Transcript of "What Determines the Stock Price and the Informative ....doc"
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6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
What Determines the Stock Price and the Informative Efficiency: The
Omitted Information Frequency
Li-Wei Chen1 Yi-Yin Yen2 Cheng-Tao Hsieh3
1. Li-wei Chen
Title: Doctorial student, Department of Accountancy, National Taipei University
(NTPU), Taiwan, R.O.C.
E-mail: ericchen@ms53.url.com.tw
2. Yi-Yin Yen
Title: Assistant Professor, Department of Accounting Information, National
Taipei College of Business, Taiwan, R.O.C.
E-mail: yenyen888@yahoo.com.tw
3. Corresponding author: Cheng-Tao Hsieh
Title: Associate Professor, Center for General Education, National Pingtung
University of Science and Technology, Taiwan, R.O.C.
E-mail: dao@mail.npust.edu.tw
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Abstract
This paper investigates the determination of the price system of the stock market.
Different from previous studies, we emphasize the concept of the “observational
frequency” of information. This paper allows each informed investor to observe
~ ~ ~
more than one kind of information. There are I kinds of information x1 , x2 ,...x I
available in a competitive stock market. Since there exists information asymmetry
among investors, the market information are respectively observed f1 , f 2 ,..., f I times
by N constant risk-averse traders to form a more precise estimate for the expected
~
value of the risky asset, v , to buy the shares to maximize their own expected utility,
and then to determine the stock market equilibrium simultaneously. Our main
findings are as follows.
First, we propose that the equilibrium price, trading quantity, and the expected
utility of investors depend not only on realized value of the information but also on
the observational frequencies and the precisions of the market information. The
competitive equilibrium price is equal to the rational expectations equilibrium price,
which aggregates all the market information according to their observational
frequencies and the precisions of the market information. Second, the fully-informed
economy equilibrium is a special case of the competitive equilibrium (or the rational
expectations equilibrium) only when the observational frequencies of all the market
information are just equal and it serves as a sufficient statistic for all the market
information about the intrinsic value of the risky asset. Finally, we prove that the
heterogeneity in the observational frequency of information is impossibility for
informative efficiency. Since the observational frequencies among the market
information are not uniform, the equilibrium price still aggregates the market
information but will not break down as the case described by Grossman (1976).
Keywords: Sufficient Statistic, Observational Frequency of Information, Price
Informativeness, Informative Efficiency, Rational Expectations Equilibrium.
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Notation:
There are N traders in the stock market. We use the following notation.
xi ~ ~i : the private information. Where v denotes the true value of the risky
~ = v +ε ~
~ ~
asset; and ε i ~ N (0, Λ i ) denotes the error term of the information xi .
−1
Λ i ≡ Var −1 (ε i ) : the precision of the information ~i .
~ x
f i : the observational frequency of the information ~i .
x
O : the initial endowment of the risky asset.
r : the absolute risk aversion of investor.
~
w : the wealth gained from trading.
~c
p : the competitive equilibrium price
~e
p : the rational expectations equilibrium price.
p a : the (artificial) fully-informed economy equilibrium price.
1. Introduction
The role of prices in aggregating and conveying information is central to the
study of the allocation of resources in a competitive economy. The most frequently
asked questions in the stock market are “What determines the stock price?”, “How
informationally efficient is the market price?”, and “Is the equilibrium price a
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sufficient statistic for the market information?”
Literatures on stock trading are extensive. Economists commonly model the
price formation process with which each informed investor observes only one piece
of information (Hence the observational frequency of information is homogeneous.) ,
and the investors invest and revise their beliefs until a market-clearing price is
established. The traditional economic theory in the stock market holds that the
information observed by investors would affect the market equilibrium. The market
equilibrium price aggregates the market information and does provide a sufficient
statistic to all the private information in the market.
The market price which can transmit private information is initially modeled by
Lintner (1969). He analyzes an economy in which the beliefs of traders are
exogenous. This leads to a characterization of the equilibrium price as the weighted
average of these beliefs. (See also Rubinstein, 1975; Verrecchia, 1980)
Grossman (1976) analyzes an economy in which each trader observes only one
piece of information about the true value of the risky asset, and claims that the
rational expectations equilibrium price reveals all of the market information to all
traders, and it is a sufficient statistic of the market information, namely the market
equilibrium price can transmit all the market information. A major limitation of this
result is that when traders take the price as given, they have no incentive to acquire
any information when the market is free of noise (such as the supply shock, see
Diamond and Verrecchia, 1981). Under this situation, the private information is a
redundancy to investors, and both the number of informed traders and the
informativeness of price would be reduced.
Under quite general conditions, Grossman (1981) shows that equilibrium exists
and the equilibrium price completely aggregates and reveals the private information
of traders in the economy when the market is complete.
Scheinkman and Weiss (1986), Huffman (1987), Dumas (1989), and Campbell
and Kyle (1993), consider competitive models in which investors with homogeneous
information trade since they have different preferences and constraints. Kyle (1985),
Admati and Pfleiderer (1988), and Foster and Viswanathan (1990) consider
noncompetitive models of stock trading in which some investors have superior
information about the stock value and trade strategically to maximize their profits.
Despite their dominance in economic theories, the previous models have
limitations for the observational frequency of information being omitted, since they
assume that each informed trader observes only one piece of information. (See
Grossman, 1976; Grossman, 1978; Grossman, 1981; Grossman and Stiglitz, 1980;
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Hellwig, 1980; Bray, 1981; Paul, 1993; Baigent, 2003).
To make remedy, we model the traders with multiple sources of information. We
are the first to introduce the concept of the “observational frequency” of the market
information to a competitive stock market and to assume that the informed traders
could observe not only one piece of information but also several ones. Hence the
observational frequencies of information are divergent among the market
information.
This paper extends the existing competitive models and captures two types of
heterogeneity. First, we consider the observational frequency of information is not
uniform and there exists information asymmetry among investors. This differs from
the noisy rational expectations model, which commonly introduces informed and
uninformed trading to study the information asymmetry. We will focus on the
relationship among the observational frequency of information, the market
equilibrium price, the trading quantity, and the value of information. Second, we
assume that the market information is also heterogeneous in precision.
~ ~ ~
Assuming there are I sources of information x1 , x2 ,...x I with different precisions
in the market 1 , which are respectively observed f1 , f 2 ,..., f I times by N risk-averse
traders, and 0 ≤ f i ≤ N , i = 1,2,...I . If f i = 0 , it implies that no investor has ever
~
observed the information xi . To give an example, there are three informed investors
~
A, B, and C in the economy. The investor A observes the information x1 ; The
~ ~
investor B observes the information x1 and the information x2 ; The investor C
~
x2 and the information ~
x3 . Then the observational
observes the information
~ ~ ~
frequencies of the information x1 , x2 , and x3 are f1 = 2 (observed by A and B);
f 2 = 2 (observed by B and C); and f 3 = 1 (observed by C) respectively.
