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Investor Sentiment, 52-week High Bias, and Idiosyncratic Volatility
Puzzle
ABSTRACT
We study how limits of arbitrage, investor sentiment, and anchoring bias affect the cross-
section of stock returns. We predict and document that high idiosyncratic volatility stocks
that are far from (close to) their 52-week highs have low (high) returns due to arbitrage risk
and anchoring bias. Consistent with Baker and Wurgler (2006), we document that high
idiosyncratic volatility stocks have low (high) returns when investor sentiment is high (low).
We further predict and document that high idiosyncratic volatility stocks that are far from
(close to) their 52-week highs have low (high) returns when investor sentiment is high (low).
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I. Introduction
The debate on the nature of the relationship between idiosyncratic volatility and
future returns remains ongoing. While extant literature provides potential explanations to
the idiosyncratic volatility puzzle documented in Ang et al. (2006), the empirical evidence is
mixed. This study argues that subjecting market participants to behavioral biases such as the
anchoring bias and investor sentiment can go a long way in helping us understand this
puzzle.
In a recent study, George and Hwang (2004) demonstrate that traders do anchor on
the 52-week high price. They suggest that stocks whose current price are far away from their
52-week high prices are overpriced because investors are unwilling to sell those stocks,
whereas stocks whose price are close to their 52-week high prices are underpriced because
traders are reluctant to bid up the price of those stocks1.
In this study, we report evidence that anchoring on the 52-week high price may have
significant effects on the cross-sectional relationship between idiosyncratic volatility and
future returns. We start with a simple observation. If stocks that move away from their 52-
week high prices are overpriced and those that move close to their 52-week high prices are
underpriced as in Gorge and Hwang (2004), then there should, all else equal, exist an
asymmetric relationship between idiosyncratic volatility and future returns for stocks that
drift far away from their 52-week high and those that are near their 52-week high. We posit
that if idiosyncratic volatility does proxy for arbitrage risk as in Ali, Hwang and Trombley
(2003)2, then any overpricing (underpricing) engendered by anchoring on the 52-week high
1 See Page 2,146 in George and Hwang (2004) for more detailed explanation. 2 See also Wurgler and Zhuravskaya (2002) and Mendenhall (2004)
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should be stronger for high idiosyncratic volatility stocks. Therefore, we expect a strong
negative (positive) relationship between idiosyncratic volatility and future returns for
stocks that move away from (close to) their 52-week high prices.
To investigate this proposition empirically, we perform a series of tests. First, we test
whether subjecting investors to behavioral biases such as the anchoring bias reveals any
asymmetric relationship between idiosyncratic volatility and future returns. To this end, we
use two-way portfolio sorts and cross-sectional Fama-MacBeth regressions. We form
quintile portfolios based on George and Hwang (2004) measure of nearness to 52-week high
(GH) first, and then on our measure of idiosyncratic volatility. Consistent with our
hypothesis, we find that while there exist a strong negative relationship between
idiosyncratic volatility and future returns for stocks that belong to our lowest GH portfolio
(far from 52-week high), this relationship appears to be strongly positive for stocks that
belong to our highest GH portfolio (close to 52-week high).
After investigating the robustness of our results to a battery of control variables and
to the well-known January seasonality of returns, we turn our attention to the persistence of
our findings overtime. We report evidence that the negative (positive) relationship between
idiosyncratic volatility and future returns for stock that are far from (close to) their 52-week
high price persist for up to 6 post-formation months. Further, we also find that investigating
the behavior of the cumulative returns differential between high and low idiosyncratic
volatility stocks within each of our GH portfolios over a period of 3 years following portfolio
formation confirms the hypothesized underreaction to news caused by anchoring bias.
We then consider the role of investor sentiment as in Baker and Wurgler (2006).
Looking at the effect of investor sentiment on the relationship between idiosyncratic
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volatility and future returns, we report evidence confirming the proposition of Baker and
Wurgler (2006) that high idiosyncratic volatility stocks earn significantly low (high)
subsequent returns when sentiment is high (low). We then build on the following
suggestions from Baker and Wurgler (2006). When the investor sentiment is high, it is
expected that investors exhibit a preference for lottery-like stocks, which increases the
demand for speculative investments (high idiosyncratic volatility stocks). However, if the
investor sentiment is low, investors are likely to exhibit a preference for non-speculative
assets, creating a shift in demand from high idiosyncratic volatility stock to low idiosyncratic
volatility stocks. We investigate the proposition that these demand shifts for speculative
investments in periods of high versus low investor sentiment should be asymmetrically
reflected in the relationship between idiosyncratic volatility and future returns for both
stocks that move close to their 52-week high price and those that move away from their 52-
week high price. Understanding this latest proposition is rather simple and intuitive.
First, during periods of high investor sentiment, investors are likely to bid up the price
of high idiosyncratic volatility stocks (increase in speculative demand), a behavior that
generates an overpricing of these stocks. However, according to George and Hwang (2004),
stocks whose price far from (close to) their 52-week high prices are overpriced
(underpriced) because investors with anchoring bias are reluctant to sell (buy) these stocks.
Therefore, the increased speculative demand of those high idiosyncratic volatility stocks
that also move away from (close to) their 52-week high price in periods of high sentiment
will reinforce (offset) the original overpricing (underpricing) caused by the anchoring bias.
Therefore, we predict and find strong evidence that there exist a negative (no) relationship
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between idiosyncratic volatility and future returns for stocks whose current price far from
(close to) their 52-week high price in periods of high investor sentiment.
Next , during low investor sentiment periods, the decreased speculative demand of
those high idiosyncratic volatility stocks that also move away from (close to) their 52-week
high price in periods of low sentiment will offset (reinforce) the original overpricing
(underpricing) caused by the anchoring bias. Here, we also predict and find robust evidence
that there exist no (positive) relation between idiosyncratic volatility and future return for
stocks whose current price far from (close to) their 52-week high price in periods of low
investor sentiment.
We finally investigate the effect of arbitrage risk on the return predictive power of
anchoring bias. Unlike George and Hwang (2004) who explain the profitability of short-term
(6 and 12 months) momentum strategies based on the anchoring bias of the 52-week high
price, we focus on the stock’s following month return after controlling for arbitrage risk. We
find that anchoring bias appears to be stronger when arbitrage risk is high. However, when
arbitrage risk is low, there appear to exist a return reversal in the following month.
Overall, the results obtained from both portfolio sorts and regression exercises
confirm our hypotheses and are consistent with George and Hwang (2004) and Baker and
Wurgler (2006). While presenting a new challenge to classical views of the cross-section of
stock prices, the evidence we report shed new lights on the relationship between
idiosyncratic volatility and future returns. Although we pursue a goal similar to that of the
previous studies attempting to understand the idiosyncratic volatility puzzle, we adopt a
fundamentally different perspective. Our results do not only build on recent studies, but also
complement earlier work on investor sentiment (Baker and Wurgler (2006)) and anchoring
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bias (George and Hwang (2004)) among others. To the best of our knowledge, this study is
the first of its kind attempting to understand the implications of the interplay between such
a cognitive bias and arbitrage risk for the idiosyncratic volatility puzzle.
The remainder of this paper is organized as follows. Section II discusses motivation
and hypothesis development. Section III describes our sample and discusses the definition
of the key variables we use in our tests. Section IV presents and discusses the results of our
empirical investigations. In Section V, we summarize the study and conclude with some
general observations.
II. Motivation and Hypothesis Development
Following the influential work of Ang et al. (2006) documenting a negative
relationship between idiosyncratic volatility and future returns, researchers have been
relentless in their efforts to provide possible explanations to this anomaly. Among candidate
explanations reported in the finance literature are those based on uncertainty (Johnson
(2004)), illiquidity (Bali and Cakici (2008) and Han and Lesmond (2011)), growth options
(Cao, Simin, and Zhao (2008) and Chen and Petkova (2012)), coskewness (Chabi-Yo and
Yang (2009)), short-sale constraints (Nagel (2005) and George and Hwang (2011)) and one-
month return reversal (Fu (2009) and Huang, Liu, Rhee, and Zhang (2010)). Researchers
such as Jiang, Xu, and Yao (2009) and Wong (2011) have also documented the role of
earnings shocks, expected idiosyncratic skewness (Boyer, Mitton, and Vorkink (2010)),
investor attention (George and Hwang (2011)), maximum daily return (Bali, Cakici, and
Whitelaw (2011)), retail trading proportion (Han and Kumar (2013)), financial distress
(Avramov, Chordia, Jostova, and Philipov (2013)), average variance beta (Chen and Petkova
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(2012)), and prospect theory (Bhootra and Hur (2014) in helping us understand better this
volatility-return relationship.
While the empirical results reported on this issue are mixed, what remains clear is
that this anomaly persists and is still evident in asset prices today. Nonetheless, a growing
body of the literature in finance builds on the evidence reported in the psychology literature
to foster our understanding of the behavior of asset prices. Of particular interest to this study
are the findings of George and Hwang (2004) who develop a trading strategy based on a
nearness of current price to its 52-week high. They attribute the success of their investment
strategy to the “adjustment and anchoring” bias of Tversky and Khaneman (1974)3, and
argue that this bias causes investors to underreact to positive (negative) information about
stocks for which current prices are near (far from) their 52-week high prices. An interesting
fact documented in George and Hwang (2004) is that stocks whose current prices are far
from their 52-week high are overpriced because investors are unwilling to sell those stocks
and those whose current prices are near their 52-week high prices are underpriced because
investors are reluctant to bid the price of those stocks higher. 4
Putting these strands of the literature together, we form the following two hypotheses. First,
If (1) stocks whose current price are far from (near) their 52-week high prices are overpriced
(underpriced) due to the anchoring bias (as shown by George and Hwang (2004)), and (2)
idiosyncratic volatility proxies for arbitrage risk (as suggested by Ali, Hwang and Trombley
(2003)), then we should expect the negative relationship between idiosyncratic volatility
3 Several other studies document the robustness of this cognitive predisposition. Among others are Russo and Schoemaker (1989), and Qu, Zhou and Luo (2008). 4 While George and Hwang (2004) explain momentum using anchoring bias, we show in this study that the anchoring bias of 52-week high combined with arbitrage risk sheds new light on the cross-sectional relation between idiosyncratic volatility and future returns.
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and future returns to be concentrated in stocks that are the far away from their 52-week high
prices. On the other hand, for stocks that move close to their 52-week high prices, we should
expect a positive relationship between idiosyncratic volatility and future returns. Our first
hypothesis is stated as follows.
Hypothesis 1: The negative relationship between idiosyncratic volatility and future
returns is concentrated in stocks whose current prices are far from their 52-week high
prices. Similarly, for stocks whose current prices are close to their 52-week high prices,
there exist a strong positive relationship between idiosyncratic volatility and future
returns.
