52-Week High Anchoring and Skewness Preferences
By
Benjamin M. Blaua, R. Jared DeLisleb, and Ryan J. Whitbyc
Abstract: Behavioral finance appears to be at odds with traditional asset pricing theory. While asset pricing models
are built on the foundation of rationality among investors, a broad literature in psychology and a growing
literature in finance suggest that behavioral biases affect the decision making of individuals. In this study,
we argue that unsystematic pricing errors induced by behavioral biases may cancel each other out and result
in the presence of rational asset prices. In particular, we test whether the return premium associated with
preferences for lottery-like characteristics is subsumed by behavioral biases from anchoring on the 52-week
high. Using traditional methods and a number of proxies for lottery characteristics, results show that the
lottery return premium is only observed in those stocks furthest away from the 52-week high. For those
stocks closest to the 52-week high, no such premium exists.
Keywords: Lotteries, Anchoring, Skewness, Behavioral Biases
aBlau is an Associate Professor in the Department of Economics and Finance, in the Jon M.
Huntsman School of Business at Utah State University, Logan Utah, 84322. Email:
[email protected]. Phone: 435-797-2340. cDelisle is an Assistant Professor in the Department of Economics and Finance, in the Jon M.
Huntsman School of Business at Utah State University, Logan Utah, 84322. Email:
[email protected]. Phone: 435-797-0885 cWhitby is an Associate Professor in the Department of Economics and Finance, in the Jon M.
Huntsman School of Business at Utah State University, Logan Utah, 84322. Email:
[email protected]. Phone: 435-797-9495.
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INTRODUCTION
Traditional asset pricing theory suggests that, in a mean-variance framework, various types
of risk will be correctly priced. However, anomalies tend to persist as a number of stock
characteristics are shown to carry with them significant return premia. Some of these anomalies
are explained both theoretically and empirically by behavioral biases. For instance, Barberis and
Huang (2008) use cumulative prospect theory to show that some investors tend to overweight the
tails of return distributions. When investors subjectively assign a higher probability to events with
objectively lower probabilities, the result is excess demand for positively skewed assets. Such
demand might create contemporaneous price premiums and subsequently lead to lower expected
returns than those predicted by the standard expected utility model. Empirical tests seem to confirm
that lottery-like stocks exhibit significant, negative alphas (Mitton and Vorkink (2007), Kumar
(2009), Boyer, Mitton, and Vorkink (2010), and Bali, Cakici, and Whitelaw (2011)).
The tension between these types of behavioral biases and traditional theory can be palpable.
While the theory provides a meaningful framework to think about how assets are priced, assuming
rational (arbitrage-free) prices seems to ignore all the evidence that humans often suffer from
systematic biases that impact their reasoning, judgement, and decision making (see Kahneman and
Tversky (2000)). In this paper, we approach the tension from a different perspective. We assume
that while arbitrage-free (zero-alpha) pricing is a necessary condition for rationality, it is not
sufficient. While perfectly rational agents are one avenue to achieve arbitrage-free prices, this
condition can also be achieved when biases are present, but unsystematic (see Fama (1970)). It is
possible that unsystematic errors may cancel each other out and result in what appears to be rational
prices. We test this proposition by examining how preferences for skewness influence asset prices
while accounting for the potential presence of anchoring bias.
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Like skewness preferences derived from probability weighting functions (Barberis and
Huang, 2008), anchoring causes individuals to incorrectly assess the tails of a distribution.
However, while probability weighting leads to an overweighting of the tails of the distribution,
anchoring results in a distribution that is too tight around a particular anchor. Kahneman, Slovic,
and Tversky (1982) define an anchor as “an initial value that is adjusted to yield the final answer.”
Prior research suggests that anchors often play an important role in the decision making process
of individuals. Tversky (1974) uses experimental results to illustrate that the assessed subjective
probability distributions of individuals suffering from anchoring bias is too tight. The anchoring
bias can be so strong that even arbitrary numbers can be influential. For example, Ariely,
Lowenstein, and Prelec (2003) find that when participants are asked to write down the last two
digits of their social security number prior to answering a question, those numbers, even though
arbitrary, have an anchoring effect on their responses. While the effects of anchors are well
documented (see Kristensen and Gärling (1997)), anchors are often just common reference points
used as benchmarks or rules of thumb.
In the finance literature, the 52-week high has been used as one example of a common
reference point that might act as an anchor. This reference point has the potential to create an
anchor that could influence subsequent decisions made with respect to that stock. For example, if
a stock is very close to the 52-week high, then an investor might anchor on the 52-week high and
therefore assess the subjective probability of the returns of that stock too tightly. This common
benchmark for stock prices has the potential to influence the purchase or sale of shares, and if
systematic, could influence the average return of stocks that fall either near or far away from those
anchors. George and Hwang (2004) find that anchoring to the 52-week high explains a large
3
portion of the momentum premium. Baker, Pan, and Wurgler (2012) find that prior stock-price
peaks act as reference points and affect several aspects of mergers and acquisitions.
The contrast between preferences for skewness and anchoring on the 52-week high form
the basis of our empirical analysis. Specifically, we examine whether the return premium of
lottery-like stocks is affected by the distance from the 52-week high. Does the tightening of
subjective probabilities associated with anchoring negate the overweighting of the tails of the
return distributions associated with preferences for lotteries? Or is any impact of anchoring
subsumed by preferences for skewness? Our empirical tests show that the 52-week high indeed
acts as an anchor and meaningfully impacts the return premium associated with lottery-like
securities. Using a variety of proxies for lottery-like characteristics, we find that, while the
negative return premium associated with these characteristics exists in stocks farthest away from
the 52-week high, the return premium does not exist in the stocks closest to the 52-week high.
This result is consistent with the idea that anchors can result in tighter assessed probabilities. Said
differently, as investors focus on the 52-week high as an anchor, they underweight the probability
of moving away from that anchor, even for stocks that have positive skewness. This finding is
generally robust to each of our proxies for lottery-like characteristics, which include the indicator
variable used to identify lottery stocks from Kumar (2009), expected idiosyncratic skewness from
Boyer, Mitton, and Vorkink (2010), and the maximum daily return from the previous month
detailed in Bali, Cakici, and Whitelaw (2011). Moreover, we find similar results whether
examining multivariate regressions on our entire sample, on subsamples sorted by the level of
anchoring, or in a portfolio setting using equal- or value-weighting. Thus, stocks with heavily
skewed return distributions that are near their 52-week high do not appear to have an anomalous
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skewness premium. This finding is consistent with the existence of behavioral biases as well as
how two prevalent biases might cancel each other out and therefore resemble rationality.
RELATED LITERATURE AND MOTIVATION
Our study is based on the asset pricing literature in two key areas: the preferences for
lottery-like characteristics, such as skewness or maximum daily returns, and the biases associated
with anchoring. The literature regarding lottery preferences focuses on how or why investors price
stocks that resemble lotteries. The most common lottery-like characteristic is skewness. In general,
this literature demonstrates that investors are positive skewness-seeking and that skewness carries
a negative price of risk (i.e. investors are willing to pay a premium for positively skewed stocks,
which leads to low future returns). Thus, it deviates from the traditional mean-variance
optimization framework that is rooted deeply in the finance literature (e.g. Markowitz, 1952, 1959;
Sharpe, 1964). For example, Brunnermeier and Parker (2005) and Brunnermeier, Gollier, and
Parker (2007) create theoretical models based in rational optimal expectations, where investors
must evaluate the trade-off between favorable beliefs and the costs of holding those beliefs, which
predict skewness preferences in investors. Mitton and Vorkink (2007) construct a model with
investors that hold heterogeneous beliefs; a portion of the investors are mean-variance optimizers
while the remainder are skewness-preferring. Their model shows that, in the equilibrium, investors
hold positively skewed, undiversified portfolios and skewness of returns is a priced risk.
