Download Spillover Effect of Disagreement

Two Trees with Heterogeneous Beliefs:
Spillover Effect of Disagreement∗
Bing Han†, Lei Lu‡, and Yi Zhou§
September 8, 2014
∗
We have benefited from comments from and discussions with Hagen Kim (CICF discussant), Wei Xiong
and conference participants at the 2014 CICF. Lei Lu acknowledges financial support from the National
Natural Science Foundation of China (#71271008)
†
Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada
M5S 3E6, Email: [email protected], Office: 416-946-0732, Fax: 416-478-5433.
‡
Department of Finance, Guanghua School of Managemente, Peking University, 5 Summer Palace Road,
Beijing, China 100871, Email: [email protected], Office: 86(10)6276-7227, Fax: 86(10)6275-3590.
§
Department of Finance, College of Business, Florida State University, Rovetta Business Bldg, 353, 821
Academic Way, P.O.Box 3061110, Tallahassee, Florida 32306-1110, Email: [email protected], Office:
850-644-7865, Fax: 850-644-4225.
1
Two Trees with Heterogeneous Beliefs: Spillover Effect
of Disagreement
Abstract
We study a model in which two groups of investors have different beliefs about the
expected dividend growth rates of two stocks. The state price density is affected by investor
disagreements for both stocks, especially that for the larger stock. The model predicts a
positive relation between investor disagreement about one stock and the expected return as
well as return volatility of the other stock, even when the fundamentals of two stocks are
uncorrelated. This spillover effect is stronger from a large stock to a small stock than vice
versa. These novel model implications are supported by empirical analysis.
JEL Classification: D53, G12
Keywords: Heterogenous Beliefs, Disagreement, Multiple Stocks, Size, Expected Return,
Volatility
1
Introduction
The traditional asset pricing literature is predominately built on representative agent models.
A growing body of research has recently highlighted that investor heterogeneity can play an
important role in understanding the behavior of asset prices (see, e.g., Basak (2005) for a
recent survey). These studies have focused on the case of a single risky asset and examined
how investor disagreement for one firm affects its own stock’s trading, volatility, and pricing.
Disagreement about either fundamental or extraneous variables (e.g., dividends, earnings,
sunspot, or rare events) can generate higher stock return volatility, larger trading volume
and predicts future stock return.
In this paper, we contribute to the heterogenous belief literature by studying stock
prices in a Lucas-type exchange economy with multiple risky assets. We extend Cochrane,
Longstaff, and Santa-Clara (2008) (hereafter CLS) to an economy in which the two investors
have different beliefs about the growth rate of dividend of two Lucas trees.1 Our model
features incomplete information about the dynamics of dividend process (growth rate of dividend payment is unknown). Investors have different priors about the dividend growth rates,
and Bayesian update their beliefs based on realized dividends. Due to the difference in prior,
investors obtain different estimates and they “agree to disagree” about the unobservable
dividend growth rates.
We first derive the equilibrium quantities. Using the technique of Cuoco and He (1994)
and Basak and Cuoco (1998), we construct a representative agent with the wealth-weighted
power utility function. Utilizing Clark-Ocone Malliavine derivatives, we derive the expected
stock return and return volatility. Then, we examine the relation between the disagreement
1
CLS use a two-tree model to explain the serial correlation of stock returns and the predictability of stock
returns by price-dividend ratios. Their work is further generalized. Eberly and Wang (2009) extend the CLS
setup to a production economy and analyze firm’s investment and Tobin’s Q. In Branger, Schlag, and Wu
(2011), the dividend drift is a known constant for one stock, but stochastic for the second stock. Martin
(2013) generalizes the CLS setup to an arbitrary number of trees and apply it to study rare disasters. These
papers use representative-agent models, while we study the effects of investors disagreement on expected
return and volatility in a model with heterogenous investors.
1
about the fundamental of one stock and the expected return or volatility of another stock.
The two stocks differ in their size, or the share each stock’s dividend contributes to the
aggregate dividend. We find that there is a “spillover” of the disagreement effect, above and
beyond the effect of disagreement on own stock which is the focus of prior research. Investors’
disagreement about the fundamental of one stock would affect the expected return as well
as the return volatility of the other stock, even when the fundamentals of the two stocks are
uncorrelated. The spillover effect is stronger from the large stock to the small stock than
from the small stock to the large stock. In other words, disagreement about a large-size
stock has a stronger effect on the expected return and volatility of a small-size stock. For
a small stock, the disagreement of other (larger) stock matters more for its expected return
and volatility than the disagreement of the own stock.
Our paper has two new theoretical contributions. First, we uncover a positive relation
between investors’ disagreement about the fundamental of stock 1 and the return volatility of
stock 2, and this effect is stronger when stock 1 is larger in size than stock 2. The intuitions
are as follows: the representative agent’s utility is a weighted average of the two investors,
and the state price density or pricing kernel depends on the relative weight assigned to the
two investors in the representative agent’s utility. This relative weight is stochastic and
depends on the investors’ disagreements for both stocks. We show that higher disagreements
among investors lead to a more volatile relative weight and hence more volatile pricing kernel,
which in turns implies higher stock return volatility. In addition, the state price density also
depends on aggregate endowment, whose volatility is proportionally related to the dividend
share. Therefore, the stock volatility is also a function of dividend share. Since the state
price density is more heavily influenced by investors’ disagreement for the larger stock, such
disagreement has a larger impact on the return volatility of the other (smaller) stock. Second,
similar intuitions also predict a positive relation between the disagreement of stock 1 and
the expected return of stock 2, and this effect is stronger when stock 1 is larger in size than
stock 2. These two model predictions distinguish our paper from the existing literature.
2
We test the two predictions of our model using the dispersion of analyst forecasts as the
proxy of investor disagreement. Consistent with our model’s unique prediction, we find a
positive and significant relation between a stock’s expected return and the analyst forecast
dispersion for the industry leader (the firm in the same industry with the largest market
capitalization). We also find a positive and significant relation between the return volatility
of a stock and the analyst forecast dispersion for the industry leader. Further confirming the
spillover effect of investor disagreement, we sort stocks into 50 portfolios each month based
on book-to-market ratio, and find a positive cross-sectional relation between a stock’s return
next month and the current dispersion of analyst forecast for the largest stock that belongs
to the same book-to-market sorted portfolio.
Our paper contributes to a large literature on heterogeneous beliefs or disagreement on
fundamental and nonfundamental variables.2 Buraschi, Trojani, and Vedolin (2013) and
Chabakauri (2013) are closest related to our paper in terms of model setup, but they examine different economic questions. Buraschi, Trojani, and Vedolin (2013) consider an economy
with two trees and two investors with different beliefs. They focus on the impact of disagreement on variance and correlation risk premia. They find that investors’ disagreement helps
explain the differential pricing of index and individual equity options. Chabakauri (2013)
study dynamic equilibrium in a Lucas economy with two stocks, two heterogeneous constant
relative risk aversion investors facing margin and leverage constraints. They find a positive
relationship between the amount of leverage in the economy and magnitudes of stock return
correlations and volatilities. We are the first to study, both theoretically and empirically,
the impact of investor disagreement for one stock on the expected return and volatility of
another stock. We find that there is a “spillover” of the disagreement effect, and such effect
is asymmetric across stocks depending their size.
2
The earlier models include Williams (1977), Abel (1990), Harris and Raviv (1993), Detemple and
Murthy (1994), and Basak (2000). The recent developments (mostly in continuous-time framework) include Scheinkman and Xiong (2003), Buraschi and Jiltsov (2006), Li (2007), David (2008), Dumas, Kurshev,
and Uppal (2009), Xiong and Yan (2010), Banerjee and Kremer (2010), Banerjee (2011), Chen, Joslin, and
Tran (2010, 2012), Buraschi, Trojani, and Vedolin (2013), Chabakauri (2013), and Croitoru and Lu (2014).
3
There is an extensive theoretical literature that implies that difference in beliefs should
lead to a positive risk premium (e.g., Varian (1985), Abel (1989), David (2008)). Empirically, Boehme, Danielsen, Kumar, and Sorescu (2009) and Avramov, Chordia, Jostova, and
Philipov (2009) document a positive relation between disagreement and expected stock return. However, Chen, Hong, and Stein (2002) and Diether, Malloy, and Scherbina (2002)
report a negative relation, which is consistent with the Miller (1977) hypothesis when shortsale constraints are present.3 Most recently, Carlin, Longstaff, and Matoba (2014) document
a positive risk premium for disagreement in the mortgage-backed security market, a liquid
essentially free from short-sale constraints. All of these papers analyze the impact of investor
disagreement about the fundamental of one risky asset on its own expected return.4 We examine the effects of disagreement about one stock (especially a large stock) on the expected
return and return volatility of other stocks (especially smaller stocks) - a ”spillover” effect
of disagreement.
The rest of the paper is organized as follows. Section 2 describes the model and solves
the dynamic equilibrium. Section 3 numerically illustrates the dispersion spill-over effect and
