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CONTINUOUS TRADING WITH
HETEROGENEOUS BELIEFS
AND NO NOISE TRADING
Albert S. Kyle and Tao Lin
Fuqua School of Business
Duke University
Version 1: August 2000
Version 2: September 2001
Current rough draft: June 2, 2002
We would like to thank Vish Viswanathan and Hui Ou-Yang for helpful comments and
suggestions. Any remaining errors are ours. Please do not quote without permission.
ABSTRACT
This paper presents a continuous-time asset-pricing model with symmetrically informed
overconfident, traders who stubbornly disagree about the importance of independent time-varying
signals of future dividend growth. Each trader believes one signal is more important than the
others, but traders agree to disagree about which signal is most important. Assuming exponential
utility and normally distributed random variables, a simple closed-form solution exists in which
the price is a linear combination of the current dividend and the average signal. Despite the
absence of noise trading, each trader takes a time-varying position proportional to the difference
between the current value of the average signal and the signal he believes to be most important.
Although learning does not occur, modest but economically meaningful differences in beliefs can
persist for many decades without being easily detected statistically.
-1-
1. INTRODUCTION
This paper presents a continuous-time asset-pricing model with symmetrically informed
overconfident, traders who stubbornly disagree about the importance of independent time-varying
signals of future dividend growth. Each trader believes one signal is more important than the
others, but traders agree to disagree about which signal is most important. Traders do not share a
common prior. Assuming exponential utility and normally distributed random variables, a simple
closed-form solution exists in which the price is a linear combination of the current dividend and
the average signal. Despite the absence of noise trading, each trader takes a time-varying position
proportional to the difference between the current value of the average signal and the signal he
believes to be most important. Although learning does not occur, modest but economically
meaningful differences in beliefs can persist for many decades without being easily detected
statistically.
The model makes it possible to distinguish four distinct properties of traders’ expectations, which
we call “aggregate optimism,” “relative optimism,” “aggregate overconfidence,” and “relative
overconfidence.” Aggregate optimism, which occurs when the average belief about the long-term
level of dividends is too high, makes prices high and returns low, but does not lead to trading
volume. Relative optimism, which occurs when there are differences in beliefs about the long
term level of dividends, induces traders to hold different positions in equilibrium. It does not lead
to trading volume. Aggregate overconfidence, which occurs when traders on average believe that
signals about future dividend growth are more important than in the correct model, leads to
excess volatility and mean reversion in prices, but not to trading volume.
Relative
overconfidence, which occurs when traders disagree about the importance of different signals of
future dividend growth, leads to trading volume, because each trader believes he can make a
trading profit at the expense of other traders. This trading volume is associated increased market
depth.
Our model is related to the traditional rational expectations literature. Each trader is rational in
that he applies Bayes Law consistently. Traders process information differently because they
have different beliefs. Even though there is no noise trading in our model, the effects of
overconfidence lead to properties similar to noise trading models such as Grossman and Stiglitz
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(1980), Campbell and Kyle (1993), or Wang (1993). Our model is different from these models
and similar to Diamond and Verrecchia (1981) and Foster and Viswanathan (1993) in that
different informed traders have symmetrically different, non-nested information. The symmetry
makes the model more tractable and avoids the problem of forecasting the forecasts of others
when traders do not observe all signals. He and Wang (1995) have a discrete time model with an
infinite number of traders. Arora and Ou-Yang (2000) adopt a similar strategy and offer a
continuous time model with differential information. Unlike all of these rational expectations
models, out model does not assume a common prior. In this respect, ours is similar to Harris and
Raviv (1993), Odean (1998), and Kyle and Wang (1998). Our general equilibrium model is
similar to Gatev and Ross (2000) and Brennan and Xia (1998). Although we do not address
learning in the model, the framework is one in which the econometrics of learning can be
discussed in a precise manner. We show that small differences in beliefs can persist for long
periods of time when learning from past experience is introduced.
The plan of this paper is as follows. Section 2 lays out the continuous-time model. Section 3
shows that a simple closed-form solution for the equilibrium exists.
Section 4 discusses
aggregate optimism, relative optimism, aggregate overconfidence, and relative overconfidence.
Section 5 show that market liquidity is higher with more relative overconfidence. Section 6
discusses correlations of levels and changes of dividends and prices. Section 7 discusses learning
about model parameters.
Section 8 concludes with a discussion of the advantages of the
continuous-time approach.
2. THE MODEL
We consider a continuous-time economy with N infinitely-lived competitive investors with
additively separable exponential utility. Each trader maximizes Et 
 