The informed investors observe and utilize some kinds of the market
information to form a more precise estimate for the expected value of the risky asset,
~
v , and to decide how many shares to invest to maximize his own utility. This
1
Throughout this paper we will put a ~ above a symbol to emphasize that it is a random variable.
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generates informative trading and price movement in the stock market. The
equilibrium price and the trading volume can be derived by solving the equilibrium
of the economy.
We show that the equilibrium price aggregates all the market information
according to not only the precision but also the observational frequency of each kind
of information. The equilibrium price, the trading quantity, and the expected utility
are functions of the information values, the observational frequencies and the
precisions of all the market information.
We also explore the conditions under which the equilibrium price could be a
sufficient statistic for the market information. We find that the price informativeness
is affected by the observational frequency of the market information.
The remainder of the paper is structured as follows: section 2 lays out the basic
economic environment to be analyzed. Especially, we are the first to consider the
impact of the observational frequencies of the market information and we prove the
lemma of additive property of trading volume. In section 3, we solve for the
competitive market equilibrium, the rational expectations equilibrium, and the fully-
informed economy equilibrium of the model to discuss how the equilibrium price,
the information value, and the trading quantity are influenced by the investor’s risk
coefficient, the precisions and the observational frequencies of all the market
information. In section 4, we establish a set of necessary and sufficient conditions
for the equilibrium price to be a sufficient statistic for the market information about
the true value of the risky asset. Conclusions and further suggestions are provided in
section 5.
2. The Model
2.1 Information Structure Assumptions
This is a two-period model. Assuming that the true value of the risky asset is
−1 ~ −1
distributed normally with zero mean 2 and its variance is Λ v , i.e. v ~ N (0, Λ v ) .
~
Following Titman and Trueman (1986), we assume that the prior distribution of v is
2
This assumption for zero mean of intrinsic value of risky asset is just for simplicity and will not influence our
analysis result.
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diffuse ( Λ v is trivial).
In addition to the equilibrium price ~ , there are kinds of information,
p I
available in the stock market. Each kind of information is related to the
~ , ~ ,..., ~
x1 x2 xI
true value of the risky asset and is an unbiased estimate of that true value.
Specifically,
xi ~ ~i ,
~ = v + ε i = 1,2,..., I
(1)
~
There is a noisy term , ε i ~ N (0, Λ i ) , which deters traders from learning the true
−1
~ −1 ~ ~
value of v . And the Var (ε i ) ≡ Λ i can be seen as the precision of the information xi
~ ~ ~
. Assuming that the noisy terms ε 1 , ε 2 ,..., ε I are jointly normally distributed and their
~ ~
covariance are zero, i.e. Cov (ε i , ε j ) = 0 ∀i ≤ I , j ≤ I , i ≠ j .
~
In addition to the price of the risky asset per share p , the informed traders
utilize various kinds of information available in the market to infer the true value of
the risky asset, make trading decisions, and maximize their own utility in the
uncertain circumstance. In the current period (before trading starts), each investor
~
searches for an information set θ which is a subset of all the market information,
~ ~ ~ ~ ~
i.e. θ ⊆ {x1 , x2 ,..., xI , p} .
~
After a trader observes the information set θ , he becomes informed to infer the
3
Grossman(1976) assume the precision is homogeneous, we generalized the assumption to that the precision is
heterogeneous.
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true value of risky asset and decides to trade shares qθ in the firm’s security
competitively. The trading size chosen by an investor is determined endogenously.
With the sign convention that purchases (buy orders; demand) are positive i.e. qθ > 0 ;
and sales (sell orders; supply) are negative, i.e. qθ < 0 . The only information that the
market broker can observe is the total number of orders to buy or sell at any given
time. In doing this, the market equilibrium is determined.
~
Let Cθ represent the pecuniary cost function of the information set θ with
precision Λθ . It is reasonable to assume that the information cost is an increasing
~ ∂Cθ
function of the precision of that information set θ , i.e. ∂Λ > 0 . As noted above, the
θ
∂Cθ
trading quantity is not a component of this cost function. i.e. ∂q = 0 .
θ
2.2 The Trading Assumption
Consider an economy where there are N traders with constant Arrow-Pratt
U ' ' (W )
measure of absolute risk aversion (CARA). − U ' (W ) = r , and U ' (W ) > 0 > U ' ' (W ) .
Without loss of generality, we assume that the initial endowments of the risky asset
(or total outstanding shares of stock) are O in the market.
The literature arising out of Kyle (1984) studies the polar case in which
informed investor’s incentive to trade on information is only mitigated by the effects
of trading on price. In that, the informed trader is assumed to be risk-neutral and
plays Cournot-Nash equilibrium. Consider the case in which informed traders have
imprecise information about the terminal payoff of the security and are trading in a
highly liquid financial market. In this case the informed traders would have to take a
large position in the security exposing them to a high degree of risk in order to
significantly move price. Thus for liquid financial markets, assuming that the
informed traders ignore their effects on price and concern about risk aversion is the
more reasonable of the two polar case. Hence in our paper, we will examine the
opposite polar case in which informed traders act competitively but are risk averse.
We begin our analysis with a theoretical examination of the operation of the
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competitive stock market. Being different to previous work, such as Kyle (1984) and
Grossman (1976), our model analyzes the market equilibrium on the basis of the
trading behavior induced when some sources of information are observed by the
investor.
For simplicity, we consider a representative trader with homogeneous
expectations. Thus the expectation of the true value of the risky asset by any trader
who has observed the same information is assumed to be equal (on average).
~
Let random wθ be the gain of financial wealth of any investor who has observed
~ ~ ~ ~
the information set θ . Let g (v | θ ) be the probability density of v , given θ = θ .
~
The objective of any trader who has observed the information set θ is to
maximize his own expected utility derived from financial wealth, i.e. Max
+∞
~ ~ ~
EU (Wθ ) ≡ ∫ U (Wθ ) g (v | θ )dv . The objective function can be written:
−∞
~ ~
Max EU (W ) = EU [(v − p )q | θ ]
qθ θ θ (2)
Where E is the expectations operator conditioned on the investor’s information.
Let ~ be the share price of the risky asset. Shares of the stock are perfectly divisible
p
and are traded at no cost in a competitive stock market.
2.3 The Determination of Market Trading
Traders act competitively after they receive the private information about the
stock’s true value in the stock market. This informational trading gives rise to
demand /supply for the risky asset.
Proposition 1:
Under the economy defined in section 2.1 and section 2.2, the informative
~
trading volume qθ for the risky asset conditioned on private information set θ
depends on the risk attitude of investors r , the expected price change (capital gain
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per share, E [ v − p | θ ] ) and the conditional precision of the information observed,
~
~
var −1 (v | θ ) . Specifically, we could derive the informative trading function of the
~
risky asset depending on the information set θ = θ as:
1
qθ = var −1 (v | θ ) E [ v − p | θ ]
~ ~
(3)
r
See appendix A for proof.