derives two testable hypotheses. Empirical test results are presented in Section 4. Section
5 concludes the paper. The Appendix contains the proof of Propositions and a summary of
key model parameters and variables.
2
The Model
We extend the two-tree model of CLS to an economy where two investors (i = A, B)
have identical preferences and endowments but different beliefs about the expected dividend growth rates of the two Lucas trees (or firms). There is a single consumption good
3
Gallmeyer and Hollifield (2008) study the effects of a market-wide short-sale constraint in a dynamic
economy with heterogeneous beliefs. They show that imposing the constraint reduces the stock price if the
optimistic investors’ intertemporal elasticity of substitution (IES) is less than one and increases the stock
price if the optimist’s IES is greater than one.
4
At the portfolio level, Yu (2011) documents that expected market return is negatively related to disagreement measured bottom-up from individual-stock analyst forecast dispersions, arguing that an increase
in market disagreement manifests as a drop in discount rate.
4
(the numeraire) and the economy has a finite horizon
[0, T ]. The uncertainty
is generated
by a four-dimensional Brownian motion, ω =
on a filtered probω1 , ωµ1 , ω2 , ωµ2
ability space
Ω, F, {Ft } , P . After setting up the model in details, we solve the
equilibrium and derive novel model implications on the effects of the disagreement about the
expected dividend growth rate of one stock for the expected return and return volatility of
the other stock.
2.1
Cash Flow and Information Structure
In our economy, the aggregate endowment is composed of two dividend streams, ε = D1 +D2 ,
where D1 and D2 denote the dividend generated by trees 1 and 2, respectively. The dividend
process Dj (j = 1, 2) follows a Geometric Brownian motion with stochastic drift given by:
dDj (t) = Dj (t) [µj (t) dt + σj dωj (t)] ,
(1)
dµj (t) = κj (πj − µj (t)) dt + σµj dωµj (t) ,
where µj and σj are the expected growth rate and volatility of dividend Dj , respectively.
πj is the long-run mean of expected growth rate, κj > 0 is the mean-reverting parameter,
and σµj denotes the
volatility of expected growth rate. The uncertainty vector ω =
is a four-dimensional Brownian motion under the objective probaω1 , ωµ1 , ω2 , ωµ2
bility measure P . Without loss of generality, we assume that the four Brownian motions ω1 ,
ωµ1 , ω2 , and ωµ2 are uncorrelated.5
The investors (i = A, B) observe the realizations of dividend process, Dj , but have incomplete information about the fundamental risk ωj . They can exactly estimate σj from the
quadratic variation of Dj , and update their estimates of µj after observing the realizations
of Dj . These assumptions are standard in the literature of incomplete information and het5
We can consider a general case where the dividend processes of the two stocks are correlated. However,
to distinguish the effect of investors’ heterogeneity about dividend growth of one stock on the return of
another stock from the correlation effect between the two stocks, we assume the dividend processes of the
two stocks are uncorrelated.
5
erogeneous beliefs (e.g., Detemple and Murthy 1994; Scheinkman and Xiong 2003; Buraschi
and Jiltsov 2006; David 2008; Dumas, Kurshev, and Uppal 2009; Xiong and Yan 2010).
Following the same literature, we assume that investors have different initial prior beliefs
B
about µj , that is, µA
j (0) 6= µj (0). Each investor, under his subjective probability measure
P i , employs the standard filtering theory (see, e.g., Liptser and Shiryaev 1977) to form his
posterior beliefs about µj : µij (t) ≡ E i [µj (t) |Ft ], which follows the dynamics given by:
dµij (t) = κj πj − µij (t) dt +
ωji
R
1 t
σj 0
dDj (s)
Dj (s)
−
vji (t)
dωji
σj
(t) ,
(2)
µij
(s) ds is investor i’s innovation process of Dj under his
h
2 i
probability measure, and vji (t) ≡ Eti µij (t) − µj (t)
is the conditional variance of investor
where
(t) =
i’s belief about µj . The standard filtering theory implies that vji is a deterministic function:
dvji (t)
dt
=
2
σµj
−
2κj vji
(t) −
vi (t) 2
j
σj
.
(3)
Remark 1 (Learning) We assume that investors have identical prior conditional variance
of their beliefs about µj (see, e.g., Basak 2005), that is, vjA (0) = vjB (0) ≡ vj (0). Under this
assumption, the conditional variance of the belief about µj is the same for the two investors
at any time: vjA (t) = vjB (t) ≡ vj (t) (∀t). The stationary conditional variance of the belief
q
−
dvj (t)
2
2
is vj =
= 0 (e.g., Scheinkman and Xiong
κj + (σµ j/σj ) − κj σj2 which satisfies dt
2003; Buraschi and Jiltsov 2006; Dumas, Kurshev, and Uppal 2009; Xiong and Yan 2010).
−
By replacing vj (t) with vj in Equation (2), the dynamics of investor i’s belief about the
expected dividend growth rate, µij , becomes
dµij (t) = κj πj − µij (t) dt +
−
vj
dωji
σj2
(t) ,
(4)
which means that investor i’s posterior estimate of µj follows a mean-reverting process with
−
the inverse precision of
vj
.
σj2
By manipulating investor i’s perceived shocks to the dividend growth rate ωji (t), we derive
6
its dynamics given by:
dωji
(t) =
1
σj
dDj (t)
Dj (t)
−
µij
(t) dt = dωj (t) +
µj (t)−µij (t)
dt,
σj
(5)
and thus the relation between two investors’ perceived innovations to dividend growth rates
is given by:
dωjB (t) = dωjA (t) + gj (t) dt,
(6)
where investor disagreement gj is defined as the (normalized) difference in beliefs about
expected dividend growth µj :
gj (t) =
B
µA
j (t)−µj (t)
σj
Its dynamics satisfies:
dgj (t) =
1
σj
B
dµA
j (t) − dµj (t) = −φj gj (t) dt,
(7)
−
where φj =
vj
σj2
gj (t) = gj (0)
+ κj > 0. Equation (7) implies that gj is a deterministic function,6 satisfying:
σj2
−
σj2 +vj t
. Therefore, if investor A is more optimistic at time t = 0, that is,
gj (0) > 0, he will remain more optimistic than investor B over time. The disagreement gj
plays a key role for state price density (Equation(25)), stock valuation (Equation(30)) as
well as in determining the expected stock return and return volatility (Equation(33)).
Applying the Ito’s lemma, the aggregate endowment ε = D1 + D2 follows a stochastic
process given by:
dε (t) = ε (t) [(f (t) µ1 (t) + (1 − f (t)) µ2 (t)) dt + f (t) σ1 dω1 (t) + (1 − f (t)) σ2 dω2 (t)] (8)
= ε (t)
f (t) µi1 (t) + (1 − f (t)) µi2 (t) dt + f (t) σ1 dω1i (t) + (1 − f (t)) σ2 dω2i (t) .
Under the objective probability measure (the first line), the drift term of the aggregate
endowment is the weighed-average of expected growth rates of dividends 1 and 2, where
6
If the two investors have different initial conditional variance of their beliefs, the stationary variance will
be different for two investors and then the disagreement will follow a stochastic process (e.g., Buraschi and
Jiltsov 2006).
7
the weight is the share of the corresponding tree’s dividend in the aggregate endowment.
There are two components in the diffusion term: the first is the product of the share of
dividend 1, f =
D1
,
D1 +D2
by its volatility, σ1 , and the second is the product of the share
of dividend 2, (1 − f ), by its volatility, σ2 . The second line describes the dynamics of the
aggregate endowment from the perspective of investor i (i = A, B) by manipulating the
relation between dωji (t) and dωj (t).
Since dividend process D1 and aggregate endowment process ε are stochastic, the dividend share of the first tree, f , follows a stochastic process given by:
df (t) = f (t) (1 − f (t)) µ1 (t) − µ2 (t) − f (t) σ12 + (1 − f (t)) σ22 dt
(9)
+ f (t) (1 − f (t)) (σ1 dω1 (t) − σ2 dω2 (t)) .
After a positive (resp. negative) shock to tree 1’s dividend, its share will increase (resp.
decrease).
Since investors have incomplete information about fundamental risk ωj and learn about
µj over time, they have different perception about the dynamics of dividend share too. From
the perspective of investor i, the share of first dividend follows the dynamics:
df (t) = f (t) (1 − f (t)) µi1 (t) − µi2 (t) − f (t) σ12 + (1 − f (t)) σ22 dt
(10)
+ f (t) (1 − f (t)) σ1 dω1i (t) − σ2 dω2i (t) .
2.2
Investment Opportunities
Investors face two fundamental risks ω1 and ω2 in the economy, which are related to the two
dividend processes D1 and D2 , respectively. To make the market (dynamically) complete,
investors continuously trade in three assets: one riskless asset (bond) in zero net supply, and
two stocks 1 and 2 (claims on the two dividends D1 and D2 respectively), each in net supply
of one unit. The bond price, B, and the prices of two stocks S1 and S2 follow the stochastic
8
processes given by:
dB(t)
B(t)
dS1 (t)
S1 (t)
= r (t) dt,
(11)
= µS1 (t) dt + σS11 (t) dω1 (t) + σS12 (t) dω2 (t)
(12)
= µiS1 (t) dt + σS11 (t) dω1i (t) + σS12 (t) dω2i (t) ,
dS2 (t)
S2 (t)
= µS2 (t) dt + σS21 (t) dω1 (t) + σS22 (t) dω2 (t)
(13)
= µiS2 (t) dt + σS21 (t) dω1i (t) + σS22 (t) dω2i (t) ,
where the objective expected return and return volatility of two stocks are endogenously
determined in equilibrium. Investors observe the historical prices of two stocks and have the
same estimates about their volatilities due to the quadratic variation-covariance of S1 and
S2 , but they have different estimates of µSj since they have different perceptions about the
innovation ωj . Investors A’s and B’s subjective expected stock returns are denoted by µA
Sj
and µB
Sj , respectively. From Equation (11), we can derive the following relation between the
subjective and objective expected stock returns for investor i = A, B
µiS1 (t) − µS1 (t) = σS11 (t)
µi1 (t)−µ1 (t)
σ1
+ σS12 (t)
µi2 (t)−µ2 (t)
,
σ2
µiS2 (t) − µS2 (t) = σS21 (t)
µi1 (t)−µ1 (t)
σ1
+ σS22 (t)
µi2 (t)−µ2 (t)
,
σ2
(14)
By the definition of the diagreement gj between the two investors for stock j = 1, 2, Equation
(14) implies that the differences in the subjective expected stock returns between the two
investors depend on the dispersion in beliefs about dividend growth rate:
B
µA
S1 (t) − µS1 (t) = σS11 (t) g1 (t) + σS12 (t) g2 (t) ,
(15)
B
µA
S2 (t) − µS2 (t) = σS21 (t) g1 (t) + σS22 (t) g2 (t) .
Note that difference in the subjective expected return of one stock arises not just due to
investors’ disagreement about its own fundamental, but also because of their disagreement
9
about the fundamental of the other stock.
Because there are three traded assets and two fundamental risks in the economy, the
market is dynamically complete under investor i’s subjective probability measure P i . There
exists an individual-specific state-price density ξ i for investor i, which statisfies the following
dynamics:
dξ i (t) = −ξ i (t) r (t) dt + θ1i (t) dω1i (t) + θ2i (t) dω2i (t) ,
(16)
where r is the riskfree rate, and θji is the market price of fundamental risk j perceived by
investor i satisfying that:





