Beauty Contests, Risk Shifting, and Bubbles ∗

H. Henry Cao and Hui Ou-Yang

This Version: February 28, 2013

∗Corresponding author: Cao is with Cheung Kong Graduate School of Business (CKGSB); e-mail: hn-cao@ckgsb.edu.cn. Ou-Yang is with Nomura Securities and CKGSB; e-mails: hui.ou-yang@nomura.com. We thankPeter DeMarzo, Jingzhi Huang, Ron Kaniel, Jing Liu, Hong Liu, Jun Liu, Harold Zhang, and seminar participantsat the 2009 AFA Meetings, CKGSB, HKU, the LBS-LSE-Oxford Adam Smith Asset Pricing Conference, ShanghaiUniversity of Finance and Economics, Tsinghua University, and 2009 China International Conference in Finance forhelpful comments.

Beauty Contests, Risk Shifting, and Bubbles

Abstract

In a beauty contest model, the current stock price depends on investors’ beliefs of interme-

diary prices and the final stock payoff. The equilibrium price will be the same as if there

exists a consensus investor and all investors have the same beliefs as the consensus investor.

Investors who perceive the lowest risk in one period tend to bear more of the financial risk in

that period. As investors who perceive the lowest risk can vary across different periods, the

overall perception of risk is reduced in an economy with dynamic trading, which increases

market liquidity. If investors with lower risk perception of the intermediary prices are more

optimistic about the intermediary prices, the consensus investor will have optimistic expec-

tation about the final stock payoff, and this effect combined with the reduction in market

liquidity will result in a bubble. On the other hand, if investors with lower risk perceptions

of the intermediary prices are more pessimistic about the intermediary prices, the consensus

investor will have pessimistic expectation about the final stock payoff, and it can offset the

reduction in liquidity which result in a negative bubble. We further characterize conditions

under which dynamic trading with heterogeneous beliefs have no effect on the stock price or

market liquidity.

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1 Introduction

In a well known metaphor, when investors have heterogeneous beliefs, Keynes (1936) views

the stock market akin to a beauty contest in which the contestants have to pick the one

that others will pick as the prettiest. Implicit in this argument is that investors care about

not only the dividend flows received by holding the stock but also the gains from trading

with other investors at intermediary prices. Therefore, investors need to forecast the crowd’s

forecasts to formulate their own trading strategies.

The Keynes beauty contest intuition can affect asset prices through two channels: the

cash flow effect and the discount rate effect. In terms of the cash flow effect, investors can

differ in their conditional expectations of the future cash flows of a stock. When investors

believe that pessimism in the crowd will prevail in the market in the future, they may sell

the stock at a very low price today, resulting in a negative bubble. When investors believe

that optimism in the crowd will dominate in the future, they may buy the stock at a very

high price, resulting in a (positive) bubble. In terms of the discount rate effect, investors can

differ in their perception of conditional risks across time. At any time period, investors with

lower perceptions end up bearing more risk. As a result, the perceived risk in the market

reduces when an investor can trade dynamically and believes that he can shift risk to others

at times when his perception of risk is higher than that of the crowd. It follows that the

stock price will increase.

This paper formalizes Keynes’ intuition and analyzes the effects of heterogenous beliefs

on stock prices. We show that even if all investors agree on the expectation of a stock payoff,

as long as they disagree on the risk of the stock payoff, a bubble can still arise due to a

lower risk premium in the stock price. More generally, both bubbles and negative bubbles

can occur depending on the size and sign of the cash flow effect. We further characterize

conditions under which heterogeneous beliefs have no effect on the stock price.

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Specifically, we analyze the effects of future public information disclosure on stock prices

and market liquidity in an economy with divergent interpretation of public information. We

consider a market with a continuum of risk-averse investors with CARA utility of the same

risk aversion. There are T +1 trading dates at time 0, 1, · · · , T − 1 and consumption occurs

at time T . At time t = 1, ..., T − 1, a public signal yt is announced. In the basic model,

there is one risky stock and one riskfree asset available for trading. Investors disagree about

the stock payoff and disagree on how to interpret public information to be released in future

trading sessions. Investors may disagree about both the mean of the public information and

the variance-covariance matrix of the public information and the stock payoff. As a result,

investors have different beliefs about the expectations and the risks of the intermediary

prices. We analyze how dynamic trading, due to the disclosure of future public signals,

affects the stock price and the market liquidity at time 0.

Let yT ≡ v, y(1, t) ≡ (y1, ..., yt), t = 1, ..., T . We show that the equilibrium price is

characterized by the existence of a representative investor such that his belief of y(1, T )

aggregates the beliefs of all investors. The representative investor’s expectation of y(1, T ) is

the precision weighted average of all investors and his precision of y(1, T ) is the average

precision. Such a characterization also implies that the price of the risky stock in the

dynamic trading economy is equivalent to the price of the risky stock in a static economy

in which investors can directly trade on artificially introduced risky stocks with pay off

yt, t = 1, ..., T − 1 in zero supply in addition to the risky stock and risk free asset.

Given the equilibrium price, we characterize the necessary and sufficient conditions under

which dynamic trading due to difference of opinion has no effect on the stock price. Due

to the assumption of joint normality, without loss of generality, we can write y(1, T − 1) =

αy(1,T−1)i+βy(1,T−1)iv+ϵy(1,T−1)i for investor i, where αy(1,T−1)i is a constant vector, βy(1,T−1)i is

a coefficient vector, ϵy(1,T−1)i is random vector and independent of v from the view of investor

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i. We show that if and only if there exist a coefficient vector βy(1,T−1), such that βy(1,T−1)i =

βy(1,T−1) for all i, i.e., all investors agree on βy(1,T−1)i, then differences of opinion has no effect

on the stock price. However, when investors disagree about βy(1,T−1)i, dynamic trading will

affect the stock price. This result also explains why earlier researchers in the literature did

not find any price effect because they all assume that the signals follow a simple form of

yt = v + ϵyti and investors disagree about the mean and precision of ϵyti with the additional

assumption that the variance covariance matrix of ϵy(1,T−1) is diagonal.1 Intuitively, following

Duffie and Huang (1985), dynamic trading can be viewed as an alternative way of trading

on the differences of y(1, T − 1). Consider an economy in which investors can trade directly

on y(1, T − 1). We show that the price of v and demand of v is the same in the static

economy and those in the dynamic economy. In the static economy, if investors agree on

β(y(1,T−1)i, the introduction of y(1, T − 1) is equivalent to the introduction of y∗(1, T − 1) ≡

y(1, T − 1) − βy(1,T−1)v = αy(1,T−1)i + ϵ(y(1,T−1)i. Since y∗(1, T − 1) is orthogonal to v and

CARA utility investors have no wealth effect, trading on y∗(1, T − 1) will have no effect on

the valuation of v. If follows that the dynamic trading due to differential interpretation of

y(1, T − 1) will also has no effect on the price of v.

More generally, when investors have different βy(1,T−1)is, additional trading sessions will

cause the stock price to change. We consider two scenarios under this general case: (i),

investors agree about the mean of the stock payoff and the public information but disagree

on βy(1,T−1)i. As a result, investors have the same expectation of intermediary prices but

different beliefs on the risks of intermediary prices; (ii) investors disagree about mean of

the stock payoff and the public information, and also disagree about βy(1,T−1)i. As a result,

investors have different beliefs about both the expectation and the risk of intermediary prices.

In the first scenario, investors have the same expectation about the intermediary prices

1See for example Kim and Verrecchia (1990, 1991); Brennan and Cao (1996, 1997); Brennan, Cao, Strong and Xu(2005); Cao and Ouyang (2009).

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but different beliefs on the intermediary price risk. Investors, who perceive the lowest risk in

one period, tend to hold more shares of the stock in that period, bearing more of the stock

risk. As investors have different views on risks in different periods, investors who perceive

high price risks in one period may perceive low price risks in the next. As a result, risk is

shifted around to those who perceive it less, and the overall risk perception in the stock is

reduced, which leads to a higher stock price.