t
e t  RAct dt  , where the

time preference parameter and the degree of risk aversion RA are the same for all traders. There
is a single risky asset and a risk-free asset with constant interest rate r (representing an intertemporal production technology). The risky asset has constant per capita supply y , and there is
no noise trading.
The risky asset generates cash flows Dt which depend on N signals
denoted S j ,t . We think of Dt as earnings which are fully paid out. The N traders agree that the N
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signals follow Ornstein-Uehlenbeck processes with identical mean reversion parameter S,
identical standard deviation  s , and identical mean equal to zero:
dSi ,t   s Si ,t dt   s dBi ,t
(1)
Here Bi ,t denotes standard Brownian motions, independent across signals i.
The traders disagree about how the signals are related to the dividend process. In particular, each
trader has a symmetrically different model of dividend growth dDi,t, according to which the j-th
trader believes the publicly observed dividends follow the process


dD j ,t   D  Dt  D j   w1S j ,t  w2  Si ,t  dt   D dBD,t
i j


(2)
Traders agree about value of the mean reversion parameter of dividends  D (with  D  0 ), the
volatility of dividends  D (with  D  0 ), the weight on the important signal w1 , and the weight
on the less important signals w2 . They disagree about which signal is most important. The idea
that the j-th trader believes that the j-th signal is more important than the other signals is captured
by the assumption
w1  w2  0
(3)
Traders also disagree about the level to which the dividend mean-reverts, with the j-th trader
believing that the dividend reverts to Dj. The standard Brownian motion BD,t is distributed
independently from the signals Si,t..
For benchmarking purposes, we assume that the true model is one in which all signals have equal
weight, denoted w , and dividends revert to a steady-state level D . Thus, the true dividend
process dDt follows


dDt    D  Dt  D   w Si ,t  dt   D dBD ,t
i


(4)
The true model is the special case of the model with different beliefs, in which the traders agree
that w1  w2  w0 , and in which D j  D for all j.
There are two different sets of assumptions that we make about the observability of the signals.
First, we can assume that the j-th signal Sj,t is private information observed only by the j-th trader.
Second, we can assume that all the signals are public information. We show below that in a
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symmetric equilibrium, the equilibrium price will not depend on which assumption is made,
because the price reveals all useful information about other signals. If the signals are all public
information, we can think of them as published analysts’ forecasts of dividend growth.
Otherwise, we can think of them as private forecasts of dividend growth. The forecasts are
unbiased if w0  1 . The assumption that w1  w2  0 captures the idea that each trader has a
“favorite” analyst, whose forecast he believes to be more significant than the forecasts of the
other analysts. Notice that the assumptions about the signals imply that the analysts’ forecasts are
de-meaned (with traders agreeing on how to demean the forecasts), that the forecasts are
detrended for the drift associated with dividends being above or below the long-term mean, and
that the signals are orthogonalized relative to each other.
There is some apparent modeling inelegance in the assumption that the dividend process is “hardwired” to the information process. In particular, if we change the weights w1 and w2 , not only
does the information structure change, but the univariate dividend process changes also. For
example, with larger values of w1 and w2 , there is more autocorrelation and variance in levels of
the dividend process.
While this may seem inelegant from a modeling perspective, the
reasonableness of this assumption is ultimately an empirical question. Traders who put a great
deal of confidence in analysts forecasts may in fact believe that levels of earnings trend more than
traders who put little weight on analysts’ forecasts. Notice that because of our continuous-time
formulation, it is important that traders agree on the instantaneous standard deviations of the
signals  s and the standard deviation of the dividend  D . Traders can estimate the instantaneous
variances
1
N
with
 D
N
n 1
t  ( n 1) t
arbitrarily
large
precision
using
sample
variances
of
the
form
 Dt  nt  / t , where the sample variances are calculated with many
2
observations over small intervals (i.e., N t is small and N is large). But parameters defining the
drift (like w1 and w2 ) cannot be estimated accurately using data over short time periods, and this
makes it reasonable to assume that traders disagree about these values.
Let Vj,t , the “risk neutral” value of the risky asset, denote the expected present value of dividends
(discounted at the risk-free rate) according to the j-th trader’s model. It is straightforward to
show that the risk neutral value is a modified version of Gorden’s growth formula
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   r u t

V j ,t  E j ,t   e   D  u  du 
u t

1
1
1
1
 Dj 
Dt  D j  

r
r  D
r  D r  S


 w1S j ,t  w2  Si ,t  .
i j


(5)
Similarly, let Vt denote the expected present value of dividends (discounted at the risk-free rate)
according to the true model. We change D j to D and change w1 and w2 to w to obtain
   r u t