 µiS1 (t) − r (t)   σS11 (t) , σS12 (t)   θ1i (t) 
,
=



 


σS21 (t) , σS22 (t)
θ2i (t)
µiS2 (t) − r (t)
(17)
which means that the equity premium perceived by investor i is related to the market price
of two fundamental risks perceived by him, θ1i and θ2i .
By manipulating Equations (15) and (17), we find that the differences in the market price
of fundamental risks perceived by investors are equal to investor disagreement, i.e., dispersion
of beliefs about the corresponding dividend growth rate scaled by dividend volatility:
θ11 (t) − θ12 (t) = g1 (t) , θ21 (t) − θ22 (t) = g2 (t) .
(18)
We will derive explicit expressions for the market prices of fundamental risks θji in Equation
(27).
2.3
Investors’ Optimization Problem
At initial time t=0, investor i (= A, B) is endowed with αji > 0 share of stock j (= 1, 2) such
that αjA + αjB = 1, and thus the initial wealth of investor i is W i (0) = α1i S1 (0) + α2i S2 (0).
i
i
Investor i chooses a nonnegative consumption process ci and portfolio π i = (πBi , πS1
, πS2
) to
hR
i
T
maximize his lifetime expected utility of consumption, Eti 0 e−βt ui (ci (t)) dt , subject to
intertemporal budget constraints. The wealth of investor i satisfies W i (T ) > 0 and follows
10
the dynamics:
dW i (t) = W i (t) r (t) − ci (t) dt
(19)
i
+ πS1
(t)
µiS1 (t) − r (t) dt + σS11 (t) dω1i (t) + σS12 (t) dω2i (t)
i
+ πS2
(t)
µiS2 (t) − r (t) dt + σS21 (t) dω1i (t) + σS22 (t) dω2i (t) .
Using the martingale techniques (e.g., Cox and Huang 1987; Karatzas, Lehoczky, and
Shreve 1987), each investor’s dynamic optimization problem can be recast into a static
optimization problem (e.g., a static budget constraint), that is,
maxi Eti
i
hR
T
c (.),π (.)
where ui (ci (.)) =
ci (.)1−γ
.
1−γ
0
i
RT
e−βt ui ci (t) dt s.t. 0 ξ i (t) ci (t) dt ≤ W i (0) ,
(20)
Investors have the same coefficient of risk aversion, γ and time
discount factor, β, but have different prior beliefs about the expected dividend growth µj . Eti
denotes the time-t conditional expectation operator under investor i’s probability measure.
The static budget constraint implies that the present value of investor i’s lifetime consumption stream can not exceed his initial wealth W i (0). By solving investor i’s optimal
problem, the first-order condition for consumption ci (t) implies
(ui )0 ci (.) = y i ξ i (t) ,
(21)
where ξ i is the state-price density from the perspective of investor i which will be determined
in Equation (24), and y i the Lagrangian multiplier for investor i’s static budget constraint.
At the optimum consumption, investor i’s static budget constraint holds with equality, that
RT
is, 0 ξ i (t) ci (t) dt = W i (0).
2.4
Equilibrium
Definition 1 (Equilibrium) An equilibrium is a price system (r, S1 , S2 ) and consumptionportfolio processes (ci , π i ) (i = A, B) such that: (i) investors choose their optimal consumption–
11
portfolio strategies given the price system (r, S1 , S2 ). (ii) the security price processes are
consistent across investors. (iii) the goods and two stock markets clear, i.e.,
cA (t) + cB (t) = ε (t) ,
(22)
A
B
πS1
(t) + πS1
(t) = 1,
B
A
(t) = 1,
(t) + πS2
πS2
i
where πSj
is the portfolio holdings (in percentage) of stock j by investor i.
Following the technique of Cuoco and He (1994) and Basak and Cuoco (1998), we introduce a “representative” investor with the following utility function defined by:
U (c; λ) = max uA cA (t) + λ (t) uB cB (t) ,
(23)
cA +cB =c
where λ (.) > 0 is a state-dependent weight describing the relative importance of investor B
in the representative agent’s utility.
In equilibrium, the market of consumption goods clears and each investor solves his
optimal problem (Equation (21)). The representative investor consumes the aggregate endowment and his optimal solution associated with Equation (23) implies that
λ (t) =
(uA )0 (cA (t))
(uB )0 (cB (t))
y A ξ A (t)
= B B .
y ξ (t)
Proposition 1 summarizes the equilibrium results.
Proposition 1 (Equilibrium) In equilibrium, the state-price of density perceived by investor i is
ξ A (t) =
1
yA
1+λ(t)1/γ
ε(t)
γ
, ξ B (t) =
1
y B λ(t)
1+λ(t)1/γ
ε(t)
γ
,
(24)
where the relative weight λ (t) of investor B follows the stochastic process given by:
dλ(t)
λ(t)
= −g1 (t) dω1A (t) − g2 (t) dω2A (t) ,
12
(25)
where y i is the Lagrangian multiplier for investor i’s static budget constraint and such that
RT
investor i’s static budget constraint holds with equality, that is, 0 ξ i (t) ci (t) dt = W i (0).
The optimal consumption is given by
cA (t) = (1 − ψ (t)) ε (t) , cB (t) = ψ (t) ε (t) ,
where ψ (t) =
λ(t)1/γ
1+λ(t)1/γ
(26)
is the consumption share of investor B.
Equation (24) implies that investors have different state-price densities and the individualspecific state-price density is driven by the relative weight of two investors. The state-price
density ξ i is related to market prices of risks perceived by investor i, θ1i and θ2i , and disagreements g1 and g2 .
Equation (25) characterizes the dynamics of the relative weight of investor B. The
relative weight in the economy is driven by the two disagreements g1 and g2 , and the two
fundamental risks perceived by investor A, dω1A and dω2A . The volatility of the relative
weight of investor B increases with investors’ disagreement about expected dividend growth
rate. This is intuitive: without loss of generality suppose B is more pessimistic about the
expected growth rate of dividend 1 than investor A, then he will invest less in stock 1. After
a positive (negative) shock to the dividend growth of stock 1, investor B underperforms
(outperforms) and thus his relative weight in the economy decreases (increases). The bigger
the investor disagreement g1 is, the more the relative weight of investor B would fluctuate,
depending on whether the dividend growth of stock 1 experiences a positive or negative shock.
Therefore, the volatility of relative weight of investor B increases with the disagreement g1 .
Same argument can be applied to analyze the effect of disagreement g2 on relative weight of
investor B.
Equation (26) implies that the consumption of investor B increases with his relative
weight in the economy and the aggregate endowment. The intuition is as follows: (1) a
positive shock to dividend leads to a higher aggregate endowment, investors will expect that
13
