Nominal Price Illusion †
Justin Birru* and Baolian Wang**
May 2015
Abstract
We explore the psychology of stock price levels and provide evidence that investors
suffer from a nominal price illusion in which they overestimate the “room to grow”
for low-priced stocks relative to high-priced stocks. While it has become increasingly
clear that nominal price levels influence investor behavior, why prices matter to
investors is a question that as of yet has gone unanswered. We find widespread
evidence that investors systematically overestimate the skewness of low-priced stocks.
In the broad cross-section of stocks, we find that investors substantially overweight
the importance of price when forming skewness expectations. Asset pricing
implications of our findings are borne out in the options market. A zero-cost option
portfolio strategy that exploits investor overestimation of skewness for low-priced
stocks relative to high-priced stocks generates significant abnormal returns. Finally,
investor expectations of future skewness increase drastically on days that a stock
undergoes a split to a lower nominal price. Empirically, however, future physical
skewness actually decreases following splits.
†
The authors are grateful for valuable comments received from Yakov Amihud, Nicholas Barberis,
Brian Boyer, Kalok Chan, Zhi Da, David Hirshleifer, Nishad Kapadia, Bingxin Li, Laura Liu, Xuewen
Liu, Abhiroop Mukherjee, Sophie Ni, Kasper Nielsen, Mark Seasholes, Rik Sen, Bruno Solnik, John
Wei, Joakim Westerhold, Chu Zhang, Feng Zhao, and seminar participants at AFA 2014, Yale Whitebox
Advisors Graduate Student Conference 2013, Asian FA 2013, FMA Asia 2013, FMA Europe 2013,
FMA 2013, HKUST, HK Baptist University, The Ohio State University, Sun Yat-Sen University and
Tsinghua. We also thank Patrick Dennis for providing the computational code. All errors are the
responsibility of the authors.
*Assistant Professor, Fisher College of Business, The Ohio State University. Email:
birru.2@fisher.osu.edu.
**Assistant Professor, Fordham School of Business, Fordham University. Email:
bwang46@fordham.edu.
1
1. Introduction
The level of a firm's stock price is arbitrary as it can be manipulated by the firm
via altering the number of shares outstanding. Nevertheless, it has become clear that
nominal prices influence investor behavior. For example, individuals tend to hold
lower-priced stocks than institutions.1 Schultz (2000) finds additional evidence of
investor price-level preferences, documenting an increase in the number of small
shareholders following a split to a lower price level, while Fernando, Krishnamurthy,
and Spindt (2004) find that IPO offer price plays a strong role in determining investor
composition. Finally, Green and Hwang (2009) find particularly strong evidence that
investors categorize stocks based on price. They show that similarly priced stocks
move together; after a stock split, splitting stocks experience increased comovement
with low-priced stocks, and decreased comovement with high-priced stocks.
Firms appear to be well aware of the important role that nominal prices play in
influencing investor perceptions, as they frequently engage in the active management
of share price levels in an apparent effort to cater to investor demand. For instance,
despite the lack of a rational explanation, firms have proactively managed share prices
to stay in a relatively constant nominal range since the Great Depression (Weld,
Michaely, Thaler, and Benartzi, 2009). Baker, Greenwood, and Wurgler (2009) find
that investors have time-varying preferences for stocks of different nominal price
levels, and that firms actively manage their share price levels to maximize firm value
by catering to these time-varying investor preferences. Dyl and Elliot (2006) also find
evidence that firms manage share prices to appeal to the firm's investor base in an
effort to increase the value of the firm. The rationale for investors’ focus on nominal
1
See Gompers and Metrick (2001), Dyl and Elliot (2006), Kumar and Lee (2006), and Kumar (2009).
2
prices is not well understood, as past work has focused on the implications of these
preferences while only hypothesizing about the potential underlying drivers. In short,
while past research shows that nominal prices clearly influence the behavior of
investors, why prices matter to investors is an as of yet unanswered question.
The lack of empirical evidence has not dissuaded speculation as to why investors
are influenced by nominal prices. For example, Kumar (2009) states that “as with
lotteries, if investors are searching for ‘cheap bets’, they are likely to find low-priced
stocks attractive.” Green and Hwang (2009) hypothesize that “investors may perceive
low-priced stocks as being closer to zero and farther from infinity, thus having more
upside potential.” While Baker, Greenwood, and Wurgler (2009) state that “One
question that the results raise, and that we leave to future work, is why nominal share
prices matter to investors...Perhaps some investors suffer from a nominal illusion in
which they perceive that a stock is cheaper after a split, has more ‘room to grow’, or
has ‘less to lose’.”
There also exists a great deal of anecdotal evidence that investors believe
low-priced stocks have more room to grow. For example, a number of mutual fund
families offer “low-priced” stock funds that primarily invest in stocks trading below a
specified price per share (the cutoff for meeting the low-priced definition varies by
fund, but is typically in the $15-$35 range). 2 This is often viewed as a marketing
gimmick designed to appeal to investor psychology. 3 The notion that low-priced
2
Fidelity’s Low-priced Stock Fund has operated since 1989, with an explicit strategy of investing in
stocks priced below $35 per share. Royce’s Low-Price Fund has been in operation since 1993, with the
strategy of investing in stocks priced below $25. A now defunct low-priced fund was launched by
Robertson Stephens in 1995. Recently, Perritt launched its own version in 2012, with a strategy of
investing in stocks priced below $15.
3
Media sources that have referred to low-priced funds as gimmicks are too plentiful to cite, but include
3
stocks have more upside potential is often reinforced by the funds themselves. The
long-time manager of Fidelity’s Low-Priced Stock Fund argues that “it’s easier for a
$4 stock to go to $8 than for a $40 stock to go to $80.” 4 While the Perritt Low Priced
Stock Fund claims that stocks priced under $15 have “plenty of room to move.” 5
In this paper, we provide evidence that investors indeed exhibit a psychological
bias in the manner in which they relate nominal prices to expectations of future return
patterns. Specifically, we find evidence that investors suffer from the illusion that low
price stocks “have more upside potential.” In doing so, we identify one potential
driver of investor demand shifts that have been shown to lead to supply responses
from corporations.
In attempting to assess expectations of upside potential, the natural variable to
focus on is skewness. Empirically, we rely on the options market to extract investor
skewness expectations. A key insight of our analysis is the use of option-implied
risk-neutral skewness (RNSkew), which is a market-based ex-ante measure of
investors' expectations. By utilizing risk-neutral skewness extracted from option
prices, we are able to circumvent the need for a long time series of returns to estimate
skewness; instead we can assess how market expectations of an asset's future
skewness change on a daily basis. In doing so, we follow a number of recent papers in
The New York Times, The Washington Post, Mutual Fund Observer, Morningstar, Motley Fool, and
even the manager of Fidelity’s Low-Priced Fund: “I was an analyst at Fidelity when everyone was
asked for ideas for new funds…They accepted this one and added the low-price provision. It’s a bit of a
gimmick.” “Fidelity’s Secret Weapon.” Kiplinger’s January 2002, p 63-65.
4
“Fidelity’s Secret Weapon.” Kiplinger’s January 2002, p 63-65.
5
http://www.prnewswire.com/news-releases/perritt-capital-management-launches-perritt-low-priced-sto
ck-fund-plowx-248540111.html.
4
inferring investor expectations from option-implied skew. 6 Importantly, in our
analyses we either hold constant the firm and examine high frequency (day-over-day)
changes in RNSkew, or hold time constant while examining differences in
option-implied skew in the cross-section of firms.
Our empirical analysis consists of three tests, each of which provides independent
evidence that investors overestimate the skewness of low-priced stocks. First,
examining the entire cross-section of stocks, we find that when forming skewness
expectations, investors substantially overweight the importance of price relative to its
true observed relationship with physical skewness. Second, we document mispricing
in option portfolios that is consistent with investor overestimation of skewness for
low-priced stocks. Third, we find that investor expectations of skewness drastically
increase (decrease) on the date of a stock split (reverse split) to a lower (higher) price.
We additionally present evidence that the findings are driven by investor
expectational errors regarding the upside, but not downside potential of low-priced
stocks. Below we discuss the findings in more detail.
Our initial analysis focuses on the cross-section of all stocks. While there is a
relatively strong univariate inverse relationship between price and physical skewness,
this relationship is driven by the correlation of price with other firm characteristics.
After controlling for firm characteristics such as size, there remains no significant
relationship between price and physical skewness. However, there remains a quite
strong inverse relationship between price and RNSkew even after controlling for firm
characteristics. That is, in forming expectations of skewness, investors overweight the
importance of price relative to its observed relationship with physical skewness (or
6
See Dennis and Mayhew (2002), Han (2008), Bali and Murray (2013), and Conrad, Dittmar, and
Ghysels (2013), among others.
5
rational model-predicted expected skewness measures that we also employ in our
analysis). Investors appear to mistakenly extrapolate the observed univariate
relationship between price and physical skewness when pricing options. The evidence
supports the idea that investors overestimate the lottery-like properties of low-priced
stocks.
Utilizing option open interest and volume data, we also find evidence that
investors display increased optimism toward low-priced stocks. Specifically, the ratio
of call to put open interest and volume is substantially higher for low-priced stocks
than it is for high-priced stocks. Past work shows investor preferences for lottery-like
assets (e.g., Barberis and Huang, 2008; Kumar 2009; Boyer, Mitton, and Vorkink,
2010; Bali and Murray, 2013; and Boyer and Vorkink, 2014); we build upon this
evidence by showing that investors also have a preference for utilizing the leverage
benefits options provide to take lottery-like bets on these lottery-like stocks.
We complement the cross-sectional results by undertaking a second, separate test
of the hypothesis that investors suffer from a nominal price illusion. Specifically, we
explore the asset-pricing implications of investor biased beliefs regarding nominal
prices. Overestimation of expected skewness for low-priced stocks relative to
high-priced stocks suggests the potential overpricing of a portfolio of OTM calls on
low-priced stocks relative to a similar portfolio of options for high-priced stocks.
Following Bollen and Whaley (2004) and Goyal and Sarreto (2009) we estimate the
returns to option portfolios and find that the overpricing of call options increases as
underlying stock price decreases. The results are consistent with relative investor
overestimation of skewness for low-priced stocks relative to high-priced stocks. To
ascertain whether RNSkew changes are driven by investor beliefs regarding upside
6
potential, downside potential, or some combination of both, we examine put and call
returns separately. We find the underlying stock price is only related to the returns of
OTM calls, but not OTM puts, consistent with the effect coming from biased investor
expectations regarding the upside potential rather than downside potential of the stock.
We later discuss additional tests that further support the explanation. As an added
benefit, the option portfolio methodology does not rely upon extracting RNSkew from
option prices and therefore also serves as a robustness check of the earlier results.
Finally we explore investor expectations around stock splits, as this is a setting
with few, if any, confounding influences, and therefore allows the ability to examine
the effects of large, exogenous price changes in a relatively clean setting. 7 We first
find that investor expectations of skewness drastically increase on the day that a stock
splits to a lower price level. On the day of a split to a lower price we find that
skewness expectations (RNSkew) increase by over 40%. In sharp contrast to this
increase reflecting a rational increase in expected skewness, we find a substantial
physical skewness decrease following stock splits. This is not surprising given that
splits occur after a long run-up in price, and therefore periods of high past skewness,
making the observed expected skewness increase all the more surprising. Importantly,
we find no such increase in RNSkew on the day of the stock split announcement. The
increase of RNSkew around the ex-date rather than the announcement date is
7
Many potential motives for stock splits have been suggested and explored, including signaling
(Brennan and Copeland, 1988; McNichols and Dravid, 1990; and Ikenberry, Rankine, and Stice, 1996),
and liquidity arguments (Muscarella and Vetsuypens, 1996; and Angel, 1997). However, splits do not
seem to be correlated with future corporate profitability (Lakonishok and Lev 1987; and Asquith,
Healy, and Palepu, 1989), nor is it clear that splits increase liquidity (Conroy, Harris and Benet, 1990;
Schultz, 2000; and Easley, O’Hara, and Saar, 2001). In contrast to information or microstructure
motives, the prevailing view is that firms split their shares to return prices to a normal trading range
(Baker and Gallagher, 1980; Lakonishok and Lev, 1987; Conroy and Harris, 1999; Dyl and Elliot, 2006;
and Weld, Michaely, Thaler, and Benartzi, 2009), and thus provide a clean laboratory to examine the
effect of nominal prices on investor expectations. In addition, by examining the effect of the
information free ex-date, our tests allow us to exclude any potential signaling effect.
7
consistent with investors reacting only to the change in stock price, and inconsistent
with an information signaling story. We find similar evidence of investor
expectational errors in a much smaller sample of reverse splits; on the date of a
reverse split RNSkew decreases drastically. In contrast to investor expectations, future
physical skewness actually increases. The evidence is consistent with investors
assigning greater upside potential (and/or lower downside potential) to stocks trading
at lower prices.
As a further effort to ascertain whether the effect is driven by investor beliefs
regarding upside or downside potential, we also examine changes in implied
volatilities for calls and puts around the split. Again, consistent with biased investor
expectations reflecting biased beliefs regarding upside potential, rather than downside
potential, we find that the difference between out of the money and at the money
implied volatility substantially increases for call options after a split, suggesting that
out of the money calls become more expensive. There is no accompanying change in
this difference for put options. The evidence suggests that the skewness results do in
fact reflect investors assigning greater upside potential to stocks trading at lower
prices, with no accompanying change in beliefs regarding downside potential. Overall,
the evidence is consistent with investors suffering from a nominal price illusion in
which they overestimate the “cheapness” or “room to grow” of low-priced stocks
relative to high-priced stocks.
The paper proceeds as follows. Section 2 discusses methodology, and introduces
the data. Section 3 presents evidence of investor nominal price biases in the
cross-section of stocks. Section 4 examines option trading. Section 5 assesses
8
asset-pricing implications of investor nominal price bias. Section 6 presents the stock
split analysis, and Section 7 concludes.
2. Risk-neutral skewness and data
2.1. Risk-neutral skewness
In order to examine whether nominal share price is systematically related to
investors’ misperception of skewness, we need measures of both investor expected
skewness as well as rational unbiased measures of expected skewness. We use
risk-neutral skewness implied from option prices to capture investor expectations of
skewness. We employ two primary measures of unbiased expected skewness. The
first measure we utilize is ex-post realized skewness (Skew) which we calculate using
daily return data over a one-year period. As a second measure, we employ the
expected skewness measure (E(Skew)) of Boyer, Mitton, and Vorkink (2010) which
incorporates all relevant past and current information in order to formulate a best
prediction of future skewness. We use the model-free methodology of Bakshi,
Kapadia, and Madan (2003) 8 to measure risk-neutral skewness (RNSkew).
Risk-neutral skewness is a prominent variable in our analysis, utilized to capture
changing investor expectations of asymmetry in return distributions. Because the
option prices from which risk-neutral moments are extracted are updated daily, they
reflect an up-to-date measure of investor ex-ante expectations.
Bakshi, Kapadia, and Madan (2003) show that the risk neutral skewness is
8
The model-free risk neutral measure of Bakshi, Kapadia, and Madan (2003) has been widely used in
the literature (Dennis and Mayhew, 2002; Han, 2008; Bali and Murray, 2013; and Conrad, Dittmar, and
Ghysels, 2013, among others).
9
RNSkewi ,t (t ) =
{
EtQ ( R(t ,t ) − EtQ [ R(t ,t )])3 }
{E (R(t,t ) − E [R(t,t )]) }
Q
t
2 3/2
Q
t
rt
rt
e Wi ,t (t ,t ) − 3µi ,t (t ,t )e Vi ,t (t ,t ) + 2 µi ,t (t ,t )
=
,
(1)
3
[e rt Vi ,t (t ,t ) − µi ,t (t ,t ) 2 ]3/2
where i, t, and τ represent stock, current time, and time to maturity, respectively. r is
the risk free rate. EtQ (.) is the expectation under the risk-neutral measure. R (t ,t ) is the
return from time t to t + t , and µi ,t (t ,t )=
e rt − 1 −
e rt
e rt
e rt
Vi ,t (t ,t ) −
Wi ,t (t ,t ) −
X i ,t (t ,t ) .
