Belief Risk and the Cross-Section of Stock

Returns∗

Rajna Gibson Brandon† and Songtao Wang‡

March 26, 2015

Abstract

We examine whether belief risk arising from stochastic fluctuations in the average

belief of investors is priced. We construct a market-wide belief measure with analysts’

EPS forecast data and an EPS forecasting model. Market-wide belief primarily cap-

tures the average subjective opinion of institutional investors and represents a source

of commonality in stock returns. The average return on stocks with high exposure to

belief risk is 6.35%/year higher than that of stocks with low exposure. This positive

∗We thank Yakov Amihud, Tim Baldenius, Michael Brennan, Stephen Brown, Francois Degeorge, Jerome

Detemple, and Jeffrey Wurgler for helpful comments and discussions, as well as participants of the 2013

NFA Annual Conference and the finance seminar at Shanghai Jiao Tong University. The financial support of

the Swiss National Science Foundation and the NCCR-Finrisk Project C1 “Credit Risk and Non-Standard

Sources of Risk in Finance” is greatly acknowledged. All errors are ours.†Rajna Gibson Brandon is the Swiss Finance Institute (SFI) Chaired Professor of Finance at the Geneva

Finance Research Institute, University of Geneva, Geneva, Switzerland. Email: rajna.gibson@unige.ch‡Songtao Wang is Assistant Professor at the Antai College of Economics and Management, Shanghai Jiao

Tong University, Shanghai, China. Email: wangsongtao@sjtu.edu.cn

1

relation between belief risk and expected stock returns holds after accounting for tra-

ditional risk factors and is prominent among large-cap stocks.

Keywords: Analysts’ EPS Forecasts; Asset pricing; Belief Risk; Commonality.

JEL codes: G11; G12.

2

I Introduction

Heterogeneity in investor beliefs plays an important role in explaining the formation of stock

prices and it is well known that asset pricing models incorporating such heterogeneity are

able to better account for stylized facts characterizing stock returns and to rationalize the

existence of trading volume and of stock market bubbles and crashes. So far, the dominant

stream of the literature on heterogeneous beliefs has been studying the impact of the disper-

sion in investor beliefs and its ability to overcome the empirically documented limitations of

representative agent pricing models1. While individuals, whether they are investors, portfo-

lio managers, financial analysts or economists, are known to disagree, it is also interesting

to observe that financial markets thrive at conforming and relying on consensus forecasts,

for instance, to extract information regarding market pessimism or optimism for trading

and portfolio allocation purposes. This explains the popularity of various retail investors’

sentiment measures developed both in academic (described in the literature review section

below) and in business settings to guide investors buying and selling decisions. Hence, the

first moment of the distribution of investors’ heterogenous beliefs seems to matter as well.

Motivated by this last observation, another less explored stream of the literature on het-

erogeneous beliefs theoretically examines the impact of the average belief of investors on

the formation of asset prices. In particular, Jouini and Napp (2007) develop a model in

which investors possess heterogeneous beliefs about the growth rate of aggregate wealth.

They show that, in the heterogeneous beliefs setting, the equilibrium asset price positively

1The role of the dispersion in investor beliefs has been studied in particular by Harrison and Kreps

(1978), Varian (1985), Harris and Raviv (1993), Detemple and Murthy (1994), Zapatero (1998), Basak

(2000), Scheinkman and Xiong (2003), Buraschi and Jiltsov (2006), Li (2007), Pavlova and Rigobon (2007),

Dumas, Kurshev, and Uppal (2009), Xiong and Yan (2010).

3

depends on the average belief defined as the risk tolerance weighted average of the individual

beliefs when investors are cautious: an optimistic average belief increases the equilibrium

risky asset price. Likewise, Kurz and Motolese (2011) derive a very similar result, namely

that the price of a risky asset is a linear function of the equally-weighted average of investors’

beliefs about the asset’s payoffs, with a positive beta coefficient for the average belief factor.

When average or consensus beliefs about the prospects of financial markets are shown

to matter and positively affect stock prices, it seems worthwhile to explore what happens

when unexpected shocks alter those average beliefs and, in particular, whether they depress

stock prices in a similar way as unexpected liquidity shocks do. Yet, little is currently known

regarding the impact of unexpected average belief shocks on stock returns. Let us define the

market-wide belief measure, and call it the “market-wide belief”, as the cross-sectional mean

of the average beliefs of investors about the future earnings of all risky stocks. One could

view this belief measure as representing the average level of optimism or pessimism currently

held by market participants regarding the short term earnings prospects of a representative

stock. If the market-wide belief measure represents an omitted source of commonality, will its

innovations depress stock prices? This main question lies at the core of our study. To answer

this question, we first show that investors’ beliefs and thus the market-wide belief evolve

stochastically over time. In Kurz and Motolese’s (2011) model, individual investors’ beliefs

about future asset payoffs are assumed to follow a stochastic AR(1) process. Along the same

vein, De Long et al. (1990) assume that noise traders’ expectations regarding asset returns

are subject to the impact of not fully predictable and stochastic sentiment. Empirically,

this conjecture is also supported by the fact that other sentiment indices developed by,

for example, Baker and Wurgler (2006, 2007) and Barone-Adesi et al. (2013) fluctuate

stochastically over time. Second, we document that the market-wide belief represents a

4

source of commonality affecting individual stocks’ average beliefs.

The finding that the market-wide belief is a random source of commonality motivates

us next to posit and test our main hypothesis, namely whether market-wide belief risk,

thereafter referred to as “belief risk” arising from stochastic fluctuations in the market-wide

belief, is priced in stock returns:

The Belief Risk Hypothesis: Stocks with Higher Exposure

to Belief Risk Earn a Higher Expected Return.

This empirical study focuses on the U.S. stock market and relies on the actual EPS and

analyst EPS forecast data provided by I/B/E/S (Institutional Brokers’ Estimate System) to

construct a market-wide belief measure. First, we adopt the econometric model proposed

by Brown and Rozeff (1979) to objectively forecast each stock’s EPS, and then compute the

average belief of investors for a stock as the mean analyst EPS forecast minus the one derived

from the Brown and Rozeff (1979) model. The market-wide belief is defined as the cross-

sectional mean of the price-scaled average beliefs across all sample stocks. Innovations in the

market-wide belief are then estimated as the residuals of an AR(6) model that simultaneously

eliminates the macro-economic components from the market-wide belief’s evolution. To test

our main hypothesis, we finally form portfolios based on the sensitivity of each stock’s excess

returns to innovations in the market-wide belief.

Our main findings can be summarized as follows: the average return delivered by stocks

with high exposure to belief risk is significantly higher than that of stocks with low exposure,

this positive relation being particularly strong in magnitude for large-cap and low to middle

book-to-market stocks. An investment strategy that is long in stocks with high exposure to

belief risk and short in stocks with low exposure to belief risk yields a significant alpha of

5

6.23%/year with the Fama and French (1993) model and 6.10%/year with the Carhart (1997)

model, meaning that the traditional three- and four-factor models cannot fully explain this

pattern in stock average returns. We also show that accounting for liquidity risk by using

the Pastor and Stambaugh (2003) liquidity risk factor does not meaningfully affect these

abnormal returns. These results are robust to: i) an alternative EPS forecasting model; ii)

value-weighted stock portfolios; iii) accounting for divergence of analysts’ forecasts; iv) a sub-

sample analysis. Finally, we examine the determinants of a stock’s exposure to belief risk and

find that it increases with the stock’s market beta, volatility, turnover rate, and sale-to-asset

ratio and decreases with size, book-to-market, momentum, and analyst coverage.

This paper provides several contributions to the growing literature on the impact of

investors’ heterogeneous beliefs on stock returns:

First, the results show that stochastic fluctuations in the market-wide belief is a priced

source of risk distinct from other sources of systematic risk accounted for by standard asset

pricing models. The issue of whether the risk arising from these stochastic fluctuations in

the market-wide belief is cross-sectionally priced in stock returns, although economically

important, has so far been neglected in the empirical asset pricing literature and this study

attempts to bridge that gap.

Second, this paper provides yet another potential explanation for the equity premium

puzzle documented first by Mehra and Prescott (1985): part of the excess equity premium

may represent a compensation for investors who have to bear systematic belief risk.

Finally, we show that the market-wide belief measure that we construct primarily captures

the average subjective opinions of an important category of investors, namely institutional

investors. So far, most sentiment measures, as for instance, the closed-end fund discount (Lee

6

et al., 1991) or the Baker and Wurgler (2006) sentiment index, were designed to reflect small

investors’ sentiment. O’Brien and Bhushan (1990) conjecture that sell-side research analysts

act as information intermediaries for institutional investors. Moreover, studies such as Brown

et al. (2012), Chen and Cheng (2006), Costelle and Hall (2011), Fang and Kosowki (2007),

Franck and Kerl (2013), and Malmendier and Shanthikumar (2009) show that institutional

investors rely on the information provided by research analysts to make their investment

decisions. Particularly, both Franck and Kerl (2013) and Malmendier and Shanthikumar

(2009) document a positive correlation between changes in institutional investors’ equity

holdings and changes in analysts’ EPS forecasts. Based on these studies, it is reasonable to

conjecture that the market-wide belief measure that we construct using the actual EPS and

analyst EPS forecast data captures the average subjective opinions of institutional investors

rather than retail investors. Thus, it is not surprising to observe that the market-wide belief

measure is only weakly correlated with other popular sentiment indices.

II Literature review

While an abundant literature has examined the effect of the dispersion in investors’ subjective

beliefs on stock prices, the focus on the impact of the average investors’ beliefs remains quite

limited.

Jouini and Napp (2007) show that the introduction of heterogeneous beliefs in an other-

wise standard competitive complete market economy has two distinct effects: the first effect

is associated with a change of the objective expectation to the aggregate belief defined as a

weighted average of the individual subjective beliefs, and the second effect is represented by

a discount factor proportional to the belief dispersion. In their heterogeneous beliefs setting,

7

the equilibrium asset price is increasing in the aggregate belief about the growth rate of

aggregate wealth when investors are cautious while the discount factor lowers the risky asset

price.

