Belief Dispersion in the Stock Market

• Author: SULEYMAN BASAK,ADEM ATMAZ
• Published date: 01 June 2018
• Date: 01 June 2018
• DOI: http://doi.org/10.1111/jofi.12618

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 3 •JUNE 2018

Belief Dispersion in the Stock Market

ADEM ATMAZ and SULEYMAN BASAK∗

ABSTRACT

We develop a dynamic model of belief dispersion with a continuum of investors differ-

ing in beliefs. The model is tractable and qualitatively matches many of the empirical

regularities in a stock price and its mean return, volatility, and trading volume. We

find that the stock price is convex in cash-flow news and increases in belief dis-

persion, while its mean return decreases when the view on the stock is optimistic,

and vice versa when pessimistic. Moreover, belief dispersion leads to higher stock

volatility and trading volume. We demonstrate that otherwise identical two-investor

heterogeneous-beliefs economies do not necessarily generate our main results.

THE EMPIRICAL EVIDENCE ON THE effects of investors’ dispersion of beliefs on as-

set prices and their dynamics is vast and mixed. For example, several studies

find a negative relation between belief dispersion and a stock’s mean return

(Diether, Malloy, and Scherbina (2002), Chen, Hong, and Stein (2002), Goetz-

mann and Massa (2005), Park (2005), Berkman et al. (2009), Yu (2011)). Others

argue that the negative relation is valid only for stocks with certain charac-

teristics (e.g., small, illiquid, worst-rated or short-sale constrained) and in fact

find a positive or no significant relation (Qu, Starks, and Yan (2003), Doukas,

Kim, and Pantzalis (2006), Avramov et al. (2009)). Existing theoretical studies

(discussed below), on the other hand, do not provide satisfactory answers for

these mixed results.

In this paper, we develop a tractable model of belief dispersion that is able to

qualitatively match many of the empirical regularities in a stock price and its

∗Adem Atmaz is with the Krannert School of Management, Purdue University.Suleyman Basak

is with the London Business School and CEPR. We thank Andrea Buffa, Georgy Chabakauri, Fran-

cisco Gomes, Christian Heyerdahl-Larsen, Ralph Koijen, Hongjun Yan,discussants Daniel Andrei,

Michael Brennan, Burton Hollifield, Philipp Illeditsch, Tim Johnson, Mina Lee, Bart Taub, as well

as seminar participants at the 2015 EFAmeetings, 2015 International Moscow Finance Conference,

2015 SFS Finance Cavalcade, 2015 Wabash River Finance Conference, 2015 WFAmeetings, 2017

Berlin-Princeton-Singapore Workshop on Quantitative Finance, London Mathematical Finance

Seminar, Marti Subrahmanyam Festschrift, Workshop in Memory of Rick Green, City University

of Hong Kong, Graduate Institute Geneva, Hong Kong Polytechnic University, London Business

School, Nanyang Technological University, University of Hong Kong, University of Hull, Univer-

sity of Lugano, University of New South Wales, University of Oxford, University of Southampton,

University of Sydney, and University of Technology Sydney for helpful comments. We have also

benefited from the valuable suggestions of Ken Singleton (the Editor) and two anonymous referees.

All errors are our responsibility. The authors do not have any conflicts of interest as identified in

the Journal of Finance’s disclosure policy.

DOI: 10.1111/jofi.12618

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mean return, volatility, and trading volume. Specifically, we develop a dynamic

general equilibrium model populated by a continuum of constant relative risk

aversion (CRRA) investors who differ in their (dogmatic or Bayesian) beliefs

and consume at a single consumption date. There are two key differences be-

tween our model and existing works, which typically employ two investors and

a continuum of consumption dates. First, rather than considering the overall

effects of belief heterogeneity, we isolate the effects of belief dispersion from

the effects of other moments and conduct comparative statics analysis with re-

spect to belief dispersion only, which results in sharp results. Second, because

our model features a continuum of investors, no investor dominates the econ-

omy in relatively extreme states, which leads to nonvanishing belief dispersion

and hence to simple uniform behavior for economic quantities. Moreover, while

dynamic models with heterogeneous beliefs are generally difficult to solve for

long-lived assets beyond logarithmic preferences (e.g., Detemple and Murthy

(1994), Zapatero (1998), Basak (2005)), our model delivers fully closed-form

expressions for the quantities of interest.1

In our analysis, we summarize the wide range of investors’ beliefs using two

sufficient measures, namely, the average bias and dispersion in beliefs, and

demonstrate that equilibrium quantities are driven by these two key endoge-

nous variables. We take the average bias to be the bias of the representative

investor. How much an investor’s belief contributes to the average bias de-

pends on her wealth and risk attitude. Investors whose beliefs are supported

by actual cash-flow news become relatively wealthier through their invest-

ment in the stock and therefore contribute more to the average bias. This leads

to fluctuations in the average bias such that following good (bad) cash-flow

news, the view on the stock becomes relatively more optimistic (pessimistic).

