Book-To-Market Ratio and Skewness of Stock Returns

THE ACCOUNTING REVIEW American Accounting AssociationVol. 88, No. 6 DOI: 10.2308/accr-505242013pp. 2213–2240

Book-to-Market Ratio and Skewness of StockReturns

Xiao-Jun Zhang

University of California, Berkeley

ABSTRACT: This study demonstrates that stocks with low book-to-market ratios, also

known as glamour stocks, have significantly more positive skewness in their return

distributions compared to the return distributions of value stocks with high book-to-

market ratios. The premium (discount) investors apply to these glamour (value) stocks

also correlates significantly with the difference in return skewness. These findings

suggest that the value/glamour-stock puzzle is partially explained by investor preference

for positive skewness in stock returns. Such preference for skewness, which is

consistent with investors having inverse S-shaped utility functions, is observed in such

consumer behaviors as lottery purchases and gambling. This paper further documents

significant predictive power of accounting-based measures, such as the book rate of

return, with respect to the skewness of stock returns.

Keywords: accounting conservatism; value/glamour stocks; skewness; book-to-marketratio; growth.

Data Availability: Data are available from sources identified in the paper.

I. INTRODUCTION

Stocks with high (low) prices relative to their fundamentals, such as book values and

earnings, are often referred to as glamor (value) stocks. By comparing 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 relatively extreme

positive (negative) skewness in their return distributions. The skewness of a stock return

distribution reflects its degree of asymmetry, with positive (negative) skewness indicating a longer

or fatter right (left) tail, which represents the likelihood of extremely large holding gains (losses).1

The analysis further reveals that a significant portion of the documented premium (discount) that

I thank Malachy English, John Harry Evans III (senior editor), Clifton Green, Sanjay G. Kallapur (editor), Charles Lee,James Ohlson, James Ryans, two anonymous reviewers, and seminar participants of the 2010 JAAF Conference forhelpful comments and suggestions.

Editor’s note: Accepted by Sanjay G. Kallapur.

Submitted: January 2011Accepted: June 2013

Published Online: June 2013

1 See Appendix B for alternative measurements of skewness.

2213

investors apply to glamour (value) stocks is driven by investor preference for positive skewness in

return distributions. Similar preference for positive skewness has also been observed in consumer

behavior with respect to lottery purchases and gambling.

Prior literature demonstrates that value stocks tend to outperform glamour stocks in subsequent

stock returns (Basu 1977; Rosenberg et al. 1985; Fama and French 1992). This phenomenon, often

labeled as the value/glamour anomaly, is considered puzzling because it contradicts the predictions

of the efficient market hypothesis and the capital asset pricing model (CAPM) of Sharpe (1964) and

Lintner (1965). In searching for the cause of this anomaly, some investigators conclude that

investor mispricing plays a major role (Lakonishok et al. 1994; La Porta 1996; Piotroski 2000).

Others question whether the documented correlation is indeed spurious (Kothari et al. 1995; Chan

et al. 1995). The persistence of this book-to-market phenomenon also leads many researchers to

conclude that the value premium may serve as compensation for risk or transaction costs (Fama and

French 1995; Berk et al. 1999; Zhang 2005; Xing 2008; Penman and Reggiani 2012).

This paper provides an alternative explanation along the lines of risk and reward analysis.

However, rather than analyzing investment risk in terms of the second moment of a return

distribution, this study explores the implications of investor preference for skewness. I posit that the

book-to-market ratio of a firm correlates with the skewness of its payoff, which in turn affects the

pricing of its stock.2 This hypothesis stems from the accounting principles that govern the

measurement of book value. Unlike stock price, which is determined by the market, the book value

of a firm is measured according to Generally Accepted Accounting Principles (GAAP). Some

fundamental principles of GAAP, such as the realization principle and the conservatism principle,

cause an enterprise’s book value to differ systematically from its market values. GAAP accounting

rules generally mandate that firms measure book value and earnings conservatively when the level

of uncertainty associated with the relevant future cash flows is high. Probable losses from future

operations are typically recognized, but not unrealized gains. With such a conservatively biased

accounting system, book value is more likely to capture a firm’s downside risk, 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 reflects a firm’s upside relative to its

downside, as discussed in more detail in Section II. In a statistical sense, I hypothesize that the

book-to-market ratio correlates with the expected skewness of future stock returns. In turn, such

difference in skewness affects the average stock returns.

How skewness affects asset pricing is a long-standing issue in finance and economics.

Although the mainstream asset pricing models, such as CAPM, make no prediction regarding the

effect of skewness on pricing, some studies move away from the mean-variance framework to offer

insights on this issue. Early papers in this vein conclude that stock coskewness, not skewness,

matters in asset pricing. Coskewness represents the degree to which stock return skewness varies

with the overall skewness of a well-diversified investment portfolio (Rubinstein 1973; Kraus and

Litzenberger 1976). Empirical studies on coskewness yield mixed results on the power of

coskewness to explain cross-sectional stock returns (e.g., Friend and Westerfield 1980; Harvey and

Siddique 2000; Hung et al. 2004; Post et al. 2008). More recently, Barberis and Huang (2008)

analyze an equilibrium asset pricing model based on the cumulative prospect theory of Tversky and

Kahneman (1992). A key feature of the Barberis and Huang (2008) model is that, in equilibrium,

idiosyncratic stock return skewness is priced. The central prediction of their model is that more

positively skewed stocks, ceteris paribus, earn lower average returns. Similar predictions are made

by Mitton and Vorkink (2007) who study an expected utility model in which some investors have

2 Payoff refers to return to investors both in terms of stock returns as well as in terms of fundamentals such asearnings and cash flows. See Appendix A for further discussions, as well as Section III for relevant test results.

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The Accounting ReviewNovember 2013

convex, skewness-loving preferences. Likewise, Brunnermeier et al. (2007) analyze a setting in

which investors choose their beliefs to maximize the discounted value of expected future utility

flows. The Barberis and Huang (2008) prediction, together with the correlation between

book-to-market ratio and skewness of stock return distribution, provides the foundation for a

hypothesis linking book-to-market ratio with average stock return via the impact of skewness on

asset pricing.

To test this hypothesis, I first compare the skewness of future return distributions among firms

with different book-to-market ratios. The sample comprises the entire set of U.S. firms traded on

NYSE, AMEX, and NASDAQ from 1963 to 2010. Following Fama and French (1992), I sort firms

into deciles each year based on their book-to-market ratios. Firms in the top three deciles are

classified as having high book-to-market 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 greater skewness in their return distributions compared to value stocks. After

removing 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. This finding is consistent with the conjecture that the book-to-market

ratio has significant predictive power with respect to the skewness of future stock returns.

Next, I examine whether such difference in return skewness, coupled with investors’ affinity

for skewness, contributes to the observed average return difference between value and glamour

stocks. I use two sets of tests to address this issue. The first set examines the correlation between

expected excess skewness and expected stock return. I use both ex post portfolio return skewness

and ex ante portfolio return skewness to proxy for expected return skewness. Cross-sectional return

regression results show that the skewness measures have significant negative correlations with

realized stock returns. Moreover, after controlling for skewness, both the magnitude and the

significance of the estimated coefficient on the book-to-market ratio are substantially reduced.

The second set of tests partitions firms in the high and the low book-to-market groups into

subsets according to the different projected payoff skewness, based on their investment growth

patterns. Analysis of the main hypothesis suggests that the skewness of future earnings and cash

flows results primarily from the skewness in firm investment growth. Under conservative

accounting, high investment growth translates into a low book rate of return on firm operating

assets (ROA) (Zhang 1998; Rajan et al. 2007). Accordingly, I compare the portfolio return

distributions of the following two sets of firms. In the first set, low- (high-) book-to-market firms

also have more (less) investment growth, as measured by ROA. In the second set, low

book-to-market firms have less investment growth than 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 is almost identical across the two sample sets. However, the difference

in projected payoff skewness is quite substantial. Test results reveal that the first set of firms

shows a highly significant difference in return skewness, as well as in average stock return,

between the high and the low book-to-market firms. In contrast, among firms in the second

sample set, the differences in the mean and 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 potential, then their payoffs are

less likely to have the desired degree of positive skewness. As a result, investors are less willing

to pay a premium for such stocks.