Kurz and Motolese (2011) develop a model in which investors differ in their beliefs about

future asset payoffs. By assuming that individual investors’ beliefs follow a stochastic AR(1)

process, they show that the equilibrium asset price is positively related to the average belief

of investors about future prospects of asset payoffs.

On the empirical side, Diether et al. (2002) show a negative cross-sectional relation be-

tween the dispersion in investor beliefs approximated by the disagreement among analysts’

earnings forecasts and expected stock returns, supporting Miller’s (1977) view that the dis-

persion in investor beliefs is priced at a premium in the presence of short-sale constraints.

Using the diversity in analysts’ forecasts measure of BKLS (1998), however, Doukas et al.

(2006) obtain an opposite result, and their finding is consistent with the predictions of mod-

els of Williams (1977), Mayshar (1983), and Epstein and Wang (1994) who posit that the

dispersion in investor beliefs is a priced source of risk. Anderson et al. (2005) provide further

evidence that heterogeneity in investor beliefs is a priced risk factor and show that incor-

porating investors’ beliefs can improve the performance of traditional asset pricing models.

What makes our paper different from these empirical studies is that we examine the im-

pact of the average belief of investors (i.e. the first moment of the distribution of investors’

heterogeneous beliefs) while those authors instead explore the impact of the dispersion in

investor beliefs (i.e. the second moment of the belief distribution).

Baker and Wurgler (2006) study how the aggregate sentiment of investors affects the

cross-section of stock returns and find that the cross-section of future stock returns is condi-

8

tional on beginning-of-period investor sentiment. When sentiment is estimated to be high,

stocks that are attractive to optimists and speculators and at the same time unattractive to

arbitrageurs - small stocks, younger stocks, growth stocks, unprofitable stocks, non-dividend

paying stocks, high volatility stocks, and distressed stocks - tend to earn relatively low

subsequent returns. Conditional on low sentiment, however, these cross-sectional patterns

disappear. The difference between Baker and Wurgler (2006) and our study is twofold: first,

as will be seen in Section VI.A, the sentiment index developed by Baker and Wurgler (2006)

captures the aggregate opinion of retail investors while our market-wide belief measure pri-

marily reflects institutional investors’ opinions; secondly, and more importantly, Baker and

Wurgler (2006) examine the cross-sectional predictability of stock returns conditional on

investor sentiment, while our aim is to study whether the risk associated with stochastic

fluctuations in the average belief of institutional investors is priced.

Ben-Rephael et al. (2012) also examine the relation between investor sentiment and

stock returns, but at the market level. Using the aggregate net flows from bond funds to

equity funds in the USA as a proxy for investor sentiment, they document a significantly

positive contemporaneous relation between monthly aggregate net flows to equity funds and

stock market excess returns and that about 85% of these price changes are reversed within

four months while the rest is reversed within ten months. Edelen and Warner (2001) and

Goetzmann and Massa (2003) obtain similar findings using higher frequency (daily) data.

Our study also relates to Lee et al. (1991) who show that stocks and closed-end funds

with high sensitivity to investor sentiment earn an extra return as a compensation for this

additional source of risk. In contrast to Lee et al. (1991), we examine a different measure

of investors’ beliefs that primarily captures the average opinion of institutional investors.

9

Furthermore, the aim of Lee et al. (1991) is to solve the closed-end fund puzzle, an issue left

unexplored in our paper.

III Data

In light of the difficulties raised by collecting data on investors’ direct opinions, we will use

analysts’ forecasts as a proxy for institutional investors’ opinions.

The analyst forecast data are taken from the Institutional Brokers’ Estimate System

(I/B/E/S) Summary History database that contains the summary statistics for analysts’

forecasts and the date when the forecast was last confirmed to be accurate. These data are

usually disclosed on the third Tuesday of each month.2

I/B/E/S collects two categories of analyst forecast data: one concerns EPS (Earnings

Per Share) and another concerns DPS (Dividends Per Share). DPS is sensitive to a firm’s

dividend payout policy whose impact is not easy to control for in empirical studies. More

importantly, the analyst DPS forecast data only have a short history and the analyst coverage

for DPS forecasts is also low. For these reasons, we use the analyst EPS forecast data in the

following empirical analysis.3

To construct the market-wide belief measure, we also need the actual EPS data. The

actual EPS data provided by I/B/E/S are called the ‘Street’ EPS since they are tracked

by financial analysts and followed by investors. COMPUSTAT provides the data of another

category of actual EPS known as the GAAP EPS reported in firms’ financial statements.

2Diether et al. (2002) provide a detailed description of the I/B/E/S database3If the payout ratios of firms are stable over time, the empirical results obtained with either the EPS or

the DPS forecasts should be similar.

10

Bradshaw and Sloan (2002) find that there exists a large and growing gap between the

‘Street’ EPS data and the GAAP EPS data since the former excludes cost items such as

‘non-recurring’ and ‘no-cash’ charges.4

The ‘Street’ EPS data are quantitatively consistent with analysts’ EPS forecasts and

hence used to construct the market-wide belief measure although the GAAP EPS data have

a longer history. The actual EPS and analyst EPS forecast data reported by I/B/E/S have

different periodicities: quarterly, semi-annually, annually, etc. This study uses the quarterly

EPS data for the following reasons: first, the analyst coverage for quarterly EPS forecasts

is higher (thus reflecting the opinions of a broader community of analysts and investors);

second, in the accounting literature, the econometric models developed to forecast earnings

are mainly intended for quarterly EPS.

Stocks used to construct the market-wide belief measure are those with fiscal quarters

ending in the months of March, June, September, and December since the majority of stocks

in financial markets belong to this category. To be included in the construction of the

market-wide belief measure, stocks should also meet other two criteria: i) have no less than

30 consecutive observations of quarterly EPS during the period March 1983 through June

2009; ii) have the analyst EPS forecast and the model-implied EPS forecast for at least one

quarter during the period August 1990 through August 2009.

Stock data such as prices, returns, trading volumes, the number of outstanding shares,

etc. are collected from the Center for Research in Securities Prices (CRSP) Monthly Stocks

Combined File that includes stocks traded on NYSE, AMEX, and Nasdaq. Only ordinary

common shares (with CRSP share code 10 or 11) are used in this study. In addition, to

4The difference between the ‘Street’ and GAAP earnings has been discussed in Ciccone (2002), Cote and

Qi (2005), and Zhang and Zheng (2011)

11

be included in the portfolio performance analysis below, stocks should have more than 24

quarters of return observations during the period August 1991 through August 2009.5 The

accounting data and, in particular, the book values of stocks’equity, the asset values, the debt

values, the dividends, and the sales are from the COMPUSTAT-CRSP merged database.

IV Empirical methodology

In this section, we first explain how to construct the market-wide belief measure using the

actual EPS and analyst EPS forecast data. We then show that it is a source of commonality

and abstract from its predictable and macro-economic components to construct the belief

risk factor. We also discuss the accuracy of the market-wide belief measure. Finally, we

show how to construct portfolios to test the belief risk hypothesis.

A Econometric EPS forecasting models

In the accounting literature, earnings forecasting is an important research topic, and many

models have been developed to undertake this task. In this study, the benchmark model

used to forecast quarterly EPS is the linear time-series model proposed by Brown and Rozeff

(henceforth BR) in 1979, which takes the following form:

E(Qs) = δ +Qs−4 + φ(Qs−1 −Qs−5) + θεs−4 (1)

where Qs−k is the EPS for quarter s− k and εs−4 is the EPS shock experienced over quarter

s− 4. Typically, the trend term δ and the coefficient φ are both positive, and the coefficient

5The sample period for the EPS data is longer because more historical data are needed for forecasting

quarterly EPS.

12

θ is negative. An advantage of the BR model is that it contains an autoregressive compo-

nent Qs−1−Qs−5 reflecting the positive autocorrelations in seasonal differences of quarterly

earnings at the first three lags and a moving average component εs−4 reflecting the negative

autocorrelation at the fourth lag.6 Moreover, the BR model also captures the seasonality

characteristics in quarterly earnings data. The main reason behind the choice of the BR

model is that, as Bathke and Lorek (1984) and Callen et al. (1996) show, it yields better

earnings forecasts than other linear time-series models and neural network models despite

the fact that quarterly earnings data are financial, seasonal, and non-linear.

For robustness purposes, we will also use the Seasonal Random Walk with Drift (hence-

forth SRWD) model to forecast quarterly EPS:

E(Qs) = δ +Qs−4 (2)

Despite its simplicity, the SRWD model has been often used in previous studies such as Sadka

(2006) and Konchitchki et al. (2013) who use it to estimate unexpected earnings shocks.

In both cases, for each stock, the forecast of the one-quarter ahead EPS is derived using

the coefficients estimated with 30 quarters of actual EPS observations.

B Market belief measure

Let Ei,jt (EPSs) denote investor j’s forecast of the EPS of stock i for quarter s conditional on

the information available up to time t and Ei,mt (EPSs) denote the forecast derived from an

econometric model, where t can be any time after the EPS for quarter s − 1 is known and

before the EPS for quarter s is publicly disclosed. Investor j’s belief gi,jt about the EPS of

6Griffin (1977) and Foster (1977) document the existence of these autocorrelations in seasonal differences

of quarterly earnings at the first four lags.