Vt  Et   e   D  u  dt 
u  t

1
1
1
1
 D
Dt  D  
w Si ,t .

r
r  D
r  D r  S i
(6)
3. EQUILIBRIUM
To calculate the equilibrium for the model with disagreement, we conjecture that the equilibrium
price Pt is the average of the traders’ risk neutral valuations, less a constant to account for risk
aversion in the presence of positive supply. The conjectured equilibrium price is therefore the
linear function of dividends Dt and signals S j ,t given by
Pt 
1
N
V
i ,t
 ky
i

1
r
1
N
 Di 
i
1
r  D
1
1


1
 Dt  N  Di   r   r  
i


D
S

1
N
w1 
N 1
N
w2   Si ,t  ky ,
(7)
i
where k is an still-to-be-determined constant.
THEOREM: An equilibrium of the conjectured form exists (with k defined below).
OUTLINE OF PROOF: While traders agree on the constant variance of the returns process (in
dollars per share), traders disagree about the drift, with each trader believing that the expected
return per share is a linear function of the difference between his favorite signal and the average
of all traders signals. Furthermore, each trader believes that this difference follows a linear
process (Ornstein-Uehlenbeck process), which defines a state variable for expected returns. With
exponential utility and normal random variables, the solution for the trader’s optimal portfolio
strategy (as discussed by Merton and Campbell and Kyle [1993]) is a linear function of the state
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variable, which usually involves a hedging component based on the correlation between
innovations in the state variable and innovations in returns. Under our symmetric assumptions,
however, the hedging component of asset demands is zero because innovations in returns are
uncorrelated with innovations in the state variable. This leads to a simple closed-form expression
for each trader’s asset demand. Markets clear because the state variables always sum to zero
across traders (since they are by definition deviations from their own cross sectional average
across traders).
PROOF: Let dQ j ,t denote the return on a fully levered portfolio long one share of the risky
asset, calculated using the j-th investor’s beliefs about the dividend process. Letting dPj ,t denote
the j-th traders beliefs about the evolution of prices (implied by his beliefs about the evolution of
dividends), it is straightforward algebra to show
dQ j ,t  dPj ,t  Dt t  rPt t
 rky t 

D 
w  w2 


D j  N1  Di  t  1
S j  N1  Si ,t  t


r  D 
r  D 
i
i


1
1
1
 D dBD 
r  D
r  D r  S
(8)
 N1 w1  NN1 w2   S  dBi,t .
i
ˆ denote the deviation of the j-th trader’s steady state level of dividends from
Let the constant D
j
the average of all traders:
Dˆ j  D j  N1  Di
(9)
i
Similarly, let the stochastic process Sˆ j ,t denote the deviation of the j-th trader’s “favorite” signal
from the average of all traders’ signals:
Sˆ j ,t  S j ,t  N1  Si ,t
(10)
i
Notice that Sˆ j ,t is a state variable which defines the j-th trader’s investment opportunity set.
The j-th trader’s investment opportunity dQ j ,t and the state variable Sˆ j ,t have a simple bivariate
linear structure (as studied by Merton and Campbell and Kyle [1993]). We can write


dQ j ,t  0  1Sˆ j ,t t   Q dBQ
(11)
dSˆ j ,t   S Sˆ j ,t t   S dBS
(12)
-7-
where  j ,0 , 1 ,  Q , and  S are constants defined by
 j ,0  rky 
1 
D ˆ
Dj
r  D
(13)
w1  w2
r  D
1  2  1 
 D  
Q 

r  D 
 r  S 

Q   S
2

1
N
(14)
w1 
N 1
N
w2 
2

 S2 


1
2
(15)
N 1
N
(16)
and dBQ ,t and dB j , S ,t are standardized Brownian motions defined by
dBQ,t 
1  1
1
1
 D dBD 

Q  r  D
r  D r  S
dB j ,S ,t 
1
S

 N1 w1  NN1 w2  S  dBi,t 
i



 S  dB j ,t  N1  dBi ,t 

i j

(17)
(18)
These formulas are obviously greatly simplified by the symmetry assumptions. Furthermore, it is
easy to show that (as a result of the symmetry assumptions) the Brownian motions dBQ ,t and
dB j , S ,t are independently distributed. From Merton or Campbell and Kyle 1993], the j-th traders
asset demand, denoted y j ,t , is given by
y j ,t 
0  1Sˆ j ,t
rRA Q2
(19)
To implement this demand, the j-th trader must observe Sˆ j ,t , the deviation of the j-th signal from
the average signal. This implementation can obviously occur if all signals are public information.
Alternatively, if signals are private information, then the average signals can be extracted from
the publicly observed price and the publicly observed dividend. Define the constant k by
k  RA Q2
In terms of exogenous variables, the j-th trader’s asset demand is given by
-8-
(20)
y j ,t  y 
 r   D   D Dˆ j   w1  w2  Sˆ j ,t 
2