On the other hand, consistenty with empirical studies, we construct our belief

dispersion measure as the cross-sectional standard deviation of investors’ dis-

agreement. This enables us to reveal the dual role of belief dispersion. First, we

uncover a novel role of belief dispersion in that it amplifies the average bias, so

that the same good (bad) news leads to more optimism (pessimism) when dis-

persion is higher. Second, we show that belief dispersion indicates the degree

to which the average bias fluctuates, and hence indicates the extra uncertainty

investors face.

Turning to our model implications, we first show that in the presence of belief

dispersion, stock price is convex in cash-flow news, indicating that stock price

is more sensitive to news in relatively good states. This result also implies that

the increase in stock price following good news is more than the decrease fol-

lowing bad news, consistent with empirical evidence (Basu (1997), Xu (2007)).

Convexity arises because, the better the cash-flow news, the higher the extra

boost to the stock price that comes from elevated optimism. Consequently, the

stock price increases (decreases) with belief dispersion when the view on the

1We explicitly obtain all quantities of interest in closed form in all of the economic settings

considered in the paper, with the exception of trading volume in the multiple-stock setting of

Section V.

Belief Dispersion in the Stock Market 1227

stock is relatively optimistic (pessimistic), also consistent with empirical evi-

dence (Yu (2011)). Our model further implies that the stock price may increase

and its mean return may decrease in investors’ risk aversion in relatively bad

states. This is because in a more risk-averse economy, investors have less ex-

posure to the stock, which in bad times limits wealth transfers to pessimistic

investors, and hence leads to a relatively optimistic view on the stock, a higher

stock price, and a lower mean return.

We next examine the widely studied relation between belief dispersion and

a stock’s mean return. Since dispersion represents the extra uncertainty in-

vestors face, risk-averse investors demand a higher return to hold the stock

when dispersion is higher. However, dispersion also amplifies optimism and

pushes the stock price up further following good news, leading to a lower mean

return in those states. When the view on the stock is relatively optimistic,

the second effect dominates and we find a negative dispersion-mean return

relation. As discussed earlier, empirical evidence on this relation is mixed,

with some studies finding a negative relation while others find a positive or

no significant relation. Our model generates both possibilities. In particular,

we demonstrate that this relation is negative when the view on the stock is

relatively optimistic and positive otherwise. Diether, Malloy, and Scherbina

(2002) provide support for our finding by documenting that, while there is an

optimistic bias overall, the negative effect of dispersion becomes stronger for

more optimistic stocks. Similar evidence is also provided by Yu (2011).

We further find that stock volatility increases monotonically in belief disper-

sion, consistent with empirical evidence (Ajinkya and Gift (1985), Anderson,

Ghysels, and Juergens (2005), Banerjee (2011)). This is because the average

bias in beliefs fluctuates more, and hence so does the stock price, when belief

dispersion is higher. In addition to belief dispersion, investors’ Bayesian learn-

ing process also increases fluctuations in the average bias, and hence leads to

higher stock volatility because all investors become relatively more optimistic

(pessimistic) following good (bad) news due to belief updating. Our closed-form

stock volatility expression allows us to disentangle the effects of belief dis-

persion and Bayesian learning and yields the novel testable implication that

Bayesian learning induces less stock volatility when belief dispersion is higher.

Moreover, we find that stock trading volume is increasing in belief dispersion,

consistent with empirical evidence (Ajinkya, Atiase, and Gift (1991), Bessem-

binder, Chan, and Seguin (1996), Goetzmann and Massa (2005)). This finding

is intuitive since when dispersion is higher, investors with relatively different

beliefs, who also have relatively higher trading demands, are more dominant.

We also find a positive relation between stock volatility and trading volume due

to the positive effect of dispersion on both quantities, which is also consistent

with empirical evidence (Gallant, Rossi, and Tauchen (1992), Banerjee (2011)).

In additional analysis we demonstrate that most of our results above do not

necessarily obtain in an economy that is identical to ours except that it is popu-

lated by two rather than a continuum of investors. In particular, we show that,

in such an economy,the stock price is no longer convex in cash-flow news across

all states of the world, and higher belief dispersion can actually lead to a lower