A tail of two worlds: Stock crashes, market contexts, and expected returns *

Transcription

1 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 USA Tel: 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: fengwu@polyu.edu.hk 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) (the corresponding author) gratefully acknowledges financial support from the General Research Fund (GRF) No. B-Q50N of the University Grants Committee of Hong Kong. This paper also benefits from the support from GRF No. B-Q42A. The authors thank Turan Bali, Jeff Callen, Agnes Cheng, Louis Cheng, Timothy Chue, Libing Fang, Allaudeen Hameed, Rudy Hirschheim, David Hirshleifer, Joanna Ho, Kose John, Oliver Li, Jin- Chai Lin, Mujitaba Mian, James Ohlson, Baochun Peng, Avanidhar (Subra) Subrahmanyam, Kam Ming Wan, John Wei, Wayne Yu, Kevin Zhu, and participants at the 2017 China Finance Annual Meeting for helpful comments. The usual disclaimer applies. 0

2 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 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 18 th century when Smith (1776) acknowledged the motive of questing for social status. 1

3 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 referencedependent 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 2

4 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 (highbeta/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. 3

5 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 4

6 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 riskfree asset F with constant payoff f 0 and a risky asset which delivers a low payoff x 2 0 in State 2 that occurs with probability 2 and a high payoff x 1 x 2 G, G 0 in State 1 that occurs with probability Assume x 2 f x 2 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 x 2 (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: s s s (1) p x x x 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 x2 G f x2 f payoff in the market. The market average payoff is m1 in State1 and m2 in 2 2 State 2. Because x 2 f x 2 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). 5

7 times (State 2) and higher than the market payoff in good times (State 1), such that x x f x2 G f m and x2 G m1. 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., x2 f x2 G f / x2 ( x2 G) / ( ) ), investors who adopt salience-related thinking perceive the 2 2 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 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): p 1 1 x1 2 2x2 ( 1x1 2x2) 2(1 2) G, (2) x [ x (1 ) G] x x where 1 (1 2) G. x 2 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 salienceadjusted 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 6

8 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 payoff m2 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 2 causes it to acquire a larger salience weight. Therefore, 0. Another key implication of the m 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 m 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 x 2 in the downside state unchanged, the asset s price in Equation (2) is influenced by the market context (i.e., the average market payoff m 2 ) as follows: p m G 0. (3) m2 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) x 2, we obtain the following: p m x m x G (4) 7

9 Thus, the market context effect on asset prices tends to be stronger for more severe crash states with smaller x 2 values. We compare the market context effects at different crash levels; at each crash level, x2 is constant. We are interested in how the market context works, given a certain level of crash (high or low). For a lower asset payoff x 2, p m 2 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 (1 2)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 ( x2 m2), such that implying that (1 ) G ( x m ), 0, (5) x2 1 ( x m ). (6) xg 2 2 We can verify that 0 m x G of salience weight. Therefore, 2 2 and 0, consistent with the implications m x x G m p x x x m x x. (7) 2 ( ) 2( )( 2 2) ( ) 2 (1 ) x2 x2 8

10 Given a certain level of x 2 in the asset s downside state, a better-performing market (i.e., p 2 p higher m 2 ) leads to a lower asset price, because 0, and m x m smaller x 2 ( p 0 m x x is more negative for a ), so the effect of the market context m2 on asset price p is more prominent in more severe crash states with lower levels of payoff x 2. 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. 9

11 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 smallprobability 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 saliencebased 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 noncausal 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 10

12 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 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 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 10 th 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 11

13 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. 12

14 subsample. The bottom quintile has a low market context return of % per day, and the top quintile has a relatively higher market context return of %, inducing a difference of % 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. 13

15 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 valueweighted 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 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 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. 14

16 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 bps per month. Although its magnitude is slightly lower than the corresponding spread of bps in Panel A, it enjoys a higher significance level, with a t-statistic of 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 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. 15

17 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 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 16

18 be 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 All the MKTCON coefficients in different regression models are statistically significant with t-statistics of no less than 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 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 17

19 of different crash levels at the same probability. In our setting, a larger probability indicates a lessrare 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 , smaller than the corresponding values of for the 2%-ETL case and for the 1%-ETL case. 18

20 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 lessinstitutional-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 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 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 noinstitutional-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. 19

21 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 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). 20

22 beta quintiles is bps per month, and the Fama-MacBeth coefficient of beta is 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 (t-statistic = ). 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 in Table III, Panel C to ). 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 21