In the second scenario, investors have different expectation about the intermediary prices

and different beliefs about the intermediary price risk. In this case, the risk reduction effect

still holds and this alone will increase the stock price. When investors, who are the most

optimistic about the intermediary prices, also perceive the lowest risks of the intermediary

prices, they will push the price up further. On the other hand, when investors, who are the

most pessimistic about the intermediary prices, perceive the lowest intermediary price risk,

they will pull the price down.

We define the intrinsic value for investor i as the the price that would occur if the

economy is populated by investors who all agree with i. Following Harrison and Kreps (1978

We say a bubble exists when the stock price is higher than the highest intrinsic value among

investors and a negative bubble occurs when the stock price is lower than the lowest intrinsic

value in the market.2 Contrary to Harrison and Kreps, who show that a bubble always exists

we show that both a bubble and a negative bubble can exist in the economy. Numerically,

consider an economy with two trading period economy (T = 2) with two groups of investors,

I and II. Group I investors believe that y1 is unrelated to the final stock payoff but have

zero variance on y1. On the contrary, group II investors believe that y1 is related to the final

stock payoff but have finite variance on y1. Then the time 1 price will be a function of y12In Harrison and Kreps (1978) in which investors are risk neutral and short-sales constraints are present, investors

have the highest expectation of next period dividend and price determines the price. However, these investors cansale the stock who became more optimistic in the future. The resale option causes the price to be higher than theprice that would obtain in the absence of dynamic trading. Notice that the price in a static economy would coincidewith the highest intrinsic value among investors.

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and the time zero stock price will be group I’s expectation of time 1 stock price since group

I investors have zero variance on y1. As a result, the time zero stock price will be simply

a linear function of group I’s expectation of the public signal y1. When group I investors

have very high or very low expectations of y1, we will have a bubble or negative bubble

accordingly.

We further define market liquidity as the units of supply needed for the price to change

by one unit in the current period. We show that the market liquidity always increases when

additional trading sessions are introduced. Intuitively, the ability to shift risk around in

the future make investors more willing to bear risk now and thus the market liquidity in

the current period is higher. Moreover, for additional trading sessions to have no effect on

market liquidity, the necessary and sufficient conditions are the same as those for the price

not to change, that is, we must have all investors to agree on βY i.

This paper represents perhaps the first study on how more risk is shifted to the investors

who perceive it less under heterogeneous beliefs. Risk shifting results in a lower perceived

risk in the market. The higher the divergence of investors’ views on the risks of stock payoffs

and public signals, the lower the perceived risk or the higher the stock price. As a result,

the liquidity premium of holding the stock drops.

Our results shed light on potential explanations for the empirical findings of Diether,

Malloy, and Sherbina (2002) and Goetzmann and Massa (2004). For example, Goetzmann

and Massa find that the dispersion of opinion of the investors in a stock, which is proxied by

investors’ age, profession, or income, is positively related to contemporaneous returns and

negatively related to its future returns. They also find evidence that dispersion of opinion

aggregates across individual stocks and generate a market-wide factor that affects stock

returns.

Our paper is related to the seminal paper of Harrison and Kreps (1978), who show that a

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bubble can form due to resale options in a market with risk-neutral investors and short-sales

constraints.3 In their model, the stock is held by those investors who are the most optimistic

about the stock payoff in every future state. Even if an investor has the highest expectation

or valuation about the stock payoffs at time 0, he does not necessarily have the highest

valuation for the stock in every future state. As a result, there is a resale option as investors

can buy the stock expecting that they can sell it to investors who are more optimistic in

some future states with a positive probability. Consequently, the stock price is higher than

the buy-and-hold value of all investors, or a bubble arises.

The Harrison-Kreps (1978) model and all of its extensions focus on the expectation part

of stock prices. For example, this literature illustrates that under heterogeneous expecta-

tions, the price of a stock can be higher than the valuation of the most optimistic investor.

In addition, the assumption of risk neutrality also makes it difficult to address how hetero-

geneous beliefs about risk affect stock prices.

A large literature following Harrison and Kreps (1978) has further analyzed the effects

of heterogeneous beliefs on various behaviors of stock prices. Scheinkman and Xiong (2003)

solve the resale option value in close form. They obtain an interesting result, that is, the

bubble is only mildly affected by transaction cost, as investors can refrain from trading

frequently when transaction cost increases. In an important insight, Allen, Morris, and Shin

(2006) demonstrate that under heterogeneous expectations due to private information, the

law of iterated expectations may not hold for the average expectations. They show that this

insight helps to explain over-reaction to (noisy) public information in a myopic model. Cao

and Ou-Yang (2009a) show that under certain conditions on investors’ expectations of other

investors’ conditional expectations of future dividends, bubbles can arise without short-sales

3In a static model, Miller (1977) argues that short-sales constraints can bias the stock price upward as investorswith pessimistic views are sidelined, or only optimistic investors participate in the market. Jarrow (1980) shows thatMiller’s argument holds in a single risky security world but additional conditions are required in a multiple securityeconomy.

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constraints.4

The rest of this paper is organized as follows. Section 2 describes the basic model.

Section 3 extends the model to differential priors and multiple stocks. Section 4 concludes

the paper. The appendix contains technical proofs.

2 Basic Model

We consider a T trading session model, with a time line of 0, 1, · · · , T . There is one risk free

asset and one risky stock available for trading. It is assumed that the financial market is

populated by investors with the population size normalized to one, each indexed by i where

i ∈ [0, 1]. At time 0, we assume that each investor is endowed with xi units of the stock and

zero units of the bond. Without loss of generality, the interest rate is taken to be zero. The

stock payoff at time T is v. The per capita supply of the stock is a positive number denoted

by x.

To obtain closed form solutions, we assume that each investor i has a negative exponential

utility function, − exp(−γWTi), where γ is his risk aversion coefficient andWTi is his terminal

wealth or consumption. We assume that for all investors, v is normally distributed and

investor i believes that the unconditional mean of v is µvi, the unconditional variance of v

is Σvi, and the unconditional precision of v is Πvi = 1/Σvi.

Investors first trade in session 0. In session t, t = 1, 2, · · · , T − 1, public signal yt is

revealed. Let µyti denote investor i’s expectation of yt at time 0. To simplify the notations,

let yT ≡ v. Let

y(t, T ) = [yt, yt+1, · · · , yT−1, yT ]′, t = 1, ..., T.

4Asset bubbles may also arise in other settings, such as those of Allen and Gorton (1993), Allen, Morris, andPostlewaite (1993), Spiegel (1998), Abreu and Brunnermeier (2003), and DeMarzo, Kaniel, and Kremer (2008). Otherstudies that employ differences of opinion include Varian (1989), Detemple and Murthy (1994), Morris (1996), Odean(1998), Zapatero (1998), Basak (2000), Duffie, Garleanu, and Pedersen (2002), Kyle and Lin (2002), Viswanathan(2002), Hong and Stein (2003), Qu, Starks, and Yan (2003), Buraschi and Jiltsov (2006), Hong, Scheinkman, andXiong (2006), and David (2008).

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Investor i’s variance-covariance matrix and precision matrix of y(t + 1, T ) at time t are

denoted by Σy(t+1,T )ti and Πy(t+1,T )ti, respectively. Let

PT ≡ v, Rt = Pt − Pt−1 for t = 1, 2, · · · , T,

R(t, T ) = [Rt, Rt+1, · · · , RT ]′ for t = 1, · · · , T.

We characterize the equilibrium prices and demands in the multiple trading session economy

as follows.