In summary, this paper shows that value and glamour stocks differ in their return skewness and

documents how this difference affects average stock returns. Findings from this study highlight the

promise of analyzing investor skewness-seeking behavior to better understand the pricing of capital

assets. The hypothesis put forth in this paper does not exclude other explanations for the

Book-to-Market Ratio and Skewness of Stock Returns 2215


book-to-market phenomenon. Firm growth, for example, can increase skewness as well as cause

investors to over-react. Nonetheless, investors’ skewness-seeking behavior is uniquely able to

explain differences in skewness across book-to-market portfolios as well as investors’ preference

for growth stocks. This study makes two contributions to the literature. First, documenting how

value and glamour firms differ in their return skewness provides a novel perspective with which to

analyze the book-to-market effect. It links the phenomenon to the fundamental issue of potential

convexity in investors’ utility functions. Second, early empirical studies on the issue of skewness

and asset pricing focus mainly on coskewness, yielding mixed results (e.g., Harvey and Siddique

2000; Hung et al. 2004; Post et al. 2008). More recent works on idiosyncratic skewness provide

more supportive, albeit still mixed, evidence on the equilibrium pricing of skewness (e.g., Kumar

2005, 2009; Zhang 2005; Kapadia 2006; Xing et al. 2010; Green and Hwang 2010; Boyer et al.

2010; Bali et al. 2011).3 This paper adds to the literature by providing new evidence supporting the

hypothesis that capital asset pricing is significantly affected by investor skewness-loving behavior.

The present study is further distinguished in two important respects. First, in prior studies, the

correlation between book-to-market and skewness is not explored; rather, book-to-market is used as

a control variable that presumably captures risk unrelated to skewness. As shown in this paper, the

book-to-market anomaly is significantly driven by skewness. Without controlling for the relation

between the book-to-market ratio and the stock return skewness, it is likely that earlier studies

underestimate the correlation between skewness and expected stock return. Second, prior studies

show that predicting return skewness and coskewness is a very challenging task (Singleton and

Wingender 1986; Harvey and Siddique 1999). This paper contributes to the literature by

demonstrating how researchers can use accounting-based measures, such as the book rate of return

and the book-to-market ratio, to predict skewness in stock returns.4

II. HYPOTHESIS DEVELOPMENT AND RESEARCH DESIGN

Correlation between Book-to-Market Ratio and Skewness

The first part of the analysis proposes a negative correlation between the book-to-market ratio

of a firm and the skewness of its stock returns. Development of this proposition is based on two

observations: (1) the bias in accounting stemming from the conservatism principle and the

realization principle, and (2) the skewness in a firm’s payoff resulting from the options embedded in

its operations.

The conservatism principle and the realization principle of accounting cause the measurement

of assets and earnings to bias downward. There are two types of accounting conservatism:

conditional and unconditional (Beaver and Ryan 2005). With unconditional conservatism,

intangible assets, such as growth potential, brand name, market share, and R&D, are ignored on the

balance sheet. With conditional conservatism, book value is written down under sufficiently

adverse circumstances, but is not written up under favorable conditions. The lower-of-cost-or-

3 Xing et al. (2010) conclude that the risk-neutral skewness measure of Bakshi et al. (2003) does not havesignificant correlation with future stock returns. In contrast, Boyer et al. (2010) document evidence suggestingthat idiosyncratic skewness and returns are negatively correlated. Kumar (2009) and Bali et al. (2011) providemore direct evidence that individual investors prefer stocks with lottery-like features. Kumar (2005) furthershows that institutional investors generally do not exhibit the same preference.

4 Conrad et al. (2009) use options prices to estimate the ex ante skewness of individual stocks. Their estimationmethod, developed in Bakshi et al. (2003), has the advantage of not relying on a cross-sectional sample toestimate skewness. The disadvantage, however, is that it involves strong assumptions in estimating the risk-neutral moments. The noisy nature of their measure is evident from the fact that, when applied to technologystocks, the measure does not show positive skewness during the bubble period. Xing et al. (2010) also examinethe Bakshi et al. (2003) measure of skewness and find no correlation between it and expected stock returns.

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market method of accounting for inventories serves as good example.5 Because of this conservative

tendency embedded in these two types of accounting rules, the book value of a firm often reflects

more of the downside of the firm, e.g., how much the firm is worth in liquidation, as opposed to the

upside potential of a firm from its future operations.6 In contrast, market value reflects both the

upside and the downside without the bias in accounting.

Another important characteristic of a firm’s payoff is the skewness attributable to the options

embedded in its operations. Simply put, when it becomes clear that a firm’s business plan will not

be fruitful, the manager can often liquidate the operation in a timely fashion to curtail further losses.

On the other hand, when market conditions appear favorable, management can exercise the option

to expand operations (Robichek and Van Horne 1967). As a result, the payoff from a firm’s

operations is often positively skewed.

The combined effects of the above-described accounting and operating factors is that the book

value and the market value tend to reflect two different aspects of a firm’s positively skewed payoff.

The book value captures more of the downside of a firm, while the market price reflects the

expected value. Because of this, the ratio of the two measures, i.e., the book-to-market ratio,

correlates with the degree of skewness in the payoff, similar to the way in which the difference

between mean and median can be used to measure skewness (Pearson 1895).7,8 That is:

H1: Stocks with low book-to-market ratios have more positive skewness, or less negative

skewness, in their return distributions as compared to stocks with high book-to-market

ratios.

There are countervailing forces that may reduce the magnitude of the above-described

correlation. For example, when a firm’s book-to-market ratio is high, the risk of bankruptcy is often

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 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 the return

distribution. The overall skewness of the distribution is a product of the combination of these two

forces. The distressed situation of high book-to-market firms makes the probability of high long-

term returns rather low, reducing the potential impact of the second force. Nonetheless, it is possible

that some of the firms with high bankruptcy risk may have high positively skewed return

distributions. Given these counteracting forces, it is interesting to empirically examine the

correlation between book-to-market ratio and the skewness in return distribution.

5 When the market price of a piece of inventory exceeds its historical cost, the inventory is valued at the historicalcost because the increase in inventory value since purchase is deemed unreliable. However, if there is anindication that the inventory’s net realizable value is below its cost, even though such indications can be asunreliable as any indication for a value increase, then firms are required to mark the inventory down to reflectsuch potential loss.

6 Although both types of conservative rules can cause the book value to reflect more on a firm’s downside, thereare some subtle differences. Compared to conditional conservatism, unconditional conservative rules cause thebook-to-market ratio to correlate more with the difference in the asset composition across industries. Hence,unconditional conservatism is more likely to capture inter-industry differences while conditional conservatismreflects more intra-industry differences.

7 Appendix A contains a more detailed illustration of this point.8 The link between book-to-market ratio and skewness also depends on the way skewness is measured. For further

discussion of the various skewness measures, see Appendix B.



Correlation between Skewness and Expected Stock Return

The second part of the analysis concerns the correlation between skewness and expected stock

return. Analysis of this issue requires deviating from the mean-variance framework of asset pricing

(Rubinstein 1973; Kraus and Litzenberger 1976; Mitton and Vorkink 2007; Barberis and Huang

2008). Studies on this topic all draw on insights obtained from analyzing individual behavior in

lottery purchases and gambling. These observations suggest that investors prefer more positively

skewed distributions. However, prior research has incorporated skewness preference into asset

pricing in different manners.

One line of studies introduces skewness-loving utility functions while maintaining their

concavity (Rubinstein 1973; Kraus and Litzenberger 1976; Harvey and Siddique 2000). The

implication is that investors are still universally risk-averse, and therefore all hold well-diversified

portfolios. In such economies, only a security’s coskewness, measured by the correlation between

the stock’s skewness and the skewness of the market portfolio, is priced.

A second line of research replaces the universal risk-aversion assumption by introducing some

convexity into the utility function (Friedman and Savage 1948; Markowitz 1952; Mitton and

Vorkink 2007; Barberis and Huang 2008). Friedman and Savage (1948) examine a continuous

utility function with inversed S-shaped segments. This smooth step-shaped utility function captures

the notion that some consumption is not divisible (Kwang 1965). As a result, increasing wealth

beyond certain threshold levels can increase utility substantially by qualitatively improving an

individual’s socioeconomic status. When the level of wealth places an individual at the bottom of a

‘‘hill’’ in the utility function, and an investment offers the opportunity for a significant ‘‘step-up’’ in

consumption without excessive downside risk, the individual becomes more likely to invest. The

individual will not fully diversify, since doing so may reduce the skewness effect. Kahneman and

Tversky (1979) developed the prospect theory to incorporate this and other aspects of human

behavior, such as loss aversion, under conditions of uncertainty. Barberis and Huang (2008) study

equilibrium asset pricing based on the cumulative prospect theory (Tversky and Kahneman 1992).

These authors show that rather than fully diversifying in equilibrium, some investors hold more

stocks with positively skewed return distributions. As a result, idiosyncratic skewness is priced in

equilibrium. More positively skewed stocks, ceteris paribus, earn lower average returns.