2
6
24
Bakshi,
Kapadia, and Madan (2003) further show that, Vi ,t (t ,t ) , Wi ,t (t ,t ) and X i ,t (t ,t ) can be
extracted from OTM options, and are defined as
{
Q
=
Vi ,t (t ,t ) E=
e − rt R(t ,t ) 2 }
t
+∫
S (t )
0
∫
∞
S (t )
2(1 − ln[ K / S (t )]
C (t ,t ; K )dK
K2
2(1 + ln[ K / S (t )]
P(t ,t ; K )dK ,
K2
(2)
6 ln[ K / S (t )] − 3(ln[ K / S (t )]) 2
Wi ,t (t ,t ) E=
e R(t ,t ) } ∫
C (t ,t ; K )dK
=
S (t )
K2
Q
t
{
− rt
-∫
S (t )
0
3
6 ln[ K / S (t )] + 3(ln[ K / S (t )]) 2
P (t ,t ; K )dK ,
K2
{
Q
=
X i ,t (t ,t ) E=
e − rt R(t ,t ) 4 }
t
+∫
S (t )
0
∞
(3)
12 ln[ K / S (t )]2 − 4(ln[ K / S (t )])3
C (t ,t ; K )dK
∫S (t )
K2
∞
12 ln[ K / S (t )]2 + 4(ln[ K / S (t )])3
P(t ,t ; K )dK .
K2
(4)
Ideally, Vi ,t (t ,t ) , Wi ,t (t ,t ) and X i ,t (t ,t ) should be calculated based on a continuum
of European options with different strikes. However, in reality, only a limited number
of options are available for each stock/expiration combination and individual equity
options are not European. To accommodate the discreteness of options strikes, we
10
follow Dennis and Mayhew (2002) to estimate the integrals in expressions (2) to (4)
using discrete data. 9
Price per se should not be mechanically related to RNSkew since RNSkew is
homogeneous of degree zero with respect to the underlying price, that is, altering the
underlying price will increase or decrease the numerator and denominator of equation
(1) by the same proportion. However, options for stocks with different price may have
different strike structures, potentially imposing a systematic bias to the calculation of
RNSkew.
Dennis and Mayhew (2002) examine two potential sources of bias in RNSkew
estimation. The first arises due to the use of discrete strike prices, and the second
arises from the potential asymmetry in the domain of integration. Dennis and Mayhew
(2002) show that the bias in RNSkew is negative and increasing in absolute
magnitude when the relative option strike interval (option strike increment/underlying
stock price) increases. 10 In practice, standard stock option strike prices are in
increments of $2.50 for strikes at or below $25, $5.00 for strikes above $25 but below
$200, and $10 for strikes above $200. However, Dennis and Mayhew (2002) show
that the bias in RNSkew induced by the option strike interval is quite small. The bias
is approximately -0.01, -0.05 and -0.07 when the relative option strike intervals
(option strike incremental/underlying stock price) are 2%, 5% and 10%, respectively.
Dennis and Mayhew (2002) also investigate the potential bias arising due to an
asymmetric domain of integration. They show that RNSkew will be biased upward
when there is a lesser number of OTM puts relative to OTM calls and will be biased
9
10
We thank Patrick Dennis for providing us the code.
Dennis and Mayhew (2002) use simulations to evaluate the bias of option discreetness. Specifically,
they choose the underlying stock price to be $50 and evaluate the magnitude of bias induced by option
strike increments from $1 to $5.
11
downward when there is a greater number of OTM puts relative to OTM calls.
However, Dennis and Mayhew (2002) show that the bias is essentially zero if there
are at least two OTM puts and two OTM calls. As a result, we require at least two
OTM put options and at least two OTM call options. 11 Finally, we standardize
RNSkew to 30 days by linearly interpolating the skewness of the option with
expiration closest to, but less than 30 days, and the option with expiration closest to,
but greater than 30 days. If there is no option with maturity longer than 30 days
(shorter than 30 days), we choose the longest (shortest) available maturity. 12
2.2. Data
IvyDB’s OptionMetrics database provides data on option prices, volume, open
interest, implied volatility (IV) and Greeks for the period from January 1996 to
December 2012. IVs and Greeks are calculated using the binomial tree model of Cox,
Ross, and Rubinstein (1979). We include options on all securities classified as
common stock. To minimize the impact of data errors, we remove options missing
best bid or offer prices, as well as those with bid prices less than or equal to $0.05. We
also remove options that violate arbitrage bounds, options with zero open interest, and
options for which we cannot calculate RNSkew or Skew. Unless stated explicitly, we
also exclude options with special settlement arrangement, 13 and options for which the
11
In the cross-sectional analysis that follows, our sample has an average of four OTM puts and four
OTM calls for each stock analyzed. The depth of the option chain is also similar in our sample for puts
and calls. The average maximum (minimum) call delta is 0.318 (0.098). The average minimum
(maximum) put delta is -0.311 (-0.09).
12
All results are robust to the use of 60-day or 100-day skewness.
13
OptionMetrics defines an option as having a standard settlement if 100 shares of the underlying
security are to be delivered at exercise and the strike price and premium multipliers are $100 per tick.
For options with a non-standard settlement, the number of shares to be delivered may be different from
100, and additional securities and/or cash may be required.
12
underlying stock price is lower than $10. 14 The mid-quote of the best bid and best
offer is taken as the option price. Data on stocks is from Center for Research in
Security Prices (CRSP). The stock split data used later in the study is obtained from
the CRSP distribution file. We define stock splits as events with a CRSP distribution
code of 5523. Specifically, we define regular splits as those with a split ratio of at
least 1.25 to 1, and reverse splits as those with a ratio below 1. We obtain company
accounting information from Compustat.
For both the large cross-sectional sample (all optionable stocks) and the stock split
sample, we only include observations for which we are able to calculate RNSkew and
future Skew. The full stock sample includes 203,974 firm-month observations. To
mitigate the effect of outliers, we winsorize all continuous variables at the 0.5% level.
3. Nominal price and skewness: All optionable stocks
3.1. Characteristics of stocks with different nominal prices
In this section we test our main hypothesis that investors suffer from a nominal
price illusion. We do so by examining the entire cross-section of stocks. We first
explore the relationship between nominal price and skewness in the cross-section of
stocks, and then examine whether this is consistent with the relationship between
price and RNSkew that investors price into options. As we are interested in isolating
only the effect of nominal price on investor behavior, we first examine how price is
correlated with firm characteristics. Table 1 reports the summary statistics of stocks
that are sorted into price quintiles based on stock prices at the end of month t-1.
14
We exclude stocks with prices less than $10 to mitigate the effects of microstructure noise. This
excludes 6.7% of the sample. As we show in the Online Appendix, our results are qualitatively
similarly if we instead include all stocks.
13
On average, our sample stocks have a large dispersion in stock price. The average
stock price for the lowest quintile is 14.505 while the average price for the highest
quintile is 74.106. Relative to high priced stocks, low priced stocks are smaller, have
higher betas, lower book-to-market ratios, worse past performance and are more likely
to list on NASDAQ. Table 1 also shows that low priced stocks have higher past
volatility, lower past skewness, and lower trading volume and liquidity.
[Insert Table 1 here]
The last rows of Table 1 examine the relationship between price and our main
measures of skewness. Our first measure of unbiased skewness is future physical
skewness calculated from daily returns from month t+1 to month t+12. 15, 16 We also
utilize a measure of expected skewness (E(Skew)). E(Skew) is a forward looking
measure of skewness first introduced by Boyer, Mitton, and Vorkink(2010) that
incorporates all relevant past information to best form a future prediction of skewness.
This methodology utilizes the parameters from a cross-sectional regression of
skewness on lagged firm characteristics in order to estimate expected skewness.
Specifically, we use the following model specification to construct our E(Skew)
variable,
E(Skew t+1,t+12 ) = α + β 1 Log(Price t-1 ) + β 2 Beta t-1 + β 3 Log(Size t-1 ) + β 4
Log(B/M t-1 ) + β 5 Leverage t-1 + β 6 Volatility t-12,t-1 + β 7 Skewness t-12,t-1 + β 8 R t-12,t-1 +
15
In the reported results, we calculate volatility and skewness from the raw returns. Moments of raw
returns are likely to be affected by outliers. Therefore, we also verify that the results hold if we
calculate volatility and skewness based on logged returns. The results are robust to this specification,
and are available in the Online Appendix.
16
The results are also robust to using physical skewness calculated using only a 30-day horizon. These
results can be found in the Online Appendix.
14
β 9 R t-60,t-13 + β 10 Log(Volume t-1 ) + β 11 Log(Illiq t-1 ) + β 12 NASDAQ t-1 +Industry Fixed
Effects+ ε.
(5)
Both future realized skewness and expected skewness from the model of Boyer,
Mitton, and Vorkink (2010) are decreasing in nominal stock price. 17 However, the
magnitude of the cross-sectional relationship between each of these measures and
price is much smaller than that between price and investor expectations reflected in
RNSkew. To see this, note that the unconditional variance of realized skew is actually
larger than that of the RNSkew measure (Table 7), however, conditional on price, the
difference in RNSkew between the top and bottom price quintiles is more than twice
that of the difference for future realized skew or E(Skew). The magnitude of the
univariate relationship between price and RNSkew provides preliminary evidence that
investors perceive the relationship between price and future skewness to be much
stronger than the true ex-post relationship realized in the data, as well as much
stronger than is predicted by a rational model of expected skewness incorporating all
relevant current and past information. While the univariate evidence is consistent with
the notion that investors overweight the informativeness of price when predicting
future skewness, we next employ a multivariate analysis to control for firm
characteristics potentially correlated with both skewness and price.
3.2. Nominal price and skewness: Fama-MacBeth regressions
We use Fama-MacBeth regressions to analyze the cross-sectional relationship
between our various skewness measures and price, while controlling for a number of
variables that the literature finds to be important in explaining skewness. Motivated
17
All of our results are robust to using size tercile dummies as in Boyer, Mitton, and Vorkink (2010).
Results are available in the Online Appendix.
15
by Dennis and Mayhew (2002), we include beta to control for systematic risk. Beta is
calculated using the past 60 months of excess return data.
The leverage effect predicts that decreases in equity value will result in volatility
increases that are larger than the decreases in volatility that occur after increases in
equity value. This asymmetry implies that the implied volatility of out-of-money puts
is higher than the implied volatility of out-of-money calls. While existing empirical
findings do not support the leverage effect (e.g., Dennis and Mayhew, 2002; Bakshi,
Kapadia, and Madan, 2003), we nevertheless include leverage in the model.
We also control for firm size, past skewness, volume, R t-12,t-1 , R t-60,t-13 , and a
dummy variable indicating whether a firm is listed on NASDAQ. Chen, Hong, and
Stein (2001) motivate the inclusion of firm size, past volatility, R t-12,t-1 and volatility,
as they find that each of these variables is significantly correlated with future
skewness. 18 We also include past skewness in the model, as Boyer, Mitton, and
Vorkink (2010) show that skewness is persistent. Finally, following Boyer, Mitton,
and Vorkink (2010) we include a dummy variable for NASDAQ firms.
Past work typically finds a positive relationship between RNSkew and valuation
in the cross-section. For example, Bali and Murray (2013), Conrad, Dittmar, and
Ghysels (2013), and Boyer and Vorkink (2014) find that RNSkew is negatively
related to future asset returns, and Friesen, Zhang, and Zorn (2012) explicitly
document that low B/M is associated with higher RNSkew in the cross-section. While
this positive documented relationship between valuation and RNSkew goes in the
opposite direction of our predicted negative relationship between price and RNSkew,
18
Boyer, Mitton, and Vorkink (2010) find that allowing for a nonlinear relationship between firm size
and skewness can fit the data better. They do so by using two dummy variables indicating small firms
and medium-sized firms based on the NYSE breakpoints. Our results are similar if we adopt their
non-linear methodology (see Table A5 of the Online Appendix).
16
we nevertheless control for multiple measures of valuation to isolate the effect of
nominal price from valuation. Specifically, we include both book-to-market and
assets-to-price per share. 19 We also include industry fixed effects to control for
industry heterogeneity. Industry definitions are based on the Fama-French 48 industry
classification scheme.
[Insert Table 2 here]
Table 2 reports the results of monthly Fama-MacBeth regressions of our different
measures of skewness regressed on the beginning of period independent variables.
Standard errors are corrected for heteroskedasticity and autocorrelation up to 12 lags
(Newey and West, 1987). We separately examine RNSkew, Skew, RN skew premium
(RNSkew-Skew), and the RN expected skew premium (RNSkew-E(Skew)). 20 For each
dependent variable, we analyze three different models: a univariate specification that
includes only log nominal price, a multivariate specification including all control
variables, and finally a multivariate specification that also includes industry fixed
effects.
As expected, the first three columns of the table show that there is a strong inverse
relationship between price and RNSkew. The fourth column shows that there exists a
univariate relationship between price and Skew, albeit one which is weaker than the
relationship between price and RNSkew. However, after controlling for other
firm-level determinants of skew, price ceases to be significant at the 10% level in
explaining Skew and the magnitude of the effect diminishes by over 66%. While there
19
We have also used earnings/price ratio and sales/price ratio (both measured in natural logarithm) as
additional controls for valuation. The results are similar with these other measures of valuation.
20
Our measure of skew premium is analogous to the manner in which the literature measures variance
risk premium. However, the relationship between RNSkew and Skew may not be linear. We address
this in a more structural way later in this section.
17
is a significant univariate relationship between price and physical skew, this
relationship is driven by the correlation of price with other firm characteristics. Once
other firm characteristics are accounted for, price no longer has an economically or
statistically significant relationship with physical skew. However, the relationship
between price and RNSkew is quite strong, even in the presence of controls for the
other firm characteristics. The results suggest that when pricing options, investors
mistakenly extrapolate the observed univariate relationship between price and
physical skew, and fail to realize that this relationship is driven by other factors.
In terms of economic magnitude, from column 3, a 100% increase in price is
associated with a 0.151 (0.693*0.218) decrease in RNSkew, which is equal to 16.9%
of one standard deviation of the RNSkew variable. In contrast, from column 6, a 100%
increase in price is associated with a 0.034 (0.693*0.049) decrease in Skew, which is
equal to only 2.5% of one standard deviation of the Skew variable and is not
statistically significant. The remaining columns show that the difference in this
relationship is statistically and economically significant for RNSkew relative to Skew
and the E(Skew) measure of Boyer, Mitton, and Vorkink (2010). The multivariate
results confirm the earlier univariate results. The negative coefficient implies that the
lower is price, the larger is RNSkew relative to Skew or E(Skew), and this difference
shrinks as price increases. In short, investors seem to substantially overestimate the
skew of low-priced stocks. Expectations of future return distribution asymmetry are
biased, as investors allow price to play an irrationally large role in the shaping of
beliefs.
Aside from price, five other variables enter significantly and in a consistent
direction in explaining both the RN skew premium and the RN expected skew premium.
18
Specifically, stocks that are small, low book-to-market, low leverage, low past skew
stocks, or high long-run return have higher RNSkew relative to expected skew or
future realized skew. High RNSkew relative to physical skew may reflect
expectational errors but also potentially indicates a lower risk premium. If high
book-to-market stocks are riskier than low book-to-market stocks, the negative
coefficient for book-to-market may be a reflection of a book-to-market risk premium.
Leverage is negatively related to RNSkew while having a positive relationship with
Skew. This is potentially consistent with the leverage effect discussed above. Large
firm size predicts low RNSkew, but does not have predictive power for future Skew.