Theorem 1 There exists a consensus investor who believes that y(t+ 1, T ) follows a multi-

variate normal distribution at time t. The consensus investor’s updates his conditional beliefs

as a Bayesian and his expectation, variance-covariance matrix, and precision matrix of y(t+

1, T ) are denoted by µy(t+1,T )tc, Σy(t+1,T )tc, and Πy(t+1,T )tc, respectively. They are given by

µy(t+1,T )tc ≡ Etc[y(t+ 1, T )] = Π−1y(t+1,T )tc

∫i

Πy(t+1,T )tiµy(t+1,T )tidi, (1)

Πy(t+1,T )tc =

∫i

Πy(t+1,T )tidi, (2)

Σy(t+1,T )tc ≡ Vartc[y(t+ 1, T )] = Π−1y(t+1,T )tc, (3)

µy(t+1,T )ti ≡ Eti[y(t+ 1, T )], (4)

Πy(t+1,T )ti = Σ−1y(t+1,T )ti, (5)

Σy(t+1,T )ti ≡ Varti[y(t+ 1, T )]. (6)

Given the beliefs of the consensus investor, there exists an equilibrium in which the prices

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(Pt) and demands (Dti) for the stock at time t = 0, 1, · · · , T − 1 are described as

Pt = µvtc − γΣvtcx, (7)

Dti =1

γ

T∑s=t+1

ΠRt+1RstiµRsti, (8)

µRsti ≡ Eti[Rs], s = t+ 1, · · · , T, (9)

ΠR(t+1,T )ti = Var−1ti [R(t+ 1, T )]. (10)

Here µvtc is the consensus investor c’s conditional expectation of v at time t and corre-

sponds to the last element in the (1× (T − t)) column vector µy(t+1,T )tc, Σvtc is c’s conditional

variance of v at time t and corresponds to the ((T−t)×(T−t)) element in Σy(t+1,T )tc matrix,

and ΠRt+1Rsti is the corresponding element of the first row of ΠR(t+1,T )ti.

The price is the same as if all investors share the same belief as the consensus investor.

The first term represents the conditional expectation of the consensus investor while the

second term represents the liquidity premium of the consensus investor at time t. Investor

i’s demand function is more complicated. To understand the intuition of investor i’s demand,

we can consider the special case of T = 2. The first term is driven by i’s demand due to

immediate expected price appreciation. The next term depends on the co-precision of the

immediate return and future returns. For example, suppose that the return at time 2 and the

return at time 1 are negatively correlated, then the co-precision will be positive. Investors

will be willing to buy more today as they can unwind their position when the next period

price is high and hold on to their positions when the next period price is low. On the other

hand, when the return at time 2 and time 1 are positively correlated, investors will be buying

less today as they have the opportunity to buy more in time 1 when the return is high, and

when the return is low, their loss from the initial holding is also limited.

In the special case in which we can write v = µvi+∑T−1

t=1 (yt−µyti)+ηvi, and y1, y2, ...yT−1, ηv

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are independent for all i, then the returns will be independent across all periods and we the

hedging demands will disappear and we have the following expression:

Dti =1

γΠRt+1Rt+1tiµRt+1ti, (11)

Here the myopic demand is optimal and we will revisit this case later in a numerical example.

It is intriguing that the consensus investor in the dynamic trading economy has such a

simple form. Duffie and Huang (1985) had argued that dynamic trading can be viewed as

a way to complete the market. An alternative to dynamic trading is to add more assets to

make the market more complete. To understand better the construction of the consensus

investor, we consider a static economy in which new assets with zero net supply with payoffs

of (y1, ..., yT−1) are introduced to the economy at time 0.. In this static economy, investors

can trade directly on T risky assets with payoffs y(1, T ) ≡ (y1, ..., yT−1, yT ≡ v). Investors

trade at time 0 and the market is closed afterwards. Consumption occurs at time T . Let

xy ≡ (0, ..., 0, x) denote the T × 1 supply vector of the y(1, T ). In the following theorem we

show that the price of v obtained in the static economy coincides with that in the dynamic

trading economy.

Theorem 2 In the static economy with the introduction of the affiliated assets y1, ..., yT−1,

there exists a consensus investor c whose belief coincides with the consensus investor in the

dynamic trading economy at time 0.

µy(1,T )c ≡ Etc[y(1, T )] = Π−1y(1,T )c

∫i

Πy(1,T )iµy(1,T )idi

Πy(1,T )c =

∫i

Πy(1,T )idi

Σy(1,T )c ≡ Varc[y(1, T )] = Π−1y(1,T )c

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Py(1,T ) = µy(1,T )c − γΣy(1,T )cxy (12)

Dy(1,T )i =1

γΠy(1,T )i(µy(1,T )i − Py(1,T )) (13)

where µy(1,T )i,Σy(1,T )i,Πy(1,T )i, µy(1,T )c,Σy(1,T )c,Πy(1,T )c are defined in Theorem 1.

The price of v is the same in Theorem 2 as that in Theorem 1 because the supply of y(1, T )

is zero except in v and the right lower corner element of Σy(1,T )c is simply Σvtc. Theorem 2

is a very general result as there is no constraint on the information structure of the public

signals. It indicates, in the case without new supply shocks or new investors in the future, the

stock price in any dynamic models with public signals in the CARA/normality framework

can be obtained in the static economy when investors can trade on public signals directly. It

should be noted that the final allocation in the static economy and dynamic economy remains

different although the price of the stock are the same. The equivalence result indicates that

the consensus investor in the dynamic trading economy has the same belief as that in a

static economy with direct trading on public signals at time 0. 5 It follows that the effects

of dynamic trading on stock price boils down to how the beliefs of the consensus investor

are affected. As the normal distribution is summarized by the expectation and variance-

covariance matrix, the effect of dynamic trading on stock prices can be decomposed into two

components in equation (7): the first component represents the expectation effect and the

second term represent the liquidity effect. Let λt ≡ −∂Pt/∂x = Σvtc. Following Kyle (1985),

1/λt denote the market liquidity at trading session t.

Earlier literature with asymmetric information or heterogenous beliefs has shown that

dynamic trading in the presence of additional trading sessions have no effect on the stock price

or market liquidity. In the following proposition, we characterize necessary and sufficient

conditions for disagreements on public information to have no effect on the stock price. We

5For other aggregation results, see Admati (1985), DeMarzo and Skiadas (1998), and Biais, Bossaerts, and Spatt(2010) under asymmetric information and Jouini and Napp (2007) under heterogeneous beliefs.

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show that the earlier literature has made assumptions that satisfy these irrelevance conditions

and that is why they were not able to show that dynamic trading in the presence of public

information has no effect on the stock price. Without loss of generality, for each investor i,

we can write

y(1, T ) = αy(1,T )i + βy(1,T )iv + ϵy(1,T )i

Proposition 1 The necessary and sufficient condition for dynamic trading to have no effect

on stock price or market liquidity is that for all investor i, there exists a constant vector

βy(1,T−1)i = βy(1,T−1)

Proposition (1) holds because when βy(1,T−1)i = βy(1,T−1), trading directly on y(1, T − 1)

is equivalent to trading directly on y∗(1, T − 1) ≡ y(1, T − 1) − βy(1,T−1)v. However, since

y∗(1, T − 1) is orthogonal to v for all investors, trading on y∗(1, T − 1) will have no effect on

their holdings in v. As a result, the introduction of y∗(1, T − 1) will have no effect on the

stock price of v and it follows that the introduction of y(1, T − 1) in the static economy will

have no effect on the stock price. Finally, the equivalence of the consensus investor in the

dynamic economy and the static economy with the addition of y(1, T − 1) indicates that the

irrelevance result will hold when βy(1,T−1)i = βy(1,T−1).

We now analyze the effects of additional trading sessions on market liquidity and stock

price due to differences of opinion on public signals.

2.1 Market Liquidity

When investors are risk averse, a supply shock of the stock will have a price impact. The

inverse of the price impact (measured by the derivative of the equilibrium price with respect

to the stock supply) is a measure of market liquidity. In this subsection, we analyze the

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effects of dynamic trading on market liquidity. As noted earlier, let

λt ≡ −∂Pt/∂x = Σvtc.

Then 1/λt measures the market liquidity at trading session t. We are interested in how the

addition of future trading opportunity affect market liquidity. That is we compare time zero

market liquidity in an economy in which investors can trade in sessions 1 to t− 1 versus an

economy in which investors can only trade in sessions 1 to t. We have the following result.

Proposition 2 Market liquidity increases with the number of future trading sessions. Fur-

thermore, for additional trading sessions to have no effect on market liquidity, there must

exist a vector βy(1,T−1) such that βy(1,T−1)i = βy(1,T−1) for almost all i.