there are more goods to consume in the future; and (2) a positive fundamental shock reduces
the wealth of investor B (again assuming B is more pessimistic) relative to A, and thus not
only his relative weight in the economy decreases but also he expects less goods available
for future consumption. Therefore, investor B’s consumption become more volatile with the
disagreements of g1 and g2 .
Proposition 1 provides equilibrium results of individual-specific state-price density ξ i ,
which plays an important role in pricing stocks. Applying Ito’s lemma to ξ i (Equation
(24)) and equating its drift and diffusion coefficients to those of Equation (16), we can
characterize riskfree rate and market prices of two dividend shocks. The results are presented
in Proposition 2.
Proposition 2 (Market Prices of Fundamental Risks and Riskfree Rate) Investor’s
individual-specific market prices of fundamental risks, θji (t) (i = A, B; j = 1, 2), are given
by:
θ1A (t) = γf (t) σ1 + ψ (t) g1 (t) , θ1B (t) = γf (t) σ1 − (1 − ψ (t)) g1 (t) ,
(27)
θ2A (t) = γ (1 − f (t)) σ2 + ψ (t) g2 (t) , θ2B (t) = γ (1 − f (t)) σ2 − (1 − ψ (t)) g2 (t) ,
and the riskfree interest rate is equal to
B
A
B
r (t) = β + γ f (t) (1 − ψ (t)) µA
1 (t) + ψ (t) µ1 (t) + 1 − f (t) (1 − ψ (t)) µ2 (t) + ψ (t) µ2 (t)
2
2
2
2
1 γ−1
(f
(t)
σ
)
+
((1
−
f
(t))
σ
)
+
ψ
(t)
g
(t)
+
g
(t)
.
(28)
− γ(γ+1)
1
2
1
2
2
2 γ
The market price of fundamental risk for tree 1, θ1i (t), is composed of two components.
The first component is standard and is equal to the product between the coefficient of risk
aversion, γ, and the volatility of aggregate endowment caused by fundamental risk 1, f (t) σ1 .
This is different from the case of one-tree model where the aggregate endowment equals the
dividend and then the volatility of endowment equals the volatility of dividend. In the two14
tree model, the volatility of aggregate endowment has two items, f (t) σ1 and (1 − f (t)) σ2 ,
which capture the share-weighted volatility of each dividend.
The second component in the expression for the market price of risk is the product of the
other investor’s consumption share and the disagreement about fundamental risk 1. Thus,
when an investor consumes a smaller share of aggregate endowment, he faces greater exposure
to price moving in the direction of the other investor’s beliefs. The intuition for the effect of
investor disagreement on the market price of fundamental risk 1 is as follows: when investors
have different beliefs about fundamental risk 1, risk is transferred from the more pessimistic
investor to the more optimistic who hold more shares of stock 1. The second component is
positive for optimistic investor (ψ (t) g1 (t)) since he bears more risk, while it is negative for
pessimistic investor (− (1 − ψ (t)) g1 (t)) since the risk is transferred to optimistic investor
and thus he faces less risk. Using the same economic mechanism, we can analyze the market
price of fundamental risk 2, θ2i (t).
The riskfree interest rate is composed of four components. The first item captures investors’ time preference (e.g., time discount factor β). The second item is the dividend-share
average of the consumption-share average of investors’ estimates about the expected growth
rate of two dividends. This term reflects the wealth effects on consumption. The third item
captures the effect of investors’ precautionary saving: as the volatility of aggregate endowments, (f (t) σ1 )2 + ((1 − f (t)) σ2 )2 , increases, investors will face more uncertainty and thus
have more incentive to save for future consumption, and reduces the equilibrium interest
rate. The last item captures the effect of disagreement: when investors disagree more about
the expected dividend growth rate, the two effects arise: (1) substitution effect: investors’
consumption becomes more volatile and thus they have higher precautionary savings motive,
therefore the interest rate decreases; and (2) wealth effect: when a positive shock to dividend
1 or 2 occurs, the more optimistic investor’s relative wealth increases and thus his relative
weight in the economy also increases, leading to a higher expected dividend growth rate
by the representative investor. Expectation of more goods available for future consumption
15
leads to more consumption today and higher current interest rate.
hingthere might be some problem here ***As a special case, if we set f = 1, the two-tree
model becomes the one-tree model and thus the riskfree rate becomes (e.g., Basak 2000;
David 2008)
γ(γ+1) 2 1 γ−1
2
2
B
+
σ
ψ
(t)
g
(t)
+
g
(t)
. (29)
r (t) = β + γ (1 − ψ (t)) µA
1
2
1
1 (t) + ψ (t) µ1 (t) −
2
2 γ
***
In our two-tree model with heterogenous beliefs, both investor disagreements gj and the
dividend share f influence the individual-specific market prices of fundamental risks and
state-price density. First, the disagreements g1 and g2 affect the diffusion coefficients of
relative weight λ, which in turn affects the state-price density ξ A . Second, the dividend
share f affects the diffusion coefficients of aggregate endowment ε (Equation (8)) and thus
affects the state-price density ξ A . Therefore, gj and f should both affect the stock prices, the
expected stock returns and return volatilities. Proposition 3 summarizes the results. The
proof, based on applications of the Mallavin calculus and the Clark-Ocone formula, can be
found in the Appendix.
Proposition 3 (Stock Price, Stock Volatility, and Expected Stock Return) The price
of stock j, Sj , is7
Sj (t) =
ε(t)γ
A
γE
(1+λ(t)1/γ ) t
hR
T −β(s−t)
e
t
1+λ(s)1/γ
ε(s)
γ
i
Dj (s) ds ,
(30)
the drift terms for the dynamics of the stock prices under objective probability measure in
7
In equilibrium, the stock price can be calculated under the probability measure of either investor. To be
consistent with the dynamics of relative weight, we calculate the stock price from the perspective of investor
A (the stock price is the same if we calculate from the perspective of investor B).
16
Equations (12) and (13) are given by:
µSj (t) = r (t) + γ [f (t) σ1 σSj1 (t) + (1 − f (t)) σ2 σSj2 (t)]
σ
(t) A
B
+ Sj1
µ
(t)
−
(1
−
ψ
(t))
µ
(t)
+
ψ
(t)
µ
(t)
1
1
1
σ1
σ
(t) B
µ2 (t) − (1 − ψ (t)) µA
.
+ Sj2
2 (t) + ψ (t) µ2 (t)
σ2
(31)
and the diffusion terms for the stock prices in Equations (12) and (13) are given by:
EtA
σS11 (t) = σ1 +
θ1A
(t) −
RT
t
−

