52-Week High Anchoring and Skewness Preferences ANNUAL MEETINGS/2018-Milan...Keywords: Lotteries, Anchoring, Skewness, Behavioral Biases a Blau is an Associate Professor in the Department

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.


[email protected]. Phone: 435-797-0885 cWhitby is an Associate Professor in the Department of Economics and Finance, in the Jon M.


[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

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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

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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).


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.


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

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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


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






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


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






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


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.


(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)


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


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.


(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)


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.


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


respectively.


(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)


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







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.


(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)


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







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


respectively.


(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)


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)

Documents

52-Week High Anchoring and Skewness Preferences ANNUAL MEETINGS/2018-Milan...Keywords: Lotteries, Anchoring, Skewness, Behavioral Biases a Blau is an Associate Professor in the Department