To understand this proposition, let’s consider the followings. First, Ali, Hwang and
Trombley (2003) characterize stocks with high idiosyncratic volatility as those with the
highest arbitrage risk. Next, George and Hwang (2004) document the overpricing
(underpricing) of stocks that are far from (near) their 52-week high prices, a phenomenon
explained by anchoring bias. Finally, put together, it is easy to see that for those already
overpriced (underpriced) stocks by the anchoring bias, the overpricing (underpricing)
should be even more pronounced in presence of high idiosyncratic volatility than low
idiosyncratic volatility. The resulting spread in overpricing (underpricing) between high
and low idiosyncratic volatility stocks should therefore yield strong negative (positive)
subsequent returns only for stocks that are far from (near) their 52-week high prices. We
summarize this interrelationship between anchoring bias and arbitrage risk (idiosyncratic
volatility) in the table below.
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Relation between Anchoring Bias and Limits of Arbitrage GH1
(Stocks far from their 52-week high) GH5
(Stocks close to their 52-week high)
These stocks are overpriced because investors are reluctant to sell these stocks due to anchoring bias
These stocks are underpriced because investors are reluctant to buy these stocks due to anchoring bias
IVOL1 (Low Arbitrage
Risk)
Overpricing is less pronounced due to low arbitrage risk
Underpricing is less pronounced due to low arbitrage risk
IVOL5 (High Arbitrage
Risk)
Overpricing is more pronounced due to high arbitrage risk
Underpricing is more pronounced due to high arbitrage risk
IVOL5 – IVOL1
(Our Prediction)
The resulting overpricing spread between IVOL5 and IVOL1 yields negative relation between IVOL and future returns
The resulting underpricing spread between IVOL5 and IVOL1 yields positive relation between IVOL and future returns
Note: This table shows the relation between anchoring bias and limits of arbitrage. GH (George and Hwang Ratio) is current price/52-week high price. GH1 (GH5) is lowest (highest) quintile portfolio on GH. IVOL1 (IVOL5) is lowest quintile portfolio
on (IVOL).
Next, we consider the role of investor sentiment in the cross-sectional relationship
between idiosyncratic volatility and future returns when conditioned by the anchoring bias
of the 52-week high price. Baker and Wurgler (2006) argue that sentiment drives the relative
demand for speculative investments. They suggest that while high sentiments increase the
demand for speculative stocks, low sentiments have the opposite effect, reducing the
demand for speculative investments. They also predict low (high) subsequent returns for
these speculative investments in periods of high (low) investor sentiment. Considering that
Kumar (2009) characterizes high idiosyncratic volatility stocks as lottery-like, hence
speculative investments, it is possible that the predictions we make in our previous
hypothesis be specific to periods of high or low investor sentiment.
To understand and formalize our expectation of the role of investor sentiment, let’s
consider the followings. First, during periods of high sentiment, investors are likely to bid up
the price of high idiosyncratic volatility stocks, which confirms the increase in speculative
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demand for these stocks as suggested by Baker and Wurgler (2006). The increased demand
in favor of high idiosyncratic volatility stocks that also move away from their 52-week high
prices will reinforce the overpricing caused by the anchoring bias. However, for high
idiosyncratic volatility stocks that are near their 52-week high prices, the increase of
speculative demand should offset the underpricing caused by anchoring bias. Altogether, in
periods of high investor sentiment, we expect a negative (no) relation between idiosyncratic
volatility and future return for stocks whose price far from (close to) their 52-week high
prices.
Similarly, during low sentiment periods, the decrease in speculative demand of high
idiosyncratic volatility stocks that are also close to their 52-week high prices should
reinforce the underpricing caused by the anchoring bias. However, for high idiosyncratic
volatility stocks that are far from their 52-week high prices, this demand shift should offset
the overpricing caused by anchoring bias. Overall, we expect that in periods of low investor
sentiment, a positive (no) relation between idiosyncratic volatility and future returns for
stocks whose price close to (far from) their 52-week high prices. Our second hypothesis is
stated as follows.
Hypothesis 2: The negative relationship between idiosyncratic volatility and future
returns is concentrated in stocks whose current prices are far from their 52-week high
prices and specific to periods of high investor sentiment. Similarly, for stocks whose
current prices are close to their 52-week high prices, there exist a strong positive
relationship between idiosyncratic volatility and future returns only in periods of low
investor sentiment.
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The table below summarizes this role of investor sentiment.
Note: This table shows the relation among investor sentiment, anchoring bias and limits of arbitrage. The investor sentiment is from Baker and Wurgler (2006). GH (George and Hwang Ratio) is current price/52-week high price. GH1 (GH5) is lowest (highest) quintile portfolio on GH. IVOL1 (IVOL5) is lowest quintile portfolio on (IVOL).
III. Data
Our sample covers the period from January 1965 to December 2012. We obtain daily
and monthly returns, prices and shares outstanding for all the stocks traded on the NYSE,
AMEX, and NASDAQ from the Center for Research in Security Prices (CRSP). We limit our
sample to firms with common share code 10 and 11 and stocks worth $5 or more each month
following Jiang, Xu and Yao (2009).5 This approach is common in the finance literature and
is often used to eliminate the effects of small and illiquid stocks.6 We obtain monthly Fama-
5We find consistent results using a sample without price restriction. 6Jiang, Xu, and Yao (2009) argue that eliminating stocks with prices less than $5 helps avoiding market microstructure related issues.
Relation between Investor Sentiment, Anchoring Bias and Limits of Arbitrage
Low Sentiment High Sentiment
GH1 GH5 GH1 GH5
Low sentiment decreases speculative demand for high IVOL stocks => Overpricing of high IVOL stocks in GH1 caused by the anchoring bias is mitigated => No relation between IVOL and future return is predicted for stocks in GH1
Low sentiment decreases speculative demand for high IVOL stocks => Underpricing of high IVOL stocks in GH5 caused by the anchoring bias is strengthened => Positive relation between IVOL and future return is predicted for stocks in GH5
High sentiment increases speculative demand for high IVOL stocks => Overpricing of high IVOL stocks in GH1 caused by the anchoring bias is strengthened => Negative relation between IVOL and future return is predicted for stocks in GH1
High sentiment increases speculative demand for high IVOL stocks => Underpricing of high IVOL stocks in GH1 caused by the anchoring bias is mitigated => No relation between IVOL and future return is predicted for stocks in GH5
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French factors returns, NYSE market capitalization decile breakpoints, and monthly risk-free
rates from Kenneth French’s website7.
For each firm, we also compute the book to market ratio (BTM) using additional
information collected from Compustat. Book-to-market is defined as the ratio of fiscal year-
end book equity plus the balance sheet deferred taxes in the prior year to market equity in
December of that year. As is common in the literature, we define firm size as the logarithm
of market capitalization. We also follow Amihud (2002) in computing a measure of illiquidity
for every stock in our sample and for every month. Our illiquidity measure (ILLIQ) is
therefore defined as the ratio of a stocks’ absolute monthly return to its dollar trading
volume.
The 52-week high price of a stock is the highest closing price of the stock during the
previous 52 weeks, as reported in the CRSP daily files. We follow George and Hwang (2004)
in the identification of the 52-week high prices and the computation of their measure of
nearness to the 52-week high price. That is, we first make sure we adjust our price variables
for stock splits and dividends using the CRSP price adjustment factor. We then compute the
measure of nearness to the 52-week high price (GH)8 at the end of every month for every
stock in our sample as the ratio of the stock’s current price over its 52-week high price. It is
given by:
𝐺𝐻 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑃𝑟𝑖𝑐𝑒
52−𝑊𝑒𝑒𝑘 𝐻𝑖𝑔ℎ 𝑃𝑟𝑖𝑐𝑒 (1)
7This data can be found at the following address: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/. 8 We refer to the George and Hwang (2004) measure of nearness to 52-week high as the GH ratio or simply GH in the remainder of this paper.
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The GH reaches its maximum at 1 when a stock’s month end price is the 52 week-high
price. As suggested in George and Hwang (2004), stocks with high GH are those for which
good news recently arrived in the market, and those with low GH are those for which bad
news recently arrived in the market.
Our idiosyncratic volatility measure is obtained using monthly stock returns
following Lehmann (1990a), and Malkiel and Xu (2002), Bali and Cakici (2008). Every month
t for each firm i, we regress stock i’s monthly excess returns on the monthly Fama and French
factors (Fama and French, 1993, 1996) over the previous 24 to 60 months as available. That
is, for every month t, we estimate the following equation from which we save the standard
deviation of the residuals (휀𝑖,𝑡) :
𝑅𝑖,𝑡 − 𝑟𝑓,𝑡 = 𝛼𝑖 + 𝛽𝑖(𝑅𝑚,𝑡 − 𝑟𝑓,𝑡) + 𝜃𝑖𝑆𝑀𝐵𝑡 + 𝛿𝑖𝐻𝑀𝐿𝑡 + 휀𝑖,𝑡 (2)
Where 𝑅𝑖,𝑡 is the rate of return on stock i on month t, 𝑟𝑓,𝑡 is the risk free rate on month t,
𝑅𝑚,𝑡, 𝑆𝑀𝐵𝑡, 𝐻𝑀𝐿𝑡 are the return of market, Size, Book-to-Market factors on month t,
respectively. Finally, 휀𝑖,𝑡 is the residual of stock i on month t. We then compute idiosyncratic
volatility as the standard deviation of the residuals from these monthly regressions.9
IV. Results
A. Idiosyncratic Volatility and Future Returns
We report results from univariate sorts on idiosyncratic volatility (IVOL) for the
entire sample period in Table I. V1 (V5) is portfolio of stocks in the bottom (top) quintile of
9 We use monthly idiosyncratic volatility because it is more consistent than daily idiosyncratic volatility. But, the results using daily idiosyncratic volatility are qualitatively similar.
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IVOL. Following Bali and Cakici (2008), we form our quintile portfolio based on CRSP, NYSE
and Equal-Market Share breakpoints. As pointed in Bali and Cakici (2008), the choice of
breakpoint and weighing scheme play an important role in the relationship between
idiosyncratic volatility and future returns. Consistent with their results (See Table 3 of Bali
and Cakici (2008)), we find that when portfolios are formed using CRSP breakpoints, both
the value and equal weighted return differentials between our highest (V5) and lowest (V1)
idiosyncratic volatility portfolios are negative yet statistically insignificant: -0.11% (t-stat =
-0.36) and -0.18% (t-stat = -0.76) respectively. We also find both corresponding Fama-
French Alpha (FF-Alpha) to be negative: -0.28 and -0.41 yet only significant when returns
are equally weighted; t-statistics = -1.58 and -2.51 respectively.
