controlling ETL Controlling Beta Controlling Size Controlling B/M Controlling Momentum Controlling Illiquidity Controlling I/A Controlling OP Controlling ROE

0.1206 0.1149 0.1649 0.1234 -0.0554 0.1879 0.1228 0.0923 0.0127

[2.19] [3.24] [4.53] [3.37] [-1.51] [5.10] [3.33] [2.53] [0.32]

Panel B: Alpha Spreads (After Controlling ETL) between Top and Bottom C-COMVT Quintile Portfolios

CAPM Alpha

Spread

3-Factor Alpha

Spread

4-Factor Alpha

Spread

5-Factor Alpha

Spread 5-Factor Alpha Spread

7-Factor Alpha

Spread

4-Factor Alpha

Spread

6-Factor Alpha

Spread

(FF3 + UMD) (FF3 + UMD +

LIQ)

(FF5) (FF5 + UMD + LIQ) (HXZ4) (HXZ4 + UMD +

LIQ)

0.0086 0.1496 0.0815 0.0860 0.2378 0.1776 0.1374 0.1180

[0.16] [2.84] [1.48] [1.55] [4.18] [3.05] [2.13] [1.84]

(Continued)

63

Table BIII – Continued

Panel C: Coefficients of C-COMVT from Fama-MacBeth Regressions After Controlling ETL and Different Explanatory Variables

Controlling

ETL

Controlling

ETL, Beta

Controlling

ETL, Beta,

ln(Size),

ln(B/M)

Controlling

ETL, Beta,

ln(Size),

ln(B/M), Mom

Controlling

ETL, Beta,

ln(Size),

ln(B/M), Mom,

Illiq

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

OP

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

OP, Mom, Illiq

Controlling

ETL, Beta,

ln(Size), I/A,

ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

ROE

Controlling

ETL, Beta,

ln(Size),

ln(B/M), I/A,

ROE, Mom,

Illiq

12.5411 10.0214 10.8771 4.9932 4.9652 10.1723 4.1717 7.0796 5.6190 1.2773

[4.27] [3.36] [3.98] [1.97] [1.89] [3.80] [1.62] [2.70] [2.16] [0.49]