Idiosyncratic Risk, Investor Base, and Returns

Transcription

Idiosyncratic Risk, Investor Base,
and Returns
Doina C. Chichernea, Michael F. Ferguson, and Haimanot Kassa∗
Using four different proxies for a firm’s investor base we demonstrate that idiosyncratic risk
premiums are larger for neglected stocks and smaller or economically insignificant for visible
stocks. Since neglected stocks have greater idiosyncratic volatility (IV), the total IV risk premium
(price × quantity) for neglected stocks will be greater than that of visible stocks. Additionally,
we find a positive size effect and negative beta effect after controlling for IV. Overall, our results
provide strong support for Merton’s theory that market segmentation induced by incomplete
information is an important component of the influence of IV in the cross-section of returns.
The role of idiosyncratic volatility (IV) in the cross-section of returns has generated a rapidly
growing empirical literature. Standard asset pricing models, such as the Capital Asset Pricing
Model (CAPM), predict that perfectly diversified investors are able to eliminate IV. As such,
there will be no idiosyncratic risk premium in equilibrium. However, recent empirical evidence
contradicts this prediction.1 A variant of the standard asset pricing model developed by Merton
(1987) demonstrates that IV can be priced in equilibrium if some investors are under-diversified
and do not hold the market portfolio.2
Merton’s (1987) basic intuition is that information about securities is costly to acquire. Therefore, it is neither optimal nor even plausible for investors to track every security in the market.
These investors only follow a subset of the securities available in the market and construct
their investment portfolios from these known securities. Since investors hold under-diversified
portfolios, they demand compensation for securities’ IV. The market clears, but the presence of
incomplete information generates an equilibrium in which risky assets earn an additional premium (relative to the complete information case), reflecting the interaction of three separate stock
characteristics: 1) IV, 2) relative market size, and 3) breadth of the shareholder base (i.e., what
proportion of investors follow a particular stock).
Merton’s (1987) model is widely cited in two research streams that have developed independently. The literature on “neglected stocks” focuses on the impact of investor recognition on firm
The authors thank Marc Lipson (Editor), Steve Slezak, seminar participants at the 2011 Midwest Finance Association,
and an anonymous referee for helpful comments.
∗
Doina C. Chichernea is an Assistant Professor in the Neff Department of Finance in the College of Business and
Innovation at the University of Toledo in Toledo, OH. Michael F. Ferguson is an Associate Professor in the Department of
Finance in the Carl H. Lindner College of Business at the University of Cincinnati in Cincinnati, OH. Haimanot Kassa
is an Assistant Professor in the Department of Finance in the Farmer School of Business at Miami University in Oxford,
OH.
1
Fu (2009), Spiegel and Wang (2005), and Malkiel and Xu (2006) find that idiosyncratic volatility is positively correlated
with expected stock returns at the firm level. This contrasts with the findings of Ang et al. (2006, 2009) who note a
negative relationship between IV and expected returns, which they call a “puzzle.” See Section I for a detailed discussion
of this literature.
2
Alternatively, Barberis and Huang (2001) develop a dynamic asset pricing model based on prospect theory in which
investors are loss averse over fluctuations in the prices of the individual stocks that they own and also obtain a positive
relation between expected returns and idiosyncratic volatility.
Financial Management • xxx 2015 • pages 1 - 27
2
Financial Management r xxx 2015
value. In contrast, Merton’s (1987) theory provides the primary nonbehavioral justification for
investigating the pricing of IV in the cross-section of returns. Surprisingly, although Merton’s
(1987) model has been used as a theoretical motivation for investigating each of these effects,
their interaction has not received much attention in the literature.3 This paper examines the effect
of investor base on the correlation between IV and returns. We exploit the fact that this interaction
between the visibility of a stock (i.e., how widely followed it is) and the pricing of its IV is unique
to Merton’s (1987) model, thus providing an empirically testable implication that can distinguish
Merton’s (1987) under-diversification model from alternative explanations for the pricing of IV.
If Merton’s (1987) intuition is correct, we should be able to determine that not only is IV
priced in the cross-section of returns, but also that it is priced conditionally, depending upon each
stock’s visibility. Therefore, the main questions that we address in this paper are: 1) is expected
IV indeed positively related to expected returns in the cross-section, and 2) if so, is there any
evidence that the market segmentation induced by informationally incomplete markets is strong
enough to produce the observed results? In other words, is the correlation between IV and returns
constant across stocks, or is IV priced conditionally as a function of the stock characteristics (in
particular, the completeness of investor base)?
We find that expected IV is positively correlated with expected stock returns. More importantly,
our results strongly reject the hypothesis that the pricing of IV is independent of the stock’s investor
base and provide support for the idea that the pricing of IV is conditional upon a stock’s visibility.
This leads us to conclude that the market segmentation induced by costly information acquisition
is strong enough to be, at least partly, responsible for the documented role of IV in the cross-section
of returns.
Using an EGARCH-M estimate for IV, we find a positive relationship between expected IV
and expected returns. In the interest of robustness, we construct four proxies for investor base
that categorize stocks in terms of their degree of visibility (an inverse measure of information
acquisition costs): breadth of institutional ownership, number of analysts following a stock,
number of shareholders, and advertising expenses. Although these proxies are produced from
different sources and cover different samples (both in terms of time periods and stocks covered),
we find that the results for each of the four proxies paint a consistent picture. As anticipated,
neglected stocks are, in general, smaller and less liquid, and have higher returns and higher IV
relative to more visible stocks. Double sorts and Fama-MacBeth (1973) regressions conditioned
on the magnitude of the investor base indicate that: 1) IV is positively related to returns in the
cross-section, and 2) the IV premium is decreasing in the visibility of the stock. Specifically,
we find that the IV premium is larger for neglected stocks, and smaller or even economically
insignificant for the most visible stocks. For neglected stocks, long-short portfolios sorted on IV
generate a significant risk-adjusted return between 2.57% and 6.10% per month depending upon
the investor base proxy. However, for visible stocks, such a strategy does not generate statistically
and economically significant returns. Moreover, after controlling for IV, we find that larger stocks
are characterized by larger returns. This final point lends support to the hypothesis advanced by
Merton (1987) that the well-documented size effect is actually a manifestation of an omitted
variable problem.
3
An important exception is Bodnaruk and Ostberg (2009). They use a comprehensive database of Swedish individual
investor shareholdings to construct measures of the shareholder base and build a proxy for the shadow cost of incomplete
information as described by Merton (1987). They report that expected returns are negatively related to the shareholder
base after controlling IV. In addition, Lehavy and Sloan (2008) investigate the relation between investor recognition and
stock returns and touch on the fact that this relation is stronger for stocks with greater idiosyncratic volatility.
Chichernea, Ferguson, & Kassa r Idiosyncratic Risk, Investor Base, and Returns
3
Our results contribute to the literature in several ways. First, our results confirm that there
is a positive relation between full sample EAGARCH-M estimated IV and expected returns
(conditional upon investor base). In addition, we are the first to document that the pricing
of IV is conditional on a stock’s investor base for US equities. Moreover, we find that after
controlling for investor base and IV, there is a positive relation between firm size and expected
returns, as predicted by Merton (1987), who posits that the well-known size effect is likely
due to an omitted variable (IV). Additionally, Merton (1987) theoretically demonstrates that
under incomplete information, the correlation between market beta and returns is not necessarily
positive, as it depends upon both the market risk premium and the average shadow cost of
incomplete information. After controlling for IV, we find a negative beta/return relationship. This
is consistent with Merton’s (1987) model if the incomplete information effect is relatively strong.
Our results highlight Merton’s (1987) intuition that the phenomenon of neglected stocks leads
directly to the pricing of IV. This implies that the interaction between the investor base and the
IV premium should not be omitted from future analysis of either phenomenon. Finally, while our
results cannot speak to the validity of alternative explanations, they provide support for the notion
that the degree of segmentation induced by incomplete information is strong enough to generate
a correlation between IV and expected returns.
The rest of the paper is organized as follows. Section I presents a short overview of Merton’s
(1987) theoretical model and some related empirical studies. Section II describes the data and
methodology employed. Section III reports our empirical results, while Section IV provides our
conclusions.
I. Merton (1987) and Related Literature: The Interaction Between IV,
Firm Size, and Investor Base
Merton (1987) shows that if investors only follow a subset of securities (or for some other
reason, like high information or transaction costs, they are restricted from investing in the whole
market portfolio), risky assets will be valued below their full information equilibrium price.
As such, they will have higher expected returns than they would under complete information.
The incremental equilibrium expected return on security k will be correlated with the shadow
cost of incompletediffusion
of
information
among investors and the relation takes the following
form: E (Rk ) − E Rk∗ = λk E Rk∗ /R , where E (Rk ) is the expected return on firm k under
incomplete information, E Rk∗ is the expected return on firm k under full information, R is the
risk-free rate, and λk is the shadow cost of incomplete information, which is defined as:
λk = δσk2 xk
1 − qk
,
qk
(1)
where δ is the coefficient of risk aversion, σk2 is the IV of security k, xk is the fraction of the
market portfolio invested in security k (in other words, the market value of the firm relative to the
aggregate market value), and qk is the shareholder base of the firm relative to the total number of
investors in the market (the fraction of all investors who follow security k).
Equation (1) is the source of both the neglected stocks literature (i.e., λk is a function of
qk ) and the IV/return literature (i.e., λk is a function of σk2 ). While it is true that the model
predicts both of these separate effects ceteris paribus, the “all else equal” is important. The
shadow cost of incomplete information, λk , is a strictly positive function of IV only if its other
components are held constant across firms. If firm size and IV are negatively related, then the
4
unconditional relation between the risk premium and IV may be positive, negative, or zero. In fact,
the literature uniformly agrees that this is indeed the case. Larger stocks have lower IV estimates.