This paper adopts the latter approach in which investor preference for skewness is inherently

related to risk-taking behavior. The analysis of skewness enables researchers to develop the link

between book-to-market ratio and stock returns. Accounting conservatism implies that the book-to-

market ratio correlates with skewness, but 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 the market portfolio. The issue of market portfolio choice is problematic

when investors do not fully diversify in equilibrium, as illustrated in Barberis and Huang (2008).

Nonetheless, for the purpose of comparison, I repeat tests based on coskewness and discuss the

results in Section V.

I perform two sets of tests to examine Hypothesis 2, which explores the second part of this

study’s theme:

H2: Investors’ preference for skewness contributes significantly to the difference in expected

stock returns between value and glamour stocks.

In the first set of tests, I calculate both ex ante and ex post return skewness of the book-to-market

portfolios and combine these measures of skewness with the book-to-market ratios in cross-

sectional return regressions. If skewness contributes to the value/glamour anomaly, then the

magnitude and the significance of the estimated coefficient on the book-to-market ratio should

decrease with the inclusion of the skewness measure in the regression. Second, to create a more

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significant divergence between the book-to-market ratio and the expected skewness, I sort firms

based on other accounting measures that predict future payoff skewness. If skewness preference

partially drives the correlation between the book-to-market ratios and expected stock returns, then

we should observe an increase in the future hedged portfolio return when firms are further

partitioned based on other accounting measures. This additional portfolio-formation step increases

the difference in the expected skewness between the long- and the short-portfolios, while

maintaining the level of difference in the average book-to-market ratios. More detailed discussion

of these tests and their results will be presented in Section IV.

Measuring Skewness in Return Distribution

Skewness in the distribution of stock returns can be measured based on either firm-specific,

time-series return data, or on cross-sectional return data. This study focuses on cross-sectional

return skewness, as opposed to the time-series return skewness of individual firms (Friend and

Westerfield 1980; Harvey and Siddique 2000). This analysis mitigates the fact that, with a 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 likelihood of successfully identifying the next Google-like

‘‘take-all’’ winner in an emerging industry or market, is slim. Therefore, for most firms with more

positively skewed ex ante return distributions, the skewness will not become observable in the

firm’s ex post time-series return data. In other words, for most firms with a reasonably greater

chance of a high level of success 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. However, this issue 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 positive skewness in the expected return

distribution. Even for firms whose ex post returns do show a positive skewness, such as Microsoft,

measuring 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 time Microsoft had clearly established itself as the

dominant player in its field, it was widely considered to be a mature firm rather than a growth firm.

Because of 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 presented in

Section V.

One difficulty in measuring skewness is that investor preference for skewness is not monotonic

(Barberis and Huang 2008). When selecting stocks, risk-loving investors are attracted only to stocks

with a considerably skewed payoff with significant upside potential. 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 proportion

of ‘‘noise’’ firms. In turn, this poses a challenge to empirically detecting abnormal skewness based

on an observed sample distribution.9

9 Consider sample firms, denoted as Y, such as those with low book-to-market ratios, that include a subset of firmswith extreme positive return skewness, denoted as X, and other firms whose returns are more symmetricallydistributed, denoted as E. That is, the population is made up of two sub-populations with differentcharacteristics: Y ¼ X [ E. Inferring the distribution of X from the observed distribution of Y is difficult. Eventhough X is positively skewed, the skewness of Y is not clear-cut. The overall skewness of the sample populationdepends on the relative locations of the mean returns of the expected returns of X and E, as well as the relativesample size of E and X.



To mitigate this 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 hedged portfolio eliminates the impact of any common noise terms that affect the

returns of both the long and short stocks. In addition, the hedged portfolio return has the property

that if the long and the short stocks have a similar degree of skewness, the distribution of the hedged

return will be symmetrical. If, however, the two groups of firms have differing skewness, the

difference will be clearly shown in the skewness of the hedged returns.

Another method is to try to purge the impact of noise from the measurement of skewness. Let Ydenote the sample firms that include a subset of firms with extreme positive return skewness,

denoted as X, and other firms whose returns are more symmetrically distributed, denoted as E. Let

c* denote a benchmark level of sample skewness if X and E were random samples from the same

pool. Subtracting this benchmark level of skewness (c*) from the observed sample skewness of Y,

denoted as cy, would yield a measure of excess sample skewness (c¼ cy� c*), reflecting the excess

skewness of X. To measure benchmark skewness c*, we must control for correlation between

sample mean and sample skewness (Bryan and Cecchetti 1999). When the distribution of the

underlying population is skewed and/or has a positive kurtosis, the sample skewness generally has a

positive correlation with the sample mean.10 This follows because drawing an extreme positive

return is likely to increase both the sample mean and sample skewness. I use the random sampling

method to assess the degree of correlation between sample skewness and other sample moments.

Based on the distribution of this random sample, excess skewness is then estimated for firms within

each book-to-market portfolio.

III. BOOK-TO-MARKET AND SKEWNESS OF FUTURE STOCK RETURNS

Sample Description

The sample is constructed based on all domestic common stocks traded on NYSE, AMEX, and

NASDAQ.11 Since return and financial data are obtained from the merged CRSP/Compustat

dataset, I restrict the sample period to 1963–2010 to avoid the sample selection problem of

Compustat data availability during the earlier years. Several key statistics for the sample firms are

provided in Table 1. SIZE is the natural logarithm of the market value of equity on the last trading

date of June of each year. BM is the book-to-market ratio, calculated as the book value at the end of

the previous fiscal year divided by the market value at the end of June of the current calendar year.

To ensure the availability of accounting data on the portfolio formation date, I require a minimum

time gap of four months between the fiscal year-end and the June 30th portfolio formation date.

This means that for firms with fiscal years ending in March to June, I will use the book value from

their previous fiscal year. Raw stock return (RET) is the buy-and-hold return over a 12-month

period, starting from the first trading day in July of each year. Market-adjusted return (MAR) is

calculated as raw return minus the corresponding beta-adjusted market return according to CAPM.

The beta of each firm is estimated based on data from a minimum of 12 monthly stock returns over

the preceding five-year period. To avoid any potential problem introduced through the process of

beta estimation, all tests are repeated with MAR calculated by simply subtracting the market index

return from the corresponding raw return. All test results remain qualitatively unchanged. I use a

10 Kurtosis is defined as l4/r4 � 3, which is also known as excess kurtosis due to the subtraction of 3. l4 is thefourth moment about the mean, and r is the standard deviation.

11 Stocks primarily traded on OTCBB and Pink Sheets are excluded from the analysis. These stocks are subject todifferent disclosure requirements and are largely avoided by most investors. When these stocks are included inthe analysis, all test results remain qualitatively unchanged, with relevant test variables retaining the same sign aswell as the same level of significance.

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12-month return window in the main tests to maximize the number of observations with non-

overlapping return windows. Admittedly, this choice is arbitrary given the unknown and possibly

variable lengths of time for firms to realize their upside potential. I present test results with longer

holding periods of up to five years in Section V.

The choice between equal weighting and 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 that favors 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

disproportionately large number of stocks with positive skewness when such investors exhibit

skewness-loving behavior in addition to risk aversion. Since returns of smaller stocks are more

likely to show extreme skewness, an equal-weighting scheme may better capture investor

skewness-seeking behavior. For these reasons, I conduct test results based on both the value-

TABLE 1

Descriptive Statistics

Panel A: Descriptive Statistics

Variable Mean Std. Dev. Q1 Median Q3 Skewness Kurtosis

SIZE 4.61*** 1.93*** 3.19*** 4.48*** 5.92*** 0.08*** �0.21***

BMa,b 0.81*** 0.69*** 0.38*** 0.67*** 1.04*** 0.32*** 6.51***

RET 0.16*** 0.57*** �0.15*** 0.07*** 0.34*** 0.18*** 44.67***

MAR �0.00 0.58*** �0.32*** �0.07*** 0.19*** 0.10*** 39.92***

Panel B: Pearson (Upper Triangle) and Spearman (Lower Triangle) Correlation Coefficients

Variable SIZE BM RET MAR

SIZE �0.31*** �0.04 �0.04*

BMa,b �0.30*** 0.06*** 0.06***

RET 0.04 0.09*** 0.94***

MAR 0.02 0.09*** 0.91***

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively.The sample contains 170,700 observations from all common stocks traded on NYSE, AMEX, and NASDAQ, excludingthe OTC Bulletin Board/Pink Sheet stocks, between June 28, 1963 and June 30, 2010. Stock price and financial data areobtained from the CRSP and Compustat datasets. Observations with missing book-to-market ratio (BM), calculated usingthe market value at the end of June of each year and the book value at the end of the previous fiscal year, are excludedfrom the analysis. Non-missing beta on the last trading date in June, calculated using a minimum of 12 monthly returnsfrom the preceding five years based on CAPM, is also required. Descriptive statistics are calculated for each sample year.This table reports the average over all sample years.a 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.b To reduce the impact of the extreme book-to-market ratios due to the misalignment of the measurement dates of book

value and market value, as well as the effect of accounting conservatism, BM is winsorized at the top and the bottom 1percent.