To examine the effects of skewness, Barberis and Huang (2008) produce an asset allocation
model using a unique feature of cumulative prospect theory (CPT): probability weighting. Tversky
and Kahneman (1992) introduce a modification to their original prospect theory (Kahneman and
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Tversky, 1979) where agents apply a weighting function to real probabilities to obtain a weighted
probability used to evaluate expected outcomes. Under this revised model, the CPT, Tversky and
Kahneman show individuals overweight small probabilities which results in extremely risk seeking
behavior when faced with improbable gains.1 Barberis and Huang (2008) find that the probability
weighting feature of CPT results in some investors holding undiversified portfolios with assets
that have positively skewed return distributions. The lottery-like characteristics of these assets
(large gains with very low probabilities) make them desirable to the investors who overweight the
tails of the probability distribution. Thus, these investors contemporaneously bid up the price of
the positively skewed securities and lower the expected returns. Given their results, Barberis and
Huang suggest that incorporating probability weighting into models can assist in explaining asset
pricing anomalies such as option implied volatility skews, the diversification discount, IPO returns,
private equity premiums, and momentum returns. To this end, De Giorgi and Legg (2012) include
probability weighting in their asset pricing model and demonstrate it generates smaller (larger)
equity premiums for positively (negatively) skewed assets.
In addition to the theoretically-based literature, there exists an extensive literature
providing empirical evidence of a negative price associated with skewness. For example,
consistent with the theoretical predictions, Mitton and Vorkink (2007) and Goetzmann and Kumar
(2008) use datasets of investor trading accounts and find individual investors hold undiversified
portfolios containing stocks with high levels of idiosyncratic skewness. Kumar (2009) finds
individual investors prefer stock with lottery-like characteristics, such as positive skewness, and
1 Kunreuther, Novemsky, and Kahneman (2001) find that their subjects treat a probability of 1/100,000 the same as
the probability of 1/10,000,000, which is empirically consistent with the predictions of CPT. Further supporting the
use probability weighting functions, Teigen (1974a, 1974b, 1983) finds that individuals’ sum of construed
probabilities of a set of outcomes exceeds one.
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these stocks typically underperform non-lottery stocks.2 Realizing historical idiosyncratic
skewness is unstable, Boyer, Mitton, and Vorkink (2010) create a measure of expected
idiosyncratic skewness and find it is negatively related to future abnormal returns. In addition,
their measure partially explains the negative relation between future returns and idiosyncratic
volatility documented by Ang, Hodrick, Xing, and Zhang (2006). Bali, Cakici, and Whitelaw
(2011) use the maximum daily return during a month (MAX) as a proxy for positive skewness
given that skewness is not very persistent. They find that MAX has a negative relation with future
returns. Using the risk-neutral distribution of returns constructed from options data, Conrad,
Dittmar, and Ghysels (2013) observe that stocks with high option-implied skewness earn low
future abnormal returns relative to stocks with low option-implied skewness. Consistent with
Barberis and Huang’s conjecture, Mitton and Vorkink (2010) and Green and Hwang (2012)
demonstrate skewness plays a significant role in the diversification discount and the IPO returns
puzzle, respectively. Additionally, Schneider and Spalt (2015), DeLisle and Walcott (2016), and
Schneider and Spalt (2016) show skewness impacts acquisitions in various dimensions, such as
target selection, acquisitions premiums, method of payment, and post-acquisition returns. Taken
altogether, there is substantial evidence that skewness is negatively priced in the cross-section of
returns and is related to several documented anomalies.
The literature is also rich with studies examining the psychological concept of anchoring.
Anchoring refers to the process of making adjustments away from an anchor, but the adjustments
are biased towards the anchor and do not sufficiently move away from it. An anchor may come
from the formulation of the problem to be solved, a computation made along the process to solving
2 Although, both Kumar (2005) and Kumar (2009) show institutional investors are skewness-averse. Autore and
DeLisle (2016) find similar evidence by demonstrating institutional investors require deeper discounts to place
seasoned equity offering shares with high skewness.
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the problem, or a random value that has nothing to do with problem. Kahneman, Slovic, and
Tversky (1982) give several examples of anchoring documented in previous studies. Two of them
highlight the insufficient adjustment associated with anchoring. One study examines the framing
of the question, where the anchor is embedded in the problem: two groups of students are given
the same multiplication problem and asked to estimate the product in five seconds. The difference
between the two groups is that the problem’s factors are arranged in different orders, one ascending
and the other descending (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 versus 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1). The
anchoring hypothesis suggests that the subjects will read the problem from left to right and anchor
on the relative size of the first few numbers. Consistent with that hypothesis, the group with
ascending (descending) numbers had a median estimate of 512 (2,250).
Another study Kahneman, Slovic, and Tversky describes uses completely random
numbers as an anchor. Subjects spin a wheel to determine a number between 1 and 100 and asked
to estimate certain quantities as a percentage. The random number generated by the wheel
systematically biases the subjects estimate towards that number. Similarly, Ariely, Loewenstein,
and Prelec (2003) find their subjects, after having them write down the last two digits of their
social security number, anchor to that random number when estimating the price of a bottle of
wine. It is this tendency for individuals to anchor that inspires George and Hwang (2004) to
investigate distance from the 52-week high stock prices and the relation to the momentum
phenomenon (Jegadeesh and Titman, 1993).
Even in a weak-form efficient market, the past 52-week high should not carry any
information about the future prospects of the stock. Yet, George and Hwang (2004) find that an
investing strategy based on distance to the 52-week high explains most of the returns from the
traditional momentum strategy. Their findings imply investors anchor to a stock’s 52-week high
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when valuing the stock, and do not sufficiently adjust above the 52-week high. This downward
bias in the valuation leads to the stock price drifting up over time instead of a relatively quick
adjustment. Lee and Piqueira (2016) suggest short sellers are aware of this phenomenon, as short
selling is negatively related to the nearness to the 52-week high. Using stock indices instead of
individual stocks, Du (2008) and Li and Yu (2012) find similar effectiveness in 52-week high
strategies. Sapp (2010) shows an anaoglous strategy applied to mutual funds is also successful in
predicting future returns. There is evidence of anchoring in the options market as well, as Driessen,
Lin, and Van Hemert (2013) find option-implied volatility decreases when the underlying stock
price approaches the 52-week high and increases if a new 52-week high is reached (i.e. the stock
price breaks through the historical 52-week high). Additionally, Heath, Huddart, and Lang (1999)
demonstrate how employees use their firm stock’s 52-week high as a reference point to exercise
their stock options.
The extensive literature in these two areas motivates us to investigate if the negative return
premium assoicated with skewness is robust to anchoring to the 52-week high price. If investors
perceive the 52-week high price as an anchor that is difficult to break through (e.g. investors bias
a stock’s valuation downward toward this anchor), then the 52-week high phenomenon may
subsume the skewness premium effect. In other words, a stock with a highly positively skewed
return distribution would not be a candidate to receive a skewness premium if it were close to the
52-week high because the weight the investor puts on the probability the stock will cross the 52-
week high anchor is severly diminished. Conversely, if the current stock price is far from the 52-
week high, there is a lot of “room” for the stock price to jump up. Thus, the skewness premium
effect should be strong far from the 52-week high and weak close to the 52-week high. This is the
focus of our study.
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The closest study to ours is that of An, Wang, Wang, and Yu (2015) which examines lottery
stock returns conditioned on capital gains overhang (CGO). Grinblatt and Han (2005) model
momentum using the price the asset is purchased at as reference point. Thus, the CGO measures
how far the current price is from the reference point. This is important in cumulative propect theory
because the loss aversion properties the theory generates predict different behavior by investors
when CGO is positive than when it is negative. Our study differs because our hypothesis relies
solely on anchoring behavior and not on loss aversion. Thus, our investigation does not require
any estimation of purchase prices to establish reference points.