More generally, given that size and investor base are firm specific and vary cross-sectionally,
this interaction must be accounted for in empirical tests. Our analysis of Equation (1) highlights
the importance of cross-sectional variation in the pricing of IV (not just the magnitude of IV).
Ignoring the material difference between IV and IV premia (amount of risk × price of risk) could
be a source of the contradictory results reported in previous studies. For example, it is clear from
Equation (1) that a large degree of IV does not necessarily imply a large IV premium. In particular,
the risk component may be offset by a relatively small firm size or a large investor base.
A number of theories make similar predictions regarding the sign of the relation between IV
and returns, but they start from fundamentally different assumptions about investors’ behavior
and ultimately have different implications regarding the meaning of this relation.4 As we can see
from Equation (1), it is the interaction between the factors that differentiates the predictions of
Merton’s (1987) market segmentation model from alternative explanations.
An important strand of the empirical literature alludes to Merton’s (1987) theoretical model
as justification for examining the (unconditional) relationship between IV and returns. Using
firm level data, Fu (2009), Spiegel and Wang (2005), and Malkiel and Xu (2006) find that IV
can positively predict expected stock returns. However, several studies that have found a positive
correlation between expected IV and returns have used forward-looking data, which may induce
a look-ahead bias in the results (Fink, Fink, and He, 2012; Guo, Kassa, and Ferguson, 2014). In
contrast, Ang et al. (2006, 2009) find a negative relationship, using estimates based on lagged
(realized) volatility. Bali and Cakici (2008) argue that the Ang et al. (2006) result is sensitive to
different weighting schemes and the estimation of IV with daily versus monthly return data. Huang
et al. (2010) suggest that it relates to the short horizon return reversal anomaly. Bali, Cakici, and
Whitelaw (2011) find that the negative effect of IV is driven by its close relation with the maximum
daily return in a month, proxying for demand for lottery-like stocks. Jiang, Xu, and Yao (2009)
hypothesize that firms with high price volatility tend to be opaque in their earnings disclosures.
Han and Lesmond (2011) argue that microstructure noise factors have a substantial effect on
the realized variance measure. We conclude from the preceding discussion that there is no firm
consensus in the literature concerning the cross-sectional relationship between IV and returns.
Alternatively, there is a parallel literature based on Merton’s (1987) intuition that examines the
relationship between investor base and returns. This strand of the literature is often referred to as
the “investor recognition” or neglected stocks hypothesis and generally implies that firms with a
smaller shareholder base have higher expected returns. There is ample empirical support for this
hypothesis. For example, event studies indicate that events that increase investor recognition lead
to increases in security values. Examples include listings on exchanges (Kadlec and McConnell,
1994; Foerster and Karolyi, 1999), initiation of analyst coverage (Irvine, 2003), addition to stock
indices (Shleifer, 1986; Chen, Noronha, and Signal, 2004), reduction of the minimum unit of
trading (Amihud, Mendelson, and Uno, 1999), hiring of investor relations firms (Bushee and
Miller, 2012), and increases in advertising expenditures (Grullon, Kanatas, and Weston, 2004).
4
The relation may reflect behavioral biases. For example, if we take IV to be a relevant limit to arbitrage, this may imply that
the relationship between IV and returns is positive among relatively undervalued stocks and negative among overvalued
stocks (Cao, 2008). Along these lines, Pontiff (2006) makes the point that of the two main categories of arbitrage costs
(transaction costs and holding costs), idiosyncratic volatility, a holding cost, represents the greater arbitrage cost. As
such, “idiosyncratic volatility is the single largest barrier to arbitrage.” Alternatively, if we consider IV to be a measure
of information uncertainty, Zhang (2006) finds that greater information uncertainty predicts higher expected returns
following good news and lower expected returns following bad news and argues that this is inconsistent with the notion
that information uncertainty is a cross-sectional risk factor that requires compensation in the form of higher stock returns.
5
There is also additional empirical support in the institutional ownership literature. Early research
by Arbel, Carvell, and Strebel (1983) finds a negative correlation between institutional ownership
and future returns, but Chen, Hong, and Stein (2002) uncover evidence of a positive relation
between the change in the number of institutional holders and future stock returns. Lehavy and
Sloan (2008) reconcile this inconsistent evidence in Chen et al. (2002) by referencing the investor
recognition hypothesis. They demonstrate that changes in investor recognition are autocorrelated
and, after controlling for this autocorrelation, changes in investor recognition are negatively
related to future stock returns. Bodnaruk and Ostberg (2009) use a comprehensive database
of Swedish individual investor shareholdings to construct measures of shareholder base and
provide convincing evidence that expected returns are negatively related to the shareholder base.
Additionally, Barber and Odean (2008) find that stocks that garner attention are disproportionately
purchased by individual investors and, subsequently, earn poor returns.
II. Data and Methodology
We start with NYSE/AMEX/NASDAQ listed securities from the Center for Research in Security
Prices (CRSP) monthly data file with Share Codes 10 or 11 from July 1963 to December 2012.
We use an EGARCH-M algorithm to construct the IV measure and we require at least 60 months
of prior return history, as well as convergence of the algorithm. In order to be able to obtain the
necessary control variables, we require that the book value of common equity from the previous
fiscal year be available from Compustat (we use the CRSP/Compustat merged database to identify
the firms).
Since there are no publicly available data describing the investor base directly, we build four
alternative proxies for investor base. The proxies come from different sources, have different
frequencies, and cover different time periods. The four proxies are: 1) breadth of institutional
ownership (obtained from Thomson Reuters 13F filings quarterly data from 1980 to 2012),
2) analyst coverage (the number of analysts is obtained from Institutional Brokers’ Estimate
System (I/B/E/S) monthly data from 1976 to 2012), 3) the number of shareholders (obtained
from Compustat annual data from 1975 to 2012), and 4) advertising expenses (obtained from
Compustat annual data from 1971 to 2012). We provide detailed explanations of the methodology
used to construct our main variables of interest (IV and investor base) next.
A. Measuring IV
IV is unobservable; therefore, the practice in the literature is to consider the errors from a factor
model (e.g., CAPM or the Fama and French, 1996, three-factor model) as the idiosyncratic part of
the returns and to use the standard deviations of these errors as a proxy for IV. However, previous
studies indicate that volatility is time varying and asymmetric and the standard deviation from the
error term may not capture these properties. ARCH-type models have been proposed to address
the asymmetry and dynamic nature of volatility (Engle, 1982; Bollerslev, 1986; Nelson, 1991;
Engle and Ng, 1993). Thus, to capture the asymmetric and dynamic nature of volatility and the
fact that (according to our null hypothesis) conditional volatility determines mean returns, we use
an EGARCH-M process to model the variance of the errors.
We describe the monthly return process by the Fama and French (1996) three-factor model as
illustrated in Equation (2) including the conditional standard deviation of the errors (IV) as one
of the regression variables (EGARCH-M process). The error term corresponds to innovations
in firm-specific cash-flows and the conditional distribution of the error term εit (conditional on
6
time t – 1 information) is assumed to be normal with mean of zero and variance σit2 . The IV of
individual stocks is the square root of the conditional variance σit2 , which is a function of the past
p periods of residual variance and the q periods of shocks as estimated by Equation (3):
Rit − r f = αi + βi (Rmt − r f ) + si SMBt + αi HMLt + δi σit + εit , εit ∼ N (0, σit ) ,
(2)
εi,t−k − (2/π )1/2 .
σ
i,t−k
(3)
lnσit2 = ait +
p
j=1
2
bi, j lnσi,t−
j +
εi,t−k
+γ
ci,k θ
σi,t−k
k=1
q
We restrict p and q between one and three, and permutations of these orders yield nine different
EGARCH-M specifications for each stock. Of these nine combinations, we select the one that
converges with lowest the Akaike Information Criterion (AIC). After completing this step, 17%
of the stocks in our initial sample are eliminated as the EGARCH-M estimation algorithm did not
converge. The modal p is one and the modal q is three (the median values for both p and qare two).
Our choice for the lowest AIC is not driven solely by the desire to find a model that fits the data best.
Given that we are modeling firm-specific risk, it is quite possible that the volatility process follows
a different specification from firm to firm. Additionally, considering several models for each stock
increases the chances that the algorithm will converge and produce meaningful estimates. Using
the EGARCH-M specification described, we obtain monthly estimates of conditional IV for each
stock each month.
We should mention that the idiosyncratic volatilities obtained here are in-sample estimates
using the full sample of returns available for each stock. This methodology is similar in spirit
to the approach in Fu (2009) and Spiegel and Wang (2005) and, to a lesser extent, Huang et
al. (2010), which has been subject to some criticism in the literature. For example, Guo et al.
(2014) and Fink et al. (2012) have found that using the full sample to estimate IV incorporates
forward-looking information. As such, it may induce a look-ahead bias that influences the relation
between returns and IV. However, our objective is to estimate IV as close to the unobservable
“true” IV as possible, as we are trying to understand the correlation between IV and returns.
As Fink et al. (2012, p. 520) note, the “use of the full information set [is] appropriate when
testing theoretical models such as Merton (1987) and Malkiel and Xu (2006).” This is precisely
our objective. While the look-ahead bias is problematic for creating trading strategies or running
predictive tests, it presents less of an issue when the purpose of the analysis is to understand the
true role of IV in the cross-section of returns.
B. Measuring the Completeness of Investor Base
The theoretical concept of investor base described by Merton (1987) should capture the fraction
of investors in the market who are informed about a certain security. While this is difficult to
measure directly, there are at least two different categories of proxies that should, in theory, be
correlated with this concept. We can consider measures correlated with the visibility of a stock.
The idea is that people are likely to be better informed about more visible stocks and, as such, are
more likely to consider holding them in their portfolios. Alternatively, it seems natural to assume
that the number of investors who follow a security is increasing in the number of investors who
actually own that security. Therefore, ownership of a security is a good proxy for knowledge of
that security.