Variable Definitions:SIZE ¼ logarithm of the market value of equity on the last trading day in June of each year;BM ¼ book-to-market ratio;RET ¼ buy-and-hold annual return from July to June; andMAR ¼ market-adjusted return calculated as raw return adjusted for stock beta and the market risk premium.



weighting and the equal-weighting methods. As discussed in Appendix B, the findings are robust to

the choice of weighting method.

There are several alternative ways to measure skewness. Since investor skewness-loving

behavior arises mainly from the effect of extreme outcomes, I use the following quantile-based

skewness in main tests:

Skewness ¼ P90þ P10� 2�Median

P90� P10;

where P90 and P10 are the 90th and the 10th percentiles of the distribution, respectively. Tests

based on a number of alternative skewness measures yield similar results that are discussed in

Appendix B.

Table 1 shows that the sample is representative of the general population of firms traded on

NYSE, AMEX, and NASDAQ. The median book-to-market ratio equals 0.67, consistent with the

existence of a significant conservative accounting bias. Note that both the raw return and the market

adjusted return are positively skewed, as shown in Panel A. This is consistent with the pattern

documented in the literature regarding the skewness of long-term stock returns. Kurtosis, or excess

kurtosis, of the sample stock returns is significantly greater than zero, which is the level of a normal

distribution. These two characteristics of the return distribution cause both the mean and the

skewness of any return sample to be positively correlated, as discussed and documented in the next

subsection.

Panel B of Table 1 reports the Pearson and the Spearman correlations among SIZE, book-to-

market (BM), and stock returns (RET and MAR). Consistent with the findings of Rosenberg et al.

(1985) and Fama and French (1992), book-to-market has significant positive Pearson and Spearman

correlations with stock return. The correlation between SIZE and stock return, however, is much

weaker. SIZE exhibits marginally significant negative Pearson correlation with future stock return.

However, the Spearman correlation shows the opposite sign.

Return Skewness of Book-to-Market Portfolios

Table 2 reports test results from comparisons of the skewness of stock returns of firms in

different book-to-market portfolios. Each year, firms are sorted into ten equal-sized 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. This sequential sorting controls 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 remaining firms are

placed in the medium-book-to-market group.

The mean and 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 category have an average return

skewness of 0.11. Consistent with the prediction of H1, the return skewness of low book-to-market

firms exceeds the return skewness of high book-to-market firms by 0.04, which is significant at the

p , 0.01 level.

Investors’ projected stock returns are driven, fundamentally, by the performance of the firm.

Next I compare the skewness of several accounting-based performance measures for the different

book-to-market portfolios. More specifically, in Table 2, Panel B I compare the skewness of growth

rates in sales (Compustat data item SALE), income before extraordinary items (Compustat data

item IBOCM), total assets (Compustat data item AT), and operating cash flows (Compustat data

2222 Zhang


item OANCF).12,13 Each of these measures reflects one particular aspect of business operations and,

hence, is not as comprehensive as stock returns in gauging the return on investment to investors.

However, these measures constitute an important supplement to stock return results, as they suffer

less from correlation with mean stock return. Moreover, these non-price measures avoid the

potential problem of market mispricing. Consistent with the findings from the stock return analysis

in Panel A of Table 2, low book-to-market firms also have greater skewness in the distributions of

all the accounting-based performance measures.

Table 2 also confirms that high book-to-market firms on average outperform firms with low

book-to-market ratios. Firms with high book-to-market on average earn a MAR of 2.15 percent. In

contrast, the average return of firms in the low book-to-market category trails the benchmark by

TABLE 2

Stock Return Skewness of Book-to-Market Portfolios

Panel A: Skewness of Portfolio Stock Returns

Book-to-Market

MAR

SkewnessMean Median

Low �5.14%*** �13.66%*** 0.15***

Medium 0.12% �5.43%*** 0.12***

High 2.15%* �3.51%*** 0.11***

High � Low 7.29%*** 10.15%*** �0.04***

Panel B: Skewness of Growth in Fundamentals

Book-to-Market Skew(GREV) Skew(GINC) Skew(GTA) Skew(GCFO)

Low 0.12*** 0.04 0.07*** �0.06**

Medium 0.03*** �0.18*** 0.02*** �0.12***

High �0.03** �0.22*** �0.01*** �0.13***

High � Low �0.15*** �0.26*** �0.08*** �0.07**

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively.Panel A reports the mean, the median, and the skewness of market-adjusted stock returns for portfolios with low,medium, and high book-to-market ratios. Market-adjusted return (MAR) is calculated as buy-and-hold annual returnadjusted for stock beta and the market risk premium. Each year, firms are sorted into decile portfolios based on SIZE,which is the logarithm of the market value of equity on the last trading day in June of each year. Within each size decilefirms are then sorted into ten portfolios based on their book-to-market ratios. Firms in the top three book-to-marketdeciles are placed in the high book-to-market portfolio. Firms in the lowest three deciles are placed in the low book-to-market portfolio. The remaining firms are placed in the medium book-to-market portfolio. The mean portfolio return, aswell as the median and the skewness, is calculated for each portfolio. Panel A reports the time-series average of the mean,the median, and the skewness of the equal-weighted portfolio stock returns. Panel B reports the skewness of one-year-ahead growth in fundamentals for each book-to-market portfolio. Growth in revenue (Compustat data item SALE),income before extraordinary items (Compustat data item IBCOM), total assets (Compustat data item AT), and cash flowfrom operations (Compustat data item OANCF) are denoted as GREV, GINC, GTA, and GCFO, respectively.

12 When data item OANCF is not available, cash flow from operations is calculated as funds from operations(FOPT) adjusted for changes in working capital.

13 To avoid the small-denominator problem, especially when calculating the growth rate of earnings and cash flowswhen the denominator value may be close to 0, the growth rate of any given variable, x, is calculated as gt(x)¼ (xt

� xt�1)/(jxtj/2 þ jxt�1j/2).



5.14 percent.14 A hedged 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 of 7.29 percent.15

Note that the cumulative prospect theory predicts that investors place more weight on the extremes

of the return distributions (Tversky and Kahneman 1992). The low abnormal return of glamour

stocks is consistent with investors’ preference for stocks with high upside potentials. The excess

abnormal return of value stocks, on the other hand, is consistent with investors’ distaste for stocks

with significant downside risk as predicted by loss-aversion.

The difference in average stock return between high and low book-to-market firms, as reported

in Table 2, raises the issue of whether the documented difference in return skewness might be

caused by the difference in mean returns. Bryan and Cecchetti (1999) show that if the underlying

distribution of a random sample is skewed, and/or has a high degree of kurtosis, then the sample

mean and sample skewness will be positively correlated. This implies that when comparing the

skewness of different samples, we must control for this mechanical aspect of the skewness measure,

which varies with the sample mean.

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.16 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. Descriptive statistics of this random portfolio return sample are presented in Panel A of

Table 3. The mean portfolio return, for both the raw return as well as for the market-adjusted return,

is very close to the population mean reported in Table 1. Portfolio return skewness is positive and

significant. Panel B of Table 3 shows the Pearson and Spearman correlations between the mean and

the skewness of return distribution. Both measures show a significant positive correlation between

sample mean and sample skewness. Bryan and Cecchetti (1999) document a similar pattern using

inflation data for the period 1947–1995.

The above sample correlation between mean and skewness provides a useful benchmark against

which to 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,

then 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 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 of

14 To appreciate the impact of accounting conservatism, note that if I replace book value in the calculation of thebook-to-market ratio with the cash balance of the firm, the resulting return difference between the high and thelow group shrinks to 2.4 percent, and the difference in skewness between the two groups becomes insignificantas well. This highlights the importance of having the difference between the book value and the market valuecapture unrealized future growth potential. Note that Simutin (2010) documents significant future stock returndifferences when firms are grouped based on excess cash, measured as the residual cash holding from aregression of cash holdings on firm characteristics such as book-to-market, size, R&D-to-sales, cash-flow-to-assets, etc.

15 Because of the large sample size, modifying the t-tests for population asymmetry based on Johnson (1978) andChen (1995) has negligible impact on the significance of the test statistics.

16 Following Fama and French (1992), firms in the top three size-deciles are classified as large firms. Firms in thebottom three size-deciles are classified as small firms. The remaining firms are combined as the medium-sizedgroup. Random sampling is conducted within each size group, rather than size decile, to ensure sufficient firmsare available from which 50 stocks can be randomly selected.

2224 Zhang


firms with mean returns closest to the sample mean. The average skewness of these random

portfolios is then designated the benchmark level of skewness.17 The untabulated test results show

that the average difference in mean return between the sample portfolio and the benchmark

portfolios equals 0.03 percent, indicating that the matching is very close. This benchmark level of

skewness is subtracted from the skewness of the book-to-market portfolio to get the adjusted, 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.

TABLE 3

Adjusted Stock Return Skewness of Book-to-Market Portfolios

Panel A: Portfolio Return Statistics, Average of 144,000 Random Portfolios

Variable Mean Std. Dev. Skewness Kurtosis

RET 0.15*** 0.52*** 0.18*** 4.44***

MAR �0.00 0.53*** 0.11*** 4.07***

Panel B: Correlation between the Mean and the Skewness of Random Portfolio Market-Adjusted Returns

Panel Data By Year

Pearson 0.29*** 0.35***

Spearman 0.25*** 0.34***

Panel C: Mean, Skewness, and Adjusted Skewness for High, Medium, and Low Book-to-Market Firms

Book-to-market

MAR

Skewness Adj. SkewnessMean Median

Low �4.95%*** �13.45%*** 0.15*** 0.05***

Medium 0.24% �5.51%*** 0.12*** �0.01

High 2.23%* �3.55%*** 0.11*** �0.03***

High � Low 7.18%*** 9.90%*** �0.04*** �0.08***

Hedged Return 7.18%*** 10.05%*** �0.06*** �0.09***

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively.Panels A and B report the correlation between the mean and skewness of stock returns of randomly formed 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 skewness of these stock returns are then calculated. Market-adjustedreturn (MAR) is calculated as buy-and-hold annual return (RET) adjusted for stock beta and the market risk premium.Panel A presents several key statistics of these portfolio returns. Panel B shows the Spearman and Pearson correlationcoefficients between sample mean and sample skewness. These random portfolio returns are then used to adjust thereturn skewness of portfolios formed randomly based on the book-to-market ratios (BMs). The results are reported inpanel C.

17 I also form benchmarks by pooling together all random portfolios whose return differs by less than 0.1 percentfrom the sample portfolio’s mean return. The result is almost identical. The matching method used in the textgenerates far less difference in mean return between the sample and benchmark portfolios.



Once I subtract the benchmark level of skewness, the adjusted skewness levels show an even

clearer monotonic pattern across different book-to-market groups. Low book-to-market firms have a

significant positive excess skewness of 0.05, while high book-to-market firms show a significant

negative excess skewness of�0.03. The difference between the two is significant at the p , 0.01 level.

Clearly, not adjusting for the mechanical correlation between sample mean and sample skewness

understates the difference in expected skewness across the various book-to-market firm groups.

I also randomly select 50 long/short pairs of stocks, from the high and low book-to-market

groups, respectively, and calculate the skewness and the adjusted skewness of the hedged returns.

Common factors affecting both the long and short groups of stocks are purges from the hedge

return; thus, it is likely to more correctly reflect the difference in expected skewness between the

high and low book-to-market groups. If the long and short groups have the same degree of

skewness, then the hedged return will be symmetrically distributed. However, as shown in Panel C

of Table 3, the mean skewness of the hedged return equals�0.06, which is highly significant (at the

p , 0.01 level). The adjusted skewness equals�0.09, also significant at the p , 0.01 level. This

provides further support for the conjecture that high book-to-market firms have a more negatively

skewed return distribution.

As an additional analysis, I calculate the skewness and adjusted skewness of the hedged portfolio

returns based on 100 random portfolio formations to assess the skewness of the hedged portfolio returns

with 50 stocks in the long/short portfolios. More specifically, I calculate the difference in average return

of long and short portfolios and the skewness of this hedged portfolio’s returns based on 100 random

portfolio formations within each size group in each year. The untabulated result shows that the mean

skewness and adjusted skewness over the 48 sample years equals�0.03 and�0.06, both significant at

the p , 0.01 level. The overall skewness and adjusted skewness calculated based on the entire panel

data are even greater, at�0.05 and�0.08, respectively. 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 degree of downside risk.

IV. SKEWNESS AND STOCK RETURNS

Results reported in Section III indicate that firms with different book-to-market ratios have

differing skewness in stock returns. Such differences in skewness can cause equilibrium stock

returns to differ across these firm groups as a result of investor preference for positively skewed

returns. In this section, I examine the extent to which this difference in return skewness, coupled

with investor preference for skewness, contributes to the observed mean return difference between

value and glamour stocks.

To analyze the impact of skewness on stock returns, I use two empirical proxies for expected

stock return skewness, each with its own advantages and disadvantages. The first proxy is the expost excess portfolio return skewness as calculated in Section III. The underlying assumption is that

investor skewness expectation is unbiased, such that the average 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 are not available to investors ex ante. The second proxy, which is

based on ex ante portfolio return skewness calculated based on stock returns during the previous

fiscal year, avoids this problem, but is relatively noisy. Given these advantages and limitations, I

present test results for both proxies.

Correlation between Ex Post Skewness and Stock Returns

I regress the average portfolio stock return on its adjusted skewness and check whether the

magnitude of return residuals varies across the three book-to-market groups. The results are

reported in Table 4.

2226 Zhang


TABLE 4

Characteristic Return Regression

Panel A: With Adjusted Ex Post Return Skewness

Dependent Variable: MAR

Model I Model II Model III Model IV

Intercept 0.01 0.04** 0.02 0.02

BM 0.02*** 0.01 0.01

SIZE �0.01* �0.01*** �0.01** �0.01**

Adj. ex post SKEW �0.09*** �0.08*** �0.07***

Adj. ex post STD �0.11**

Adj. ex post KURT �0.00

R2 (%) 1.93 2.11 2.59 4.50

Panel B: With Adjusted Ex Ante Return Skewness



Intercept 0.01 0.02 �0.01 �0.01

BM 0.02*** 0.02* 0.02*

SIZE �0.01* �0.01** �0.01* �0.01*

Adj. ex ante SKEW �0.09*** �0.09*** �0.09***

Adj. ex ante STD 0.01

Adj. ex ante KURT �0.00

R2 (%) 1.93 2.09 2.85 2.91

Panel C: With Ex Ante Return Skewness



Intercept 0.01 0.05*** 0.01 0.00

BM 0.02*** 0.02** 0.02**

SIZE �0.01* �0.01** �0.01* �0.01*

ex ante SKEW �0.08*** �0.06*** �0.07***

ex ante STD 0.01

ex ante KURT �0.00

R2 (%) 1.93 1.99 2.63 2.68

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively, adjustedfor clustering within each year-SIZE-BM group.

This table reports the cross-sectional regression results based on 43,200 portfolio returns. BM and SIZE are the meanbook-to-market and size of each portfolio, respectively. Market-adjusted return (MAR) is calculated as buy-and-holdannual return adjusted for stock beta and the market risk premium. Adj. ex post SKEW is the adjusted portfolio returnskewness calculated as the raw return skewness minus the average return skewness of mean-return-and-size-matched,random portfolios. Adj. ex ante SKEW is the adjusted return skewness calculated based on returns in the preceding fiscalyear. Similarly, Adj. ex ante (ex post) STD and KURT are the adjusted portfolio return standard deviation and kurtosisduring the current and the preceding fiscal years



The estimated regression coefficient on the adjusted skewness is negative and highly

significant. Panel A of Table 4 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, which is consistent with the prediction of H2, indicates that the difference in skewness

among book-to-market groups is an important driver underlying the return differences among the

various book-to-market firm groups.

To check whether the return difference between the high and low book-to-market groups is

caused by other aspects of the return distributions, such as variance, rather than by skewness, 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 deviations

for the low, medium, and high book-to-market groups are 0.61, 0.49, and 0.47, respectively. That

is, the group with the highest variance (i.e., the low book-to-market firms) actually has the lowest

average return. Ang et al. (2006) document a similar pattern. This pattern, on the other hand, is

consistent with the argument that high variance in payoff increases skewness as well as stock price.

Nonetheless, it is still of concern that the documented correlation between adjusted skewness

and mean portfolio return, as reported in Models I to III of Panel A, Table 4, could result from

correlation between return and other moments of the distribution. To analyze this conjecture, I

include adjusted return standard deviation, and adjusted return kurtosis in the regression and report

the results in Model IV of Panel A, Table 4. Consistent with the findings of Ang et al. (2006), there

is a significant negative correlation between adjusted standard deviation and mean portfolio return.

Kurtosis, however, exhibits no significant correlation with mean return. The correlation between

skewness and mean return remains negative and highly significant.