Consider the basic case when an investor has observed the information set
~
θ = {~k } to form a more precise estimate for the expected value of the risky asset, v .
x ~
~ ~ ~ ~ ~
Recall the equation (1): xk = v + ε k , Where v ~ N (0, Λ v ) ε k ~ N (0, Λ−k1 )
−1
and is
~
independent of v .
Given xk , we could prove that the conditional expectation and the inverse of the
variance of the true value of the risky asset are equal to be xk and Λ k respectively
(see Appendix B for proof). Written as:
~
E (v | xk ) = xk (4)
~
Var −1 (v | xk ) = Λ k (5)
~
By equations (4) and (5) , The posterior distribution of v conditioned on xk is
−1
normally distributed with mean xk and variance Λ k , where Λ k means the
conditional accuracy given the information xk . Specifically, when information xk is
observed, the informative trading volume can be written as:
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Λ k ( xk − p )
qk = (6)
r
Substituting equation (6) back to the objective function (2) gives
EU ( w | xk ) = Λ k ( xk − p ) − Ck
2
~
(7)
2r
In the following section, we assume that any investor could observe various
kinds of information. Before further discussion, we first prove the additive property
of the informative trading.
Lemma 1:
Under the economy defined in section 2.1 and section 2.2, the informative
trading volume for the risky asset of an investor who has observed the information
~ ~
set {xi , x j } is equal to the sum of the informative trading volume of an investor who
~
has observed the information xi and that of an investor who has observed the
~
information x j . Specifically, qi , j = qi + q j . More generally,
q1, 2,...,I = q1 + q2 + ... + qI (8)
See Appendix C for proof
3. Market Equilibrium and Comparative Analysis
3.1 Competitive Equilibrium vs. Rational Expectations Equilibrium
Let us now consider the equilibrium of the model defined in section2.1 and
section 2.2. Since the informative trading for the risky asset depends on the
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information observed, it is natural to think of the market clearing equilibrium (price
and trading quantity) and the optimal expected utility as functions of the market
~ ~ ~
information x1 , x2 ,..., x I . It implies that, different market information about the true
value of the risky asset leads to different market trading behavior of the risky asset
and gives different market equilibrium regimes.
In the proposition 2, we shall prove that the equilibrium price is a linear
function of all the market information and the initial endowments of the risky asset.
It reflects the observational frequencies and the precisions of all the market
~c
information and we find the competitive equilibrium price p is equal to the rational
~e
expectations equilibrium price p . Further, if the observational frequency of the
~c ~e
market information is homogeneous, then p (or p ) degenerates to the fully-
a
informed economy equilibrium price p .
Proposition 2:
Under the Walrasian market defined in section 2.1 and section 2.2, the rational
~e
expectations equilibrium price p is equivalent to the competitive equilibrium price
p which aggregates all the relevant information ~1 , ~2 ,...~I about a risky asset
~c x x x
~e
according to their corresponding observational frequencies and the precisions. p or
~c
p is determined by
I
~ rO ∑fΛ~
x
~e = ~c = x −
i i i
p p I ~≡
x i =1
∑
i =1
fi Λi ,
∑
I
fi Λi
(9a)
i =1
I
~
p p ~ ~
~c = ~e = v + ε − rO
~
∑ fΛε i i i
I
ε ≡ i =1
∑ i =1
fi Λi ,
∑
I
fi Λi
(9b)
i =1
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See Appendix D and Appendix E for proof
Equation (9a) reflects that the market competitive equilibrium price is a
weighted average of all the market information with weights of the corresponding
observational frequencies and the precisions of that information ( f1Λ 1 , f 2 Λ 2 ,..., f I Λ I ) .
All the traders’ expectations of future price of risky asset enter the price function.
This implies that the informationally efficient price system aggregates diverse
~ ~ ~
information ( x1 , x2 ,..., x I )by an auction mechanism with traders 4 , and completely
characterizes the price movement around a trade under a competitive market. Also,
the competitive equilibrium price or the rational expectations equilibrium price of
the stock market is in negatively related to the risk aversion of investors and the
total number of outstanding shares O .
For ignoring the effect of the observational frequency of information, Grossman
(1976) and related works argued that the market equilibrium price is a simple
average of the market information collected by traders and proved that the market
equilibrium price is a sufficient statistic for all the market information. However, we
show that the Grossman’s result is only a special case in which the aggregate
precision of each kind of information is uniform. i.e. f1Λ1 = f 2 Λ 2 = ... = f I Λ I .
Also, the properties of the equilibrium price in equations (9a, 9b) are
characterized as below:
~ rO
~ c ) = E ( ~ e ) = E[v ] −
E( p p I
∑fΛ
i =1
i i
(10)
I
~ ~
∑f i
2
Λi
Var ( ~ c ) = Var ( ~ e ) = Var ( ~ ) = Var (v ) + Var (ε ) , Var (ε ) =
p p x ~ i =1
I (11)
(∑ f i Λ i ) 2
i =1
4
Certainly, there is no information revelation without trade.
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∂p e ∂p c Λ x
= = I k k
∂f k ∂f k (12)
∑ fi Λ i i =1
I
∂p
∑f k xk
= i =1
(13)
∂Λ k I
∑
i =1
fiΛi
The relation between the equilibrium price and the precision of information is
affected by the observational frequency and the value observed of the information.
The equation (10) implies that the expected value of the competitive (or rational
expectations) equilibrium price is not larger than the expected intrinsic value of the
~
risky asset v . If the total outstanding shares of risky asset, O , or the risk aversion
coefficient of investors, r is equal to zero and the sum of the aggregate precision of
I
the market information ∑fΛ
i =1
i i rises to relatively large then that expected price
would approach to the intrinsic value of the risky asset. Under this situation, the
market competitive equilibrium price or the rational expectations equilibrium price
could be an unbiased estimator of the intrinsic value of the risky asset. Although
raising both the observational frequency and the precision of an alternative source of
information (such as non-financial information) will increase the sum of the
I
aggregate precision of information ∑fΛ
i =1
i i and the investors will become better
informed, if the specific information (such as a financial report) conveys a smaller
portion of the total information, then it could result decline in the value relevance of
that financial report in the meantime. Recently, the proliferations of alternative
information have informed investors better, but the annual corporate financial report
still convey a smaller portion of the total information. Hence, the value relevance of
that report declines.
The equation (11) appears that the price of the risky asset tends to exhibit
excess volatility relative to its intrinsic value.