13

stock i for quarter s is defined as the difference between Ei,jt (EPSs) and Ei,m

t (EPSs):7

gi,jt = Ei,jt (EPSs)− Ei,m

t (EPSs) (3)

A positive gi,jt implies that investor j is optimistic relative to an econometrician about the

EPS of stock i for quarter s. The average of individual beliefs among investors, denoted by

Zit, is equal to:

Zit =

1

M

M∑j=1

gi,jt

=1

M

M∑j=1

[Ei,j

t (EPSs)− Ei,mt (EPSs)

]= E

i

t(EPSs)− Ei,mt (EPSs) (4)

where M is the number of investors for stock i and Ei

t (EPSs) is the average forecast of

investors.8 Even if provided with the same set of information, investors may still form

distinct beliefs about future EPS since they treat the information in different ways, and Zit

reflects the average belief of the M investors: the higher Zit, the more optimistic the investors.

We use the average of analysts’ EPS forecasts provided by I/B/E/S as a proxy for Ei

t(EPSs),

and Ei,mt (EPSs) is estimated with the time-series models proposed in Section IV. A.

For stocks with fiscal quarters ending in March, June, September, and December, the

actual EPS are released respectively in the second half of April, July, October, and January.

Analysts’ EPS forecasts are usually disclosed in the middle of each month. For a stock,

as time moves towards next quarter’s EPS release date, analysts’ forecasts will gradually

contain more public information about next quarter’s actual EPS of the stock so that Zit

7Jouini and Napp (2007) and Kurz and Motolese (2011) define an investor’s belief in a similar way.8In Jouini and Napp (2007) and Xiong and Yan (2010), Zi

t is the risk tolerance or wealth-weighted average

belief. However, data on the weights of individual risk tolerances and wealth are empirically difficult to collect

and compute. Thus, we rely on the equally-weighted average belief.

14

constructed with those forecast data is more likely to reflect objective information rather

than analysts’ subjective judgment. For this reason, we only use in this study the analyst

EPS forecast data disclosed in February, May, August, and November, that is, when analysts

possess the least information about next quarter’s EPS. This procedure shall enable us to

focus on studying the impact of the most subjective opinions regarding stocks’ quarterly

EPS.

To enable comparison across stocks, we scale each Zit by the stock price observed at the

end of previous month Pit−1. We define the market-wide belief, denoted by Zm

t , as the

cross-sectional average of the price-scaled Zit for all stocks in the sample:

Zmt =

1

N

N∑i=1

Zit

P it−1

(5)

where N is the number of sample stocks.9 By definition, Zmt can be interpreted as a measure

of the average belief of investors about the earnings of a stock representative of the overall

economy, a positive Zmt indicating that investors are optimistic. It is worth noting that Zm

t

only captures investors’ average subjective belief about short-term earnings.

INSERT FIGURE 1

The top graphs in Fig. 1 plot the time series of Zmt constructed with the BR and

SRWD models during the period August 1990 through August 2009. The market-wide belief

fluctuates over time and declines sharply during economic recession periods such as the

9The number of stocks used to construct Zmt varies from 602 to 1629, with an increasing trend over time

during the sample period due to the fact that more stocks have been covered by analysts. In order to avoid

the effect of outliers, we exclude the top and bottom 5% of values of Zit in the construction of Zm

t . Another

robust measure of market belief is the median of Zit, and empirical results obtained with that measure are

quantitatively similar and available upon request.

15

dot.com bubble burst at the beginning of this century and the 2007-09 subprime mortgage

crisis. Panel A of Table 1 reports summary statistics regarding the market-wide belief

variable. From this table, we see that investors were mostly optimistic during the sample

period and that the distribution of the market-wide belief variable is left-tailed, meaning

that investors can, as suggested by Fig. 1, sometimes also become very pessimistic.

INSERT TABLE 1

C Commonality in belief

An important assumption made in this study for developing the belief risk hypothesis is that

the movement in Zmt is a source of commonality affecting individual stocks’ average beliefs.

To test whether there is commonality in average belief among individual stocks, we regress

changes in the average belief for each individual stock on the changes in the market-wide

belief, i.e.10

CZit = αi + βi,1CZ

mt−1 + βi,2CZ

mt + βi,3CZ

mt+1 + β

′

XXi,t + εi,t (6)

where CZit is, for stock i, the change from time t− 1 to t in the average belief Zi

t, and CZmt

is the concurrent change in the cross-sectional average of the same variable (i.e. the market-

wide belief Zmt ). One lag and one lead of change in the market-wide belief are included to

capture any lagged adjustment in commonality. Xi,t is the set of changes in the following

variables: the growth rate in industrial production index, the growth rate in consumer price

index, the growth rate in employment, the federal funds rate, and the NBER economic

recession dummy that equals 1 when the economy is in a recession or 0 otherwise. These

10Only stocks with as least 30 observations of Zit are considered in the examination of commonality in

belief.

16

variables are included as control variables in the regression to ensure that the covariation

in Zit, if it exists, is not driven solely by the associated macroeconomic information used

by financial analysts to forecast EPS. In each individual regression, the dependent variable

stock is excluded when computing the market-wide belief.

Cross-sectional averages of time-series slope coefficients shown in Table 2 reveal the ex-

istence of commonality in individual stocks’ average beliefs. For example, in the BR case

and when the impact of macroeconomic information is controlled for, the average value of

the estimated coefficients βi,2’s for the contemporaneous change in the market-wide belief is

0.737 with a t-statistic of 7.23. About two-thirds of these individual βi,2’s are positive while

17.84% exceeds the 5% one-tailed critical value. The average value of βi,1’s, although small

in magnitude, is also positive and significant. As revealed by its t-statistic, the combined

contemporaneous, lag, and lead beta coefficient, labeled ‘Sum’, is highly significant. Com-

monality in average belief among individual stocks is further slightly stronger in the SRWD

case.

INSERT TABLE 2

D Belief risk factor

When forecasting quarterly EPS, besides the firm specific information contained in historical

earnings data, financial analysts may also use publicly available information about macroe-

conomic factors that drive variations in stocks’ earnings. If this is true, then Zmt is not a

pure subjective belief measure and empirical results obtained with Zmt about the pricing of

belief risk are possibly driven by the cross-sectional differences in the sensitivity of excess

stock returns to fluctuations in macroeconomic factors. As can be seen from Panel B of

17

Table 1, Zmt is indeed strongly correlated with a set of macroeconomic variables such as the

growth rate in industrial production index, the growth rate in the consumer price index,

the growth rate in employment, the federal funds rate, and the NBER economic recession

dummy and the correlations coincide with our expectations: increases in industrial produc-

tion, consumer price, employment, and the federal funds rate are accompanied by positive

market-wide belief shocks while the economic recession decreases the market-wide belief.11

Another issue preventing us from directly using Zmt in the empirical analysis is that it is

autocorrelated and thus partially predictable. Fig. 2 plots the autocorrelation and partial

autocorrelation functions of Zmt . In the BR (SRWD) case, the autocorrelations at the first

eight (three) lags are significantly positive.

INSERT FIGURE 2

To remove the macroeconomic and predictable components of Zmt , we thus run the fol-

lowing linear regression:12

Zmt = αz +

6∑i=1

ϕiZmt−i + β1IPIt + β2CPIt +

β3EMPLt + β4RATEt + β5DUMt + εz,t (7)

where Zmt−i is the lagged market-wide belief in quarter t − i, IPIt is the growth rate in

industrial production index, CPIt is the growth rate in consumer price index, EMPLt is the

growth rate in employment, RATEt is the federal funds rate, DUMt is the NBER economic

recession dummy that equals 1 when the economy is in a recession or 0 otherwise, and εz,t

11Baker and Wurgler (2006) use similar macroeconomic variables, a difference is that we also use the

federal funds rate - a factor that has been shown to strongly influence the economy.12The lag order is determined by fitting the time-series observations of the market-wide belief into an

autoregressive model.

18

is the normally distributed error term. In the following empirical analysis, we will rely on

the innovations in Zmt denoted by Bt estimated from Eq. (7) as the variable of interest and

we will call this risk source belief risk.

E Private information and biased analysts’ forecasts

Analysts’ EPS forecasts may reflect not only analysts’ subjective opinions and the public

information conveyed by historical earnings data and macroeconomic variables, but also

the private information about future EPS possessed by financial analysts. Such private

information, if it exists, will bias Zmt as a measure of investors’ average subjective belief.

Fig. 3 plots the time series of the ratios of stocks used to construct Zmt from each of the

ten size deciles (relative to all sample stocks). As can be seen from this figure, stocks from

the large size deciles account for a high total percentage, for instance, the ratio of stocks

from the top five size deciles exceeds 75% of the entire sample in each sample month.

INSERT FIGURE 3

The Securities and Exchange Commission (SEC) passed the Selective Disclosure and

Insider Trading Regulation on August 10, 2000, which prohibits the selective disclosure of

material non-public information by issuers to privileged individuals. This regulation, called

Regulation Fair Disclosure (FD), states that “when an issuer, or persons acting on its behalf,

discloses material non public information to certain enumerated persons (in general, securities

market professionals and holders of the issuers securities who may well trade on the basis

of the information), it must make public disclosure of that information” (SEC 2000). The

public disclosure should be made “simultaneously” for an intentional selective disclosure and

19

“promptly” for a non intentional selective disclosure by filling out Form 8-K or through any

other medium capable of mass and unbiased distribution (SEC 2000).13

The two facts discussed above should mitigate the impact of private information on

Zmt . First, private information is much less of a concern for large-cap stocks. Second, under

regulation FD, the probability that financial analysts acts on the basis of private information

about future EPS is substantially reduced. Consequently, the private information component

of Zmt should be almost negligible and Zm

t can be considered as a reliable measure of investors’

subjective average belief.

There is evidence in the literature that analysts issue systematically biased EPS forecasts.

De Bondt and Thaler (1985, 1987, 1990), LaPorta (1996), Dechow and Sloan (1997), Capstaff

et al. (1998), and Brown (2001) show that analysts are usually optimistic about annual and

long-term EPS forecasts, and O’Brien (1988) and Matsumoto (2002) show that analysts

become slightly pessimistic as the forecasting horizon declines. However, this empirical

analysis relies on innovations in the market-wide belief, which, by definition, are invariant

to the impact of a persistent bias in analysts’ EPS forecasts.