Correlation between Ex Ante Skewness and Stock Returns

The ex post skewness measure in Panel A of Table 4 may incorporate certain spurious

correlations between ex post return moments. To reduce this risk, I replace ex post skewness with exante portfolio return skewness calculated using stock returns from the preceding year in Panel B of

Table 4. The ex ante skewness measure demonstrates a strong negative correlation with stock

returns, similar to the regression results using the ex post skewness measure. This lends further

support to H2, which predicts that skewness contributes significantly to the observed difference in

mean returns among book-to-market groups of firms.

Inclusion of the ex ante skewness measure substantially reduces, but does not completely

remove, the statistical significance of the correlation between book-to-market and stock return.

Contrary to the results using ex post standard deviation and kurtosis, ex ante standard deviation

shows a positive, but insignificant, correlation with stock returns. A positive correlation is

consistent with the popular belief that variance should increase, not decrease, expected returns.18

To avoid potential noise introduced during the process of estimating the adjusted standard

deviation, skewness, and kurtosis, I repeat the regressions with the raw ex ante return moments. The

results are reported in Panel C of Table 4. Consistent with the hypothesis that skewness contributes

significantly to cross-sectional variance in stock returns, ex ante skewness exhibits a significant,

negative correlation with subsequent stock returns.

18 To further gauge how much of the book-to-market returns are explained by skewness, I regress the portfolioreturns on the ex ante adjusted skewness. Untabulated result shows that the residual return difference between thehigh and the low book-to-market groups, after controlling for skewness, equals only 2.08 percent (significant atthe p , 0.10 level).

2228 Zhang


Ex Ante Proxy for Skewness Using Other Accounting-Based Measures

If investor preference for positive skewness is a major driver of the book-to-market effect, then

we should observe a more significant difference in stock returns as the variation in the fundamental

skewness between portfolios increases. Published research shows that past time-series return

coskewness has little predictive power over future return skewness (Singleton and Wingender 1986;

Harvey and Siddique 1999). In this section, I present results using ex ante accounting-based

measures and the book-to-market ratio to forecast return skewness.

The analysis in Section II shows that the skewness in a firm’s payoff is closely related to the

growth option associated with firm assets. Ohlson (1995) obtains the following relationship

between a firm’s market value and its book value:

MVt ¼ BVt þ q�1E½xatþ1� þ q�2E½xa

tþ2� þ � � � ;

where q is the discount rate and xa denotes abnormal earnings, calculated as earnings minus the

beginning book value multiplied by the required rate of return. 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

Company, which had a book-to-market ratio of 0.11 in October 2012. Even though the firm is relatively

mature, with limited growth potential, conservatism in accounting reduces its book-to-market ratio.

The preceding observation suggests that separating firms based on their investment growth may

identify firms with more strongly skewed payoff distributions. Zhang (1998) and Rajan et al. (2007)

show that, with conservative accounting, the book rate of return on operating assets (ROA) becomes a

comprehensive and effective measure, which can be used to infer growth in a firm’s investment. This

measure has the advantage of measuring investment growth in both tangible assets and intangible

assets, such as goodwill, brand name, and technology. High growth in investments, especially in

intangible assets, depresses a firm’s ROA.19 Thus, grouping firms based on ROA, in addition to book-

to-market, can improve the separation of firms with different degrees of skewness in future payoffs.

Next, I further partition firms in both book-to-market groups based on ROA. Those firms 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 less likely to have excess positive skewness. I

group those two subsets of firms into one test group, and create the following measure:

BMa ¼LOW if BM is low and ROA is low

MEDIUMHIGH if BM is high and ROA is high

:

8<:

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 may

provide important insight regarding the effects of skewness and book-to-market. If the book-to-

market ratio is the driving factor, then we should still observe a significant difference in average

stock returns between these two groups of firms, despite the fact that they may not show a

significant difference in their return skewness. On the other hand, if skewness were a more

fundamental driving factor, then we would observe less difference in stock returns when comparing

these firms on the following measure:

19 Xing (2008) uses the growth rate of capital expenditure (CAPEX) to gauge investment growth, and obtains asimilar correlation between growth and return. Compared with CAPEX, using ROA has the advantage ofincorporating the effect of investment in intangible assets. The disadvantage, however, is that the low-ROAgroup also include firms with poor performance. This effect is mitigated by incorporating the book-to-marketratio in the analysis.



BMb ¼LOW if BM is low and ROA is high

MEDIUMHIGH if BM is high and ROA is low

:

8<:

The test results are reported in Table 5. As Panel A shows, the BMa measure significantly increases

the difference in skewness between the high and low groups. The raw and the adjusted skewness of

the low-BMa group exceed those of the high-BMa group by significant 0.10 and 0.15 margins,

respectively. The average portfolio return of the high-BMa group exceeds that of the low-BMa

group by a significant 10.06 percent. Untabulated test results also show that the high- and the low-

BMa groups differ significantly in the skewness of the growth rates in fundamentals such as the

revenues, earnings, and cash flows.

Table 5, Panel B shows that the partition based on BMb provides a quite different picture. The

difference in the average book-to-market ratios between the high and low book-to-market groups is

1.35, which is actually slightly greater than that of the value of 1.21 for the BMa partition.

Therefore, if book-to-market is a fundamental driver of return difference, then we would expect the

BMb partition to yield a more significant difference in the average stock return between groups.

However, the average return difference is only 2.98 percent, which is not statistically significant.

The difference in the skewness of returns between these two groups of firms is not statistically

significant, nor are the differences in revenue growth skewness and other fundamentals. These

results confirm my conjecture that although some firms may have significantly lower book-to-

market ratios than others, if their book rates of return indicate rather limited growth potential, then

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.

Panel C of Table 5 reports the test results from cross-sectional characteristic regression with

both skewness and ROA as independent variables. Model II shows that ROA has significant

predictive power for future stock returns. However, when skewness is added to the regression in

Model III, the coefficient on ROA is no longer significant. The estimated coefficient on skewness is

negative and highly significant.

Adding an additional screen based on ROA helps draw a more focused picture of a firm’s true

upside potential relative to its downside risk. For instance, a firm’s book-to-market ratio can be low

under two scenarios: (1) when the firm is fast-growing and the market price is high, and (2) when

the firm is significantly reducing its book value. The first scenario is more likely to have significant

upside potential as reflected in return skewness. Combining ROA with book-to-market separates the

two cases, thereby providing a better prediction of future return skewness.

This study does not claim that investors necessarily use book-to-market or ROA to forecast

return skewness. 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.20 The untabulated test results show that stock returns of low book-to-market firms

significantly exceed those of high book-to-market firms in the portfolio formation year, consistent

with investors’ preference for such stocks.21

20 For example, according to the Associated Press, shares of Apple Inc. soared after an analyst upgraded the stockbased on her belief that the iPhone’s growth potential has been underestimated. ‘‘We believe Apple is emergingas the clear leader in the battle over the mobile Internet,’’ said Morgan Stanley analyst Kathryn Huberty, in aresearch note. She sees mobile Internet, where people use their cell phones to go online, as the next biggestmarket opportunity in the technology sector. With a market size of four billion cell phone users that companiescould connect to the Internet, ‘‘it is 40 times the opportunity to get 100 million PC users online in the 1990s,’’Huberty said.

21 Analyst forecasts of mean EPS growth might also reflect the skewness in growth. The untabulated test resultsshow that firms with high analyst-projected growth indeed have more positively skewed return distributions.