Forming portfolios based on NYSE breakpoints, we find that the value (equal)
weighted return differential between V5 and V1 is positive (negative) yet statistically
insignificant: 0.06% (t-stat = 0.25) and -0.03% (t-stat = -0.14) respectively. We also find that
the behavior of the FF-Alphas in this case replicates that of the FF-Alphas obtained when
portfolios are formed based on CRSP breakpoints. That is, the FF-Alpha for value and equal
weighted returns appear to be both negative: -0.13 and -0.29, yet only statistically significant
when returns are equally weighted; t-statistics = -0.94 and -2.47 respectively. Using Equal
Market Share to form our quintile portfolios, we find that both the value and equal weighted
return differentials between V5 and V1 are positive but also statistically insignificant: 0.17%
(t-stat = 0.77) and 0.12% (t-stat= 0.60) respectively. In this final scenario, we find both FF-
Alphas to be negative yet statistically insignificant: -0.01 (t-stat = -0.01) and -0.18 (t-stat = -
1.63) for value and equal weighted returns respectively.
[Insert Table I Here]
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Overall, the results obtained from Table I closely replicate the findings of Bali and
Cakici (2008)10. Our results confirm their proposition that the choice of weighing scheme
and breakpoints all play an important role in the relationship between idiosyncratic
volatility and future returns. We therefore move to provide, in the following sections,
evidence of the role of anchoring bias on the idiosyncratic volatility puzzle under these
various setups.
B. The Role of the 52-week High Price Anchor
The finance literature has struggled with the puzzling findings of Ang, Hodrick, Xing,
and Zhang (2006, 2009) that high idiosyncratic volatility stocks earn low future returns.
Although this puzzle has been extensively investigated in the literature, an important and
widely accepted aspect of the behavior of investors has proven to be absent from this debate.
In this section, we investigate the role of anchoring bias as it pertains to the 52-week high
price on the idiosyncratic volatility puzzle.
George and Hwang (2004) suggest that stocks whose current prices are close to their
52-week highs are underpriced because investors are reluctant to bid up the prices of these
stocks whereas stocks whose current prices are far away from their 52-week high prices are
overpriced because investors are unwilling to sell these stocks due to the anchoring bias.
Building on the findings in George and Hwang (2004), our primary hypothesis is that among
stocks whose current prices are far away from their 52-week high prices, high idiosyncratic
volatility stocks are more overpriced than low idiosyncratic volatility stocks whereas among
10 Our findings do not exactly replicate the findings of Bali and Cakici (2008) because of $5 price restriction and the sample period difference.
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stocks whose current prices are close to their 52-week high prices, high idiosyncratic
volatility stocks more underpriced than low volatility stocks, because of arbitrage risk;
proposition consistent with the view of high idiosyncratic volatility as a proxy for arbitrage
risk as in Ali, Hwang and Trombley (2003). Therefore, our primary hypothesis predicts a
negative (positive) relationship between idiosyncratic volatility and future returns for
stocks that move far away from (close to) their 52-weeh high prices.
To allow variations in idiosyncratic volatility (IVOL) to be unrelated to our measure
of nearness to 52-week high price (GH), we employ a double sorting portfolio approach.
First, at the end of every month, we rank our stocks based on their respective GH ratio and
form quintile portfolios. We then subdivide each GH quintile into five portfolios on the basis
of the stocks’ respective idiosyncratic volatilities. We obtain 25 GH-IVOL portfolios. We also
set our breakpoints for each IVOL quintile portfolio using CRSP, NYSE and Equal Market
Share. Table II presents results obtained from this exercise.
[Insert Table II Here]
To demonstrate the dispersion of stocks in our GH portfolios, Panel A of Table II
reports both value and equal weighted GH. While the average GH of stocks in our lowest
value and equal weighted GH portfolios (GH1) are 0.51 and 0.52 respectively, those of our
highest GH portfolio (GH5) are both equal to 0.96. This implies that stocks in GH1 have
current prices close to half of their respective 52-week high prices whereas those in GH5 are
on average very close to their respective 52-week high prices.
In Panel B of Table II, we present results obtained after forming our portfolios based
on CRSP breakpoints. We find that, after controlling for GH, both value and equal weighted
return differentials between the V5 and V1 are negative and strongly significant only for
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stocks that belong to the lowest and the second lowest GH quintiles (GH1 and GH2). For
stocks that belong to our highest GH quintile (GH5), we find strong positive value and equal
weighted return differentials between V5 and V1. Specifically, we find that for stocks that are
the farthest away from their 52-week high prices (those that belong to GH1), the value
(equal) weighted return differential between V5 and V1 is -1.03% (-1.15%) per month with
a t-statistic of -3.57 (-4.55). However, for stocks that are the closest to their 52-week high
prices (those that belong to GH5), the value (equal) weighted return differential between V5
and V1 is 0.75% (0.81%) per month with a t-statistic of 2.99 (3.80). In addition, we find the
corresponding FF-Alphas to follow a similar pattern both in sign and significance.
Perhaps even more interestingly, we find in Panels C. and D. of Table II that forming
portfolios based on NYSE or Equal Market Share breakpoints does not change the pattern in
the volatility-return relationship that we document for stocks that are far from (near) their
52-week high prices. That is, irrespective of the choice of breakpoints to employ in the
formation of portfolios, the idiosyncratic volatility puzzle exists only for stocks that are far
from their 52-week high prices (low GH stocks). However, for stocks that are close to their
52-week high prices, there exists, consistent with classical finance theories, a strong positive
relationship between idiosyncratic volatility and future returns. Figure 1 below summarizes
our findings graphically for the two-way sort on nearness to 52-week high and then
idiosyncratic volatility for the various breakpoint choices. It is clear, as depicted by the solid
and clear bars that while the relationship between volatility and future returns is negative
for GH1, it is positive for GH5, irrespective of weighing scheme.
These results we report so far lend support to our primary hypothesis that if
idiosyncratic volatility does indeed proxy for arbitrage risk, there is a negative (positive)
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relationship between idiosyncratic volatility and future returns only for stocks that are far
from (close to) their 52-week high prices.
Figure 1: Two-way sorts: Future Returns by Nearness to 52-week high (GH) and Idiosyncratic Volatility (IVOL).
GH1(GH5) is our lowest (highest) quintile GH portfolio, and V1(V5) is our lowest (highest) quintile IVOL portfolio. The solid
bars are the value-weighted return differential between V5 and V1 within each GH group (IVOL5-IVOL1|VW), and the clear
bars are the equal weighted return differential between V5 and V1 within each GH group (IVOL5-IVOL1|EW)
We now turn our attention to the investigation of the robustness of our results after
controlling for other known drivers of the volatility-return relationship. To provide such
evidence, we perform a series of firm-level Fama-MacBeth cross-sectional regression tests
that allow us to control for other variables. We perform these tests separately for our lowest
and highest quintile GH portfolios. Each month from January 1965 to December 2012, we
run firm-level Fama-MacBeth cross-sectional regressions of stock returns in month t+1 on
the lagged explanatory variables in month t. The full cross-sectional regression specification
takes the following form:
-0.95
-0.45
0.05
0.55
1.05
GH1 GH2 GH3 GH4 GH5
Ave
rage
Ret
urn
s
NYSE Breakpoints
-0.8
-0.3
0.2
0.7
1.2
GH1 GH2 GH3 GH4 GH5
Ave
rage
Ret
urn
s
GH Portfolios
Equal Market Share Breakpoints
IVOL5-IVOL1|VW
IVOL5-IVOL1|EW
-1.2
-0.7
-0.2
0.3
0.8
GH1 GH2 GH3 GH4 GH5
Ave
rage
Ret
urn
s
CRSP Breakpoints
19
𝑅𝑖,𝑡+1 = 𝛼𝑡 + 𝛽1IVOL𝑖,𝑡 + 𝛽2REV𝑖,𝑡 + 𝛽3MAX𝑖,𝑡 + 𝛽4BETA𝑖,𝑡 + 𝛽5BTM𝑖,𝑡 + 𝛽6SIZE𝑖,𝑡 + 𝛽7MOM𝑖,𝑡
+ 𝛽8SKW𝑖,𝑡 + 𝛽9ILLIQ𝑖,𝑡 + 𝑒𝑖,𝑡+1
Where the dependent variable, 𝑅𝑖,𝑡+1 is the return of stock i in month t+1. The lagged
explanatory variables, computed in month t, include idiosyncratic volatility (IVOL), monthly
stock return (REV), maximum daily return (MAX), stock’s beta (Beta), book-to-market ratio
(BTM), the natural log of market capitalization (SIZE), the holding period return from month
t-12 to month t-2 (MOM), the idiosyncratic skewness (SKW) and illiquidity (ILLIQ).
Huang, Liu, Rhee, and Zhang (2010; HLRZ hereafter) suggests that the idiosyncratic
volatility puzzle is attributable to the short-term reversals in returns documented in
Jegadeesh (1990), Lehmann (1990b), and Lo and MacKinlay (1990)11. They find that in cross-
sectional regressions of future returns of stocks on idiosyncratic volatility that control for
previous month’s return, the coefficient on idiosyncratic volatility is no longer statistically
significant; Hence our control for the short-term reversal of monthly stock return (REV). We
also control for maximum daily return (MAX) because Bali, Cakici, and Whitelaw (2011; BCW
hereafter) finds a significant negative relationship between stocks’ maximum daily return in
a given month and their returns in the following month. These authors also find that after
controlling for MAX in the cross-sectional regressions of future returns on idiosyncratic
volatility, the coefficient on volatility is insignificant in some specifications or even
significantly positive in others.
Table III reports results from our Fama-MacBeth cross-sectional regressions. Models
(1) and (2) are estimated using stocks in our lowest quintile GH portfolio (GH1) whereas
Models (3) and (4) are estimated using stocks in our highest quintile GH portfolio (GH5). In
11 Fu (2009) also documents a similar role of return reversals in the negative volatility-return relationship
20
Models (1) and (3), we investigate, in univariate regression settings, the existence of the IVOL
puzzle in our sample. The results are striking. While Model (1) shows that there is a strongly
significant and negative (coefficient estimate= -0.164 and t-stat= -5.50) relationship
between idiosyncratic volatility and future returns for stocks in GH1, Model (3) shows that
this relationship is indeed strongly positive (coefficient estimate= 0.136 and t-stat= 3.67)
when stocks are close to their 52-week high prices. This piece of evidence reinforces our
belief that anchoring on the 52-week high and arbitrage risk do indeed explain the
idiosyncratic volatility puzzle.