 1  1
2
2
2
N 1

rRA   D  


N w1  N w2   S


r


S 



(21)
ˆ and Sˆ are both by definition deviations from averages across traders, both sum to
Since D
j
j ,t
zero across traders. Thus, markets clear because average demand equals per capital supply. This
completes the proof of the theorem.
4. OPTIMISM AND OVERCONFIDENCE
Let us discuss optimism and overconfidence by comparing the equilibrium price and returns
process with the benchmark case where all traders believe the true model. The actual returns
process (assuming traders have incorrect models), denoted dQt , is given by
dQt  dPt  Dt t  rPt t
 rky t 

N 1
1
D  1


N w1  N w2  w  1
D

D

t

N  i
N  S i , t  t



r  D  i
r  D

 i

1
1
1
 D dBD 
r  D
r  D r  S
(22)
 N1 w1  NN1 w2   S  dBi ,t .
i
An econometrician who observes a very long history of prices and dividends (and inferred the
average signal from a linear combination of price and dividend) can estimate accurately the
parameters of the actual returns process. Note that the true returns process is not relevant for
calculating the equilibrium, because the equilibrium depends only on traders beliefs, not on
whether the beliefs are correct. To convert the formula for the the actual returns process dQt into
the j-th trader’s perceived returns process dQ j ,t , three changes are made. First, the true steadystate dividend mean D is replaced by the j-th trader’s belief D j . Second, the average signal
1
N
S
i
is replaced by Sˆ j , the deviation of the j-th trader’s signal from the average signal.
Third, the coefficient on the average signal is changed from   N1 w1  NN1 w2  w to w1  w2 .
The innovations in the actual returns process are identical to the innovations in the j-th trader’s
perceived returns process. Because of our continuous time formulation, the j-th trader’s perceives
variances and covariances (innovations) correctly, while he perceives means (drifts) incorrectly.
-9-
Traders (on average) exhibit “aggregate optimism” if the average forecast of the steady state level
of dividends is greater than the true steady state level, i.e. if
1
N
D  D
i
(23)
i
Aggregate optimism makes the price higher than it would be if traders believed the true model.
As a result, actual expected returns are correspondingly lower.
The j-th trader exhibits “relative optimism” if he believes the steady state level of dividends is
ˆ  0 or, equivalently,
higher than the average trader, i.e., if D
j
Dj 
1
N
D
i
(24)
i
A trader exhibits relative pessimism if the inequality is reversed. A trader with relative optimism
holds a larger position (on average) than a trader with relative pessimism, because he believes
expected future dividends will be higher. Since the market consists of relative optimists and
relative pessemists who are, on average, neither optimistic nor pessemistic (by definition),
relative optimism and pessimism do not affect the actual returns process.
Traders exhibit “aggregate overconfidence” if the average weight on on signals is greater than the
true weight, i.e. if
1
N
w1  NN1 w2  w
Traders exhibit “aggregate underconfidence” if the inequality is reversed.
(25)
When there is
aggregate overconfidence, the equilibrium price puts a greater weight on signals than would be
the case if all traders believed the true model. As a result, price changes respond more to
innovations in signals than in the true model. Aggregate overconfidence increases the volatility
of returns relative to the true model. In addition, expected returns become a linear function of the
average signal, with expected returns low when the average signal is high, and vice versa. In
other words, aggregate overconfidence leads to excess volatility and mean reversion of prices.
Traders exhibit “relative overconfidence” if w1  w2 , i.e. the j-th trader believes the j-th signal is
more important in determining dividend growth than the other signals. Note that (unlike relative
optimism), our symmetry assumptions imply that all traders are relatively overconfidenct.
- 10 -
Relative overconfidence leads to trading volume. Let trading volume dy 2j ,t be measured as
variance per unit of time. Trading volume is given by
dy j ,t 
2
 w1  w2   2
2
S