High past Skew predicts high future Skew in the cross-section, while having predictive
power of the opposite sign for RNSkew. A risk premium based interpretations would
need to argue that larger firms and more positively skewed firms are riskier than
smaller firms and less positively skewed firms. While we cannot completely exclude
these possibilities, it is not consistent with most of the existing studies.
4. Nominal price and option trading
The evidence presented thus far is consistent with investors overestimating the
lottery-like qualities of low-priced stocks. Mitton and Vorkink (2007), and Barberis
and Huang (2008) incorporate investor preferences for skewness into models
explaining stock returns, while Kumar (2009) finds empirical evidence that retail
investors prefer stocks with lottery features. Given the past evidence and general view
that investors exhibit a preference for skew, and the evidence presented here that
investors perceive low-priced stocks to have high skew, we next examine investor
trading behavior in the options market regarding low-priced stocks. We find evidence
19
that investors exhibit increased optimism, as well as gambling-like behavior toward
these lottery-like assets. We do so by examining the ratio of call to put volume and
open interest.
The ratio of call to put option trading volume is commonly regarded as a
sentiment measure, with more call option trading volume indicating optimism (see for
example, Lemmon and Ni, 2014). Given the documented investor preference for skew,
if price does affect investor perceptions of upside potential, we should expect to see a
negative correlation between stock price and the call-put volume ratio. Option volume
reflects both option writing and position closing. Open interest measures the total
existing position, and thus can potentially better measure investor beliefs. Thus, we
also examine the relationship between nominal price and the call to put open interest
ratio.
We define our volume ratio (VolRatio) and open interest ratio (OIRatio) as
VolRatio=log (1+ call option volume)-log (1+ put option volume),
OIRatio=log (1+ call option open interest)-log (1+ put option open interest).
We add one to deal with instances of zero trading volume or open interest.
[Insert Table 3 here]
The results in Table 3 show that price is strongly negatively related to the
call-to-put volume ratio and open interest ratio. That is, investors are more optimistic
about low-priced stocks, and take more lottery-like bets on low-priced stocks. The
evidence is consistent with investors perceiving low-priced stocks to be lottery-like
and with investors possessing more optimistic perceptions of the upside potential of
low-priced stocks relative to high-priced stocks. We next examine the asset-pricing
implications of investor biases regarding nominal prices.
20
5. Asset pricing implications
Expectational errors regarding the upside potential of low relative to high-priced
stocks will also have asset pricing implications. One implication of investor
overestimation of skewness of low-priced relative to high-priced stocks is that call
options will be more overvalued for low-priced stocks than for high-priced stocks. A
portfolio taking a long position in calls of high-priced stocks and a short position in
calls of low-priced stocks should therefore make positive returns. This will be
particularly true for OTM options. To test this hypothesis we construct both
delta-hedged and non-hedged call portfolios and examine whether call portfolio
returns are systematically related to underlying stock price. Our methodology and
analysis largely follows that of Goyal and Saretto (2009). This analysis also doubles
as a robustness check of the earlier results relying on RNSkew.
To be clear, the purpose of this analysis is to illustrate the existence of mispricing
in the options market in a manner that is consistent with investor overestimation of
skew for low-priced stocks. As a means to this end, we utilize a portfolio strategy as a
tool to identify whether mispricing exists. In undertaking this analysis, we measure
option prices as the midpoint of the bid and ask quotes. Whether the mispricing can be
profitably exploited in practice once option trading costs are considered is a separate
question, and is beyond the scope of this paper.
Portfolios are formed on the expiration Friday (or the previous trading day if
Friday is a public holiday) of the month, and the option portfolio strategies are
initiated on the first trading day (typically a Monday) after the expiration Friday of the
month. On each portfolio formation day, we sort all stocks with available options into
21
quintiles based on the stock price on the portfolio formation day, and choose only
options expiring within one month. For each option series, we construct portfolios
which are held until option expiration. All the portfolios are equal weighted. As in
Goyal and Saretto (2009), we use the absolute position value as the reference
beginning price to calculate portfolio return. For delta-hedged portfolios we use the
following formula to calculate returns, 21
T
Rcall =
(cT − c0 ) − D 0 ( ST + ∑ Dt e r (T −t ) − S0 )
t =1
.
| D 0 S0 − c0 |
(6)
Relative to ATM and ITM options, OTM options better reflect investor skewness
beliefs. Thus, we focus on the OTM options. As a comparison, we also report results
for ATM options. We define option moneyness following Bollen and Whaley (2004).
ATM options are defined as call options with delta greater than 0.375 and not greater
than 0.625. OTM options are call options with delta above 0.02 and not greater than
0.375. Options with absolute delta below 0.02 are excluded due to the distortions
caused by price discreteness.
[Insert Table 4 here]
Table 4 reports the results of the portfolio return analysis by price quintile. Panel
A reports monthly average returns for delta-hedged portfolios, while Panel B displays
results for non-delta hedged portfolios. In addition to reporting the raw return of the
delta-hedged option portfolios, we follow Goyal and Saretto (2009), and also report
risk-adjusted returns using model (7).
R put ,t − Rcall =
a + β ' Ft + ε t
21
(7)
We consider stock splits and use the adjustment factor given by OptionMetrics for the adjustment.
22
To obtain risk-adjusted returns, we regress the portfolio returns on a linear pricing
model consisting of the three Fama-French factors and the momentum factor (Fama
and French, 1993, Carhart, 1997), 22 and an additional aggregate factor reflecting the
average call return of S&P 500 index options. The average call return of S&P 500
index options may capture the compensation to jump risk (Pan, 2002). We use the
same method (Equation (6)) to calculate the average call return of S&P 500 index
options by moneyness. The call index option returns are matched to the call individual
stock option returns with the same moneyness. The intercept from the regression can
be interpreted as mispricing relative to the factor model. We refer to the adjusted
return as the five-factor adjusted return.
For both delta and non-delta hedged portfolios, the portfolio return increases when
moving from the lowest to highest price quintile. In Panel A, the call returns for the
delta-hedged sample increase from -2.901% in the lowest price quintile to -0.908% in
the highest price quintile. The five-factor adjusted call return increases from -2.02%
in the lowest price quintile to 0.368% in the highest price quintile. The high minus
low portfolio raw return and the five-factor adjusted return is 1.993% and 2.387%,
respectively. Both are statistically significant at 1% level. The results indicate that
OTM calls are substantially more overpriced for low-priced stocks than for
high-priced stocks. We also find some supportive evidence from ATM options. For
ATM options, the raw and five-factor adjusted return of the call portfolio increases as
a function of the underlying stock price. The magnitude of the highest minus lowest
price quintile difference in call portfolio returns is larger for OTM options than for the
ATM options, consistent with the view that skewness misperception most greatly
22
We compound the daily factor return to get monthly factor returns to match the timing of the option
strategy.
23
affects the price of the OTM options. The patterns are similar and of larger magnitude
for the non-delta hedged analysis in Panel B.
[Insert Table 5 here]
Lastly, we attempt to isolate the effect of price on returns to examine whether
other factors might be driving the observed relationship between price and option
returns. To examine this question we rely on Fama-MacBeth regressions to control for
other variables the literature finds to affect option pricing. Columns 1-3 of Table 5
confirm the positive relationship between OTM call option returns and price
documented in Table 4.
In Table 5 we also test two further predictions of our main hypothesis. First, if the
relationships documented thus far are driven by investor overestimation of upside
potential and not downside potential, then we should expect to only see an effect for
calls, but not for puts. In columns 4-6 we examine whether there is a relationship
between put returns and price. The put portfolios are formed with OTM options,
defined as those with delta greater than 0.375 and not greater than 0.02. Columns 4-6
of Panel A and Panel B in Table 5 confirm that there is no significant relationship
between put portfolio returns and the price of the underlying, consistent with the
effects documented being driven by investor overestimation of upside potential, rather
than downside potential.
A further implication of investor overestimation of skewness of low-priced
relative to high-priced stocks is that relative to put options, call options will be more
overvalued for low-priced stocks than for high-priced stocks. We formally test this in
columns 7-9, and confirm that this difference is increasing in price and that this
relationship is statistically significant. Importantly, the results confirm that the
24
abnormal returns we find are completely driven by the mispricing of calls, and not
puts. After controlling for other variables which are known to be related to option
returns, we find that OTM call returns are increasing in the nominal price of the
underlying stock, while OTM put returns are unrelated to nominal price. This is
consistent with investors overestimating the upside potential of low-priced stocks.
[Insert Table 6 here]
While the results are consistent with investor overestimation of skew for
low-priced stocks leading to overpricing of low-priced OTM call options, the tests
cannot definitively attribute overpricing of low-priced OTM calls to investor
overestimation of skew. To further support a causal story, we next examine a
double-sort of returns on both price and RNSkew. If investor overestimation of skew
for low-priced stocks is driving our results, we would expect to see the effect show up
most strongly in the high RNSkew quantiles. Specifically, options of low-priced
stocks with high RNSkew are potentially overvalued, while options of high-priced
stocks with high RNSkew are less likely to be overvalued according to our story. On
the other hand, within the low RNSkew quantile we are unlikely to see substantial
overestimation of skew, and therefore would not expect to see as large of an effect.
We utilize independent sorts to ensure that the sample of low and high-priced stocks
and spread in high and low price is similar across RNSkew terciles and is consistent
with the analysis in Table 4. Table 6 examines this hypothesis. Consistent with the
above arguments, we find that the difference in OTM call returns between top and
bottom price quintiles is large and statistically significant for the highest RNSkew
tercile and smaller and insignificant for the lowest RNSkew tercile. While not proving
causality, the results are consistent with a causal story.
25
A further question is whether the abnormal returns that this strategy generates
result from irrational beliefs on the part of investors, or whether options on low-priced
stocks are more difficult for dealers to hedge, and therefore trade at a premium. A
dealer hedging story is not consistent with our finding of an effect only for call
options on low-priced underlying stocks, but not put options, as it cannot explain why
only call options on low-priced stocks, but not put options on low-priced stocks would
be difficult to hedge. 23 The evidence presented here is consistent with the effects we
see being driven by biased beliefs regarding the upside potential, but not downside
potential of the stock. In section 6 we provide additional evidence that investor biased
beliefs are driven solely by biased beliefs regarding upside potential and not downside
potential.
6. Nominal price and skewness: stock splits
In this section we examine the effect of nominal price on investor skewness
expectations in a setting where nominal price changes are seemingly exogenous to
changes in expectations of future return distributions. The prevailing view is that
stock splits are motivated by an effort to return prices to a normal trading range
(Baker and Gallagher (1980), Lakonishok and Lev (1987), Conroy and Harris, (1999),
Dyl and Elliot (2006), and Weld, Michaely, Thaler, and Benartzi (2009)). Regardless
of the rationale for splitting, the ex-date is information free, as any potential
information revelation occurs at the announcement date. Stock splits therefore provide
23
One hedging story that does predict an asymmetry in hedging costs is put forth by Ofek, Richardson,
and Whitelaw (2004). Shorting costs, particularly the fact that short sales constraints are more binding
for low-priced stocks, will make hedging OTM puts more difficult relative to OTM calls, particularly
for low-priced stocks. This effect, however, works in the opposite direction of our results.
26
a clean environment to examine the effect of nominal price changes on investor
expectations. 24
[Insert Figure 1 here]
Figure 1 provides a preview of the main split results. RNSkew is plotted against
days relative to ex-date. From Figure 1, it is evident that stocks undergoing splits
experience a large jump in RNSkew on the ex-date, while those undergoing reverse
splits experience a large decrease in RNSkew that again occurs precisely on the
ex-date. 25
6.1. Skewness around regular splits
Table 7 displays summary statistics for the full sample of stocks alongside
summary statistics for the split sample and the reverse split sample. 26 The regular split
sample has 1,940 observations, and the reverse split sample has 130 observations. 27
The average pre-split price of regular splits is 79.982, which is more than double the
price of an average optionable stock. The average split ratio is 1.985. The average
post-split stock price is 39.534. The average pre-split price of a stock undergoing a
reverse split is 10.984, which is much lower than the average stock price. The average
split ratio is 0.258, resulting in an average post-split price of 38.534.
Not surprisingly, relative to the large sample, regular splits are larger, have higher
market valuation ratios (lower B/M), and higher past performance (both R t-12,t-1 and
24
Recent papers by Green and Hwang (2009) and Baker, Greenwood, and Wurgler (2009) also use
stock splits as an instrument to test behavioral theories, arguing that the lack of a relationship between
splits and firm fundamentals allows for a particularly clean experimental setting.
25
The confidence intervals are quite narrow, especially for regular splits. In the Online Appendix, we
show a version of Figure 1 with confidence intervals.
26
Please refer to Table A1 in the Appendix for detailed definitions of all variables.
27
Unsurprisingly, a significant number of stocks in the reverse split sample have prices lower than $10.
We include options on these stocks in the reverse split analysis.
27
R t-60,t-13 ), while reverse splits are smaller, have lower market valuation ratios, and
lower past performance. Furthermore, regular splits have similar past volatility as
compared to the larger sample, while the past volatility of reverse splits is much
higher than the average optionable stock. The average RNSkew for the full sample,
regular split sample, and reverse split sample is -0.634, -0.732 and 0.553, respectively.
Future physical skewness (Skew) of these three groups of stocks is 0.231, 0.163 and
0.487, respectively.
[Insert Table 7 here]
Panel A of Table 8 more rigorously examines whether risk neutral skewness
expectations are affected by split-induced changes in nominal price. Panel A1
explores changes in risk neutral skewness around the ex-date. The results indicate that
investor expectations are greatly affected by the nominal change in price. RNSkew
increases from -0.732 to -0.464 on the day of the stock split. The effect is
economically large and statistically significant, and persists in the weeks and months
after the split. That investors respond immediately to a split-induced change in price is
not unexpected, as substantial investor ex-date responses have been documented in
the equity literature. Schultz (2000) finds that small shareholders are very active on
the day of a split. He documents a large and immediate increase in small shareholders
at the ex-date, as net small trade buy volume increases from slightly above zero in the
day prior to the split to about two million shares on the ex-date. Furthermore, ex-date
price responses are also seen in the equity market as there is a large abnormal return
accruing to stocks on the ex-date, a finding first documented by Grinblatt, Masulis,
and Titman (1984).
[Insert Table 8 here]
28
As previously mentioned, past work finds that splits do not seem to be motivated
by factors correlated with firm fundamentals. To further verify that our results reflect
a response to the change in price, rather than a change in fundamentals, we examine
whether there is a skewness response at the announcement date. If splits signal
changes in fundamentals and the change in RNSkew is driven by this change in
investors’ information set, then we should expect to see an effect at the announcement
date rather than the ex-date. Panel A2 shows that this is not the case. In Panel A2, we
examine the change of RNSkew around the announcement date. To avoid the effect of
the ex-date, we only examine the period before the ex-date in the announcement date
analysis. Indeed, we see no statistically or economically significant effect on the date
of the announcement. There is some increase in RNSkew beginning on the day after
announcement, but the magnitude is much smaller than the change at the actual date
of the split. Rather, RNSkew reaches its max on the day of the ex-date. 28 The
evidence suggests that nominal changes in price levels around stock splits affect
investor beliefs about the future distribution of returns.
A potential story not excluded by the lack of an effect on the announcement date
is that there is a change in liquidity on the ex-date. It is possible that the increase in
RNSkew on the ex-date could reflect a decrease in risk premium driven by an increase
in liquidity. The previous literature typically finds little evidence that splits increase
liquidity, for instance Schultz (2000), Easley, O’Hara, and Saar (2001), Maloney and
Mulherin (1992), Gray, Smith, and Whaley (2003), and Conroy, Harris, and Benet
28
In unreported results, we examine the change of RNSkew around the announcement date separately
for options with maturity before the ex-date and options with maturity after the ex-date. We find that
RNSkew does not change around the announcement date if the options used to calculate RNSkew
expire before the ex-date. This finding also suggests that the RNSkew change around stock splits is not
driven by release of new information.