Understandably, we expect readers to inquire about similar results after controlling
for other variables known in the finance literature to help explain this puzzle. Models (2) and
(4) take on this task for stocks in GH1 and GH5 respectively. We find that including a set of
control variables (REV, MAX, BETA, BTM, SIZE, MOM, SKW and ILLIQ) does not change our
results. For stocks that are far from their 52-week high prices, we find in Model (2) that the
relationship between idiosyncratic volatility and future returns remains negative and
strongly significant (coefficient estimate= -0.084 and t-stat= -5.70). However, Model (4)
shows that for stock that are near their 52-week high prices, idiosyncratic volatility remains
positively and significantly related to future returns (coefficient estimate= 0.056 and t-stat=
2.67). Further, most of the control variables have expected signs: the coefficients on REV and
firm size (SIZE) are negative and statistically significant, while coefficients on MOM and
idiosyncratic skewness are positive and statistically significant.12
12 As long as investors prefer skewness, a negative relation between skewness and future return is expected. However, while Bhootra and Hur (2014) and Bali, Cakici, and Whitelaw (2010) document a positive relation between returns and lagged idiosyncratic skewness at the firm level, Boyer, Mitton, and Vorkink (2010) find a negative relation between returns and expected idiosyncratic skewness at the firm level as well as the portfolio level.
21
[Insert Table III Here]
To summarize, the results obtained two-way sorts and regression tests suggest that
even after accounting for other important variables, it remains clear, as we hypothesized,
that the negative volatility-return relationship documented by Ang et al. (2006) exists only
in stocks that are far from their 52-week high prices. More importantly, our results are
consistent with the main hypothesis that there is a negative (positive) relationship between
idiosyncratic volatility and future returns for stocks whose current price are far from (close
to) their 52-week high prices.
Peterson and Smedema (2011) provide robust evidence to the fact that the negative
relationship between idiosyncratic volatility and future returns is particular to every month
of the year other than the month of January. Their results find support in other studies such
as Doran, Jiang, and Peterson (2012) and Bhootra and Hur (2014). We turn in the next
section to investigation of the robustness of our findings to this apparent January seasonality
in returns.
C. Controlling for the January Effect
In view of the evidence presented by Peterson and Smedema (2011) and Bhootra and
Hur (2014) among others on the role of the January effect on the IVOL puzzle, it is important
to ensure that our findings are not a simple artifact of the well documented January
seasonality in stock returns. We start with an investigation of the existence of the January
effect in our sample. In Table IV, we report average value and equal weighted monthly
returns of quintile portfolios formed on IVOL for January and Non-January months.
22
We find that for the month of January, both value and equal weighted return
differentials between V5 and V1 are positive (2.38% and 3.18% per month respectively) and
strongly significant with a t-statistic of 2.58 and 3.82 respectively; we also note that the
corresponding FF-Alphas appear to be positive and significant: 0.94% (t-stat= 2.36) and
1.56% (t-stat=2.42) respectively. For Non-January months on the other hand, we find the
value (equal) weighted return differential between V5 and V1 to be negative -0.32% (-
0.49%) per month yet only marginally statistically significant when returns are equally
weighted: t-statistics of -1.11 and -1.86 respectively. Here, we also find the corresponding
FF-Alphas to be negative (-0.42 and -0.60 respectively) and strongly significant (t-statistics
of -2.26 and -3.81 respectively).
[Insert Table IV Here]
Overall, the results reported in Table IV are generally consistent with prior studies.
We find a positive relationship between idiosyncratic volatility and future returns in the
month of January. However, for Non-January months, we find a generally negative
relationship between idiosyncratic volatility and future returns, irrespective of the weighing
scheme employed. We now turn our attention to the examination of the role of the nearness
to the 52-week high on the idiosyncratic volatility puzzle for January versus Non-January
months. To this end, we replicate our analysis for Table II, discriminating this time between
January and Non-January months. The results are reported in Table V and summarized in
Figure 2 below.
[Insert Table V Here]
Panel A (B) of Table V reports both value and equal weighted return differentials
between V5 and V1 for January (Non-January) months and for each GH portfolio. Consistent
23
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
GH1 GH2 GH3 GH4 GH5
Eq
ual
-Wei
ghte
d R
etu
rns
GH Portfolios
with previous studies, we find for both GH1 and GH5 that there exists a strong positive
relationship between idiosyncratic volatility and future returns in the month of January.
However, in Non-January months, we find for stocks that are far from their 52-week high
that both value and equal weighted return differentials between V5 and V1 are negative and
strongly significant (-1.33% and -1.46% per month with t-statistics of -4.49 and -5.59
respectively). Conversely, we find that, for stocks that are near their 52-week high prices
both value and equal weighted return differentials between V5 and V1 are positive and
strongly significant (0.62% and 0.69% per month with t-statistics of 2.37 and 3.12
respectively). Figure 2 shows the results of Table V graphically. It is clear, as depicted by the
solid bars that in Non-January months, while the relationship between volatility and future
returns is negative for GH1, it is positive for GH5. However, for January months (clear bars),
it is just as evident that the relationship between volatility and future returns is consistently
positive across the GH portfolios.
Figure 2: Two-way sorts: Future Returns by Nearness to 52-week high (GH) and Idiosyncratic Volatility (IVOL). GH1 (GH5) is our lowest (highest) quintile GH portfolio, and V1 (V5) is our lowest (highest) quintile IVOL portfolio. The solid bars are returns differential between V5 and V1 in Non-January months (IVOL5-IVOL1|NON-JAN), and the clear bars are the returns differential between V5 and V1 during the months of January (IVOL5-IVOL1|JAN)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
GH1 GH2 GH3 GH4 GH5
Val
ue-
Wei
ghte
d R
etu
rns
24
For robustness, we also perform regression tests separately for our extreme GH
portfolios (GH1 and GH5). In these regressions, we also discriminate between January and
Non-January months. We present the results of this exercise in Table VI. Models (1) through
(4) are estimated for our lowest GH portfolio (GH1) while Models (5) through (8) are
estimated for our highest GH portfolio (GH5).
For our lowest GH portfolio (GH1), those stocks that are far from their 52-week high
prices, we find that while idiosyncratic volatility is negatively related to future returns in
Non-January months (Model (2)), it is positively related to future returns in the month of
January (Model (1)). Specifically, the coefficient estimate on our idiosyncratic volatility
measure is positive and significant in January (coefficient estimate= 0.256 and t-stat= 3.02)
but negative and strongly significant in Non-January months (coefficient estimate= -0.202
and t-stat= -6.51). When we include other control variables, we find that while the coefficient
estimate on our idiosyncratic volatility measure remains positive (0.027) in January, it loses
its significance (t-stat= 0.58). However, we also find that the coefficient estimate on our
idiosyncratic volatility measure remains negative and strongly significant in Non-January
months (coefficient estimate= -0.094 and t-stat= -6.34).
[Insert Table VI Here]
Interestingly, for our highest GH portfolio (GH5), we find that idiosyncratic volatility
is unwaveringly positively related to future returns in both January (Models (5)) and Non-
January months (Models (6)). Similarly, the inclusion of control variables appears to have no
effect on our findings. Specifically, for stocks in GH5, we find that the coefficient estimate on
our idiosyncratic volatility measure is positive and significant in January (coefficient
estimate= 0.406 and t-stat= 4.09), but also positive and strongly significant in Non-January
25
months (coefficient estimate= 0.112 and t-stat= 2.91). When we include our control
variables, we find similar results. In January, the coefficient estimate on our idiosyncratic
volatility measure remains positive (0.203) and significant (t-stat= 4.68). In Non-January
months, the coefficient estimate on our idiosyncratic volatility measure remains positive and
significant (coefficient estimate= 0.043 and t-stat= 2.07).
Overall, the evidence we present in Tables V and VI suggest that the findings we
document in this study on the role of anchoring bias on the IVOL puzzle are indeed robust to
the well documented January seasonality in stock returns. In the next section, we turn our
focus to the examination of the persistence of our results.
D. Examining the persistence of our results and Underreaction to News
The results we report up to this point suggest a systematic overpricing (underpricing)
of high IVOL stocks that move away from (close to) their 52-week high prices due to
anchoring bias and arbitrage risk. In this section, we examine the persistence of our results
over time. To this end, every month t, we form quintile portfolios based on our measure of
nearness to the 52-week high price (GH). Within each of these quintile portfolios formed on
GH, we also form quintile portfolios based on IVOL. For each of our GH portfolios, we then
report the post-formation average value (equal) weighted return differentials between our
highest and lowest IVOL portfolios.
The results of this exercise are reported in Table VII. In Panel A (B), we report the
average value (equal) weighted post-formation returns differentials between V5 and V1
within each of our GH portfolios over the period going from t+2 to t+6 following the portfolio
formation at time t. We remind the readers that our all previous results focus on the post-
26
formation month t+1, and we do not repeat these results here. We find in Panel A (B) that for
our lowest GH portfolio (GH1), the value (equal) weighted return differentials between our
highest and lowest idiosyncratic volatility portfolios (V5-V1) are significantly negative for
up to 6 months following the portfolios formation months. Remarkably, a look at our highest
GH portfolio (GH5) also suggests that the positive relationship between idiosyncratic
volatility and future returns that we document in this study for stocks that are close to their
52-week high appears to persist over time.
[Insert Table VII Here]
Figure 3 shows an alternative representation of the results we report in Table VII up
to 12 months following portfolio formation. Specifically, Figure 3 depicts both value and
equal weighted return differentials between V5 and V1 for our lowest (solid bars – GH1) and
highest (clear bars – GH5) GH portfolios. We also depict with the dotted line, the average
value and equal weighted return differentials between V5 and V1 for our entire sample. As
is evident in these graphs, both value and equal weighted return differential between V5 and
V1 for our lowest GH portfolio (GH1) appear to be negative (solid bars) up to 12 months
following portfolio formation. On the other hand, the value and equal weighted returns
differentials between V5 and V1 for our highest GH portfolio (GH5) appear to be positive up
to 12 months following portfolio formation.13
The apparent persistence of our results raises a question of the validity of the
underreaction to information argument we build on in this study. That is, as George and
Hwang (2004) suggest, if nearness to the 52-week high is seen as the arrival of good news in
13 Table VII reports average value and equal weighted return differentials between V5 and V1 up to 6 months following portfolio formation because the return differentials are not statistically significant from month t + 7 even if they show the persistence up to month t + 12.
27
-1.5
-1
-0.5
0
0.5
1
1.5
T+2 T+3 T+4 T+5 T+6 T+7 T+8 T+9 T+10T+11T+12
Val
ue-
wei
ghte
d R
etu
rns
-1.5
-1
-0.5
0
0.5
1
1.5
T+2 T+3 T+4 T+5 T+6 T+7 T+8 T+9 T+10T+11T+12
Eq
ual
-Wei
ghte
d R
etu
rns
IVOL5-IVOL1|GH1
IVOL5-IVOL1|GH5
IVOL5-IVOL1
Figure 3: Two-way sorts: Future Returns by Nearness to 52-week high (GH) and Idiosyncratic Volatility (IVOL). The solid bars are both value and equal weighted return differential between V5 and V1 for lowest GH group (IVOL5-IVOL1|GH1), and the clear bars are both value and equal weighted returns differential between V5 and V1 for our highest GH group (IVOL5-IVOL1|GH5). The dotted line is value (equal) weighted return differential between V5 and V1 for the
entire sample.
the market, then the underpricing of stocks that are near their 52-week high price occurs
because investors underreact to this new (positive) information and are reluctant to bid up
the price of these stocks. Similarly if distance to the 52-week high is seen as the arrival of bad
news in the market, then the overpricing of stocks that are far from their 52-week high prices
occurs because investors also underreact to this piece of (bad) information and are unwilling
to sell those stocks.