2230 Zhang


V. SUPPLEMENTARY TEST RESULTS

Different Return Horizons

All test results reported thus far are based on one-year-ahead returns. However, it may take

longer for a firm to fully realize its upside potential. I repeat the tests with annualized returns for

TABLE 5

Skewness Measure Based on Book-to-Market and Return on Assets

Panel A: Portfolios Sorted by BMa

MAR

Skewness Adj. Skewness BMtMean Median

Low �9.98%*** 21.49%** 0.22*** 0.14*** 0.22***

Medium 0.85% �6.26%** 0.15*** 0.02 0.87***

High 0.08% �4.72%*** 0.12*** �0.01 1.43***

High � Low 10.06%*** 16.77%*** �0.10*** �0.15*** 1.21***

Panel B: Portfolios Sorted by BMb

MAR

Skewness Adj. Skewness BMtMean Median

Low �1.47% �8.45%*** 0.14*** 0.02 0.27***

Medium �0.80% �7.35%*** 0.13*** 0.01 0.79***

High 1.51% �7.12%*** 0.14*** 0.02 1.62***

High � Low 2.98% 1.33% 0.00 0.00 1.35***

Panel C: Characteristic Return Regression Results


Model I Model II Model III

Intercept 0.01 �0.03 0.03

BM 0.02*** 0.01* 0.00

SIZE �0.01* �0.00 �0.01*

ROA 0.13** 0.05

Adj. ex post SKEW �0.09***

R2 (%) 1.93 2.99 3.26

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively.This table reports the mean, the adjusted skewness, and several other key statistics of portfolio returns when portfoliosare formed based on book-to-market (BM) and return on assets (ROA). SIZE is the logarithm of the market value ofequity on the last trading day in June of each year. Market-adjusted return (MAR) is calculated as buy-and-hold annualreturn adjusted for stock beta and the market risk premium. In Panel A, The low BMa group includes firms from thebottom three BM deciles and the bottom three ROA deciles. The high BMa group includes firms from the top three BMdeciles and the top three ROA deciles. The remaining firms are in the medium group. In Panel B, the low (high) BMb

groups consist of firms in the bottom (top) three BM deciles and the top (bottom) three ROA deciles. Panel C reports thecross-sectional regression results based on portfolio returns. BM, SIZE, and ROA are the average book-to-market ratio,SIZE, and ROA of each portfolio, respectively. Adj. ex post SKEW is the adjusted portfolio return skewness calculated asthe raw return skewness minus the average return skewness of mean-return-and-size matched, random portfolios.



horizons of three years and five years and report the results in Panel A of Table 6.22 The differences

in the mean and the skewness between the high and the low book-to-market groups show similar

patterns across the three return horizons. Note that tests for three-year and five-year return horizon

are based on smaller firm-year samples with non-overlapping return windows to ensure efficiency

of the significance tests. To reduce the potential impact of any survivorship bias on long-window

return tests, firms with missing return data due to bankruptcy are assumed to earn zero abnormal

return above their respective benchmarks.

TABLE 6

Return Skewness for Different Holding Horizons

Panel A: Return Skewness for Holding Periods of One, Three, and Five Years

Book-to-Market

Mean

Skewness Adj. SkewnessMean Median

One Year

Low �4.95%*** �13.45%*** 0.15*** 0.05***

Medium 0.24% �5.51%*** 0.12*** 0.00

High 2.23%* �3.55%*** 0.11*** �0.03***

High � Low 7.18%*** 9.90%*** �0.04*** �0.08***

Three Years

Low �12.40%*** �12.60%*** �0.01 0.08***

Medium �4.81%*** �4.31%*** �0.04** 0.01

High �3.30%* �2.46%*** �0.06*** �0.03***

High � Low 9.10%*** 10.14%*** �0.05*** �0.11***

Five Years

Low �12.12%*** �11.49%*** �0.04** 0.09***

Medium �4.63%*** �3.32%** �0.11*** �0.01

High �2.86%* �1.75% �0.09*** �0.02*

High � Low 9.26%*** 9.74%*** �0.05*** �0.11***

Panel B: Persistence of Annual Return Skewness Difference for Years þ1, þ3, þ5, and þ10

High � Low Year þ1 Year þ3 Year þ5 Year þ10

BM 1.04*** 0.77*** 0.57*** 0.39***

Mean MAR 7.18%*** 7.44%*** 5.92%** 2.36%

Adj. Skewness �0.08*** �0.07*** �0.05** �0.03

*, **, *** Indicate two-tailed statistical significance at 10 percent, 5 percent, and 1 percent levels, respectively.This table reports the annualized mean, skewness, and adjusted skewness of stock returns for the three book-to-market(BM) portfolios. Each year, within each SIZE group, 100 portfolios are constructed from each book-to-market firm groupby randomly selecting 50 firms within the BM group for each portfolio. The annualized mean, skewness, and adjustedskewness of the stock returns are then calculated for each portfolio. This process is repeated for each year from 1963 to2010 with non-overlapping return windows. Panel A shows the returns with holding periods ranging from one year tofive years. Panel B shows the average return for sample years tþ1, tþ3, tþ5, and tþ10, respectively.

22 Returns are annualized to increase comparability across different return horizons. Without annualizing thecumulative returns, for all three book-to-market portfolios, both the mean and the skewness of portfolio returnsare positive for the three-year and the five-year return horizons.

2232 Zhang


In addition, I calculate the annual hedged return, as well as adjusted skewness, for years tþ1

through tþ10.23 The results are presented in Panel B of Table 6. The difference in the average book-

to-market ratio between the high and low groups of firms gradually shrinks as the horizon expands.

The hedged return is also reduced to 5.92 percent in year 5, and to 2.36 percent in year 10. The

difference in adjusted skewness decreases gradually with the hedged return, from�0.08 in year 1,

to �0.07 in year 3, �0.05 in year 5, and �0.03 in year 10.

Coskewness

As discussed in Section II, most of the literature on skewness and returns maintains the

assumption of uniform risk aversion. The implication is that the coskewness of stocks matters in

cross-sectional stock return tests. Following Harvey and Siddique (2000), I calculate each

individual stock’s return coskewness based on monthly stock returns over the prior five years.

Untabulated test results show that firms with low past coskewness have a lower mean return

compared to firms with high past coskewness; the difference, however, is insignificant. Consistent

with the findings of Harvey and Siddique (1999), a stock’s past coskewness shows little predictive

power with respect to future coskewness. The difference in future coskewness between the high-

and the low-coskewness groups, estimated based on realized monthly returns in years tþ1 to tþ5, is

insignificant. In addition, I sort stocks based on ex post coskewness over (tþ1, tþ5) to examine the

extent to which coskewness might correlate with average stock returns. The average return for the

low ex post coskewness group is 2.11 percent, whereas the average return for the high-coskewness

group is 1.52 percent. The difference (0.59 percent) is not statistically significant.

Subsample Analysis Based on Firm Age

Accounting tends to be more conservative for firms in the early growth phase of their life cycle

due to aggressive expensing of investments. Such cross-sectional variation in the degree of

conservatism can affect the correlation between the book-to-market ratios and stock return

skewness.24 To test this conjecture, I divide the sample into two subsamples based on firm age,

measured as the number of years the firm has been on Compustat. This measure, of course, has the

disadvantage that not all firms go public or have Compustat coverage initiated at the same age.

Untabulated results show that book-to-market indeed has more significant correlation with both

the future mean return as well as return skewness for younger firms. More specifically, among

younger firms, the difference in the mean and the skewness of stock returns between the high and

the low book-to-market groups is 8.86 percent and �0.09, respectively. In contrast, among older

firms, the difference in the mean and the skewness of stock returns is 5.57 percent and �0.04,

respectively. The differences between these two groups are significant at p , 0.01.25,26

23 In contrast to Panel A, the test in Panel B is based on tracking the same set of firms over tþ1 to tþ10.24 I thank one of the referees for recommending this test.25 Note that there is no statistically significant difference between the younger-firm and the older-firm subsamples

in terms of the difference in the average book-to-market ratio across the high and the low book-to-marketportfolios. This is because the sorting based on book-to-market is done independently of the sorting based onfirm age.

26 It can also be argued that the accounting for financial services companies is less conservative, compared to theaccounting for most manufacturing firms, due to the application of fair value accounting to a significant portionof banks’ assets. Consistent with this view, when the sample is divided into financial and nonfinancial firms, thedifferences in the mean and the skewness between the high and the low book-to-market groups are significantlygreater for nonfinancial firms.



Other Supplementary Test Results

To assess robustness of the test results, I conduct an extensive set of sensitivity checks. These

tests examine the issues of bankruptcy risk, investor mispricing, alternative measurements of

skewness, as well as other variations in the test design. Details of these tests and their results are

presented in Appendix B. In summary, the main results are robust to these alternative

considerations.

VI. CONCLUSION

This paper documents that glamour stocks have significant excess positive skewness in their

return distributions compared with value stocks. This implies that in evaluating investment portfolio

performance any systematic pattern in skewness needs to be closely monitored. Investment

strategies that aim to exploit market mispricing may inadvertently expose investors to significant

skewness risk. Results also show that a simple measure based on the book-to-market ratio and the

book rate of return has significant predictive power with respect to future return skewness. This

provides a new perspective on accounting conservatism, a basic accounting principle that has long

been criticized for reducing information quality (Hendriksen and Van Breda 1992). By tying

accounting treatment to the underlying level of uncertainty, a conservative accounting system can

provide researchers with useful information to assess a firm’s relative downside.

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APPENDIX A

Correlation between Book-to-Market Ratio and Skewness

Suppose that a firm possesses an asset with liquidation value b . 0. The firm engages in a

value-enhancing activity that may fail. If the operation fails, then the manager utilizes the

abandonment option and sells the asset for b. If the operation is successful, then the firm expands

and generates a payoff exceeding b. The final cash flow generated by firm assets equals c ¼Max(b,x), where x is the cash flows from operations.

