Electronic copy available at: http://ssrn.com/abstract=2383340

Competitive Trading and Endogenous Learning of

Asymmetrically Informed Investors∗

Yu Liu Hao Wang Lihong Zhang†

February 13, 2014

Abstract

Information asymmetry between privately informed investors, interacting with pub-

lic information transparency, significantly affects trading and learning behaviors, price

formation, information revelation, and market efficiency. Private information asymmetry-

induced strategic trading behaviors explain the asymmetric U-shape patterns of intra-

day stock trading volume, return, and volatility. Market depth tends to rise (fall) at

market opening (closure) with private information asymmetry increasing. A non-zero

degree of private information asymmetry leads to minimal informational trading profit

and maximal market efficiency. Disclosure policy should consider its effects on both

private information asymmetry and public information transparency in optimizing ef-

ficiency and investor protection.

JEL Classification: G14, G18.

Keywords: Private information asymmetry, strategic trading, endogenous learning,

market micro-structure, information disclosure.

∗We would like to thank Jiangze Bian, Ming Guo, Burton Hollifield, Neil Pearson, Hao Zhou, and par-ticipants of the CFAM 2013 for helpful discussions. All errors are ours. The authors acknowledge fundingsupport from the National Natural Science Foundation of China (Grant No. 71272023 and 71071086).†Yu Liu, School of Economics and Management, