Book-to-Market Ratio and Skewness of Stock Return
Xiao-Jun Zhang*
Haas School of Business
University of California, Berkeley
Current Version: October 2010
* Comments and suggestions from James Ohlson and Charles Lee are greatly appreciated.
Book-to-Market Ratio and Skewness of Stock Return
Abstract
Stocks with low book–to-market ratio, also known as glamour stocks, are shown
to have significant excess positive skewness in their return distributions compared with
value stocks. The premium (discount) investors apply to glamour (value) stocks also
correlates significantly with the difference in return skewness. These findings suggest
that the value/glamour-stock puzzle can be partially explained by investors’ preference
for positive skewness in stock returns. Such a preference for skewness, which is
consistent with investors having S-shaped utility of wealth functions, has been observed
in consumer behaviors such as lottery purchase and gambling.
Key words: Book-to-market ratio; Skewness; Accounting conservatism; Growth; Capital
asset pricing
1. INTRODUCTION AND SUMMARY OF FINDINGS
Glamour (value) stocks refer to stocks whose market values are relatively high
(low) compared to certain benchmarks such as earnings, book value, or free cash flows.
By studying the skewness of the cross-sectional return distribution of value versus
glamour stocks, this paper documents evidence suggesting that glamour (value) stocks
are more likely to include firms with extreme positive (negative) skewness in their return
distributions. 1 Analysis of stock returns further shows that a significant portion of the
documented premium (discount) that investors apply to glamour (value) stocks is driven
by investors’ preference for positive skewness in return distribution, similar to consumers’
preference for lotteries and gambling.
Prior studies have established that value stocks tend to outperform glamour stocks
in subsequent stock returns (e.g., Basu 1977, Rosenberg, Reid, and Lanstein 1985, Fama
and French 1992). It is unclear why such return difference exists.
Some believe that
investor mispricing plays an important role (Lakonishok, Shleifer, and Vishny 1994,
LaPorta 1996, Dechow and Sloan 1997, Rozeff and Zaman 1998, Piotroski 2000, Jiang,
Lee, and Zhang 2005). Others question whether the documented explanatory power of
the book-to-market ratio is spurious (Kothari, Shanken, and Sloan 1995, Chan, Jegadeesh,
and Lakonishok 1995). The persistence of this book-to-market effect also leads many
researchers to believe that the value premium might be a compensation for changes in
risk or transaction cost (Fama and French 1995, Berk, Green, and Naik 1999, Ali,
Huwang, and Trombley 2003, Petkova and Zhang 2005). This paper provides another
explanation along the lines of risk and reward analysis. However, instead of analyzing
1
Skewness reflects the degree of asymmetry in the distribution of a random variable.
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investment risk in terms of the second moment of a return distribution, this study
explores the implication of investors’ preference for skewness. I posit that the book-tomarket ratio of a firm correlates with the skewness of its payoff, which in turns affects
the pricing of its stock.
The hypothesis stems from the accounting principles which govern the
measurement of book value. Unlike the stock price, which is determined by the market,
the book value of a firm is measured according to the Generally Accepted Accounting
Principles (GAAP). Some fundamental principles of GAAP, such as the conservatism
principle, cause companies’ book values to differ significantly from their market values.
These accounting rules, loosely speaking, mandate that firms measure book value (and
earnings) conservatively when the level of uncertainty associated with the future cash
flows from an asset is high. Probable losses from future operations are often recognized,
but not unrealized gains. With such a biased accounting system, the book value is more
likely to capture a firm’s downside as opposed to its full upside potential. In contrast, the
market value of a firm fully reflects its upside potential as well as its downside risk. As a
result, the book-to-market ratio correlates with a company’s upside relative to its
downside (this point will be discussed in more detail in the next section). In a statistical
sense, I hypothesize that the book-to-market ratio correlates with the expected skewness
in a stock’s future payoff. Such skewness, in turn, may affect investors’ valuation of the
stock.
The issue of how the skewness affects asset pricing has long been studied in
finance and economics. Although the mainstream asset pricing models, such as the
Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), do not make
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any prediction regarding the effect of skewness on pricing, other studies have ventured
away from the mean-variance framework to offer insights on this issue. Earlier papers
conclude that a stock’s coskewness, not its skewness, matters in asset pricing.
Coskewness represents the degree to which a stock’s skewness varies with the overall
skewness of an investor’s well-diversified portfolio (e.g., Rubinstein 1973, Kraus and
Litzenberger 1976). Empirical studies on coskewness yield mixed results as to the power
of coskewness in explaining cross-section stock returns (e.g., Friend and Westerfield
1980, Lim 1989, Sears and Wei 1985, Harvey and Siddique 2000, Hung, Shackleton, and
Xu 2004, Smith 2006, Post, Vliet, and Levy 2008). More recently, Barberis and Huang
(2008) study an equilibrium asset pricing model based on the cumulative prospect theory
developed by Tversky and Kahneman (1992). A key feature of their model is that, in
equilibrium, idiosyncratic stock return skewness is priced. The central prediction of
Barberis and Huang (2008) is that positively skewed stocks, ceteris paribus, earn lower
average returns. Similar predictions are also made by Mitton and Vorkink (2007), who
study an expected utility model in which some investors have convex, skewness-loving
preferences, and by Brunnermeier, Gollier, and Parker (2007) who analyze a setting
where investors choose their beliefs to maximize the discounted value of expected future
utility flows. This prediction, together with the correlation between the book-to-market
ratio and the skewness of the payoff distribution, provides the foundation of the
hypothesis which links the book-to-market ratio with the average stock return via the
impact of skewness on asset pricing.
To test the above hypothesis, I first compare the skewness of future return
distributions among firms with different book-to-market ratios. The sample covers the
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entire set of U.S. firms traded on NYSE, AMEX, and NASDAQ from 1963 to 2006.
Following Fama and French (1992), each year firms are sorted into deciles based on their
book-to-market ratios. Firms in the top three deciles are classified as having high bookto-market (BM) ratios, while firms in the bottom deciles are classified as having low
book-to-market ratios. Firms with low book-to-market, i.e., glamour stocks, are shown to
have significantly larger skewness in their return distribution compared to that of the
value stocks. If we further remove any mechanical correlation between sample skewness
and sample mean, the resulting measure of excess return skewness shows a more striking
difference between firms in different book-to-market portfolios.
Such a finding is
consistent with the conjecture that the book-to-market ratio has significant predictive
power for the skewness of future stock payoffs.
The next issue is whether such difference in return skewness, coupled with
investors’ skewness-loving attitudes, might contribute to the observed average return
difference between value and glamour stocks. I conduct two sets of tests to analyze this
issue.
The first set of tests examines the correlation between the expected excess
skewness and the expected stock return. I use both the ex post portfolio return skewness
and the ex ante portfolio return skewness to proxy for the expected return skewness.
Cross-sectional return regression shows that the skewness measures have significant
negative correlations with realized stock returns. Moreover, after controlling for the
skewness, both the magnitude and the significance of the estimated coefficient on the
book-to-market ratio are substantially reduced.
In the second set of tests I partition firms in the high and the low book-to-market
groups into subsets according to the different projected payoff skewness based on their
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investment growth patterns. Analysis of the main hypothesis suggests that the skewness
of a firm’s future payoff results primarily from the skewness in firms’ investment growth.
With conservative accounting, high investment growth transforms into low book rate of
return on a firm’s operating assets (Zhang 1998, Rajan, Reichelstein, and Soliman 2007).
Accordingly, I compare the portfolio return distributions of two sets of firms. In the first
set, low (high) book-to-market firms also have more (less) investment growth, as
measured by the book rate of return on operating assets. In the other set, low book-tomarket firms have less investment growth compared to high book-to-market firms. The
magnitude of the difference in the average book-to-market ratio between the high and the
low book-to-market portfolios are quite comparable across the two sample sets. However,
the difference in the projected payoff skewness is quite substantial. Test results reveal
that the first set of firms shows a highly significant difference in the return skewness as
well as average stock return between the high and the low book-to-market firms. In
contrast, among the second set of firms, the differences in the mean and the skewness of
the return distributions between the high and low book-to-market portfolios are
insignificant. These test results strongly suggest that, even though some firms may have
significantly lower book-to-market ratios compared to others, if their investment growth
pattern indicates rather limited upside potentials, their payoffs are less likely to have the
desirable degree of positive skewness. As a result, investors are less willing to pay a
premium for such stocks.
In summary, this paper proposes and tests how value and glamour stocks differ in
their return skewness, and how that affects the average stock return. Findings of this
study highlight the promise of analyzing investors’ skewness-seeking attitude to better
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understand the pricing of capital assets. I would like to point out that the skewness effect
does not exclude other explanations for the book-to-market phenomenon. Firm growth,
for instance, can increase skewness as well as cause investors to overreact. Nonetheless,
the skewness hypothesis is unique in explaining the difference in skewness across bookto-market portfolios. The findings of this study make several contributions to the
literature. By documenting how value and glamour firms differ in their return skewness,
I provide a new perspective for analyzing the book-to-market effect. In addition, the
study documents a significant negative correlation between the expected skewness and
the mean stock return, consistent with the prediction of Barberis and Huang (2008). Prior
studies on the issue of skewness have documented significant difficulty in forecasting
future coskewness as well as skewness (Singleton and Wingender 1986, Harvey and
Siddique 1999, Chen, Hong, and Stein 2001). This paper demonstrates that screening
firms using accounting-based measures, such as the book rate of return and the book-tomarket ratio, provides a rather effective way to separate firms with different expected
return skewness. Lastly, but not least, the paper suggests that the conservative bias in our
accounting system, which has been widely criticized for tarnishing the representational
faithfulness of financial reports, may provide investors with useful information to assess
a firm’s upside potential relative to its downside risk. 2
2. HYPOTHESIS DEVELOPMENT AND RELATED LITERTURE
2.1. Correlation between book-to-market and skewness
2
For more information on the conservatism principle, see Statement of Financial Accounting Concepts No.
2, 1975, Financial Accounting Standards Board, Stamford, CT.
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The first part of the hypothesis proposes a negative correlation between the bookto-market ratio of a firm and the skewness of its payoff.
This conjecture is developed
based on two observations: (1) the bias in accounting due to the conservatism principle,
and (2) the skewness in a firm’s payoff as a result of the options embedded in its
operation.
The conservatism principle of accounting, loosely speaking, mandates that firms
measure assets (and earnings) conservatively by anticipating potential losses but not
unrealized gains. As a result, intangible assets such as growth potential, brand name,
market share, and R&D are often marked down to zero on the balance sheet. The lowerof-cost-or-market method of accounting for inventories serves as another good example.
When the market price of a piece of inventory exceeds its historical cost, the inventory is
often valued at the historical cost because the increase in inventory value (since purchase)
is deemed to be unreliable. However, if there is an indication that the inventory’s net
realizable value is below its cost, even though such indications can be as unreliable as
any indication for a value increase, firms are required to mark the inventory down to
reflect such potential loss. Because of this conservative tendency in accounting, the book
value of a firm often reflects the downside of the firm, e.g., how much the firm can get in
liquidation. In contrast, the market price reflects the average of both the upside and the
downside.
Another important characteristic of a firm’s payoff is its skewness due to the
options embedded in its operation. Simply put, when it is clear that a firm’s business
plan is not going to be fruitful, the manager can often liquidate the operation in a timely
fashion to curtail further losses. In contrast, when the market condition turns favorable,
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the managers can exercise the option to considerably expand. As a result, the payoff of a
firm’s operation is often positively skewed.
Combining the effects of the above accounting and operation factors, an
interesting result emerges. The book value and the market value of a firm tend to reflect
two different aspects of a firm’s positively skewed payoff: the book value captures the
downside while the market price reflects the mean. Because of this, the difference
between the two measures, i.e., the book-to-market ratio, is affected by the degree of
skewness in the payoff similar to the way the difference between the mean and the
median can be used to measure the skewness (Pearson 1895). 3
The following setup can be used to illustrate the idea more precisely. Suppose
that a firm possesses an asset whose liquidation value is b>0. Assume that the firm will
engage in value enhancing activity, which may fail. If the operation fails, the manager
utilizes the abandonment option and sells the asset for b. However, if the operation is
successful, the firm will expand and generate a payoff exceeding b. The final cash flow
generated by the firm’s asset equals:
,
where x is the payoff from operation.
Consider two scenarios. Under the first scenario, denoted as i, the payoff xi
equals
3
The link between the book-to-market ratio and the skewness certainly depends on the way skewness is
measured. For further discussion of the different skewness measures, see Section 5.2.
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Where y>b, k>0, and y-k<b. In this case, the mean payoff equals
Next consider the second scenario, j, with a similar payoff pattern:
Suppose that g>1. It is easy to see that
and
The idea is simple: the option to expand on the upside, as well as the option to contract
on the downside, leads to a skewed distribution of payoff.
Note that because of accounting conservatism, the book value of the firm’s assets
equals b. As shown in Barberis and Huang (2008), investors discount cash flows more
under the first scenario (i.e., i) since the distribution of cash flow is relatively more leftskewed. Therefore,
Note that regardless of the discount rate, the skewness of payoff from j exceeds that of i,
that is,
.
Therefore, with conservative accounting, the book-to-market ratio and the skewness of
the stock return distribution have a negative correlation. 4 Some related predictions are
also made in Duffee (2002) based on the assumption that positive changes in firm value
are more likely to be related to intangible assets. A key difference between this paper
4
Small or zero book values often cause problems in the calculation of the market-to-book ratio. Therefore,
in developing and testing the hypothesis, I use the book-to-market ratio instead of the market-to-book ratio.
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and Duffee (2002) is that we emphasize that book value is significantly affected by the
conservatism principle, and hence reflects more of the downside of future payoff. In
contrast, Duffee (2002) models book value as “asset in place” and the market premium as
a derivative of the book value. As a result, Duffee (2002) does not offer any directional
prediction as to how book-to-market correlates with skewness. 5
There are also counter forces which may reduce the magnitude of the above
correlation.
For example, when a firm’s book-to-market ratio is high, the risk of
bankruptcy becomes significant. On the one hand, the extreme low return in bankruptcy
creates a negative skewness in return, strengthening the negative correlation between the
book-to-market ratio and the skewness. On the other hand, since the negative stock
return is capped at -1, whereas the upside is potentially unlimited, such an asymmetry can
add a certain degree of positive skewness in return distribution. The overall skewness of
the distribution would be the combination of the above two forces. The distressed
situation of the high book-to-market firms makes the possibility of extreme long-term
high return rather low. Therefore, overall, high book-to-market firms may still have a
more negatively skewed return distribution.
Nonetheless, given these counteracting
forces, it is interesting to empirically examine the correlation between the book-to-market
ratio and the skewness in return distribution.
2.2. Correlation between skewness and average stock return
The second part of the hypothesis concerns the correlation between the expected
skewness and the average stock return. Analysis of this issue requires deviation from the
5
Empirical tests in Duffee (2002), which are based on the time-series property of individual stock returns,
do not detect the negative correlation between expected skewness and expected return.
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mean-variance framework of asset pricing (e.g., Rubinstein 1973, Kraus and Litzenberger
1976, Harvey and Siddique 2000, Mitton and Vorkink 2007, Barberis and Huang 2008).
Studies on this issue all build upon the same insights obtained from analyzing individual
behavior in lottery purchase and gambling. That is, ceteris paribus, investors prefer more
positively skewed distributions. However, the ways these papers incorporate skewness
preference into asset pricing differ significantly.
One line of studies introduces skewness-loving utility functions while maintaining
their concavity (e.g., Rubinstein 1973, Kraus and Litzenberger, 1976, Harvey and
Siddique 2000, Vanden 2006). The implication is that investors are still universally risk
averse, and therefore all hold well diversified portfolios. In such economies only a
securities’ coskewness, which is measured by the correlation between the stock’s
skewness and the skewness of the market portfolio, is priced.
Another line of research abolishes the universal risk aversion assumption by
introducing some convexity into the utility function (e.g., Friedman and Savage 1948,
Markowitz 1952, Mitton and Vonkink 2007, Barberis and Huang 2008). The step-wise
S-shaped utility function studied in Friedman and Savage (1948) serves as a good
example (Figure 1). The step-wise shape captures the idea that certain consumption is
not divisible (Kwang 1965). 6
As a result, increase in wealth that is beyond certain
threshold levels can increase someone’s utility substantially by moving him/her into a
qualitatively different socioeconomic status. When the wealth of an individual places
her/him at the bottom of a “hill” in the utility function, and an investment offers him/her
the chance of a significant “step-up” in consumption without too much downside risk,
6
For example, we cannot purchase half of the education of a good private school when we could only
afford half of the tuition.
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she/he is more likely to invest.7 The individual will not fully diversify since doing so can
reduce the skewness effect. Kahneman and Tversky (1979) developed the prospect
theory to incorporate this and other aspects of human behavior under uncertainties, such
as loss aversion, which are inconsistent with the prediction of the expected utility theory.
Barberis and Huang (2008) study the equilibrium asset pricing based on the cumulative
prospect theory (Tversky and Kahneman 1992). They show that some of the investors do
not fully diversify in equilibrium.
Instead these investors hold more stocks with
positively skewed return distributions.
A key feature of their models is that, in
equilibrium, idiosyncratic skewness is priced. Positively skewed stocks, ceteris paribus,
earn lower average returns.
This paper follows the second approach. On a conceptual level, I believe that
investors’ preference for skewness is inherently related to risk-taking behavior (Mitton
and Vorkink 2007, Barberis and Huang 2008).
More importantly, the analysis of
skewness enables researchers to develop the link between the book-to-market ratio and
stock returns. Accounting conservatism implies that the book-to-market ratio correlates
with the skewness. However, a similar prediction cannot be made for coskewness. By
studying the skewness as opposed to the coskewness of individual stocks, my results are
also less sensitive to the choice of market portfolio. The issue of market portfolio is
problematic when investors do not fully diversify in equilibrium, as illustrated in Barberis
and Huang (2008).
Nonetheless, for the purpose of comparison, I also perform tests
based on coskewness and present the results in Section 5.3.
7
Such a significant increase in consumption could come in various forms, such as moving to a bigger
house, having an earlier retirement, or even having your own tropical island,
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2.3. Measuring skewness in return distribution
The skewness in the distribution of stock returns can be measured based on either
firm-specific, time-series return data or cross-sectional return data. This study focuses on
the cross-sectional return skewness as opposed to the time-series return skewness of
individual firms (Friend and Westerfield 1980, Sears and Wei 1985, Lim 1989, Harvey
and Siddique 2000). This approach is taken primarily to mitigate one significant
drawback associated with the time-series approach: the expected stock return skewness of
many firms may not be uncovered from the ex post time-series return data.
When investors bet on the upside potential of a firm, the success rate of such bets
is usually very low. The chance of successfully identifying the next Microsoft, i.e., the
“take-all” winner in an emerging industry (or market), is slim. This means that, for most
of the firms whose ex ante return distributions are more positively skewed, the skewness
will never become observable in the firm’s ex post time series return data. In other words,
for most of the firms with a chance of being extremely successful ex ante, the promise of
success will not materialize. Simply observing the ex post realized returns of a firm is
not likely to uncover the anticipated upside of such a stock. Such an issue, however, is
less of a problem for tests based on the skewness of cross-sectional return data. As long
as a few successful firms do experience phenomenal returns ex post, the cross-sectional
return distribution will reveal the positive skewness in the expected return distribution.
Even for the firms whose ex post returns do show a positive skewness, such as
Microsoft, measuring the expected future return skewness based on a firm’s realized
return skewness can be problematic. By the time the upside potential is finally realized,
it is likely that the growth potential of the firm has also changed. For example, by the
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time Microsoft clearly established itself as the dominant player in its field, it was widely
considered to be more of a mature firm rather than a growth firm.
The time-series approach of skewness measure is also very sensitive to
idiosyncratic shocks to stock returns.
This is an important reason which causes the
observed persistence in the time-series return skewness to be low. In contrast, measuring
skewness based on cross portfolio stock returns is less sensitive to idiosyncratic shocks.
Because of all the above reasons, this paper measures skewness based on cross-sectional
return data. Test results based on time-series return skewness, as part of the sensitivity
analysis, are also presented in Section 5.3.
Early empirical studies on the issue of skewness and asset pricing focus mostly
on coskewness, yielding mixed results (e.g., Harvey and Siddique 2000, Hung,
Shackleton, and Xu 2004, Smith 2006, Post, Vliet, and Levy 2008). More recent works
on idiosyncratic skewness also provide mixed evidence on the equilibrium pricing of
skewness (e.g., Conrad, Dittmar, and Ghysels 2009, Xing, Zhang, and Zhao 2010). This
paper adds to the literature by providing new evidence, based on cross-sectional
skewness analysis, supporting the hypothesis that capital asset pricing is significantly
affected by investors’ skewness-loving attitude. Zhang (2005) and Kapadia (2006) also
examined the cross-section skewness of stock return.
In Zhang (2005), the cross
sectional skewness among firms from difference industries (as well as from different size
and book-to-market groups) are used to predict individual return skewness. Kapadia
(2006) explores whether the monthly cross-sectional return skewness of all listed stocks
can explain the idiosyncratic volatility puzzle documented by Ang, Hodrick, Xing, and
Zhang (2006). The findings of this paper corroborate the findings of Zhang (2005) and
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Kapadia (2006) in supporting the relevance of skewness in affecting the expected stock
returns. The current study also distinguishes itself in two important aspects. First, in
prior studies, including Zhang (2005) and Kapadia (2006), the correlation between bookto-market and skewness has not been explored. Book-to-market was used as a control
variable in cross sectional return tests. As shown in this paper, the book-to-market effect
is significantly driven by skewness. Without controlling for this effect, these studies are
likely to under-estimate the correlation between skewness and expected stock return.
Second, the current study examines the ex post skewness in realized stock returns as test
of difference in the expected stock return skewness. Prior studies have only examined
the ex post mean return to provide evidence on the pricing of skewness. 8
Another issue with the measurement of skewness is that investors’ preference for
skewness is not monotonic (Barberis and Huang 2008). When selecting stocks, riskloving investors are only attracted to stocks with considerably skewed payoff and
significant upside potentials. Stocks with moderate positive skewness may very well be
discounted by investors because of risk aversion. This implies that our sample firms are
likely to include firms with extreme skewness, as well as a significant portion of “noise”
firms. This, in turn, poses a challenge to empirically detecting abnormal skewness based
on observed sample distribution. Consider some sample firms (denoted as Y) such as
those with low book-to-market ratios, which include a subset of firms with extreme
positive return skewness (denoted as X), and other firms whose returns are more
8
Conrad, Dittmar, and Ghysels (2009) use option prices to estimate individual stock’s ex ante skewness.
The advantage of their approach, compared to ours, is that their method does not rely of cross-sectional
sample to estimate skewness. The disadvantage, however, is that their estimation method, developed in
Bakshi, Kapadia and Madan (2003), involves strong assumptions and estimates risk-neutral moments. The
noisy nature of their measure can be seen from the fact that their measure, when applied to technology
stocks, does not show positive skewness during the bubble period. Xing, Zhang, and Zhao (2010) use the
same measure but report finding no negative correlation between skewness and stock return.
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symmetrically distributed (denoted as E). That is, the population is made up of two subpopulations with different characteristics:
. Inferring the distribution of X
based on the observed distribution of Y is difficult. Even though X is positively skewed,
the skewness of Y is not clear-cut. The overall skewness of the sample population
depends on the relative locations of the mean returns of E and X (i.e., the expected return
of X and E), as well as the relative sample size of E and X.
To mitigate the above problem, I calculate the skewness of the hedged return,
which is the return from holding a long/short pair of stocks from different book-to-market
groups.
The return distribution of a hedge portfolio eliminates the impact of any
common noise terms which affect the returns of both the long and the short stocks. In
addition, the hedged portfolio return has an interesting property: if the long and the short
stocks (from different book-to-market groups) have a similar degree of skewness, the
distribution of the hedged return will be symmetrical. If, however, the two groups of
firms have difference skewness, the difference will be clearly shown in the skewness of
the hedge returns.
Another method I use is to try to purge the impact of E by estimating what the
sample skewness of Y would be if X and E are random samples from the same pool. Let
γ* denote this benchmark level of sample skewness. By subtracting this benchmark level
of skewness (γ*) from the observed sample skewness of Y (denoted as γy), we get a
measure of excess sample skewness (γ= γy –γ*) which hopefully indicates the excess
skewness of X. To measure the benchmark skewness (i.e., γ*) properly, we need to
control for the correlation between sample mean and sample skewness (Bryan and
Cecchetti 1999). When the distribution of the underlying population is skewed and/or
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has a positive kurtosis, the sample skewness generally has a positive correlation with the
sample mean. 9 The reason is intuitive: A drawing of extreme positive return is likely to
increase both the sample mean and the sample skewness. In this paper I use the random
sampling method to assess the degree of correlation between sample skewness and other
sample moments. Based on distribution of this random sample, the excess skewness is
then estimated for firms within each book-to-market portfolio.
3. BOOK-TO-MARKET AND SKEWNESS OF FUTURE PAYOFF
3.1. Sample
The sample is constructed based on all domestic common stocks traded on NYSE,
AMEX, and NASDAQ. 10
Since the return and financial data is obtained from the
merged CRSP/COMPSTAT dataset, I restrict the sample period to 1963-2006 to avoid
the sample selection problem of COMPUSTAT data during the earlier years. The final
sample includes all firm-year observations with non-missing book-to-market ratios.
Several key statistics for the sample firms are provided in Table 1. SIZE is the
logarithm of the market value of equity on the last trading date of June in each year. BM
is the book-to-market ratio, calculated as the book value at the end of the previous fiscal
year deflated by the market value at the end of June of the current calendar year. To
ensure the availability accounting data on the portfolio formation date, I require a
minimum time gap of four months between fiscal year end and the June 30th portfolio
formation date. This means that for firms with fiscal years ending in March, April, May
9
Kurtosis is defined as μ4/σ4 -3 where μ4 is the fourth moment about the mean and σ is the standard
deviation.
10
Stocks primarily traded on OTCBB and Pink Sheets are excluded from the analysis. These stocks are
subjected to different disclosure requirements and are largely avoided by most investors. All test results
remain qualitatively unchanged with the inclusion of these stocks in the analysis.
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and June, the book value from their previous fiscal year, instead of the most recent fiscal
year, will be used. Raw stock return (RET) is the buy-and-hold return over a 12-month
period, starting from the first trading day in July each year. Market adjusted return (MAR)
is calculated as raw return minus the corresponding beta-adjusted market return
according to the Capital Asset Pricing Model (CAPM). The beta of each company is
estimated based on a minimum of 12 monthly stock return data over the prior 5-year
period. To avoid any potential problem introduced in the process of beta estimation, all
tests were repeated with MAR calculated by simply subtracting the market index return
from the corresponding raw return. All our test results remain qualitatively unchanged. I
use a 12-month return window in the main tests to maximize the number of observations
with non-overlapping return windows. Admittedly, such a choice is arbitrary given the
unknown and possibly variant length of time for firms to realize its upside potentials. In
Section 5.4 I present test results with longer holding periods, ranging from three years to
10 years.
The choice of equal- or value-weighting usually has a significant impact on the
magnitude of portfolio returns. Even though the value-weighting method is generally
preferred due to the various issues associated with small stocks (Fama 1998), there is an
important conceptual reason in favor of the equal-weighting method. As shown in Mitton
and Vorkink (2007) and Barberis and Huang (2008), investors often forgo diversification
in favor of holding a disproportionally large number of stocks with positive skewness
when they exhibit skewness-loving behavior in addition to risk aversion. Since returns of
smaller stocks are more likely to have extreme skewness, an equal-weighting scheme
may better capture investors’ skewness-seeking behavior. Because of these reasons, I
- 18 -
present test results based on both the value-weighting and the equal-weighting methods.
As can be seen from the tables, the findings are rather robust to the choice of the
weighting method. This robustness reflects the fact that tests in this paper focus on the
relation between the skewness and the mean of portfolio return, as opposed to the
magnitude of the mean portfolio return.
There are several alternative ways to measure skewness.
Since investors’
skewness-loving attitude arises mostly from the effect of extreme outcomes, I use the
following quantile-based skewness in main tests:
90
10
90
2
10
Where P90 and P10 are the 90th and the 10st percentiles of the distribution. Test results
based on several alternative skewness measures are qualitatively similar and are
presented and discussed in Section 5.2.
Table 1 shows that the sample is rather representative of the general population of
firms traded on NYSE, AMEX, and NASDAQ. The median firm size is about $85
million. The median book-to-market ratio equals 0.61, consistent with the existence of
significant conservative accounting bias. Note that both the raw return and the marketadjusted return are positively skewed, as shown in Panel A. This is consistent with the
pattern documented in prior literature regarding the skewness of long term stock returns.
The Kurtosis of the sample stock returns is significantly larger than 0, which is the
Kurtosis level for a Normal distribution. These two characteristics of the return
distribution cause the mean and skewness of any return sample to be positively correlated,
as we will discuss and document next.
- 19 -
3.2. Return skewness for book-to-market portfolios
3.2.1. Comparing sample skewness
Table 2 reports the test results from comparing the skewness of stock returns of
firm in different book-to-market portfolios. Each year firms are sorted into 10 equalsized groups based on their market capitalization at the end of June. Then, within each
size group, firms are placed into deciles based on their book-to-market ratios. Such a
sequential sorting is done to control for the effect of firm size when analyzing the effect
of book-to-market. Following the practice of Fama and French (1992), firms in the top
three deciles are classified as the high book-to-market group, while firms in the bottom
three deciles are classified as the low book-to-market group. The rest of the firms are
placed in the medium book-to-market group.
The mean and the skewness of the portfolio stock returns are calculated for each
group. Table 2 reports the time series mean across sample years, with t-statistics
calculated based on the time series standard deviation (Fama and MacBeth 1973). All
three book-to-market portfolios have significant positive skewness in stock returns.
Firms in the low book-to-market category have an average return skewness of 0.15,
whereas firms in the high book-to-market category have an average return skewness of
0.12. Consistent with the prediction of my hypothesis, the return skewness of the low
book-to-market firms exceeds the return skewness of the high book-to-market firms by
0.03, which is significant at the 1% level.
Table 2 also confirms that high book-to-market firms on average outperform
firms with low book-to-market ratios (Basu 1977, Fama and French 1992). Firms with
high book-to-market on average earn a market-adjusted return of 2.95%. In contrast, the
- 20 -
average return of firms in the low book-to-market category trails the market index by
5.59%. A hedge portfolio formed by buying firms with high book-to-market ratios and
shorting those with low book-to-market ratios yields an average annual return 8.54%. In
the value weight case, the difference is less (5.78%), but still significant.
The difference in the average stock return between the high and the low book-tomarket firms raises the issue of whether the documented difference in return skewness
might somehow be caused by the difference in the mean return. Bryan and Cecchittee
(1999) offer an insight as to how sample mean and skewness might be correlated. They
show that if the underlying distribution of a random sample is skewed, and/or has a high
degree of Kurtosis, the sample mean and the sample skewness will be positively
correlated. This implies that when comparing the skewness of different samples, we need
to control for this mechanical aspect of the skewness measure which varies with the
sample mean.
3.2.2. Adjusted return skewness for book-to-market portfolios
To get a precise measure of the correlation between sample mean and sample
skewness, I perform the following test on a large stock return sample constructed from
random portfolios. Each year I randomly select 50 stocks from each size group to form a
portfolio, and record the mean as well as the skewness of the portfolio stock returns. This
random stock selection procedure is repeated 1,000 times for each size group to generate
a sample of mean and skewness of portfolio returns. Some descriptive statistics of this
random portfolio return sample are presented in Panel A of Table 3. The mean portfolio
return, both for the raw return as well as the market-adjusted return, is very close to the
- 21 -
population mean reported in Table 1. The portfolio return skewness is positive and
significant.
Panel B of Table 3 shows the Pearson and the Spearman correlation between
portfolio mean return and the skewness of return distribution. Both measures show a
significant positive correlation between sample mean and sample skewness. Bryan and
Cecchittee (1999) document a similar pattern using the inflation data from 1947 to 1995.
The above sample correlation between mean and skewness provides a useful
benchmark against which we can measure the excess return skewness of any portfolio.
For example, if the low book-to-market portfolio contains a subset of firms with low
mean returns and an abnormally high level of skewness, such abnormal skewness would
cause the skewness of the sample to deviate from the benchmark level. In other words,
after controlling for the change in the mean return, any excess skewness observed may
indicate the existence of an abnormal level of skewness associated with a particular
portfolio. Next I apply this method to measure the excess skewness of book-to-market
portfolios.
Within each book-to-market group I randomly select 50 firms to form a portfolio.
The mean and skewness of the portfolio stock returns are then calculated. Based on the
average portfolio return, each portfolio is matched with ten random portfolio samples
from the same size group (as described above) with mean returns closest to the sample
mean. The average skewness of these random portfolios is then used as the benchmark
level of skewness. 11 The untabulated test results show that the average difference in mean
return between the sample portfolio and the benchmark portfolios equals 0.03%,
11
I also form benchmarks by pooling together all random portfolios whose return differs by less than 0.1%
from the sample portfolio’s mean return. The result is almost identical. The matching method used in the
text generates far less difference in the mean return between the sample and the benchmark portfolios.
- 22 -
indicating that the matching is very close. Such a benchmark level of skewness is
subtracted from the skewness of the book-to-market portfolio to get the abnormal (or
excess) level of return skewness. Note that each book-to-market portfolio contains 50
firms so that the sample size is consistent with the random sample from which the
benchmark distribution of skewness is obtained. This process is repeated 100 times for
each book-to-market group, within each size group, in each year. The results are reported
in Panel C of Table 3.
Once we subtract the benchmark level of skewness, the abnormal skewness levels
show an even clearer monotonic pattern across different book-to-market groups. Low
book-to-market firms have a significant positive abnormal skewness of 0.06, while the
high book-to-market firms show a significant negative abnormal skewness of -0.03. The
difference between the two is significant at the 1% level. Clearly, not adjusting for the
mechanical correlation between sample mean and sample skewness will understate the
difference in the expected skewness across the different book-to-market firm groups.
3.2.3. Skewness of the hedged portfolio returns
I also randomly select 50 long/short pairs of stocks, from the high and the low
book-to-market groups respectively, and calculate the skewness of the hedge returns.
The distribution of this hedge return has the interesting property of purging the effects of
any common factors affecting both the long and the short group of stocks, thus being
more likely to give a clearer picture of the difference in expected skewness between the
high and the low book-to-market groups. If the long and short groups have the same
degree of skewness, the hedge return will be symmetrically distributed. However, as
- 23 -
shown in Panel C of Table 3, the mean skewness of the hedge return equals -0.04, which
is highly significant (at the 1% level). This provides further support for the conjecture
that high book-to-market firms have more negatively skewed return distribution.
As an additional analysis, I calculate the skewness of the hedged portfolio returns
based on 100 random portfolio formations. The purpose is to assess the skewness of the
hedged portfolio returns, with a reasonable amount of stocks (50 in this case) in the
long/short portfolios. More specifically, after the difference in average return of long and
short portfolios is obtained, I calculated the skewness of this hedged portfolio returns
based on 100 random portfolio formations within each size group, in each year. The
untabulated result shows that the mean skewness over the 44 sample years equals -0.03,
significant at the 1% level. The overall skewness calculated based on the entire panel
data is even larger, at -0.05. This result indicates that running a hedge fund based on the
book-to-market strategy, although earning a positive return on average, may expose
investors to a significant amount of negative skewness in the hedged portfolio returns.
3.4. Skewness of accounting-based performance measures
In Table 4 I compare the skewness of several accounting-based performance
measures, including the skewness of the growth rates in sales (Compustat data item #12),
income before extraordinary items (Compustat data item #237), total assets (Compustat
data item #6), and operating cash flows (Compustat data item #308). 12 Each of them
measures one particular aspect of the business operation, and hence is not as
comprehensive as the stock return in gauging the return of investment to investors.
12
When data #308 is not available, cash flow from operation (CFO) is calculated as fund from operation (#110)
adjusted for changes in working capital.
- 24 -
However, these measures provide an important supplement to the stock return results as
they do not suffer as much from their correlation with the mean stock return. Moreover,
these non-price measures avoid the potential problem of market mispricing.
Consistent with the findings from the stock return analysis, low book-to-market
firms also have a relatively more positive skewness in the distribution of all the
accounting-based performance measures. One interesting thing to note is that, unlike the
other measures, the growth rate in income has an average negative skewness. This is
presumably caused by the conservative bias in accounting (Zhang 2000). The other
measures presented in the table, i.e., the growth rates of revenues (GREV), total assets
(GTA) and cash flow from operations (GCFO) in particular, are less susceptible to the
different kinds of accounting reserves.
4. SKEWNESS AND STOCK RETURN
Results reported in the prior section indicate that firms with different book-tomarket ratios have different skewness in stock returns. Such difference in skewness, in
turn, may cause the equilibrium stock return to differ across these firm groups as a result
of investors’ preference for positively skewed returns. In this section I examine the
extent to which such difference in return skewness, coupled with investors’ skewnessloving attitudes, might contribute to the observed mean return difference between value
and glamour stocks. I conduct several tests on this issue.
To examine the impact of skewness on stock return, we need empirical proxies for
the expected stock return skewness. Two such proxies are used in this study, each with
its advantages as well as drawbacks. The first proxy is the ex post excess portfolio return
- 25 -
skewness as calculated in Section 3. The underlying assumption is that investors’
skewness expectation is unbiased, so that the average of the ex post return skewness
approximates the expected skewness. The advantage of using this proxy is that it is
robust to the alternative specifications of the skewness expectation models.
The
disadvantage, however, is that it is based on ex post data which is not available to
investors ex ante. The second proxy, which is based on ex ante portfolio return skewness,
avoids this problem. However, such a proxy has its limitations as well, the main one
being that it could be too noisy. Given the advantages and limitations, I present results
from tests based on both measures. In Section 4.1, I present test results based on the first
proxy. Test results based on the ex ante skewness proxy are presented in Section 4.2.
4.1. Correlation between ex post skewness and stock return
I regress the average portfolio stock return on its adjusted skewness, and check
whether the magnitude of return residuals vary across the three book-to-market groups.
The result is reported in Table 5.
The estimated regression coefficient on the adjusted skewness is negative and
significant. Panel A of Table 5 shows that once the adjusted ex post skewness measure is
added to the return regression, the estimated coefficient on book-to-market loses its
statistical significance. This finding indicates that the difference in skewness among
book-to-market groups is an important driver underlying the return differences among the
different book-to-market firm groups.
Could the return difference between the high and low book-to-market groups be
caused by other aspects of the return distributions, such as variance, rather than skewness?
- 26 -
To answer this question I compare the variance of the three book-to-market portfolios. If
it were the case that the difference in return variance (much of which can be idiosyncratic)
caused the average return to differ across book-to-market groups, we would expect the
firms with the largest return variance to have the highest average return. However, the
untabulated results show that the average standard deviation for the low, medium, and
high book-to-market groups is 0.61, 0.49, and 0.47 respectively. That is, the group with
the highest variance (i.e., low book-to-market) actually has lowest average return. Ang,
Hordrick, Xing, and Zhang (2006) also document a similar pattern. This pattern, on the
other hand, is consistent with our argument that high variance in payoff increases the
skewness as well as the stock price.
Nonetheless, it is still a concern that the documented correlation between the
adjusted skewness and the mean portfolio return, as reported in Models I and II of Panel
A, might be a result of the correlation between return and other moments of distribution,
such as the standard deviation. To analyze this conjecture, I include the adjusted return
standard deviation, and the adjusted return kurtosis in the regression. The result is
reported in Model III of Panel A. Consistent with the findings of Ang, Hordrick, Xing,
and Zhang (2006), there is a significant negative correlation between adjusted standard
deviation and the mean portfolio return. Kurtosis, however, does not exhibit any
signification correlation with mean return. The correlation between skewness and mean
return remains negative and highly significant.
- 27 -
4.2. Correlation between ex ante skewness and stock return
Using the ex post skewness measure, as in Panel A, often involves a high risk of
incorporating certain spurious correlation between ex post return moments. To reduce
that risk, I replace ex post skewness with the ex ante portfolio return skewness calculated
using stock returns during the prior year. The regression result is reported in Panel B of
Table 5. The ex ante skewness measure also demonstrates a strong negative correlation
with stock return, similar to the regression result using the ex post skewness measure.
This lends further support to our hypothesis that skewness contributes significantly to the
observed difference in mean return among book-to-market groups of firms.
Including the ex ante skewness measure does not completely remove the
statistical significance of the correlation between book-go-market and stock return.
However, the significance level is greatly reduced and the magnitude is substantially
reduced as well. Contrary to the results with ex post standard deviation and kurtosis, the
ex ante standard deviation shows a positive, yet insignificant correlation with stock return.
Such a positive correlation is consistent with the popular belief that variance should
increase, not decrease, expected return.
To avoid any potential noise introduced during the process of estimating the
adjusted standard deviation, skewness, and kurtosis, I also repeat the regressions with the
raw ex ante return moments. The results are reported in Panel C of Table 5. Consistent
with the hypothesis that skewness contributes significantly to the cross-sectional variance
in stock returns, ex ante skewness exhibits a significant, negative correlation with
subsequent stock return.
- 28 -
4.3. Ex ante proxy for skewness using accounting-based measures
If investors’ preference for positive skewness is a major driver of the book-tomarket effect, we should observe more significant difference in stock returns if we can
somehow increase the variation in the fundamental skewness between portfolios. Prior
research has shown that past time-series return coskewness has little actual predictive
power over future return skewness (Singleton and Wingender 1986, Harvey and Siddique
1999). In this section I present results which use ex ante accounting-based measures, in
addition to the book-to-market ratio, which could help forecast return skewness.
4.3.1. Return on assets (ROA) and skewness
The analysis in Section 2 shows that the skewness in a firm’s payoff is closely
related to the growth option associated with firm assets. Ohlson (1995) demonstrates the
following relationship between a firm’s market value and its book value:
+…
Where xa denotes the amount of abnormal earnings, calculated as earnings minus the
beginning book value times the discount rate. The above equation shows that firms with
similar book-to-market ratios may have quite different projected growth patterns with
respect to their expected future payoffs. To appreciate this point, consider a mature firm
such as Kellogg. Its market-to-book ratio is currently a hefty 7.8. Even though the firm
is quite mature, with limited growth potential, conservatism in accounting causes its
book-to-market to be rather low.
The above observation suggests that, if we can further separate firms based on
their investment growth, we should be able to better filter out firms with more significant
- 29 -
skewed payoff distribution. Zhang (1998) and Rajan, Reichelstein, and Soliman (2007)
show that, with conservative accounting, the book rate of return on operating assets
(ROA) becomes a rather comprehensive and effective measure based on which we can
infer the growth in a firm’s investment in both tangible and intangible assets. High
growth in investments, especially in intangible assets, depresses a firm’s ROA. 13 Thus,
grouping firms based on ROA, in addition to book-to-market, can help us improve the
separation of firms with different degrees of skewness in future payoffs.
Within the high- and the low- book-to-market group of firms, I further partition
firms based on ROA. Those with low book-to-market and low ROA are more likely to
have positively skewed distribution. Similarly, those with high book-to-market and high
ROA are the least likely to have excess positive skewness in distribution. I group those
two subsets of firms into one test group, and create the following skewness measure:
In contrast, the difference in skewness between firms with low book-to-market
but high ROA, and those with high BM and low ROA, is not clear. However, tests on
these two sets of firms can provide us with important insight regarding the effects of
skewness and book-to-market. If the book-to-market ratio is the driving factor, we
should still observe significant difference in the average stock return between these two
groups of firms despite the fact that they may not have a significant difference in their
13
Xing (2008) used the growth rate of capital expenditure (CAPEX) to gauge investment growth, and
obtained similar correlation between growth and return. Compared with CAPEX, using ROA has the
advantage of incorporating the effect of investment in intangible assets. The disadvantage, however, is that
low ROA firms also included firms with poor performance. This effect is mitigated by incorporating the
book-to-market ratio into the SKW measure.
- 30 -
return skewness. On the other hand, if skewness is a more fundamental driving factor,
we would observe less difference in stock return when comparing these firms:
The test results are reported in Table 6. As Panel A shows, the SKW measure
significantly increases the difference in the skewness between the high and the low group.
The raw and the adjusted skewness of the high SKW group exceeds the low SKW group
by a significant 0.09 and 0.14 margin, respectively. The difference in the skewness of
revenue growth also increases significantly from the book-to-market groupings, to 0.40.
In addition, the average portfolio return of low SKW group exceeds that of the high SKW
group by a significant 11.80%.
The partition based on BM’ provides a quite different picture, as shown in Panel
B. Note that the difference in the average book-to-market ratios between the high- and
the low-BM’ groups is 1.40, which is actually slightly larger than that of the SKW
partition (1.28). Therefore, if book-to-market is a fundamental driver of return difference,
we would expect the BM’ partition to yield more significant difference in the average
stock return between groups.
However, that is not the case.
The average return
difference is only 2.87% with equal-weighting, and 2.22% value-weighted respectively,
not statistically significant. The difference in the skewness between these two groups of
firms reveals the reason: the two groups are not statistically different in the skewness of
return. Although the difference in the revenue growth skewness is significant, the
magnitude is much less compared to that of the SKW partition. These results confirm our
conjecture: even though some firms may have significantly lower book-to-market ratios
- 31 -
compared to others, if their investment growth rates indicate rather limited growth
potentials, their payoff is less likely to have the desirable degree of positive skewness.
As a result, investors are less willing to pay a premium for such stocks.
In sum, the book-to-market ratio is likely to be a noisy measure of a firm’s true
upside potential relative to its downside risk. Adding an additional screen based on ROA
can help us get a more focused picture. For instance, a firm’s book-to-market ratio can
be low for two reasons: (1) when the firm is fast-growing and the market price is high, (2)
when the firm is significantly reducing its book value. Comparing the two scenarios, the
first one is more likely to have significant upside potential as reflected in return skewness.
Combining ROA with book-to-market (BM) separates the two cases, hence giving us a
better prediction of future return skewness.
I would like to point out that it is not the intention of this paper to argue that
investors necessarily use book-to-market or asset growth, per se, to forecast return
skewness. In fact, I believe that book-to-market reflects perceived skewness as a result
of investors’ liking of firms with high growth potentials and bidding their price high
relative to their book values.
Investors often assess the growth potential of a firm by
observing the degree of innovation in its products, its technological advancement, and the
potential size of the market for its products.14
The untabulated test results show that
stock return of low book-to-market firms significantly exceeds that of high book-to-
14
For example, according to the Associated Press, shares of Apple Inc. soared recently after an analyst
upgraded the stock on her belief that the iPhone's growth potential has been underestimated. "We believe
Apple is emerging as the clear leader in the battle over the mobile Internet," said Morgan Stanley analyst
Kathryn Huberty, in a research note. She sees mobile Internet, where people use their cell phones to go
online, as the next biggest market opportunity in the technology sector. With a market size of 4 billion cell
phone users that companies could connect to the Internet, “it is 40 times the opportunity to get 100 million
PC users online in the 1990s,” Huberty said.
- 32 -
market firms in year 0 (i.e., the portfolio formation year), consistent with investors
bidding high on stocks with high growth potentials. 15
4.3.2. Return on assets (ROA) and investment growth
An alternative explanation of the finding in the prior section is that ROA reflects
firms’ cost of capital. As a result firms with low cost of capital have low subsequent
stock return ((Berk, Green, and Naik 1999, Xing 2008, Chen, Novy-Marx, and Zhang
2010). The skewness hypothesis of this paper corroborates with the above Q-theory
based hypothesis in that the two effects are likely to reinforce each other in generating the
widely documented correlation between growth and return. 16 However, the two views
also differ fundamentally: in the Q-theory low cost of capital (ROA) induces high
subsequent asset growth. In contrast, the correlation goes the other way around in the
skewness hypothesis, i.e., high growth potential leads investors to lower required rate of
return. The Q-theory based hypothesis is silent with respect to what causes the cost of
capital to be low for low book-to-market firms. The skewness hypothesis provides such
an explanation, which accounts for the negative correlation between book-to-market and
realized return skewness documented in this paper.
In Table 7 we test the subsequent investment return of portfolios formed based on
the book rate of return on operating assets (ROA). Contrary to the cost of capital
hypothesis, the low ROA group actually has a significantly lower investment growth rate
(as measured in the growth rate of capital expenditure, as well as growth in total assets).
15
Analysts’ forecast of the mean EPS growth might also reflect, to some extent, the skewness in growth.
The untabulated test results show that firms with high analyst projected growth indeed have more positively
skewed return distributions.
16
Tobin (1969).
- 33 -
In year t+1, the CAPEX grew at an average of -3% for the low ROA group. In contrast,
the average CAPEX growth rate equals 19% for the high ROA firms.
Significant
difference is observed in years -1, year 0, and Year +2 as well.
In addition, the time-series growth pattern of CAPEX does not lend much support
to the cost-of-capital effect either. For low ROA firms, the rate of CAPEX growth was
actually at the lowest level in years 0 and 1, when the ROA is relatively low. In other
words, the results suggest firms actually grow CAPEX more when ROA is relatively high.
This contradicts the prediction that lower ROA (cost-of-capital) induces more investment
growth. Test results based on the growth rate of total assets as well as the growth rate of
total revenues show a very similar pattern.
In Panels D to F I compare the skewness of the growth rate of CAPEX, total
assets, and revenue, across firms with different ROA.
The skewness hypothesis
conjectures that the low ROA group is more likely to contain firms with more positive
skewness in payoff.
Such firms will subsequently experience skewed growth in
investment. However, since low ROA firms are also likely to include firms that are not
doing well, the average growth rate of this group of firms may be low (as we confirm in
Panels A to C). The high growth of a subset of successful firms will increase the
skewness of the growth rate for this group of firms. Indeed, this is what we find. As
shown in Panels D to F, the skewness in the growth rates of CAPEX, total assets, and
revenues are significant larger for the low ROA group. These findings provide strong
support for the skewness hypothesis.