In the interest of thoroughness, we consider two proxies from each category. As proxies for
the visibility of a stock we consider: 1) the number of analysts following the stock and 2) the
7
firm’s advertising expenses. Individual investors’ holding data would provide the ideal data for
building an ownership proxy; however, these type of data are not currently available for the US
market.5 Therefore, we use alternative ownership proxies, such as: 1) the breadth of institutional
ownership and 2) the number of shareholders. We describe each proxy in more detail below.
1. Breadth of Institutional Ownership
This variable is derived from the Thomson CDA/Spectrum 13F institutional transaction quarterly data, covering the period from 1980 to 2012. Following Chen et al. (2002) and Lehavy and
Sloan (2008), we define the breadth of institutional ownership as the ratio of the number of 13F
filers that hold a long position in that stock to the total number of 13F filers in the sample for
that quarter. This variable is highly autocorrelated. As such, the previous literature has accounted
for this problem by considering changes in, instead of levels of, institutional ownership. Initially,
Chen et al. (2002) find evidence of a positive correlation between the change in the number of
institutional holders and future stock returns, and point out that this is inconsistent with Merton’s
(1987) model. However, Lehavy and Sloan (2008) report that changes in investor recognition are
also autocorrelated and that after controlling for this autocorrelation, a negative relation emerges.
It should be noted though, that Merton’s (1987) model implies a connection between the level
(i.e., not the change) of investor base and expected returns. Since our methodology employs this
variable mainly as a cross-sectional sorting mechanism, we are less concerned about the autocorrelation issue, and choose to use the level of the breadth of institutional ownership in order
to stay closer to Merton’s (1987) original theoretical construct. The frequency of the 13F filings
is quarterly. As such, we transform this measure into a monthly measure by assuming that the
breadth of ownership stays constant during the months of a quarter.6
2. Analyst Coverage
The data on analyst coverage are obtained from the I/B/E/S that provides monthly analysts
forecasts covering the period from 1976 to 2012. Following Diether, Malloy, and Scherbina
(2002), for each stock in CRSP, we set the coverage in any given month equal to the number of
I/B/E/S analysts who provide fiscal Year 1 earnings estimates that month. Analyst coverage is
associated with more recognizable firms (Moskowitz, 2005) and it will be correlated with the
visibility of a stock. This variable has also been shown to improve the speed with which a stock’s
price responds to information (Hong, Lim, and Stein, 2000).
3. Number of Shareholders
The number of shareholders is an annual measure obtained directly from the Compustat annual
files and has been used in previous literature as a proxy for attention/recognition (Moskowitz,
2005). In order to transform this variable into a monthly variable, we assumed that the number
of shareholders remains constant throughout the year. We assigned the yearly measure at the
beginning of year t to each of the 12 months in year t. The data covers 1975-2012.
5
Bodnaruk and Ȫstberg (2009) obtained access to a comprehensive database of Swedish individual investor shareholdings
and used it to construct measures of investor base and the shadow cost of incomplete information (Merton, 1987).
6
We also used interpolation and assumed that the variable changes linearly throughout the months of the quarter. The
results are qualitatively similar and are available upon request.
8
4. Advertising Expense
Advertising expense provides another measure of recognition that has been shown to affect
investor’s portfolio choices (Cronqvist, 2006), as well as a firm’s liquidity and breadth of ownership. Based on the idea that investors are more likely to buy what they know, Grullon et al. (2004)
find that firms with greater advertising expenditures have a larger number of both individual and
institutional investors. This suggests that advertising expenditures are likely to be correlated with
the investor base concept described by Merton (1987). Since this is an expense item coming from
the income statement, we transform the measure into a monthly measure by assigning the yearly
amount at the end of year t to each of the 12 months of year t.
Each of the four proxies described above have several common characteristics that influenced
our choice of methodology: 1) they are all inherently sticky (autocorrelated) variables, 2) aside
from the advertising expense, they are not continuous, but categorical variables, and 3) they are
all highly correlated with the market capitalization of the stocks considered. These characteristics
would present difficulties if these variables were used directly as control variables in crosssectional tests (autocorrelation, insufficient cross-sectional variation, or multicollinearity). Thus,
we use them primarily as cross-sectional sorting tools to group stocks into visible/neglected
categories. We then test within these categories for significantly different behavior of the relation
between IV and returns.
C. Control Variables
We consider control variables that have been previously shown to be significant in the crosssection of returns. Specifically, we follow Fama and French (1992) to build proxies for beta,
size, and book-to-market variables (for size, we also construct relative size as the percentage
of the overall capitalization of the market in an attempt to follow the theoretical construct described in Merton, 1987).7 Following Fink et al. (2012), we also include the systematic risk
betas from the Fama-French (1992) model denoted as MKTBETA, SMBBETA, and HMLBETA,
estimated based on a rolling 60-month basis. Since liquidity has been shown to affect mean
returns (Amihud and Mendelson, 1986; Brennan and Subrahmanyam, 1996; Chordia, Subrahmanyam, and Anshuman, 2001) and it shares some common characteristics with IV (Spiegel
and Wang, 2005), we consider dollar volume as a proxy for the liquidity characteristic of a
firm. Additionally, we include a liquidity beta measure as a control in our multivariate tests,
estimated using the Pastor and Stambaugh (2003) liquidity factor on a 60-month rolling basis denoted as PSBETA. As robustness check, we also used DOLARVOLUME and TURNOVER
as proxies for liquidity in all of our tests, and the results are qualitatively unchanged (untabulated). We also consider cumulative returns over the previous six months as a control for
momentum (Chichernea and Slezak, 2013, and McLean, 2010, show that IV and momentum
may be related). In addition, Huang et al. (2010) demonstrate that lagged return (as a control
for reversals) is an important variable in cross-sectional return regressions. As such, we include it as a control variable in our tests.8 Our empirical findings are discussed in the following
section.
7
We present the relative size proxy (x) in Table I (Descriptive Statistics). The results in the following tables refer to the
classic market cap proxy. These results are robust to using the scaled version of size.
8
We thank an anonymous referee for these suggestions.
9
III. Empirical Results
Once we obtain proxies for each of the variables of interest, our approach is relatively straight
forward. We begin by documenting the overall sample characteristics and investigate whether the
four proxies for investor base paint a consistent picture in terms of the characteristics of stocks
assigned to visible/neglected categories. Next, we examine whether there is any evidence that the
correlation between idiosyncratic risk and returns is conditional upon the completeness of the
investor base as predicted by Merton’s (1987) equilibrium model.
A. Sample Characteristics
The summary statistics for the main variables used in our tests are presented in Table I.
In Panel A, we report the descriptive statistics for the complete sample of stocks traded on
NYSE/AMEX/NASDAQ from July 1963 to December 2012 for which we were able to obtain IV
estimates and controls. We further restrict this sample depending upon data availability for each
investor base proxy and report the descriptive statistics in Panel B.
The average monthly return in the overall sample is 1.27%, with a mean excess return (raw
return minus one-month T-bill rate) of 0.84% and an average IV of 11.48%. The summary
statistics for the control variables are presented.
In Panel B of Table I, we observe that the average (median) breadth is 5.53% (2.46%), the
average (median) number of analysts following is 7.63 (5.17), the average (median) advertising
expense is $47.95 ($3.78) million, and the average (median) number of shareholders is 10,228
(2,280). The second set of columns in Panel B provides the sample coverage for these four
investor base proxies. The richest samples in terms of the average number of firms per crosssection are obtained when Breadth of Ownership and Number of Shareholders are used as proxies
for investor base (2,706 and 2,608 average number of firms per cross-section, respectively). The
other two variables, Advertising Expenses and Number of Analysts, cover relatively fewer firms
per cross-section (860 and 1,745, respectively), but the sample coverage starts a little earlier
(1971 and 1976, respectively). The frequency of these proxies varies from monthly to yearly
(see discussion in Section II on how we transform these into monthly variables). The advantage
of using four different samples is that they serve as robustness checks on each other. However,
there is an unavoidable selection bias when using these proxies. Any of the four samples will be
tilted toward relatively large stocks (which are more likely to have analyst coverage, to be owned
by institutions, and to report advertising expenses and the number of shareholders). However, a
sample biased toward larger, better known firms should generally work against our finding any
evidence of market segmentation.
We use each of our four proxies for investor base to categorize stocks based on their visibility
in order to investigate the “reasonableness” of our proxies and to demonstrate any significant
differences between what we deemed as neglected (narrower investor base) versus “visible”
(broader investor base) stocks. The results are reported in Table II.
The four proxies paint a consistent picture of the characteristics of visible/neglected stocks.
As expected, neglected stocks are, in general, smaller and less liquid, and have higher returns relative to the more visible stocks. This is consistent with the investor recognition hypothesis that has been documented in previous studies. Additionally, neglected stocks have a
considerably higher IV than visible stocks. For example, in the breadth sample (Panel A of
Table II), the average IV in the neglected stock quintile is more than twice the average IV
in the visible stock quintile (17.33% vs. 8.13%). We find similar patterns in the other three
samples.
10
Table I. Descriptive Statistics
Panel A presents the descriptive statistics for the comprehensive sample of stocks traded on NYSE, AMEX,
or NASDAQ from July 1963 to December 2012. RETURN is the percentage monthly return. EXRET is the
monthly excess return (raw return minus one-month T-bill rate). BETA and BE/ME are estimated based
on the definitions in Fama and French (1992). The market value of equity (ME) is the product of the
monthly closing price and the number of outstanding shares. The relative size (x) is calculated as the ratio
of the market value of equity for a particular firm to the total market value. The expected idiosyncratic
volatility (EIV) is the one-month ahead expected idiosyncratic volatility estimated through an EGARCH-M
specification applied to the Fama-French (1996) three-factor model. DVOL is the average of dollar volume
calculated over the past 36 months beginning in second to the last month, following Chordia et al. (2001).