F Belief risk sensitivity based stock portfolios

For each stock, we run the following regression:

ri,t − rf,t = αi + βi,mMKTt + βi,BBt + εi,t ∀i (8)

where ri,t is the return of stock i, rf,t is the 1-month risk-free interest rate, MKTt is the

excess market return, and Bt is the belief risk factor. The coefficient βi,B, called belief

13Irani and Karamanou (2003) provide a detailed discussion of ‘Regulation Fair Disclosure’ and its impact

on analysts’ earnings forecasts.

20

beta, measures the sensitivity of stock i’s excess returns to innovations in the market-wide

belief (i.e., stock i’s exposure to belief risk). MKTt is included as a control variable in the

regression so that the empirical results about the pricing of belief risk obtained are not driven

by stocks’ exposures to pure stock market risk. At the beginning of each month of March,

June, September, and December during the period December 1997 through September 2009,

stocks are sorted into five equal portfolios based on their β̂B estimated with observations

in the preceding 24 quarters: stocks with β̂B in the first quintile are sorted into the first

portfolio, stocks with β̂B in the second quintile are sorted into the second portfolio, and so

forth.14 Portfolios are held for three months, and the portfolio return is calculated as the

equally-weighted average of the returns of all stocks held in the portfolio.15

V Empirical results

In this section, we first present the main empirical results on the cross-sectional effect of

belief risk on stock returns, and then conduct various robustness tests to provide further

support on the pricing of belief risk.

A Main results

Table 3 reports summary statistics of monthly returns delivered by the portfolios formed on

belief betas: minimum, maximum, mean, standard deviation, skewness, and kurtosis. The

14The belief beta in Eq. (8) is estimated with the prior 24 quarters of the market belief innovations

data that are available for the time period between February 1991 and November 2009, this means that

the estimated belief beta is available starting in December 1996, the date as of which we can form stock

portfolios.15The results obtained with winsorized stock returns are similar, and they are available upon request.

21

returns on the highest and lowest belief beta portfolios have similar volatilities. For all the

five portfolios, the return distribution is left-skewed with heavy tails, indicating that they

suffer infrequent yet large losses.

INSERT TABLE 3

The relation between belief risk and expected stock returns is strictly positive, and port-

folios composed of stocks with higher belief betas deliver higher returns. Specifically, the av-

erage return on the highest belief beta portfolio is 0.988%/month, 0.529% (i.e. 6.35%/year)

higher than the one on the lowest belief beta portfolio, and the difference is statistically

significant at the 5% level. This result provides preliminary support for the belief risk hy-

pothesis.

A.1 Double-sorting by size and belief beta

In order to examine whether the return pattern across belief beta quintiles captures a size

effect in stock returns, we double-sort stocks based on their market capitalizations and their

belief betas. At the beginning of each month of March, June, September, and December

during the period December 1997 through September 2009, stocks are sorted into five equal

portfolios based on their market capitalizations at the end of previous month. Within each

size quintile, we run a time-series regression of excess stock returns in the preceding 24

quarters on the market factor and the belief risk factor, and stocks are then sorted into five

further equal portfolios based on their belief betas.

INSERT TABLE 4

22

The results in Panel A of Table 4 reveal that the positive relation between belief risk and

expected stock returns prevails within four out of the five size quintiles. For example, within

the fourth size quintile, the average return on the highest-minus-lowest belief beta portfolio

is 0.710%/month with a t-statistic of 2.09, and within the second and fifth size quintiles,

the average returns are respectively 0.744%/month and 0.769%/month and statistically sig-

nificant at the 1% level. These results imply that size effect cannot by itself explain the

cross-sectional variations in the returns of portfolios formed on stock’s exposure to belief

risk.

It is noticeable that the cross-sectional effect of belief risk is stronger for large-cap stocks.

Within the two largest size quintiles, the average return on the highest belief beta portfolio is

over 150% higher than the one on the lowest belief beta portfolio, but the difference between

the average returns on the highest and lowest belief beta portfolios is smaller within the other

three size quintiles and even negative within the smallest size quintile. Diether et al. (2002)

find that stocks covered by financial analysts are usually issued by large firms, and Fig. 3

illustrates that the majority of stocks used to construct the market-wide belief measure have

large market capitalizations (75% of them belong to the five largest size deciles). Thus, the

market-wide belief measure constructed with the analyst EPS forecast data in this study

primarily reflects investors’ subjective belief about the earnings of large firms and will be

more relevant for the analysis of the cross-sectional effect of belief risk on the returns of

large-cap stocks. Incidentally, these stocks are also the ones that are the most prevalent

within the portfolios held by institutional investors.

23

A.2 Double-sorting by book-to-market and belief beta

We also examine whether the return pattern across belief beta quintiles captures a book-to-

market effect in stock returns by double-sorting stocks based on their book-to-market ratios

and belief betas. At the beginning of each month of March, June, September, and December

during the period December 1997 through September 2009, stocks are sorted into five equal

portfolios based on their book-to-market ratios, and within each book-to-market quintile,

stocks are sorted into five further equal portfolios based on their belief betas estimated with

observations in the preceding 24 quarters. The book value of equity is calculated as the

COMPUSTAT book value of stockholders’ equity, plus the balance sheet deferred taxes and

investment tax credit (if available), minus the book value of preferred stock. Depending on

availability, we use redemption, liquidation, or par value (in that order) to estimate the book

value of preferred stock. To ensure that the book value of equity is already known to the

market before the returns that it is used to explain, we match the book value of equity for

all fiscal years ending in calendar year y − 1 with returns starting in July of year y. The

book value of equity is then divided by the market value of equity at the end of previous

month to form the book-to-market ratio.

We observe from Panel B of Table 4 that the highest belief beta portfolio delivers a higher

average return than the lowest belief beta portfolio within all the book-to-market quintiles

and the return pattern is more pronounced for stocks with low book-to-market ratios. The

difference between the average returns on the highest and lowest belief beta portfolios exceeds

0.50%/month within the first three book-to-market quintiles and is statistically significant at

the 1% level within the second and third book-to-market quintiles. Based on these results, we

can say that the value premium puzzle is exacerbated in the presence of belief risk. Finally,

24

stocks with low book-to-market ratios usually tend to have large market capitalizations,

thus, the results in this subsection are consistent the findings in Section V.A.1.

B Regression results

Fama and French (1996) show that sorting stocks on variables such as the book-to-market

ratio, the earnings-to-price ratio, or the cash-flow-to-price ratio can produce a strong ordering

of returns across deciles. However, they also argue that estimates of three-factor time-series

regressions indicate that the three-factor model captures these patterns in average returns.

Along these lines, we conduct similar tests to see if the return patterns observed in Tables

2 and 3 can be explained by conventional risk factors. Table 5 reports the risk-adjusted

performance (alphas) of the five portfolios formed on belief betas, evaluated with the Fama

and French (1993, FF) three-factor model, the Carhart (1997) four-factor model and the

Carhart model augmented with the Pastor and Stambaugh (2003) liquidity risk factor16:

ri,t − rf,t = αi + βi,mMKTt + βi,sSMBt + βi,hHMLt + εi,t (9)

ri,t − rf,t = αi + βi,mMKTt + βi,sSMBt + βi,hHMLt + βi,uUMDt + εi,t (10)

ri,t − rf,t = αi + βi,mMKTt + βi,sSMBt + βi,hHMLt + βi,uUMDt + βi,lLIQt + εi,t (11)

where ri,t is the return of portfolio i, rf,t is the 1-month risk-free interest rate, MKTt is the

excess market return, SMBt is the excess return of small-cap stocks over large-cap stocks,

HMLt is the excess return of value stocks over growth stocks, UMDt is the excess return

of prior month winning stocks over losing stocks, LIQt is the excess return of high liquidity

beta stocks over low liquidity beta stocks, and εi,t is the normally distributed error term.

INSERT TABLE 516We thank Lubos Pastor and Robert F. Stambaugh for making the data publicly available.

25

The alphas delivered by portfolios formed on the basis of their belief betas exhibit sim-

ilar cross-sectional patterns as the average raw returns of those portfolios discussed so

far. For example, in the FF case, the alpha strictly increases in portfolio’s exposure to

belief risk. The lowest belief beta portfolio delivers a marginally significant negative al-

pha of -0.246%/month while the highest belief beta portfolio delivers a positive alpha of

0.273%/month with a t-statistic of 1.83, and the alpha of the highest-minus-lowest belief

beta portfolio is 0.519%/month and statistically significant at the 1% level, suggesting that

an investment strategy that is long in the highest belief beta portfolio and short in the lowest

belief beta portfolio will deliver a significant yearly alpha of 6.23% that cannot be explained

by the three FF risk factors. The results in Panels B and C of Table 5 suggest that the

pattern observed in the alphas cannot be explained neither by the presence of the Cahart

momentum nor of the Pastor and Stambaugh liquidity risk factors.

C Robustness tests

The findings so far strongly support the belief risk hypothesis in that stocks with higher

exposures to belief risk earn higher expected and abnormal returns. However, it is possible

that these results are driven by model mis-specifications or alternative explanations. To

address these concerns, we next perform a series of robustness tests.

C.1 Value-weighted portfolios

While the empirical analysis in Section V.A relies on equally-weighted portfolios, we obtain

similar results for value-weighted portfolios.

INSERT TABLE 6

26

Panel A of Table 6 reports the average returns and alphas delivered by value-weighted

portfolios formed on belief beta. As documented for the case of equally-weighted portfolios,

the average return and alpha delivered by a value-weighted portfolio are increasing functions

of the exposure of this portfolio to belief risk. In fact, the difference between the (risk-

adjusted) returns on the highest and lowest belief beta value-weighted portfolios is even

larger and more significant. For example, the average return of an investment strategy that

is long in the highest belief beta portfolio and short in the lowest belief beta portfolio is

0.865%/month (10.38%/year) with a t-statistic of 2.79 and about 64% higher than the one

obtained with equally-weighted portfolios. This result is not surprising given that, as shown

in Panel A of Table 4 and as discussed in Section V.A.1, the cross-sectional effect of belief

risk is stronger for large-cap stocks. Finally, the positive relation between belief risk and

expected stock returns also holds within most of the size and book-to-market quintiles.