- 34 -
4.3.3. Time series analysis of return premium
Following Fama and French (1996), I also construct a time series of return
premium for skewness (SKW). Each year I sort firms based on SKW and construct a
series of monthly returns based on the average stock return for firms within each SKW
group. Panel A of Table 8 shows some key statistics of the skewness premiums, together
with the factor return to book-to-market factor (HML) obtained from the website of Ken
French. Since the market premium and HML are calculated using the value-weighting
method, I also use value-weighting in constructing the SKW return series. The average
monthly return to SKW equals 0.81%. In comparison, the average return to HML equals
0.46%.
Panel B of Table 8 reports the results of regressing HML on the SKW factor
returns, and vice versa. The second column shows that after controlling for HML, SKW
factor still has a significant positive alpha (0.30%). In contrast, after controlling for
SKW, HML has a negative alpha of -0.01%. As a comparison, I also construct a calendar
time return premium series for BM’.
Since both SKW and BM’ are formed from
subsamples of the book-to-market partition, it is interesting to see how regression results
change when we replace SKW factor with BM’. Panel C of Table 8 shows that the
conclusion is reversed. After controlling for BM’, HML still has a significant positive
alpha of 0.25%. In contrast, when regressing BM’ on HML, the residual becomes
insignificant (0.11%).
Collectively, the evidence presented in Section 4 indicates that investors’
preference for positive return skewness explains a significant portion of the observed
premium (discount) associated with glamour (value) stocks.
- 35 -
5. ADDITIONAL TESTS
To assess the robustness of the test results, I perform several sensitivity checks.
The results are summarized as follows.
5.1. Bankruptcy score
One of most widely accepted explanations for the correlation between the bookto-market ratio and stock returns is that the book-to-market ratio captures the risk of
financial distress (Fama and French 1996, Campbell, Hilscher, and Szilagyi 2008). The
argument is that when the book value of a firm is close to its market value, investors
often consider the firm to have a significant risk of bankruptcy, and demand high return
to compensate for bearing such risk. Prior studies on bankruptcy risk provide mixed
support to this claim. Griffin and Lemmon (2002) document that default risk is not
confined to high book-to-market firms: More firms in the top O-score quintile, which is
based on the bankruptcy risk score developed by Ohlson (1980), have low book-tomarket rather than high book-to-market. Dichev (1998) also reports low correlation
between book-to-market and both O-scores and Altman Z-scores (Altman 1968). Fama
and French (2006) report a negative correlation between bankruptcy risk and returns. To
test whether the skewness measure provides any additional explanatory power, I include
the O-score in the characteristic regression of Section 4.1.
Panel A of Table 9 reports the results. Consistent with prior findings, SIZE shows
significant negative correlation with stock return while book-to-market shows significant
positive correlation with stock return (Model I).
- 36 -
The second column shows that
estimated coefficient on the O-score is not significant.
This is consistent with the
findings of Dichev (1998) and Fama and French (2006). The last column shows that,
with the O-score, the adjusted skewness measure remains negative and highly significant.
Although bankruptcy can be one of the reasons causing stock return to be skewed,
there are other factors that lead to skewness in returns. Hence we do not expect the Oscore to completely eliminate the predictive power of skewness. This would be
particularly true when change in the probability of financial distress is less of an issue, for
example, when the book-to-market ratio is low. Thus, we would expect skewness to be a
more significant predictor of stock return, compared with O-score, for firms with high
book-to-market ratios.
I repeat regression for firms with high, medium, and low book-to-market ratios
separately. The results are reported in Panel B of Table 9. For stocks with high book-tomarket ratios (i.e., when financial distress risk is a relatively more important concern),
the estimated coefficient on the O-score is positive and marginally significant, indicating
that high O-score on average increases the expected stock return. In contrast, for the
group of firms with low book-to-market ratio (i.e., when financial distress is less of an
issue), the estimated coefficient on O-score becomes insignificantly different from zero.
This result confirms our conjecture: the risk of financial distress can explain some of the
correlation between book-to-market and stock return, especially when such risk is high.
However, for firms with low book-to-market ratios, the correlation between book-tomarket and stock return is caused more by the difference in return skewness.
Lack of correlation between bankruptcy risk and returns could also be caused by
the noise in measuring the bankruptcy risk. Begley, Ming, and Watts (1996) update the
- 37 -
O-score model coefficient using data from the 1980s. I repeat the analysis with such
updated O-score coefficients. The results are essentially the same as what are reported in
Table 9. Vassalou and Xing (2004), and Hilligeist, Keating, Cram, and Lundstedt (2002)
use the Black-Scholes-Merton (BSM) measure of bankruptcy probability, and show that
such a measure provides significant improvement over O-score and Z-score due to its
incorporation of market price data. Such an advantage comes at the expense of smaller
sample size (due to data requirement and non-convergence in model estimation). I repeat
our tests using the BSM score. The results are similar to the test results with O-score,
with the exception that with BSM score added to the regression, the estimated coefficient
on book-to-market is no longer significant. This is likely to be caused by the fact that
BSM score itself is estimated based on market price.
5.2. Alternative measures of skewness
Besides the skewness measure used in the main tests, which is based on 10th and
the 90th percentile distributions, I also conduct tests based on the following alternative
skewness measures:
(1) Standardized Moment Skewness:
(2) Pearson Skewness:
- 38 -
(3) Bowley Skewness:
3
1
2
3
1
Table 10 reports the results using these alternative skewness measures.
All these
measures show the same pattern: low book-to-market firms have more positively skewed
returns, consistent with the findings reported based on the basic skewness measure. The
strong result with P-Skew supports the conjecture that the book-to-market ratio correlates
with skewness in a similar way as the difference between the mean and the median
captures skewness in the Pearson Skewness measure.
5.3. Coskewness
As discussed in Section 2, most prior research on skewness and return maintains
the assumption of uniform risk aversion. The implication is that the coskewness of
stocks matters in cross-sectional stock return tests. I conduct several tests regarding the
return coskewness and report the results in Table 11.
Panel A reports the results obtained from sorting firms based on the individual
stock return coskewness.
Following Harvey and Siddique (2000), I calculate each
individual stock’s return coskewness based on monthly stock return over the past 5 years.
Panel A shows that firms with low past coskewness have a lower mean return compared
to firms with high past coskewness. The difference, however, is insignificant. This lack
of return difference could be due to two reasons: either coskewness does not correlate
with average stock returns, or past coskewness is a poor proxy for expected stock return
coskewness in the future.
- 39 -
To shed further light on the above two possible causes, I calculate each firm’s ex
post realized return coskewness and report results in Panel A. Ex post coskewness is
estimated based on realized monthly returns in years t+1 to t+5. Consistent with the
findings of Harvey and Siddique (1999), a stock’s past coskewness shows very little
predictive power with respect to future coskewness. The difference in future coskewness
between the high and the low groups is insignificant.
In order to examine the extent to which coskewness might correlate with average
stock return, I sort stocks based on ex post coskewness over (t+1, t+5) and report the
result in Panel B. That is, Panel B shows that return difference if we can have perfect
foresight regarding all firms’ future coskewness. The average return for the low
coskewness group is 2.11%, whereas the average return for the high coskewness group is
1.52%. However, the difference (0.59%) is not statistically significant.
5.4. Different return horizon
All the test results reported so far are based on one-year-ahead future returns.
Common sense suggests that it may take longer for a firm to fully realize its upside
potential. I repeat our tests for different return horizons of 1 year, 3 years, 5 years, and
10 years. The results are summarized in Panel A of Table 12.
Comparing the results for the four different holding periods, several patterns
emerge. First, consistent with the conjecture, the average return skewness increases as
the holding period lengthens. Skewness of the low BM group increases from 0.15 for the
one-year horizon to 0.22 for the 10-year horizon. Second, the average return differences
between the high and the low group increases monotonically. For instance, with equal
- 40 -
weighting, the return difference for the one-year horizon is 8.33%. This return difference
increases to 33.88%, 66.14%, and 133.42% for the 3-, 5-, and 10-year horizons
respectively. Third, the difference in the skewness between the high and the low groups
always shows the same pattern. That is, consistent with our prediction, firms with low
book-to-market tend to have more positively skewed returns
I also calculate the annual hedge return, as well as the adjusted skewness, for
years from t+1 to t+10. The result is presented in Panel B of Table 12. The difference in
the average book-to-market ratio between the high and low group of firms gradually
shrink as the horizon expands (Nissam and Penman 2001). The hedge return is also
reduced to 6.10% in year 5, and to 2.51% in year 10. Consistent with the hypothesis of
this study, the difference in the adjusted skewness decreases gradually with the hedge
return, from -0.09 in year 1, to -0.07 in year 3, -0.06 in year 5, and -0.03 in year 10.
5.5. Other sensitivity test results
I also performed several other sensitivity tests, including (1) using the market
price at fiscal year end, instead of the market price in June each year, to calculate the
book-to-market ratio; (2) using 20, 30, or 80 stocks, instead of 50 stocks, in each random
portfolio formation; (3) increasing the size of the random sample from 1,000 observations
each size-year to 10,000; (4) restricting sample year to 1973 and beyond so that firms
from all three exchanges (i.e., NYSE, AMEX, and NASDAQ) are presented in all sample
years.
(5) calculating the market-adjusted return (MAR) as raw return minus the
corresponding return of the market index All results are robust to these variations in the
test design.
- 41 -
6. CONCLUSION
This paper documents that glamour stocks have significant excess positive
skewness in their return distributions compared with value stocks. This finding has
several implications: (1) it indicates that the book-to-market ratio and the asset growth
correlates with the skewness in a firm’s payoff; (2) the skewness in payoff is priced by
investors; and (3) that investors’ preference for skewness might have contributed to the
documented premium (discount) applied to glamour (value) stocks.
In evaluating the performance of investment portfolios, any systematic pattern in
skewness needs to be closely monitored. Many investment strategies which aim at
exploiting market mispricing may actually expose investors to significant skewness risk.
This paper shows that a simple measure based on the book-to-market ratio and the asset
growth rate has significant predictive power for future return skewness.
The findings of this paper also provide an interesting perspective regarding
accounting conservatism. Such a basic accounting principle has long been criticized for
reducing information quality (Hendriksen and Ban Breda 1992). This paper suggests that,
by tying the accounting treatment to the underlying level of uncertainty, a conservative
accounting system can provide investors with useful information to assess a firm’s upside
potential relative to its downside risk.