RET(–7:–2) represents the cumulative average return over the past six months (skipping a month). Skewed
variables are reported as the natural logarithms. Panel B describes the summary statistics for each of the four
proxies for investor base. We report these separately since each of these four sets of statistics are calculated
based on different samples, depending upon the coverage of that particular proxy (the last columns of Panel
B describes the coverage of each variable including the time period covered, the frequency of the variable,
and the average number of firms in a cross-section). Breadth of institutional ownership is the ratio of the
number of 13F filers that hold a long position in that stock to the total number of 13F filers in the sample
for that quarter. NoAnalysts is the number of IBES analysts who provide fiscal year one earnings estimates
each month. Advertising Expenses and Number of Shareholders come directly from the Compustat annual
files and are winsorized at the 1% level to eliminate the influence of outliers.
Variables
Mean
Std Dev
Median
Q1
Q3
Panel A. Descriptive Statistics Overall Sample (July 1963 to December 2012, on average, 3,248 firms
per cross-section)
RETURN (%)
EXRET (%)
EIV (%)
BETA
Ln(BE/ME)
Ln(ME)
Relative Size (x) (%)
Ln(x)
Ln(DVOL)
RET(–7:–2) (%)
1.269
0.841
11.484
1.301
−0.476
11.734
0.044
−5.388
10.693
7.862
13.244
13.244
6.576
0.328
0.865
1.978
0.182
1.978
2.433
35.630
0.262
−0.166
9.977
1.285
−0.407
11.642
0.008
−5.480
10.771
3.358
−5.718
−6.147
7.191
1.094
−0.963
10.280
0.002
−6.842
8.961
−11.741
Panel B. Investor Base Proxies
Mean Std Dev Median
Breadth (%)
5.533
8.390
NoAnalysts
7.633
6.986
AdvExp ($ mill)
47.947 139.675
NoShareholders (Thou) 10.228 24.737
2.462
5.172
3.775
2.280
6.769
6.340
13.917
1.536
0.072
13.102
0.027
−4.020
12.491
20.689
Sample Coverage
Q1
Q3
Time
Period
Frequency Avg. No.
Firms
0.822 6.439 1980-2012 Quarterly
2.156 11.088 1976-2012 Monthly
0.716 21.459 1971-2012 Yearly
0.931 7.078 1975-2012 Yearly
2,706
1,745
860
2,608
In Table III, we determine that the general characteristics of the correlation matrix are consistent
with the results documented in the previous literature for many of the control variables.
We focus our discussion on three variables of interest: 1) IV, 2) investor base, and 3) size.
Regardless as to the proxy involved, the correlation between expected IV and returns is positive.
Although the traditional size effect is evident (negative correlation between size and returns),
there is also a strong negative correlation between IV and size (small stocks have larger IV). As
11
Table II. Descriptive Statistics per Category of Investor Base
At the end of every month t, stocks are ranked by the respective completeness of the investor base measure
(five categories, with the lowest one as the most neglected stocks). The four panels represent the four
different proxies for the investor base (the use of each proxy dictates a different sample). The table presents
the average characteristics of the portfolios obtained for returns, expected idiosyncratic volatility, beta,
book-to-market, size, dollar volume, and past cumulative returns (see Table I for a description of these
variables).
Completeness of Investor Base
1 (Neglected)
2
3
4
5 (Visible)
Panel A. Breadth of Institutional Ownership as Proxy for Investor Base
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVol)
RET(–7:–2) (%)
1.524
17.331
1.419
9.542
−0.347
8.151
7.014
1.331
14.960
1.448
10.755
−0.425
9.829
6.960
1.238
12.815
1.394
11.859
−0.568
11.384
8.032
1.320
10.764
1.297
13.020
−0.745
12.887
10.046
1.241
8.125
1.077
14.946
−0.870
14.852
9.477
Panel B. Number of Analysts as Proxy for Investor Base
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVol)
RET(–7:–2) (%)
1.332
13.606
1.405
11.018
−0.376
10.121
8.364
1.320
12.247
1.374
11.765
−0.525
11.151
9.339
1.335
11.038
1.318
12.567
−0.666
12.176
9.330
1.255
9.874
1.233
13.476
−0.745
13.240
8.413
1.129
8.333
1.082
14.933
−0.809
14.749
7.208
Panel C. Advertising Expenses as Proxy for Investor Base
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVol)
RET(–7:–2) (%)
1.407
16.026
1.459
9.755
−0.469
8.548
8.547
1.227
13.647
1.423
10.739
−0.430
9.745
8.040
1.187
12.125
1.383
11.547
−0.439
10.725
8.315
1.152
10.838
1.302
12.517
−0.507
11.914
8.206
1.159
8.469
1.103
14.485
−0.755
13.993
7.685
Panel D. Number of Shareholders as Proxy for Investor Base
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVol)
RET(–7:–2) (%)
1.566
14.733
1.420
10.512
−0.461
9.488
10.568
1.574
13.987
1.411
10.934
−0.488
10.045
9.901
1.464
13.088
1.370
11.390
−0.483
10.577
9.124
1.351
12.026
1.317
12.159
−0.549
11.541
8.746
1.174
8.895
1.093
14.042
−0.550
13.566
7.452
noted in our discussion following Equation (1), this is consistent with Merton’s (1987) argument
that the size effect documented in the literature may be a manifestation of an omitted variable
(IV). Our multivariate tests will further investigate this issue.
The univariate relationship between our proxies of investor base and returns does not present
consistent signs across our different samples. However, all of the investor base proxies are
12
Table III. Correlation Table
This table presents the time-series means of the cross-sectional Pearson correlations within each of the four
samples obtained by using the four proxies for investor base (since each sample has different coverage, we
report separate correlations for each). The four correlation numbers in each cell represents, respectively,
the correlation obtained based on the sample built using the breadth of ownership, number of analysts,
advertising expenses, and the number of shareholders. Variable definitions are presented in Table I.
RETURN
EIV
Ln
BETA Ln(ME) (BE/ME)
Ln
(DVOL)
RET(–7:–2) InvBase
Panel A. Advertising Expense Sample
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVOL)
RET(–7:–2) (%)
AdExpense
1
0.081
−0.008
−0.014
0.029
−0.014
0.007
0.003
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVOL)
RET(–7:–2) (%)
Breadth
1
0.082
−0.012
−0.011
0.030
−0.011
0.013
0.001
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVOL)
RET(–7:–2) (%)
NoAnalysts
1
0.051
−0.007
−0.011
0.022
−0.010
0.021
−0.005
1
0.380
−0.460
0.081
−0.339
−0.035
−0.256
1
−0.421
1
0.073 −0.414
−0.272
0.927
−0.020
0.097
−0.371
0.583
1
−0.429
−0.251
−0.131
1
0.125
0.489
1
0.006
1
1
0.142
0.710
1
0.034
1
1
0.142
0.764
1
−0.008
1
1
0.129
0.444
1
−0.005
1
Panel B. Breadth of Ownership Sample
1
0.397
−0.452
−0.017
−0.315
−0.055
−0.325
1
−0.420
1
−0.043 −0.312
−0.263
0.920
−0.033
0.120
−0.395
0.799
1
−0.354
−0.263
−0.168
Panel C. Number of Analysts Sample
1
0.462
−0.454
−0.031
−0.280
−0.053
−0.297
1
−0.489
1
−0.055 −0.268
−0.307
0.901
−0.022
0.108
−0.345
0.799
1
−0.322
−0.266
−0.167
Panel D. Number of Shareholders Sample
RETURN (%)
EIV (%)
BETA
Ln(ME)
Ln(BE/ME)
Ln(DVOL)
RET(−7:–2) (%)
NoShareholders
1
0.092
−0.006
−0.017
0.031
−0.016
0.010
−0.003
1
0.393
−0.462
0.000
−0.331
−0.033
−0.236
1
−0.417
1
−0.029 −0.331
−0.263
0.922
−0.017
0.099
−0.370
0.530
1
−0.371
−0.255
−0.001
(Continued)
13
Table III. Correlation Table (Continued)
Breadth
NoAnalysts
NoShareholders
AdExpense
Panel E. Investor Base Proxy Variables
Breadth
NoAnalysts
NoShareholders
AdExpense
1
0.779
0.773
0.778
1
0.554
0.566
1
0.738
1
negatively correlated with IV and positively correlated with size. This is consistent with the
results reported in Table II that neglected stocks tend to be small stocks with higher IV. Panel E
of Table III reports the correlation between the four proxies and it varies from 55% (Number of
Analysts and Number of Shareholders) to 78% (Breadth of Ownership and Number of Analysts)
suggesting that while they are not identical measures, they capture similar characteristics. Overall,
Tables II and III indicate that the four proxies for investor base generate subsamples that are
consistent with our intuition regarding the characteristics of neglected and visible stocks. On
average, neglected stocks are smaller, highly volatile, less liquid, and have higher returns.
B. Portfolio Sorts
We begin our formal analysis by sorting stocks into quintile portfolios based on expected IV. In
Panel A of Table IV, we sort stocks into quintile portfolios on expected IV and report the value
weighted Fama and French (1996) risk-adjusted returns (i.e., alphas) for each quintile portfolio.9
The procedure is repeated for each of the four samples determined by the use of a particular
investor base proxy and the overall sample.
Panel A of Table IV indicates that those stocks with high IV generally have higher risk-adjusted
returns. The average monthly difference in risk-adjusted returns (i.e., alphas) between the high
and low IV stocks varies from 2.08% (for the sample determined by Number of Shareholders)
to 0.99% (for the sample determined by Number of Analysts) and is statistically significant. The
correlation between IV and returns is not sample specific (all four samples produce qualitatively
similar results). While results from these sorts confirm a statistically and economically significant
positive relationship between IV and returns, Merton’s (1987) model further predicts that the
difference in risk-adjusted returns between high and low IV stocks should be more pronounced
in the neglected stocks (i.e., low investor base category) than visible stocks (i.e., high investor
base).