 1 
r 2 R A2   D2  


 r  S 

2
 N1

2
w1  NN1 w2   S2 


2
dt
(26)
The square root of trading volume is proportional to the degree of relative overconfidence, as
measured by w1  w2 , proportional to the variance of signals  S2 , inversely proportional to the
squared degree of risk aversion RA2 , and inversely proportional to the dollar variance of price
changes. Note that differences in relative optimism make traders hold different average positions,
but do not make traders trade once the position is established. Relative overconfidence induces
traders to trade, but does not affect their average position. Relative overconfidence affects a
trader’s perceived returns process, but not the true returns process (which is affected by aggregate
overconfidence). Without relative overconfidenc (or underconfidence), trading volume is zero:
traders follow different buy-and-hold strategies based on different degrees of relative optimism
and pessimism.
A trader is “bullish” if he holds a position larger than the average endowmwent, and “bearish” if
he holds a position less than the average endowment. According to this definition of bullishness,
the market is always a mixture of bulls and bears. Relatively optimistic traders tend to be bullish
more often than they are bearish, but a relatively pessimistic trader becomes bullish when there is
relative overconfidence and his relative signal Sˆ j ,t becomes large enough.
Another interesting benchmark case occurs when there is relative optimism and relative
overconfidence, but the degree of aggregate optimism and the degree of aggregate
overconfidence is the same as when all traders believe the true model. In this benchmark case,
there may be relative optimists and relative pessimists, but the average belief about the steadystate level of dividends is correct.
Similary, the j-th relatively overconfident traders may
overweight the j-th signal relative to the true model, but the overweighting is exactly offset by the
underweighting of the same signal by the other n-1 traders. In this case, even though no trader
believes the true model, the equilibrium price is the same as it would be if all traders believed the
true model, because the inaccuracies in their models exactly cancel out in the trading process. In
- 11 -
this benchmark case, the actual returns process is the same as if all traders believe the true model
because beliefs are correct on average, even though all traders are equally relatively
overconfident and no trader’s beliefs are actually correct.
5. MARKET LIQUIDITY
Let us define market liquidity as the slope of the net demand curve of all investors combined. If
all signals are public information, the model (as developed above) does not have a well-defined
demand curve, because it is a zero probability event for the price to be different from what the
traders expect on the basis of the signals. We could, however, think of the dependence of price
on per capital supply y as defining a demand curve. The slope of the inverse aggregate demand
curve defined this way is
dy i ,t
 dP

i
1
1

k
RA Q2
(27)
This equation says that a permanent increase in per capita supply reduces prices by an amount
inversely proportional to the degree of risk aversion and price volatility.
Consider, however, the case where the j-th signal is private information to the j-th trader.
A
demand curve depending on the market price can be constructed by extracting the average signal
from price and dividend and substituting into the formula for the equilibrium quantity demanded
by the j-th trader. The slope of the aggregate inverse aggregate demand scheduled obtained from
this approach is given by
dy i ,t
 dP
i

 w1  w2 
1
1
rRA  r   S   N w1  NN1 w2 
2
Q
Market liquidity is proportional to the degree of relative overconfidence w1  w2 .
(28)
When
w1  w2  0 , there is no relative overconfidence, and markets are infinitely illiquid. Thus, in the
same way that relative overconfidence creates trading volume, it also creates market depth.
6. CORRELATION OF DIVIDENDS WITH PRICES
Our models provides a simple intuitive explanation for why there can simultaneously be a large
correlation between the levels of dividends and prices and a small correlation between changes in
- 12 -
dividends and changes in prices. This difference emerges when the mean reversion parameter of
signals  S is much greater than the mean reversion parameter of dividends  D , i.e. when trders
trade on the basis of short-term forecasts about a dividend process which itself is very persistent.
It is straightforward to show that the correlation between changes in dividends and changes in
prices, denoted c, is given by
c
1
1/ 2
2 
  1 2
2 S
N 1
1
1  

w

w


  r   S  N 1 N 2  D2 


(29)
Similarly, the correlation between levels of dividends and levels of prices, denoted d, is given by
d
1
1/ 2
2 
  1 2
2 D  S
N 1
1
1  