- 42 -
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Table 1: Descriptive statistics
The sample is constructed from all domestic common stocks traded on NYSE, AMEX and NASDAQ
(excluding OTC Bulletin Board/Pink Sheet), between June 28th, 1963 and June 30th, 2006. Stock price and
financial data are obtained from the CRSP and COMPSTAT datasets. Observations with missing book-tomarket ratio (BM), calculated using the market value at the end of June each year and the book value at the
end of the previous fiscal year, are excluded from the analysis. 17,18 Non-missing BETA on the last trading
date in June, which is calculated using a minimum 12 monthly return data from the prior 5 years, is also
required. SIZE is the logarithm of the market value of equity on the last trading day in June each year.
RET is the buy-and-hold annual return from July to June. Market adjusted return (MAR) is calculated as
raw return adjusted for stock BETA and market risk premium.
Variable
N
Mean
STDEV
Q1
Median
Q3
SIZE
BM
RET
MAR
151,717
151,717
151,717
151,717
4.57
0.79
0.18
-0.00
2.13
0.73
0.70
0.69
3.03
0.33
-0.17
-0.33
4.44
0.61
0.08
-0.05
6.02
1.03
0.38
0.22
Skewness Kurtosis
0.54
-0.13
0.69
9.00
0.59
135.84
0.53
134.58
17
To ensure the availability of accounting data, book value is obtained from the prior fiscal year which ends at least
four months before portfolios are formed.
18
Because of the misalignment of the measurement dates of the book value and the market value, as well as the effect
of accounting conservatism, the max (min) of the book-to-market ratio exceeds 100 (-300). To reduce the impact of the
extreme book-to-market ratios, BM is winsorized at the top and the bottom 1%.
- 48 -
Table 2: Return skewness of book-to-market portfolios
This table reports the mean and the skewness of market-adjusted stock returns for portfolios with low,
medium, and high book-to-market ratios. Each year firms are sorted into decile portfolios based on SIZE.
Within each size group firms are then sorted into three portfolios based on their book-to-market ratios. The
mean portfolio return, as well as the skewness, is calculated for each portfolio. This table reports the time
series average of the equal-weighted and the value-weight portfolio returns, as well as the skewness. *, **,
and *** indicate two-tailed statistical significance at 10, 5, and 1.
Book-to-market
Low
Medium
High
High-Low
Mean
Equal-weight
Value-weight
-0.92%
-5.59%***
1.14%
0.27%
4.04%**
2.95%*
5.78***
8.54%***
- 49 -
Skewness
0.15***
0.13***
0.12***
-0.03***
Table 3: Adjusted return skewness and fundamentals of book-to-market
portfolios
Panels A and B document the correlation between the mean and the skewness of stock returns of randomly
selected portfolios. Each year 1,000 random portfolios are constructed from all firms in each of the large,
medium and small size-groups with 50 stocks in each portfolio. The mean and the skewness of these stock
returns are then calculated. Panel A presents some key statistics of these portfolio returns. Panel B shows
the Spearman and the Pearson correlation between sample mean and sample skewness. These random
portfolio returns are then used to adjust the return skewness of portfolios formed randomly based on the
book-to-market ratios. The results are reported in panel C. *, **, and *** indicate two-tailed statistical
significance at 10, 5, and 1 percent levels.
Panel A: Portfolio return statistics, average of 132,000 random portfolios
Variable
Mean
STD
RET
MAR
0.16***
0.00
0.49***
0.50***
Skewness
0.21***
0.13***
Kurtosis
4.34***
3.92***
Panel B: Correlation between the mean and the skewness of random portfolio return (MAR)
Pearson
Spearman
Panel data
By year
0.29***
0.24***
0.35***
0.32***
Panel C: Mean, skewness and excess skewness for high, medium and low book-to-market firms
MAR
Book-to-market
Skewness
Adj. Skewness
Equal-weight
Value-weight
-0.80%
Low
-5.21%**
0.06***
0.15***
***
0.18%
Medium
0.46%
0.13
-0.02
3.11%**
High
3.12%*
0.12***
-0.03***
3.91**
High-Low
8.33%**
-0.03***
-0.09***
Hedge Return
8.33%**
6.14***19
-0.04***
-0.09***
19
The market capitalizations of the long-side stocks are used as the weights in calculating the valueweighted mean hedge return.
- 50 -
Table 4: Fundamentals of book-to-market portfolios
This table reports the skewness of fundamentals for each book-to-market portfolio. Growth in revenue
(Compustat data item #12), income before extraordinary items (Compustat data item #237), total assets
(Compustat data item #6), and cash flow from operations (Compustat data item #308) are denoted as
GREV, GINC, GTA, and GCFO respectively. 20 *, **, and *** indicate two-tailed statistical significance at
10, 5, and 1 percent levels.
Book-to-market
Low
Medium
High
High-Low
Skew(GREV)
***
0.36
0.19***
0.06***
-0.30***
Skew(GINC)
***
-0.52
-0.34***
-1.67***
-1.15.***
20
Skew(GTA)
***
0.39
0.22***
0.05***
-0.34***
Skew (GCFO)
0.21***
0.13***
0.01
-0.19***
When data #308 is not available, cash flow from operation (CFO) is calculated as fund from operation (#110)
adjusted for changes in working capital.
- 51 -
Table 5: Characteristic return regression
This table reports the cross sectional regression result based on 396,000 random portfolio returns. BM,
SIZE are the mean book-to-market and size of each portfolio. Adj. ex post Skew is adjusted ex post
portfolio return skewness. Adj. ex ante Skew is the adjusted return skewness during the prior fiscal year.
Similarly, Adj. ex ante (ex post) STD (KURT) are the adjusted ex ante or ex post portfolio return standard
deviation and kurtosis. *, **, and *** indicate two-tailed statistical significance at 10, 5, and 1 percent levels,
adjusted for clustering within each year-SIZE-BM group.
Panel A: With adjusted ex post return skewness
Intercept
BM
SIZE
Adj. ex post SKEW
Adj. ex post STD
Adj. ex post KURT
R-square
Dependent Variable: MAR
Model II
0.02
0.01
-0.01**
-0.09***
Model I
-0.00
0.03***
-0.01*
2.54%
3.02%
Model III
0.03
0.01
-0.01**
-0.07***
-0.14**
-0.00
5.16%
Panel B: With adjusted ex ante return skewness
Intercept
BM
SIZE
Adj. ex ante SKEW
Adj. ex ante STD
Adj. ex ante KURT
R-square
Model I
-0.00
0.03***
-0.01*
Dependent Variable: MAR
Model II
0.01
0.02*
-0.01*
-0.08***
2.54%
2.79%
Model III
0.03
0.02*
-0.01*
-0.08***
0.02
-0.00
2.87%
Panel C: With ex ante return skewness
Intercept
BM
SIZE
Ex ante SKEW
Ex ante STD
Ex ante KURT
R-square
Model I
-0.00
0.03***
-0.01*
Dependent Variable: MAR
Model II
0.02
0.02**
-0.01*
-0.05**
2.54%
2.89%
- 52 -
Model III
-0.00
0.02**
-0.01
-0.07***
0.03
-0.00
3.08%
Table 6: Skewness measure based on book-to-market and ROA
This table reports the mean, the adjusted skewness, and some other key statistics of portfolio returns when
portfolios are formed based on book-to-market (BM) and return on asset (ROA). In Panel A, group 1
contains firms from the bottom three BM deciles and the bottom three ROA deciles. Group 3 contains
firms from the top three BM deciles and the top three ROA deciles. The rest of the firms are in group 2. In
Panel B, the low (high) BM’ groups consist of firms in the bottom (top) three BM deciles and the top
(bottom) three ROA deciles. *, **, and *** indicate two-tailed statistical significance at 10, 5 and 1 percent
levels.
Panel A: Portfolio sort by SKW
Mean
EW
VW
High SKW
-11.55%***
-3.33%*
0.20***
Adj.
Skewness
0.13***
Skew
(GREV)
0.40***
0.14***
Skewness
BMt
Medium SKW
-0.09%
0.99%
0.14***
0.02
0.23***
0.88***
Low SKW
Low - High
0.26%
11.80%***
3.63%*
6.96%**
0.11***
-0.09***
-0.01
-0.14***
-0.00
-0.40***
1.42***
1.28***
Skew
(GREV)
BMt
0.14***
Adj.
Skewness
0.03
0.26***
0.25***
Panel B: Portfolio sort by BM’
Mean
Skewness
Low BM’
EW
-1.76%
VW
-0.08%
Medium BM’
-099.%
0.20%
0.13***
0.02
0.25***
0.75***
High BM’
High - Low
1.11%
2.87%
2.14%
2.22%
0.14***
0.00
0.04
0.02
0.10***
-0.16**
1.65***
1.40***
- 53 -
Table 7: Average investment growth of ROA portfolios
This table reports the mean and the skewness of the growth rate in capital expenditure (CAPEX, Compustat
data #128), total assets (TA, Compustat data item #6), and revenues (REV, Compustat data item #12) for
each ROA portfolios. ROA is the book rate of return of operating assets, calculated as operating income
(Compustat data item #178), minus tax, dividend by the average amount of net operating assets. Net
operating assets are estimated as the sum of the book value of common equity (Compustat data item #60)
plus the book value of interest bearing debts (Compustat data items #9, #34, plus preferred stocks, #130).
Each year portfolios are formed based on the distribution of ROA among firms within the same size deciles.
* **
, , and *** indicate two-tailed statistical significance at 10, 5 and 1 percent levels.