In Panel B of Table IV, we report value weighted risk-adjusted returns for five-by-five double
sorts by investor base and IV. At the beginning of each month, we divide all of the firms into
quintiles based on the investor base proxy. Within each investor base group, we sort the stocks into
quintiles based on IV. Value weighted Fama and French (1996) three-factor alphas are presented
for each of the 25 portfolios. For every one of the four samples, the correlation between IV and
returns is monotonically decreasing as we move from the most neglected to the most visible
stocks category. For example, in the sample determined by Breadth of Ownership as a proxy for
investor base, a portfolio strategy that is long in high IV stocks and short in low IV stocks earns
9
We value weight our portfolios based on the previous month log size, in order to mitigate the effect of influential large
outliers in terms of size. Results remain qualitatively unchanged when we use the raw size variable to weight our returns.
We get qualitatively similar results for equally weighted portfolios as well.
14
Table IV. Alphas of Portfolios Sorted by Idiosyncratic Volatility (IV) and Investor
Base
The table reports Fama and French (1996) alphas, with Newey-West (1987) corrected t-statistics in parenthesis (using six lags). Panel A presents the alphas of value weighted portfolios for our complete sample
(before merging with the investor base proxies), as well as within each of the fours samples determined by
the use of the respective proxy for investor base (we use ln(Size)t–1 for the weights). Each month, we sort
stocks into quintiles based on the level of their expected IV for that month. We then form value weighted
portfolios and calculate the alphas relative to the Fama-French (1992) three factors for each quintile (the last
column presents the alphas of the difference between extreme quintiles). For Panel B, we first sort the stocks
based on the investor base measure (using each of the four proxies) and then, within each quintile, we sort
the stocks based on the expected IV for that month. Then, we form value weighted portfolios and calculate
the alphas relative to the Fama-French (1992) three factors for each category (the last column presents the
alphas of the difference between extreme quintiles of expected IV). We report the results for each of the four
samples determined by the use of the respective proxy for investor base.
Panel A. Simple Sorts by IV
IV Rank
Breadth
NoAnalysts
AdvExp
NoShareholders
1 (Low)
2
3
4
5 (High)
High – Low
−0.16
(−2.30)
−0.10
(−1.52)
−0.27
(−3.20)
−0.21
(−3.28)
−0.35
(−4.88)
−0.24
(−3.53)
−0.35
(−4.30)
−0.35
(−5.29)
−0.39
(−5.30)
−0.24
(−3.56)
−0.45
(−5.23)
−0.39
(−5.70)
−0.21
(−2.11)
−0.12
(−1.47)
−0.27
(−2.67)
−0.17
(−1.90)
1.80
(5.96)
0.88
(3.95)
1.60
(6.19)
1.87
(6.65)
1.96∗∗∗
(5.76)
0.99∗∗∗
(3.96)
1.87∗∗∗
(6.46)
2.08∗∗∗
(6.63)
Panel B. Subsequent Sorts by Investor Base Proxy and IV
1 (Low)
2
5 (High)
High − Low
4.69
(9.66)
2.51
(5.44)
0.95
(3.31)
0.59
(2.83)
0.42
(2.36)
6.10∗∗∗
(11.57)
3.35∗∗∗
(6.69)
1.48∗∗∗
(4.33)
0.68∗∗∗
(2.82)
0.28
(1.33)
4
5 (High)
High − Low
−0.05
(−0.36)
−0.09
(−0.79)
0.03
(0.33)
2.02
(5.57)
1.10
(3.69)
0.79
(4.03)
2.57∗∗∗
(6.44)
1.50∗∗∗
(4.46)
1.06∗∗∗
(4.62)
3
4
I. Breadth of Institutional Ownership
Low Breadth
(Neglected)
2
3
4
High Breadth
(Visible)
−1.41
(−10.09)
−0.84
(−9.12)
−0.53
(−6.35)
−0.09
(−1.05)
0.13
(1.56)
−1.15
(−7.17)
−0.88
(−7.20)
−0.48
(−5.19)
−0.21
(−2.46)
0.06
(0.78)
−0.65
(−3.84)
−0.66
(−5.19)
−0.40
(−4.51)
−0.09
(−0.99)
0.04
(0.47)
0.45
(1.90)
0.09
(0.52)
0.03
(0.26)
0.08
(0.78)
0.09
(1.04)
IV Rank
1 (Low)
2
3
II. Number of Analysts
Low NoAnalysts
(Neglected)
2
3
−0.55
(−7.09)
−0.40
(−5.16)
−0.27
(−3.30)
−0.85
(−7.36)
−0.37
(−4.44)
−0.21
(−2.56)
−0.50
(−4.58)
−0.32
(−2.97)
−0.12
(−1.39)
(Continued)
15
Table IV. Alphas of Portfolios Sorted by Idiosyncratic Volatility (IV) and Investor
Base (Continued)
IV Rank
1 (Low)
2
3
4
5 (High)
High – Low
0.01
(0.08)
0.04
(0.37)
0.40
(1.93)
0.28
(1.34)
0.40∗
(1.66)
0.20
(0.86)
0.41
(1.88)
0.17
(1.15)
−0.05
(−0.33)
−0.19
(−1.34)
0.11
(0.88)
3.68
(8.76)
2.08
(5.97)
1.03
(3.64)
0.65
(2.41)
0.22
(1.12)
4.99∗∗∗
(10.57)
2.89∗∗∗
(7.70)
1.52∗∗∗
(5.10)
1.05∗∗∗
(3.62)
0.09
(0.50)
3.09
(8.49)
2.54
(7.79)
1.88
(6.30)
1.33
(4.72)
0.14
(0.66)
4.14∗∗∗
(10.12)
3.08∗∗∗
(8.39)
2.26∗∗∗
(6.83)
1.58∗∗∗
(4.87)
0.01
(0.06)
II. Number of Analysts
−0.00
(−0.05)
0.08
(0.97)
4
High NoAnalysts
(Visible)
−0.14
(−1.71)
−0.05
(−0.82)
−0.06
(−0.64)
−0.02
(−0.27)
III. Advertising Expenses
Low AdExpense
(Neglected)
2
3
4
High AdExpense
(Visible)
−1.31
(−12.11)
−0.82
(−7.78)
−0.50
(−5.53)
−0.40
(−3.75)
0.12
(1.30)
−1.16
(−7.62)
−0.83
(−6.36)
−0.46
(−4.59)
−0.26
(−2.52)
0.05
(0.49)
−0.75
(−5.09)
−0.46
(−3.55)
−0.20
(−1.55)
−0.27
(−2.21)
−0.06
(−0.49)
IV. Number of Shareholders
Low NoShareholders
(Neglected)
2
3
4
High NoShareholders
(Visible)
−1.05
(−11.86)
−0.54
(−6.91)
−0.38
(−4.56)
−0.25
(−3.25)
0.12
(1.29)
−0.62
(−7.04)
−0.49
(−5.68)
−0.44
(−4.61)
−0.27
(−3.31)
0.03
(0.50)
−0.39
(−4.14)
−0.30
(−3.23)
−0.31
(−3.43)
−0.27
(−3.30)
0.00
(0.01)
0.08
(0.59)
0.03
(0.30)
−0.01
(−0.09)
−0.21
(−2.04)
−0.18
(−1.70)
∗∗∗
Significant at the 0.01 level.
∗
∗∗
a statistically and economically significant average alpha of 6.10% per month in the neglected
stock category (i.e., the 1st quintile). However, the economic and statistical significance of a
similar strategy decreases monotonically as we move to the more visible stocks. The next four
quintiles produce an alpha of 3.35%, 1.48%, 0.68%, and 0.28%, respectively, and the 0.28% return
difference in the most visible stock category (i.e., the last quintile) is not statistically significant.
We obtain qualitatively similar results for the remaining three samples (see Panels B.II, B.III, and
B.IV of Table IV).
Overall, the results from Table IV indicate that the positive relation between IV and returns
is primarily driven by neglected stocks. This is consistent with Merton’s (1987) prediction that
if investors follow a subset of securities, risky assets will be valued below their full information
values and, as such, will have higher expected returns. A more precise picture of the magnitude
and behavior of this relationship can be established via multivariate analysis.
16
C. Fama-MacBeth Regressions
In this subsection, we utilize Fama-MacBeth (1973) regressions to investigate the relationship
between IV, investor base, and expected stock returns. First, we confirm that there is a positive
correlation between stock returns and expected IV in Table V. Then, we investigate this relation
further within each investor base quintile portfolio and find that it is significantly weaker in the
visible stock portfolio (Tables VI and VII). Finally, as a robustness check, we demonstrate that
our results are unchanged when we use contemporaneous realized IV as an alternative measure
for volatility (Table VIII).
The results in Table V indicate that the positive relation between expected IV and expected
stock returns is robust to different model specifications (i.e., including different control variables)
and across different samples.
For example, Model 2 of Panel A (the Breadth sample) indicates that after controlling for commonly used stock characteristics, the Fama-MacBeth (1973) regression coefficient on expected
IV is 2.49 with a Newey-West (1987) adjusted t-statistics of 12.12. In Model 3, we report that this
result is robust to including MKTBETA, SMBBETA, and HMLBETA as measures of systematic
risk, along with the Pastor and Stambaugh (2003) factor PSBETA as a proxy for liquidity. The
signs and significance of the coefficients for the controls for systematic risk are consistent with
the results obtained in the previous literature (e.g., Fink et al., 2012). PSBETA does not seem to
be significant in most of our specifications, which may be the due to the nature of our samples
(which, as noted above, are tilted toward larger, more liquid stocks). The results are qualitatively
similar across the four samples considered (see Panels B, C, and D of Table V) and the coefficient
on IV is relatively stable and economically significant. For example, a 1% increase in IV would
increase expected returns by about 1.73% to 2.74% per month depending upon the sample and
the model specification used. Additionally, the table indicates that including IV in the regression
increases the explanatory power of our models in each of the four samples.