C.2 Alternative EPS forecasting model

All the results obtained above rely on the market-wide belief measure constructed with

the BR model. To understand if these results are specific to the choice of EPS forecasting

model, we conduct a similar empirical analysis, but with another market-wide belief measure

constructed with the SRWD model. Table 7 summarizes the results.

INSERT TABLE 7

Clearly, the cross-sectional effect of belief risk on stock returns remains although the dif-

ference between the average returns on the highest and lowest belief beta portfolios becomes

smaller and less significant. As discussed in Section IV.A, the SRWD model is less accurate

than the BR model in forecasting quarterly EPS, so the market-wide belief constructed with

27

the SRWD model is a biased measure of investors’ subjective opinions and the belief beta

may thus capture other factors in addition to a stock’s mere exposure to the risk arising from

stochastic fluctuations in investors’ subjective average belief. Forming portfolios based on

the noisier belief beta estimated from the SRWD model will therefore yield less significant

results. Like in the BR case, the results obtained with value-weighted portfolios are slightly

stronger.

C.3 Divergence of opinion

Diether et al. (2002) show a negative cross-sectional relationship between the divergence of

opinion among investors approximated by the dispersion in analysts’ earnings forecasts and

expected stock returns, supporting Miller’s (1977) view that divergence of opinion is priced

in the presence of short-sale constraints. Stocks with higher exposure to belief risk may also

have lower divergence of opinion so that the return pattern across belief beta quintiles is

driven by the cross-sectional effect of the divergence of opinion on stock returns. We address

this concern by double-sorting stocks into 5× 5 portfolios based on the dispersion in analysts’

EPS forecasts scaled by the absolute value of the mean earnings forecast at the end of the

previous month and on their belief betas estimated with observations in the preceding 24

quarters. Table 8 reports average portfolio returns within each dispersion quintile. Stocks

with the zero mean earnings forecast are discarded.

INSERT TABLE 8

When looking at Table 8, we observe that the highest belief beta portfolio delivers a

significantly higher average return than the lowest belief beta portfolio within all but the

fourth dispersion quintiles. Thus, the return pattern across belief beta quintiles is clearly

28

not driven by the cross-sectional effect of the divergence of opinion. It can furthermore be

seen in Table 8 that unlike in Diether et al. (2002), there exists no clear return pattern

across dispersion quintiles, this might be due to the different data and time periods used in

this study.

C.4 Subsample analysis

Table 9 shows the results about the cross-sectional effect of belief risk on stock returns

for two separate subsample periods: the first one covers the period from December 1997

to November 2003 and the second one extends from December 2003 to November 2009.

During the first subsample period, the difference between the average returns on the highest

and lowest belief beta portfolios is 0.683%/month with a t-statistic of 1.99, and during the

second subsample period, the difference, although smaller in magnitude (0.374%/month), is

also statistically significant at the 5% level. While it remains more pronounced for large-

cap stocks during both subsample periods, the cross-sectional effect of belief risk varies over

time within the book-to-market quintiles: it is stronger for low book-to-market stocks during

the first subsample period but stronger for high book-to-market stocks during the second

subsample period. We further observe in Table 9 that average portfolio returns were much

higher during the first than during the second subsample period, which is due to the fact

that the second subsample period nests the 2007-09 financial crisis when stocks suffered large

losses.

INSERT TABLE 9

As discussed in Section IV.C, the Securities and Exchange Commission (SEC) passed

Regulation FD on August 10, 2000, and the regulation prohibits the selective disclosure

29

of material non public information by issuers to privileged individuals. Consequently, the

private information became a relatively less relevant issue for analysts’ EPS forecasts as of

August 10, 2000, which is prior to the beginning date of the second subsample period. If our

results were driven mainly by the cross-sectional differences in the sensitivity of stock excess

returns to fluctuations in the private information possessed by financial analysts, the cross-

sectional effect of belief risk should have been noticeably weaker during the second subsample

period. However, this is not what we observe and it suggests that the difference between

the average returns on the highest and lowest belief beta portfolios cannot be regarded

as compensation for the risk associated with the fluctuations in financial analysts’ private

information.

VI Further discussion

A Market-wide belief vs. other investor sentiment measures

A variety of investor sentiment measures have been proposed in the literature17. In this

subsection, we discuss three well-known sentiment measures, and compare them with the

market-wide belief measure constructed in this study.

Closed-end funds are investment firms that issue a fixed number of shares traded on stock

exchanges. The closed-end fund discount is the first sentiment measure used for comparison,

and it is calculated as the difference between the net asset value of a fund’s actual security

holdings and the fund’s market price. Lee et al. (1991) argue that if closed-end funds

are disproportionately held by retail investors, the average discount on closed-end funds

17Baker and Wurgler (2007) provide a discussion of some of these measures.

30

may represent a small investor sentiment measure, with the discount increasing when retail

investors become bearish.

Mutual fund flows were also used as a sentiment measure in previous studies such as

the ones by Brown et al. (2002) and by Ben-Rephael et al. (2012). Investors move their

money into and out of mutual funds with different levels of risk, and the changes in mutual

fund flows should reflect investors’ sentiment about market conditions. Following Baker

and Wurgler (2007), and relying on the monthly net flows data of eight equity-oriented

categories of mutual funds provided by the Investment Company Institute, we adopt the

principal component analysis approach to extract two main components from the changes in

mutual fund flows, which together can explain about 87% of variations in net flows within

the eight categories of mutual funds18. These two principal components are used as the

second proxy for investor sentiment.

Finally, we use the Baker and Wurgler (2006) index as the third sentiment proxy. The

authors construct their composite sentiment index based on the common variation in six

underlying proxies for investor sentiment: the closed-end fund discount; the NYSE share

turnover (the ratio of reported share volume to average shares listed from the NYSE Fact

Book); the number of IPOs; the average first day returns on IPOs; the equity share in

new issues; the dividend premium (the log difference of the average market-to-book ratios

of payers and non-payers). They start by estimating the first principal component of the

six proxies and their lags. This yields a first-stage index with 12 loadings, one for each of

the current and lagged proxies. Then, they calculate the correlation between the first-stage

18The eight equity-oriented categories of mutual funds include: “Aggressive Growth”, “Growth”, “Bal-

anced”, “Growth and Income”, “Sector”, “Income Equity”, “Income Mixed”, and “Asset Allocation”. We

thank Jeffrey Wurgler for sharing his mutual fund flows data (until May 2006) with us.

31

index and the current and lagged values of each of the proxies. Finally, they define the

sentiment index as the first principal component of the correlation matrix of six variables –

each respective proxy’s lead or lag, whichever has the higher correlation with the first-stage

index – rescaling the coefficients so that the index has unit variance.

INSERT TABLE 10

Table 10 shows that the correlations between the market-wide belief measure and all

the three sentiment measures described above are rather low. The correlation between the

closed-end fund discount and the market-wide belief measure is negative, suggesting that

the discount is low when investors are optimistic, this coincides with our expectation. The

market-wide belief measure is negatively although weakly correlated with the Baker and

Wurgler sentiment index. The graphs in the various panels of Fig. 4 further suggest that

there is no clear common pattern to be distinguished in the pairwise evolution of these

sentiment measures over time.

INSERT FIGURE 4

Although the market-wide belief measure and the above cited sentiment measures have

all been proposed to capture investors’ subjective opinions, they differ along several dimen-

sions. First, the data used to construct these various measures are not the same. We use

analysts’ EPS forecasts to construct the market-wide belief measure while the other senti-

ment measures are constructed with the data on the closed-end fund discount, mutual fund

flows, share turnover, etc. Second, the underlying estimation methods are different. Third,

and most importantly, these sentiment measures capture the opinions of different categories

of investors. Baker and Wurgler (2007), Ben-Rephael et al. (2012), and Lee et al. (1991)

32

argue that their sentiment measures mainly capture the opinions of retail investors or noise

traders who, on average, are less sophisticated. As discussed in Section I, the market-wide

belief measure that we construct, by contrast, primarily represents the ave belief of insti-

tutional investors. Due to these differences, it is not surprising that the market-wide belief

measure and these other sentiment measures are only weakly correlated.

B Understanding a stock’s exposure to belief risk

Finally, we examine how stock and firm specific characteristics affect excess stock returns’

exposures to belief risk. For this purpose, we run panel data regressions with time fixed effect

of individual belief betas on the following lagged stock and firm characteristics: the market

beta of stock returns estimated using the data over the period between 36 and 1 months

prior to t; the stock’s market capitalization in the month prior to t; the book-to-market

ratio; the accumulative return over the 11-month period between 12 and 2 months prior to

t; the annualized standard deviation of stock returns over the 12-month period between 12

and 1 months prior to t; the average stock turnover rate over the 12-month period between

12 and 1 months prior to t; the firm’s debt-to-book ratio; the firm’s sale-to-asset ratio; the

firm’s dividend-to-book ratio; the number of years between the stock’s first appearance on

the CRSP and t; and the number of financial analysts covering the stock in the month prior

to t. The accounting data from the fiscal year ending in year y−1 are matched to belief betas

from July of year y through June of year y+ 1. These selected variables represent important

firm characteristics such as their size, their maturity, their leverage, their dividend policy,

their growth opportunities, and most of them were also used in the studies by Diether et

al. (2002) and by Baker and Wurgler (2006). All the regressors are standardized so that we

33

can compare their powers in explaining the cross-sectional variations of stocks’ exposures to

belief risk.