DATA AND METHODOLOGY
The data used throughout the study is obtained from a variety of sources. From the Center
for Research on Security Prices (CRSP), we gather daily and monthly returns, prices, trading
volumes, shares outstanding, etc. From Compustat, we obtain annual balance sheet data in order
to obtain the book-value of equity. Finally, we obtain estimates of expected idiosyncratic
skewness, which is used in Boyer, Mitton, and Vorkink (2010), directly from Brian Boyer.
Following the existing literature, we use three proxies for lottery characteristics. Kumar
(2009) using an indicator variable to classify stocks that are most likely to resemble lotteries.
Lottery is equal to unity if, during a particular month, a stock has idiosyncratic skewness above
the median, idiosyncratic volatility above the median, and a closing share price below the median.
We note that idiosyncratic measures of skewness and volatility are obtained from the residuals of
a daily four-factor model. Stated differently, we estimate a four-factor model, where the factors
are described in Fama and French (1993) and Carhart (1997). From these regressions, we obtain
residual returns to estimate the idiosyncratic moments of the return distribution. Given the need to
have a sufficiently large number of observations when accurately estimating the higher moments
10
of the return distribution, we use a rolling six-month window from a particular month. E[IdioSkew]
is the expected idiosyncratic skewness, which is calculated as the predicted values of a regression
described in Boyer, Mitton, and Vorkink (2010).3 Maxret is the maximum daily return for stock i
during month t. 52-Week is the 52-week high price while Anchor is the difference between the 52-
week high price and the current (end-of-the-month) share price. Beta is the CAPM beta obtained
from estimating a standard daily CAPM data using a six-month rolling window. Size is the end-
of-month market capitalization (in $Billions). B/M is the book-to-market ratio for each stock in
each month. Illiquidity is the Amihud (2002)measure of illiquidity, which is the ratio of the
absolute value of the daily return scaled by dollar volume (in $Millions).
Table 1 reports statistics that summarize the data used throughout the analysis. From the
table, we find that approximately 22.4% of stocks are classified as Lottery stocks. Furthermore,
the average stock in our sample has expected idiosyncratic skewness of 1.1635, a maximum daily
return of 7.89%, a 52-week high price of $35.50, a distance from the 52-week high of $7.39, a beta
of .8511, market capitalization of $1.71 billion, a book-to-market ratio of 0.4276, and Amihud’s
(2002) measure of illiquidity of 9.4178. The other summary statistics are reported in the table.
EMPIRICAL RESULTS
Cross-Sectional Multivariate Tests – A Fama and MacBeth (1973) Approach
To examine the relation between preferences for lottery stocks and anchoring, we estimate
the following equation in a Fama and MacBeth (1973) framework.
3 As described in Boyer, Mitton, and Vorkink (2010 pgs. 175-176), expected idiosyncratic skewness is obtained by
estimating the predicted values from cross-sectional regressions of idiosyncratic skewness in month t on a number of
independent variables, which include lagged idiosyncratic skewness, lagged idiosyncratic volatility, and a vector of
firm-specific variables that include stock return momentum, share turnover, and a number of indicator variables that
control for firm size, industry classification, and exchange listing.
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Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t +
β6Lotteryi,t + εi,t+1
The dependent variable in this regression is the monthly return for stock i in month t+1. The
independent variables include Beta, which is the CAPM beta obtained from estimating a standard
daily CAPM model using a six-month rolling window. Size is the natural log of end-of-month
market capitalization in billions of dollars. B/M is the natural log of the book-to-market ratio for
each stock in each month. Momentum is the cumulative return from month t-12 to t-2 for each
stock. Illiquidity is Amihud’s (2002) measure of illiquidity, which is the ratio of the absolute value
of the daily return scaled by dollar volume in millions. Lottery is the indicator variable capturing
lottery-like stocks (Kumar (2009)).
To better understand the relationship between anchoring and lottery stocks, we sort stocks
into quintiles based on their distance from the 52-week high. Stocks farthest away from their 52-
week high are labeled as low anchor stocks (quintile 1) and stocks near their 52-week high are
labeled as high anchor stocks (quintile 5). We then estimate equation (1) for each quintile of stocks
based on anchoring. Column [1] of Table 2 reports the estimates from the regression on low anchor
stocks. With respect to the control variables, we find a negative return premium associated with
Beta and positive return premia associated with book-to-market ratios and momentum. These
results support findings in the prior literature (Frazzini and Pedersen (2014), Fama and French
(1992 and 1996), and Jegadeesh and Titman (1993)).4 Furthermore, we find a negative and
significant relation between lottery stocks and next-month returns for low anchor stocks. The
coefficient on Lottery is -0.5149 and is significant at the 5 percent level and suggests that, after
4 We note that in columns [2] and [4], we find a positive return premium associated with Illiquidity, which is
consistent with Amihud and Mendelson (1986). Further, columns [3] through [5] show that Size is negatively
associated with future returns, which is consistent with findings in Banz (1981) and Fama and French (1992).
(1)
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controlling for factors that have been shown to influence the predictability of stock returns, lottery
stocks underperform non-lottery stocks by more than 50 basis points per month. Similar results
are found in Column [2], which reports the results from estimating equation (1) for the second
quintile sorted by anchoring. However, in the latter quintiles, we start to see a difference in the
estimated coefficient on Lottery for stocks in the higher anchoring quintiles. For instance, the
coefficient on Lottery in Column [3] is -0.2543, but is not significantly different from zero. In
fact, we find that the coefficient on Lottery is strictly monotonic across each of the five increasing
quintiles. The results for the high anchor stocks, found in the fifth quintile (Column [5]), shows an
estimated coefficient on Lottery of -0.0156 with a t-statistic of -0.08. This finding demonstrates
the relation between investor preferences for lottery-like stocks and the propensity to anchor on
the 52-week high. The well documented return premium associated with preferences for lotteries
is apparently offset by an anchoring effect that is measured by the distance from the stock’s 52-
week high. In other words, preferences for skewness weaken as stocks approach their 52-week
high.
Next, we continue our analysis of anchoring and lottery preferences but instead of
examining Lottery preferences as measured by Kumar (2009), we examine expected idiosyncratic
skewness as detailed in Boyer, Mitton, and Vorkink (2010) (E[IdioSkewi,t]). Table 3 estimates the
following regression in a Fama and MacBeth (1973) framework and includes all of the control
variables detailed above in reference to the estimation of equation (2).
Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t +
β6E[IdioSkewi,t] + εi,t+1
Table 3 is similar to Table 2 in that it presents the results for the each of the anchoring quintiles
starting with the low anchor stocks in Column 1. Once again, we see that the coefficients on the
(2)
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proxy for lottery-like stocks, in this case E[IdioSkew], are generally increasing across anchoring
quintiles with a low of -0.4505 (t-statistic = -2.12) in Column [1] to a high of -0.0953 (t-statistic =
-0.59) in Column [5]. This is true after holding constant the return premiums associated with Beta,
market capitalization, book-to-market ratios, momentum, and illiquidity. We are careful to note
that the findings in Table 3 are weaker than the findings in Table 2. For instance, in column [4],
we observe a negative and significant coefficient on the variable of interest. However, we generally
find that the negative return premium associated with expected idiosyncratic skewness is driven
by stocks that are further away from their 52-week high. Once again, it appears that the return
premium associated with lottery-like stocks is offset by an anchoring effect that is apparent when
stocks are near their 52-week high.
Table 4 replicates the analysis of Tables 2 and 3, but uses MaxRet which is defined as the
maximum daily return for a particular stock during month t as described in Bali, Cakici, and
Whitelaw (2011). In particular, we estimate the following equation using pooled stock-month
data.
Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t +
β6Reversali,t + β7MaxReti,t + εi,t+1
In addition to the control variables included in the previous equations, equation (3) follows Bali,
Cakici, and Whitelaw (2011) and includes return reversals to account for the inherent mean
reversion associated with stock returns. The results from estimating equation (3) are reported in
Table 4. As before, Column [1] reports the results for the low anchor quintile and Column [5]
reports the results for the high anchor quintile. Once again, the coefficients on the control variables
produce estimates that are similar in sign to those in previous research. More importantly, we see
that in low anchor stocks exhibit a large and significant negative return premium associated with
(3)
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lottery like preferences. MaxRet has an estimated coefficient of -10.0290 that is significant at the
1 percent level with a t-statistic of -5.79. In economic terms, the magnitude of this coefficient
suggests that a one standard deviation increase in MaxRet is associated with a 90 basis point
reduction in next-month returns. In contrast, the coefficient on MaxRet in the high anchor quintile
is only -1.463 and indistinguishable from zero. Furthermore, the coefficients are increasing
monotonically across increasing quintiles. The results from tests in Tables 2 through 4 seem to
suggest that no matter how lottery-like characteristics are measured, the negative return premium
associated with preferences for lotteries are offset by the nearness to the 52-week high.
Portfolio Tests – Raw Returns
To continue our analysis, we examine the performance of portfolios sorted by both
anchoring and lottery preferences. Table 5 reports next-month raw returns by double sorted
portfolios Stocks are first sorted into quintiles based on their proximity to the 52-week high, which
is our anchoring variable. Within each quintile, stocks are then sorted into lottery and non-lottery
portfolios, based on the classification in Kumar (2009). Differences between lottery and non-
lottery and high and low anchor portfolios are then calculated and reported with corresponding t-
statistics. Panel A of Table 5 reports results for equally weighted portfolios and Panel B reports
results for value-weighted portfolios. Although there are some significant differences between
portfolios in Table 5, the relation between anchoring and the lottery return premium is not as clear.
A few results are noteworthy. First, the difference between returns for lottery vis-à-vis non-lottery
stocks in Panel A is most negative in stocks that are farthest from their 52-week high. However,
this difference is not significant. In general, the differences are increasing although not
monotonically. When examining the value-weighted portfolios in Panel B, we find that next-month
returns are significantly lower for lottery stocks compared to non-lottery stocks in quintile 1. The
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difference is 82 basis points per month and is reliably different from zero (t-statistic = -2.70). None
of the other differences are significant although they are generally increasing across increasing
quintiles. The results in Table 5 provide some evidence supporting the idea that the negative return
premium associated with lottery stocks is driven by stocks that are further from their 52-week high.
We note, however, that the evidence is weak at best. Therefore, we perform a similar portfolio
analysis using expected idiosyncratic skewness (Table 6) and the maximum daily return (Table 7).
Table 6 is similar to Table 5, but instead of a binary choice between lottery and non-lottery
stocks we sort stocks into quintiles based on expected idiosyncratic skewness during the second
sort. The horizontal sort is based on the nearness to the 52-week high while the vertical sort is
based on increasing idiosyncratic skewness. Similar to the previous table, Panel A reports results
for equally-weighted portfolios while Panel B reports results for value-weighted portfolios. Panel
A shows that the negative return premium associated with idiosyncratic skewness stocks is greatest
in Column [1], which identifies the stocks that are farthest away from the 52-week high. The
difference between the fifth and first quintile of idiosyncratic skewness stocks is -0.0104 with a t-
statistic of -2.63. This difference is both economically and statistically significant. Highly skewed
stocks underperform low skewed stocks by more than 12% per year. As we move across the table
toward the high anchor stocks, we see a monotonic reduction in the negative return premium
associated with expected idiosyncratic skewness. The high-minus-low difference for the stocks
closest to the 52-week high is only -0.0013 and is not reliably different from zero. This highlights
the fact that the negative return premium associated with preferences for lotteries is mitigated by
the stock’s nearness to the 52-week high. We interpret these findings as evidence that anchoring
to the 52-week high appears to offset the behavior associated with lotteries. Panel B shows the
results from the value-weighted portfolios. Here, we do not find supportive evidence of our
16
hypothesis as the high-minus-low difference is neither increasing nor decreasing across increasing
Anchor quintiles. Again, the results in Table 6 only provide weak evidence of our hypothesis.
Next, we examine the performance of portfolios sorted first by Anchor and then by the
maximum daily return. MaxRet quintiles are in the rows and anchoring quintiles are in the
columns. Results are reported in Table 7. Focusing on the bottom row in each panel, we find that
strong evidence that the negative return premium is driven by stocks that are farthest from their
52-week high. For instance, Panel A (equal-weighted portfolios) shows that the difference between
the high MaxRet portfolio and the low MaxRet portfolio is -0.0128 suggesting a return premium
of nearly 130 basis point per month. The high-minus low difference are increasing monotonically
across columns. For stocks that are closest to the 52-week high, the return premium associated
with MaxRet is 0.0018 (t-statistic = 1.06). We note that the difference in differences is 0.0146 (t-
statistic = 5.01). Similar results are reported in Panel B of Table 7, which reports the results from
the value-weighted portfolios instead of equal-weighted portfolios. Here, the high-minus-low
difference is -0.0114 in Column [1] and 0.0025 in Column [5]. As before, the difference in
differences is 0.0140 and reliably different from zero (t-statistic = 4.26). These findings again
support the idea that the return premium associated with lottery-like stocks is driven by stocks that
are farthest from their 52-week high.
Portfolio Tests – Multifactor Analysis
Since a variety of risk factors have been shown to influence the expected return of stock
returns it is important to try and control for some of these factors in a multivariate setting. Table
8, 9, and 10 report our findings for the following regression:
Returnp,t – Rf,t = α + βMRP(MRPt) + βSMB(SMBt) + βHML(HMLt) + βUMD(UMDt) + εp,t
(4)
17
The dependent variable is the excess return (or the return in excess of the risk-free rate, which is
approximated by the yield on one-month U.S. Government Treasury Bills) for each portfolio.
Following Fama and French (1993) and Carhart (1997), the independent variables include MRP,
which is the market risk premium, or the excess return of the market less the risk-free rate. SMB
is the small-minus-big or size return factor while HML is the high-minus-low or value return factor.
UMD is the up-minus-down or momentum factor. We report the alphas from estimating the four-
factor model for each of the double-sorted portfolios – first by our measure of Anchor and then by
one of our three proxies for lottery-like characteristics. The next three tables take the same format
as those in Tables 5 through 7. In essence, Table 8 controls for various risk factors and, therefore,
provides robustness for Table 5; Table 9 provides robustness for Table 6; and Table 10 provides
robustness for Table 7.
Table 8 shows the four-factor alphas across double-sorted portfolios – first by Anchor, then
by Kumar’s (2009) classification for lottery stocks. Results in Table 8 are qualitatively similar to
the corresponding results in Table 5. In particular, Panel B of the table shows that in the stocks
that are farthest away from their 52-week high, the negative return premium is the most significant.
For example, in Column [1], the difference in four-factor alphas between lottery and non-lottery
stocks is 69 basis points per month. Column [5] reports that this difference is only 6 basis points.
These results support our findings in Table 5 but only provide weak evidence of our hypothesis as
the findings in Panel A (the equally-weighted portfolios) are not conclusive.
Table 9 reports the four-factor alphas by double-sorted portfolios, where the second sort is
based on Boyer, Mitton, and Vorkink’s (2010) measure of expected idiosyncratic skewness. The
four factor results in Table 9 are somewhat weaker than the corresponding results in Table 6 as
Panel A does not provide significant, negative return premia in the earlier columns. However, we
18
still find that the differences between extreme E[IdioSkew] portfolios are generally increasing
across increasing Anchor portfolios (across Columns [1] through [5]). Panel B provides no such
pattern across columns. Again, Table 9 provides only marginal evidence supporting our hypothesis
that the negative return premium associated with lottery-like stocks is driven by stocks that are
farthest from their 52-week high.