By equation (12), the response of the competitive equilibrium price or the
rational expectations equilibrium price of the stock market to the observational
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frequency of the specific information f k is determined by both the precision and the
realization value of that information, and it is positively (negatively) related to the
observational frequency of that information when a good news, i.e. xk > 0 , (bad
news, i.e. xk < 0 ) is observed. Recent empirical researches also tend to support a
positive association between the disclosure level of information and the price of
equity capital (Botosan, 1997).
~
Empirically, the elasticity of the equilibrium price to a specific information xk
is an indicator of the value relevance of that information, denoted η k .
∂p xk f k Λ k xk2
ηk ≡ = (14)
∂xk p ∑ f i Λ i xi − rO
~
By equation (14), the value relevance of the information xk is positively related
to f k , Λ k , xk , r , and O .
2
We calculate the difference between the precision of the intrinsic value
~ ~ ~
conditioned on the full information x1 , x2 ,..., x I and that of conditioned on the single
price information, and gives:
I I
Var −1[v | ~1 , ~2 ,..., ~I ] - Var −1[v | p ] = ∑∑ ( f i − f j ) Λ i Λ j ≥ 0
~ x x ~ 2
x (15)
i =1 j =1
See Appendix F for proof.
~ ~ ~ ~
The precision of v conditioned on the full information x1 , x2 ,..., x I is not less
~
than that of v conditioned on the observed equilibrium price. Usually, the
observational frequency of the market information is not homogeneous, then
~ x x x ~ ~ ~
Var −1[v | ~1 , ~2 ,..., ~I ] > Var −1[v | p ] or Var[v | x1 , x2 ,..., xI ] < Var[v | p ] .
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~a
The (artificial) fully-informed economy equilibrium price, denoted by p , is
that each trader gets to observe the whole information available in the market and
then forms his trading decision for the risky asset (see Grossman, 1981, pp. 548).
~a
The p is a special case of the competitive (or rational expectations) equilibrium
price when the observational frequency of market information is homogeneous. We
shall now explore the fully-informed economy equilibrium price.
Corollary:
Under fully-informed economy, the observational frequency of market
~e ~c ~a ~e ~c ~a
information is homogeneous, then p or p degenerates to p . That is p = p = p ,
and
I
[∑ Λ i ~i ] − rO
x
~a =
p i =1
I (16)
∑ Λi
i =1
By equation (16), the fully-informed economy equilibrium price is a weighted
average according to the associated precisions of all the market information.
Proof: Since each trader posses full information {~1 , ~2 ,..., ~I } ,
x x x hence
f1 = f 2 = ... = f I = N , recall the additive property of informative trading volume, the
above equation (16) is easy to show.
Proposition 3:
Only when the observational frequency of market information is homogeneous,
i.e. f1 = f 2 = ... = f I , the equilibrium price is a sufficient statistic for market
information.
Usually, the observational frequency of market information is not
homogeneous. It implies that the current price can not serve as a sufficient estimator
of the intrinsic value of the risky asset; so that the traders still have incentives to
collect private information other than the equilibrium price to improve their
estimate. The competitive markets will tend to be informationally efficient but will
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not break down as the case described by Grossman (1976).
3.2 Comparative Analysis
Inserting equation (9a) back to equations (6) and (7), we obtain the optimal
trading quantity and the optimal expected utility generated by observing the specific
~
information xk
I
Λ ( x − p* ) Λ k
∑ f Λ (x i i k − xi ) + rO
q = k k
*
k
= i =1
I (17)
r r
∑ i =1
fi Λi
I
Λ ∑ f Λ (x
i i k − xi ) + rO
EU = k [ i =1
*
k I
]2 − C ( Λ k ) (18)
2r
∑fΛ i =1
i i
Equations (17) and (18) imply that both the optimal trading for the risky asset
and the optimal expected utility gained by an investor who has observed the
~
information xk reveal the aggregate differences between the specific information
~ ~
observed xk and the other information observed xi , i = 1,2,..., I , i ≠ k . The larger the
differences, the more active trading behavior an investor will adopt, and thus it leads
to higher levels of the optimal trading volume and the expected utility (the value of
that information).
Investors trade among themselves because they have different valuations about
the intrinsic value of the risky asset. Thus the informative trading behavior is closely
linked to the underlying heterogeneous among investors. With different model, our
result is similar as the result proposed by Wang (1994). Policies that eliminate
information dispersion will reduce the value of information and the liquidity for a
firm's securities.
This also implies that the trading volume and the value of information convey
important information about factors which price the risky asset. Our analysis has
valuable empirical applications. For example, one empirical difficulty is how to
identify the nature of the heterogeneity such as information realized, the precision ,
the observational frequency of the information, and so forth which are not directly
observable. Our result suggests that examining the joint behavior of the price, the
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trading volume, and the value of information can help one to learn about the
underlying heterogeneity among risky assets.
The preceding analysis yields several comparative results. The first involves the
precision of the specific information (such as the precision of brand, earning et al.)
for inferring the intrinsic value of a firm. To see how an increase in the precision of
the specific information affects its value (the utility gained when that information is
observed). Equation (18) could be differentiated with respect to Λ k or simply using
the envelop theory. Doing so yields
2
I
∂EU k * ∑ f i Λ i ( xk − xi ) + rO
1 i =1 r
= = 2 (qk ) 2 >0
*
(19)
∂Λ k 2r I
2Λ k
∑ fi Λi
i =1
~
The result shows that the value of the information xk increases as the precision
of that information increases. Intuitively, this is because an increase in the precision
of information increases the expected financial wealth. On the one hand, the greater
the precision of information, the more weight investors place on that information,
and hence the greater the utility gained. On the other hand, the greater the precision
of information, the less uncertainty the investors face over the nature of the risky
asset, and therefore the smaller the risk premium is.
Second, to discuss how an information value is to be influenced by the risk-
averse attitude of investors, equation (18) could be differentiated with respect to r or
simply using the envelop theory. Doing so yields:
2
I
∑ f i Λ i ( xk − xi ) + rO
∂EU k − Λ k i =1
*
(q * ) 2
= = − k <0 (20)
∂r 2r 2 I
2Λ i
∑ fi Λ i
i =1
By equation (20), the higher the risk aversion of the investor is, the lower the
value of the information to the trader is. This is because the greater the coefficient of
risk aversion of the investor is, the more the risk premium is.
Third, to discuss how the informative trading quantity to be influenced by the
observational frequency of the specific information, equation (17) could be
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differentiated with respect to f k , and gives
Λ − rOΛ k
∂qk = k I
*
≤0
r
∂f k (∑ f i Λ i ) 2 (21)
i =1
Equation (21) shows that the increase of the observational frequency of the
specific information f k will crowd out its optimal trading quantity correspondingly
because of the existence of initial risky asset endowment. If there does not exist the
*
initial endowments of risky asset O , the informative trading quantity qk will be
fixed. The crowding out effect means that an investor observing a given piece of
information trades less aggressively on that information as the number of investors
who observe that piece of information increase (Paul, 1993). We find that when there
is no initial endowments of risky asset, the crowding out would have not any effects.