29
(1990) all find that spreads as a percentage of price increase following splits,
suggesting lower liquidity. In the Online Appendix we report results documenting that
this is also true for our sample of splits. We find that, consistent with the past research,
relative spreads and relative effective spreads increase in the post-split period,
suggesting that liquidity changes cannot explain the observed RNSkew change we see.
Further alleviating the concern of a relationship between RNSkew and liquidity
changes, we find that in the cross-section of splits there is no statistically significant
relationship between RNSkew changes and liquidity changes. The results are reported
in the Online Appendix.
What prevents smart money from anticipating this increase in RNSkew and
trading ahead of the ex-date? 29 It is well known that substantial limits to arbitrage
exist in options markets (for example, options are relatively illiquid as compared to
equities, additionally one would want to have a delta-hedged position to hedge the
risk of price changes in the underlying, but this requires nearly constant
rebalancing). 30 The extent of limits to arbitrage in the options market is perhaps most
clearly illustrated by the recent finding that end-user demand for options plays a
strong role in influencing options prices (Bollen and Whaley, 2004; and Garleanu,
Pedersen, and Poteshman, 2009). 31 Despite the limits to arbitrage in the options
market, in the case of the stock split evidence discussed earlier, Panel A2 of Table 8
does show that the RNSkew does begin to increase slightly in the period between the
29
An analogous effect occurs contemporaneously in the equity market. There is a large abnormal
return accruing to stocks on the ex-date (Grinblatt, Masulis, and Titman, 1984), suggesting that even in
the more liquid stock market, the market lacks the ability to fully absorb the demand from investors
who prefer lower-priced stocks.
30
Buraschi and Jackwerth (2001), Coval and Shumway (2001), and Jones (2006) show that there are
limits to arbitrage between options and stocks, and that options are not simply redundant assets.
31
Cao and Han (2013) also clearly illustrate limits to arbitrage in the options market.
30
announcement date and ex-date, consistent with some effort to arbitrage this
phenomenon.
The lack of an effect on the announcement day suggests that changes in investor
expectations are not driven by expectations of changes in fundamentals. However, to
completely rule out the possibility that the observed change in RNSkew reflects a
rational expectation of change in future expected skewness, we assess whether
physical skewness does change in the period following splits. A priori, expectations of
post-split increases in skewness seem especially hard to rationalize given that splits
occur after a run-up in stock price and therefore a period of above-average skewness.
The last row of Panel B in Table 8 displays the change in physical skew. In
contrast to the expected increase in skewness exhibited in the options market, physical
skewness actually decreases in the period after the split. The decrease is substantial,
with daily skewness in the year following a split decreasing by over 50% (from 0.384
to 0.151, t = 8.93) relative to the year leading up to the split. That physical skewness
actually decreases following splits, makes the evidence of investor expectations of
increased upside potential for recently split stocks all the more compelling.
To further ensure that the change in physical skewness that we observe post-split
does not reflect a change in fundamentals, we compare splitting firms to a matched
sample of non-splitting firms. The matching firms are similar in that they have
experienced a similar price run-up, and are of similar size, book-to-market, and
similar past skewness. A detailed discussion of the matching procedure is documented
in the table description. 32 These are firms that can reasonably be expected to have
32
The results are similar if we vary the matching method.
31
equal expectations ex-ante of undergoing a split. After matching, our sample firms
reduce to 1,520 due to missing Compustat or CRSP data.
Alleviating the concern that splitting stocks undergo a change in fundamentals, we
find no difference in future skewness between the split sample and the matched
sample. The lack of a difference in skewness provides reassurance that the post-split
decrease in skewness is not an unpredictable artifact of the split. The findings provide
evidence strongly supporting the theory that a nominal price bias leads investors to
attribute irrationally high skewness to low-priced stocks relative to high-priced stocks.
6.2. Skewness around reverse splits
[Insert Table 9 here]
As an additional test of our hypothesis, we examine the RNSkew response around
reverse splits. The results are reported in Table 9. Despite a much smaller sample size,
the results are quite clear. Consistent with nominal prices inducing a bias in investor
expectations, we find that RNSkew drastically decreases on the day that prices
increase due to a reverse split taking effect (Panel A of Table 9). 33 Panel B of Table 9
confirms that the future physical skewness changes do not reflect the risk neutral
changes in expectations. In fact, we find that in contrast to the observed decrease in
risk-neutral skewness, physical skewness increases substantially in the year following
a reverse split (from 0.552 to 0.674, t=2.09).
33
We are not able to compare the RNSkew change around the announcement date, as the
announcement dates in CRSP are missing for most reverse splits.
32
6.3. Robustness
In this robustness section we address two concerns. The first regards the
estimation of physical skew. Physical skew is sensitive to tail events and typically
requires a long time series of return data for estimation. Even with a sufficiently long
time series, it is still subject to a peso problem: that is, extremely good or bad events
do not occur as frequently as rationally expected, leading to an error in the measured
physical skew. The problem may be more severe for the reverse split sample than the
regular split sample, as the reverse split sample is much smaller. We attempt to
address this fear by examining a much larger sample of splits. We extend the analysis
of physical skew to all splits occurring after 1963. In this sample, we have 11,005
regular splits (a five-fold increase in sample size relative to the earlier analysis) and
1,607 reverse splits (a 12-fold increase in sample size relative to the earlier analysis).
This much larger split sample confirms the earlier results; physical skew drastically
decreases following stock splits. The results are available in the Online Appendix.
A second concern is that somehow the split process mechanically induces a
change in RNSkew as a result of the necessary adjustment of the strike price by the
split factor. While it is not at all clear how this would induce a change in RNSkew, we
are fortunate to encounter a natural experiment during our sample period that we
exploit to rule out this possibility. Historically, option contracts have been adjusted
accordingly by the split factor. For example, if a stock undergoes a 2 for 1 split, an
option with a strike price of 50 calling for the delivery of 100 shares would be divided
into two options with a strike price of 25, each calling for the delivery of 100 shares.
Effective September 4, 2007, the Options Clearing Corporation (OCC) adopted a
new rule to govern the post-split administration of options contracts. At the time,
33
strike prices were denominated in 1/8th increments which led to small inaccuracies in
adjustments for certain stock splits. For example, an option contract with a strike price
of $40 for which the underlying stock experienced a 3 for 2 split should have a new
strike of $26.67, however, because strikes are only denominated in 1/8ths, the new
strike would instead be $26.625. To avoid this inaccuracy, the new OCC rule effective
September 4, 2007 leaves option contracts untouched, and instead recalculates the
stock price to the hypothetical price it would trade had the split not occurred. The new
rule affected all splits except those with 2 for 1 or 4 for 1 splits, and extended until
February 12, 2010, at which time decimalization of strike prices was set to take place
(for more details of the rule change refer to Memos 23484 and 26853 of OCC). We
exploit this window of time in which strike prices were unadjusted by splits in order
to confirm that our results are not somehow mechanically driven by adjustments to the
strike price itself.
[Insert Table 10 here]
The empirical findings we document are robust in both the pre and
post-rule-change periods. Table 10 shows the effect of splits on RNSkew for the
subsample of splits occurring between September 4, 2007 and February 12, 2010. The
results are consistent with those from our earlier analysis. 34 The sample is quite small
for multiple reasons. First, 2 for 1 and 4 for 1 splits were exempt from the rule change.
Second, there were relatively few splits occurring during this time period, which is not
surprising given the generally poor market performance during this period. Finally,
because 2 for 1 and 4 for 1 splits are excluded, the sample is tilted toward very large
34
Because the sample is not large, we also examine median changes to ensure that the results aren’t
driven by a few outliers. The median change ranges from 1.822 to 2.343, larger than the mean change.
All the median changes are significant at the 1% level (based on Wilcoxon ranked-sign test).
34
splits, the average split in the rule change sample is slightly greater than 6 for 1.
Because investors react more strongly to larger price changes, the magnitude of the
effect seen here is actually much larger than in the entire sample of splits. The results
for the regular splits are overall consistent with the results over the entire sample
period, suggesting that the relationship between RNSkew and price is not driven by
options market microstructure concerns. 35
In summary, we find sharp changes in RNSkew precisely at the date that a stock
splits to a lower price. In stark contrast to investor expectations of post-split skewness
increases, we find a drastic decrease in physical skewness following a split. The
evidence is further strengthened by supportive results among the smaller sample of
stocks undergoing reverse splits. The evidence is consistent with nominal price levels
biasing investor expectations of future return distributions.
6.4. Increased upside or decreased downside?
The observed option-implied skew increase following splits could reflect an
increase in the perceived upside potential for the stock, or a decrease in the perceived
downside potential for the stock, or a combination of both. To answer this more
precise question we separately examine puts and calls. Specifically, we examine
changes to OTM put and OTM call implied volatilities.
Investor perception of increased upside potential will manifest itself in
option-implied skew through the increased valuation of OTM calls. Perception of
35
To further exclude the possibility of a mechanical relationship, we also examine a sample of ETF
splits. Typically, ETFs try to mimic the return of the whole market or a very broad group of stocks. It is
very unlikely that investors would think lower priced ETFs have more upside potential, especially
considering that when an ETF splits, there is no split of the underlying index. We find no change in
RNSkew for this sample. Results are available in the Online Appendix.
35
decreased downside potential will show up in option-implied skew via decreased
valuations of OTM puts. Intuitively, implied volatility changes allow us to identify
valuation changes in order to assess whether it is changing beliefs regarding upside
potential or downside potential (or some combination of the two) that drives the
observed RNSkew changes.
To examine how the relative valuations of puts and calls change, we compare
OTM implied volatilities to ATM implied volatilities. Specifically, we focus on the
difference in OTM and ATM implied volatilities, and examine whether this quantity
changes after the split. Table 11 displays the results. Panel A examines the case of
regular splits, and Panel B examines reverse splits. 36
[Insert Table 11 here]
The results are quite clear. Panel A1 documents that the difference in OTM and
ATM implied volatilities substantially increase following the split. That is, OTM calls
become more expensive following the split, reflecting investors’ increased perception
of upside potential. However, the same is not true for puts. Panel A2 shows that while
on average OTM puts do seem to get slightly cheaper following a split, this change is
small, and generally not statistically significant. The smaller sample of reverse split
results (Panel B of Table 11) are also consistent with the regular split evidence. After
price increases due to a reverse split, OTM calls become relatively cheaper, reflecting
investor beliefs of decreased upside potential, however, there is no significant change
in the case of puts. The evidence suggests that the skew change we see does in fact
reflect changes in investor perceptions regarding upside potential, with no
36
We have slightly more observations for this analysis than for the RNSkew estimation as we do not
require a minimum number of OTM options as we do in the RNSkew analysis, nor is it necessary for
all OTM options to have the same maturity as is the case in the RNSkew analysis.
36
accompanying change in investor perceptions regarding downside potential. This
evidence complements the earlier option return results that look differentially at OTM
calls and OTM puts, and provides additional evidence that the general effects we see
are driven by investor biased beliefs regarding the upside potential, but not downside
potential, of low-priced stocks.
7. Conclusion
We provide the first evidence that investors link nominal share price to return
skewness and systematically overestimate the skewness of low-priced stocks relative
to high-priced stocks. The evidence presented is consistent with investors suffering
from the illusion that low-priced stocks have more upside potential.
Analysis of option volume data provides further evidence that investors exhibit
greater optimism toward low-priced stocks than high-priced stocks. Finally, investor
overweighting of nominal prices in forming return distribution expectations has
asset-pricing implications. We find that OTM call options of low-priced stocks are
more overvalued than those of high-priced stocks.
Firms have long engaged in the costly management of share price through stock
splits, despite nominal prices lacking real economic content. Recent work provides
evidence that investors view stocks of similar price as sharing similar attributes.
Green and Hwang (2009) find that investors categorize stocks by nominal price, while
Baker, Greenwood, and Wurgler (2009) find that firms exploit the fact that nominal
prices matter to investors by increasing the supply of low-priced securities when
investors are willing to pay a premium for them. While it is clear that nominal prices
matter to investors, it has thus far been less clear why prices matter. We provide the
37
first empirical insight into why prices matter to investors. Specifically, we find
evidence that is consistent with the notion that investors view low-priced stocks as
“cheap” assets with “more room to grow”.
38
Appendix
Table A1. Definition of variables
Variable name Description
Price
Log (Price)
Beta
Stock price at the end of month t-1
Natural log of stock price at the end of month t-1
Calculated using monthly return data over the past 5 years, following Fama
and French (1992)
Log (Size)
Size is the product of stock price and number of shares outstanding,
calculated at the end of month t-1
Log (BM)
Book-to-Market is the ratio of book value of common equity to market
value of common equity, and is matched to CRSP data following Fama and
French (1992). Book equity of common equity is calculated following
Fama and French (2008). It is equal to total assets (Compustat data item 6),
minus liabilities (181), plus balance sheet deferred taxes and investment tax
credit (35) if available, minus preferred stock liquidating value (10) if
available, or redemption value (56) if available, or carrying value (130).
Negative BM observations are deleted.
Leverage
Leverage is the ratio of total liabilities divided by total assets, and is
matched to CRSP data following Fama and French (1992).
Past Volatility Volatility is calculated using past daily returns over the past year. Daily
standard deviation is annualized by multiplying by the square root of 252.
Past Skewness Skewness is calculated using daily returns from month t-12 to month t-1
R t-12,t-1
R t-12,t-1 is the cumulative return from month t-12 to month t-1
R t-60,t-13
R t-60,t-13 is the cumulative return from month t-60 to month t-13
Log (Volume) Natural log of total shares traded in month t-1
Log (ILIIQ)
Natural log of the illiquidity measure of Amihud (2002) calculated using
daily data in month t-1
NASDAQ
A dummy variable equal to 1 if the firm is listed in NASDAQ, and 0
otherwise.
RNSkew
Risk-neutral skewness implied from option prices, calculated following
Bakshi, Kapadia, and Madan (2003).
Skew
Future skewness, calculated using future one year daily returns from month
t+1 to month t+12
39
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42
Figure 1. RNSkew around stock splits
This figure shows RNSkew around stock splits. The solid line with square markers represents reverse splits while the solid line
without square markers represents regular splits. The graph plots RNSkew against trading days relative to the split. Day 0 is the
ex-date. RNSkew is calculated following Bakshi, Kapadia, and Madan (2003). The sample period is from January 1996 to
December 2012.
0.80
0.60
0.40
0.20
0.00
-0.20
-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5
0
5 10 15 20 25 30 35 40 45 50 55 60
-0.40
-0.60
-0.80
-1.00
Day Relative to Split
Regular Split
Reverse Split
43
Table 1. Summary statistics by price quintiles
This table reports summary statistics for the main variables of interest, sorted on price at the end of month t-1. We include all
stocks for which RNSkew and Skew can be calculated and the underlying stock price is greater than $10. We calculate the mean
of the below variables month by month, and reported the time series average of the cross sectional means. Beta is calculated
following Fama and French (1992); for each month, we use the past 60 months’ monthly excess return data and CRSP value
weighted excess market return to calculate Beta. Size is market capitalization (price times shares outstanding). B/M is the ratio
between book value of common equity and market price. Leverage is the ratio of total liabilities to total assets. Past Volatility and
Past Skewness are calculated using past one year daily returns. R t-12,t-1 is the cumulative return from month t-12 to month t-1.
R t-60,t-13 is the cumulative return from month t-60 to t-13. Volume is the total number of shares traded in the last month. ILLIQ is
measured following Amihud (2002). NASDAQ is a dummy variable which is equal to 1 if the firm is listed in NASDAQ and 0
otherwise. RNSkew is calculated following Bakshi, Kapadia, and Madan (2003). Skew is calculated using daily data from month
t+1 to month t+12. E(Skew) is the expected future physical skewness, calculated following Boyer, Mitton, and Vorkink (2010),
specifically using equation (5) from the text. The sample period is from January 1996 to December 2011.