As investors have more opportunities to trade in the future, the stock price is less sensitive

to the stock supply. When investors have disagreements on βy(1,T−1)i, each investor feels that

he can adjust his position with someone with a lower risk perception in the future. As a

result, the perceived risk of holding the stocks is reduced and the market becomes more

liquid.

2.2 Stock Price

While the reduction of market liquidity would increase stock price at time zero, dynamic

trading can also change the consensus investor’s expectation. As a result it is not clear how

the stock price will change in the presence of dynamic trading. We have the following result:

Proposition 3 (i) When investors agree on the expectation of the stock payoff and public

signals but disagree on βy(1,T−1)i, dynamic trading will cause the stock price to increase. (ii)

When investors disagree on the expectation of the stock payoff and public signals and disagree

on βy(1,T−1)i, dynamic trading has an ambiguous effect on the stock price.

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Result (i) holds because when all investors agree on the expectations of the stock payoff

and the public signals, the consensus investors’ expectation of the stock payoff and the

public signals will be the same as that of all investors since his expectation is the weighted

average of all investors. Therefore, dynamic trading will not affect the consensus investor’s

expectations. This result combined with the reduction of liquidity premium implies that the

stock price will increase. Result (ii) holds because the consensus’ investor’s expectation of

the stock payoff can either increase or decrease. If investors who are pessimistic (optimistic)

about the stock returns have low risk perceptions, they will drive the consensus investor’s

expectation down (up) and causes the price to drop (rise).

2.3 Asset Bubble and Dynamic Trading

Harrison and Kreps (1978) show that in the presence of short sale constraints, dynamic

trading results in a asset bubble. They define the bubble as the difference between the price

in the dynamic setting and the highest intrinsic value among investors, which coincides with

the price that would obtain in the static setting economy. In our model, the highest intrinsic

value is different from the stock price in the static economy. We provide different definitions

of asset bubble using either the highest intrinsic value or the stock price in the static economy

as a bench mark. Let Pi denote the price that would obtain if all investors share the same

belief of investor i, which we define as the intrinsic value of investor i. Let PM ≡ maxiPi,

then PM denote highest intrinsic value among investors. Let PN ≡ miniPi, then PN denote

lowest intrinsic value among investors.Let Ps denote the price that would obtain in the static

economy.

Definition When P0 > PM , we say dynamic trading result in a bubble of Type I. When

P0 < PN , we say dynamic trading result in a negative bubble of Type I. When P0 > Ps, we

say dynamic trading result in a bubble of Type II. When P0 < Ps, we say dynamic trading

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result in a negative bubble of Type II.

We have the following results regarding asset bubbles in a dynamic setting.

Proposition 4 When investors have same expectations of v, y(1, T − 1), dynamic trading

gives a bubble of type II and may result in a bubble of type I but cannot give rise to a negative

bubble of type I. When investors have different expectations of v, y(1, T −1), dynamic trading

can result in either positive or negative bubbles of type I and II depending on the parameters

of the economy.

Our results differs from Harrison and Kreps in that asset bubble occurs in dynamic

trading under some rather restrictive conditions, namely all investors must have the same

expectations. In this case the reduction of liquidity always increases the stock price. However,

in general, either positive or negative bubble can occur. This is because without short sale

constraints and risk neutrality, the intermediary prices are influenced more by by those who

perceive lower immediate risk. Since both pessimistic and optimistic investors can affect

intermediary prices, and the intermediary prices in turn determine current price, whether a

positive or a negative bubble results depends on the weight and risk perception of optimistic

and pessimistic investors. Compared to Harrison and Kreps, our results are closer to the

Keynes metaphor that prices today are influenced by crowd perception in the future, not

just the most optimistic investors.

To get more intuition on how dynamic trading affects stock price and market liquidity,

we analyze a special case with two groups of investors and two trading periods.

2.3.1 A Dynamic Two Trading Period Example

In the two trading period economy, investors trade in time 0 and 1. Liquidation of stocks

and and consumption occurs in time 2. There are two types of investors , type I and II. We

16

further assume that that

v = µvi + (y − µyi) + ηi, i = I, II. (14)

where signal y arrives at time 1 (for notational convenience, since there is only one public

signal, we have dropped the subscript 1 in this subsection). For investor of type i, i = I, II,

µvi is i’s expectation of v, µyi is i’s expectation of y, and η is a noise term with mean zero.

Investors believe that y and η are independent and normally distributed and disagree about

the variances of y and η. Investors of type i believe that the variances of y and ϵ, and v are

σ2yi, σ

2ηi, σ

2vi respectively and that σ2

vi = σ2yi + σ2

etai. As shown in equation (11), under such

an assumption, the hedging demands will be zero and the myopic demand is optimal, which

greatly simplify the analysis.

Notice that here the reverse regression of y on v is

y = µyi + βyi(v − µvi) + ϵyi, (15)

where

βyi =σ2yi

σ2vi

(16)

ϵyi =σ2ηi

σ2vi

(y − µyi)−σ2ηi

σ2vi

ηi (17)

At time one, investor i’s demand for stock is given by

D1i =µvi + (y − µyi)− P1

γσ2ηi

.

For the market to clear, we must have

x =µvI + (y − µyI)− P1

2γσ2ηI

+µvII + (y − µyII)− P1

2γσ2ηII

.

17

We thus have

P1 = µv1c + (y − µyc)− γσ2ηcx,

where

σ2ηc =

2σ2ηIσ

2ηII

σ2ηI + σ2

ηII

is the harmonic average of the variance of η among investors, and

µv1c ≡σ2ηIIµvI + σ2

ηIµvII

σ2ηII + σ2

ηI

,

µyc ≡σ2ηIIµyI + σ2

ηIµyII

σ2ηI + σ2

ηII

.

Notice that the capital gains from trading at time 1, v−P1, is independent of y. Therefore

the capital gains from trading at time 1 is independent from capital gains from trading at

time 0. Since investors have CARA utility function, there is no additional hedging demand

at time 0, and the optimal demand at time 0 is simply the myopic demand. Let σP1i denote

investor i’s variance of P1 at time 0. We then have σ2P1i = σ2

yi. Investor i’s demand for stock

at time zero is given by

D0i =µv1c + (µyi − µyc)− γσ2

ηc − P0

γσ2P1i

.

Market clearing leads to

x =D0I +D0II

2

=µv1c + (µyI − µyc)− γσ2

ηcx− P0

2γσ2P1I

+µv1c + (µyII − µyc)− γσ2

ηcx− P0

2γσ2P1II

.

The equilibrium stock price at time 0 is then given by

P0 = µv0c − γσ2ηcx− γσ2

ycx = µv0c − γσ2v0cx,

18

where

µv0c = µv1c + (µyI − µyII )

(1

1 + σ2yI/σ2

yII

− 1

1 + σ2η1/σ2

η2

),

σ2yc =

2σ2yIσ

2yII

σ2yI + σ2

yII

, σ2v0c = σ2

ηc + σ2yc.

2.3.2 Price and Liquidity in the Static Economy without Dynamic Trading

Without Dynamic trading it is straight forward to derive the equilibrium demands for CARA

utility investors. Let Dsi denote the demand for group i investors in the static economy, and

Ps the price in the static economy. Then the optimal demand for group i investors are

Dsi =µvi − Ps

γσ2vi

Market clearing leads to

x =DsI +DsII

2

=µvI − Ps

2γσ2vI

+µvII − Ps

2γσ2vII

.

and the equilibrium price in the static economy is

Ps = µvsc − γσ2vscx

and

µvsc ≡σ2vIIµvI + σ2

vIµvII

σ2vII + σ2

vI

σ2vsc =

2σ2vIσ

2vII

σ2vI + σ2

vII

Comparing the prices and liquidity in the dynamic and static economy allows us to

determine the effects of dynamic trading. From Proposition 2, dynamic trading always

19

increases liquidity and thus we have σvsc > σv0c. The liquidity effect alone will always

increase the stock price.