It is interesting to note that in addition to the relation between IV and stock returns, the
regression results in Models 2 and 3 demonstrate that beta is negatively correlated with returns.
This result is also consistent with Merton’s (1987) prediction, who argues that in the presence
of incomplete information, investors demand an additional premium to hold risky assets relative
to CAPM. Specifically, under incomplete information, Merton (1987) derives the following
equilibrium expected return for a security k:
E (Rk ) = R f + βk E(R M ) − R f + λk − βk λ M ,
(4)
where λk − βk λ M is the additional premium relative to CAPM and is determined by the shadow
cost of incomplete information for security k (λk ) and the weighted average shadow cost of
incomplete information over all securities (λ M ). This result implies that beta can have a negative
or positive correlation with returns depending upon whether the market risk premium is less than
or greater than the average shadow cost of incomplete information.10 Thus, the regression results
from Models 2 and 3 of Table V imply that the shadow cost of incomplete information is greater
than the market risk premium resulting in a negative beta coefficient.11
Finally, Table V indicates that size is positively correlated with returns after controlling for
IV. Fu (2009) reports a similar result using in-sample EGARCH estimates of IV. This result
∂k ( E(Rk )−R f )
The partial derivative with respect to beta becomes
= E(R M ) − R f − λ M , where E(R M ) − R f is the market
∂k βk
risk premium, and λ M is the weighted average shadow cost of incomplete information.
10
11
This is consistent with results obtained by Fink et al. (2012). We thank the anonymous referee for suggesting this
interpretation.
17
Table V. Fama-MacBeth Regressions
The table presents the time-series averages of the slopes in the cross-sectional regressions using standard
Fama and MacBeth (1973) methodology (the regressions are run using the whole universe of stocks). Several
models are considered. The t-statistics reported in parentheses are Newey-West corrected t-statistics using
six lags. The dependent variable is the percentage monthly return. BETA_FF92 and Ln(BEME) are beta
and book-to-market estimated following Fama and French (1992). Ln(ME) is the log of ME at time t – 1.
EIV is the expected conditional idiosyncratic volatility estimated through an EGARCH-M model, and it is
multiplied by 10. RETt–1 is percentage lag return. RET(–7:–2) represents the percentage cumulative returns
over the past six months (skipping one month). We got the variables MKTBETA, SMBBETA, HMLBETA, and
PSBETA from a rolling 60-month window regression of excess return on excess market return, SMB, HML,
and Pastor and Stambaugh (2003) liquidity factor. The last column reports the average adjusted R-squares
of the cross-sectional regressions.
MODEL1
MODEL2
MODEL3
Panel A. Breadth Sample
Intercept
BETA_FF92
Ln(ME)
Ln(BEME)
RETt−1
RET(–7:–2)
2.156∗∗∗
(3.701)
−0.100
(−0.310)
−0.086∗∗
(−2.142)
0.297∗∗∗
(3.263)
−0.043∗∗∗
(−10.666)
0.005∗∗
(2.183)
−3.849∗∗∗
(−4.960)
−1.310∗∗∗
(−5.406)
0.296∗∗∗
(7.115)
0.553∗∗∗
(6.666)
−0.040∗∗∗
(−10.805)
0.008∗∗∗
(3.952)
2.486∗∗∗
(12.122)
0.041
0.062
EIV
MKTBETA
SMBBETA
HMLBETA
PSBETA
Adj. R2
MODEL1
MODEL2
BETA_FF92
Ln(ME)
Ln(BEME)
1.806∗∗∗
(3.498)
−0.092
(−0.343)
−0.076∗∗
(−2.119)
0.280∗∗∗
(3.270)
−3.327∗∗∗
(−5.215)
−1.143∗∗∗
(−5.361)
0.238∗∗∗
(6.600)
0.441∗∗∗
(5.558)
MODEL2
MODEL3
Panel B. Number of Analysts Sample
−5.785∗∗∗
(−10.217)
0.361∗∗∗
(10.450)
0.510∗∗∗
(6.952)
−0.044∗∗∗
(−11.272)
0.008∗∗∗
(4.777)
2.547∗∗∗
(12.660)
−0.519∗∗∗
(−4.739)
−0.278∗∗∗
(−4.690)
0.345∗∗∗
(4.482)
0.040
(0.560)
0.068
MODEL3
Panel C. Advertising Expenses
Intercept
MODEL1
2.043∗∗∗
(3.509)
−0.055
(−0.192)
−0.090∗∗
(−2.362)
0.193∗∗
(2.176)
−0.038∗∗∗
(−8.497)
0.008∗∗∗
(3.374)
−1.382∗
(−1.846)
−0.854∗∗∗
(−3.850)
0.115∗∗∗
(2.811)
0.310∗∗∗
(3.752)
−0.036∗∗∗
(−8.279)
0.010∗∗∗
(4.910)
1.732∗∗∗
(9.152)
0.054
0.068
MODEL1
MODEL2
−2.563∗∗∗
(−4.330)
0.154∗∗∗
(4.252)
0.255∗∗∗
(3.601)
−0.042∗∗∗
(−9.467)
0.010∗∗∗
(5.552)
1.780∗∗∗
(9.674)
−0.378∗∗∗
(−3.280)
−0.230∗∗∗
(−3.811)
0.271∗∗∗
(3.672)
0.002
(0.021)
0.078
MODEL3
Panel D. Number of Shareholders
−4.466∗∗∗
(−8.312)
0.257∗∗∗
(7.499)
0.391∗∗∗
(5.297)
2.389∗∗∗
(4.465)
−0.035
(−0.122)
−0.110∗∗∗
(−2.870)
0.309∗∗∗
(3.341)
−3.957∗∗∗
(−5.740)
−1.256∗∗∗
(−5.647)
0.294∗∗∗
(7.623)
0.580∗∗∗
(6.642)
−5.720∗∗∗
(−10.786)
0.352∗∗∗
(10.370)
0.545∗∗∗
(6.716)
(Continued)
18
Table V. Fama-MacBeth Regressions (Continued)
MODEL1
MODEL2
MODEL3
Panel C. Advertising Expenses
RETt– 1
RET(–7:–2)
−0.061∗∗∗
(−11.503)
0.002
(0.879)
−0.058∗∗∗
(−11.476)
0.004∗∗
(2.198)
2.412∗∗∗
(12.908)
0.047
0.067
EIV
MKTBETA
SMBBETA
HMLBETA
PSBETA
Adj. R2
MODEL1
MODEL2
MODEL3
Panel D. Number of Shareholders
−0.062∗∗∗
(−11.876)
0.005∗∗∗
(3.118)
2.466∗∗∗
(13.467)
−0.422∗∗∗
(−4.051)
−0.372∗∗∗
(−6.089)
0.282∗∗∗
(3.779)
0.129∗
(1.888)
0.075
−0.049∗∗∗
(−11.636)
0.005∗∗
(2.245)
−0.046∗∗∗
(−11.359)
0.007∗∗∗
(3.884)
2.685∗∗∗
(14.164)
0.043
0.064
−0.049∗∗∗
(−11.706)
0.007∗∗∗
(4.690)
2.738∗∗∗
(14.745)
−0.494∗∗∗
(−4.925)
−0.275∗∗∗
(−4.952)
0.306∗∗∗
(4.404)
0.030
(0.429)
0.071
∗∗∗
∗
∗∗
is consistent with the notion that the well-known size effect documented in the literature is a
manifestation of an omitted variable problem (namely, IV).
The novel argument in our paper is that consistent with Merton’s (1987) hypothesis, there
is a difference in the pricing of IV across neglected and visible stocks. In order to test this
hypothesis, for each of the four samples, we divide the universe of stocks into quintiles based
on the investor base measure and assign a dummy variable for each quintile. We define dummy
variables to identify the differences between coefficients of IV across categories. Our base group
is the visible stocks. We allow both the intercept and the coefficient of IV to vary across groups
(by including regular dummies and interaction dummies, respectively). The results are presented
in Table VI.
The coefficient on IV in Table VI represents the pricing of IV in the control group that, in
our case, is the group of visible stocks. Our variables of interest are the interaction dummies
(I20 , I40 , I60 , and I80 ) that capture the differences in the pricing of IV across investor base
categories (relative to the visible stock category, which is the base group). The interaction dummies
are monotonically increasing in magnitude indicating that the importance of IV increases as the
visibility of the stocks decreases. The difference between the neglected groups and the visible
group (I20 and I40 ) is significant for all of the specifications in all four samples considered
supporting the argument that IV carries a greater premium among neglected stocks.
One other thing to note in Table VI is the significance of the raw dummies designed to allow the
intercept to vary across investor base categories. Since our models may not be comprehensive and
the intercept may catch any omitted variables, it is important to allow it to change across groups
(the results indicate that there may be other differences in the pricing of neglected stocks, aside
from the differential IV effect). Unexpectedly, the coefficient on the dummies has a negative sign
indicating that after controlling for IV and its interaction effects, the neglected stocks have lower
returns.
EIV
RET(–7:–2)
RETt– 1
Ln(BEME)
Ln(ME)
BETA_FF92
D20
D40
D60
D80
Intercept
MODEL2
2.876∗∗∗
(3.820)
−0.317∗
(−1.902)
−0.992∗∗∗
(−4.295)
−2.894∗∗∗
(−7.831)
−4.778∗∗∗
(−11.905)
−1.057∗∗∗
(−4.460)
−0.112∗∗
(−2.552)
0.440∗∗∗
(5.531)
−0.039∗∗∗
(−10.622)
0.009∗∗∗
(4.706)
1.308∗∗∗
(5.925)
−0.077∗
(−1.815)
0.392∗∗∗
(5.540)
−0.042∗∗∗
(−11.088)
0.009∗∗∗
(5.516)
1.344∗∗∗
(6.092)
1.656∗∗
(2.466)
−0.338∗∗
(−2.076)
−1.075∗∗∗
(−4.93)
−3.022∗∗∗
(−8.582)
−4.894∗∗∗
(−12.776)
Panel A.