Price Portfolio
Price
Log (Price)
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
RNSkew
Skew
E(Skew)
Lowest
14.505
2.659
1.684
13.652
-1.029
0.188
0.616
0.157
0.120
1.060
11.799
-19.869
0.599
-0.231
0.370
0.373
2
22.125
3.086
1.414
14.194
-1.129
0.198
0.523
0.176
0.212
1.267
11.809
-20.458
0.497
-0.469
0.265
0.300
3
31.214
3.429
1.277
14.733
-1.261
0.198
0.479
0.216
0.307
1.473
11.968
-21.078
0.418
-0.709
0.207
0.229
4
43.574
3.762
1.103
15.267
-1.310
0.209
0.420
0.231
0.363
1.483
12.047
-21.653
0.319
-0.796
0.173
0.173
Highest
74.106
4.231
1.012
15.968
-1.426
0.217
0.382
0.286
0.522
1.583
12.215
-22.434
0.236
-0.929
0.109
0.103
44
Table 2. The relationship between nominal price and skewness: Fama-MacBeth regressions
For each month, the following cross-sectional regression is run.
DVi ,t =α t + β1 Log ( Pricεi ,t ) + X i ,t + ε i ,t ,
where DV is one of the following: RNSkew, Skew, RNSkew-Skew, or RNSkew-E(Skew). RNSkew is calculated following
Bakshi, Kapadia, and Madan (2003). Skew is calculated using daily data from month t+1 to month t+12. E(Skew) is a forward
looking skewness measure constructed following the method of Boyer, Mitton, and Vorkink (2010), specifically using equation (5)
from the text.
The independent variables include log price, beta, log of firm market capitalization, log B/M, leverage, past volatility, past
skewness, R t-12,t-1 , long run return, log volume, log ILLIQ, a NASDAQ dummy indicating listing exchange, and industry dummies.
Beta is calculated following Fama and French (1992); for each month, we use the past 60 months’ monthly excess return data and
CRSP value weighted excess market return to calculate Beta. Size is market capitalization (price times shares outstanding). B/M is
the log ratio of book value of common equity to market equity. Leverage is the ratio of total liabilities to total assets. Past
Volatility and Past Skewness are calculated using past one year daily returns. R t-12,t-1 is the cumulative return from month t-12 to
month t-1. R t-60,t-13 is the cumulative return from month t-60 to t-13. Volume is the total number of shares traded in the last month.
ILLIQ is measured following Amihud (2002). Industry is defined based on the 48 Fama-French industry classification. We
include all stocks for which RNSkew and Skew can be calculated and the underlying stock price is higher than $10. The sample
period is from January 1996 to December 2011. Reported t-statistics are Newey-West adjusted with 12 lags. Reported adj-R2 is
the average adj-R2 across all the months.
45
RNSkew
Log (Price)
(1)
-0.437***
(-15.72)
Log (AT/Price)
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
Constant
Industry FE
Adj-R2
Obs.
0.865***
(8.51)
No
0.085
192
(2)
-0.213***
(-4.88)
0.022*
(1.74)
-0.005
(-0.62)
-0.105***
(-6.96)
-0.052***
(-4.05)
-0.217***
(-8.47)
0.307***
(3.27)
-0.020***
(-4.03)
-0.051***
(-3.99)
0.003
(0.80)
0.038
(1.51)
0.025
(1.15)
0.014*
(1.71)
1.621***
(8.15)
No
0.163
192
Skew
(3)
-0.218***
(-5.28)
0.057***
(5.82)
-0.009
(-1.42)
-0.098***
(-6.84)
-0.073***
(-6.15)
-0.287***
(-11.95)
0.298***
(3.50)
-0.020***
(-4.27)
-0.047***
(-4.39)
0.005
(1.20)
0.026
(1.17)
0.022
(1.15)
0.028***
(3.79)
1.566***
(7.92)
Yes
0.217
192
(4)
-0.150***
(-7.42)
0.741***
(9.84)
No
0.007
192
(5)
-0.059*
(-1.89)
0.030***
(2.71)
-0.012
(-0.62)
0.002
(0.12)
0.007
(0.60)
0.167***
(4.49)
0.249*
(1.77)
0.023**
(2.51)
-0.092***
(-4.22)
-0.015***
(-4.42)
-0.025
(-1.13)
0.018
(0.90)
0.006
(0.25)
0.971***
(4.07)
No
0.032
192
RNSkew-Skew
(6)
-0.049
(-1.48)
0.039***
(3.86)
-0.021
(-1.17)
-0.009
(-0.62)
0.000
(0.01)
0.096*
(1.82)
0.238*
(1.68)
0.019*
(1.94)
-0.086***
(-4.00)
-0.017***
(-4.14)
-0.010
(-0.48)
0.024
(1.18)
0.001
(0.05)
1.006***
(4.03)
Yes
0.087
192
(7)
-0.286***
(-10.83)
0.124
(1.32)
No
0.014
192
(8)
-0.154***
(-3.84)
-0.007
(-0.57)
0.007
(0.39)
-0.106***
(-4.98)
-0.059***
(-3.03)
-0.384***
(-10.55)
0.058
(0.27)
-0.043***
(-4.03)
0.041
(1.39)
0.018***
(3.17)
0.063***
(2.41)
0.007
(0.35)
0.008
(0.39)
0.650*
(1.85)
No
0.052
192
RNSkew-E(Skew)
(9)
-0.170***
(-4.20)
0.018
(1.23)
0.013
(0.70)
-0.089***
(-4.47)
-0.073***
(-3.65)
-0.383***
(-8.31)
0.060
(0.30)
-0.038***
(-3.60)
0.039
(1.46)
0.021***
(3.56)
0.036
(1.44)
-0.002
(-0.09)
0.026
(1.23)
0.560
(1.62)
Yes
0.105
192
(10)
-0.262***
(-9.84)
0.033
(0.33)
No
0.037
192
(11)
-0.132***
(-3.29)
0.011
(0.90)
-0.012
(-0.66)
-0.066***
(-4.22)
-0.084***
(-4.04)
-0.295***
(-8.38)
0.302**
(2.56)
-0.070***
(-9.92)
0.051**
(2.05)
0.020***
(3.17)
0.069**
(2.63)
0.027
(1.18)
0.055***
(2.96)
0.145
(0.76)
No
0.140
192
(12)
-0.137***
(-3.58)
0.057***
(5.79)
0.002
(0.09)
-0.045***
(-3.36)
-0.118***
(-6.44)
-0.357***
(-8.87)
0.292***
(2.68)
-0.046***
(-6.29)
0.051**
(2.25)
0.020***
(3.35)
0.044*
(1.82)
0.023
(1.19)
0.084***
(4.54)
-0.044
(-0.21)
Yes
0.249
192
46
Table 3. Nominal price and option trading
For each month, we run the cross-sectional regression
DVi ,t =α t + β1 Log ( Pricεi ,t ) + X i ,t + ε i ,t ,
where DV is either call relative to put option trading volume ratio (VolRatio), or the call relative to put option
open interest ratio (OIRatio). VolRatio=log (1+ volume of call options)-log (1+ volume of put options).
OIRatio=log (1+ options interests of call options)-log (1+ options interests of put options). The independent
variables include log price, beta, log of firm market capitalization, log of B/M, leverage, past volatility, past
skewness, R t-12,t-1 , R t-12,t-13 , log volume, log ILLIQ, a NASDAQ dummy indicating listing exchange, and
industry dummies. Beta is calculated following Fama and French (1992); for each month, we use the past 60
months’ monthly excess return data and CRSP value weighted excess market return to calculate Beta. Size is
market capitalization (price times shares outstanding).B/M is the ratio of book value of common equity to
market equity. Leverage is the ratio of total liabilities to total assets. Past Volatility and Past Skewness are
calculated using past one year daily returns. R t-12,t-1 is the cumulative return from month t-12 to month t-1.
R t-60,t-13 is the cumulative return from month t-60 to t-13. Volume is the total number of shares traded from the
last month. ILLIQ is measured following Amihud (2002). Industry is defined based on the 48 Fama-French
industry classification. We include all stocks for which RNSkew and Skew can be calculated and the underlying
stock price is greater than $10. The sample period is from January 1996 to December 2011. Reported t-statistics
are Newey-West adjusted with 12 lags. Reported adj-R2 is the average adj-R2 across all the months.
VolRatio
(1)
(2)
(3)
-0.264*** -0.384*** -0.399***
Log (Price)
(-13.31)
(-19.06)
(-16.67)
-0.006
-0.005
Log (AT/Price)
(-0.51)
(-0.56)
-0.011
-0.018*
Beta
(-1.07)
(-1.86)
0.103*** 0.083***
Log (Size)
(8.77)
(7.26)
0.007
-0.006
Log (BM)
(0.73)
(-0.70)
-0.045
-0.082***
Leverage
(-1.33)
(-2.90)
-0.062
-0.129*
Past Volatility
(-0.95)
(-1.74)
0.015**
0.017***
Past Skewness
(2.52)
(3.15)
0.181***
0.168***
R t-12,t-1
(7.60)
(8.35)
0.013*** 0.009***
R t-60,t-13
(3.07)
(3.06)
-0.085*** -0.077***
Log (Volume)
(-4.92)
(-4.84)
0.041**
0.034**
Log (ILLIQ)
(2.52)
(2.32)
0.020
0.019
NASDAQ
(1.49)
(1.62)
1.582*** 2.338*** 2.453***
Constant
(12.09)
(10.19)
(9.77)
No
No
Yes
Industry FE
0.030
0.071
0.135
Adj-R2
192
192
192
Obs.
OIRatio
(4)
(5)
-0.113*** -0.228***
(-7.80)
(-14.51)
-0.037***
(-4.27)
-0.011
(-1.21)
0.103***
(11.54)
0.027***
(3.03)
-0.052
(-1.37)
-0.230***
(-3.17)
0.010
(1.61)
0.218***
(7.88)
0.006**
(2.08)
-0.067***
(-4.61)
0.060***
(4.19)
0.016*
(1.82)
0.919*** 1.929***
(10.13)
(10.60)
No
No
0.009
0.061
192
192
(6)
-0.247***
(-15.53)
-0.052***
(-5.76)
-0.016*
(-1.89)
0.088***
(9.75)
0.023***
(2.76)
-0.023
(-0.73)
-0.280***
(-3.57)
0.013**
(2.18)
0.205***
(8.11)
0.003
(1.34)
-0.062***
(-4.25)
0.053***
(4.18)
0.011
(1.31)
2.041***
(10.62)
Yes
0.122
192
47
Table 4. Option trading strategy
Portfolios are formed on the expiration Friday (or last trading day if Friday is a public holiday) of the month,
and the option portfolio strategies are initiated on the first trading day (typically a Monday) after the expiration
Friday of the month. OTM options are call options with delta greater than 0.02 and not greater than 0.. As in
Bollen and Whaley (2004), we exclude options with absolute delta below 0.02 due to the distortions caused by
price discreteness. On each portfolio formation day, we choose the options for any given moneyness that will
expire within one month. On each portfolio formation day, all stocks with available options are sorted into
quintiles based on the stock price on the portfolio formation day. For each option series, we construct
delta-hedged and non-delta hedged portfolios and hold them until option expiration. As in Goyal and Sarreto
(2009), we use the absolute value of the position as the reference price to calculate the delta-hedged return.
Specifically, the formula is
T
Rcall =
(cT − c0 e rT ) − D 0 ( ST + ∑ Dt e r (T −t ) − S0 e rT )
t =1
,
| D 0 S0 − c0 |
We also report the average abnormal return adjusting for Fama-French three factors, the Carhart (1997)
momentum factor and a factor reflecting the market level call return. The portfolio returns are equal-weighted.
The sample extends from January 22, 1996 to December 21, 2012. There are 203 months in total.
Panel A. Delta-hedged portfolio returns
Price
OTM
Call
Call adjusted
ATM
Call
Call adjusted
Lowest
2
3
4
Highest
Highest - Lowest
-2.901***
(-6.67)
-2.020***
(-6.97)
-2.734***
(-6.41)
-1.974***
(-7.15)
-2.208***
(-5.14)
-0.506**
(-2.60)
-1.296***
(-3.17)
-0.270
(-0.98)
-0.908*
(-1.94)
0.368
(1.15)
1.993***
(5.53)
2.387***
(7.25)
-0.373
(-1.34)
0.104
(0.58)
-0.033
(-0.13)
0.372***
(3.66)
-0.007
(-0.03)
0.356**
(2.20)
0.101
(0.40)
0.537***
(3.18)
0.043
(0.16)
0.484**
(2.37)
0.416**
(2.25)
0.380**
(2.07)
Panel B. Non-delta hedged portfolio returns
Price
OTM
Call
Lowest
-21.498***
(-4.70)
Call adjusted -20.423***
(-6.40)
ATM
Call
2.484
(0.59)
Call adjusted -0.361
(-0.19)
2
3
4
Highest
Highest - Lowest
-27.849***
(-6.45)
-27.453***
(-8.77)
-25.099***
(-5.32)
-24.518***
(-6.93)
-18.928***
(-3.24)
-17.049***
(-3.87)
-10.190
(-1.08)
0.914
(0.17)
11.308**
(2.55)
21.338***
(4.25)
5.855
(1.41)
2.257
(1.25)
6.192
(1.46)
2.382
(1.26)
5.502
(1.21)
1.722
(0.82)
8.889
(1.57)
5.095**
(2.01)
6.405*
(1.92)
5.457**
(2.35)
48
Table 5. Option trading strategy: Fama-MacBeth regressions
For each month, we run the cross-sectional regression
DVi ,t =α t + β1 Log ( Pricεi ,t ) + X i ,t + ε i ,t ,
where DV is either delta-hedged OTM call return, OTM put return or the difference between OTM call and
OTM put return (OTM Call – OTM Put). OTM options are call options with delta greater than 0.02 and not
greater than 0.375 and put options with delta greater than -0.375 and not greater than -0.02. As in Bollen and
Whaley (2004), we exclude options with absolute delta below 0.02 due to the distortions caused by price
discreteness. We choose the put and call options for any given moneyness that will expire within one month.
Delta-hedged option return is calculated from the first trading day (typically a Monday) after the expiration
Friday of the month until option expiration. As in Goyal and Sarreto (2009), we use the absolute value of the
position as the reference price to calculate the delta-hedged return. Specifically, the formulas are
T
Rcall =
(cT − c0 e rT ) − D 0 ( ST + ∑ Dt e r (T −t ) − S0 e rT )
t =1
| D 0 S0 − c0 |
T
, R put =
( pT − p0 e rT ) − D 0 ( ST + ∑ Dt e r (T −t ) − S0 e rT )
t =1
| D 0 S0 − p0 |
.
If there is more than one option series for one stock-moneyness group, we calculate the equal-weighted
delta-hedged option return. The independent variables include log price, beta, log of firm market capitalization,
log of B/M, leverage, past volatility, past skewness, R t-12,t-1 , R t-60,t-13 , implied volatility, Log (volume), R t-12,t-1 ,
Log (ILLIQ), a NASDAQ dummy indicating listing exchange, and industry dummies. Beta is calculated
following Fama and French (1992); for each month, we use the past 60 months’ monthly excess return data and
CRSP value weighted excess market return to calculate Beta. Size is market capitalization (price times shares
outstanding). B/M is the ratio of book value of common equity to market equity. Leverage is the ratio of total
liabilities to total assets. Past Volatility and Past Skewness are calculated using past one year daily returns.
R t-12,t-1 is the cumulative return from month t-12 to month t-1. Long run return is the cumulative return from
month t-60 to t-13. Volume is the total number of shares traded from the last month. ILLIQ is measured
following Amihud (2002). Industry is defined based on the 48 Fama-French industry classification. We include
all stocks for which we have both OTM call and OTM put options and the underlying stock price is greater than
$10. The sample period is from January 22, 1996 to December 21, 2012. There are 203 months in total.