w

w


  r   S  N 1 N 2  S  D2 


(30)
These expressions differ only by the presence of the factor  D  S in the expression for d.
Suppose traders choose signals with a half-life of one or two years, similar to the horizon of
analysts forecasts. Then  S is likely to have a value like 0.5 or 1.0. Suppose innovations in
dividends persist for many years, even decades. Then  D is likely to be as small as 0.05 or 0.1.
Thus, it is not unreasonable for
 D  S to be as small as 0.10. This difference has an
economically significant tendency to make d much larger than c.
7. LEARNING
In our model of optimism and overconfidence, the optimistic or overconfident traders never learn
that their overconfidence is not justified. The j-th trader holds fixed prior beliefs (with infinite
ˆ , and all traders hold the same
precision) about his degree of relative optimism or pessimism D
j
fixed beliefs about the overconfidence parameters w1 and w2 . Traders do not adjust their beliefs
over time as new information arrives (in the form of signals, prices, and dividend news). The
standard argument made against models of overconfidence is that traders should eventually learn
that their overconfidence is not justified. We address this argument by showing that modest but
economically significant and realistic degrees of optimism and overconfidence, held with tight
priors, can survive for many decades, even hundreds of years. Our basic point is an appropriately
- 13 -
modified version of familiar material. We take a Bayesian approach, which becomes more
intuitive when Sharpe ratios are used.
Let us review the familiar material first. Suppose that there is no relative overconfidence. Then
the average signal does not forecast returns, and returns have the form
dQt  dPt  Dt t  rPt t
(31)
 Q dt   Q dBQ ,t .
Now consider the problem faced by and Bayesian econometrician-trader who knows the volatility
parameter  Q with perfect certainty, but does not know the risk premium parameter Q . Define
the Sharpe ratio, denoted Q , by Q  Q  Q . The Sharpe ratio shows up as the drift of the
returns process when scaled by the standard deviation:
dQt /  Q  Q dt  dBQ ,t .
(32)
Let Q ,t denote the econometrician’s prior mean at date t. If the econometrician began with a
diffuse (normal) prior and based his prior on H years of historical data, his prior mean at date t is
the historical average
Q ,t
1

H
t

dQt /  Q .
(33)
tH
The statistic Q ,t can be thought of as an OLS regression coefficient obtained by regressing
dQt /  Q on a vector of ones.
X X 
t

t H
1 dt  H and X Y 
In the familiar formula
t

Q ,t   X X  X Y , we have
1
dQt /  Q .
tH
With a diffuse prior, his posterior variance (which is the variance of the residuals in the OLS
regression) is 1 H , i.e. the use of Sharpe ratios denominates the posterior precision (reciprocal of
variance) in units of time. Even if the econometrician came up with his prior at time t using
different methods, we can still think of the prior variance as being measured in units of time.
Going forward from date t, suppose that the econometrician learns from future returns. Given his
prior distribution Q ,t at date t, his posteror distribution at date t+T has mean
- 14 -
Q,t T 
t T

1 
H


 Q,t  dQt /  Q 
H T 
t

(34)
and variance 1  H  T  . Conditional on knowing both the prior mean Q ,t and the true Sharpe
ratio Q , the posterior mean at t+T is distributed
Q,t T
N ( HHT Q ,t  HTT Q ,
T
 H T 2
)
(35)
Since the posterior mean is a time-weighted average of the prior mean and the truth, we can see
that it takes T  H years for the expected revision of the prior to take in half-way to the truth.
Now suppose the trader’s prior Sharpe ratio is a modest value given by Q,t  0.10 and the prior
precision is such that the trader view it as a two-standard deviation event to learn that the true
Sharpe ratio is Q  0 . This implies H  1  0.05   400 . Then it takes 400 years for the
2
trader’s expected revision is his prior to take it from 0.10, to 0.05 (half-way to the truth). If the
trader view learning the true prior was less than zero as a one-standard deviation event, the
revision is expected to take place in 100 years. It takes a long time for a confident belief in a
modest Sharpe ratio to be revised significantly.
Now let us apply the same approach to learning by traders in our model. There are two issues
which make the analysis different. First, our model is framed in such a way that traders have
different beliefs about forecasts of dividends, and their differences in beliefs about prices and
returns derives solely from their differences in beliefs about dividends. For this reason, traders
learn about the accuracy of their forecasting ability by observing dividends, and price
observations are irrelevant to the learning process. In fact, since traders disagree about dividends,
they would prefer to trade futures contracts on dividends to trading the risky asset itself. We
implicitly assume that enforcability issues prevent trading in futures contracts on dividends.
Second, when traders are relatively overconfident (but not when they are optimistic or
pessimistic), their forecasts of dividend growth (and therefore Sharpe ratios) are changing over
time according to the deviation of their signal from the average. This introduces randomness into
the rate at which traders learn. A trader learns quickly when his signal deviates greatly from the
average across all traders, and learns slowly when his signal is near the mean.
Consider first the case of a relatively optimistic trader. To focus on optimism, suppose that there
is no aggregate overconfidence and no relative overconfidence, i.e. w1  w2  w . If we add back
- 15 -
in the mean reversion term for dividends, subtract out the growth rate expected from signals, and
scale by the standard deviation of dividends, we obtain dy j ,t defined by
dy j ,t 
dD j ,t   D  Dt  D j  dt  w Si ,t
i
D