INSERT TABLE 11

Table 11 displays the regression results. The belief beta decreases with size, book-to-

market ratio, and momentum and increases with return volatility, implying that smaller,

growth, less performing and more volatile stocks face higher exposures to belief risk. The

inverse relationship between the book-to-market ratio and a stock’s belief beta suggests that

belief risk may exacerbate the well known value premium since the so far neglected exposure

to belief risk is stronger for growth stocks. Young stocks and stocks with lower analyst

coverage have higher exposures to belief risk, this result is consistent with the finding about

the relation between size and belief beta since firms issuing these stocks are usually small-

sized. A high turnover rate increases a stock’s exposure to belief risk as well: frequently

traded stocks are, not surprisingly, more sensitive to innovations in investors’ average belief.

The sale-to-asset ratio has a significant positive impact on the belief beta, making these

large revenues generating firms more sensitive to belief risk regarding their future EPS.

VII Conclusion

In this paper, we use the actual EPS and analyst EPS forecast data provided by I/B/E/S to

construct the market-wide belief measure defined as the cross-sectional average of individual

beliefs for all sample stocks, with individual belief being defined as the mean analyst EPS

forecast minus the forecast derived from an econometric EPS forecasting model. We then

test the Belief Risk Hypothesis by examining whether belief risk – the risk arising from

34

stochastic fluctuations in the market-wide belief – is priced in the cross-section of stock

returns. We find that an investment strategy that is long in stocks with high exposure to

belief risk and short in stocks with low exposure to belief risk earns an average yearly return

of 6.35%. This positive relation between belief risk and expected stock returns persists after

accounting for traditional risk factors and for portfolios double-sorted on the basis of their

size or book-to-market ratio characteristics. These results are robust to: i) an alternative

EPS forecasting model; ii) value-weighted stock portfolios; iii) accounting for divergence

of analysts’ forecasts; iv) a sub-sample analysis. Finally, we find that a stock’s exposure

to belief risk increases with its return volatility, turnover rate, and sale-to-asset ratio and

decreases with its market capitalization, book-to-market ratio, momentum, age, and analyst

coverage.

Our findings suggest that fluctuations in the average belief of investors display common-

ality and are a priced source of risk in stocks’ average and excess returns. Thus, our results

are closely related to Lee et al. (1991) who show that stocks and closed-end funds with high

sensitivity to investor sentiment earn an extra return as compensation for this extra risk.

The main difference between this study and Lee et al. (1991) is that we rely on a sentiment

measure expressing the average belief of a different category of investors: the market-wide

belief measure captures the average opinions of institutional investors who follow analysts’

EPS forecasts while the closed-end fund discount proposed by Lee et al. (1991) mainly cap-

tures the average opinions of retail investors. In that sense, this study complements the one

undertaken by Lee et al. (1991).

The question as to whether the risk arising from stochastic fluctuations in the average

belief (i.e. belief risk) of institutional investors – the largest category of investors active in

35

U.S. financial markets – is priced is economically relevant, but has so far not been addressed

empirically. It would be interesting in the future to examine whether belief risk is also priced

in other asset classes and in other countries. Finally, developing a theoretical model that

endogeneizes belief risk into an extended asset pricing framework seems a promising area

for future research that could shed light on its ability to rationalize some well-known asset

pricing puzzles.

36

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42

Figure 1

The top graphs plot the time series of the market-wide belief variables constructed respec-

tively with the Brown and Rozeff (1979, BR) model and the Seasonal Random Walk with

Drift (SRWD) model, and the bottom graphs plot innovations in market-wide belief that

are estimated as the residuals of the linear regression model proposed in Section III.B. The

data covers the sample period August 1990 through August 2009.

1990 1995 2000 2005 2010

−0.

004

−0.

002

0.00

00.

002

0.00

4

Market Belief

BR

1990 1995 2000 2005 2010

−0.

015

−0.

010

−0.

005

0.00

00.

005

Market Belief

SRWD

1990 1995 2000 2005 2010

−0.

003

−0.

001

0.00

10.

003

Innovations in Market Belief

BR

1990 1995 2000 2005 2010

−0.

004

−0.

002

0.00

00.

002

Innovations in Market Belief

SRWD

43

Figure 2

The top left graph plots the autocorrelation function (ACF) of the market-wide belief variable

constructed with the Brown and Rozeff (1979, BR) model, and the top right graph plots the

partial autocorrelation function (Partial ACF) of the same market-wide belief variable. The

bottom graphs are for the market-wide belief variable estimated with the Seasonal Random

Walk with Drift (SRWD) model.

0 5 10 15

−0.

20.

00.

20.

40.

60.

81.

0

Lag

AC

F

Z_BR

5 10 15

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF

Z_BR

0 5 10 15

−0.

20.

00.

20.

40.

60.

81.

0

AC

F

Z_SRWD

5 10 15

−0.

40.

00.

20.

40.

60.

8

Par

tial A

CF

Z_SRWD

44

Figure 3

This figure plots the time-series of the ratios of stocks used to construct the market-wide

belief measure from each of ten size deciles to all sample stocks. In each month for the

market-wide belief measure to be constructed, a stock is assigned to one of ten deciles based

on its market capitalization at the end of previous month. Q1 denotes the decile of the

smallest stocks and Q10 denotes the decile of the largest stocks.

Time Period

Ratio

1990 1995 2000 2005 2010

0.0

0.1

0.2

0.3

0.4

0.5

0.6 Q 1

Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q 9Q10

45

Figure 4

The graphs (from top left to bottom right) plot the time series of the market-wide belief

variables constructed with the Brown and Rozeff (1979, BR) model and the Seasonal Random

Walk with Drift (SRWD) model, the Backer and Wurgler (2006) sentiment index, the closed-

end fund discount, and the first two principal components of changes in mutual fund flows.

BR Market Belief

1990 1995 2000 2005 2010

-0.002

0.000

0.002

SRWD Market Belief

1990 1995 2000 2005 2010

-0.015

-0.010

-0.005

0.000

BW Sentiment Index

1990 1995 2000 2005 2010

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Closed-End Fund Discount

1990 1995 2000 2005 2010

0.0

0.5

1.0

1.5

1st PC of Mutual Fund Flows

1990 1995 2000 2005 2010

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

2nd PC of Mutual Fund Flows

1990 1995 2000 2005 2010

-0.5

0.0

0.5

1.0

1.5

2.0

46

Table 1

Summary Statistics and Correlation Matrix

Panel A reports summary statistics of the market-wide belief variables ZBR and ZSRWD con-

structed with the Brown and Rozeff (1979) model and the Seasonal Random Walk with Drift

model: minimum, median, maximum, proportion of positive market-wide belief (PPMB),

standard deviation, skewness, and kurtosis. Panel B reports the correlation matrix of ZBR,

ZSRWD, and the following macroeconomic variables: the growth rate in Industrial Production

Index (IPI); the growth rate in Consumer Price Index (CPI); the growth rate in Employment

(EMPL); the Federal Funds Rate (RATE); and the NBER economic recession dummy that

equals 1 if the economy is in a recession or 0 otherwise (DUM).

Panel A: Summary Statistics

Minimum Median Maximum PPMB Std Dev Skewness Kurtosis

(%) (%) (%) (%)

ZBR -0.367 0.070 0.369 74.026 0.133 -0.807 4.590

ZSRWD -1.357 0.089 0.357 70.130 0.303 -2.808 12.321

Panel B: Market Belief and Macroeconomic Variables

IPI CPI EMPL RATE DUM ZBR ZSRWD

IPI 1.000

CPI 0.283 1.000

EMPL 0.895 0.318 1.000

RATE 0.531 0.490 0.612 1.000

DUM -0.640 0.174 -0.468 -0.150 1.000

ZBR 0.559 0.285 0.495 0.297 -0.476 1.000

ZSRWD 0.823 0.478 0.659 0.384 -0.594 0.7568 1.000

47

Table 2

Market-wide Commonality in Belief

Changes in the average belief for each individual stock are regressed in time series on changes

in the market-wide belief for all stocks in the sample:

CZit = αi + βi,1CZ

mt−1 + βi,2CZ

mt + βi,3CZ

mt+1 + εi,t

CZit = αi + βi,1CZ

mt−1 + βi,2CZ

mt + βi,3CZ

mt+1 + β

′

XXi,t + εi,t

where ’C’ denotes a change in the variables it precedes and Xi,t is the set of changes in

macroeconomic variables used as control variables. In each individual regression, the market-

wide belief excludes the dependent variable stock. Cross-sectional averages of time series

slope coefficients are reported with t-statistics in parentheses. ’Concurrent’, ’Lag’, and ’Lead’

refer, respectively, to the same, previous, and next quarter observations of the market-wide

belief. ’% positive’ reports the percentage of positive slope coefficients, while ’% + significant’

gives the percentage with t-statistics greater than +1.645 (the 5% critical level in a one-tailed

test). ’Sum’ aggregates coefficients for concurrent, previous, and next observations of the

market-wide belief. Coefficients for control variables are not reported.

BR SRWD—————————— ——————————–

Concurrent 0.657 0.737 0.703 0.653

(7.14) (7.23) (10.16) (8.63)

% positive 67.89 66.16 71.18 69.55

% + significant 19.67 17.84 32.79 24.85

Lag 0.242 0.280 0.158 0.106

(2.75) (3.00) (2.57) (1.37)

% positive 52.19 51.89 52.24 50.81

% + significant 8.46 7.85 10.49 9.78

Lead 0.092 0.101 0.086 0.081

(1.03) (0.99) (1.52) (1.35)

% positive 55.66 55.86 51.83 51.12

% + significant 9.58 9.68 11.10 9.88

Sum 0.990 1.117 0.948 0.840

(6.35) (6.47) (8.64) (6.75)

Control Variables Yes Yes

Adjusted R2 0.039 0.047 0.072 0.087

48

Table 3

Summary Statistics of Monthly Portfolio Returns

This table reports summary statistics of monthly portfolio returns: minimum, maximum,

mean, standard deviation, skewness, and kurtosis. At the beginning of each month of March,

June, September, and December during the period December 1997 through September 2009,

we run a time-series regression of excess stock returns in the preceding 24 quarters on the

market factor and the belief risk factor, and stocks are sorted into five equal portfolios

based on belief beta (βB). Portfolios are held for three months, and the portfolio return is

calculated as the equally-weighted average of the returns of all stocks in the portfolio. The

t-statistic is adjusted for autocorrelation by the Newey-West method.