Reported t-statistics are Newey-West adjusted with 12 lags. Reported adj-R2 is the average adj-R2 across all the
months.
49
Table 5 (continued)
Panel A. Delta Hedged
Log (Price)
(1)
0.010*
(6.01)
RNSkew
RNVar
RNKurt
Log
Beta
Log (Size)
Log (BM)
Leverage
Past
Past
Rt-12,t-1
Rt-60,t-13
Log
Log (ILLIQ)
NASDAQ
Industry FE
Adj-R2
Obs.
No
0.006
203
OTM Call
(2)
0.011**
(4.37)
0.001
(0.57)
0.101**
(2.64)
-0.000
(-0.32)
0.000
(0.19)
0.001
(0.44)
-0.007*
(-4.01)
0.001
(0.60)
-0.002
(-0.51)
-0.017*
(-3.07)
0.001
(0.71)
0.002
(1.13)
-0.000
(-0.47)
0.002
(0.94)
-0.004*
(-1.83)
-0.003*
(-1.90)
No
0.062
203
(3)
0.011**
(4.83)
0.001
(0.45)
0.101**
(2.74)
-0.000
(-0.30)
-0.000
(-0.15)
-0.001
(-0.42)
-0.007*
(-4.45)
0.001
(1.08)
-0.001
(-0.14)
-0.017*
(-3.11)
0.000
(0.50)
0.001
(0.50)
-0.000
(-0.78)
0.003
(1.37)
-0.004*
(-1.76)
-0.003*
(-1.89)
Yes
0.133
203
(4)
0.005*
(4.02)
No
0.004
203
OTM Put
(5)
(6)
0.005
0.005*
(1.48)
(1.74)
0.004** 0.004**
(2.14)
(2.29)
0.042
0.053
(0.96)
(1.23)
0.001** 0.001**
(2.45)
(2.63)
0.000
0.000
(0.54)
(0.18)
0.001
0.000
(0.67)
(0.25)
-0.007* -0.007*
(-4.38)
(-4.33)
0.000
-0.000
(0.11)
(-0.24)
0.002
0.005
(0.35)
(1.29)
-0.013* -0.012*
(-2.11)
(-2.03)
-0.001* -0.001*
(-2.56)
(-2.98)
-0.001
-0.002
(-0.70)
(-1.11)
0.000
0.000
(0.63)
(0.56)
-0.000
-0.000
(-0.04)
(-0.16)
-0.006* -0.006*
(-3.22)
(-3.62)
-0.002
-0.002
(-1.25)
(-1.27)
No
Yes
0.057
0.134
203
203
OTM Call - OTM Put
(7)
(8)
(9)
0.004* 0.006* 0.006**
(3.27)
(2.05)
(2.02)
-0.003
-0.004*
(-1.76)
(-2.00)
0.059
0.048
(1.51)
(1.16)
-0.001
-0.001*
(-2.57)
(-2.79)
-0.000
-0.000
(-0.15)
(-0.32)
-0.001
-0.001
(-0.27)
(-0.54)
0.000
-0.001
(0.15)
(-0.31)
0.001
0.001
(0.31)
(0.89)
-0.004
-0.006
(-0.61)
(-0.98)
-0.005
-0.005
(-0.70)
(-0.75)
0.002* 0.002**
(2.34)
(2.26)
0.004*
0.003
(1.78)
(1.29)
-0.000
-0.000
(-0.79)
(-1.02)
0.002
0.003
(0.85)
(1.22)
0.002
0.002
(0.94)
(0.86)
-0.000
-0.001
(-0.21)
(-0.57)
No
No
Yes
0.003
0.045
0.115
203
203
203
50
Table 5 (continued)
Panel B. Non-delta-hedged
Log (Price)
(1)
0.089**
(2.14)
RNSkew
RNVar
RNKurt
Log(AT/Price)
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
Rt-12,t-1
Rt-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
Industry FE
Adj-R2
Obs.
No
0.004
203
OTM Call
(2)
0.186***
(5.17)
-0.227***
(-7.97)
-1.311**
(-2.37)
-0.009***
(-2.94)
0.001
(0.04)
0.037*
(1.77)
-0.150***
(-6.79)
0.015
(0.69)
-0.143**
(-2.58)
-0.251**
(-2.36)
-0.013*
(-1.78)
0.034
(0.62)
-0.006
(-1.37)
-0.024
(-0.54)
-0.128***
(-3.16)
-0.040
(-1.21)
No
0.048
203
(3)
0.178***
(5.21)
-0.222***
(-8.79)
-1.174**
(-2.27)
-0.009***
(-2.70)
0.034
(1.46)
0.021
(1.25)
-0.153***
(-7.67)
-0.006
(-0.32)
-0.204***
(-3.04)
-0.170**
(-1.97)
-0.016**
(-2.52)
0.015
(0.29)
-0.008
(-1.57)
-0.054*
(-1.73)
-0.156***
(-5.35)
-0.032
(-1.14)
Yes
0.134
203
(4)
-0.011
(-0.59)
No
0.004
203
OTM Put
(5)
-0.073
(-1.10)
0.222***
(10.54)
2.198***
(4.71)
0.007***
(3.18)
-0.005
(-0.27)
0.020
(0.90)
-0.064**
(-2.28)
0.012
(0.63)
0.069
(1.08)
0.055
(0.58)
-0.002
(-0.25)
-0.020
(-0.89)
0.010**
(2.17)
0.045
(1.50)
-0.020
(-0.73)
-0.043**
(-2.06)
No
0.050
203
(6)
-0.079
(-1.52)
0.215***
(11.24)
2.209***
(4.72)
0.008***
(3.76)
-0.022
(-1.06)
0.015
(0.97)
-0.064**
(-2.49)
0.020
(1.42)
0.143**
(2.11)
0.068
(0.73)
-0.002
(-0.24)
-0.028
(-1.22)
0.010***
(2.75)
0.043
(1.59)
-0.021
(-0.79)
-0.041**
(-2.15)
Yes
0.131
203
OTM Call - OTM Put
(7)
(8)
(9)
0.099* 0.259***
0.257***
(1.90)
(3.09)
(3.66)
-0.448*** -0.437***
(-12.25)
(-14.21)
-3.509*** -3.383***
(-5.84)
(-5.69)
-0.016*** -0.017***
(-4.51)
(-4.91)
0.006
0.056
(0.17)
(1.49)
0.017
0.005
(0.50)
(0.22)
-0.087**
-0.089**
(-2.04)
(-2.37)
0.003
-0.026
(0.11)
(-1.02)
-0.213**
-0.347***
(-2.34)
(-3.32)
-0.306*
-0.237
(-1.79)
(-1.64)
-0.011
-0.014
(-0.96)
(-1.42)
0.054
0.043
(0.77)
(0.64)
-0.016**
-0.018***
(-2.08)
(-2.68)
-0.068
-0.097**
(-1.17)
(-2.11)
-0.108**
-0.136***
(-2.02)
(-3.16)
0.004
0.010
(0.09)
(0.27)
No
No
Yes
0.005
0.062
0.156
203
203
203
51
Table 6. Double Sorts: Option trading strategy
Portfolios are formed on the expiration Friday (or last trading day if Friday is a public holiday) of the month, and the option
portfolio strategies are initiated on the first trading day (typically a Monday) after the expiration Friday of the month. OTM
options are call options with delta greater than 0.02 and not greater than 0.. As in Bollen and Whaley (2004), we exclude options
with absolute delta below 0.02 due to the distortions caused by price discreteness. On each portfolio formation day, we choose the
options for any given moneyness that will expire within one month. On each portfolio formation day, all stocks with available
options are sorted into quintiles based on the stock price on the portfolio formation day. For each option series, we construct
delta-hedged and non-delta hedged portfolios and hold them until option expiration. As in Goyal and Sarreto (2009), we use the
absolute value of the position as the reference price to calculate the delta-hedged return. Specifically, the formula is
T
Rcall =
(cT − c0 e rT ) − D 0 ( ST + ∑ Dt e r (T −t ) − S0 e rT )
t =1
,
| D 0 S0 − c0 |
We also report the average abnormal return adjusting for Fama-French three factors, the Carhart (1997) momentum factor and a
factor reflecting the market level call return. The portfolio returns are equal-weighted. The sample extends from January 22, 1996
to December 21, 2012. There are 203 months in total.
Panel A. Delta hedged portfolio returns
RNSkew Tercile
Low
Medium
High
Lowest
-1.327**
(-2.35)
-2.173***
(-3.88)
-2.007***
(-4.23)
2
-1.680***
(-3.17)
-1.865***
(-4.25)
-1.753***
(-3.85)
Price Quintile
3
-1.671***
(-3.68)
-1.948***
(-4.45)
-1.670***
(-3.68)
4
-1.243***
(-3.19)
-1.288***
(-3.25)
-0.874**
(-2.06)
Highest
-0.674
(-1.62)
-0.859**
(-2.22)
-0.391
(-0.97)
Highest - Lowest
0.653
(1.44)
1.314***
(3.23)
1.616***
(4.72)
Highest
16.827**
(2.51)
1.517
(0.19)
-27.888***
(-7.08)
Highest - Lowest
6.958
(1.49)
13.379**
(1.97)
12.984***
(3.23)
Panel B. Non-delta hedged portfolio returns
RNSkew Tercile
Low
Medium
High
Lowest
9.869
(1.09)
-11.862*
(-1.67)
-40.872***
(-6.52)
2
2.494
(0.32)
-13.023**
(-2.19)
-41.153***
(-8.81)
Price Quintile
3
-0.123
(-0.02)
-17.902***
(-3.05)
-48.656***
(-11.95)
4
5.465
(0.71)
-11.197*
(-1.78)
-38.165***
(-9.77)
52
Table 7. Summary statistics
This table reports summary statistics for three different samples. “All optionable stocks” includes all stocks for
which RNSkew and Skew can be calculated and the underlying stock price is greater than $10. Regular splits
and reverse splits are the split stocks that are also covered by OptionMetrics. Regular splits and reverse splits are
defined as splits with split ratio greater than or equal to 1.25-to-1 and lower than 1-to-1, respectively. Beta is
calculated following Fama and French (1992); for each month, we use the past 60 months’ monthly excess
return data and CRSP value weighted excess market return to calculate Beta. Size is market capitalization (price
times shares outstanding). B/M is the ratio between book value of common equity and market price. Leverage is
the ratio of total liabilities to total assets. Past Volatility and Past Skewness are calculated using past one year
daily returns. We reach annualized volatility by multiplying daily standard deviation by the square root of 252.
Momentum is the cumulative return from month t-12 to month t-1. Long-run return is the cumulative return
from month t-60 to t-13. Volume is the total number of shares traded from the last month. ILLIQ is measured
following Amihud (2002). NASDAQ is a dummy variable which is equal to 1 if the firm is listed in NASDAQ
and 0 otherwise. RNSkew is calculated following Bakshi, Kapadia, and Madan (2003). Skew is calculated using
daily data from month t+1 to month t+12. E(Skew) is expected skewness. We calculate E(Skew) following the
method of Boyer, Mitton, and Vorkink (2010), specifically using equation (5) from the text. The sample period
is from January 1996 to December 2011.
Variables
Price
Log (Price)
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
RNSkew
Skew
E(Skew)
Price after split
Split ratio
All optionable stocks
(N=203,974)
Mean
STDEV
37.285
33.454
3.436
0.576
1.300
0.818
14.778
1.486
-1.216
0.870
0.203
0.184
0.483
0.235
0.218
1.093
0.300
0.889
1.357
2.818
12.005
1.280
-21.146
1.753
0.414
0.493
-0.634
0.893
0.231
1.346
0.249
0.350
NA
NA
NA
NA
Regular Splits
(N=1,940)
Mean STDEV
79.982
85.484
4.238
0.498
1.311
0.998
15.126
1.379
-1.774
0.940
0.185
0.193
0.499
0.271
0.429
0.921
1.324
2.261
2.234
3.957
11.679
1.190
-21.538
1.527
0.440
0.497
-0.732
0.848
0.163
1.066
0.100
0.348
39.534
18.691
1.985
1.201
Reverse Splits
(N=130)
Mean STDEV
10.984
14.735
1.519
1.515
1.848
1.233
12.866
2.188
-0.152
1.346
0.261
0.240
0.836
0.520
0.604
1.569
-0.247
0.769
-0.093
1.347
12.826
2.069
-19.281
2.731
0.392
0.490
0.553
0.942
0.487
1.176
0.337
0.275
38.534
46.313
0.258
0.189
53
Table 8. RNSkew around regular stock splits
Regular stock splits are events with a CRSP distribution code of 5523 and a split ratio greater than or equal to
1.25-for-1. RNSkew is calculated following Bakshi, Kapadia, and Madan (2003) and is detailed in the paper.
Panel A reports changes in RNSkew and Panel B reports the change in physical skewness. For RNSkew, we
report the change in RNSkew around the ex-date and the announcement date. For the analysis of RNSkew
change around announcement dates, to avoid the effect of ex-date, we only examine the period before the
ex-date. We use a paired t-test to gauge statistical significance. Different windows are chosen. For example, for
(-3, 0) the before column displays RNSkew on day -3 relative to the event date, and the after column displays
RNSkew on day 0 relative to the event date (i.e., the date of the event). For each given window, we require
RNSkew to be available on both days. In Panel B, for each split stock, we require the matched stock and the
split stock to be in the same size quintile, BM quintile, momentum quintile and skewness quintile. We also
require that the skewness difference between the matched stock and the split stock is lower than 0.25. If more
than one stock satisfies the above criteria, we choose the one with the smallest price difference with the split
stock. All the information used for matching is available at the time of split. B/M is the ratio of book value of
common equity to market price. R t-12,t-1 is the cumulative return from month t-12 to month t-1. Size is equal to
the log market capitalization at the end of month t-1. Pre-matching skewness is calculated using daily returns
from month t-11 to month t-1. Post-matching skewness is calculated using daily returns from month t+1 to
month t+12. The sample period is January 1996 to December 2012.
Panel A. RNSkew change
Window
N
Before
After
Change
Panel A1. Ex-date
(-10, 0)
1847
-0.734 -0.474
0.260***
(-5, 0)
1898
-0.705 -0.469
0.236***
(-3, 0)
1917
-0.706 -0.468
0.238***
(-1, 0)
1940
-0.732 -0.464
0.268***
(-1, 1)
1936
-0.733 -0.461
0.272***
(-1, 3)
1926
-0.733 -0.481
0.252***
(-1, 5)
1916
-0.730 -0.502
0.229***
(-1, 10)
1901
-0.730 -0.452
0.277***
Panel A2. Announcement
(-10, 0)
1674
-0.863 -0.872
-0.009
(-5, 0)
1707
-0.876 -0.869
0.007
(-3, 0)
1722
-0.881 -0.865
0.016
(-1, 0)
1749
-0.883 -0.860
0.022
(-1, 1)
1727
-0.880 -0.837
0.043**
(-1, 3)
1663
-0.878 -0.800
0.078***
(-1, 5)
1615
-0.882 -0.785
0.096***
(-1, 10)
1403
-0.879 -0.722
0.158***
Panel B. Physical skewness change: Matched sample analysis (N=1,520)
Before
After
Treated
Matched
Dif.
t
Treated Matched
Dif.
Log (BM)
-1.771
-1.762
-0.008
-0.43
Log (Price)
4.145
4.049 0.096*** 14.47
R t-12,t-1
1.242
1.114 0.128***
2.81
Log (Size)
15.096
15.261 -0.165***
-4.67
Skew
0.384
0.386
-0.273
-0.85
0.151
0.190 -0.039
t
11.06
10.68
11.20
13.02
13.10
11.38
10.00
12.02
-0.36
0.29
0.78
1.23
2.28
3.68
4.14
6.14
t
-1.23
54
Table 9. RNSkew around reverse stock splits
Reverse stock splits are events with a CRSP distribution code of 5523 and a split ratio less than 1-for-1.