The difference of σ2vsc and σ2

v0c is

σ2v0c − σ2

vsc =2(σ2

yI/σ2yII − σ2

ηI/σ2ηII)

2σ2yIσ

2yIIσ

2ηIσ

2ηII

(σ2yI + σ2

yII)(σ2ηI + σ2

ηII)(σ2yI + σ2

yII + σ2ηI + σ2

ηII)

There is no liquidity effect when σ2yI/σ

2yII − σ2

ηI , that is when the relative risk ratio are

equalized across the two periods among the two groups of investors, there will be no risk

shifting and dynamic trading in the future has does not reduce liquidity in the current period

However the difference between the expectation of the consensus investor in the static

and dynamic economy is ambiguous.

We have

µv0c − µvsc = A(µvI − µvII) +B(µyI − µyII),

where

A ≡ 1

1 + σ2ηI/σ

2ηII

− 1

1 + σ2vI/σ

2vII

and

B ≡ 1

1 + σ2yI/σ

2yII

− 1

1 + σ2ηI/σ2

ηII

Notice that when

σ2ηI

σ2ηII

>σ2yI

σ2yII

,

we have

σ2ηI

σ2ηII

>σ2yI + σ2

ηI

σ2yII + σ2

ηII

=σ2vI

σ2vII

In the dynamic economy, when investors of group I will perceive relatively more price risk

in the second period than that in the first period. As a result, A < 0 and B > 0. That is in

20

the dynamic setting, the consensus investor will put more weight on group I’s expected stock

value in the first period and less weight on their expected stock value in the second period.

Intuitively, investors who face relatively lower risk in the period will have more influence in

determining the price in that period. It is the relative risk ratio in each period that matters

in the comparison between the price in the static and dynamic setting. If the relative ratio

remains the same, there will be no price or liquidity effects.

To Summarize, we have the following results with respect to market liquidity and stock

price.

Corollary 1 (i) When σyI/σyII = σηI/σηII , dynamic trading has no effect on market liquid-

ity or stock price. (ii) When σyI/σyII = σηI/σηII , market liquidity will increase. In this case

if investors have the same expectations of v and y, price will always increase. If investors

have different expectations, then price can either increase or decrease.

Similarly, in general, asset bubbles or negative bubbles can occur in the two period

economy

Proposition 5 When investors have same expectations of v, y, dynamic trading gives a

bubble of type II and may result in a bubble of type I but cannot give rise to a negative

bubble of type I. When investors have different expectations of v, y, dynamic trading can

result in either positive or negative bubbles of type I and II depending on the parameters of

the economy.

3 Conclusion

We show that the impact of heterogeneous beliefs on the stock price can be decomposed into

two components: the expectation effect and the liquidity premium effect. Dynamic trading

always reduces the liquidity premium. Different investors perceive different levels of risk

21

across time and states. In some states, when some investors perceive high risks, they believe

that other investors perceive low risks, so that they can share risks with one another. As a

result, all investors believe that they can achieve better risk sharing under dynamic trading.

In equilibrium, the liquidity premium decreases. The expectation effect arises when investors

have different expectations about the stock payoff and the public signals. When investors,

who are relatively more optimistic about the next period stock return, also believe that their

beliefs are more precise, they tend to push the stock price up and a positive bubble can

result. When investors, who are relatively pessimistic about the next period stock return,

have more precise beliefs, they tend to pull the stock price down. The pessimistic expectation

effect can more than offset the reduction in liquidity premium and can generate a negative

bubble.

22

Appendix

Before proving Theorem 1, we first present the following lemma.

Lemma 1 Let X be an M dimensional random vector, Y be an N dimensional random

vector, and X and Y are multi-variate normally distributed. Let ΣX,Y denote the variance-

covariance matrix and ΠX,Y the precision matrix of (X ′, Y ′)′. Let ΠX,Y (X,X) be the M×M

submatrix corresponding to X. Let ΣX|Y and ΠX|Y denote the variance-covariance matrix and

the precision matrix of X conditional on Y , respectively. We have that ΠX|Y = ΠX,Y (X,X)

Proof: Let ΣX ,ΣY denote the variance-covariance matrix of X, Y , ΣXY denote the covari-

ance matrix of X and Y , and ΣY X denote the transpose of ΣXY . From the block matrix

inversion formula, we have

ΠX,Y (X,X) = [ΣX − ΣXYΣ−1Y ΣY X ]

−1 = [ΣX|Y ]−1 = ΠX|Y . (18)

Proof of Theorem 1: The proof is divided into four parts:

1. Verify the definition of the consensus investor’s conditional expectation in equation (1)

and variance in equation (3) of signals y(t+ 1, T ) is consistent for all t, i.e., the law of

iterated expectation holds for the consensus investor;

2. the consensus investor’s conditional expectation and variance of returns R(t+1, T ) can

be defined similarly as in equations (1) and (3);

3. the demand function of investor i at session t is defined as in equation (8);

4. the price function (7) and investor’s demand function (8) constitutes an equilibrium.

Part 1 For 0 ≤ t1 ≤ t2 ≤ T , there are two ways to define the consensus investor’s conditional

expectation and variance of y(t2+1, T ) at session t2. One way is to define them as in equations

23

(1) and (3), the other is to derive them from the conditional expectation and variance at

session t1. We need to show the conditional expectation and variance defined using these

two methods are the same. Here only the special case of t1 = 0 is proved and the general

case can be done in the same way.

Hereafter, we drop the time label t in notations in session 0. For example, the consensus

investor expectation of the stock payoff in session 0 is denoted as µvc rather than µv0c. For

the conditional precision matrix, be the definition in equation (3) we have

Πy(t+1,T )tc =

∫i

Πy(t+1,T )tidi

=

∫i

Πy(1,T )i(y(t+ 1, T ), y(t+ 1, T ))di

= Πy(1,T )c(y(t+ 1, T ), y(t+ 1, T ))

= (Varc[y(t+ 1, T )]− Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]Covc[y(1, t), y(t+ 1, T )])−1

= (Vartc[y(t+ 1, T )])−1

Here, the first and third equations come from the definition in equation (3), the second from

Lemma 1, the fourth from the block matrix inversion formula, and the last from the usual

definition of conditional variance. So, we can define the consensus investor’s conditional

variance of y(t+ 1, T ) in session t either directly from the definition in equation (3) or from

the unconditional variance of y(1, T ) in session 0.

Denote

A ≡∫i

(Πy(1,T )i(y(1, t), y(1, t))µy(1,t)i +Πy(1,T )i(y(1, t), y(t+ 1, T ))µy(t+1,T )i

)di,

B ≡∫i

(Πy(1,T )i(y(t+ 1, T ), y(1, t))µy(1,t)i +Πy(1,T )i(y(t+ 1, T ), y(t+ 1, T ))µy(t+1,T )i

)di.

It’s easy to verify∫iΠy(1,T )iµy(1,T )idi =

[AB

].

24

On one hand, we can define the conditional expectation of y(t+ 1, T ) in session t using

the consensus investor’s unconditional expectation and variance of y(1, T ):

µy(t+1,T )tc = µy(t+1,T )c + Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)](y(1, t)− µy(1,t)c)

= µy(t+1,T )c − Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]µy(1,t)c

+ Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]y(1, t) (19)

By definition of µy(1,T )c in equation (1),

µy(1,t)c = Varc[y(1, t)]A+ Covc[y(1, t), y(t+ 1, T )]B

µy(t+1,T )c = Covc[y(t+ 1, T ), y(1, t)]A+Varc[y(t+ 1, T )]B

So,

µy(t+1,T )c − Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]µy(1,t)c

= Covc[y(t+ 1, T ), y(1, t)]A+Varc[y(t+ 1, T )]B

− Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)](Varc[y(1, t)]A+ Covc[y(1, t), y(t+ 1, T )]B)

= (Varc[y(t+ 1, T )]− Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]Covc[y(1, t), y(t+ 1, T )])B

= Vartc[y(t+ 1, T )]B

We can verify the following equation holds for any j ∈ [0, 1] or j = c, using block matrix

inversion formula:

Πy(1,T )j(y(t+1, T ), y(1, t))Varj[y(1, t)]+Πy(1,T )j(y(t+1, T ), y(t+1, T ))Covj[y(t+1, T ), y(1, t)] = 0.

