Ex Ante Skewness and Expected Stock Returns
First Draft: March 2007
This Draft: January 11, 2012
Abstract
We use a sample of option prices to estimate the ex ante higher moments of the underlying individual
securities’ risk-neutral returns distribution. We find that individual securities’ risk neutral volatility,
skewness, and kurtosis are strongly related to future returns. Specifically, we find a negative relation
between past volatility and subsequent returns in the cross-section. We also find that ex ante more
negatively (positively) skewed returns are associated with subsequent higher (lower) returns, while
ex ante kurtosis is positively related to subsequent returns. We analyze the extent to which these
returns relations represent compensation for risk and find evidence that, even after controlling for
differences in co-moments, individual securities’ skewness matters.
1 Introduction
Models implying that investors consider higher moments in returns have a long history in the
literature. Researchers such as Rubinstein (1973), Kraus and Litzenberger (1976) and Kraus
and Litzenberger (1983) develop models of expected returns which incorporate skewness. In
these models, the higher moments which are relevant for individual securities are co-moments
with the aggregate market portfolio. Subsequently, empirical work provided evidence that
higher moments of the return distribution are important in pricing securities. Consistent
with the models’ focus on co-moments, the tests in these papers ask whether a security’s co-
skewness or co-kurtosis with the market is priced; historical returns data are typically used to
measure these co-moments. For example, Harvey and Siddique (2000) explore both skewness
and co-skewness and test whether co-skewness is priced, and Dittmar (2002) tests whether
a security’s co-skewness and co-kurtosis with the market portfolio might influence investors’
expected returns.
Other recent papers have suggested that additional features of individual securities’ pay-
off distribution may be relevant for understanding differences in assets’ returns. For ex-
ample, Ang, Hodrick, Xing, and Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009)
document that firms’ idiosyncratic return volatility contains important information about fu-
ture returns. The work of Barberis and Huang (2008), Brunnermeier, Gollier, and Parker
(2007), and the empirical evidence presented in Mitton and Vorkink (2007) and Boyer, Mit-
ton, and Vorkink (2010) imply that the skewness of individual securities may also influence
investors’ portfolio decisions. Xing, Zhang, and Zhao (2010) find that portfolios formed by
sorting individual securities on a measure which is related to idiosyncratic skewness gen-
erate cross-sectional differences in returns. Green and Hwang (2009) use the approach of
Zhang (2006) and find that IPOs with high expected skewness (’lottery’ stocks) experience
significantly greater first-day returns, followed by substantially greater negative abnormal
returns in the subsequent three to five years.
We therefore have two strands in the existing literature: (1) models and empirical re-
sults that emphasize the importance of higher moments as they affect stochastic discount
factors, (2) models and empirical evidence that focus on the higher moment characteristics
of individual securities. In this second strand of the literature, researchers have proposed
both behavioral and rational models. For example, Barberis and Huang (2008) argue that
investors with cumulative prospect theory preferences demand securities with highly skewed
payoffs, such as IPO stocks. Brunnermeier, Gollier, and Parker (2007) develop a model of op-
timal (as opposed to rational) beliefs which also predicts that investors will overinvest in the
most highly (right) skewed securities, with the consequence that those securities will have
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lower subsequent average returns. They also show that, while there is a rational expectations
solution to their model, it represents a knife-edge case. Mitton and Vorkink (2007) introduce
a rational model where investors have heterogeneous preferences for skewness and show that
idiosyncratic skewness can impact prices. Chabi-Yo, Ghysels, and Renault (2010) also present
a model where heterogeneity of beliefs across rational investors can generate a pricing kernel
which depends on idiosyncratic moments.
In this paper, we examine the importance of higher moments using a new approach. We
exploit the fact that if option and stock prices reflect the same information, then it is possi-
ble to use options market data to extract estimates of the higher moments of the securities’
(risk-neutral) probability density function. Our method has several advantages. First, option
prices are a market-based estimate of investors’ expectations. Many authors, including Bates
(1991), Rubinstein (1994) and Jackwerth and Rubinstein (1996) have argued that option mar-
ket prices can capture the information of market participants. Second, the use of option prices
eliminates the need of a long time series of returns to estimate the moments of the return dis-
tribution; this is especially helpful when trying to forecast the payoff distribution of relatively
new firms or during periods where expectations, at least for some firms, may change relatively
quickly. Third, options reflect a true ex ante measure of expectations; they do not give us, as
Battalio and Schultz (2006) note, the “unfair advantage of hindsight.” As Jackwerth and Ru-
binstein (1996) state, “not only can the nonparametric method reflect the possibly complex
logic used by market participants to consider the significance of extreme events, but it also
implicitly brings a much larger set of information . . . to bear on the formulation of probability
distributions.”
We begin with a sample of options on individual stocks, and test whether cross-sectional
differences in estimates of the higher moments of an individual security’s payoff extracted
from options are related to subsequent returns. Consistent with the Ang, Hodrick, Xing, and
Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009) findings for physical measures of
idiosyncratic volatility, we find a negative relation between risk-neutral volatility and subse-
quent returns. We also document a significant negative relation between firms’ risk-neutral
skewness and subsequent returns - that is, more negatively skewed securities have higher
subsequent returns. In addition, we find a significant positive relation between firms’ risk-
neutral kurtosis and subsequent returns. These relations persist after controlling for firm
characteristics, such as beta, size, and book-to-market ratios, and adjustment for the Fama
and French (1993) risk factors.
We examine the extent to which these relations between risk-neutral higher moments and
subsequent returns are determined by co-moments with the market portfolio. We measure
co-moments using the approaches of Harvey and Siddique (2000) and Bakshi, Kapadia, and
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Madan (2003), and then decompose total moments into co-moments – such as co-skewness –
and idiosyncratic moments. We find that the relation between idiosyncratic higher moments,
particularly idiosyncratic skewness, and subsequent returns persists, even after controlling
for differences in covariance, co-skewness and co-kurtosis.
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Our results are consistent with models such as Brunnermeier, Gollier, and Parker (2007),
and Barberis and Huang (2008), which predict that investors will trade off the benefits of
diversification and skewness, holding more concentrated positions in skewed securities, and
resulting in a negative relation between idiosyncratic skewness and expected returns. These
results are also consistent with the empirical evidence in Mitton and Vorkink (2007), who
examine the choices of investors in a sample of discount brokerage accounts and find that
investors appear to hold relatively undiversified portfolios and accept lower Sharpe ratios
for positively skewed portfolios and securities. These papers focus on physical moments of
returns, in contrast to the risk-neutral moments that we examine. Consequently, we ana-
lyze the relation between our risk-neutral estimates of skewness and estimates formed from
historical returns. We find a positive and statistically significant relation between these esti-
mates; however, we find comparatively little evidence that the relation between risk-neutral
moments and subsequent returns in our sample is driven by this relation; that is, after con-
trolling for differences in physical moments, the predictive relation between risk-neutral mo-
ments and subsequent returns continues to hold. In contrast, after controlling for differences
in risk-neutral moments, we find no clear pattern in returns for portfolios which differ in
physical skew.
The remainder of the paper is organized as follows. In section 2, we detail the method we
employ for recovering measures of volatility, skewness, and kurtosis, following Bakshi, Ka-
padia, and Madan (2003) and we discuss the data (and data filters) used in our analysis. In
Section 3 we focus on testing whether estimates of the ex ante higher moments of the payoff
distribution obtained from options data are related to the subsequent returns of the underly-
ing security. In Section 4, we analyze the extent to which the relations between option-based
ex ante higher moment sorts and subsequent returns are due to investors seeking compensa-
tion for higher co-moment risk, rather than idiosyncratic moments. We examine in Section 5
the relation between risk-neutral and physical distributions, and in particular the comparison
of portfolio sorts based on skewness under both measures. We conclude in Section 6.
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In robustness checks, we also explore a stochastic discount factor approach and consider several alternative
specifications of the stochastic discount factor, both parametric and non-parametric. We find results similar to
those obtained from the decomposition of higher moments. These results are available in a companion document
containing supplementary material: see BLANK (2011).
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2 Data and Computing Ex Ante Risk-Neutral Moments
We wish to examine the relation, if any, between features of the risk-neutral density func-
tion and the pricing of stocks. In this section we describe the data and the methods used to
compute ex ante estimates of volatility, skewness, and kurtosis.
Our data on option prices are from Optionmetrics (provided through Wharton Research
Data Services). We begin with daily option price data for all out-of-the-money calls and puts
for all stocks from 1996-2005. Closing prices are constructed as midpoint averages of the
closing bid and ask prices.
Data on stock returns are obtained from the Center for Research in Security Prices (again
provided through Wharton Research Data Services). We employ daily and monthly returns
from 1996-2005 for all individual securities covered by CRSP with common shares outstand-
ing. Risk free rates are the continuously compounded yield computed from the bank discount
yields on secondary market three month Treasury Bills taken from the Federal Reserve Re-
port H.15. Finally, we obtain balance sheet data for the computation of book-to-market ratios
from Compustat and compute these ratios following the procedure in Davis, Fama, and French
(2000).
We begin by calculating higher moments of firms’ risk neutral probability distributions. In-
tuitively, a risk neutral probability distribution is computed so that today’s fair (i.e. arbitrage-
free) price of an asset is equal to the discounted expected value of the future payoffs of the
asset, where the discount rate used is simply the riskfree rate. Thus, under the risk neutral
measure, all financial assets in the economy have the same expected rate of return, regard-
less of their risk. In contrast, if we use the actual (or physical) probability distribution of
the asset’s payoffs and assume that investors are risk-averse, assets which have more risk in
their distribution of payoffs should have a greater expected rate of return (and so lower prices)
than less risky assets. The relation between risk-neutral and physical probabilities therefore
depends on the price of risk; risk-neutral probabilities subsume, or incorporate, the effects of
risk, since the prices from which they are calculated embed investors’ risk preferences.
Like the physical density, the risk neutral density has first, second, third and fourth mo-
ments, respectively mean, variance, skewness and kurtosis. All densities are extracted from
options and are therefore conditional and for a given horizon. In a risk-neutral density, the
mean should correspond to the risk free rate at a given time with a particular maturity.
To estimate the higher moments of the (risk-neutral) density function of individual securi-
ties, we use the results in Bakshi and Madan (2000) and Bakshi, Kapadia, and Madan (2003).
Bakshi and Madan (2000) show that any payoff to a security i can be constructed and priced
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