RNSkew is calculated following Bakshi, Kapadia, and Madan (2003) and is detailed in the paper. Panel A
reports changes in RNSkew and Panel B reports the change in physical skewness. We report the change in
RNSkew around the ex-date. A paired t-test is used to gauge statistical significance. Different windows are
chosen. For example, for (-3, 0) the before column displays RNSkew on day -3 relative to the event date, and
the after column displays RNSkew on day 0 relative to the event date (i.e., the date of the event). For each given
window, we require RNSkew to be available on both days. In Panel B, for each split stock, we require the
matched stock and the split stock to be in the same size quintile, BM quintile, momentum quintile and skewness
quintile. We also require that the skewness difference between the matched stock and the split stock is lower
than 0.25. If more than one stock satisfies the above criteria, we choose the one with the smallest price
difference with the split stock. All the information used for matching is available at the time of split. B/M is the
ratio of book value of common equity to market price. R t-12,t-1 is the cumulative return from month t-12 to month
t-1. Size is equal to the log market capitalization at the end of month t-1. Pre-matching skewness is calculated
using daily returns from month t-11 to month t-1. Post-matching skewness is calculated using daily returns from
month t+1 to month t+12. The sample period is January 1996 to December 2012.
Panel A. Change of RNSkew
Window
N
Before
(-10, 0)
123
0.255
(-5, 0)
126
0.320
(-3, 0)
127
0.332
(-1, 0)
130
0.553
(-1, 1)
128
0.566
(-1, 3)
128
0.566
(-1, 5)
128
0.566
(-1, 10)
126
0.570
Panel B. Change of physical skewness (N=61)
Before
Treated Matched
Dif
Log (BM)
-0.509
-0.401
-0.108*
Log (Price)
1.507
2.223 -0.716***
R t-12,t-1
-0.184
-0.153
-0.030
Log (Size)
13.566
13.321
0.245**
Skew
0.552
0.538
0.013
After
-0.426
-0.434
-0.455
-0.470
-0.408
-0.407
-0.423
-0.321
t
-1.76
-8.06
-1.61
2.07
0.76
Change
-0.681***
-0.754***
-0.787***
-1.023***
-0.974***
-0.974***
-0.989***
-0.891***
Treated
0.674
After
Matched
0.731
t
-5.11
-6.00
-6.18
-7.81
-7.94
-7.81
-8.08
-7.75
Dif
t
-0.057
-0.24
55
Table 10. New rule period: Regular splits
This table reports the change of RNSkew around regular stock splits for the new rule period. Before
September 4, 2007, option strike prices were adjusted to the nearest 1/8th to undo the effect of stock splits.
The new rule effective September 4, 2007 leaves option contracts untouched, and instead recalculates the
stock price to the hypothetical price it would trade had the split not occurred. The new rule affected all
splits except 2 for 1 or 4 for 1 splits, and extended until February 12, 2010. Regular stock splits are events
with a CRSP distribution code of 5523 and a split ratio greater than or equal to 1.25-for-1. The new rule
period is from September 4, 2007 to February 12, 2010. We exclude 2-for-1 and 4-for-1 splits. RNSkew is
calculated following Bakshi, Kapadia, and Madan (2003). We report the change in RNSkew around the
ex-date and the announcement date. A paired t-test is used to gauge statistical significance. Different
windows are chosen. For example, for (-3, 0) the before column displays RNSkew on day -3 relative to the
event date, and the after column displays RNSkew on day 0 relative to the event date (i.e., the date of the
event). For each given window, we require RNSkew to be available on both days.
Window
(-10, 0)
(-5, 0)
(-3, 0)
(-1, 0)
(-1, 1)
(-1, 3)
(-1, 5)
(-1, 10)
N
22
22
22
22
21
22
22
22
Before
-0.613
-0.413
-0.381
-0.606
-0.606
-0.606
-0.606
-0.606
After
1.186
1.186
1.186
1.186
1.226
1.273
1.286
1.307
Mean Change
1.799***
1.599***
1.567***
1.792***
1.833***
1.879***
1.893***
1.913***
t
6.45
5.73
5.69
6.72
7.32
7.12
7.62
7.63
56
Table 11. Implied volatility change around stock splits
This table reports the change of implied volatility (measured in percent) around stock splits, both regular
splits (Panel A) and reverse splits (Panel B). Regular stock splits are events with a CRSP distribution code
of 5523 and a split ratio greater than or equal to 1.25-for-1. Reverse stock splits are events with a CRSP
distribution code of 5523 and a split ratio less than 1-for-1. Implied volatility is obtained from
OptionMetrics. ATM options are defined as call options with delta higher than 0.375 and not higher than
0.625 and put options with delta higher than -0.625 and not higher than -0.375. OTM options are call
options with delta greater than 0.02 and not greater than 0.375 and put options with delta greater than
-0.375 and not greater than -0.02. As in Bollen and Whaley (2004), we exclude options with absolute delta
below 0.02 due to the distortions caused by price discreteness. We report changes in the difference between
the implied volatility of OTM call options and ATM call options (Panel A1 and Panel B1) and the
difference between the implied volatility of OTM put options and ATM put options (Panel A2 and Panel
B2), around the ex-date. A paired t-test is used to gauge statistical significance. Different windows are
chosen. For example, for (-3, 0) the before column displays IV differences on day -3 relative to the event
date, and the after column displays IV differences on day 0 relative to the event date (i.e., the date of the
event). For each given window, we require the IV difference to be available on both days. The sample
period is January 1996 to December 2012.
Panel A. Regular splits
Window
N
Panel A1: OTM call - ATM call
(-10, 0)
2171
(-5, 0)
2181
(-3, 0)
2193
(-1, 0)
2196
(-1, 1)
2196
(-1, 3)
2195
(-1, 5)
2193
(-1, 10)
2191
Panel A2: OTM put - ATM put
(-10, 0)
2171
(-5, 0)
2181
(-3, 0)
2193
(-1, 0)
2196
(-1, 1)
2196
(-1, 3)
2195
(-1, 5)
2193
(-1, 10)
2191
Before
After
Change
t
-1.493
-1.675
-1.677
-1.699
-1.699
-1.709
-1.703
-1.698
-0.597
-0.591
-0.647
-0.657
-0.780
-0.302
-0.216
0.189
0.896***
1.084***
1.030***
1.042***
0.919***
1.407***
1.487***
1.888***
2.73
3.27
3.29
3.75
3.04
4.64
4.70
5.94
2.335
2.401
2.551
2.508
2.508
2.508
2.508
2.512
2.122
2.121
2.121
2.121
2.206
2.260
2.398
2.622
-0.213
-0.281
-0.430**
-0.387**
-0.303
-0.248
-0.110
0.110
-1.01
-1.38
-2.28
-2.01
-1.65
-1.21
-0.52
0.55
57
Table 11 (continued)
Panel B. Reverse splits
Window
N
Panel A1: OTM call - ATM call
(-10, 0)
197
(-5, 0)
199
(-3, 0)
199
(-1, 0)
200
(-1, 1)
200
(-1, 3)
197
(-1, 5)
196
(-1, 10)
193
Panel A2: OTM put - ATM put
(-10, 0)
197
(-5, 0)
199
(-3, 0)
199
(-1, 0)
200
(-1, 1)
200
(-1, 3)
197
(-1, 5)
196
(-1, 10)
193
Before
After
Change
t
5.335
6.864
5.029
9.346
9.346
8.846
8.931
8.641
-1.171
-1.289
-1.289
-1.288
-4.480
0.059
-1.051
-0.149
-6.505***
-8.152***
-6.318***
-10.634***
-13.825***
-8.787***
-9.982***
-10.128***
-3.11
-3.67
-2.90
-5.04
-6.88
-4.12
-5.11
-5.03
1.053
-2.570
2.073
2.036
2.036
1.590
0.984
0.945
-0.585
-0.663
-0.663
-0.679
-0.438
-0.481
-0.665
-1.745
-1.638
1.907
-2.735
-2.716
-2.474
-2.071
-1.649
-2.690
-0.73
0.87
-1.25
-1.33
-1.17
-0.98
-0.85
-1.31
58
Online Appendix for “Nominal Price Illusion”
The Online Appendix reports the robustness tests mentioned in the paper but not reported
in the paper.
Figure A1 shows RNSkew around stock splits (both regular splits and reverse splits). It is
the same as Figure 1, but now also shows the confidence interval.
Table A1 to Table A5 report robustness tests of the Fama-MacBeth regression analysis of
RNSkew-Skew and RNSkew-E(Skew). In Table A1, we include stocks priced lower than $10.
In Table A2, we calculate physical volatility and physical skewness from log returns (instead
of raw returns). In Table A3, we standardize RNSkew to 60 days or 100 days, instead of 30
days as is done in the paper. In Table A4 we use physical skewness estimated over a 30-day
window. In Table A5 we include two firm size tercile dummy variables instead of including
the continuous measure of firm size.
Table A6 reports factor loadings for the option portfolios
Table A7 reports the analysis of physical skewness change around stock splits for a much
larger sample: 1963 to 2012.
Table A8 and A9 examine liquidity changes around the ex-date.
Table A10 reports the analysis of RNSkew change for ETF splits.
59
Figure A1. RNSkew around stock splits
This figure shows RNSkew around stock splits. The solid line with square markers represents reverse splits
while the solid line without square markers represents regular splits. The four dashed lines represent 95%
confidence intervals. The graph plots RNSkew against trading days relative to the split. Day 0 is the ex-date.
RNSkew is calculated following Bakshi, Kapadia, and Madan (2003). The sample period is from January 1996
to December 2012.
0.80
0.60
0.40
0.20
0.00
-0.20
-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5
0
5 10 15 20 25 30 35 40 45 50 55 60
-0.40
-0.60
-0.80
-1.00
Day Relative to Split
Regular Split
Reverse Split
60
Table A1. The relationship between nominal price and skewness: Fama-MacBeth regressions (including
stocks with price lower than or equal to $10)
The model specification is the same as Table 2 except we no longer exclude stocks under $10.
Log (Price)
(1)
-0.292***
(-10.16)
AT/Price
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
Constant
Industry FE
Adj-R2
Obs.
0.146
(1.63)
No
0.019
192
RNSkew-Skew
(2)
-0.158***
(-3.67)
-0.009
(-0.78)
0.024
(1.35)
-0.110***
(-5.09)
-0.057***
(-2.96)
-0.364***
(-11.00)
-0.063
(-0.30)
-0.043***
(-4.51)
0.053*
(1.71)
0.019***
(3.18)
0.081***
(3.26)
0.018
(0.89)
0.009
(0.43)
0.749**
(2.31)
No
0.055
192
(3)
-0.173***
(-3.89)
0.016
(1.24)
0.024
(1.45)
-0.093***
(-4.64)
-0.073***
(-3.99)
-0.361***
(-9.09)
-0.033
(-0.16)
-0.039***
(-3.99)
0.048*
(1.72)
0.020***
(3.69)
0.055**
(2.25)
0.011
(0.58)
0.026
(1.20)
0.662**
(2.13)
Yes
0.104
192
(4)
-0.263***
(-9.85)
0.035
(0.35)
No
0.037
192
RNSkew-E(Skew)
(5)
-0.133***
(-3.29)
0.011
(0.90)
-0.012
(-0.64)
-0.067***
(-4.27)
-0.084***
(-4.06)
-0.295***
(-8.41)
0.291**
(2.54)
-0.070***
(-10.03)
0.052**
(2.07)
0.020***
(3.18)
0.069**
(2.62)
0.027
(1.18)
0.054***
(2.94)
0.154
(0.81)
No
0.140
192
(6)
-0.138***
(-3.58)
0.057***
(5.83)
0.002
(0.11)
-0.045***
(-3.40)
-0.119***
(-6.47)
-0.358***
(-8.90)
0.283**
(2.66)
-0.046***
(-6.34)
0.051**
(2.26)
0.021***
(3.35)
0.045*
(1.81)
0.022
(1.18)
0.084***
(4.53)
-0.036
(-0.17)
Yes
0.249
192
61
Table A2. The relationship between nominal price and skewness: Fama-MacBeth regressions (physical
volatility and skewness measures calculated from logged daily returns)
The model specification is the same as Table 2, except we now calculate physical volatility and physical
skewness from logged daily returns, rather than raw returns.
Log (Price)
AT/Price
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
Constant
Industry FE
Adj-R2
Obs.
RNSkew-Skew
(2)
-0.152***
(-3.40)
-0.001
(-0.10)
0.002
(0.11)
-0.171***
(-6.78)
-0.101***
(-5.16)
-0.373***
(-9.74)
0.342
(1.36)
-0.030***
(-3.01)
0.024
(0.64)
0.020**
(2.43)
0.119***
(4.88)
0.031
(1.65)
0.047*
(1.68)
0.949***
1.576***
(8.87)
(4.48)
No
No
0.032
0.077
192
192
(1)
-0.425***
(-14.78)
(3)
-0.138***
(-2.96)
0.024*
(1.69)
0.014
(0.77)
-0.166***
(-7.41)
-0.098***
(-5.02)
-0.344***
(-7.67)
0.365
(1.50)
-0.023**
(-2.41)
0.018
(0.53)
0.023***
(2.87)
0.110***
(4.54)
0.029
(1.58)
0.031
(1.03)
1.503***
(4.29)
Yes
0.125
192
RNSkew-E(Skew)
(5)
(6)
-0.128***
-0.137***
(-3.14)
(-3.54)
0.012
0.057***
(0.96)
(5.89)
-0.006
0.005
(-0.33)
(0.30)
-0.072***
-0.048***
(-4.53)
(-3.60)
-0.085***
-0.125***
(-4.05)
(-6.43)
-0.301***
-0.361***
(-8.53)
(-8.99)
0.161
0.193*
(1.40)
(1.83)
-0.053***
-0.033***
(-7.28)
(-4.75)
0.049*
0.049**
(1.95)
(2.16)
0.021***
0.021***
(3.31)
(3.41)
0.080***
0.051**
(2.98)
(2.05)
0.032
0.025
(1.41)
(1.33)
0.055***
0.085***
(2.90)
(4.52)
0.829***
0.237
0.020
(8.73)
(1.25)
(0.10)
No
No
Yes
0.037
0.139
0.248
192
192
192
(4)
-0.393***
(-14.82)
62
Table A3. The relationship between nominal price and skewness: Fama-MacBeth regressions (standardizing RNSkew to 60 days or 100 days)
The model specification is the same as Table 2, except that we standardize RNSkew to 60 days or 100 days by linear interpolation.