e−β(s−t) ξ A (s)D1 (s)ψ(s)g1 (t)+γf (s)σ1 +(γf (s)−1)

(1−eκ1 (t−s) )v1 
ds
κ1 σ1
, (32)
R
A T
e−β(s−t) ξ A (s)D1 (s)ds


−
1−eκ2 (t−s) )v2
(
ds
e−β(s−t) ξ A (s)D1 (s)ψ(s)g2 (t)+γ(1−f (s))σ2 +
κ2 σ2
R
,
EtA tT e−β(s−t) ξ A (s)D1 (s)ds
Et
t

EtA
RT
t
σS12 (t) = θ2A (t) −
and

R
A T
Et
σS21 (t) =
θ1A
t
e−β(s−t) ξ A (s)D2 (s)ψ(s)g1 (t)+γf (s)σ1
(t) −
R
A T
Et
R
EtA tT
σS22 (t) = σ2 + θ2A (t) −

t

(1−eκ1 (t−s) )v−1 ds
+
κ1 σ1
e−β(s−t) ξ A (s)D2 (s)ds

,
e−β(s−t) ξ A (s)D2 (s)ψ(s)g2 (t)+γ(1−f (s))σ2 +(γ(1−f (s))−1)
R
A T
Et
t
e−β(s−t) ξ A (s)D2 (s)ds
(33)

(1−eκ2 (t−s) )v−2 ds
κ2 σ2
,
Note that although we assume that the two fundamental risks are uncorrelated, in equilibrium, the return volatility of one stock is also affected by fundamental risk of the other
tree, because the state-price density, via its dependence on the stochastic relative weight of
the two investors, is affected by two fundamental risks (see Equations (24) and (25)). In
addition, because the relative weight is a function of dispersion of beliefs for both trees, it
explains why one stock’s return volatility depends on investor disagreement about its own
fundamental as well as investor disagreement about the other stock. Equation (8) implies
that the volatility of aggregate endowment is related to dividend share f and the aggregate
17
endowment ε affects the state-price density. Therefore, the stock volatility is also a function
of f . In Section 3, we numerically analyze the effects of disagreements g1 and g2 and dividend
share f on stock volatility.
The subjective expected return of stock j from the perspective of investor i is related to
the market prices of two fundamental risks (see (17)):
µiSj (t) = r (t) + σSj1 (t) θ1i (t) + σSj2 (t) θ2i (t) .
By manipulating the relation between subjective expected stock return µiSj (t) and objective
expected return µSj (t) using Equation (14), we can derive the objective expected stock
return given in Equation (31). The risk premium of a stock under the objective probability
measure comprises of three components. The first component is equal to the product of risk
aversion coefficient and the covariance between stock return and aggregate endowment. The
other two components are consumption-share weighted average of the estimation errors of
the two investors for each tree’s expected dividend growth rate, respectively.
hR
i
T −β(s−t) A
1
A
A
A
ξ (s) c (s) ds .
In equilibrium, the wealth of investor A is W (t) = ξA (t) Et t e
Since ξ A (t) and cA (t) are functions of ε (t) and λ (t), we can derive the dynamics of investor
A’s wealth:
dW A (t) = Et dW A (t) +
−
∂W A (t)
λ (t)
∂λ(t)
∂W A (t)
ε
(t)
f
∂ε(t)
(t) σ1 dω1A (t) + (1 − f (t)) σ2 dω2A (t)
(34)
g1 (t) dω1A (t) + g2 (t) dω2A (t) .
By equating the diffusion coefficients of Equations (19) and (34), we obtain
A
A
πS1
(t) σS11 (t) + πS2
(t) σS21 (t) =
∂W A (t)
ε (t) f
∂ε(t)
A
A
(t) σS22 (t) =
πS1
(t) σS12 (t) + πS2
∂W A (t)
ε (t) (1
∂ε(t)
(t) σ1 −
∂W A (t)
λ (t) g1
∂λ(t)
− f (t)) σ2 −
(t) ,
∂W A (t)
λ (t) g2
∂λ(t)
(35)
(t) .
We can derive the portfolio holdings of stocks 1 and 2 by investor A by solving Equation
(35). Proposition 4 presents the optimal portfolio holdings by investor A, and a detailed
18
proof can be found in the appendix.
Proposition 4 (Portfolio Holdings) The optimal portfolio for investor A is

πSA


A
(t)  
 πS1

=
(t) = 
 

A
πS2 (t)
∂W A (t)
∂W A (t)
∂W A (t)
∂W A (t)
ε(t)f (t)σ1 − ∂λ(t) λ(t)g1 (t) −σS21 (t)
ε(t)(1−f (t))σ2 − ∂λ(t) λ(t)g2 (t)
σS22 (t)
∂ε(t)
∂ε(t)

σS11 (t)σS22 (t)−σS12 (t)σS21 (t)
∂W A (t)
∂W A (t)
∂W A (t)
∂W A (t)
ε(t)(1−f
(t))σ
−
λ(t)g
(t)
−σ
(t)
ε(t)f
(t)σ
−
λ(t)g
(t)
σS11 (t)
2
2
1
1
S12
∂ε(t)
∂λ(t)
∂ε(t)
∂λ(t)


.