Breadth Sample
MODEL1
MODEL2
−0.160
(−0.209)
0.124
(0.890)
−0.010
(−0.056)
−0.657∗∗∗
(−3.034)
−1.444∗∗∗
(−5.355)
−0.776∗∗∗
(−3.515)
0.052
(1.196)
0.296∗∗∗
(3.641)
−0.036∗∗∗
(−8.264)
0.010∗∗∗
(5.213)
1.239∗∗∗
(5.485)
0.084∗∗
(2.152)
0.240∗∗∗
(3.435)
−0.041∗∗∗
(−9.435)
0.010∗∗∗
(5.758)
1.298∗∗∗
(5.968)
−1.180∗
(−1.870)
0.129
(0.917)
0.024
(0.132)
−0.645∗∗∗
(−3.038)
−1.462∗∗∗
(−5.603)
Panel B.
Number of Analysts Sample
MODEL1
MODEL2
−1.182
(−1.352)
−0.303
(−1.220)
−0.566∗∗
(−2.003)
−1.772∗∗∗
(−5.775)
−2.898∗∗∗
(−7.79)
−0.855∗∗∗
(−4.170)
0.145∗∗∗
(3.072)
0.413∗∗∗
(5.640)
−0.060∗∗∗
(−11.919)
0.004∗∗
(2.336)
1.155∗∗∗
(4.423)
0.157∗∗∗
(3.646)
0.378∗∗∗
(5.525)
−0.064∗∗∗
(−12.241)
0.005∗∗∗
(3.275)
1.137∗∗∗
(4.508)
−1.948∗∗∗
(−2.656)
−0.319
(−1.324)
−0.620∗∗
(−2.294)
−1.842∗∗∗
(−6.304)
−2.891∗∗∗
(−8.070)
Panel C.
Ad Expense Sample
MODEL1
MODEL2
−4.117∗∗∗
(−5.765)
−0.046
(−0.28)
−0.223
(−1.263)
−0.390∗∗
(−2.065)
−0.744∗∗∗
(−3.728)
−1.157∗∗∗
(−5.309)
0.330∗∗∗
(7.972)
0.596∗∗∗
(6.988)
−0.046∗∗∗
(−11.262)
0.007∗∗∗
(3.860)
1.984∗∗∗
(8.768)
(Continued)
0.375∗∗∗
(9.981)
0.554∗∗∗
(6.939)
−0.049∗∗∗
(−11.649)
0.007∗∗∗
(4.707)
2.008∗∗∗
(9.029)
−5.585∗∗∗
(−9.821)
−0.101
(−0.625)
−0.295∗
(−1.729)
−0.470∗∗
(−2.519)
−0.830∗∗∗
(−4.187)
Panel D.
Number of Shareholders Sample
MODEL1
This table presents the time-series averages of the slopes in the cross-sectional regressions using standard Fama and MacBeth (1973) methodology. The
regressions are run using the whole universe of stocks. We include two sets of dummy variables to differentiate the effects across the investor base category.
The first set, Dn, contains regular dummies designed to allow the intercept to change across groups (the subscript n denotes the higher percentile level that
determines the cutoff for the respective group. The control group is the group of visible stocks, D20, denoting the group of neglected stocks). The second set
of dummies, In , are interaction dummies (EIV × Dn ) designed to allow the effect of idiosyncratic volatility to change across groups (same convention applies
in terms of the control group and the meaning of the subscripts). The dependent variable is excess monthly returns. The rest of independent variables are as
described in Table V. Numbers in parentheses are Newey and West (1987) adjusted t-statistics with six lags. Each panel presents one of the samples determined
by the use of the respective proxy for investor base.
Table VI. Fama-MacBeth Regressions with Dummy Variables for Investor Base Categories
19
0.074
−0.253∗∗∗
0.017
(0.104)
0.145
(0.785)
1.140∗∗∗
(4.392)
1.901∗∗∗
(8.132)
∗∗
∗
∗∗∗
Adj R2
PSBETA
HMLBETA
SMBBETA
MKTBETA
I20
I40
I60
I80
MODEL2
(−4.198)
0.312∗∗∗
(4.140)
0.049
(0.723)
0.080
0.020
(0.123)
0.180
(0.997)
1.182∗∗∗
(4.613)
1.930∗∗∗
(8.459)
−0.477∗∗∗
(−4.506)
Panel A.
Breadth Sample
MODEL1
MODEL2
0.078
−0.224∗∗∗
−0.021
(−0.116)
0.163
(0.840)
0.542∗∗∗
(2.809)
0.927∗∗∗
(4.326)
(−3.717)
0.265∗∗∗
(3.631)
0.007
(0.082)
0.087
−0.029
(−0.161)
0.137
(0.719)
0.527∗∗∗
(2.745)
0.935∗∗∗
(4.409)
−0.363∗∗∗
(−3.153)
Panel B.
Number of Analysts Sample
MODEL1
MODEL2
0.083
−0.321∗∗∗
0.234
(0.854)
0.549∗∗
(2.048)
1.393∗∗∗
(5.103)
1.944∗∗∗
(6.916)
(−5.373)
0.218∗∗∗
(3.039)
0.133∗∗
(2.042)
0.090
0.264
(0.982)
0.629∗∗
(2.425)
1.503∗∗∗
(5.655)
2.024∗∗∗
(7.403)
−0.327∗∗∗
(−3.217)
Panel C.
Ad Expense Sample
MODEL1
MODEL2
0.074
−0.255∗∗∗
0.422∗∗∗
(2.665)
0.649∗∗∗
(4.276)
0.829∗∗∗
(5.588)
0.976∗∗∗
(5.878)
(−4.644)
0.301∗∗∗
(4.444)
0.024
(0.355)
0.080
0.437∗∗∗
(2.804)
0.671∗∗∗
(4.533)
0.851∗∗∗
(5.727)
1.006∗∗∗
(6.010)
−0.459∗∗∗
(−4.698)
Panel D.
Number of Shareholders Sample
MODEL1
Table VI. Fama-MacBeth Regressions with Dummy Variables for Investor Base Categories (Continued)
20
21
Table VII. The Effect of Idiosyncratic Volatility (IV) within Investor Base
Categories
This table presents the coefficients on IV from Fama-MacBeth (1973) regressions that are run separately
within investor base quintiles. For each of the four samples, we divide the universe of stocks into quintiles
based on the respective proxy for visibility (the four proxies for visibility are described in Table I).
Fama-MacBeth (1973) characteristic regressions are run at individual stock levels within each investor base
quintile, using the following two cross-sectional models:
Model 1:
RET i,t = α + β1 BETA F F92i,t−1 + β2 Ln(ME)i,t−1 + β3 Ln(BEME)i,t−1 + β4 RET i,t−1 + β5 RET(−7 :
−2)i,t−1 + β 6 EIV i,t + εi,t .
Model 2:
RET i,t = α + β1 MKTBETAi,t−1 + β2 Ln(ME)i,t−1 + β3 Ln(BEME)i,t−1 + β4 RET i,t−1 + β5 RET(−7 :
−2)i,t−1 + β 6 EIV i,t + β7 SMBBETAi,t−1 + β8 HMLBETAi,t−1 + β9 PSBETAi,t−1 + εi,t .
The variable descriptions are in Table V. The table presents the time-series average of the coefficient on
EIV (β6 ). The last column presents the differences in the coefficients of IV between the neglected stocks
and the visible stocks (the time-series distribution of β6 coefficients obtained in each category is used to
test for statistical significance). Numbers in parentheses are Newey and West (1987) adjusted t-statistics
with six lags. Each panel presents the results in the sample determined by the use of the respective proxy
for investor base.
Investor Base Quintiles
1 (Neglected)
2
3
4
5 (Visible) Neglected – Visible
Panel A. Breadth of Institutional Ownership Sample
β 6 : Model 1
β 6 : Model 2
3.408∗∗∗
(15.239)
3.470∗∗∗
(15.565)
2.598∗∗∗
(9.807)
2.649∗∗∗
(10.024)
1.416∗∗∗
(7.245)
1.488∗∗∗
(8.013)
1.041∗∗∗
(5.481)
1.099∗∗∗
(6.067)
0.800∗∗∗
(3.719)
0.873∗∗∗
(4.517)
2.608∗∗∗
(11.112)
2.597∗∗∗
(11.338)
Panel B. Advertising Expenses Sample
β 5 : Model 1
β 5 : Model 2
∗∗∗
3.204
(14.632)
3.264∗∗∗
(15.159)
2.678∗∗∗
(10.583)
2.732∗∗∗
(10.979)
1.739∗∗∗
(7.662)
1.792∗∗∗
(7.865)
1.344∗∗∗
(5.703)
1.406∗∗∗
(6.065)
0.921∗∗∗
(3.675)
0.928∗∗∗
(3.785)
2.283∗∗∗
(8.371)
2.336∗∗∗
(8.349)
β 5 : Model 1
β 5 : Model 2
∗∗∗
2.398
(10.462)
2.485∗∗∗
(10.859)
1.870∗∗∗
(8.157)
1.923∗∗∗
(8.418)
1.332∗∗∗
(6.76)
1.335∗∗∗
(7.014)
0.960∗∗∗
(4.531)
1.002∗∗∗
(5.053)
0.922∗∗∗
(4.027)
0.919∗∗∗
(4.339)
1.476∗∗∗
(6.527)
1.566∗∗∗
(7.192)
β 5 : Model 1
β 5 : Model 2
∗∗∗
∗∗∗
3.029
(14.889)
3.090∗∗∗
(15.211)
∗
∗∗
2.900∗∗∗
2.721∗∗∗
2.405∗∗∗
(13.380)
(12.905)
(10.961)
2.944∗∗∗
2.779∗∗∗
2.483∗∗∗
(13.800)
(13.392)
(11.451)
1.570∗∗∗
(6.729)
1.626∗∗∗
(7.357)
1.459∗∗∗
(7.492)
1.464∗∗∗
(7.527)
22
Table VIII. The Effect of Idiosyncratic Volatility (IV) within Investor Base
Categories: Alternative IV Measure
The table presents the coefficients on IV from Fama-MacBeth (1973) regressions that are run separately
within investor base quintiles, using the contemporaneous Ang et al. (2006) realized measure of IV
multiplied by 10. For each of the four samples, we divide the universe of stocks into quintiles based on the
respective proxy for visibility (the four proxies for visibility are described in Table I). We repeat the FamaMacBeth (1973) characteristic regressions described in Table VII using the alternative measure of IV and
present the coefficients of this variable within each investor base category (and for each respective sample).