60 days
100 days
RNSkew-Skew
RNSkew-E(Skew)
RNSkew-Skew
RNSkew-E(Skew)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Log (Price)
-0.270*** -0.131*** -0.148***
-0.246*** -0.109*** -0.116***
-0.281*** -0.143*** -0.159***
-0.257*** -0.122*** -0.127***
(-11.71)
(-3.59)
(-4.11)
(-10.00)
(-2.83)
(-3.24)
(-14.04)
(-3.99)
(-4.50)
(-11.17)
(-3.11)
(-3.53)
AT/Price
-0.003
0.023
0.015
0.062***
0.001
0.026
0.020
0.065***
(-0.24)
(1.50)
(1.14)
(5.72)
(0.09)
(1.58)
(1.36)
(5.56)
Beta
0.008
0.015
-0.012
0.004
0.009
0.016
-0.010
0.005
(0.44)
(0.85)
(-0.63)
(0.21)
(0.52)
(0.91)
(-0.55)
(0.26)
Log (Size)
-0.108*** -0.091***
-0.068*** -0.047***
-0.116*** -0.101***
-0.076*** -0.057***
(-4.99)
(-4.51)
(-4.31)
(-3.54)
(-5.24)
(-4.84)
(-4.48)
(-3.84)
Log (BM)
-0.067*** -0.081***
-0.091*** -0.126***
-0.075*** -0.088***
-0.099*** -0.133***
(-3.24)
(-3.82)
(-4.13)
(-6.46)
(-3.50)
(-4.02)
(-4.23)
(-6.52)
Leverage
-0.393*** -0.394***
-0.305*** -0.368***
-0.405*** -0.407***
-0.316*** -0.382***
(-10.59)
(-8.21)
(-8.32)
(-9.01)
(-11.15)
(-8.32)
(-8.26)
(-8.90)
Past Volatility
0.111
0.107
0.355*** 0.339***
0.097
0.092
0.340*** 0.325***
(0.52)
(0.53)
(3.02)
(3.18)
(0.46)
(0.46)
(2.89)
(3.13)
Past Skewness
-0.044*** -0.039***
-0.072*** -0.047***
-0.044*** -0.038***
-0.071*** -0.046***
(-4.09)
(-3.64)
(-10.00)
(-6.30)
(-4.07)
(-3.59)
(-10.16)
(-6.36)
R t-12,t-1
0.028
0.028
0.038
0.039*
0.011
0.013
0.022
0.024
(0.96)
(1.02)
(1.56)
(1.74)
(0.39)
(0.44)
(0.87)
(1.02)
R t-60,t-13
0.019*** 0.022***
0.021*** 0.021***
0.020*** 0.022***
0.022*** 0.021***
(3.37)
(3.80)
(3.41)
(3.56)
(3.31)
(3.77)
(3.32)
(3.50)
Log (Volume)
0.071***
0.043*
0.077***
0.052**
0.061**
0.034
0.067***
0.044*
(2.82)
(1.77)
(3.10)
(2.25)
(2.43)
(1.44)
(2.68)
(1.90)
Log (ILLIQ)
0.005
-0.005
0.025
0.019
-0.005
-0.015
0.015
0.010
(0.27)
(-0.28)
(1.04)
(0.96)
(-0.27)
(-0.87)
(0.60)
(0.46)
NASDAQ
0.003
0.021
0.049*** 0.079***
-0.004
0.015
0.043**
0.072***
(0.13)
(1.02)
(2.67)
(4.32)
(-0.18)
(0.71)
(2.31)
(4.02)
Constant
0.090
0.457
0.360
-0.001
-0.048
-0.243***
0.141*
0.551*
0.471
0.050
0.045
-0.132
(1.03)
(1.42)
(1.11)
(-0.01)
(-0.24)
(9.00)
(1.74)
(1.83)
(1.46)
(0.54)
(0.19)
(-0.59)
Industry FE
No
No
Yes
No
No
Yes
No
No
Yes
No
No
Yes
Adj-R2
0.013
0.052
0.106
0.034
0.147
0.260
0.013
0.054
0.109
0.038
0.162
0.281
Obs.
192
192
192
192
192
192
192
192
192
192
192
192
63
Table A4. The relationship between nominal price and skewness: Fama-MacBeth
regressions (using 30-day physical skewness)
The model specification is the same as Table 2, except that we calculate physical skew over a 30-day horizon
rather than one-year horizon.
Log (Price)
AT/Price
Beta
Log (Size)
Log (BM)
Leverage
Past Volatility
Past Skewness
R t-12,t-1
R t-60,t-13
Log (Volume)
Log (ILLIQ)
NASDAQ
Constant
Industry FE
Adj-R2
Obs.
RNSkew-Skew
(1)
(2)
(3)
-0.425*** -0.412*** -0.436***
(-14.47)
(-9.29)
(-9.84)
0.063*** 0.099***
(4.56)
(9.46)
-0.016*
-0.017**
(-1.83)
(-2.12)
-0.231*** -0.229***
(-11.78)
(-12.78)
-0.111*** -0.138***
(-6.45)
(-8.74)
-0.317*** -0.414***
(-9.87)
(-13.84)
0.077
0.040
(0.46)
(0.26)
-0.187*** -0.184***
(-17.04)
(-16.55)
-0.040** -0.040***
(-2.16)
(-3.08)
-0.000
0.000
(-0.04)
(0.03)
-0.083** -0.109***
(-2.38)
(-3.28)
-0.206*** -0.224***
(-8.76)
(-8.87)
-0.001
0.019***
(-0.19)
(2.78)
0.638***
0.643**
0.579*
(7.22)
(1.99)
(1.79)
No
No
Yes
0.040
0.120
0.176
192
192
192
64
Table A5. The relationship between nominal price and skewness: Fama-MacBeth regressions (different
specification for size)
The model specification is the same as Table 2, except we control for size using two firm size dummy variables
rather than the continuous measure of size. In each month, stocks are sorted based on their market capitalization
(price per share times number of shares outstanding) at the end of the previous month. All stocks are sorted into
terciles based on NYSE breakpoints. Small and medium sized firm dummies indicate the small and medium
sized stocks, respectively.
RNSkew-Skew
RNSkew-E(Skew)
(1)
(2)
(3)
(4)
(5)
(6)
Log (Price)
-0.286*** -0.176*** -0.185***
-0.262*** -0.156*** -0.152***
(-10.83)
(-4.17)
(-4.57)
(-9.84)
(-3.81)
(-3.97)
AT/Price
-0.015
0.015
0.006
0.054***
(-1.18)
(1.03)
(0.51)
(5.72)
Beta
0.009
0.015
-0.013
0.000
(0.47)
(0.83)
(-0.72)
(0.00)
Log (BM)
-0.046*** -0.066***
-0.075*** -0.114***
(-2.70)
(-3.65)
(-3.89)
(-6.47)
Leverage
-0.361*** -0.360***
-0.281*** -0.345***
(-10.00)
(-7.88)
(-8.32)
(-8.73)
Past Volatility
0.207
0.168
0.430*** 0.377***
(1.19)
(1.05)
(3.72)
(3.46)
Past Skewness
-0.046*** -0.040***
-0.073*** -0.047***
(-4.49)
(-3.91)
(-9.81)
(-6.28)
R t-12,t-1
0.046*
0.044*
0.053**
0.053**
(1.67)
(1.77)
(2.14)
(2.32)
R t-60,t-13
0.020*** 0.023***
0.021*** 0.022***
(3.57)
(3.75)
(3.31)
(3.41)
Log (Volume)
0.034
0.012
0.048*
0.032
(1.29)
(0.48)
(1.83)
(1.36)
Log (ILLIQ)
0.044**
0.028*
0.052**
0.042**
(2.44)
(1.71)
(2.39)
(2.20)
NASDAQ
0.021
0.037
0.066*** 0.092***
(0.81)
(1.44)
(3.36)
(4.79)
Small size firm dummy
0.102***
0.103**
0.021
0.011
(2.85)
(2.61)
(0.68)
(0.35)
Medium size firm dummy
0.019
0.006
0.008
-0.007
(0.63)
(0.19)
(0.51)
(-0.50)
Constant
0.124
0.173
0.087
0.033
-0.032
-0.142
(1.32)
(0.51)
(0.22)
(0.33)
(-0.17)
(-0.60)
Industry FE
No
No
Yes
No
No
Yes
Adj-R2
0.013
0.052
0.106
0.034
0.139
0.250
Obs.
192
192
192
192
192
192
65
Table A6. Portfolio (both delta hedged and non-delta hedged option portfolio) factor loadings
Fama-French factors and momentum are included. S&P500 is the new option factor constructed ourselves. It is
constructed as if the SP500 index option is an individual stock option. Panel A reports factor loadings for delta
hedged call option returns. Panel B reports loadings for non-delta hedged call option returns.
Panel A. Delta hedged
Price
Lowest
Panel A1. OTM
Mktrf
-0.349***
(-5.47)
SMB
-0.184**
(-1.98)
HML
0.019
(0.23)
UMD
0.077
(1.38)
SP500
0.619***
(9.72)
Adj-R2
0.639
Panel A2. ATM
Mktrf
-0.174***
(-4.62)
SMB
-0.035
(-0.59)
HML
-0.048
(-1.33)
UMD
-0.047
(-1.33)
SP500
0.795***
(14.11)
Adj-R2
0.642
2
3
4
Highest
Highest - Lowest
-0.336***
(-5.54)
-0.217**
(-2.45)
0.115
(1.44)
0.142***
(2.69)
0.582***
(9.59)
0.660
-0.211***
(-4.92)
-0.084
(-1.35)
0.028
(0.49)
0.036
(0.95)
0.383***
(8.95)
0.413
-0.221***
(-3.65)
-0.050
(-0.57)
0.139**
(1.74)
-0.059
(-1.11)
0.750***
(12.41)
0.632
0.267***
(3.80)
-0.291***
(-2.83)
0.076
(0.82)
-0.028
(-0.46)
1.108***
(15.76)
0.626
0.616***
(8.50)
-0.106
(-1.01)
0.056
(0.59)
-0.105*
(-1.66)
0.489***
(6.76)
0.319
-0.086***
(-4.07)
-0.079**
(-2.39)
-0.023
(-0.78)
0.003
(0.17)
0.552***
(17.47)
0.580
-0.128***
(-3.80)
-0.064
(-1.20)
-0.002
(-0.03)
0.028
(0.89)
0.735***
(14.60)
0.665
-0.129***
(-3.68)
-0.093*
(-1.68)
0.009
(0.19)
-0.029
(-0.88)
0.738***
(14.03)
0.633
-0.019
(-0.45)
-0.128*
(-1.92)
0.031
(0.52)
0.002
(0.05)
0.802***
(12.60)
0.536
0.155***
(4.05)
-0.093
(-1.56)
0.079
(1.47)
0.049
(1.36)
0.007
(0.12)
0.068
66
Table A6 (continued)
Panel B. Non-delta hedged
Price
Lowest
Panel B1. OTM
Mktrf
4.108***
(5.89)
SMB
6.436***
(6.11)
HML
1.009
(1.09)
UMD
-1.228**
(-2.01)
SP500
0.127***
(8.02)
Adj-R2
0.587
Panel B2. ATM
Mktrf
2.658***
(4.77)
SMB
6.871***
(10.39)
HML
0.791
(1.38)
UMD
-1.238***
(-3.26)
SP500
0.278***
(11.65)
Adj-R2
0.807
2
3
4
Highest
Highest - Lowest
4.818***
(7.04)
2.623**
(2.54)
0.781
(0.86)
0.161
(0.27)
0.118***
(7.59)
0.543
4.493***
(5.80)
2.386**
(2.04)
0.110
(0.11)
1.274*
(1.88)
0.153***
(8.66)
0.527
5.821***
(6.04)
2.261
(1.55)
0.275
(0.21)
2.471***
(2.93)
0.192***
(8.75)
0.533
6.921***
(5.94)
-0.195
(-0.11)
-2.984*
(-1.93)
3.017***
(2.96)
0.434***
(16.38)
0.741
2.813**
(2.56)
-6.631***
(-4.00)
-3.993***
(-2.74)
4.245***
(4.42)
0.307***
(12.27)
0.622
3.480***
(6.73)
4.335***
(7.07)
0.755
(1.42)
-0.133
(-0.38)
0.290***
(13.07)
0.829
3.573***
(6.60)
3.647***
(5.68)
0.225
(0.40)
0.884**
(2.40)
0.318***
(13.70)
0.826
3.292***
(5.42)
3.135***
(4.35)
-0.106
(-0.17)
1.684***
(4.08)
0.361***
(13.88)
0.807
3.684***
(5.05)
2.437***
(2.81)
-1.941**
(-2.59)
2.587***
(5.21)
0.472***
(15.08)
0.823
1.026
(1.54)
-4.434***
(-5.60)
-2.732***
(-3.99)
3.825***
(8.43)
0.194***
(6.78)
0.595
67
Table A7. Physical skewness around stock splits: 1963-2012
Regular stock splits are events with a CRSP distribution code of 5523 and a split ratio greater than or equal to
1.25-for-1. Panel A reports changes in physical skewness for regular splits and Panel B reports the change in
physical skewness for reverse splits. We use a paired t-test to gauge statistical significance. For each split stock,
we require the matched stock and the split stock to be in the same size quintile, BM quintile, momentum quintile
and skewness quintile. We also require that the skewness difference between the matched stock and the split
stock is lower than 0.25. If more than one stock satisfies the above criteria, we choose the one with the smallest
price difference with the split stock. All the information used for matching is available at the time of split. B/M
is the ratio of book value of common equity to market price. Momentum is the cumulative return from month
t-12 to month t-1. Size is equal to the log market capitalization at the end of month t-1. Pre-matching skewness
is calculated using daily returns from month t-11 to month t-1. Post-matching skewness is calculated using daily
returns from month t+1 to month t+12. The sample period is January 1963 to December 2012.
Panel A. Physical skewness change for regular splits: Matched sample analysis (N=11,005)
Before
After
Treated Matched
Dif.
t Treated Matched
Dif.
t
Log (BM)
-1.034
-1.027
-0.007
-1.47
Log (Price)
3.596
3.452 0.144*** 41.64
Momentum
0.865
0.773 0.093***
9.88
Log (Size)
12.664
12.812 -0.148*** -17.51
Skew
0.528
0.526
0.002
1.52
0.336
0.337 -0.001 -0.10
Panel B. Physical skewness change for reverse splits: Matched sample analysis (N=1,607)
Before
After
Treated Matched
Dif.
t Treated Matched
Dif.
t
Log (BM)
-0.359
-0.346
-0.013
-0.79
Log (Price)
-0.334
0.170 -0.504*** -25.53
Momentum
-0.247
-0.237
-0.010
-0.88
Log (Size)
9.615
9.579
0.036*
1.69
Skew
0.937
0.932
0.005
1.55
1.238
1.137 0.100 1.85
68
Table A8. Liquidity change around splits
This table shows the liquidity change around stock splits, for both regular stock splits and also reverse splits. We
examine two liquidity variables, both calculated based on data from TAQ. The definitions are:
Percentage spread=2*Spread/(Bid+Ask),
Percentage effective spread=2*Effective Spread/(Bid+Ask),
where Bid, Ask, and Price are the best bid price, best ask price, and the transaction price, respectively. Liquidity
variables in the Before period is calculated from month t-11 to month t-1. Liquidity variables in the After period
is calculated from month t+1 to month t+12.
Before
Panel A. Regular splits
Percentage spread (*100)
Percentage effective spread (*100)
Panel B. Reverse splits
Percentage spread (*100)
Percentage effective spread (*100)
Treated
After
Dif.
t-stat
0.400
0.309
0.474
0.356
0.074
0.047
10.64
9.65
0.748
0.633
0.568
0.443
-0.180
-0.190
-2.35
-2.92
69
Table A9. Cross-sectional correlation: Liquidity change and RNSkew change
This table shows the correlation (from pooled data) of liquidity changes and RNSkew changes around stock
splits (regular splits and reverse splits, separately). P-values are in the parentheses.
Regular
Reverse
Relative spread
0.025
(0.33)
0.144
(0.28)
Relative effective spread
0.015
(0.57)
0.143
(0.29)
70
Table A10. RNSkew around ETF splits
This table shows the change of RNSkew around the ex-date for regular ETF splits. The table format is the same
as Panel A1 of Table 2. ETFs are securities with a share code of 73 in CRSP. The average split ratio is 2.69 for 1.
The largest split ratio is 10 for 1 and the smallest is 2 for 1.
Window
(-10, 0)
(-5, 0)
(-3, 0)
(-1, 0)
(-1, 1)
(-1, 3)
(-1, 5)
(-1, 10)
N
42
42
42
42
42
42
42
42
Before
-0.821
-0.941
-0.742
-0.810
-0.810
-0.790
-0.810
-0.810
After
-0.702
-0.702
-0.739
-0.739
-0.739
-0.742
-0.875
-0.769
Change
0.120
0.239
0.003
0.071
0.071
0.048
-0.065
0.041
t
0.89
1.90*
0.02
0.66
0.83
0.58
-0.76
0.42
71