βB———————————————————————————————————–

Low 2 3 4 High H-L t (H-L)

Minimum -20.72 -19.05 -17.19 -19.55 -21.75

Maximum 15.25 14.12 14.92 16.47 18.78

Mean 0.459 0.673 0.731 0.758 0.988 0.529 2.51

Std Dev 5.899 4.769 4.606 4.928 6.227

Skewness -0.594 -0.943 -0.898 -0.911 -0.683

Kurtosis 3.836 5.281 5.433 5.717 4.634

49

Table 4

Double Sort

Panel A reports the average monthly returns of portfolios double-sorted on size and belief

beta. At the beginning of each month of March, June, September, and December during the

period December 1997 through September 2009, stocks are sorted into five equal portfolios

based on their market capitalizations at the end of previous month. Within each size quintile,

we run a time-series regression of excess stock returns in the preceding 24 quarters on the

market factor and the belief risk factor, and stocks are sorted into five further equal portfolios

based on belief beta (βB). Similarly, Penal B reports the average monthly returns of portfolios

double-sorted on book-to-market ratio and belief beta. Portfolios are held for three months,

and the portfolio return is calculated as the equally-weighted average of the returns of all

stocks in the portfolio. The t-statistics are adjusted for autocorrelation by the Newey-West

method.

βB———————————————————————————————————

Panel A: Double Sort on Size and Belief Beta

Size Low 2 3 4 High H-L t (H-L)

Small 0.786 0.747 0.558 0.813 0.747 -0.039 -0.19

2 0.528 0.612 0.791 0.832 1.273 0.744 2.86

3 0.456 0.831 0.787 0.829 0.829 0.373 1.55

4 0.401 0.635 0.831 0.777 1.111 0.710 2.09

Large 0.199 0.447 0.647 0.643 0.968 0.769 2.81

Panel B: Double Sort on Book-to-Market Ratio and Belief Beta

B/M Low 2 3 4 High H-L t (H-L)

Low 0.065 0.369 0.404 0.593 0.586 0.522 1.47

2 0.530 0.514 0.779 0.575 1.249 0.719 2.70

3 0.540 0.696 0.744 0.710 1.176 0.636 3.03

4 0.748 0.728 0.851 0.875 0.944 0.196 1.00

High 0.707 0.791 0.760 0.982 1.012 0.305 1.49

50

Table 5Regression Results

This table reports the risk-adjusted performance (i.e. alphas) of belief beta portfolios eval-uated respectively with the Fama and French (1993) model, the Carhart (1997) model, andthe Carhart model augmented with the Pastor and Stambaugh (2003) liquidity risk factor.At the beginning of each month of March, June, September, and December during the periodDecember 1997 through September 2009, we run a time-series regression of excess stock re-turns in the preceding 24 quarters on the market factor and the belief risk factor, and stocksare sorted into five equal portfolios based on belief beta (βB). Portfolios are held for threemonths, and the portfolio return is calculated as the equally-weighted average of the returnsof all stocks in the portfolio. The t-statistics in parentheses are adjusted for autocorrelationby the Newey-West method.

Belief Beta Portfolios——————————————————————————————————————

Low 2 3 4 High H-L——————————————————————————————————————

Panel A: Fama and French (1993) Model

α (%) -0.246 -0.023 0.046 0.076 0.273 0.519(-1.65) (-0.22) (0.39) (0.64) (1.83) (2.73)

MKT 0.971 0.848 0.813 0.890 1.036 0.065(29.6) (30.7) (29.2) (28.6) (44.4) (1.90)

SMB 0.608 0.372 0.379 0.370 0.641 0.033(8.77) (5.58) (5.88) (4.52) (11.5) (0.73)

HML 0.304 0.530 0.514 0.474 0.272 -0.032(5.07) (11.0) (10.6) (10.4) (5.66) (-0.56)

R2 0.919 0.924 0.924 0.932 0.947 0.034

Panel B: Carhart (1997) Model

α (%) -0.196 0.025 0.092 0.115 0.312 0.508(-1.29) (0.24) (0.86) (1.04) (2.28) (2.38)

MKT 0.921 0.801 0.767 0.851 0.997 0.076(25.0) (30.8) (27.1) (24.7) (26.4) (1.62)

SMB 0.626 0.390 0.395 0.384 0.655 0.029(10.4) (6.87) (7.44) (5.35) (13.2) (0.67)

HML 0.271 0.498 0.483 0.448 0.246 -0.025(5.33) (12.9) (11.5) (9.96) (5.36) (-0.38)

UMD -0.085 -0.080 -0.078 -0.066 -0.066 0.019(-1.65) (-2.99) (-3.72) (-3.55) (-3.06) (0.31)

R2 0.926 0.933 0.934 0.938 0.950 0.031

Panel C: Carhart (1997) Model augmented with Liquidity Factor

α (%) -0.270 -0.010 0.026 0.059 0.240 0.510(-1.68) (-0.09) (0.25) (0.57) (1.67) (2.21)

MKT 0.914 0.798 0.761 0.846 0.990 0.076(25.1) (31.9) (28.2) (26.7) (29.2) (1.53)

SMB 0.626 0.389 0.395 0.384 0.655 0.029(10.4) (6.79) (7.45) (5.47) (13.8) (0.71)

HML 0.276 0.501 0.487 0.452 0.251 -0.025(5.72) (13.7) (13.4) (11.9) (7.14) (-0.40)

UMD -0.087 -0.081 -0.080 -0.068 -0.068 0.019(-1.77) (-3.07) (-4.05) (-3.58) (-3.24) (0.33)

LIQ 0.074 0.034 0.066 0.057 0.072 -0.002(2.09) (1.18) (3.03) (2.36) (2.16) (-0.03)

R2 0.928 0.934 0.937 0.940 0.952 0.024

51

Table 6

Value-Weighted Portfolios

Panel A reports the average monthly returns and alphas of belief beta sorted portfolios. At

the beginning of each month of March, June, September, and December during the period

December 1997 through September 2009, we run a time-series regression of excess stock

returns in the preceding 24 quarters on the market factor and the belief risk factor, and

stocks are sorted into five equal portfolios based on belief beta (βB). Panel B (Panel C)

reports the average monthly returns of portfolios double-sorted on market capitalization

(book-to-market ratio) at the end of previous month and belief beta. Portfolios are held

for three months, and the portfolio return is calculated as the value-weighted average of the

returns of all stocks in the portfolio. The t-statistics are adjusted for autocorrelation by the

Newey-West method.

βB————————————————————————————————————

Panel A: Sort on Belief Beta

Low 2 3 4 High H-L t (H-L)

Mean 0.000 0.091 0.627 0.681 0.865 0.865 2.79

FF α -0.464 0.345 0.202 0.293 0.509 0.973 3.48

Carhart α -0.457 -0.307 0.220 0.261 0.582 1.039 3.57

Liquidity α -0.452 -0.232 0.160 0.229 0.479 0.931 3.12

Panel B: Double Sort on Size and Belief Beta

Size Low 2 3 4 High H-L t (H-L)

Small 0.857 0.791 0.561 0.827 0.794 -0.063 -0.27

2 0.526 0.599 0.752 0.835 1.270 0.744 2.69

3 0.459 0.782 0.804 0.830 0.828 0.369 1.69

4 0.385 0.623 0.797 0.780 1.158 0.773 2.19

Large 0.107 0.050 0.544 0.697 0.687 0.580 2.12

Panel C: Double Sort on Book-to-Market Ratio and Belief Beta

B/M Low 2 3 4 High H-L t (H-L)

Low 0.103 0.081 0.311 0.408 0.661 0.559 1.09

2 -0.015 0.422 0.741 0.819 1.092 1.107 3.81

3 0.069 0.170 0.870 0.828 0.804 0.735 2.01

4 0.136 0.362 1.074 0.985 1.029 0.893 2.22

High -0.058 -0.286 0.754 0.757 1.152 1.210 2.19

52

Table

7

Seaso

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71

PanelC:Double-S

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Book-to-M

ark

etRatioand

BeliefBeta

B/M

Low

23

4H

igh

H-L

t(H

-L)

Low

23

4H

igh

H-L

t(H

-L)

Low

0.12

10.

220

0.36

70.

548

0.81

00.

689

1.31

0.27

00.

169

0.16

60.

442

0.77

70.

507

0.82

20.

603

0.76

80.

694

0.64

50.

944

0.34

01.

280.

424

0.43

70.

595

0.85

00.

752

0.32

70.

98

30.

757

0.62

60.

766

0.66

81.

078

0.32

11.

330.

443

0.34

00.

397

0.78

80.

998

0.55

51.

35

40.

902

0.72

40.

850

0.79

60.

890

-0.0

12-0

.07

0.34

50.

822

0.73

70.

499

1.20

70.

861

2.00

Hig

h0.

792

1.03

80.

620

0.78

51.

034

0.24

30.

900.

087

0.03

90.

485

0.55

51.

131

1.04

42.