Table 10 presents the results when the second sort is based on MaxRet. Here, the results
have the starkest contrast between portfolios. Panel A, which reports the results for equally-
weighted portfolios show that the difference in four-factor alphas between extreme MaxRet
portfolios in the low anchor quintile is -0.0091 with a t-statistic of -2.91. In economic terms, this
finding translates into a negative return premium of more than 10 percent per year. In contrast,
the difference in four-factor alphas in the high anchor portfolio is 0.0023 and is insignificantly
different from zero with a t-statistic of 1.50. We further note that the difference in differences,
located at the bottom of Column [6] is reliably different from zero.
Qualitatively similar results to those in Panel A are found in Panel B when we examine
value-weighted portfolios. For instance, the negative return premium associated with MaxRet
represents approximately 85 basis point per month in stocks that are farthest from their 52-week
high (Column [1]). To the contrary, the corresponding return premium for stocks nearest to their
52-week high (Column [5]) is positive 46 basis points per month. Again, the difference in
differences (bottom of Column [6]) is significantly difference from zero. These results confirm the
findings in Table 7 and support the idea that the negative return premium associated with MaxRet
is driven by stocks farthest from their 52-week high. Although the relation between the negative
return premium associated lottery-like stocks and anchoring to the 52-week high is not as strong
in some of our proxies of lottery stocks, the results generally show that anchoring tends to offset
19
lottery preferences for stocks near the 52-week high. These findings have important implications
as they demonstrate that, what appears to be arbitrage-free pricing might simply be
unsystematically strong behavioral biases competing against each other and resulting the
perception of rational prices.
CONCLUSION
In the case of efficient markets, asset pricing theory suggests that assets will convey
rational, or arbitrage-free, prices. According to different models, such as the Capital Asset Pricing
Model or arbitrage pricing theory, rational pricing will result in a zero-alpha condition. However,
this condition is often violated for various reasons. One such reason is that the behavioral biases
of investors might meaningfully influence demand in a way that prices might deviate away from
their theoretical price. Using cumulative prospect theory (Kahneman and Tversky (1979, 1992)),
Barberis and Huang (2008) show that investor preferences for lottery-like characteristics, such as
positive skewness in the distribution of returns, will lead to excess demand, price premiums, and
subsequent underperformance. Several studies seem to confirm this theoretical prediction (Mitton
and Vorkink (2007), Kumar (2009), Boyer, Mitton, and Vorkink (2010), and Bali, Cakici, and
Whitelaw (2011), among others). These studies seem to violate both expected utility theory and
arbitrage pricing theory as stocks with lottery payoffs tend to underperform other stocks.
In this paper, we develop and test the hypothesis that while a zero alpha outcome is a
necessary condition for rational pricing, it is not a sufficient condition. For instance, if different
behavioral biases are not systematic, then these biases might cancel each other out in a way that
zero alphas are observed but investors are not behaving rationally. In particular, we argue that,
when conditioned on reference points, or anchoring bias, investor preferences for lottery-like stock
characteristics will no longer be strong enough to meaningfully influence asset prices. Tversky
20
(1974) provides experimental evidence that, when anchoring bias is present, agents will form a
distribution around a reference point that is too tight. Using the 52-week high as a reference point
(Green and Hwang (2004) and Baker, Pan, and Wurgler (2012)), we argue that when prices are
near the 52-week high, demand for these stocks by investors with lottery preferences will no longer
be unusually high given their perception that prices cannot meaningfully move beyond the
reference point.
To test our hypothesis, we conduct a series of cross-sectional and portfolio tests and
examine the return premium associated with lottery-like stocks while conditional on the nearness
to the 52-week high. Results seem to support our hypothesis as the negative return premium is
only observed in stocks that are farthest from their 52-week high. In fact, in stocks that are closest
to this reference point, lottery stocks neither outperform nor underperform other stocks. These
results are robust to different proxies for lottery-like characteristics. In particular, we find that
these results hold when using Kumar’s (2009) lottery stock classification, Boyer, Mitton, and
Vorkink’s (2010) measure of expected idiosyncratic skewness, and Bali, Cakici, and Whitelaw’s
(2011) max return. These findings have important implications that tie together behavioral finance
and more traditional asset pricing theory. Finding arbitrage-free prices is not neccesarily
tantamount to finding rational prices.
21
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24
Table 1
Summary Statistics
The table reports statistics that describe our data. Lottery is an indicator variable that captures Lottery Stocks.
Following Kumar (2009), Lottery is equal to unity if, during a particular month, a stock has idiosyncratic skewness
above the median, idiosyncratic volatility above the median, and a closing share price below the median. We note
that idiosyncratic measures of skewness and volatility are obtained from the residuals of a daily four-factor model.
We use a rolling six-month window from a particular month. E[IdioSkew] is the expected idiosyncratic skewness,
which is calculated as the predicted values of a regression described in Boyer, Mitton, and Vorkink (2010). Maxret
is the maximum daily return for stock i during month t. 52-Week is the 52-week high price while Anchor is the
difference between the 52-week high price and the current (end-of-the-month) share price. Beta is the CAPM beta
obtained from estimating a standard daily CAPM data using a six-month rolling window. Size is the end-of-month
market capitalization (in $Billions). B/M is the book-to-market ratio for each stock in each month. Illiquidity is the
Amihud (2002) measure of illiquidity, which is the ratio of the absolute value of the daily return scaled by dollar
volume (in $Millions).
Mean Std. Deviation 25th Percentile Median 75th Percentile
[1] [2] [3] [4] [5]
Lottery 0.2237 0.4168 0.0000 0.0000 0.0000
E[idioskew] 1.1635 0.6986 0.7634 1.1148 1.5075
Maxret 0.0789 0.0951 0.0329 0.0548 0.0938
52-week 35.4995 993.7250 7.8750 17.8750 33.4375
Anchor 7.3933 173.8526 0.8800 2.8750 7.0100
Beta 0.8511 0.9350 0.3472 0.8368 1.3167
Size 1.7099 9.9758 0.0003 0.1268 0.6232
B/M 0.4276 11.9400 0.0376 0.0657 0.1080
Illiquidity 9.4178 660.2983 0.0072 0.0999 1.2157
25
Table 2
Fama-MacBeth (1973) Regressions – The Return Premium of Lottery Stocks
The table reports the results from estimating variants of the following equation using a Fama-MacBeth (1973)
regression.
Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t + β6Lotteryi,t + β7Anchori,t +
β8Lotteryi,t×Anchori,t + εi,t+1
The dependent variable is the monthly return for stock i in month t+1. The independent variables include the
following. Beta is the CAPM beta obtained from estimating a standard daily CAPM data using a six-month rolling
window. Size is the natural log of end-of-month market capitalization (in $Billions). B/M is the natural log of the book-
to-market ratio for each stock in each month. Momentum is the cumulative return from month t-12 to t-2. Illiquidity is
the Amihud (2002) measure of illiquidity, which is the ratio of the absolute value of the daily return scaled by dollar
volume (in $Millions). Lottery is the indicator variable capturing lottery-like stocks (Kumar (2009)). Anchor is the
difference between the current share price and the 52-week high price, or the distance to the anchor point. The sample
is sorted in quintiles based on Anchor, where quintile 1 (5) contains firms whose stock price is furthest from (closest
to) the 52-week high. T-statistics are obtained from Newey-West (1987) standard errors that account for three lags.
*,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. There are approximately
377,000 stock-month observations in each quintile.