 D Dˆ j
dt  dBD ,t
D
(36)
If we define  j by
 D Dˆ j
j 
D
(37)
we can think of  j as a pseudo-Sharpe-ratio. It is what the trader’s Sharpe ratio would be if
traders could trade futures contracts on dividends (conjecture to be proven). Since  j is the
ˆ scaled by a known constant, statistical inference about  is equivalent
optimism parameter D
j
j
ˆ , so we adopt similar notation. Given his prior distribution 
to statistical inference about D
j
j ,t
at date t, his posteror distribution at date t+T has mean
 j ,t T 
t T

1 
H


dy

j ,t
t 
H T 
(38)
and variance 1  H  T  . Conditional on knowing both the prior mean  j ,t and the true Sharpe
ratio, which is by definition zero, the posterior mean at t+T is distributed
 Q,t T
N ( HHT  Q ,t ,
T
 H T 2
)
(39)
Notice that as new data comes in, all traders will revise their estimates of relative optimism in the
same direction. This will actually result in a change in aggregate optimism, with degrees of
relative optimism converging towards zero. For the same reasons as discussed above, this
convergence towards zero will occur slowly when the pseudo-Sharpe-ratios are already close to
zero.
It is instructive to compare the pseudo-Sharpe-ratio with the actual Sharpe ratio in the market.
Under the same assumptions, scaled returns can be written
dQ j ,t /  Q   j ,t cdt  dBQ ,t
(40)
where c is the instaneous correlation betwenn innovations in dividends and innovations in returns.
Since c is by definition less than one (perhaps approximately 0.20), the perceived actual Sharpe
ratio is less than the pseudo-Sharpe-ratio. This undermines somewhat our argument that non-zero
- 16 -
perceived Shape ratios can persist for a long time. Because dividends are a better instrument for
learning about trading ability than returns histories, Sharpe ratios will be even lower than in the
familiar case discussed first.
Now consider learning about relative overconfidence. To focus on relative overconfidence,
suppose there is no relative optimism. Suppose further that as traders adjust w1 and w2 so that
there is no aggregate overconfidence, i.e.
1
N
w1  NN1 w2  w . If, as before, we add back in the
mean reversion term for dividends, subtract out the growth rate expected from the average signal,
and scale by the standard deviation of dividends, we obtain dy j ,t defined by
dy j ,t 
dD j ,t   D  Dt  D j  dt  w Si ,t
i
D
 w1  w2  Sˆ j ,t

D
dt  dBD ,t
(41)
The statisitical analysis changes because the relative overconfidence measure w1  w2 is
multiplied by the random factor Sˆ j ,t in the drift term. Let us define regression variables x j ,t by
2 S
x j ,t  Sˆ j ,t
(42)
S
The regression variable x j ,t is scaled so that its unconditional variance is one. Define the square
root of the expected squared pseudo-Sharpe-ratio  j by
j 
 w1  w2   S
D
(43)
2 S
The parameter  j is the the relative overconfidence measure w1  w2 scaled by a known
constant. With these definitions, dy j ,t is can be written
dy j ,t   j x j ,t dt  dBD ,t
(44)
Let  j ,t ,t T denote the OLS regression estimate of  j using dividend data and forecasting
variable Sˆ j ,t . The OLS estimate  j ,t ,t T is given by
t T
 j , t , t T 
x
t
t T
j ,t
dyD ,t
 j 
t T
x
2
j ,t
dt
t
x
t
dBD ,t
(45)
t T
x
t
- 17 -
j ,t
2
j ,t
dt
t T
and this estimate has precision

x 2j ,t dt . Note that the precison is a random variable with
t
expectation T, because x j ,t has been scaled to have unit variance. Learning takes place at a
random rate, which is high when x j ,t is far from its mean and slow when x j ,t is close to its
mean.
Given the trader’s prior distribution  j ,t at date t, his posteror distribution at date t+T has mean
H   j ,t 
 j , t T 
t T
x
t
2
j ,t
dt   j ,t ,t T
(46)
t T
H
x
2
j ,t
dt
t
t T
and precision H 

x 2j ,t dt . Conditional on knowing both the prior mean  j ,t and the true
t
pseudo-Sharpe-ratio, which is by definition zero, the posterior mean at t+T is distributed
 j , t T






