6th Global Conference on Business & Economics                  ISBN : 0-9742114-6-X




   What Determines the Stock Price...
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sufficient statistic for the ...
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Hellwig...
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diffuse ( Λ...
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true value of risky ass...
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competitive stock ma...
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per sha...
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6th Global Conference on Business & Economics                                                       ISBN : 0-9742114-6-X

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6th Global Conference on Business & Economics                                                                    ISBN : 0-...
6th Global Conference on Business & Economics                                    ISBN : 0-9742114-6-X



trading volume, a...
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differentiated with r...
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of the asset,...
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     Without loss of gene...
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sufficient statistic for the mark...
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13. Grossman, S.J. & ...
6th Global Conference on Business & Economics                                                 ISBN : 0-9742114-6-X




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6th Global Conference on Business & Economics                                                          ISBN : 0-9742114-6-...
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6th Global Conference on Business & Economics                                                 ISBN : 0-9742114-6-X



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  1. 1. 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
  2. 2. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 2 GUTMAN CONFERENCE CENTER, USA
  3. 3. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 3 GUTMAN CONFERENCE CENTER, USA
  4. 4. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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; OCTOBER 15-17, 2006 4 GUTMAN CONFERENCE CENTER, USA
  5. 5. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 5 GUTMAN CONFERENCE CENTER, USA
  6. 6. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 6 GUTMAN CONFERENCE CENTER, USA
  7. 7. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 7 GUTMAN CONFERENCE CENTER, USA
  8. 8. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 8 GUTMAN CONFERENCE CENTER, USA
  9. 9. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 9 GUTMAN CONFERENCE CENTER, USA
  10. 10. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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: OCTOBER 15-17, 2006 10 GUTMAN CONFERENCE CENTER, USA
  11. 11. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 11 GUTMAN CONFERENCE CENTER, USA
  12. 12. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 12 GUTMAN CONFERENCE CENTER, USA
  13. 13. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 13 GUTMAN CONFERENCE CENTER, USA
  14. 14. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X ∂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 OCTOBER 15-17, 2006 14 GUTMAN CONFERENCE CENTER, USA
  15. 15. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 ] . OCTOBER 15-17, 2006 15 GUTMAN CONFERENCE CENTER, USA
  16. 16. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X ~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 OCTOBER 15-17, 2006 16 GUTMAN CONFERENCE CENTER, USA
  17. 17. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 17 GUTMAN CONFERENCE CENTER, USA
  18. 18. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 18 GUTMAN CONFERENCE CENTER, USA
  19. 19. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 19 GUTMAN CONFERENCE CENTER, USA
  20. 20. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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, OCTOBER 15-17, 2006 20 GUTMAN CONFERENCE CENTER, USA
  21. 21. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X ~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. OCTOBER 15-17, 2006 21 GUTMAN CONFERENCE CENTER, USA
  22. 22. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 22 GUTMAN CONFERENCE CENTER, USA
  23. 23. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 23 GUTMAN CONFERENCE CENTER, USA
  24. 24. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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. OCTOBER 15-17, 2006 24 GUTMAN CONFERENCE CENTER, USA
  25. 25. 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. OCTOBER 15-17, 2006 25 GUTMAN CONFERENCE CENTER, USA
  26. 26. 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' OCTOBER 15-17, 2006 26 GUTMAN CONFERENCE CENTER, USA
  27. 27. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 27 GUTMAN CONFERENCE CENTER, USA
  28. 28. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X ~ ~ 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  OCTOBER 15-17, 2006 28 GUTMAN CONFERENCE CENTER, USA
  29. 29. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X  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 OCTOBER 15-17, 2006 29 GUTMAN CONFERENCE CENTER, USA
  30. 30. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X * 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. OCTOBER 15-17, 2006 30 GUTMAN CONFERENCE CENTER, USA
  31. 31. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X 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 OCTOBER 15-17, 2006 31 GUTMAN CONFERENCE CENTER, USA
  32. 32. 6th Global Conference on Business & Economics ISBN : 0-9742114-6-X ~ ~ 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 OCTOBER 15-17, 2006 32 GUTMAN CONFERENCE CENTER, USA
  33. 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 ] OCTOBER 15-17, 2006 33 GUTMAN CONFERENCE CENTER, USA
  34. 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 OCTOBER 15-17, 2006 34 GUTMAN CONFERENCE CENTER, USA
  35. 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 ) OCTOBER 15-17, 2006 35 GUTMAN CONFERENCE CENTER, USA

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