(20)

25

On the other hand, the conditional expectation at session t is defined in equation (1):

µy(t+1,T )tc

= Π−1y(t+1,T )tc

∫i

Πy(t+1,T )tiµy(t+1,T )tidi

= Π−1y(t+1,T )tc

∫i

Πy(t+1,T )ti(µy(t+1,T )i + Covi[y(t+ 1, T ), y(1, t)]Var−1i [y(1, t)](y(1, t)− µy(1,t)i))di

= Π−1y(t+1,T )tc

∫i

(Πy(t+1,T )tiµy(t+1,T )i − Πy(t+1,T )tiCovi[y(t+ 1, T ), y(1, t)]Var−1i [y(1, t)]µy(1,t)i)di

+ Π−1y(t+1,T )tc

(∫i

Πy(t+1,T )tiCovi[y(t+ 1, T ), y(1, t)]Var−1i [y(1, t)]di

)y(1, t)

= Π−1y(t+1,T )tc

∫i

(Πy(1,T )i((t+ 1, T ), (t+ 1, T ))µy(t+1,T )i +Πy(1,T )i(y(t+ 1, T ), y(1, t))µy(1,t)i)di

− Π−1y(t+1,T )tc

(∫i

Πy(1,T )i(y(t+ 1, T ), y(1, t))di

)y(1, t)

= Vartc[y(t+ 1, T )]B − Π−1y(t+1,T )c(y(t+1,T ),y(t+1,T ))Πy(1,T )c(y(t+ 1, T ), y(1, t))y(1, t)

= Vartc[y(t+ 1, T )]B + Covc[y(t+ 1, T ), y(1, t)]Var−1c [y(1, t)]y(1, t).

The equivalence of the conditional expectation of y(t+ 1, T ) in session t defined by two

methods is proved.

Part 2 In this part, we verify

Var−1tc [R(t+ 1, T )] = ΠR(t+1,T )tc =

∫i

ΠR(t+1,T )tidi

µtc[R(t+ 1, T )] = Π−1R(t+1,T )tc

∫i

ΠR(t+1,T )tiµR(t+1,T )tidi

At time t, by definition we have

Etc[v] = b′

tcy(1, t) + ctc, 1 ≤ t ≤ T

26

here, btc is a t-dimensional column vector and ctc is a scalar. So, we can express R(t+ 1, T )

as Rt+1

Rt+2...

RT

= Btcy(t+ 1, T ) + ¯Btcy(t+ 1, T ) + Ctc − γΩtcx (21)

here,

Btc =

b(t+1)c(1)− btc(1) b(t+1)c(2)− btc(2) · · · b(t+1)c(t)− btc(t)

b(t+2)c(1)− b(t+1)c(1) b(t+2)c(2)− b(t+1)c(2) · · · b(t+2)c(t)− b(t+1)c(t)...

... · · · ...−b(T−1)c(1) −b(T−1)c(2) · · · −b(T−1)c(t)

¯Btc =

b(t+1)c(t+ 1) 0 · · · 0

b(t+2)c(t+ 1)− b(t+1)c(t+ 1) b(t+2)c(t+ 2) · · · 0...

... · · · ...−b(T−1)c(t+ 1) −b(T−1)c(t+ 2) · · · 1

Ctc =

c(t+1)c − ctc

c(t+2)c − c(t+1)c...

−c(T−1)c

, Ωtc =

Σv(t+1)c − Σvtc

Σv(t+2)c − Σv(t+1)c...

−Σv(T−1)c

brc(s) is the s-th element of the r-dimensional column vector brc.

So, we have

ΠR(t+1,T )tc = (Vartc[R(t+ 1, T )])−1 = ( ¯BtcVartc[y(t+ 1, T )] ¯B′

tc)−1

= ¯B−1′

tc Πy(t+1,T )tc¯B−1tc = ¯B−1′

tc

(∫i

Πy(t+1,T )tidi

)¯B−1tc

=

∫i

¯B−1′

tc Πy(t+1,T )ti¯B−1tc di

=

∫i

ΠR(t+1,T )tidi

27

µR(t+1,T )tc = Etc[Btcy(1, t) +¯Btcy(t+ 1, T ) + Ctc − γΩtcx]

= Btcy(1, t) +¯Btcµy(t+1,T )tc + Ctc − γΩtcx

= Btcy(1, t) + Btcy(1, t) +¯BtcΠ

−1y(t+1,T )tc

∫i

Πy(t+1,T )tiµy(t+1,T )tidi+ Ctc − γΩtcx

= ¯BtcΠ−1y(t+1,T )tc

∫i

Πy(t+1,T )ti¯B−1tc (µR(t+1,T )ti − Btcy(1, t)− Ctc + γΩtcx)di+ Ctc − γΩtcx

= Π−1R(t+1,T )tc

∫i

¯B−1′

tc Πy(t+1,T )ti¯B−1tc µR(t+1,T )tidi

= Π−1R(t+1,T )tc

∫i

ΠR(t+1,T )tiµR(t+1,T )tidi

Part 3 Here, we prove the demand function (8) is optimal for investor i. Following Lemma

(1), we have ΠRt+1Rs+1ti = ΠRt+1Rs+1i for s ≥ t.

We now prove the claim in two steps. We first show that the prices and demands in the

last period T − 1 constitute an equilibrium. We then show that if the prices and demands

from t + 1 and onwards constitute a dynamic equilibrium, then the prices and demands at

period t also constitute a dynamic equilibrium. The theorem thus follows by mathematical

induction.

Notice that

µR(t+1,T )ti − µR(t+1,T )i = Covi[R(t+ 1, T ), R(1, t)]ΠR(1,t)i[R(1, t)− µR(1,t)i]. (22)

Let ΠRt+1R(1,t)i and ΠRt+1R(t+1,T )i denote the column vectors that represent the first t elements

and the last T − t elements of the (t + 1)th column vector in ΠR(1,T )i. Let Ot denote the t

dimensional zero column vector. By equation (20)

Vari[R(1, t)]ΠRt+1R(1,t)i + Covi[R(1, t), R(t+ 1, T )]ΠRt+1R(t+1,T )i = Ot. (23)

28

We thus have

Π′

Rt+1R(t+1,T )i[µR(t+1,T )ti − µR(t+1,T )i]

= Π′

Rt+1R(t+1,T )iCovi[R(t+ 1, T ), R(1, t)]ΠR(1,t)i

[R(1, t)− µR(1,t)i

]= −Π

′

Rt+1R(1,t)iVari[R(1, t)]ΠR(1,t)i

[R(1, t)− µR(1,t)i

]= −Π

′

Rt+1R(1,t)i

[R(1, t)− µR(1,t)i

]. (24)

Denote

ΠR(t+1,T )ti = [ΠRt+1ti,ΠRt+2ti, · · · ,ΠRT ti], and

ΠRs+1ti = ΠRs+1R(t+1,T )ti =

ΠRs+1Rt+1ti

ΠRs+1Rt+2ti

· · ·ΠRs+1RT ti

, s ≥ t.

It can further be shown that the equilibrium demand proposed in Theorem 1 satisfies

the following equation:

Dti =1

γ

T−1∑s=t

ΠRt+1Rs+1tiµRs+1ti

=1

γ

T−1∑s=t

ΠRt+1Rs+1iµRs+1i −1

γ

t−1∑s=0

ΠRt+1Rs+1i(Rs+1 − µRs+1i)

=1

γ

T−1∑s=0

ΠRt+1Rs+1iµRs+1i −1

γ

t−1∑s=0

ΠRt+1Rs+1iRs+1. (25)

At time T , the utility for investor i has the following form:

UTi = − exp

−γ

[W0i +

T−1∑j=0

DjiRj+1

]. (26)

29

At time T − 1, there is only one period left and the problem reduces to a maximization

problem in a static setting. Let

µRT (T−1)i ≡ E(T−1)i[RT ] = E(T−1)i [v − PT−1] ,

ΠRTRT (T−1)i ≡ Var−1(T−1)i[RT ].