In this section, we investigate this issue more closely. On the one hand, investors may
underreact to information in month t as George and Hwang (2004) argue. On the other hand,
it is also possible that investors’ actions during the holding period of month t + 1could be
28
purely an overreaction to firm-specific information.14 If stocks do overreact to firm-specific
information, then we should observe price reversals over the long run.15
Figure 4: Two-way sorts: Cumulative Returns by Nearness to 52-week high (GH) and Idiosyncratic Volatility (IVOL). The lines depicts cumulative returns of V5 (highest quintile idiosyncratic volatility portfolio) – V1 (lowest quintile idiosyncratic volatility portfolio) for GH1 (lowest quintile George and Hwang ratio (current price/52-week high price) portfolio) and GH5 (highest quintile George and Hwang ratio (current price/52-week high price) portfolio) up to 3 years after portfolio formation.
Similar to Jegadeesh and Titman (2001), we investigate the cumulative returns of V5–
V1 during the post-holding period from month t+2 to month t+36. To perform a simple test
of reaction to information hypothesis, we plot in Figure 4 both value and equal weighted
cumulative returns of V5-V1 for our lowest (GH1) and highest (GH5) GH portfolios for a
period up to 36 months after portfolio formation. We find that the value and equal weighted
cumulative returns for GH1 (GH5) decreases (increases) up to around month t + 14 and then
become stable. This evidence supports underreaction to information hypothesis because of
anchoring bias as in George and Hwang (2004).
14 While Jegadeesh and Titman (1993) document short-term underreaction to information, the long-term overreaction is well documented in Debondt and Thaler (1985, 1987, and 1990). 15 See Jegadeesh and Titman (2001) for more detailed explanation of underreaction and overreaction to information hypothesis.
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
t+1
t+2
t+3
t+4
t+5
t+6
t+7
t+8
t+9
t+1
0t+
11
t+1
2t+
13
t+1
4t+
15
t+1
6t+
17
t+1
8t+
19
t+2
0t+
21
t+2
2t+
23
t+2
4t+
25
t+2
6t+
27
t+2
8t+
29
t+3
0t+
31
t+3
2t+
33
t+3
4t+
35
t+3
6
EW_GH1 VW_GH1 VW_GH5 EW_GH5
29
E. Controlling for Investor Sentiment
Baker and Wurgler (2006) argue that sentiment drives the relative demand for
speculative investments. Specifically, they suggest that while high sentiment increases the
demand for speculative stocks, low sentiment have the opposite effect, reducing the demand
for speculative investments. Ultimately, they predict low (high) subsequent returns for these
speculative investments in periods of high (low) investor sentiment. Kumar (2009) on the
other hand characterizes high idiosyncratic volatility stocks as lottery-like stocks (See page
1896). Put together, these studies suggest that during period of high (low) investor
sentiment, we should expect an increase (decrease) in the speculative demand for high
idiosyncratic volatility stocks, which would eventually lead to lower (higher) subsequent
returns. However we postulate that conditioning this relationship on the anchoring bias of
the 52-week high as in George and Hwang (2004) will reveal the concentration of the
idiosyncratic volatility puzzle (the negative relation between idiosyncratic volatility and
future return) only for stocks that are far from their 52-week high price in periods of high
investor sentiment.
Understanding this latest proposition is rather simple and intuitive. First, during
periods of high sentiment, the increased speculative demand of those high idiosyncratic
volatility stocks that also move away from (close to) their 52-week high prices will reinforce
(offset) the original overpricing (underpricing) caused by the anchoring bias.
Next, during low investor sentiment periods, the decreased speculative demand of
those high idiosyncratic volatility stocks that also move away from (close to) their 52-week
high prices will offset (reinforce) the original overpricing (underpricing) caused by the
anchoring bias.
30
In summary, we predict a negative (positive) relationship between idiosyncratic
volatility and future returns for stocks whose current prices are far from (close to) their 52-
week high prices in periods of high (low) sentiment. We further predict no relation between
idiosyncratic volatility and future returns for stocks whose current prices are far from (close
to) their 52-week high prices in periods of low (high) sentiment.
Our investigation of these later propositions starts with an assessment of the
relationship between idiosyncratic volatility and future returns for all stocks in our sample,
discriminating between periods of high, medium and low investor sentiment. We obtain the
investor sentiment data from Baker and Wurgler (2006). However, data availability (July
1965 to December 2010) on this variable dictates the length of the sample employed in the
various tests we perform in this section. After assigning the various months to the sentiment
categories (low, medium, high), we obtain 182 months for each sentiment category. We then
form within each category and every month, quintile portfolios based on our measure of
idiosyncratic volatility and compute value and equal weighted return differentials between
our highest (V5) and lowest (V1) idiosyncratic volatility portfolios. Table VIII presents the
results obtained from this preliminary analysis.
We find in Table VIII that there generally exists a strong negative relationship
between idiosyncratic volatility and future returns (both value and equal weighted) only in
periods of high investor sentiment. Specifically, we find that when the investor sentiment is
high, the value (equal) weighted return differential between V5 and V1 is negative -1.53% (-
1.62%) per month and statistically significant with a t-statistic of -2.55 (-2.90). We also find
the corresponding FF-Alphas to be negative and strongly significant. However, when
sentiment is low, the relationship between idiosyncratic volatility and future returns
31
appears to be positive and strongly significant. The value (equal) weighted return
differential between V5 and V1 in this case is positive 1.12% (1.04%) per month and
statistically significant with a t-statistic of 2.76 (2.86). This finding is consistent with Baker
and Wurgler (2006).16
[Insert Table VIII Here]
Next, we add new findings to Baker and Wurgler (2006) after controlling for the
stocks’ nearness to their respective 52-week high prices using a double portfolio sorting
approach. We report the results of this exercise in Table IX. In Panel A, we focus on periods
of low investor sentiment. During periods of low investor sentiment, we find that while the
relationship between idiosyncratic volatility and future returns is strongly positive for
stocks that are near their 52-week high price (GH5), it appears to be flat for stocks that are
far from their 52-week high price. Specifically, when the investor sentiment is low, we find
for stocks that are far from their 52-week high price (GH1), both value and equal weighted
return differentials between V5 and V1 are negative (-0.15% and -0.08% per month) yet
statistically insignificant (t-statistics of -0.31 and -0.22 respectively). However, for stocks
that are close to their 52-week high prices, we find the value (equal) weighted return
differential between V5 and V1 to be positive 1.17% (1.33%) per month and statistically
significant with a t-statistic of 3.36 (4.30). We also find that FF-Alphas are positive and
strongly significant. Interestingly, we find below that these results reverse when the investor
sentiment is high.
16 See Table III of Baker and Wurgler (2006). They report equal weighted average portfolio returns over months in which the investor sentiment from the previous year-end is positive and months in which it is negative. They find over the period between 1963 and 2001 that high idiosyncratic volatility stocks have lower (higher) returns than low idiosyncratic volatility stocks when the investor sentiment from the previous year-end is positive (negative). However, our results are based on the investor sentiment each month using value and equal weighted returns over the period between 1965 and 2010.
32
In Panel C. we present the results obtained for periods of high investor sentiment.
Here, we find that while there exist a strong negative relationship between idiosyncratic
volatility and future returns for stocks that belong to our lowest GH portfolio (GH1), this
relationship appears to be flat for stocks that belong to our highest GH portfolio (GH5).
Specifically, when the investor sentiment is high, we find for stocks in GH1 that both value
and equal weighted return differentials between V5 and V1 are negative (-2.34% and -2.54%
per month) and strongly significant (t-statistics of -4.15 and -4.40 respectively). However,
for stocks that are close to their 52-week high prices (stocks in GH5), we find the value
(equal) weighted return differential between V5 and V1 to be negative -0.47% (-0.23%) per
month, yet statistically insignificant with a t-statistic of -0.91(-0.55). We also find that the
corresponding FF-Alphas appear to similar patterns both in sign and significance.
[Insert Table IX Here]
In summary, the findings in Table IX is consistent with our predictions, a negative
(positive) relation between idiosyncratic volatility and future returns for stocks whose
current prices are far from (close to) their 52-week high prices in periods of high (low)
sentiment and no relation between idiosyncratic volatility and future returns for stocks
whose current prices are far from (close to) their 52-week high prices in period of low (high)
sentiment.
To provide further evidence in support of the findings reported in Tables IX, we
perform a series of regression tests for which we report the results in Table X. For this
exercise, we focus, separately, on both our lowest (GH1) and highest (GH5) GH portfolios.
Within each portfolio, we discriminate between high and low investor sentiment.
Specifically, for both GH groups (GH1 and GH5), we run cross-sectional firm-level Fama-
33
MacBeth regressions of stock returns on lagged explanatory variables, differentiating
between periods of low and high investor sentiment.
In Models (1) through (4), the focus is on stocks that are far from their 52-week highs,
those that belong to our lowest GH portfolio (GH1). As shown in Models (1) and (2), the
relationship between idiosyncratic volatility and future returns appears to be flat in periods
of low investor sentiment. Specifically, we find in both models that the coefficient estimates
of our idiosyncratic volatility measure are negative (-0.061 and -0.049) yet insignificant (t-
statistics of -1.52 and -1.35 respectively). Interestingly, we also find in Models (3) and (4)
that for stocks that are far from their 52-week highs, there exist a strong negative
relationship between idiosyncratic volatility and future returns when sentiment is high. In a
univariate setting, Model (3) shows that in high sentiment periods, the coefficient estimate
of our idiosyncratic volatility variable is negative (-0.144) and strongly significant (t-stat= -
6.84). We find similar results when we account in Model (4) for a set of control variables
(REV, MAX, BETA, BTM, SIZE, MOM, SKW and ILLIQ) known to help explain the idiosyncratic
volatility puzzle. In this case, the coefficient estimate of our idiosyncratic volatility variable
remains negative (-0.118) and strongly significant (t-stat= -6.63).