Consider two scenarios. Under the first, denoted as i:

xi ¼

yþ k with probability1

4

y with probability1

2

y� k with probability1

4

;

8>>>>>>>><>>>>>>>>:

where y . b, k . 0, and y � k , b. In this case, the mean payoff equals:

E½ci� ¼3yþ k þ b

4:

Under the second scenario, j:

xj ¼

gðyþ kÞ with probability1

4

y with probability1

2

y� k with probability1

4

;

8>>>>>>>><>>>>>>>>:

where g . 1. It is evident that E[cj] . E[ci] and SKEW[xj] . SKEW[xi]. The option to expand on

the upside, as well as the option to contract on the downside, leads to a skewed distribution of

payoff.

Because of accounting conservatism, the book value of firm 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 left-skewed.27 Therefore, BMj , BMi. Note

that regardless of the discount rate, the skewness of payoff from j exceeds that of i, that is,

SKEW[Rj] . SKEW[Ri]. Therefore, with conservative accounting, the book-to-market ratio and the

skewness of stock return distribution show a negative correlation.28 Several related predictions are

set forth 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 and Duffee (2002) is

that I 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

27 Note that the premium investors apply to j further decreases its equilibrium book-to-market ratio.28 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.



‘‘assets in place’’ and the market premium as a derivative of book value. As a result, Duffee (2002)

offers no directional prediction as to how book-to-market correlates with skewness.29

APPENDIX B

Sensitivity Analysis

To assess robustness of the main results, I perform an extensive set of sensitivity checks. The

results are summarized as follows.

Bankruptcy Score

One possible explanation for the correlation between book-to-market ratio and stock returns is

that the book-to-market ratio captures the risk of financial distress. Published studies on bankruptcy

risk provide mixed support for this claim. Griffin and Lemmon (2002) document that default risk is

not confined to high book-to-market firms; that is, more firms in the top O-score quintile, which is

based on the bankruptcy risk score developed by Ohlson (1980), have low book-to-market rather

than high book-to-market ratios. Further, Dichev (1998) reports low correlation between book-to-

market and the O-scores. To test whether the skewness measure provides any additional

explanatory power, I include the O-score in the characteristic regression of Section IV. Untabulated

test results show that the estimated coefficient on the O-score is not significant, consistent with the

findings of Dichev (1998). With the O-score included in the regression, the adjusted skewness

measure remains negative and highly significant.30

Return on Assets and Investment Growth

An alternative explanation of the findings of Section IV is that ROA reflects firms’ cost of

capital. As a result, firms with low cost of capital have low subsequent stock returns (Berk et al.

1999; Xing 2008). The skewness hypothesis of this paper corroborates 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 returns (Tobin 1969). However, these two views also

differ fundamentally; under the cost of capital hypothesis, low ROA induces high subsequent asset

growth. In contrast, the correlation is the other way around in the skewness hypothesis; that is, high

growth potential leads investors to lower their required rate of return. The cost of capital 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 an explanation that accounts for the negative correlation between

book-to-market and realized return skewness documented in this paper.

I compare 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 by the growth rate of capital

expenditure (CAPEX) and growth rate of total assets. In year tþ1, CAPEX grew at an average of�3

percent for the low-ROA group. In contrast, the average CAPEX growth rate equals 19 percent for

high-ROA firms. Significant differences are observed in years�1, 0, andþ2. In addition, the time-

series growth pattern of CAPEX lends little support for the cost-of-capital effect. For low-ROAfirms, the rate of CAPEX growth is actually at its lowest level in years 0 and 1, when ROA is

29 Empirical tests in Duffee (2002), which are based on the time-series property of individual stock returns, do notdetect any negative correlation between expected skewness and expected return.

30 I also repeat the test with updated O-score based on coefficients estimated using data from the 1980s (Begley etal. 1996) as well as the distance-to-default measure (Hillegeist et al. 2002). Result remains qualitativelyunchanged.

2238 Zhang


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 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.

I also compare the skewness of the growth rates of CAPEX, total assets, and revenue, across

firms with differing ROA. The skewness hypothesis conjectures that the low-ROA group is more

likely to contain firms with more positive skewness in payoff. Such firms 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. The high growth in a

subset of successful firms should increase the skewness of the growth rate for this group of firms;

indeed, this is what I find. Untabulated test results show that the skewness in the growth rates of

CAPEX, total assets, and revenues are significantly greater for the low-ROA group. These findings

provide strong support for the skewness hypothesis.

Investor Mispricing

The literature provides evidence suggesting that investor mispricing also plays a role in causing

the book-to-market effect (Lakonishok et al. 1994; La Porta 1996; Piotroski 2000). Such mispricing

may result from simple functional fixation on earnings, or from more fundamental cognitive biases,

such as overconfidence. To test whether the skewness effect documented in this paper provides any

incremental explanatory power for the book-to-market effect, I include analyst forecast error in the

cross-sectional regression test of Table 4. If investors are somehow biased in their projection of

future firm performance, then the ex post analyst forecast error based on realized earnings should

provide a good proxy for such bias. Untabulated results show that the estimated coefficient on

analyst forecast error is indeed positive. The coefficient on skewness remains negative and highly

significant, indicating that skewness remains a significant contributor to the book-to-market effect

after controlling for potential mispricing.

Variation over Time

Although the lengthy sample period of this study adds weight to the significance of the

empirical results, it can also obscure any variation of the phenomenon over time. To analyze this

issue, I divide the sample into four decades: the 70s, the 80s, the 90s, and the 00s.31 Untabulated

results show that hedged book-to-market portfolio returns are substantially less in the 90s,

compared to the returns in the other three subsample periods. From 1990 to 1999, the average

annual book-to-market factor return equals only 1.39 percent, which is not statistically different

from 0. Further analysis reveals that the difference in return skewness between the high and the low

book-to-market portfolios is also significantly less compared to the level in the other three decades.

However, when I use both the book-to-market ratio and the book rate of return in portfolio

construction, as outlined in Section IV, the difference in the portfolio return skewness increases to

0.10, significant at the p , 0.01 level. The hedged portfolio return also increases to 6.18 percent,

significant at the p , 0.01 level. Collectively, this evidence suggests that the lower return to the

book-to-market portfolio in the 90s is mostly caused by a reduction in the correlation between the

book-to-market ratio and the expected return skewness. See Section IV for details regarding the

limitations of using the book-to-market ratio to identify firms with skewed stock returns.

31 Observations in the 60s were dropped in this analysis to ensure that each subsample period covers exactly tenyears.



In addition, I also construct a monthly factor return series based on the portfolio formation

outlined in Section IV, and compare this skewness factor return with the three factor returns of

Fama and French (1993). The skewness factor returns show a marginally significant positive

correlation with the Fama-French size factor (SMB), and significant positive correlations with the

book-to-market factor (HML), and the momentum factor (UMD). The Pearson (Spearman)

correlation coefficients are 0.08 (�0.06), 0.26 (0.29), and 0.34 (0.16), respectively. When

regressing the skewness factor returns on the HML factor returns, the intercept (i.e., alpha) equals

0.62 percent, which is significant at the p , 0.01 level, indicating that the skewness factor captures

risk beyond HML. In contrast, when regressing HML returns on the skewness factor, the intercept

(0.24 percent) is not statistically different from 0. This result also suggests that the book-to-market

phenomenon can be substantially explained by investors’ preference for skewness.

Alternative Measures of Skewness

Aside from the skewness measure used in the main tests, which is based on 10th and 90th

percentile distributions, I conduct tests based on the following alternative skewness measures: (1)

standardized moment skewness, defined as the third moment about the mean divided by the

standard deviation; (2) Pearson skewness, defined as (mean-median)/standard deviation; (3) Bowley

skewness, defined as (Q3þQ1� 2 � Median)/(Q3� Q1) where Q3 and Q1 stands for the 3rd and

the 1st quintile; (4) percentile skewness, defined as (P99þP1 – 2 �Median)/(P99� P1) where P99

and P1 are the 99th and the 1st percentile. All test results remain qualitative unchanged with these

alternative skewness measures.

Other Sensitivity Test Results

I perform several other sensitivity tests, including (1) using the market price at fiscal year-end,

instead of the market price in June of 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; (4) restricting sample year to 1973 and beyond so that firms from all three

exchanges (i.e., NYSE, AMEX, and NASDAQ) are represented in all sample years; (5) calculating

the MAR as raw return minus the corresponding return of the market index; and (6) calculating

value-weighted, rather than equal-weighted, portfolio returns. All test results are robust to these

variations in test design.

2240 Zhang


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Book-To-Market Ratio and Skewness of Stock Returns

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