4. Condition of Informative Efficiency
The issue “Is the equilibrium price a sufficient statistic for the information
about the true value of firms?” is central to the study about the price informativeness
of stock market. Under strong assumptions of homogeneous in the precision and the
observational frequency of market information, Grossman (1976) and related works
conclude that the market equilibrium price is a simple average over the market
information collected by traders and is a sufficient statistic for the market
information. While the current price is a sufficient statistic for the unknown value of
~
v , the price system may eliminate the incentives of investors for collecting the
private information. Interestingly, they present a “self-destruction” model.
In this section, we will prove that the homogeneity of the observational
frequency of market information is a sufficient condition for the market equilibrium
price to be a sufficient statistic for the information about the true value of the risky
asset. Hence, the previous conclusion that the market equilibrium price is a
sufficient statistic about the true value of the risky asset might be just an inevitable
outcome for ignoring the heterogeneity of the observational frequency of market
information.
Recall equation (3), the informative trading decisions of investors are
completely characterized by the conditional mean and variance of the intrinsic value
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of the asset, hence we have lemma 2. (Also see Bray, 1981)
Lemma 2:
~
E ( v | xk , p ) ~
E (v | p )
= ~ ~
If all the random variables are normal, and if Var (~ | x , p) E (v | p ) , then p is
v k
~
a sufficient statistic for the information xk .
Proposition 4:
If and only if
Cov ( p, xk ) ~ ~
V(p ~ ) = 1 or [Cov ( p, xk ) − V (v )] p − [V ( p) − V (v )]xk = 0 (22)
~ ~
then the random variable p is a sufficient statistic for the information xk .
See the Appendix G for proof.
Cov ( ~, ~k )
p x
It makes sense that the coefficient = 1 is analogous to the coefficient
V ( ~)
p
~ ~ ~
on p , when one regresses xk on p using OLS.
~
~ = ∑ f i Λ i xi Cov ( ~, ~ ) − V (v ) = f k ~ ) − V (v ) = ∑ f i Λ i
2
p p xk ~ V(p ~
Recall ∑ fi Λi , ∑ fi Λi , (∑ f i Λ i ) 2 , thus the
f k ∑ f i Λ i xi
condition of informative efficiency is that xk = ∑ fi 2Λi ,
∀k = 1,2,..., I . It is
obvious that the condition (22) is remote to hold for the information asymmetry
prevailing in the market, i.e. f k ≠ f i , ∀k ≠ i . Therefore, the heterogeneity of the
observational frequency of market information is another impossibility of
informative efficiency.
Proposition 5:
If the observational frequency of each piece of information is homogeneous,
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~c
that is f1 = f 2 = ... = f k , then the market competitive equilibrium price p , the rational
~e
expectations equilibrium price p and the fully-informed economy equilibrium price
~a
p are all equal and serve as a sufficient statistic for all the market information
~
about the true value of the risky asset v .
See Appendix H for detail proof.
5. Conclusion and Further Research Suggestion
When some agents behave irrationally or when some markets operate
inefficiently, opportunities exist for others to make profit. The profit-seeking
behavior of investors tend to eliminate these opportunities as some individuals
attempt to earn extra return on information collection in a perfect competitive
market, then the market equilibrium price will be affected and the private
information will be transmitted across all the agents.
Grossman (1976) argue that an informationally efficient price system
aggregates diverse information perfectly, but in doing this the price system also
eliminates the private incentive for collecting the information. We argue that they
just discuss a special case where the aggregate precision of each kind of information,
denoted by f i Λ i , is uniform. i.e. f1Λ1 = f 2 Λ 2 = ... = f I Λ I .
As the traditional model of stock market assumes that each informed trader
observes only one kind of information, (see Grossman, 1976; 1978; 1981; Grossman
and Stiglitz, 1980; Hellwig, 1980; Bray, 1981; Paul, 1993; and Baigent, 2003), we
emphasize the factor of the “Observational Frequency” of the market information
to a competitive stock market and to assume that each informed trader could observe
more than one piece of information. Hence the observational frequencies among all
the market information are heterogeneous.
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Without loss of generality, we assume that the total number of outstanding
shares (initial endowments of risky asset) are O in the market. We analyze the
~ ~ ~
operation of the price system where there are I kinds of information x1 , x2 ,...x I
available in a competitive stock market, which is to be observed f1 , f 2 ,..., f I times
respectively by N risk-averse traders with constant risk attitude. Each Informed
investor bases on the information set observed to form a more precise estimate for
~
the expected value of the firm’s true value, v , and then invests to maximize his own
expected utility. Thus the stock market equilibrium is determined. Our main findings
are as follows.
First, we show that the market equilibrium price in the above framework
aggregates all the market information according not only to the precision but also to
the observational frequency of each kind of market information. The equilibrium
price, trading quantity, and the value of information are functions of the realized
information, the precisions and the observational frequencies of all the market
information.
Second, we find that the market competitive equilibrium price is the same as the
rational expectations equilibrium price. The full-informed equilibrium price is a
special case of that competitive (or rational expectations) equilibrium price under the
situation when the observational frequencies of all the market information are
equalized. The response of the equilibrium price to an unexpected change of
information is positively related to both the observational frequency and the
precision of that information. Investors trade among themselves because they have
different valuations about the intrinsic value of the risky asset. Thus the informative
trading behaviors are closely linked to the underlying heterogeneity among
investors. Policies that eliminate information dispersion will reduce the value of
information and the liquidity for a firm's securities.
Third, only when the observational frequencies of all the market information
are indifferent, the market equilibrium price could be a statistic that sufficiently
reflects the market information about the intrinsic value of the risky asset. However,
the observational frequencies of the market information are usually not uniform.
Hence, the traders still have incentives to collect and to search for valuable
information to improve their estimate of the true value of the risky asset. Under this
situation, the market but will not break down as the case described by Grossman
(1967). Therefore, the heterogeneity of the observational frequency of the market
information is another impossibility of informative efficiency.
Finally, the previous works conclude that the market equilibrium price is a
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sufficient statistic for the market information under naively assuming that each
investor observes only one piece of information. It might be just an inevitable
outcome of their model settings for ignoring the heterogeneity of the observational
frequency of the market information. This paper changes the assumption of
traditional model in the competitive stock market and creates the new concept of
market equilibrium to point out the new determines of research. It is valuable to
reconsider the heterogeneity of the observational frequency of the information to
explore more general results in the related fields.
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References
1. Admati, Anat R., & Pfleiderer, Paul (1988). A Theory of Intraday Patterns: Volume and
Price Variability.” Rev. Financial Studies no.1,3-40
2. Baigent, G. Glenn (2003). Competitive markets and aggregate information. Eastern
Economic Journal, fall, 29, 4, pp.593-606.