For stocks that are close to their 52-week high prices, Models (5) and (6) reveal
consistent with our previous results that there exist, in low sentiment periods, a strong
positive relationship between idiosyncratic volatility and future returns. In Model (5), we
find that the coefficient estimate of our idiosyncratic volatility variable is positive (0.130)
and strongly significant (t-stat= 3.92). Including control variables in the regression as in
Model (6) does not change our results. The coefficient estimate of our idiosyncratic volatility
variable in this regression remains positive (0.142) and significant (t-stat= 2.99). However,
34
when sentiment is high, we find in Models (7) and (8) that while the coefficients estimates
of our idiosyncratic volatility variable are both negative (-0.015 in Model (7) and -0.038 in
Model (8)), they are also statistically insignificant (t-statistics of -0.55 and -1.32 for Models
(7) and (8) respectively).
[Insert Table X Here]
In summary, the negative relationship between idiosyncratic volatility and future
returns appears not only to be concentrated in stocks that move away from their 52-week
high prices (GH1), but it is also specific to periods of high investor sentiment. However,
consistent with classical finance theories, there exist a strong positive relationship between
idiosyncratic volatility and future returns only for stocks that move close to their 52-week
high prices. Perhaps even more importantly, this positive relationship between idiosyncratic
volatility and future returns for stocks that are close to their 52-week high prices appears to
be specific to periods of low investor sentiment. These results do not only shed new lights
on the idiosyncratic volatility puzzle, but also emphasize the importance of behavioral biases
such as the anchoring bias in explaining asset pricing anomalies.
F. The Effects of Arbitrage Risk on Anchoring Bias
Our last tests investigate whether arbitrage risk affects the predictability of anchoring
bias on the following month’s return.17 We report in Panel A of Table XI, quintile portfolios
are formed solely on our measure of nearness to 52-week high price (GH). We then compute
17 We appreciate the comments from an anonymous referee on the importance of discussing this issue. We also note that while George and Hwang (2004) explain the short-term momentum in 6 and 12 months after portfolio formation with the anchoring bias, the tests we perform here are specific to the predictive power of anchoring bias on the following month’s return.
35
both value and equal weighted return differentials between our highest (GH5) and lowest
(GH1) GH portfolios. We find no predictability of anchoring bias on the following month’s
returns. Specifically, when value weighted, we find the return differential between GH5 and
GH1 to be negative (-0.16% per month) but insignificant (t-stat= 0.85). However, when
returns are equally weighted, we find that the return differential between GH5 and GH1 is
positive (0.28% per month) yet also insignificant (t-stat= 1.39). Although both value and
equal weighted return differential between GH5 and GH1 appear to be insignificant, we find
the corresponding FF-Alphas to be positive (0.02% and 0.58%) yet only significant when
returns are equally-weighted.
To investigate the effect of arbitrage risk on the predictability of anchoring bias on
future returns, we follow Ali, Hwang and Trombley (2003) to sort stocks based on
idiosyncratic volatility first and then, within each idiosyncratic volatility portfolio, form
quintile portfolios based on our measure of nearness to 52-week high (GH). We then
compute for each IVOL portfolio both value and equal weighted return differentials between
GH5 and GH1.18 In Panel B, we find, after sorting stocks based on IVOL and then GH, that
when arbitrage risk is low (V1), the return differential between our highest (GH5) and lowest
(GH1) GH portfolio is negative; both value and equal weighted return differentials between
GH5 and GH1 are negative (-0.38% and -0.58% per month) and strongly significant (t-
statistics of -2.54 and -5.08 respectively). We conjecture that this result is a manifestation of
short-term reversal19 because stocks in GH5 have high returns and stocks in GH1 have low
18 Ali, Hwang and Trombley (2003) find that book-to-market effect is greater for stocks with higher idiosyncratic return volatility. 19 See Jegadeesh (1990), Lehmann (1990b) and Lo and MacKinlay (1990) for discussions on short-term reversal
36
returns during the current month. However, when arbitrage risk is high (V5), both value and
equal weighted return differentials between GH5 and GH1 are positive (1.22% and 1.17%
per month) and strongly significant (t-statistics of 4.16 and 4.88 respectively). This later
result suggests that anchoring bias appear to be stronger when arbitrage risk is high. This
finding is consistent with the proposition in George and Hwang (2004) that stocks whose
current price is close to (far from) their 52-week high prices in GH5 (GH1) are underpriced
(overpriced) due to anchoring bias.
[Insert Table XI Here]
We confirm the evidence reported in Table XI with regression tests and present the
results in Table XII. While univariate regression of future returns on nearness to 52-week
high (GH) result in a negative (-0.411) and significant (t-stat= -2.64) coefficient estimate for
our lowest idiosyncratic volatility portfolio (Model (1)), this coefficient is positive (1.959)
and strongly significant (t-stat= 9.49) for our highest idiosyncratic volatility portfolio. We
also find in Models (2) and (4) that controlling for other control variables does not change
the sign or the significance of our results in Models (1) and (3). Overall the predictability of
the anchoring bias on returns in the next month appears to depend on arbitrage risk.
[Insert Table XII Here]
V. Conclusion
The finance literature has struggled with the puzzling findings of Ang, Hodrick, Xing,
and Zhang (2006, 2009) that high idiosyncratic volatility stocks earn low subsequent
returns. The main contribution of this study is to provide evidence of the role of anchoring
bias and investor sentiment on the relationship between idiosyncratic volatility and future
37
returns. Building on idiosyncratic volatility as arbitrage risk in Ali, Hwang and Trombley
(2003) and anchoring bias of George and Hwang (2004), we hypothesize and show that if
idiosyncratic volatility proxies for arbitrage risk, then the idiosyncratic volatility puzzle
exists only for stocks that are far from their respective 52-week high prices. However, for
stocks that move close to their 52-week highs, there exists for the same reason a strong
positive relationship between idiosyncratic volatility and future returns, consistent with
classical finance theories.
In addition, we show building on the predictions of Baker and Wurgler (2006) that
while idiosyncratic volatility is negatively related to future returns only for stocks that move
away from their respective 52-week high prices, this relationship exists only when investor
sentiment is high. Similarly, the positive relationship between idiosyncratic volatility and
future returns appears to be concentrated in stocks that move close to their respective 52-
week high prices, and it exists only in periods of low investor sentiment. We also show that
these results appear to persist over time and are robust to consideration of other variables
known in the finance literature to help explain the idiosyncratic volatility puzzle as well as
the well-known January seasonality in stock returns.
Finally, the predictability of the anchoring bias on the following month’s returns
appears to depend on arbitrage risk.
Ultimately, we report novel evidence of the interrelation between arbitrage risk,
anchoring bias, and investor sentiment on the cross-sectional relation between idiosyncratic
volatility and future return. While the evidence we report in this study undoubtedly shed
new light on the idiosyncratic volatility puzzle, we also make the case in this study in favor
of a focus on specific behavioral biases such as the tendency of investor to anchor on publicly
38
available information such as the 52 week-high prices as a possible explanation for asset
pricing anomalies.
39
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43
Table I Average Monthly Returns of Portfolio Sorted on Idiosyncratic Volatility
This table reports value and equally-weighted monthly returns of the quintile portfolios formed on idiosyncratic volatility. Quintile portfolios are formed on idiosyncratic volatility every month from January 1965 to December 2012. Idiosyncratic volatilities (IVOL) are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
CRSP Breakpoints NYSE Breakpoints 20% Market Share
Value-weighted Returns
Equal-weighted Returns
Value-
weighted Returns
Equal-weighted Returns
Value-
weighted Returns
Equal-weighted Returns
V1 0.86 1.10 0.85 1.10 0.77 1.00
2 0.96 1.23 0.94 1.20 0.91 1.08
3 1.03 1.24 0.98 1.26 0.86 1.16
4 1.04 1.24 1.05 1.25 0.94 1.23
V5 0.75 0.92 0.91 1.07 0.94 1.12
V5-V1 -0.11 -0.18 0.06 -0.03 0.17 0.12
(-0.36) (-0.76) (0.25) (-0.14) (0.77) (0.60)
Alpha -0.28 -0.41 -0.13 -0.29 -0.01 -0.18
(-1.58) (-2.51) (-0.94) (-2.47) (-0.01) (-1.63)
44
Table II Average Monthly Returns of Portfolio Sorted on GH and Idiosyncratic Volatility
:CRSP Breakpoints, NYSE Breakpoints, and “Equal” Market Share Breakpoints This table reports value and equally-weighted monthly returns of the quintile portfolios formed on GH (George and Hwang Ratio: current price/52-week high price) and idiosyncratic volatility (IVOL). Quintile portfolios are formed on GH first and then another quintile portfolios are formed on IVOL within each GH portfolio each month from January 1965 to December 2012. Idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
Value-weighted Returns Equal-weighted Returns
GH1 2 3 4 GH5 GH1 2 3 4 GH5
Panel A: Average GH
0.51 0.72 0.82 0.89 0.96 0.52 0.72 0.82 0.89 0.96
Panel B: CRSP Breakpoints V1 1.11 0.95 0.91 0.91 0.71 1.4 1.36 1.27 1.07 0.86 2 0.95 0.89 0.83 0.89 0.71 1.28 1.23 1.19 1.15 0.97 3 0.79 0.95 0.77 1.12 0.91 0.98 1.06 1.19 1.2 1.12 4 0.50 0.90 0.91 1.21 1.31 0.66 1.07 1.22 1.34 1.36
V5 0.08 0.45 0.90 1.30 1.46 0.25 0.67 1.18 1.62 1.67 V5-V1 -1.03 -0.50 -0.01 0.39 0.75 -1.15 -0.70 -0.09 0.55 0.81
(-3.57) (-1.84) (-0.05) (1.40) (2.99) (-4.55) (-3.08) (-0.38) (2.38) (3.80) Alpha -1.24 -0.65 -0.24 0.25 0.65 -1.25 -0.82 -0.30 0.33 0.64
(-5.08) (-2.92) (-1.10) (1.06) (2.97) (-7.53) (-4.73) (-1.57) (1.58) (3.55) Panel C: NYSE Breakpoints
V1 1.15 1.00 0.90 0.91 0.74 1.43 1.38 1.28 1.08 0.85 2 1.09 0.94 0.96 0.85 0.64 1.44 1.32 1.22 1.11 0.94 3 1.03 0.94 0.74 1.01 0.80 1.25 1.18 1.22 1.18 0.97 4 0.79 0.96 0.75 1.15 0.99 1.06 1.16 1.19 1.20 1.18
V5 0.42 0.68 0.91 1.24 1.36 0.49 0.85 1.21 1.50 1.53 V5-V1 -0.72 -0.32 0.01 0.33 0.62 -0.93 -0.53 -0.08 0.43 0.69
(-2.65) (-1.31) (0.05) (1.47) (2.96) (-4.01) (-2.78) (-0.39) (2.19) (3.74) Alpha -0.90 -0.40 -0.17 0.18 0.53 -1.09 -0.69 -0.30 0.20 0.50
(-4.08) (-1.96) (-0.89) (0.97) (2.92) (-7.54) (-5.07) (-1.92) (1.19) (3.32) Panel D: “Equal” Market Share Breakpoints
V1 1.15 1.08 0.88 0.92 0.72 1.33 1.31 1.13 0.82 0.88 2 1.05 0.95 0.73 0.81 0.57 1.25 1.28 0.98 0.72 0.93 3 1.06 0.76 0.88 0.92 0.50 1.33 1.30 1.10 0.87 1.04 4 0.98 0.82 0.88 1.01 0.66 1.23 1.17 1.14 0.93 1.10
V5 0.61 0.65 1.01 1.14 1.32 0.67 0.96 1.37 1.40 1.55 V5-V1 -0.53 -0.42 0.13 0.22 0.60 -0.66 -0.36 0.23 0.58 0.67
(-2.15) (-1.93) (0.57) (1.84) (2.51) (-2.59) (-1.98) (1.08) (3.02) (3.38) Alpha -0.62 -0.53 -0.02 0.01 0.40 -0.89 -0.56 -0.02 0.35 0.41
(-2.91) (-2.77) (-0.10) (0.04) (1.92) (-6.57) (-3.98) (-0.12) (2.09) (2.55)
45
Table III Fama-MacBeth Cross-Sectional Regressions
Each month from January 1965 to December 2012, we run a firm-level Fama-MacBeth cross-sectional regressions of stock return in month t+1 on the lagged explanatory variables in month t. The explanatory variables include stock’s monthly idiosyncratic volatility (IVOL), monthly stock return (REV), maximum daily return (MAX), BETA, the book-to-market ratio (BTM), the log of market capitalization (Size), the holding period return from month t-12 to month t-2 (MOM), the idiosyncratic skewness (Skw), and the illiquidity measure (ILLIQ). GH1 (GH5) is the lowest (highest) quintile portfolio on GH (George and Hwang Ratio: current price/52-week high price). Newey-West (1978) adjusted t-statistics are reported in parenthesis.