(Far)
Quintile 1
Quintile 2
Quintile 3
Quintile 4
(Close)
Quintile 5
[1] [2] [3] [4] [5]
Constant 2.3560*** 3.0712*** 4.1846*** 5.8949*** 4.8724***
(3.30) (4.35) (5.49) (7.42) (8.34)
Beta -0.2241** -0.0414 0.1053 0.1183 0.0410
(-2.22) (-0.43) (1.00) (1.45) (0.47)
Size 0.0140 -0.0374 -0.1304** -0.2476*** -0.1715***
(0.30) (-0.77) (-2.42) (-4.08) (-4.09)
B/M 0.5041*** 0.6253*** 0.6610*** 0.7140*** 0.6483***
(7.96) (10.27) (10.10) (9.12) (8.64)
Momentum 0.5928*** 0.2726 0.3886** 0.1196 0.8189***
(3.04) (1.43) (2.13) (0.68) (5.07)
Illiquidity -0.0540 0.0444*** 0.0026 0.0072** -0.0007
(-0.49) (2.71) (0.26) (2.20) (-0.11)
Lottery -0.5149** -0.4936*** -0.2543 -0.1426 -0.0156
(-2.10) (-2.90) (-1.48) (-0.81) (-0.08)
26
Table 3
Fama-MacBeth (1973) Regressions – The Expected Idiosyncratic Skewness Return Premium
The table reports the results from estimating variants of the following equation using a Fama-MacBeth (1973)
regression.
Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t + β6E[IdioSkewi,t]+ β7Anchori,t +
β8E[IdioSkewi,t]×Anchori,t + εi,t+1
The dependent variable is the monthly return for stock i in month t+1. The independent variables include the
following. Beta is the CAPM beta obtained from estimating a standard daily CAPM data using a six-month rolling
window. Size is the natural log of end-of-month market capitalization (in $Billions). B/M is the natural log of the book-
to-market ratio for each stock in each month. Momentum is the cumulative return from month t-12 to t-2. Illiquidity is
the Amihud (2002) measure of illiquidity, which is the ratio of the absolute value of the daily return scaled by dollar
volume (in $Millions). E[IdioSkewi,t] is the expected idiosyncratic skewness (Boyer, Mitton, and Vorkink (2010)).
Anchor is the difference between the current share price and the 52-week high price. The sample is sorted in quintiles
based on Anchor, where quintile 1 (5) contains firms whose stock price is furthest from (closest to) the 52-week high.
T-statistics are obtained from Newey-West (1987) standard errors that account for three lags. *,**, and *** denote
statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. There are approximately 377,000 stock-month
observations in each quintile.
(Far)
Quintile 1
Quintile 2
Quintile 3
Quintile 4
(Close)
Quintile 5
[1] [2] [3] [4] [5]
Constant 4.9630*** 4.8076*** 5.9567*** 8.2186*** 6.4663***
(5.08) (5.99) (-7.12) (10.26) (9.45)
Beta -0.1806* -0.0330 0.0530 0.0769 0.0256
(-1.74) (-0.33) (0.49) (0.94) (0.27)
Size -0.0028 -0.0286 -0.1428** -0.3047*** -0.1901***
(-0.06) (-0.54) (-2.34) (-4.94) (-3.64)
B/M 1.1081*** 1.1321*** 1.1263*** 1.1153*** 1.0494***
(9.72) (11.73) (10.67) (10.44) (9.97)
Momentum 0.7830*** 0.5000*** 0.4742** 0.2836 0.7924***
(4.07) (2.63) (2.46) (1.64) (4.78)
Illiquidity -0.0024 0.0459** 0.0161*** 0.0116 -0.0050
(-0.07) (2.53) (3.67) (3.45) (-0.78)
E[IdioSkew] -0.4505** -0.2816* -0.2278 -0.4364* -0.0953
(-2.12) (-1.82) (-1.07) (-1.96) (-0.59)
27
Table 4
Fama-MacBeth (1973) Regressions – The Max Return Premium
The table reports the results from estimating variants of the following equation using a Fama-MacBeth (1973)
regression.
Returni,t+1 = β0 + β1Betai,t + β2Sizei,t + β3B/Mit + β4Momentumi,t + β5Illiquidityi,t + β6Maxreti,t + β7Reversal,t
+β8Anchori,t + β9Maxreti,t×Anchori,t + εi,t+1
The dependent variable is the monthly return for stock i in month t+1. The independent variables include the
following. Beta is the CAPM beta obtained from estimating a standard daily CAPM data using a six-month rolling
window. Size is the natural log of end-of-month market capitalization (in $Billions). B/M is the natural log of the book-
to-market ratio for each stock in each month. Momentum is the cumulative return from month t-12 to t-2. Illiquidity is
the Amihud (2002) measure of illiquidity, which is the ratio of the absolute value of the daily return scaled by dollar
volume (in $Millions). Maxret is the maximum daily return for a particular stock during month t (Bali, Cakici, and
Whitelaw (2011)). We also follow Bali, Cakici, and Whitelaw (2011) and include Reversal to account for the price
reversal. Anchor is the difference between the current share price and the 52-week high price. The sample is sorted
in quintiles based on Anchor, where quintile 1 (5) contains firms whose stock price is furthest from (closest to) the
52-week high. T-statistics are obtained from Newey-West (1987) standard errors that account for three lags. *,**, and
*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. There are approximately 377,000
stock-month observations in each quintile.
(Far)
Quintile 1
Quintile 2
Quintile 3
Quintile 4
(Close)
Quintile 5
[1] [2] [3] [4] [5]
Constant 2.8484*** 3.2873*** 4.4832*** 5.9363*** 5.0952***
(4.32) (4.89) (5.90) (7.35) (7.67)
Beta -0.1257 -0.0140 0.1340 0.1110 0.039
(-1.23) (-0.14) (1.33) (1.36) (0.46)
Size -0.0223 -0.0445 -0.1433** -0.2431*** -0.1855***
(-0.51) (-0.93) (-2.58) (-3.87) (-3.87)
B/M 0.3981*** 0.5706*** 0.6244*** 0.6983*** 0.6361***
(6.73) (9.86) (9.79) (9.05) (8.68)
Momentum 0.5342*** 0.1180 0.3214* 0.0867 0.7745***
(2.69) (0.57) (1.76) (0.48) (4.69)
Illiquidity -0.0628 0.0472*** 0.0045 0.0086** -0.001
(-0.57) (2.75) (0.39) (2.56) (-0.16)
Reversal -2.3215*** -1.5886*** -0.4260 -0.2425 0.1602
(-4.51) (-3.52) (-1.03) (-0.63) (0.40)
Maxret -10.0290*** -8.0985*** -5.4096*** -2.5098** -1.4653
(-5.79) (-5.89) (-4.41) (-2.30) (-1.64)
28
Table 5
Portfolio Analysis – The Return Premium of Lottery Stocks
The table reports 2-way portfolio sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio,
we sort by Lottery and Non-Lottery stocks. We then report next-month raw returns for each of the portfolios. The
horizontal sorts (first-stage) are based on Anchor while the vertical sorts (second-stage) are based on Lottery. Panel A
reports the results for equal-weighted portfolios while Panel B shows the results for value-weighted portfolios. Column
[6] reports the differences between extreme Anchor portfolios while the bottom row in each panel consists of the
difference between Lottery and Non-Lottery portfolios. The sample is sorted in quintiles based on Anchor, where
quintile 1 (5) contains firms whose stock price is furthest from (closest to) the 52-week high. T-statistics are reported
below each difference. At the bottom of column [6], we provide the difference-in-differences along with a
corresponding t-statistic. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Lottery
Non-Lottery
Difference
0.0061
0.0083
-0.0022
(-0.74)
0.0081
0.0088
-0.0007
(-0.22)
0.0127
0.0092
0.0035
(1.29)
0.0208
0.0113