H
H
N
 j ,t ,

t T
2
t T

 
 H  x 2 dt
2
t j ,t
 H   x j ,t dt  

t

 

(47)
As new data comes in, traders will revise their estimates of relative optimism in different
directions because the relative signals Sˆ j ,t (proportional to x j ,t ) are different for different
traders, but the revisions will over time tend make the posterior mean drift towards zero. In
addition, traders will revise their posterior variances differently. Thus, learning from a symmetric
starting point will lead to asymmetric results. For the same reasons as discussed above, this
convergence towards zero will occur slowly when the pseudo-Sharpe-ratios are already close to
zero.
The comparison of the pseudo-Sharpe-ratio for the case of relative overconfidence is similar to
the compariosn for relative optimism. Scaled returns can be written
dQ j ,t /  Q   j cx j ,t dt  dBQ ,t
- 18 -
(48)
Thus, as before, the actual Sharpe ratio  j c is less than the pseudo-Sharpe-ratio  j by a factor
equal to the c, the instaneous correlation betwenn innovations in dividends and innovations in
returns. Since traders learn faster than from dividend data than from returns data, learning from
dividends will reduce the perceived Sharpe ratio faster than it would be reduced by learning from
returns.
I
8. CONCLUSION
A summary of results about optimism and overconfidence (relative to the case where all traders
believe the true model) is as follows:
1. Aggregate optimism makes prices higher and returns lower, but does not lead traders to
believe they can forecast prices and does not lead to trading volume.
2. Aggregate overconfidence leads to excess volatility and mean reversion in prices, with returns
low when the average signal is high, and vice versa. But aggregate overconfidence does not lead
traders to believe that they can forecast prices and does not lead to trading volume.
3. Relative optimism and pessimism lead traders to hold different average positions, but do not
affect the actual returns process or trading volume.
4. Relative overconfidence has no implications for the actual returns process, but it leads traders
to trade because they believe (incorrectly) that they can forecast returns. Trading volume is
proportional to how overconfident traders are. When traders are neither relatively overconfident
nor underconfident, there is no trade once initial positions are established.
Overconfidence and optimism have effects similar to noise trading, in for example the model of
Campbell and Kyle [1993]. An increase in the mean supply from noise traders lowers prices
similar to aggregate pessimism. An increase in the variance of independent noise trading tends to
create excess volatility and mean reversion, which is similar to aggregate overconfidence. Noise
trading also leads to trading volume, similarly to relative overconfidence.
- 19 -
The dynamic model used in this paper has several advantages over a static model. In a static, two
period models, the distinction between buy-and-hold strategies and dynamic tradings strategies is
hard to capture, and this makes it harder to explain the idea that optimism leads to different buyand-hold strategies while relative overconfidence leads to dynamic trading strategies which
generate trading volume propportional to the degree of relative overconfidence. In comparison
with static models, the dynamic approach also makes it possible to explain why levels of earnings
have much higher correlations with level of prices than short-term changes in earnings have with
short-term changes in prices. This difference in correlations occurs when the half-life of dividend
innovations is much greater than the half-life of innovations in signals of future earnings growth.
The mean reversion coefficients  D and  S capture this idea in a simple and natural manner.
The use of continuous-time techniques (as opposed to discrete time) forces traders to agree on
variances even if they disagree on means. The use of linear models with normally distributed
random variables allows Bayesian learning to be related closely to OLS regressions.
The particular information structure used in this paper hard-wires information about earnings
growth to the univariate dividend process. This has two distinct advantages. First, it gives the
model a very simple, analytical solution for optimal trading strategies. It is a proved result that
these optimal trading strategies have a “myopic” structure, because it is a proved result that the
dynamic hedging component of asset demands disappears. Second, it makes it possible to show
that aggregate overconfidence leads to excess volatility and mean reversion. In a related paper,
Kyle and Lin [2002] develop a model in which the univariate dividend structure does not change
as information changes. In this model, it is shown that overconfidence leads to reductions in
volatility and to persistence in innovations to expected returns. Thus, hard-wiring the information
structure to the dividend structure in a simple manner leads to distinctly different testable
empirical predictions.
Although the model developed in this paper does not allow the traders to learn, it is shown that if
Bayesian learning is introduced, learning slows down dramatically when perceived Sharpe-ratios
become small. Thus, when perceived Sharpe ratios are small to begin with, the assumption that
traders do not learn may not be an unrealistic first approximation.
- 20 -
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