11

53

Table 8

Dispersion of Analyst Forecasts

At the beginning of each month of March, June, September, and December during the period

December 1997 through September 2009, stocks are sorted into five equal portfolios based on

the dispersion of analyst forecasts scaled by the absolute value of the mean analyst forecast

at the end of previous month. Within each forecast dispersion quintile, we run a time-series

regression of excess stock returns in the preceding 24 quarters on the market factor and

the belief risk factor, and stocks are sorted into five further equal portfolios based on belief

beta (βB). Portfolios are held for three months, and the portfolio return is calculated as the

equally-weighted average of the returns of all stocks in the portfolio. This table reports the

average monthly returns of these portfolios. The t-statistics are adjusted for autocorrelation

by the Newey-West method.

βBForecast ———————————————————————————————————–

Dispersion Low 2 3 4 High H-L t (H-L)

Small 0.917 0.914 1.014 1.039 1.688 0.772 3.20

2 0.120 0.664 0.417 0.478 0.562 0.442 2.43

3 0.281 0.384 0.674 0.685 1.077 0.797 2.71

4 0.516 0.414 0.671 0.421 0.922 0.406 1.37

Large 0.133 0.466 0.602 0.864 1.184 1.051 2.04

54

Table

9

Subsample

Analysis

Pan

elA

rep

orts

the

aver

age

mon

thly

retu

rns

ofb

elie

fb

eta

por

tfol

ios.

At

the

beg

innin

gof

each

mon

thof

Mar

ch,

June,

Sep

tem

ber

,

and

Dec

emb

erduri

ng

two

separ

ate

subsa

mple

per

iods:

one

exte

nds

from

Dec

emb

er19

97to

Nov

emb

er20

03an

dan

other

exte

nds

from

Dec

emb

er20

03to

Nov

emb

er20

09,

we

run

ati

me-

seri

esre

gres

sion

ofex

cess

stock

retu

rns

inth

epre

cedin

g24

quar

ters

onth

em

arke

t

fact

oran

dth

eb

elie

fri

skfa

ctor

,an

dst

ock

sar

eso

rted

into

five

equal

por

tfol

ios

bas

edon

bel

ief

bet

a(β

B).

Pan

elB

(Pan

elC

)re

por

ts

the

aver

age

mon

thly

retu

rns

duri

ng

the

two

subsa

mple

per

iods

ofp

ortf

olio

sdou

ble

-sor

ted

firs

ton

thei

rm

arke

tca

pit

aliz

atio

ns

(book

-

to-m

arke

t-ra

tios

)at

the

end

ofpre

vio

us

mon

than

dth

enon

bel

ief

bet

a.P

ortf

olio

sar

ehel

dfo

rth

ree

mon

ths,

and

the

por

tfol

iore

turn

is

calc

ula

ted

asth

eeq

ual

ly-w

eigh

ted

aver

age

ofth

ere

turn

sof

all

stock

sin

the

por

tfol

io.

Thet-

stat

isti

csar

ead

just

edfo

rau

toco

rrel

atio

n

by

the

New

ey-W

est

met

hod.

βB

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

–

Dec

emb

er19

97to

Nov

emb

er20

03D

ecem

ber

2003

toN

ovem

ber

2009

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

PanelA:Sort

on

BeliefBeta

Low

23

4H

igh

H-L

t(H

-L)

Low

23

4H

igh

H-L

t(H

-L)

0.79

31.

014

0.98

71.

067

1.47

60.

683

1.99

0.12

50.

332

0.47

60.

450

0.49

90.

374

2.16

PanelB:Double-S

ort

on

Sizeand

BeliefBeta

Siz

eL

ow2

34

Hig

hH

-Lt

(H-L

)L

ow2

34

Hig

hH

-Lt

(H-L

)

Sm

all

1.27

81.

325

1.03

81.

364

1.26

7-0

.011

-0.0

30.

295

0.17

00.

077

0.26

30.

227

-0.0

68-0

.31

20.

940

1.04

41.

160

1.43

81.

872

0.93

22.

170.

117

0.18

10.

422

0.22

60.

673

0.55

72.

00

30.

799

1.13

20.

883

1.29

51.

254

0.45

51.

150.

113

0.53

10.

692

0.36

40.

404

0.29

11.

39

40.

556

0.84

01.

153

0.86

11.

351

0.79

41.

420.

245

0.43

00.

510

0.69

30.

871

0.62

62.

21

Lar

ge0.

323

0.64

00.

759

0.81

31.

359

1.03

62.

140.

075

0.25

30.

535

0.47

20.

576

0.50

11.

79

PanelC:Double-S

ort

on

Book-to-M

ark

etRatioand

BeliefBeta

B/M

Low

23

4H

igh

H-L

t(H

-L)

Low

23

4H

igh

H-L

t(H

-L)

Low

0.10

60.

634

0.36

90.

946

1.29

01.

184

1.83

0.02

40.

103

0.44

00.

241

-0.1

17-0

.141

-0.8

0

20.

722

0.71

50.

982

0.68

21.

816

1.09

52.

550.

338

0.31

20.

576

0.46

80.

682

0.34

31.

31

30.

945

0.87

81.

004

0.98

21.

488

0.54

32.

070.

136

0.51

40.

483

0.43

70.

865

0.73

02.

20

41.

263

1.20

01.

289

1.32

51.

279

0.01

60.

050.

233

0.25

50.

412

0.42

50.

609

0.37

61.

78

Hig

h1.

310

1.43

71.

002

1.33

81.

563

0.25

30.

650.

103

0.14

50.

517

0.62

50.

461

0.35

71.

96

55

Table 10

Correlation Matrix of Market-Wide Belief and Investor Sentiment Variables

This table reports the correlations between the following variables: the market-wide belief

constructed with the Brown and Rozeff (1979) model (ZBR) and the Seasonal Random Walk

with Drift model (ZSRWD); the Backer and Wurgler (2006) sentiment index (SENT); the

closed-end fund discount (CEFD); and the first two principal components (PC1mff and

PC2mff ) of changes in mutual fund flows. The data covers the sample period August 1990

through August 2009.

ZBR ZSRWD SENT CEFD PC1mff PC2mff

ZBR 1.000

ZSRWD 0.757 1.000

SENT -0.195 -0.130 1.000

CEFD -0.079 -0.273 -0.203 1.000

PC1mff 0.055 0.089 -0.199 -0.329 1.000

PC2mff -0.063 0.121 -0.008 0.108 -0.296 1.000

56

Table

11

Determ

inants

ofSto

ck’s

Exposu

reto

BeliefRisk

This

table

rep

orts

the

pan

eldat

are

gres

sion

resu

lts

wit

hti

me

fixed

effec

tof

indiv

idual

bel

ief

bet

ason

lagg

edst

ock

and

firm

char

ac-

teri

stic

s:th

em

arke

tb

eta

ofst

ock

retu

rns

esti

mat

edfr

omusi

ng

the

dat

afo

rth

ep

erio

db

etw

een

36an

d1

mon

ths

pri

ortot

(βmkt)

;

the

mar

ket

capit

aliz

atio

nin

the

mon

thpri

ortot

(ME

),th

eb

ook

-to-

mar

ket

rati

o(B

E/M

E),

the

accu

mula

tive

retu

rnfo

rth

e11

-mon

th

per

iod

bet

wee

n12

and

2m

onth

spri

ortot

(MoM

),th

est

andar

ddev

iati

onof

stock

retu

rns

for

the

12-m

onth

per

iod

bet

wee

n12

and

1

mon

ths

pri

ortot

(Std

Dev

),th

eav

erag

est

ock

turn

over

rate

for

the

12-m

onth

per

iod

bet

wee

n12

and

1m

onth

spri

ortot

(Turn

over

),th

e

deb

t-to

-book

rati

o(L

ever

age)

,th

esa

le-t

o-as

set

rati

o(S

ale/

AT

),th

ediv

iden

d-t

o-b

ook

rati

o(D

IV/B

E),

the

num

ber

ofye

ars

bet

wee

n

the

stock

’sfirs

tap

pea

rance

onC

RSP

andt

(Age

),an

dth

enum

ber

ofan

alyst

sin

the

mon

thpri

ortot

(Cov

erag

e).

The

acco

unti

ng

dat

afr

omth

efisc

alye

aren

din

gin

yeary−

1ar

em

atch

edto

bel

ief

bet

asfr

omJuly

ofye

ary

thro

ugh

June

ofye

ary

+1.

All

the

expla

nat

ory

vari

able

suse

din

the

regr

essi

ons

are

nor

mal

ized

.**

*,**

,an

d*

resp

ecti

vely

den

ote

the

sign

ifica

nce

atth

e1%

,5%

and

10%

leve

ls.

12

34

56

78

βmkt

-0.1

139

-0.6

213∗

∗∗-0.8

611∗

∗∗-0.1

154

-0.1

134

-0.1

278

-0.3

149∗

-0.2

031

ME

-1.1

592∗

∗∗-1.0

525∗

∗∗-1.1

763∗

∗∗-1.1

461∗

∗∗-1.1

361∗

∗∗-1.1

513∗

∗∗-0.7

911∗

∗∗-0.4

500∗

∗

BE

/ME

-0.4

949∗

∗∗-0.6

117∗

∗∗-0.4

761∗

∗∗-0.4

507∗

∗∗-0.4

979∗

∗∗-0.4

970∗

∗∗-0.5

142∗

∗∗-0.3

518∗

MoM

-0.6

826∗

∗∗-1.1

340∗

∗∗-1.0

400∗

∗∗-0.6

516∗

∗∗-0.7

125∗

∗∗-0.6

821∗

∗∗-0.7

211∗

∗∗0.

1072

Std

Dev

1.26

35∗∗

∗

Turn

over

2.28

56∗∗

∗

Lev

erag

e-0.0

820

Sal

e/A

T0.

9217

∗∗∗

DIV

/BE

-0.1

252

Age

-1.4

765∗

∗∗

Cov

erag

e-1.5

921∗

∗∗

Adj.R

2(%

)0.

682

0.70

60.

784

0.67

70.

698

0.68

30.

729

0.57

8

#of

Obs

159,

860

159,

860

159,

860

159,

276

159,

788

159,

808

159,

860

81,6

13

57