0.0095***
(3.24)
0.0170
0.0138
0.0032
(1.51)
0.0109***
(2.78)
0.0055***
(3.01)
0.0054**
(2.06)
Panel B. Value Weighted Portfolios
Lottery
Non-Lottery
Difference
0.0012
0.0094
-0.0082***
(-2.70)
0.0055
0.0094
-0.0040
(-1.22)
0.0064
0.0107
-0.0043
(-1.33)
0.0108
0.0107
0.0001
(0.02)
0.0109
0.0100
0.0010
(0.32)
0.0097**
(2.42)
0.0006
(0.29)
0.0091***
(3.02)
29
Table 6
Portfolio Analysis – The Expected Idiosyncratic Skewness Return Premium
The table reports 2-way portfolio sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio,
we again sort stocks into quintiles based on Expected Idiosyncratic Skewness (E[IdioSkew]). We then report next-
month raw returns for each of the portfolios. The horizontal sorts (first-stage) are based on Anchor while the vertical
sorts (second-stage) are based on E[IdioSkew]. Panel A reports the results for equal-weighted portfolios while Panel
B shows the results for value-weighted portfolios. Column [6] reports the differences between extreme Anchor
portfolios while the bottom row in each panel consists of the difference between extreme E[IdioSkew] portfolios. T-
statistics are reported below each difference. At the bottom of column [6], we provide the difference-in-differences
along with a corresponding t-statistic. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels,
respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Q I (Low Skew)
Q II
Q III
Q IV
Q V (High Skew)
Q V – Q I
0.0201
0.0239
0.0172
0.0141
0.0097
-0.0104***
(-2.63)
0.0157
0.0182
0.0133
0.0091
0.0097
-0.0060**
(-1.98)
0.0139
0.0155
0.0114
0.0103
0.0093
-0.0046*
(-1.64)
0.0114
0.0138
0.0119
0.0108
0.0099
-0.0015
(-0.65)
0.0126
0.0128
0.0107
0.0114
0.0112
-0.0014
(-0.60)
-0.0075***
(-2.91)
-0.0111***
(-3.22)
-0.0065**
(-2.00)
-0.0027
(-0.81)
0.0014
(0.44)
0.0089***
(2.93)
Panel B. Value Weighted Portfolios
Q I (Low Skew)
Q II
Q III
Q IV
Q V (High Skew)
Q V – Q I
0.0102
0.0121
0.0084
0.0062
0.0082
-0.0020
(-0.55)
0.0083
0.0135
0.0113
0.0072
0.0060
-0.0023
(-0.74)
0.0118
0.0119
0.0112
0.0084
0.0077
-0.0041
(-1.40)
0.0081
0.0126
0.0112
0.0099
0.0100
0.0019
(0.82)
0.0120
0.0106
0.0107
0.0114
0.0107
-0.0013
(-0.57)
0.0018
(0.67)
0.0014
(0.43)
0.0023
(0.78)
0.0052*
(1.72)
0.0025
(0.86)
0.0008
(0.23)
30
Table 7
Portfolio Analysis – The Max Return Premium
The table reports 2-way portfolio sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio,
we again sort stocks into quintiles based on Maxret. We then report next-month raw returns for each of the portfolios.
The horizontal sorts (first-stage) are based on Anchor while the vertical sorts (second-stage) are based on Maxret.
Panel A reports the results for equal-weighted portfolios while Panel B shows the results for value-weighted portfolios.
Column [6] reports the differences between extreme Anchor portfolios while the bottom row in each panel consists of
the difference between extreme Maxret portfolios. T-statistics are reported below each difference. At the bottom of
column [6], we provide the difference-in-differences along with a corresponding t-statistic. *,**, and *** denote
statistical significance at the 0.10, 0.05, and 0.01 levels, respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Q I (Low Max)
Q II
Q III
Q IV
Q V (High Max)
Q V – Q I
0.0158
0.0203
0.0099
0.0064
0.0030
-0.0128***
(-3.44)
0.0171
0.0177
0.0136
0.0101
0.0084
-0.0087***
(-2.74)
0.0155
0.0176
0.0130
0.0116
0.0105
-0.0051*
(-1.88)
0.0130
0.0151
0.0135
0.0132
0.0129
-0.0001
(-0.05)
0.0119
0.0129
0.0123
0.0129
0.0136
0.0018
(1.06)
-0.0039
(-1.27)
-0.0074**
(-2.18)
0.0024
(0.71)
0.0065*
(1.92)
0.0107***
(3.12)
0.0146***
(5.01)
Panel B. Value Weighted Portfolios
Q I (Low Max)
Q II
Q III
Q IV
Q V (High Max)
Q V – Q I
0.0140
0.0080
0.0041
0.0029
0.0026
-0.0114**
(-2.49)
0.0153
0.0133
0.0090
0.0067
0.0057
-0.0096**
(-2.25)
0.0118
0.0132
0.0105
0.0088
0.0088
-0.0030**
(-2.19)
0.0102
0.0125
0.0117
0.0105
0.0097
-0.0005
(-0.54)
0.0085
0.0098
0.0112
0.0104
0.0110
0.0025
(1.58)
-0.0056
(-1.58)
0.0018
(0.54)
0.0071**
(2.19)
0.0075**
(2.25)
0.0084**
(2.49)
0.0140***
(4.26)
31
Table 8
Portfolio Analysis – The Return Premium of Lottery Stocks: Multifactor Regressions
The table reports the results from estimating the following equation for two-way sorted portfolios.
Returnp,t – Rf,t = α + βMRP(MRPt) + βSMB(SMBt) + βHML(HMLt) + βUMD(UMDt) + εp,t
The dependent variable is the excess return (or the return in excess of the risk-free rate) for each portfolio. Following
Fama and French (1993) and Carhart (1997), the independent variable includes MRP, which is the market risk
premium, or the excess return of the market less the risk-free rate. SMB is the small-minus-big return factor while
HML is the high-minus-low return factor. UMD is the up-minus-down factor. Portfolios are obtained from two-way
sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio, we sort by Lottery and Non-Lottery
stocks. We then report the alphas from the estimating the four-factor model for each of the portfolios. The horizontal
sorts (first-stage) are based on Anchor while the vertical sorts (second-stage) are based on Lottery. Panel A reports the
results for equal-weighted portfolios while Panel B shows the results for value-weighted portfolios. Column [6] reports
the differences between extreme Anchor portfolios while the bottom row in each panel consists of the difference
between Lottery and Non-Lottery portfolios. T-statistics, which are robust to conditional heteroscedasticity (White
(1980)) are reported below each difference. At the bottom of column [6], we provide the difference-in-differences
along with a corresponding t-statistic. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels,
respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Lottery
Non-Lottery
Difference
0.0001
(0.04)
0.0002
(0.29)
-0.0001
(-0.04)
-0.0003
(-0.09)
0.0001
(0.08)
-0.0004
(-0.11)
0.0028
(1.13)
0.0002
(0.23)
0.0026
(0.99)
0.0104***
(3.44)
0.0021**
(2.31)
0.0083***
(2.63)
0.0063**
(2.29)
0.0027**
(2.38)
0.0036
(1.21)
0.0062*
(1.67)
0.0025*
(1.88)
0.0037
(0.94)
Panel B. Value Weighted Portfolios
Lottery
Non-Lottery
Difference
-0.0064***
(-3.11)
0.0005
(0.78)
-0.0069***
(-3.20)
-0.0036
(-1.36)
-0.0007
(-1.00)
-0.0029
(-1.06)
-0.0050**
(-2.16)
-0.0000
(-0.05)
-0.0050*
(-2.03)
-0.0012
(-0.44)
-0.0004
(-0.47)
-0.0008
(-0.28)
-0.0021
(-0.76)
-0.0015
(-1.41)
-0.0006
(-0.20)
0.0043
(1.25)
-0.0020
(-1.61)
0.0063*
(1.72)
32
Table 9
Portfolio Analysis – The Expected Idiosyncratic Skewness Return Premium: Multifactor Regressions
The table reports the results from estimating the following equation for two-way sorted portfolios.