We have

ET−1[UTi]

∝ −∫RT

exp

[−γ

T−1∑j=0

DjiRj+1 −1

2(RT − µRT (T−1)i)

′ΠRTRT (T−1)i(RT − µRT (T−1)i)

]dRT

∝ −∫RT

exp

[−(γD(T−1)i − ΠRTRT (T−1)iµRT (T−1)i

)RT − 1

2ΠRTRT (T−1)iR

2T

]dRT

∝ − exp

[(γD(T−1)i − ΠRTRT (T−1)iµRT (T−1)i

)22ΠRTRT (T−1)i

]. (27)

From the above expected utility function of investor i, we then arrive at the investor’s

optimal demand for stock:

D(T−1)i = ΠRTRT (T−1)iµRT (T−1)i/γ. (28)

The market clearing condition, x =∫iD(T−1)idi, yields the equilibrium stock price:

PT−1 =

∫iΠRTRT (T−1)iµv(T−1)idi− γx∫

iΠRTRT (T−1)idi

= µv(T−1)c − γΣv(T−1)cx. (29)

That is, the stock price is the precision weighted average expectation minus the risk premium.

Notice that 1/∫iΠRTRT (T−1)idi = 1/ΠRTRT (T−1)c = Σv(T−1)c.

Suppose that the demand is optimal for trading sessions larger than t. We show that

the equilibrium demand is also optimal at session t. Then by mathematical induction, the

equilibrium demand described in the theorem is optimal for all trading sessions. Notice that

30

from equation (25), and conditional on the information at time t, we have the following

identity for Dri, r > t:

Dri =1

γ

T−1∑s=r

ΠRr+1Rs+1riµRs+1ri

=1

γ

T−1∑s=r

ΠRr+1Rs+1tiµRs+1ti −1

γ

r−1∑s=t

ΠRr+1Rs+1ti(Rs+1 − µRs+1ti)

=1

γ

T−1∑s=t

ΠRr+1Rs+1tiµRs+1ti −1

γ

r−1∑s=t

ΠRr+1Rs+1tiRs+1. (30)

Plugging in the expression for Dsi, s > t, and taking the expectation with respect to

R(t+ 1, T ) at trading session t, we have

Eti[Ui]

∝ −∫R(t+1,T )

exp

[−γ

T−1∑s=0

DsiRs+1

−1

2(R(t+ 1, T )− µR(t+1,T )ti)

′ΠR(t+1,T )ti(R(t+ 1, T )− µR(t+1,T )ti)

]dR(t+ 1, T )

= −∫R(t+1,T )

exp

[−γ

t∑s=0

DsiRs+1 −T−1∑s=t+1

Rs+1

(T−1∑r=t

ΠRs+1Rr+1tiµRr+1ti −s−1∑r=t

ΠRs+1Rr+1tiRr+1

)

−1

2(R(t+ 1, T )− µR(t+1,T )ti)

′ΠR(t+1,T )ti(R(t+ 1, T )− µR(t+1,T )ti)

]dR(t+ 1, T )

∝ −∫R(t+1,T )

exp

[−γ

t∑s=0

DsiRs+1 −T−1∑s=t+1

Rs+1

(T−1∑r=t

ΠRs+1Rr+1tiµRr+1ti −s−1∑r=t

ΠRs+1Rr+1tiRr+1

)

−1

2R(t+ 1, T )′ΠR(t+1,T )tiR(t+ 1, T ) + µ′

R(t+1,T )tiΠR(t+1,T )tiR(t+ 1, T )

]dR(t+ 1, T )

31

= −∫R(t+1,T )

exp

[−γ

t∑s=0

DsiRs+1 −T−1∑s=t+1

Rs+1

(T−1∑r=t

ΠRs+1Rr+1tiµRr+1ti −s−1∑r=t

ΠRs+1Rr+1tiRr+1

)

−1

2

T−1∑s=t

ΠRs+1Rs+1tiR2s+1 −

T−1∑s=t+1

s−1∑r=t

ΠRs+1Rr+1tiRr+1Rs+1

+T−1∑s=t

T−1∑r=t

ΠRs+1Rr+1tiµRr+1tiRs+1

]dR(t+ 1, T )

= −∫R(t+1,T )

exp

[−γ

t−1∑s=0

DsiRs+1 −

[γDti −

T−1∑s=t

ΠRt+1Rs+1tiµRs+1ti

]Rt+1

−1

2

T−1∑s=t

ΠRs+1Rs+1tiR2s+1

]dR(t+ 1, T )

∝ −∫Rt+1

exp

[−

(γDti −

T−1∑s=t

ΠRt+1Rs+1tiµRs+1ti

)Rt+1 −

1

2ΠRt+1Rt+1tiR

2t+1

]dRt+1

∝ − exp

(γDti −T−1∑s=t

ΠRt+1Rs+1tiµRs+1ti

)2

/2ΠRt+1Rt+1ti

. (31)

The optimal demand can then be determined as

Dti =1

γ

T−1∑s=t

ΠRt+1Rs+1tiµRs+1ti.

Part 4 At last, we show that the demand (8) and price function (7) constitutes an equilib-

rium, i.e., the market clearing condition holds. Notice that Rt is the differences of conditional

expectations for consensus investor c, it is thus independent across t for investor c, ignoring

the risk premium. Consequently, we have

Σvtc − Σv(t+1)c =T∑

r=t+1

ΣRrc −T∑

r=t+2

ΣRrc = ΣRt+1c, (32)

µRt+1tc = γ(Σvtc − Σv(t+1)c)x = γΣRt+1cx. (33)

32

Aggregating demands across investors, we arrive at

∫i

Dtidi =

∫i

1

γ

T−1∑s=t

ΠRt+1Rs+1tiµRs+1tidi =1

γ

T−1∑s=t

ΠRt+1Rs+1tcµRs+1tc

=1

γΠRt+1Rt+1tcµRt+1tc =

γΣRt+1tcx

γΣRt+1tc

= x (34)

Q.E.D.

Proof of Proposition 1: Regressing yt on v gets yt = dti + βtiv + εti, so we have:

y1y2...

yT−1

=

d1id2i...

d(T−1)i

+

β1i

β2i...

β(T−1)i

v +

ε1iε2i...

ε(T−1)i

= di + βiv + εi, i ∈ [0, 1].

Denote the variance and precision matrix of εi as Σεi and Πεi. By definition,

Πy(1,T )c =

∫i

Πy(1,T )idi

=

∫i

[Σy(1,T−1)i βiΣv

β′iΣv Σv

]−1

di

=

∫i

[βiΣvβ

′i + Σεi βiΣv

β′iΣv Σv

]−1

di

=

∫iΠεidi −

∫iΠεiβidi

−∫iβ

′iΠεidi Πv +

∫iβ

′iΠεiβidi

Hereafter, we denote variables with superscript T in T− session dynamic economy and

without superscript in static economy. Σεi is a symmetric positive definite matrix, so we

33

have

ΠTvc

= Πv +

∫i

β′

iΠεiβidi−∫i

β′

iΠεidi

(∫i

Πεidi

)−1 ∫i

Πεiβidi (35)

= Πv +

∫i

(βi −

(∫i

Πεidi

)−1 ∫i

Πεiβidi

)′

Πεi

(βi −

(∫i

Πεidi

)−1 ∫i

Πεiβidi

)di(36)

≥ Πv

which implies ΠTvc ≥ Πv, i.e., Σ

Tvc ≤ Σv. The equality holds if and only if all βi ≡ β =(∫

iΠεidi

)−1 ∫iΠεiβidi are the same across investors.

It’s easy to verify that ΣTvc is the same for all permutations of y1, y2, · · · , yT−1, so we can

assume the extra trading session is added between trading session T − 1 and the final value

v revealed without generality.