σS11 (t)σS22 (t)−σS12 (t)σS21 (t)
(36)
Because the investors in mour model have heterogenous beliefs about expected dividend
growth rate, they have different portfolio holdings. Since the optimistic investor expects
higher dividend growth than the pessimistic investor, he will hold more stocks than pessimistic investor. In addition, Equation (36) implies that the dividend share f affects portfolio holdings via its impact on the volatility of aggregate endowment and stock volatility.
3
Numerical Analysis
In this section, we examine the effects of the disagreements about expected dividend growth
of the two trees and the relative dividend share (“size”) on the stock expected return and
volatility. We use Monte Carlo simulations to perform a numerical analysis. To isolate the
disagreement and size effects, we assume that all parameters of the two stocks are the same
except their size and degree of investor disagreement.
Table 1 reports the parameter values. Following Dumas, Kurshev, and Uppal (2009), the
long-run mean of expected dividend growth π1 = π1 = 0.015, and the volatility of dividend
σ1 = σ2 = 0.15. The coefficient of risk aversion γ is 3, the time discount factor β is 0.1, and
the investment horizon T = 40 years. The relative weight of investor B at time t = 0,
λ(0)
,
1+λ(0)
is set to be 0.5. That is, the two investors have the same initial weight in the economy.
We assume that the consumption-share average of dividend growth estimated by investors
at time t=0 equals the actual one, that is,
1
µA
1+λ(0)1/γ j
(0) +
λ(0)1/γ
µB
1+λ(0)1/γ j
(0) = µj (0). In
this case, the “representative agent” is unbiased about the expected dividend growth, and
19
Volatility (%)
25
20
15
10
5
Small−on−Large
Large−on−Small
0
0.1
0.2
0.3
Disagreement
0.4
0.5
Return (%)
25
20
15
10
5
Small−on−Large
Large−on−Small
0
0.1
0.2
0.3
Disagreement
0.4
0.5
Figure 1: Return Volatility and Expected Return of Stock 2 against the Disagreement about
the Growth Rate of Dividend 1 Claimed by Stock 1
thus we can separate the disagreement effect from the aggregate bias effect. To make the
disagreement persistent, we set the volatility of expected dividend growth, σµj , to be 0.02
and the mean-reverting parameter κj to be 0.1. We choose the value of share of dividend
1, f, at time t = 0 to be either 0.1 (i.e., stock 1 is small, stock 2 is large) or 0.9 (stock 1 is
large, stock 2 is small): when f = 0.1, we examine the effect of a small stock’s disagreement
on the expected return and return volatility of a large stock; when f = 0.9, we examine the
effect of a large stock’s disagreement on the expected return and return volatility of a small
stock.
The upper panel of Figure 1 plots the return volatility of stock 2 against various degree
of investor disagreement g1 about the first stock’s fundamental. The lower panel of Figure 1
20
plots the objective expected return of stock 2 against various degree of investor disagreement
g1 about the first stock’s fundamental. In each panel, there are two graphs. The solid graph
correponds to the case of f = 0.9 (i.e., stock 1 is large and stock 2 is small), and the dotted
graph correponds to the case of f = 0.1. We find that the return volatility of stock 2
increases with g1 , investor disagreement for the other stock. The slope of the solid line is
higher than that of the dotted line, suggesting that the effect of a large stock’s disagreement
on a small stock’s volatility is bigger than the effect of a small stock’s disagreement on a
large stock’s volatility. Similarly, the objective expected return of stock 2 increases with
investor disagreement g1 about the first stock’s fundamental. This effect is stronger when
the first stock is larger. We sumamrize these patterns in the following hypotheses which will
be tested in the next section.
Hypothesis 1 (Stock Volatility): There exists a positive relation between the disagreement about the expected growth rate of the dividend claimed by stock 1, g1 , and the
return volatility of stock 2, and this effect is stronger when stock 1 is larger in size
than stock 2.
Hypothesis 2 (Stock Return): There exists a positive relation between the disagreement
about the expected growth rate of the dividend claimed by stock 1, g1 , and the expected
return of stock 2, and this effect is stronger when stock 1 is larger in size than stock 2.
4
Empirical Analysis
We test the two hypotheses using monthly return data for firms listed on NYSE, AMEX
and NASDAQ, obtained from the Center for Research in Security Prices (CRSP). Our proxy
for investor disagreement is the dispersion of analyst forecast, obtained from the Thomson
Reuters’ IBES Summary History file. Dispersion is measured as the standard deviation of
analyst earnings forecasts scaled by the absolute value of the mean earnings forecast. In the
theoretical analysis, the size of a firm is measured by its dividend share in the aggregate
21
endowment. In the empirical analysis, we use the standard proxy for size, the stock’s market
capitalization.
Our test methodology is the Fama-MacBeth regressions. In each month’s cross-sectional
regression, the dependent variable is next month’s stock return or realized volatility. The
independent variables include a stock’s own dispersion of analyst forecast, lagged stock return
or volatility. In the model, there are only two stocks, and the disagreement of the large
stock has more significant impact on the return and volatility of the small stock. There
are thousands of stocks in the data. To test our model’s implication that other stock’s
disagreement also matters, we need to, for each stock and in each month, identify another
(larger) stock whose dispersion of analyst forecasts will serve as a regressor in the crosssectional regressions.
In Table 3, we regress a stock’s return or volatility on the disagreement of the corresponding industry leader. For each industry classified by the 2-digit SIC code (HISMG), we
label the firm within that industry that has the largest market capitalization as the industry
leader. We record its dispersion of analyst forecast, and assign it to all firms belonging to
that industry. Table 3 indicates a positive relation between a firm’s stock volatility and its
analyst forecast dispersion, and a negative relation between stock return and own firm’s disagreement. Both relations are statistically significant and consistent with previous studies
(e.g., Diether, Malloy, and Scherbina (2002)). More importantly, Models 1 and 2 show a
significant positive relation between stock volatility and other firm’s disagreement (in this
case, the disagreement for the industry leader), consist with our first hypothesis. Models
3 and 4 show a significant positive relation between stock return and disagreement for the
industry leader), supporting our second hypothesis.
In Table 4, we perform a further test for Hypothesis 2 using another specification of
other firm’s disagreement. In each month, we sort stocks into 50 portfolios based on the
book-to-market ratio. For a given stock, we assign to it the dispersion of analyst forecasts
for the largest stock belonging to the same B/M sorted portfolio (i.e., B/M leader). Again,
22
we find a significant positive relation between a stock’s return next month and the current
disagreement for the B/M leader. At the same time, there is a negative relation between
stock return and own firm’s disagreement.
5
Conclusion
We extend CLS to an economy where two groups of investors have different beliefs about
the expected dividend growth of two stocks. The state price density in the model is affected
by investor disagreements for both stocks, especially that for the larger stock. The model
predicts a positive relation between the expected return or return volatility of one stock and
investor disagreement about the other stock, even when the two fundamentals are uncorrelated. This spillover effect is stronger when one examines the impact of disagreement about
a large stock on the return or volatility of a small stock. These novel model implications are
supported by our empirical analysis.
23
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26
Appendix: Proof of Proposition 3
The price of stock 2 can be expressed as
γ
hR
i
T −β(s−t) 1+λ(s)1/γ
ε(t)γ
A
D2 (s) ds .
S2 (t) = 1+λ(t)1/γ γ Et t e
ε(s)
(
)
We have
−
vj
A
A
dµA
j (t) = κj πj − µj (t) dt+ σj dω j (t) .
Solving the above stochastic differential equation yields
R s eκj (u−s) v−j A
κj (t−s) A
κj (t−s)
µA
(s)
=
e
µ
(t)
+
1
−
e
π
+
dω j (u) .
j
j
j
σj
t
−
eκj (t−s) vj
.The
σj
A A
Taking Mallavine derivative of dividend process µA
j , we have Dt,j µj (s) =
dend process Dj satisfies
Rt
Rt
1 2
A
A
Dj (t) = Dj (0) exp 0 µj (u) − σj du + 0 σj dωj (u)
2
divi-
A
Taking Mallavine derivative of dividend process D2 , we have Dt,1
D2 (s) = 0, and
A
Dt,2
D2
R s eκ2 (t−u) v−2 du
(s) = D2 (s) σ2 +
(u) du = D2 (s) σ2 + t
σ2
−
(1−eκ2 (t−s) )v2
.
= D2 (s) σ2 +
κ2 σ2
Rs
A A
Dt,2
µ2
t
A
A
D1 (s) = D1 (s) σ1 +
D1 (s) = 0, and Dt,1
Similarly, we have Dt,2
−
(1−eκ1 (t−s) )v1
κ1 σ1
.
Solving stochastic differential equations of λ, we have
o
n R
Rt
Rt
t
λ (t) = λ (0) exp − 0 21 g1 (u)2 + g2 (u)2 du − 0 g1 (u) dω1A (u) − 0 g2 (u) dω2A (u) .
Taking Mallavine derivative of dividend process λ, we have
A
A
λ (s) = − γ1 λ (s)1/γ gj (t) .
Dt,j
λ (s)1/γ = γ1 λ (s)1/γ−1 Dt,j
Similarly, the Mallavine derivative of aggregate endowment, ε, is
−
(1−eκj (t−s) )vj
A
A
Dt,j ε (s) = Dt,j D (s) = Dj (s) σj +
.
κj σj
27
Therefore, we have
A A
Dt,j
ξ
(s) =
A
Dt,j
1+λ(s)1/γ
ε(s)
A
= γξ (s)
γ
=γ
A λ(s)1/γ
Dt,j
1+λ(s)1/γ
−
γ−1 DA λ(s)1/γ
1+λ(s)1/γ
ε(s)
A ε(s)
Dt,j
ε(s)
t,j
ε(s)
1/γ
+ 1 + λ (s)
A 1
Dt,j
ε(s)
.
and then
A A
Dt,1
ξ
A
−
(1−eκ1 (t−s) )v1
(s) = −ξ (s) ψ (s) g1 (t) + γf (s) σ1 +
,
κ1 σ1
−
(1−eκ2 (t−s) )v2
A
A A
Dt,2 ξ (s) = −ξ (s) ψ (s) g2 (t) + γ (1 − f (s)) σ2 +
.
κ2 σ2
1/γ
λ(t)
where ψ (t) = 1+λ(t)
1/γ . Using the Clark-Ocone formula, we can derive the stock volatility
(e.g., the two diffusion items) of second stock given by:
σS2,j (t) =
θjA
(t) +
EtA
RT
(
)ds
A
A
t Dt,j ξ (s)D2 (s)
RT
A
A
Et t ξ (s)D2 (s)ds
=
θjA
(t) +
EtA
RT
t
A ξ A (s)+ξ A (s)D A D (s) ds
[D2 (s)Dt,j
]
t,j 2
.
RT
EtA
t
ξ A (s)D2 (s)ds
That is,