Returnp,t – Rf,t = α + βMRP(MRPt) + βSMB(SMBt) + βHML(HMLt) + βUMD(UMDt) + εp,t
The dependent variable is the excess return (or the return in excess of the risk-free rate) for each portfolio. Following
Fama and French (1993) and Carhart (1997), the independent variable includes MRP, which is the market risk
premium, or the excess return of the market less the risk-free rate. SMB is the small-minus-big return factor while
HML is the high-minus-low return factor. UMD is the up-minus-down factor. Portfolios are obtained from two-way
sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio, we then sort stocks into quintiles
based on Expected Idiosyncratic Skewness (E[IdioSkew]). We then report the alphas from the estimating the four-
factor model for each of the portfolios. The horizontal sorts (first-stage) are based on Anchor while the vertical sorts
(second-stage) are based on E[IdioSkew]. Panel A reports the results for equal-weighted portfolios while Panel B
shows the results for value-weighted portfolios. Column [6] reports the differences between extreme Anchor portfolios
while the bottom row in each panel consists of the difference between extreme E[IdioSkew] portfolios. T-statistics,
which are robust to conditional heteroscedasticity (White (1980)) are reported below each difference. At the bottom
of column [6], we provide the difference-in-differences along with a corresponding t-statistic. *,**, and *** denote
statistical significance at the 0.10, 0.05, and 0.01 levels, respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Q I (Low Skew)
Q II
Q III
Q IV
Q V (High Skew)
Q V – Q I
0.0109***
(4.27)
0.0153***
(4.86)
0.0096***
(3.16)
0.0075**
(2.10)
0.0048
(1.30)
-0.0061
(-1.36)
0.0051**
(2.42)
0.0086***
(3.76)
0.0035**
(2.05)
0.0006
(0.25)
0.0014
(0.86)
-0.0037
(-1.39)
0.0029*
(1.70)
0.0045***
(2.85)
-0.0001
(-0.10)
-0.0001
(-0.04)
0.0007
(0.63)
-0.0022
(-1.08)
-0.0009
(-0.78)
0.0014
(1.14)
0.0003
(0.26)
0.0002
(0.26)
0.0005
(0.48)
0.0014
(0.01)
-0.0001
(-0.06)
0.0006
(0.55)
-0.0012
(-1.15)
0.0002
(0.16)
0.0009
(0.60)
0.0010
(-0.45)
-0.0110***
(-3.61)
-0.0147***
(-4.41)
-0.0108***
(-3.36)
-0.0073*
(-1.93)
-0.0039
(-0.98)
0.0071
(1.41)
Panel B. Value Weighted Portfolios
Q I (Low Skew)
Q II
Q III
Q IV
Q V (High Skew)
Q V – Q I
-0.0008
(-0.31)
0.0019
(0.80)
-0.0007
(-0.35)
-0.0031
(-1.19)
0.0018
(0.71)
0.0026
(0.72)
-0.0037*
(-1.71)
0.0025
(1.28)
0.0003
(0.20)
-0.0034**
(-2.14)
-0.0033*
(-1.90)
0.0004
(0.14)
0.0003
(0.14)
-0.0001
(-0.05)
-0.0002
(-0.14)
-0.0014
(-0.89)
0.0001
(0.03)
-0.0002
(-0.05)
-0.0027*
(-1.94)
0.0007
(0.54)
0.0001
(0.06)
0.0003
(0.23)
0.0014
(1.44)
0.0041**
(2.42)
0.0001
(0.04)
-0.0004
(-0.36)
0.0001
(0.10)
0.0012
(1.26)
0.0017
(1.39)
0.0016
(0.57)
0.0009
(0.25)
-0.0023
(-0.88)
0.0008
(0.36)
0.0043
(1.55)
-0.0001
(-0.04)
-0.0010
(-0.22)
33
Table 10
Portfolio Analysis – The Max Return Premium: Multifactor Regressions
The table reports the results from estimating the following equation for two-way sorted portfolios.
Returnp,t – Rf,t = α + βMRP(MRPt) + βSMB(SMBt) + βHML(HMLt) + βUMD(UMDt) + εp,t
The dependent variable is the excess return (or the return in excess of the risk-free rate) for each portfolio. Following
Fama and French (1993) and Carhart (1997), the independent variable includes MRP, which is the market risk
premium, or the excess return of the market less the risk-free rate. SMB is the small-minus-big return factor while
HML is the high-minus-low return factor. UMD is the up-minus-down factor. Portfolios are obtained from two-way
sorts. We first sort stocks into quintiles by Anchor. Within each Anchor portfolio, we then sort stocks into quintiles
based on Maxret. We then report the alphas from the estimating the four-factor model for each of the portfolios. The
horizontal sorts (first-stage) are based on Anchor while the vertical sorts (second-stage) are based Maxret. Panel A
reports the results for equal-weighted portfolios while Panel B shows the results for value-weighted portfolios. Column
[6] reports the differences between extreme Anchor portfolios while the bottom row in each panel consists of the
difference between extreme Maxret portfolios. T-statistics, which are robust to conditional heteroscedasticity (White
(1980)) are reported below each difference. At the bottom of column [6], we provide the difference-in-differences
along with a corresponding t-statistic. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels,
respectively.
Panel A. Equal Weighted Portfolios
(Far)
QI
Q II
Q III
Q IV
(Close)
QV
Q V – Q I
[1] [2] [3] [4] [5] [6]
Q I (Low Max)
Q II
Q III
Q IV
Q V (High Max)
Q V – Q I
0.0051*
(1.91)
0.0106***
(3.36)
0.0003
(0.09)
-0.0018
(-0.60)
-0.0040*
(-1.79)
-0.0091***
(-2.91)
0.0055***
(2.78)
0.0067***
(3.58)
0.0029*
(1.85)
0.0012
(0.61)
0.0006
(0.32)
-0.0049*
(-1.80)
0.0039**
(2.55)
0.0061***
(4.50)
0.0021*
(1.88)
0.0016
(1.28)
0.0020
(1.46)
-0.0019
(-0.93)
0.0018
(1.47)
0.0041***
(3.60)
0.0027**
(2.50)
0.0033***
(3.19)
0.0040***
(3.31)
0.0022
(1.28)
0.0023**
(2.03)
0.0028**
(1.99)
0.0026**
(2.47)
0.0031***
(3.01)
0.0046***
(4.45)
0.0023
(1.50)
-0.0028
(-0.97)
-0.0078**
(-2.26)
0.0023
(0.66)
0.0049
(1.54)
0.0086***
(3.49)
0.0114***
(3.00)
Panel B. Value Weighted Portfolios
Q I (Low Max)
Q II
Q III
Q IV
Q V (High Max)
Q V – Q I
0.0014
(0.50)
-0.0040*
(-1.72)
-0.0069***
(-3.73)
-0.0073***
(-3.73)
-0.0071***
(-3.04)
-0.0085**
(-2.33)
0.0031
(1.37)
0.0008
(0.41)
-0.0031*
(-1.80)
-0.0039**
(-2.49)
-0.0032*
(-1.90)
-0.0063**
(-2.23)
-0.0006
(-0.38)
0.0006
(0.43)
-0.0010
(-0.76)
-0.0022**
(-1.98)
-0.0002
(-0.13)
0.0004
(0.18)
-0.0017
(-1.25)
0.0014
(1.20)
0.0008
(0.63)
0.0006
(0.55)
0.0010
(0.95)
0.0027
(1.57)
-0.0021*
(-1.92)
-0.0007
(-0.58)
0.0010
(0.91)
0.0008
(0.76)
0.0025**
(2.51)
0.0046***
(3.11)
-0.0035
(-1.16)
0.0033
(1.26)
0.0079***
(3.67)
0.0081***
(3.64)
0.0096***
(3.78)
0.0131***
(3.33)