ΠT+1vc − Πv

=

∫i

(βT+1i − βT+1)

′ΠT+1

εi (βT+1i − βT+1)di

=

∫i

(βT+1i − βT+1)

′[ΣT

εi Ci

C′i σ2

εT+1i

]−1

(βT+1i − βT+1)di

=

∫i

[βTi − βT+1(1, T − 1)βTi − βT+1(T )

]′ (ΣT

εi − CiC′i/σ

2εT+1i

)−1 − 1σ2εT+1i

(ΣTεi − CiC

′i/σ

2εT+1i

)−1Ci

− 1σ2εT+1i

C′i(Σ

Tεi − CiC

′i/σ

2εT+1i

)−1 (σ2εT+1i

− C′iΠ

TεiCi)

−1

[βTi − βT+1(1, T − 1)βTi − βT+1(T )

]di

=

∫i

(βTi − βT+1(1, T − 1))

′(ΣT

εi − CiC′

i/σ2εT+1i

)−1(βTi − βT+1(1, T − 1))di

−2

∫i

(βTi − βT+1(1, T − 1))

′(ΣT

εi − CiC′

i/σ2εT+1i

)−1Ci(βTi − βT+1(T ))di+

∫i

(βTi − βT+1(T ))2

σ2εT+1i

− C′iΠ

TεiCi

di

34

=

∫i

(βTi − βT+1(1, T − 1))

′ΠT

εi(βTi − βT+1(1, T − 1))di+

∫i

[(βT

i − βT+1(1, T − 1))′ΠT

εiCi

]2σ2εT+1i

− C′iΠ

TεiCi

di

−2

∫i

(βTi − βT+1(1, T − 1))

′ΠT

εiCi(βTi − βT+1(T ))

σ2εT+1i

− C′iΠ

TεiCi

di+

∫i

(βTi − βT+1(T ))2

σ2εT+1i

− C′iΠ

TεiCi

di

=

∫i

(βTi − βT+1(1, T − 1))

′ΠT

εi(βTi − βT+1(1, T − 1))di

+

∫i

[(βT

i − βT+1(1, T − 1))′ΠT

εiCi − (βT i − βT+1(T ))]2

σ2εT+1i

− C′iΠ

TεiCi

di

≥∫i

(βTi − βT+1(1, T − 1))

′ΠT

εi(βTi − βT+1(1, T − 1))di

≥∫i

(βTi − βT )

′ΠT

εi(βTi − βT )di

= ΠTvc − Πv

here, the last inequality comes from that βT is the solution of the following minimisation

problem

minx

∫i

(βTi − x)′ΠT

εi(βTi − x)dx.

The above equations means ΣT+1vc ≤ ΣT

vc, i.e., the consensus investor’s variance of the

stock reduces with the number of trading sessions. Q. E. D.

Proof of Proposition 2: From equations (1), (2) and (3), we know the conditional

expectation and variance of final payoff don’t change as the number of trading sessions. The

result follows immediately from equation (7). Q. E. D.

35

Proof of Proposition 3: By definition,

Σy(1,T )c =

∫iΠεidi −

∫iΠεiβidi

−∫iβ

′iΠεidi Πv +

∫iβ

′iΠεiβidi

−1

=

Σy(1,T−1)cΣy(1,T−1)c

∫i Πεiβidi

Πv+∫i β

′iΠεiβi∫

i β′iΠεidiΣy(1,T−1)c

Πv+∫i β

′iΠεiβi

Σvc

−1

with Σvc = (Πv+∫iβ

′iΠεiβidi−

∫iβ

′iΠεidi

(∫iΠεidi

)−1 ∫iΠεiβidi)

−1 and Σy(1,T−1)c = (∫iΠεidi−∫

i Πεiβidi∫i β

′iΠεidi

Πv+∫i β

′iΠεiβidi

)−1.

So the consensus investor’s expectation of the stock final value is

µvc =

[∫iβ

′iΠεidiΣy(1,T−1)c

Πv +∫iβ

′iΠεiβidi

,Σvc

]∫i

Πεi −Πεiβi

−β′iΠεi Πv + β

′iΠεiβi

µ1i

µ2i...µv

di

=

∫i

(∫iβ

′iΠεiΣy(1,T−1)cdi

Πv +∫iβ

′iΠεiβidi

− β′i

Πvc

)Πεiµy(1,T−1)idi

+µv

∫i

(Σvc(Πv + β

′

iΠεiβi)−∫iβ

′iΠεidiΣy(1,T−1)cΠεiβi

Πv +∫iβ

′iΠεiβidi

)di

Define random variables νt and πt , which are uniformly distributed on [0, 1] and νt(ω) =

µytω, [π1(ω), π2(ω), · · · , πT−1(ω)] =

(∫i β

′iΠεiΣy(1,T−1)cdi

Πv+∫i β

′iΠεiβidi

− β′ω

Πvc

)Πεω, πT (ω) = Σvc(Πv+β

′ωΠεωβω)−

∫i β

′iΠεidiΣy(1,T−1)cΠεωβω

Πv+∫i β

′iΠεiβidi

, ω ∈ [0, 1]. Because the variance matrix is orthogonal to the precision

36

matrix, we have E[πt] = 0 for 1 ≤ t ≤ T − 1 as well as E[πT ] = 1, so

P T0 − P0 = (µT

vc − γΣTvcx)− (µv − γΣvx) = (µT

vc − µv)− γ(ΣTvc − Σv)x

=

∫i

(∫iβ

′iΠεiΣy(1,T−1)cdi

Πv +∫iβ

′iΠεiβidi

− β′i

Πvc

)Πεiµy(1,T−1)idi− γ(ΣT

vc − Σv)x

=T−1∑t=1

Cov[πt, νt]− γ(ΣTvc − Σv)x. (37)

The last term is positive due to ΣTvc ≤ Σv. Specially, if βi ≡ β is the same for all i ∈ [0, 1],

we have

E

π1

π2...

πT−1

=

∫i

(∫iβ

′ΠεiΣy(1,T−1)cdi

Πv +∫iβ ′Πεiβdi

− β′

Πvc

)Πεidi

=

(∫iβ

′ΠεiΣy(1,T−1)cdi

Πv +∫iβ ′Πεiβdi

− β′

Πvc

)∫i

Πεidi

= 0.

which implies∫i β

′ΠεiΣy(1,T−1)cdi

Πv+∫i β

′Πεiβdi− β

′

Πvc= 0, we thus have Cov[πt, νt] = 0, t = 1, · · · , T −1. And

from equation (36), we get ΣTvc = Σv. So, adding trading sessions have no effect on time

0 stock price if the investors agree on the covariance of signal y and payoff v. Generally,

Cov[πt, νt] could be either positive or negative and its absolute value can be extremely large

or small depends on the absolute value of νt, which implies P T0 can be either larger or smaller

than P0. Q.E.D.

Proof of Proposition 4: The result follows from Lemma 1 and Proposition 2. Q.E.D.

Proof of Proposition 5: Similarly to equation (35), the conditional precision of final

payoff can be expressed as following when the investors have different priors:

ΠTvc =

∫i

Πvidi+

∫i

β′

iΣ−1εi βidi−

∫i

β′

iΣ−1εi di

(∫i

Σ−1εi di

)−1 ∫i

Σ−1εi βidi. (38)

37

It is obvious that part (i) and (iv) follow from equation (38) and part (iii) follow from

Lemma 1 and equation (38).

The difference between prices in dynamic and static economies is

P T0 − P0 =

T∑t=1

Cov[πt, νt]− γ(ΣTvc − Σvc)x. (39)

here, Σvc = (∫iΣvidi)

−1. The result of part (ii) immediately follows from the above equation.

Q.E.D.

Proof of Proposition 6: The price in T-period dynamic economy can be written as

P T0 =

T∑t=1

Cov[πt, νt] +

∫i

µvidi− γΣTvcx. (40)

By the same logistics in the proof of Proposition 3, P T0 depends on the absolute value of

E[µyt ], 1 ≤ t ≤ T −1, which have no effect on the prices in static economy. We can let P T0 go

to positive or negative infinity by choosing different values of E[µyt ], 1 ≤ t ≤ T − 1. Q.E.D.

Proof of Theorem 2 , The proof is similar to that of Theorem 1 and is omitted here.

Q. E. D.

38

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41