Panel A: Average growth in CAPEX (GCAPEX) of ROA portfolios
Low ROA
Medium ROA
High ROA
High-Low
Year -3
Year -2
Year -1
Year 0
Year +1
Year +2
Year +3
0.14***
0.12***
0.14***
0.00
0.12***
0.12***
0.14***
0.02
0.08***
0.11***
0.19***
0.09***
-0.02
0.10***
0.25***
0.27***
-0.03
0.08***
0.19***
0.22***
0.05***
0.08***
0.10***
0.05**
0.08***
0.08***
0.08***
0.00
Year +1
Year +2
Year +3
Panel B: Average growth in total assets (GTA) of ROA portfolios
Low ROA
Medium ROA
High ROA
High-Low
Year -3
Year -2
Year -1
Year 0
***
***
***
***
0.09
0.09***
0.11***
0.02*
0.09
0.08***
0.12***
0.03**
0.07
0.08***
0.14***
0.07***
***
0.01
0.07***
0.16***
0.15***
0.01
0.07***
0.14***
0.13***
***
0.04***
0.07***
0.11***
0.08***
0.07***
0.10***
0.06***
0.03
Panel C: Average growth in total revenues (GREV) of ROA portfolios
Year -3
Low ROA
Medium ROA
High ROA
High-Low
0.12***
0.10***
0.12***
0.00
Year -2
Year -1
Year 0
Year +1
Year +2
Year +3
0.11***
0.10***
0.13***
0.02**
***
0.05***
0.09***
0.16***
0.09***
0.08***
0.09***
0.12***
0.03***
0.08***
0.08***
0.10***
0.02***
0.08***
0.08***
0.10***
0.02***
0.09
0.10***
0.15***
0.06***
Panel D: Skewness of CAPEX growth of ROA portfolios
Low ROA
Medium ROA
High ROA
High-Low
Year -3
Year -2
Year -1
Year 0
Year +1
Year +2
Year +3
1.64***
0.93***
1.39***
-0.25**
1.61***
1.02***
1.25***
-0.36***
1.55***
0.94***
1.18***
-0.37***
1.30***
0.87***
1.26***
-0.03
1.21***
0.71***
1.08***
-0.13*
1.26***
0.76***
0.91***
-0.35***
1.31***
0.75***
0.85***
-0.46***
- 54 -
Panel E: Skewness of TA growth of ROA portfolios
Low ROA
Medium ROA
High ROA
High-Low
Year -3
Year -2
Year -1
Year 0
Year +1
Year +2
Year +3
0.79***
0.29***
0.26***
-0.53***
0.81***
0.28***
0.26***
-0.55***
0.77***
0.28***
0.28***
-0.49***
0.42***
0.21***
0.31***
-0.11***
0.24***
0.14***
0.23***
-0.01
0.22***
0.12***
0.18***
-0.04*
0.22***
0.11***
0.13***
-0.08***
Year +1
Year +2
Year +3
Panel F: Skewness of revenue growth of ROA portfolios
Low ROA
Medium ROA
High ROA
High-Low
Year -3
Year -2
Year -1
Year 0
***
***
***
***
0.54
0.19***
0.20***
-0.34***
0.55
0.19***
0.20***
-0.35***
0.53
0.18***
0.22***
-0.31***
- 55 -
0.40
0.15***
0.23***
-0.17***
***
0.36
0.11***
0.12***
-0.24***
***
0.21
0.09***
0.10***
-0.11***
0.18***
0.07***
0.07***
-0.11***
Table 8: Calendar time return analysis
This table presents the results from time-series regression with the skewness return series constructed based
on SKW. The calendar time return-series contains 528 monthly observations from July 1963 to June 2007.
Panel B reports the result from regressing the HML factor returns on SKW, and vice versa. Panels C shows
the results from similar tests with SKW replaced with BM’. *, **, and *** indicate two-tailed statistical
significance at 10, 5 and 1 percent levels.
Panel A: Descriptive statistics.
Variable
N
Mean
STD
Q1
Median
Q3
MKT-RF (%)
HML (%)
SKW (%)
528
528
528
0.48
0.46
0.81
4.36
2.90
3.73
-2.12
-1.14
-1.12
0.80
0.44
0.81
3.41
1.79
2.59
Panel B: Regression of HML on UD, and vice versa.
Dependent variable
Intercept (%)
SKW
HML
N
R-square
HML
-0.01
0.62***
528
63.4%
SKW
0.30***
1.02***
528
63.4%
Panel C: Regression of HML on UD’, and vice versa
Dependent variable
Intercept
BM’
HML
N
R-square
HML
0.25**
0.41***
528
22.4%
BM’
0.11
0.55***
528
22.4%
- 56 -
Table 9: Characteristic return regression with bankruptcy score
This table presents the results from cross-sectional return regressions with O-Score which measures the
probability of financial distress (Ohlson 1980). Other independent variables include book-to-market (BM),
SIZE, and the adjusted ex ante skewness. In Panel B separate regressions are conducted for firms with high,
medium, and low book-to-market ratios. *, **, and *** indicate two-tailed statistical significance at 10, 5,
and 1 percent levels, adjusted for within-book-to-market-group clustering (Peterson 2009).
Panel A: All sample firms
Intercept
BM
SIZE
O-score
Adj. ex ante Skew
R-square
Model I
-0.00
0.03***
-0.01*
Dependent Variable: MAR
Model II
-0.04
0.04***
0.00
0.01
2.54%
3.19%
Model III
-0.03
0.02**
-0.00
0.01
-0.10***
3.65%
Panel B: Separate regression results for firms with high, medium, or low book-to-market ratios
Dependent Variable: MAR
Low BM
Medium BM
High BM
0.13*
-0.10*
0.18**
Intercept
**
-0.02
-0.02
-0.11
BM
-0.01
0.01
-0.01
SIZE
0.04*
0.02
0.02*
O-score
***
**
-0.06**
-0.10
-0.06
Adj. ex ante Skew
9.57%
2.57%
6.99%
R-square
- 57 -
Table 10: Alternative measures of skewness
This table shows the stock return skewness, based on several alternative measures, for the three book-tomarket (BM) portfolios. Pearson Skewness (P-Skew) is defined as (mean-median)/standard deviation.
Bowley Skewness (B-Skew) is calculated as (Q3+Q1-2*Median)/(Q3-Q1). SM-Skew is calculated as the
third standardized moment, defined as μ3/σ3, where μ3 is the third moment about the mean and σ is the
standard deviation. . Each year random portfolios are constructed from each book-to-market group with 50
firms in each portfolio. The mean and the various skewness measures of stock returns are then calculated
for each portfolio. This process is repeated for each sample year from 1963 to 2006. The numbers shown
are the time series mean across all sample years. *, **, and *** indicate two-tailed statistical significance at
10, 5 and 1 percent levels.
Book-to-market
Low
Medium
High
High-Low
Mean
Value weight
Equal weight
-0.80%
-5.21%***
0.18%
0.46%
3.11%**
3.12%*
3.91**
8.33%***
Skewness
SM-Skew
B-Skew
P-Skew
0.15***
0.13***
0.12***
-0.03***
1.16***
1.07***
1.06***
-0.10***
0. 06***
0.05***
0.04***
-0.02**
0.11***
0.10***
0.09***
-0.02***
- 58 -
Table 11: Adjusted return skewness for portfolios based on coskewness
This table reports some key statistics of stock returns when portfolios are formed based on firm-specific
coskewness calculated from time-series return data. Panel A reports the test results when firms are sorted
based on their return coskewness calculated over years t-4 to t. Panel B presents test results when firms are
sorted based on ex post realized coskewness over years t+1 to t+5. *, **, and *** indicate two-tailed
statistical significance at 10, 5 and 1 percent levels.
Panel A: Firms are sorted base on ex ante individual stock return (t-4 to t) coskewness
Mean
Cross-sectional
Time-series
Co-skew
Adj. skewness
skewness
coskewness
Equal weight
Value weight
0.11%
-2.11**
Low
-2.34%
0.14***
0.03
*
**
0.19%
-1.82
Medium
0.99%
0.13***
0.01
0.94%
-2.42**
High
-1.29%
0.12***
0.01
-0.83%
0.31
Low-High
-1.05%
0.02
0.02
Panel B: Firms are sorted base on ex post individual stock return coskewness (t+1 to t+5)
Ex post
Mean
Cross-sectional
time- series
Co-skew
Adj. skewness
skewness
Equal
weight
Value
weight
coskewness
-9.19***
Low
2.11%
1.17%
0.14***
0.04
**
**
-1.84
Medium
1.23%
1.42%
0.12***
-0.01
5.51***
High
1.52%
0.25%
0.12***
-0.01
***
-14.70
0.02
0.05
Low-High
0.59%
0.92%
- 59 -
Table 12: Return skewness for different holding horizons
This table shows the mean, skewness and the adjusted skewness of stock returns for the three book-tomarket (BM) portfolios. Each year we construct 100 portfolios from each book-to-market group by
randomly selecting 50 firms within the BM group for each portfolio. The mean, skewness and adjusted
skewness of the stock return are then calculated for each portfolio. This process is repeated for years from
1963 to 2006 with non-overlapping return windows. Panel A shows the returns with holding period
ranging from 1 year to 10 years. Panel B shows the average return for sample years t+1, t+3, t+5 and t+10
respectively. *, **, and *** indicate two-tailed statistical significance at 10, 5, and 1 percent levels.
Panel A: Return skewness for holding periods of 1, 3, 5, and 10 years.
Mean
Skewness
Adj. Skewness
Book-to-market
Equal weight
Value weight
1 Year
-0.80%
Low
-5.21%***
0.06***
0.15***
***
0.18%
Medium
0.46%
0.13
-0.02
3.11%**
High
3.12%*
0.12***
-0.03***
High-Low
8.33%***
3.91**
-0.03**
-0.09***
3 Years
-5.12%
Low
-22.85%***
0.23***
0.10***
***
2.35%
Medium
0.91%
0.16
-0.01
***
*
***
12.46%
High
11.03%
0.15
-0.04***
17.58%***
High-Low
33.88%***
-0.08***
-0.14***
5 Years
-1.46%
Low
-40.77%***
0.28***
0.14***
**
**
***
8.03%
Medium
10.01%
-0.03**
0.17
25.64%***
High
25.38%**
0.18***
-0.05***
**
***
***
27.10%
High-Low
66.14.%
-0.10
-0.19***
10 Years
***
4.11%
Low
-97.36%
0.22***
0.08***
25.43%*
Medium
12.04%
0.18***
-0.01
**
**
52.82%
High
36.06%
0.13***
0.05
48.71%*
High-Low
133.42%***
-0.09***
-0.13***
Panel B: Persistence of annual return skewness difference for years +1, +3, +5, and +10
High-Low
BTM
MAR
Adj. Skewness
Year +1
Year +3
Year +5
Year +10
1.08***
8.33%**
-0.09***
0.75***
8.44%***
-0.07***
0. 58***
6.10%**
-0.06***
0.38***
2.51%
-0.03
- 60 -
Figure 1: Step-wise utility function
Source: Friedman and Savage (1948).
- 61 -