3. Botosan, Christine A. (1997). Disclosure Level and the Cost of Equity Capital. The
Accounting Review, Vol.72, No.3, July, pp.323-349.
4. Bray, M. (1981). Futures Trading, Rational Expectations, and The Efficient Markets
Hypothesis. Econometrica, May 49, 3,575-596.
5. Campbell, John Y., & Kyle, Albert S. (1993). Smart Money, Noise Trading and Stock
Price Behavior. Rev. Econ. Studies 60(January), 1-34.
6. Diamond, Douglas W. & Verrecchia, Robert E. (1981). Information Aggregation in a
Noisy Rational Expectations economy. Journal of Financial Economics 9,221-235.
7. Diamond, Douglas W. & Verrecchia, Robert E. (1991). Disclosure, Liquidity, and the
Cost of Capital”. The Journal of Finance. Cambridge: Sep.Vol.46, Iss. 4; pg. 1325, 35
pgs
8. Dumas, B. (1989). Two-Person Dynamic Equilibrium in the Capital Market. Rev.
Financial Studies 2, No.2, 157-88.
9. Foster, F. D., & Viswanathan, S. (1990). A Theory of the Interday Variation in Volume,
Variance, and Trading Costs in Securities Markets. Rev. Financial Studies, no.4,
593-624.
10. Grossman, S.J. (1976). On the Efficiency of Competitive Stock markets where Traders
Have Diverse Information”. The Journal of Finance, (May.)vol.31,no.2:573-583
11. Grossman, S.J.(1978). Further Results on The Information Efficiency of competitive
Stock markets. Journal of Economic Theory, 18, 81-101.
12. Grossman, S.J. (1981). An Introduction to the Theory of Rational Expectations under
Asymmetric Information. Review of Economic Studies, 48, 541-559.
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6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
13. Grossman, S.J. & Stiglitz, J. E. (1980). On the Impossibility of Informationally
Efficient Markets. American Economic Review, (Jun.),Vol.70,no.3, : 393-408
14. Hellwig, M.F. (1980) On the Aggregation of Information in Competitive Markets.
Journal of Economic Theory 22,477-498.
15. Huffman, George W.(1987). A Dynamic equilibrium model of asset prices and
transaction volume.” J. P. E. 95(February), pp. 138-159.
16. Kyle, A.S. (1984). Market Structure, Information, Futures Markets and Price
Formation, in G.G. Storey, A. Schmitz, and A.H. Sarris, eds: International Agricultural
Trade: Advanced Readings in Price Formation, Market Structure, and price Instability ,
Westview Press Boulder, Colo.,: 45-64.
17. Kyle, A.S. (1985). Continuous Auctions and Insider Trading. Econometrica
56(November), 1315-35.
18. Lintner, J. (1969). The Aggregation of Investors Diverse Judgments and Preferences in
Purely Competitive Stock markets. Journal of Financial and Quantitative Analysis
4,347-400.
19. Paul, Jonathan, M. (1993). Crowding Out and the Informativeness of Security Prices.
The Journal of Finance vol. xlviii no 4 sep. : 1475-1496
20. Rubinstein, M. (1975). Securities market efficiency in an Arrow-Debreu economy,
American Economic Review 65,812-824.
21. Scheinkman, Jose A., & Laurence Weiss (1986). Borrowing Constraints and Aggregate
Economic Activity. Econometrica 54 (January), pp.23-45.
22. Titman, S. & Trueman (1986). Information Quality and the Valuation of New Issues.
Journal of Accounting and Economics 8, pp. 159-172.
23. Verrecchia, R.E. (1980). Consensus beliefs, information acquisition, and market
efficiency. American Economic Review 70, 874-884.
24. Wang, J. (1994). A Model of Competitive Stock Trading Volume. Journal of Political
Economy.102, 127-168.
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6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
Appendix A:
Assuming that traders are price takers, we describe the optimization problem
below.
~
The trading volume caused when the information θ is observed by a trader is
~
denoted qθ , and that trade could gain a random financial wealth wθ and expected
~
utility EU ( w | θ ) . The expected utility is concave in qθ , i.e. U > 0 and U < 0
' ''
The objective of a representative potential trader who has observed information
~
θ is to choose trading quantity qθ to maximize his own expected utility of financial
wealth. That is:
~ ~
Max EU (W ) = EU [(v − p)q | θ ]
Max qθ θ θ ,
The utility-maximizing value of qθ is given by the following first-order condition.
~ ~
F.O.C. E[U ' (W ) ⋅ (v − p ) | θ ] = 0
~ ~ ~ ~
E[U ' (W ) ⋅ (v − p ) | θ ] = E[U ' (W ) | θ ] ⋅ E[(v − p ) | θ ] + Cov[U ' (W ), (~ − p) | θ ] = 0
~ v (A1)
By Stein’s Lemma:
~ ~ ~ ~ ~ ~ ~
Cov[U ' (W ), (v − p ) | θ ] = Cov[v , (v − p ) | θ ] ⋅ E[U ' ' (W ) | θ ]qθ = Var (v | θ ) ⋅ E[U ' ' (W ) | θ ]qθ
(A2)
Insert the equation (A.2) back to the equation (A.1), gives
~
E[v − p | θ ] ~
E[v − p | θ ]
qθ = = ~
Var[v | θ ] rVar[v | θ ]
U '' ~
−
U'
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Where E[.] and Var[.] represent the mean and variance operators, given the
information observed .
~
θ
Appendix B:
~ ~ ~ ~ ~
Given that xk = v + ε k , ε k ~ N (0, Λ k ) , v ~ N (uv , Λ v ) ,
−1 −1
~
v ~~
vv v ~k Λ−1
~x Λ−1
Let ∑ = var ~ = cov ~ ~ ~ ~ = Λ−1
v v
xk xk v xk xk v Λ−1 + Λ−k1
v
~
v | xk ~ N [uv + ∑ vk ∑ −1 ( xk − u k ), ∑ vv − ∑ vk ∑ −1 ∑ kv ]
kk kk
~ 1 1
Var (v | xk ) = ∑ vv − ∑ vk ∑ −1 ∑ kv = Λ−1 − Λ−1
kk v v −1 −1
Λ−1 =
v
Λ + Λk
v Λv + Λk
Λk
E (v | xk ) = uv + ∑ vk ∑ −1 ( xk − u k ) =0+ ( xk − 0)
kk
Λv + Λk
Since Λ v is trivial, we have
~
Var −1 (v | xk ) = Λ k (B1)
E (v | xk ) = x k (B2)
1 Λ ( x − p)
qk = var −1 (v − p | xk ) E [ v − p | xk ] = k k
~ ~
(B3)
r r
Appendix C: The additive property for demand
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~ ~
Assuming that there are information xi and x j , i ≠ j available in the stock
market, and each kind of information is an unbiased estimate of the true value of the
~ −1
risky asset. Specifically, given v ~ N (0, Λ v )
x ~ ~
xi ~ ~i , and ~ j = v + ε j
~ = v +ε
~ −1 ~ −1
Where ε i ~ N (0, Λ i ) and ε j ~ N (0, Λ j ) are both independent of v , i.e.