GH1 GH5 (1) (2) (3) (4)
IVOL -0.164 (-5.50)
-0.084 (-5.70)
0.136 (3.67)
0.056 (2.67)
REV -0.066 -0.025
(-11.00)
(-5.03)
MAX -0.134 0.013
(-8.21)
(1.13)
BETA 0.004 0.003 (0.12)
(0.07)
BTM 0.023 0.035 (0.27)
(0.49)
SIZE -0.116 -0.161 (-2.57)
(-4.25)
MOM 0.593 0.442 (2.95)
(3.31)
SKW 0.346 0.143 (7.29)
(4.84)
ILLIQ 0.003 -0.004 (0.93) (-1.14)
46
Table IV Average Monthly Returns of Portfolio Sorted on Idiosyncratic Volatility:
January versus Non-January This table reports value and equally-weighted monthly returns of the quintile portfolios formed on idiosyncratic volatility (IVOL) for January and Non-January months. Quintile portfolios are formed on IVOL each month from January 1965 to December 2012. Idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
January Non-January
Value-weighted Returns
Equal-weighted Returns
Value-weighted Returns
Equal-weighted Returns
V1 0.85 1.93 0.84 1.02
2 1.36 2.61 0.90 1.09
3 2.24 3.30 0.89 1.03
4 2.84 4.37 0.86 0.95
V5 3.23 5.11 0.52 0.54
V5-V1 2.38 3.18 -0.32 -0.49
(2.58) (3.82) (-1.11) (-1.86) Alpha 0.94 1.56 -0.42 -0.60
(2.36) (2.42) (-2.26) (-3.81)
47
Table V Average Monthly Returns of Portfolio Sorted on GH and IVOL:
January versus Non-January This table shows value and equally-weighted monthly returns of the portfolios formed on GH (George and Hwang Ratio: current price/52-week high price) and idiosyncratic volatility (IVOL) for January and Non-January months. Quintile portfolios are formed on GH first and then another quintile portfolios are formed on IVOL within each GH portfolio each month from January 1965 to December 2012. Monthly idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
Panel A : January
Value-weighted Returns Equal-weighted Returns
GH1 2 3 4 GH5 GH1 2 3 4 GH5
V1 1.40 2.09 1.73 1.47 0.00 3.67 3.67 2.86 2.08 0.71
2 3.42 2.53 2.15 1.31 0.25 4.99 3.97 3.07 1.98 0.73
3 3.60 3.74 1.8 1.67 0.27 5.63 4.57 3.16 2.24 1.12
4 2.96 4.27 2.67 1.28 0.55 5.46 5.19 3.95 2.86 1.78
V5 3.57 3.48 3.17 2.84 2.04 5.91 5.3 4.73 4.16 2.76
V5-V1 2.17 1.39 1.44 1.37 2.03 2.24 1.62 1.87 2.08 2.05
(1.95) (1.52) (1.54) (1.49) (2.50) (2.64) (2.07) (2.43) (2.86) (2.97)
Alpha 0.65 0.41 0.47 0.64 1.04 0.97 0.82 0.86 1.14 0.89
(1.01) (0.63) (0.66) (0.81) (1.45) (1.36) (0.92) (1.34) (2.22) (2.00)
Panel B : Non-January
V1 1.08 0.85 0.83 0.86 0.78 1.20 1.16 1.13 0.98 0.87
2 0.74 0.74 0.71 0.86 0.75 0.94 0.98 1.02 1.08 0.99
3 0.53 0.69 0.70 1.07 0.96 0.57 0.75 1.03 1.11 1.13
4 0.28 0.62 0.75 1.19 1.39 0.23 0.71 0.97 1.20 1.33
V5 -0.25 0.19 0.69 1.18 1.40 -0.26 0.27 0.87 1.40 1.57
V5-V1 -1.33 -0.67 -0.14 0.32 0.62 -1.46 -0.89 -0.26 0.42 0.69
(-4.49) (-2.35) (-0.48) (1.08) (2.37) (-5.59) (-3.82) (-1.09) (1.73) (3.12)
Alpha -1.41 -0.73 -0.31 0.20 0.57 -1.49 -0.95 -0.40 0.28 0.60
(-6.05) (-3.48) (-1.39) (0.88) (2.63) (-8.40) (-5.75) (-2.16) (1.43) (3.46)
48
Table VI
Fama-MacBeth Cross-Sectional Regressions : January versus Non-January
Each month from January 1965 to December 2012, we run a firm-level Fama-MacBeth cross-sectional regressions of stock return in month t+1 on the explanatory variables in month t. The explanatory variables include stock’s monthly idiosyncratic volatility (IVOL), monthly stock return (REV), maximum daily return (MAX), BETA, the book-to-market ratio (BTM), the log of market capitalization (Size), the holding period return from month t-12 to month t-2 (MOM), the idiosyncratic skewness (Skw), and the illiquidity measure (ILLIQ). GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis
GH1 GH5 (1) (2) (3) (4) (5) (6) (7) (8) Jan Non-Jan Jan Non-Jan Jan Non-Jan Jan Non-Jan
IVOL 0.256 (3.02)
-0.202 (-6.51)
0.027 (0.58)
-0.094 (-6.34)
0.406 (4.09)
0.112 (2.91)
0.203 (4.68)
0.043 (2.07)
REV -0.111 (-5.38)
-0.062 (-10.32)
-0.088 (-4.49)
-0.020 (-3.67)
MAX -0.164 (-5.21)
-0.131 (-7.33)
0.089 (2.17)
0.006 (0.46)
BETA -0.112 (-1.49)
0.015 (0.36)
-0.141 (-1.95)
0.016 (0.36)
BTM 0.766 (1.89)
-0.043 (-0.48)
0.527 (1.63)
-0.009 (-0.11)
SIZE -1.152 (-2.95)
-0.024 (-0.54)
-0.465 (-3.10)
-0.134 (-3.51)
MOM -1.460 (-1.52)
0.775 (3.76)
-0.323 (-1.37)
0.509 (3.51)
SKW 0.547 (4.08)
0.328 (6.80)
-0.124 (-1.60)
0.167 (5.30)
ILLIQ 0.024 (1.79)
0.001 (0.24)
0.026 (1.12)
-0.006 (-2.01)
49
Table VII
Average Monthly Returns in Post Holding Period Months This table reports the average monthly returns of the V5 (firms with high idiosyncratic volatility) – the V1 (firms with low idiosyncratic volatility) portfolios during the post-holding period from month t+2 to month t+6 from January 1965 until June 2012. Quintile portfolios are formed on GH (George and Hwang Ratio: current price/52-week high price) first and then another quintile portfolios are formed on idiosyncratic volatility within each GH portfolio each month t. Monthly idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis
t+2 t+3 t+4 t+5 t+6 Panel A: Value-weighted Returns
GH1 -1.10 (-3.43)
-0.79 (-2.51)
-0.95 (-3.01)
-0.66 (-2.10)
-0.60 (-1.96)
2 -0.08 (-0.26)
0.19 (0.58)
0.12 (0.39)
-0.06 (-0.17)
-0.35 (-1.07)
3 0.13 (0.40)
0.52 (1.61)
0.29 (0.93)
0.49 (1.60)
0.64 (2.06)
4 0.62 (2.11)
0.70 (2.13)
0.36 (1.08)
0.58 (1.89)
0.72 (2.36)
GH5 0.94 (3.15)
0.76 (2.54)
0.77 (2.63)
0.43 (1.43)
0.76 (2.41)
Panel B: Equal-weighted Returns GH1 -1.00
(-3.54) -0.73
(-2.57) -0.64
(-2.23) -0.60
(-2.02) -0.64
(-2.31) 2 -0.15
(-0.57) 0.12
(0.48) 0.16
(0.60) -0.06
(-0.22) -0.11
(-0.44) 3 0.29
(1.15) 0.52
(2.08) 0.25
(0.99) 0.26
(1.05) 0.38
(1.53) 4 0.68
(2.64) 0.71
(2.79) 0.50
(1.98) 0.47
(1.89) 0.55
(2.16) GH5 0.81
(3.06) 0.91
(3.63) 0.66
(2.55) 0.43
(1.69) 0.74
(2.80)
50
Table VIII
Average Monthly Returns of Portfolio Sorted on
Idiosyncratic Volatility Across Investor Sentiments
This table shows value and equally-weighted monthly returns of the quintile portfolios formed on idiosyncratic volatility (IVOL). Quintile portfolios are formed on idiosyncratic volatility each month from January 1965 to December 2010. Idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. The investor sentiment data is from Baker and Wurgler (2006) and downloaded from Wurgler’s website from July 1965 to December 2010. There are 182 months for each Low, Medium, and High sentiments. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
LOW SENTIMENT MEDIUM SENTIMENT HIGH SENTIMENT Value-
weighted Returns
Equal-weighted Returns
Value-weighted Returns
Equal-weighted Returns
Value-weighted Returns
Equal-weighted Returns
V1 0.86 1.13 0.54 0.56 1.19 1.66 2 1.28 1.60 0.62 0.57 0.96 1.53 3 1.58 1.87 0.73 0.58 0.77 1.27 4 1.86 2.11 0.95 0.74 0.31 0.90
V5 1.97 2.17 0.69 0.64 -0.34 0.04
V5-V1 1.12
(2.76) 1.04
(2.86) 0.16
(0.34) 0.08
(0.19) -1.53
(-2.55) -1.62
(-2.90)
Alpha 0.04
(0.16) 0.04
(0.23) -0.12
(-0.46) -0.17
(-0.87) -1.06
(-2.85) -1.37
(-3.98)
51
Table IX Average Monthly Returns of Portfolio Sorted on GH and Idiosyncratic Volatility