R
EtA tT

e−β(s−t) ξ A (s)D2 (s)ψ(s)g1 (t)+γf (s)σ1 +
σS2,1 (t) = θ1A (t) −
R
A T
Et
EtA
RT
t
t

(1−eκ1 (t−s) )v−1 ds
κ1 σ1
,
e−β(s−t) ξ A (s)D2 (s)ds


e−β(s−t) ξ A (s)D2 (s)

σS2,2 (t) = σ2 + θ2A (t) −
EtA
h
ψ (s) g2 (t) + γ (1 − f (s)) σ2
−
(1−eκ2 (t−s) )v2
+ (γ (1 − f (s)) − 1)
κ2 σ2
RT
t
e−β(s−t) ξ A (s)D2 (s)ds
µA
1 (t)−µ1 (t)
σ1
i
µS2 (t) = r (t) + σS21 (t) γf (t) σ1 + ψ (t) g1 (t) −
i
h
µA (t)−µ (t)
+ σS22 (t) γ (1 − f (t)) σ2 + ψ (t) g2 (t) − 2 σ2 2
= r (t) + γ [f (t) σ1 σS21 (t) + (1 − f (t)) σ2 σS22 (t)]
(t) A
B
+ σS21
µ
(t)
−
(1
−
ψ
(t))
µ
(t)
+
ψ
(t)
µ
(t)
1
1
1
σ1
σS22 (t) A
+ σ2 µ2 (t) − (1 − ψ (t)) µ2 (t) + ψ (t) µB
2 (t) .
28


ds

Proof of Proposition 4
Following the methodology of Buraschi and Jiltsov (2006), we calculate the joint distribution of aggregate endowment and relative weight of investor B.
(1) Marginal distribution of relative weight of investor B, λ (t): the stochastic relative
weight λ (t) is given by
Rs
Rs
Rs
λ (s)=λ (t) exp − t 21 g1 (u)2 + g2 (u)2 du × exp − t g1 (u) dω1A (u) − t g2 (u) dω2A (u)
p
= λ (t) exp Mλ (s, t) − Vλ (s, t)Zλ , Zλ ∼ N (0, 1) .
(2) Marginal distribution of aggregate endowment, ε (t): the aggregate endowment ε (t) is
given by
R s 2
2
A
1
1
ε (s) = ε (t) exp t f (u) µA
1 (u) + (1 − f (u)) µ2 (u) − 2 [f (u) σ1 ] − 2 [(1 − f (u)) σ2 ] du
R s
Rs
× exp t f (u) σ1 dω1A (u) + t (1 − f (u)) σ2 dω2A (u)
p
= ε (t) exp Mε (s, t) − Vε (s, t)Zε , Zε ∼ N (0, 1) .
(3) Joint distribution of λ (t) and ε (t):
log λ (s)
Mλ (s, t)
Vλ (s, t)
Covλ,ε (s, t)
∼N
,
.
log ε (s)
Mε (s, t)
Covλ,ε (s, t)
Vε (s, t)
(4) The wealth of investor A is
hR
i
1
T
W A (t) = A EtA t e−β(s−t) ξ A (s) cA (s) ds =
ξ (t)
R
T
=
EtA
t
EtA
T −β(s−t)
γ−1
e
ε(s)1−γ (1+λ(s)1/γ )
ds
t
R
γ
ε(t)−γ (1+λ(t)1/γ )
e−β(s−t) Fε (t,s)1−γ Fλ (t,s)γ−1 ds
γ
ε(t)−γ (1+λ(t)1/γ )
where
h
i1/γ
√
1/γ
= 1 + λ (t) eHλ (s,t)
,
Fλ (t, s) = Fλ (t, s, λ (t) , Zλ ) = 1 + λ (t) eMλ (s,t)− Vλ (s,t)Zλ
Fε (t, s) = Fε (t, s, ε (t) , Zε ) = ε (t) eHε (s,t) ,
where Hλ (s, t) = Mλ (s, t) −
p
p
Vλ (s, t)Zλ and Hε (s, t) = Mε (s, t) − Vε (s, t)Zε .
29
The numerator of W A (t) can be written as
hR
i
T
EtA t e−β(s−t) Fε (t, s)1−γ Fλ (t, s)γ−1 ds
R TR ∞ R ∞
= t −∞ −∞ Fε (t, s, ε (t) , Zε )1−γ Fλ (t, s, λ (t) , Zλ )γ−1 N (Zε , Zλ , ρ (t, s)) dZε dZλ ds,
where ρ (t, s) = √
Covλ,ε (s,t)
√
Vε (s,t)
Vλ (s,t)
.
Therefore, the derivative of W A (t) with respect to λ (t) is
∂W A (t)
∂λt
ε(t)γ
=
∂
EA
∂λt t
R
T −β(s−t)
e
Fε (t,s)1−γ Fλ (t,s)γ−1 ds
t
−
γ
(1+λ(t)1/γ )
ε(t)γ λ(t)1/γ−1 EtA
R
T −β(s−t)
e
Fε (t,s)1−γ Fλ (t,s)γ−1 ds
t
γ+1
(1+λ(t)1/γ )