~ ~ ~ ~ ~ ~
cov(v , ε i ) = 0 and cov(v , ε j ) = 0 . Let cov(ε i , ε j ) = ε ij
The following basic results are useful.
~ ~
cov(v , v ) = Λ−1
v
~ ~ 1 1
cov(v , ~i ) = E[(v − 0)( ~i − 0)] = E[v(v + ε i )] = E[v ⋅ v + v ⋅ ε i ] =
~ x x ~ ~
, cov(v , x j ) = Λ ,
Λv v
1 1 1 ~ ~ 1 1
cov( ~i , ~i ) =
x x + ~ ~
, cov( xi , x j ) = Λ + ε ij And cov( x j , x j ) = Λ + Λ
Λv Λi v v j
~ ~
Investors could use xi along with x j to form a more precise estimate for the expected
~
value of the firm’s terminal value, v .
Using moment generating function ,we get
~
~ | X ~ N [u + ∑ ∑ −1 ( X − u ), ∑ − ∑ ∑ −1 ∑ ] X = xi
v
~
v vX XX X vv vX XX Xv , ~
x j
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1 1 1
~
~
v ~~
vv v ~i
~x v ~j Λ v
~x Λv Λv
v ~ ~~ ~~ ~~ = 1 1 1 1
+ ε ij ,
Let ∑ = var X = Var xi = cov xi v
~ xi xi xi x j +
x ~ v ~ x j ~i ~ ~ Λv Λv Λi Λv
j xj x xjxj
1 1 1 1
Λ + ε ij +
v Λv Λv Λ j
1 1 1 1
Λ Λ + Λ + ε ij
1 1 1 Λv
∑ vv = , ∑ vX = , , ∑ Xv = v ∑ XX = v i
1 1 1 1
Λv Λv Λv + ε ij +
Λv
Λv
Λv Λ j
Λv + Λ j 1 + ε ij Λ v
ΛΛ −
v j Λv
1 + ε ij Λ v Λv + Λi
− Λ ΛvΛi
∑ −1 = v
XX
Λ v + Λ v Λ i + Λ v Λ j + Λ j Λ i 1 + Λ2vε ij + 2Λ vε ij
2 2
−
Λ2v Λ j Λ i Λ2v
~ 1 − Λ i Λ j ε ij
2
Var (v | xi , x j ) = ∑ vv − ∑ vX ∑ −1 ∑ Xv =
Λ v + Λ i + Λ j − Λ i Λ j Λ vε ij − 2Λ i Λ j ε ij
XX 2
Λ i (1 − Λ j ε ij ) xi + Λ j (1 − Λ i ε ij ) x j
E (v | xi , x j ) = u v + ∑ vx ∑ −1 ( X − u X ) =
XX
Λ v (1 − Λ i Λ j ε ij ) + Λ i (1 − Λ j ε ij ) + Λ j (1 − Λ i ε ij )
2
Since Λ v is trivial, and according the independent assumption cov(ε 1 , ε 2 ) = ε ij = 0 , We
get
~
Var −1 (v | xi , x j ) = Λ i + Λ j (C1)
~ Λi Λj
E (v | xi , x j ) = xi + xj (C2)
Λi + Λ j Λi + Λ j
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*
Thus the optimal utility EU i + j gained by any trader who has observed the
realized value xi and x j is written as:
~ [
~ ] 1 ~
EU ( wi | xi , x j ) = E v − p | xi , x j qi + j − r ( qi + j ) 2 var(v − p | xi , x j ) − C (Λ i ) − C ( Λ j )
2
*
The utility-maximizing value of qi + j is given by the following first-order
condition.
~ [ ] ~
F.O.C. E v − p | xi , x j − rqi + j var(v − p | xi , x j ) = 0 , and gives the demand
Λi Λ
qi*+ j = ( xi − p) + j ( x j − p ) = qi + q j (C3)
r r
It is very easy to see the corollary result. Since
q1, 2,...,I = q1 + q2,3,...,I = ... = q1 + q2 + ... + qI (C4)
Appendix D: The competitive equilibrium
There are N potential traders in the market, and O be the total stock shares of
~
the risky asset. Let 0 ≤ f i ≤ N , i = 1,2,...I means the frequency of information xi
being observed by investors . An equilibrium price must satisfies market clearing
condition, that is, ∑ fq = O , hence 5
f1q x1 + f 2 q x2 + ... + f I q xI = O (D1)
The equation (D1) states that the aggregate trading quantity over market
~ ~~ ~
information set x = ( x1, x2, ..., xI ) for the risky asset must equal the number of shares
5
Since the utility of investors is identical , the property of initial endowment irrelevance holds.
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outstanding O (initial endowment of risky asset).
Where and q1 , q2 ,..., qI is the trading quantity function given by equation (6), and
~c
p denote the competitive equilibrium price which solves the market clear equation.
We have market clear equation as:
I
fi Λi
∑
i =1 r
( xi − p ) = O (D2)
~c
p ,
We solves the equation (D2) for competitive equilibrium price
~ ~ ~ ~c
According to the assumption that x1 , x2 ,...x I are independent with each other, p is a
linear function of a vector of joint normal distributed variables and is also normally
distributed. We can rewrite price as a random form as
I
~* = ~ −
p x
rO ∑fΛ~
x i i i
I ~=
x i =1
∑
i =1
fiΛi ,
∑
I
fi Λi
(D3)
i =1
~ ~ ~
or equivalently (recall that xi = v + ε i )
I
~
p ~ ~
~* = v + ε − rOr
~
∑fΛε i i i
I
ε = i =1
∑ i =1
fi Λi ,
∑fΛ
I
i i
(D4)
i =1
Appendix E: The Rational Expectations equilibrium
~ ~
We first assume that p = v + ε , and once we have found an rational expectations
equilibrium price later, we will confirm that such a condition does indeed exist.