Across Investor Sentiments This table shows value and equally-weighted monthly returns of the quintile portfolios formed on GH (George and Hwang Ratio: current price/52-week high price) and idiosyncratic volatility (IVOL). Quintile portfolios are formed on GH first and then another quintile portfolios are formed on IVOL within each GH portfolio each month t from January 1965 to December 2010. Idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. The investor sentiment data is from Baker and Wurgler (2006) and downloaded from Wurgler’s website from July 1965 to December 2010. There are 182 months for each Low, Medium, and High sentiments. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
GH1 2 3 4 GH5 GH1 2 3 4 GH5
Value-weighted Returns Equal-weighted Returns
Panel A: LOW SENTIMENT
V1 1.76 1.37 0.71 0.82 0.74 1.99 1.68 1.36 1.07 0.74
2 1.98 1.45 1.03 0.92 0.72 2.27 1.71 1.56 1.32 1.05
3 1.78 1.72 1.18 1.36 1.31 2.35 1.99 1.74 1.55 1.49
4 1.91 2.05 1.56 1.71 1.74 2.14 2.24 1.93 1.74 1.80
V5 1.62 2.03 1.92 1.90 1.90 1.91 2.22 2.39 2.16 2.07
V5-V1 -0.15
(-0.31) 0.66
(1.49) 1.20
(2.84) 1.08
(2.67) 1.17
(3.36) -0.08
(-0.22) 0.54
(1.60) 1.04
(2.88) 1.09
(3.04) 1.33
(4.30)
Alpha -1.29
(-4.26) -0.35
(-1.10) 0.33
(0.90) 0.17
(0.57) 0.48
(1.59) -0.87
(-4.33) -0.17
(-0.69) 0.22
(0.78) 0.23
(0.94) 0.62
(2.61)
Panel B: MEDIUM SENTIMENT
V1 0.76 0.37 0.60 0.59 0.41 0.69 0.67 0.58 0.50 0.45
2 0.77 0.46 0.46 0.63 0.49 0.55 0.53 0.52 0.51 0.56
3 0.39 0.48 0.44 0.49 0.64 0.22 0.27 0.58 0.60 0.73
4 0.31 0.50 0.74 0.95 1.44 -0.04 0.46 0.54 0.99 1.26
V5 0.07 0.34 0.68 1.21 1.97 -0.11 0.16 0.82 1.57 1.85
V5-V1 -0.68
(-1.34) -0.03
(-0.07) 0.08
(0.17) 0.62
(1.32) 1.55
(3.35) -0.80
(-2.07) -0.51
(-1.34) 0.25
(0.65) 1.07
(2.61) 1.39
(3.52)
Alpha -0.89
(-2.40) -0.23
(-0.66) -0.16
(-0.48) 0.39
(1.22) 1.34
(4.14) -0.99
(-4.47) -0.71
(-3.09) 0.05
(0.20) 0.85
(3.05) 1.17
(4.36)
Panel C: HIGH SENTIMENT
V1 0.95 1.17 1.48 1.36 1.04 1.61 1.78 1.96 1.71 1.43
2 0.27 0.81 1.07 1.25 0.93 1.11 1.48 1.54 1.68 1.37
3 0.26 0.67 0.80 1.52 0.77 0.45 0.97 1.32 1.48 1.26
4 -0.57 0.30 0.41 1.03 0.90 0.02 0.56 1.16 1.34 1.20
V5 -1.39 -0.83 0.16 0.95 0.58 -0.93 -0.29 0.47 1.25 1.20
V5-V1 -2.34
(-4.15) -2.00
(-3.72) -1.33
(-2.35) -0.41
(-0.66) -0.47
(-0.91) -2.54
(-4.40) -2.07
(-4.42) -1.49
(-3.17) -0.46
(-0.96) -0.23
(-0.55)
Alpha -1.99
(-4.89) -1.73
(-4.70) -1.50
(-3.89) -0.45
(-1.08) -0.40
(-1.21) -2.10
(-5.51) -1.89
(-5.85) -1.73
(-5.04) -0.62
(-1.86) -0.16
(-0.57)
52
Table X
Fama-MacBeth Cross-Sectional Regressions Across Investor Sentiments Each month from January 1965 to December 2010, we run a firm-level Fama-MacBeth cross-sectional regressions of stock return in month t+1 on the lagged explanatory variables in month t. The explanatory variables include stock’s monthly idiosyncratic volatility (IVOL), monthly stock return (REV), maximum daily return (MAX), BETA, the book-to-market ratio (BTM), the log of market capitalization (Size), the holding period return from month t-12 to month t-2 (MOM), the idiosyncratic skewness (Skw), and the illiquidity measure (ILLIQ). The investor sentiment data is from Baker and Wurgler (2006) and downloaded from Wurgler’s website from July 1965 to December 2010. There are 182 months for each Low, Medium, and High sentiments. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis.
GH1 GH5 (1) (2) (3) (4) (5) (6) (7) (8) Low Sentiment High Sentiment Low Sentiment High Sentiment
IVOL -0.061 (-1.52)
-0.049 (-1.35)
-0.144 (-6.84)
-0.118 (-6.63)
0.130 (3.92)
0.142 (2.99)
-0.015 (-0.55)
-0.038 (-1.32)
REV -0.095 (-8.39)
-0.064 (-6.49)
-0.034 (-3.83)
-0.030 (-3.63)
MAX -0.130 (-3.55)
-0.186 (-7.91)
0.007 (0.30)
0.020 (1.16)
BETA 0.035 (0.81)
-0.029 (-0.55)
0.082 (1.27)
-0.058 (-0.63)
BTM 0.098 (0.70)
-0.040 (-0.27)
0.006 (0.04)
0.116 (0.85)
SIZE -0.241 (-2.92)
-0.136 (-1.54)
-0.124 (-1.73)
-0.248 (-3.52)
MOM 0.674 (2.09)
0.744 (2.67)
0.848 (4.07)
0.389 (1.65)
SKW 0.373 (3.93)
0.358 (4.09)
0.122 (2.16)
0.109 (2.78)
ILLIQ -0.002 (-1.05)
0.001 (0.26)
-0.002 (-0.48)
-0.002 (-0.49)
53
Table XI
Average Monthly Returns of Portfolio Sorted on 𝐈𝐯𝐨𝐥 and GH
This table shows value and equally-weighted monthly returns of the quintile portfolios formed on idiosyncratic volatility (IVOL) and GH (George and Hwang Ratio: current price/52-week high price). Quintile portfolios are formed on idiosyncratic volatility first and then another quintile portfolios are formed on GH within each idiosyncratic volatility portfolio each month from January 1965 to December 2012. Monthly idiosyncratic volatilities are the standard deviation of residuals from the regression of excess monthly stock returns on the contemporaneous monthly Fama-French factors using the previous 24 to 60 monthly returns (as available) each month. V1 (V5) is portfolio of stocks in the bottom (top) quintile of IVOL. GH1 (GH5) is the lowest (highest) quintile portfolio on GH. Newey-West (1978) adjusted t-statistics are reported in parenthesis. Alpha reports Fama-French three factor alpha.
Panel A : Sort on GH
Value-weighted Returns Equal-weighted Returns
GH1 0.95 0.91
2 0.90 1.08
3 0.85 1.21
4 0.97 1.28
GH5 0.80 1.20
GH5-GH1
-0.16 (0.85)
0.28
(1.39)
Alpha 0.02
(0.14)
0.58 (4.14)
Panel B : Sort on IVOL and then GH
V1 2 3 4 V5 V1 2 3 4 V5
GH1 1.01 1.14 1.02 0.88 0.19 1.37 1.43 1.28 1.03 0.41
2 1.09 0.94 1.04 0.82 0.43 1.32 1.31 1.22 1.06 0.57
3 0.76 0.76 0.84 0.83 0.53 1.08 1.20 1.22 1.25 0.83
4 0.78 1.06 1.02 1.25 1.03 0.92 1.16 1.20 1.36 1.19
GH5 0.63 0.76 1.18 1.31 1.41 0.79 0.96 1.17 1.47 1.58 GH5-GH1
-0.38 (-2.54)
-0.38 (-2.16)
0.16 (0.73)
0.43 (1.71)
1.22 (4.16)
-0.58 (-5.08)
-0.47 (-3.55)
-0.11 (-0.70)
0.44 (2.15)
1.17 (4.88)
Alpha -0.20 -0.23 0.34 0.66 1.39 -0.36 -0.22 0.17 0.69 1.41
(-1.51) (-1.48) (1.71) (2.89) (4.85) (-3.51) (-1.93) (1.32) (3.94) (6.34)
54
Table XII Fama-MacBeth Cross-Sectional Regressions
Each month from January 1965 to December 2012, we run a firm-level Fama-MacBeth cross-sectional regressions of stock return in month t+1 on the lagged explanatory variables in month t. The explanatory variables include GH (George and Hwang Ratio: current price/52-week high price), stock’s monthly , monthly stock return (REV), maximum daily return (MAX), BETA, the book-to-market ratio (BTM), the log of market capitalization (Size), the holding period return from month t-12 to month t-2 (MOM), the idiosyncratic skewness (Skw), and the illiquidity measure (ILLIQ). IVOL1 (IVOL5) is the lowest (highest) quintile portfolio on idiosyncratic volatility. Newey-West (1978) adjusted t-statistics are reported in parenthesis.
IVOL1 IVOL5 (1) (2) (3) (4)
GH -0.411 (-2.64)
-1.045 (-3.70)
1.959 (9.49)
2.939 (7.16)
REV -0.074 -0.037 (-10.05)
(-8.99)
MAX 0.004 -0.077 (0.26)
(6.20)
BETA 0.055 0.023 (1.43)
(0.71)
BTM 0.012 0.243 (0.14)
(2.91)
SIZE -0.063 -0.185 (-2.65)
(-3.25)
MOM 1.021 0.433 (4.72)
(3.98)
SKW 0.118 0.232 (5.07)
(4.77)
ILLIQ 0.001 -0.002 (0.60) (-0.49)