hR
i
T −β(s−t)
1−γ
γ−1
∂
A
E
e
F
(t,
s)
F
(t,
s)
ds
γ
ε
λ
∂λt t
t
ε(t)
h
i ,

= 1+λ(t)
1/γ γ
λ(t)1/γ−1 A R T −β(s−t)
(
)
Fε (t, s)1−γ Fλ (t, s)γ−1 ds
− 1+λ(t)1/γ Et t e
where
∂
EA
∂λt t
hR
EtA
hR
=
T −β(s−t)
e
Fε
t
T −β(s−t)
e
Fε
t
(t, s)1−γ Fλ (t, s)γ−1 ds
(t, s)
1−γ
∂
∂λt
i
γ−1 Fλ (t, s)
ds
i
hR
1/γ−1 H (s,t) i
(t, s)1−γ Fλ (t, s)γ−2 λ (t) eHλ (s,t)
e λ ds
h
iγ−2
R T −β(s−t)
1−γ Hλ (s,t)/γ
1/γ−1
γ−1 A
Hλ (s,t) 1/γ
= γ Et
e
Fε (t, s)
e
1 + λ (t) e
λ (t)
ds ,
t
=
γ−1 A
Et
γ
T −β(s−t)
e
Fε
t
and, similarly, the derivative of W A (t) with respect to ε (t) is
ε(t)γ
∂W A (t)
=
∂ A
E
∂εt t
R
T −β(s−t)
e
Fε (t,s)1−γ Fλ (t,s)γ−1 ds
t
γε(t)γ−1 EtA
R
T −β(s−t)
e
Fε (t,s)1−γ Fλ (t,s)γ−1 ds
t
+
γ
γ
(1+λ(t)1/γ )
(1+λ(t)1/γ )
h
i


R T −β(s−t)
1−γ
γ−1
∂
A
E
e
F
(t,
s)
F
(t,
s)
ds
ε
λ
t
∂ε
t
R
 t

ε(t)γ
T −β(s−t)
= 1+λ(t)
,
1/γ γ 
γε(t)−1 EtA
e
Fε (t,s)1−γ Fλ (t,s)γ−1 ds
(
)
t
+
γ
(1+λ(t)1/γ )
∂εt
where
∂
EA
∂εt t
hR
T −β(s−t)
e
Fε
t
(t, s)
1−γ
γ−1
Fλ (t, s)
i
hR
i
T −β(s−t)
−γ (1−γ)Hε (s,t)
γ−1
A
ds = Et t e
ε (t) e
Fλ (t, s)
ds .
The optimal holding of stock 2 for investor A is
A
πS2
(t) =
∂W A (t)
∂W A (t)
ε(t)[(1−f (t))σS11 (t)σ2 −f (t)σS12 (t)σ1 ]+ ∂λ(t) λ(t)[σS12 (t)g1 (t)−σS11 (t)g2 (t)]
∂ε(t)
σS11 (t)σS22 (t)−σS12 (t)σS21 (t)
30
.
Table 1: Parameters
V ariables
Symbols Numbers
P arameters f or aggregate endowment
Dividend share of asset 1 at t=0
f (0)
0.1, 0.9
Long-run mean of growth of dividend j
πj
0.015
Standard deviation of dividend j
σj
0.13
Standard deviation of growth of dividend j
σµj
0.02
Mean-reverting parameter of dividend j
kj
0.2
Investment horizon (years)
T
40
P arameters f or investors
Coefficient of Risk Aversion
γ
3
Time discount factor
β
0.1
λ(0)
0.5
Weight of investor B at t=0
1+λ(0)
Disagreement about stock 2
g2
0.25
31
V ariable
Dj
ε
f
µj
µij
kj
πj
σj
σµj
ωj
ωµj
ωji
vji
−
vj
gj
φj
B
Sj
µSj
σSjk
µiSj
ξi
r
θji
ci
β
γ
i
πSj
Wi
ψ
yi
λ
Table 2: Variable Description
Description (j = 1, 2)
Dividend j
Aggregate endowment
Share of D1
Expected growth rate of Dj
Investor i’s posterior belief about µj
Mean-reverting parameter of µj
Long-run mean of µj
Volatility of Dj
Volatility of µj
Fundamental risk (of Dj )
Brownian motion (of µj )
Investor i’s innovation (of Dj )
Conditional variance of investor i’s belief about µj
Stationary variance of investor i’s belief about µj
Disagreement between two investors about µj
Mean-reverting parameter of gj
Price of riskless asset
Price of stock j
Objective expected return of stock j
Return volatility of stock j caused by fundamental risk k
Investor i’s subjective expected return of stock j
Investor i’s state-price density
Riskfree rate
Market price of fundamental risk j perceived by investor i
Investor i’s consumption
Time discount factor
Coefficient of risk aversion
Investor i’s portfolio of stock j
Investor i’s wealth
Consumption share of investor B
Lagrangian multiplier for investor i’s static budget constraint
Relative weight of investor B
32
Table 3: The Industry Leader’s Dispersion and the Firms’ Return and Volatility
This table reports the results of monthly Fama-MacBeth regressions of the firm’s volatility and
return of the next period on this period-ending firm’s own dispersion DISP and the dispersion
of the industry leader respectively. The dependent variables are the next period volatility (Model
1 and Model 2) and return (Model 3 and Model 4) respectively. F irm DISP is measured as
the standard deviation of earnings forecasts scaled by the absolute value of the mean earnings
forecast obtained from IBES. IndustryLeader DISP is the dispersion of the largest firm in the
same industry based on the 2-digit SIC code. Volatility is the historical volatility of the last month.
Return is the equity return of the last month. The sample period is from January 1976 to December
2011. The numbers in the brackets are t-statistics. ∗ (resp. ∗∗ and ∗ ∗ ∗) denotes significance at
10% (resp. 5% and 1%) level.
F irm DISP
Industry DISP
Model 1
Volatility at t + 1
486.74***
(43.49)
29.57**
(2.52)
Volatility at t
Model 2
Volatility at t + 1
201.77***
(23.66)
19.68**
(2.37)
0.57***
(89.65)
Model 3
Return at t + 1
−10.50**
(−2.02)
9.71***
(2.94)
Model 4
Return at t + 1
−12.01**
(−2.48)
9.70***
(2.96)
0.01***
(5.24)
−0.03***
(−5.42)
0.01***
(5.42)
Return at t
Constant
0.38***
(64.58)
0.16***
(44.93)
33
Table 4: The B/M Portfolio Leader’s Dispersion and the Firms’ Return
This table reports the results of monthly Fama-MacBeth regressions of next month’s stock return
on current dispersion of analyst forecasts for the own firm and as well as that for the B/M Leader.
F irm DISP is measured as the standard deviation of earnings forecasts scaled by the absolute
value of the mean earnings forecast. In each month, we sort stocks into 50 portfolios based on the
book-to-market ratio. For a given stock, B/M Leader DISP is the dispersion of the largest stock
belonging to the same B/M sorted portfolio. The sample period is from January 1980 to September
2013. The numbers in the brackets are t-statistics. ∗ (resp. ∗∗ and ∗ ∗ ∗) denotes significance at
10% (resp. 5% and 1%) level.
F irm DISP
B/M Leader DISP
Model 1
Return at t + 1
−0.16***
(−2.92)
2.00***
(6.60)
Return at t
Model 2
Return at t + 1
−0.16***
(−2.90)
1.78***
(6.00)
−0.04***
(−7.17)
B/M
Size
Constant
1.19***
(4.70)
1.25***
(4.98)
34
Model 3
Return at t + 1
−0.20***
(−3.58)
0.79***
(3.09)
1.11***
(6.89)
0.03
(0.78)
0.03
(0.04)
Model 4
Return at t + 1
−0.19***
(−3.55)
0.79***
(3.10)
−0.04***
(−7.59)
0.99***
(6.25)
0.03
(0.89)
0.07
(0.08)