Under the economy defined in section 2.1 and section2.2, the trading quantity
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~ ~
of an investor for risky asset who has observed the information xi , p denoted qi , p ;
~ ~ ~
the trading quantity of an investor who has observed the information xi , x j , p
denoted qi , j , p Specifically,
Λi Λ Λ Λ
qi , p = ( xi − p ) − v p ; q j , p = j ( x j − p ) − v p
r r r r
Λi Λ Λ
qi , j , p = ( xi − p ) + j ( x j − p ) − v p
r r r
f1Λ1 f Λ f Λ Λ
( x1 − p ) + 2 2 ( x2 − p ) + ... + I I ( xI − p ) − N v p = O , assuming that Λ v is
r r r r
trivial
I
∑ f Λ x − rO
i i i
~e =
p i =1
I
= ~ c Q.E.D.
p
∑fΛ
i =1
i i
Appendix F:
~ ~
According assumption that v and ε i , i = 1,2,.., I are independent with each
other. We get
~ ~
~ * , v ) = E[ ∑ f i Λ i (v + ε i ) − rOt ⋅ v ] =
Cov ( p ~ ~ ∑fΛ
i i ~
⋅ Var (v ) = σ v2 = Λ−1
∑ ∑fΛ
v
fi Λi i i
Cov (v , ~k ) = E[v ⋅ (v + ε k )] = Λ−1
~ x ~ ~ ~
v
~ ~~ v ~ * Λ v
~p Λ−1
−1
v vv v
Var ~ * = Cov ~ *~ ~ * ~ * = Λ−1 −1
p p v p p v Λp
~ Λ v − Λ p Λ−p1 − Λ−1
Var[v | p] = ∑ vv − ∑ vp ∑ −1 ∑ pv =
v
pp = −1
Λv2
ΛvΛ p
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33.
6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
Var ( ~ ) − Var (v )
p ~
~* ~ ~
= ~) ,where Vap( p ) = Var (v ) + Var (ε )
Λ vVar ( p
~ ~
~ Var (ε ) Var (ε )
⇒ Var[v | p ] = ~ ~ = ~
Λ v [Var (v ) + Var (ε )] 1 + Λ vVar (ε )]
~ ~
Var −1[v | p] = Var −1 (ε ) + Λ v (F1)
I I
~
~
∑ f i Λ iε i
−1 ~
(∑ f i Λ i ) 2
Where ε = and Var (ε ) =
i =1 i =1
I I insert back to (C.1)
∑αiΛi
i =1
∑ i =1
fi2Λi
I I
Var −1[v | x1 , x2 ,..., x I ] - Var −1[v | p] = ∑∑ ( f i − f j ) Λ i Λ j >0 , if f i ≠ f j ∀i ≠ j
~ ~ 2
i =1 j =1
Equation (15) approved.
Appendix G: Condition of informative efficiency
~ ~ ~ ~ ~
Assume that random variables are joint normal and Cov (v , xk ) = Cov (v , p ) = Var (v ) ,
and traders have constant absolute risk averse.
~ ~ ~
If and only if p is a sufficient statistic for xk about v , then
~
E[v | p, xk ] ~
E[v | p]
⇔ ~ = ~
Var[v | p, xk ] Var[v | p ]
~ ~
E[v | p, xk ] − E[v | p ] ~
E[v | p ]
⇔ ~ ~ = ~
Var[v | p, xk ] − Var[v | p] Var[v | p ]
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34.
6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
~ ~~
v vv v ~*
~p v ~k Λ−1 Λ−1
~x
v v Λ−1
v
~ ~ * ~ = Λ−1 Λ−1
Var ~ * = Cov ~ *v
p p ~* ~*
p p p xk v p σ pxk , σ pxk ≡ Cov ( ~ * , ~k )
p x
~
xk xk ~
~v ~ ~*
xk p ~ ~ Λ−1 σ
xk xk v ~ )
V ( xk
pxk
Λ−1
∑ vv = Λ−1 , ∑ vθ = [Λ−1
v v Λ−1 ] , ∑θv = −1
v
v
Λv
Λ−p1 σ pxk
∑θθ =
σ pxk
V ( ~k )
x
V ( ~k ) − σ pxk
'
x
− σ Λ−p
1
−1
∑θθ = pxk
Λ−p1
σ pxk
σ pxk V ( ~k )
x
Part1
~ ~
E[v | p * , xk ] = uv + ∑ vθ ∑θθ ( X − uθ ) ; E[v | p * ] = u v + ∑ vp ∑ −1 ( p − u p )
−1
pp
~ ~* ~ ~* ~* ~
~ ~ [V ( xk ) − Cov( p , xk )] p + [V ( p ) − Cov ( p , xk )]xk
E[v | p * , xk ] = V (v )
V ( ~ * )V ( ~k ) − Cov 2 ( ~ * , ~k )
p x p x (G1)
~
~ | p * ] = V (v ) p
E[v
V ( ~)
p (G2)
~ * ~ *
(G1)-(G2)= E[v | p , xk ] − E[v | p ]
~* ~ ~* ~ ~ ~* ~* ~* ~
~ ) Cov ( p , xk )[Cov ( p , xk ) − Var ( p )] p + Var ( p )[Var ( p ) − Cov ( p , xk )]xk
= Var (v
Var ( ~ * )[Var ( ~ * )Var ( ~k ) − Cov 2 ( ~ * , ~k )]
p p x p x
~
Var (v )
= [Var ( ~ * ) − Cov ( ~ * , ~k )][Var ( ~ ) xk − Cov( ~ * , ~k ) p]
p p x p p x
Var ( ~ * )[Var ( ~ * )Var ( ~k ) − Cov 2 ( ~ * , ~k )]
p p x p x
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35.
6th Global Conference on Business & Economics ISBN : 0-9742114-6-X
∀p, xk ∈ ℜ
E[v | ~ * , ~k ] = E[v | ~ * ] ⇔ Var ( ~ * ) = Cov( ~ * , ~k )
~ p x ~ p p p x (G3)
Part2
~ ~
Var[v | p, xk ] = ∑ vv − ∑ vθ ∑θθ ∑θv ; Var[v | p ] = ∑ vv − ∑ vp ∑ −1 ∑ pv
−1
pp
~
[V (v )]2
~ ~
Var[v | p, xk ] = V (v ) − [V ( ~k ) + V ( ~ ) − 2σ pxk ]
x p
V ( ~ )V ( ~k ) − σ 2 k
p x px
(G4)
~ 2
~ ~ [V (v )]
Var[v | p ] = V (v ) − (G5)
V ( ~)
p
~ ~ ~ [V ( ~ ) − σ pxk ]2
p
(G4)-(G5)= Var[v | p, xk ] - Var[v | p ] = − [V (v )]2 ~ ≤0
V ( p )[V ( ~ )V ( ~k ) − σ 2 k ]
p x px
Var[v | ~ * , ~k ] = Var[v | ~ * ] ⇔ Var ( ~ ) = Cov ( ~, ~k )
~ p x ~ p p p x (G6)
~
E[v | p, xk ] ~
E[v | p ]
~ = ~
Var[v | p, xk ] Var[v | p ]
⇔ [Cov ( p, xk ) − V ( ~ )] ⋅ {[Cov ( p, xk ) − V (v )] p − [V ( p ) − V (v )]xk } = 0
p ~ ~
(G7)
Q.E.D.
Appendix H:
~ ~ ~
First prove that if f1 = f 2 = ... = f I , then Var ( p ) = Cov ( xk , p )
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