Model 1:
RET i,t = α + β1 BETA F F92i,t−1 + β2 Ln(ME)i,t−1 + β3 Ln(BEME)i,t−1 + β4 RET i,t−1 + β5 RET(−7 :
−2)i,t−1 + β 6 IV i,t + εi,t .
Model 2:
RET i,t = α + β1 MKTBETAi,t−1 + β2 Ln(ME)i,t−1 + β3 Ln(BEME)i,t−1 + β4 RET i,t−1 + β5 RET(−7 :
−2)i,t−1 + β 6 IV i,t + β7 SMBBETAi,t−1 + β8 HMLBETAi,t−1 + β9 PSBETAi,t−1 + εi,t .
Numbers in parentheses are Newey and West (1987) adjusted t-statistics with six lags. Each panel presents
the results in the sample determined by the use of the respective proxy for investor base.
Investor Base Quintiles
1 (Neglected)
2
3
4
5 (Visible) Neglected – Visible
Panel A. Breadth of Institutional Ownership Sample
β 6 : Model 1
β 6 : Model 2
0.036∗∗∗
(11.435)
0.036∗∗∗
(11.391)
0.029∗∗∗
(7.670)
0.030∗∗∗
(7.764)
0.021∗∗∗ 0.014∗∗∗
(6.023)
(3.475)
0.021∗∗∗ 0.014∗∗∗
(6.243)
(3.626)
0.008∗
(1.668)
0.007∗
(1.684)
0.029∗∗∗
(6.595)
0.029∗∗∗
(6.791)
Panel B. Advertising Expenses Sample
β 6 : Model 1
β 6 : Model 2
∗∗∗
0.038
(12.323)
0.039∗∗∗
(12.527)
0.031∗∗∗
(9.506)
0.032∗∗∗
(9.64)
0.022∗∗∗ 0.020∗∗∗
(6.601)
(5.721)
0.023∗∗∗ 0.020∗∗∗
(6.878)
(5.844)
0.013∗∗∗
(3.372)
0.012∗∗∗
(3.087)
0.025∗∗∗
(7.394)
0.027∗∗∗
(7.651)
β 6 : Model 1
β 6 : Model 2
∗∗∗
0.035
(9.139)
0.036∗∗∗
(9.276)
0.026∗∗∗
(6.727)
0.027∗∗∗
(6.871)
0.021∗∗∗ 0.011∗∗∗
(5.233)
(2.952)
0.022∗∗∗ 0.011∗∗∗
(5.387)
(3.047)
0.007
(1.496)
0.007
(1.603)
0.028∗∗∗
(6.824)
0.029∗∗∗
(7.089)
β 6 : Model 1
β 6 : Model 2
∗∗∗
∗∗∗
0.035
(10.549)
0.036∗∗∗
(10.574)
∗
∗∗
0.035∗∗∗
0.032∗∗∗ 0.030∗∗∗
(11.809)
(10.841)
(9.246)
0.036∗∗∗
0.033∗∗∗ 0.030∗∗∗
(11.817)
(10.956)
(9.478)
0.021∗∗∗
(6.941)
0.022∗∗∗
(7.249)
0.014∗∗∗
(5.802)
0.013∗∗∗
(5.634)
23
A potential caveat for the results in Table VI is that while the coefficient of IV and the intercept
are allowed to vary across investor base categories, the coefficients of the other control variables
are not. To the extent that there are any other differential effects at play between the visible
and neglected stocks, this may influence our results (and may be responsible for the unexpected
negative sign on the intercepts). In order to account for that, we run separate Fama-MacBeth
(1973) regressions within each category of investor base (allowing every coefficient to vary
across groups) and use the time-series distribution of IV coefficients to determine whether there
are any significant differences across the neglected and visible stocks categories. Table VII
presents the coefficients on our measure of IV from two alternative cross-sectional models for
returns within each one of our four samples.
The results from Table VII indicate that regardless of the sample or the model employed, the
coefficients of IV decrease monotonically across the investor base categories. More specifically,
the coefficient of IV for the neglected stocks is significantly higher than the coefficient for the
visible stocks categories. The last column in Table VII examines whether the coefficient on
IV within the neglected stocks category is significantly larger than the coefficient within the
visible stocks category. The results indicate that neglected stocks have significantly higher IV
coefficients than their visible counterparts across the board.
To summarize, the empirical results in Tables VI and VII strongly support the notion that the
premium per unit of IV is greater for neglected stocks than for visible stocks. Our previous results
(see Table II) also find that the magnitude of IV for neglected stocks is greater, on average, than
that of the more visible stocks. Taken together, we conclude that the IV premium (amount of
risk × pricing of risk) is larger for neglected stocks, which is consistent with the predictions of
Merton’s (1987) theoretical model.
E. Robustness Checks
We subject our results to a battery of robustness checks to determine whether our conclusions
are driven by particular assumptions we made in proxy formation or our choice of methodology.
For example, we verified that our results are not dependent on alternative weighing schemes
(Bali and Cakici, 2008), the use of raw versus abnormal returns, or the use of alternative control
variables (untabulated). One main concern is whether our conclusions are driven by our proxy for
IV, which has been subject to some debate in the literature. To alleviate this concern, we repeat
our tests using the realized volatility measure from Ang et al. (2006). We investigate whether the
coefficients on this measure vary significantly with the visibility of the stocks (i.e., we repeat
our main tests from Table VII using this alternative measure of IV). The results are presented in
Table VIII.
The results obtained using realized volatility support our conclusion that the return per unit
of IV is greater for neglected stocks than it is for visible stocks across the alternative investor
recognition samples.
Another interesting question is whether the impact of the incomplete information effect on the
IV-expected return relation has significantly changed through time. It seems likely that advances
in information technology have reduced the cost of following all stocks and, as such, has reduced
the market segmentation between the most visible and most neglected stocks. Thus, it seems
likely that the average investor base has increased over time. Moreover, the average firm size has
also increased in recent years. Finally, there is some evidence that average IV has also increased.12
12
There is some debate in the literature regarding the trend of the average idiosyncratic risk over time. For example,
Campbell et al. (2001) argue that there is an upward trend whereas Bekaert, Hodrick, and Zhang (2012) find no such
evidence after extending the earlier sample.
24
Since the shadow cost of incomplete information as described by Merton (1987) Equation (1) is
increasing in firm size and IV, but decreasing in investor base, these trends may have offsetting
effects on the IV risk premium. Consequently, the net effect over time is not obvious. Our
(untabulated) analysis supports the conclusion that the offsetting effects roughly neutralize each
other, resulting in a fairly stable IV risk premium over the full sample period.
IV. Conclusion
We conduct a comprehensive investigation of the role of IV in the cross-section of returns with
a primary focus on the interaction between the breadth of investor base and the pricing of IV. Most
of the IV literature citing Merton’s (1987) model of market segmentation omits this interaction
effect and begins from the premise that the model unequivocally predicts a positive correlation
between IV and returns. We argue that this is not the case and that focusing on the influence
that the shareholder base may have on the relationship between IV and returns can empirically
differentiate between the predictions of the model proposed by Merton (1987) and other models
proposed in the literature.
Using an EGARCH-M methodology to estimate IV and employing four different proxies for
investor base (breadth of institutional ownership, analyst coverage, the number of shareholders,
and advertising expense), we find strong support for the theoretical model proposed by Merton
(1987). Specifically, our results consistently demonstrate a significant interaction between IV and
investor base. IV risk premia are larger for neglected stocks and smaller, or even economically
insignificant, for more visible stocks. This finding arises from the interaction of two effects
that reinforce each other. First, as repeatedly noted in the extant literature, neglected stocks are
characterized by greater IV than more visible stocks. In addition, consistent with Merton (1987)
(and this is the primary contribution of our paper), we find that the price per unit of IV risk is
greater for neglected stocks than for those stocks characterized by broader investor bases. Since
the total IV risk premium is the product of the quantity of risk and its price, these two effects
combine to generate the significantly greater IV risk premia that we find. Finally, we find two
other results that are consistent with Merton’s (1987) predictions. Once we control for IV, we
find that larger stocks are characterized by larger returns. In addition, when accounting for the
incomplete information effect, we observe a negative correlation between betas and returns. The
first result is consistent with the hypothesis that the well-documented size effect is actually a
manifestation of an omitted variable (IV). The second result speaks to the theoretical proof that
once the common information assumption is relaxed, the market beta/return relation is a function
of the degree of incomplete information. If this information effect is strong enough, Merton’s
(1987) model allows for a negative market beta/return relation. Our empirical results suggest that
the incomplete information effect is severe.
Our results are the first evidence based on US stock market data validating Merton’s (1987)
equilibrium model of asset pricing in the presence of incomplete information. We find support for
the major implications of his model. The price of IV risk varies with the investor base. The IV risk
premium is a function of investor base, firm size, and the magnitude of IV. The market beta/return
relation can be negative (i.e., firms with greater exposure to market risk earn lower returns) and,
after controlling for IV, the size effect is positive (i.e., larger firms earn larger returns). One
implication of these findings is that the market segmentation induced by incomplete information
is strong enough to be, at least partially, responsible for the documented relation between IV and
returns.
25
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Idiosyncratic Risk, Investor Base, and Returns

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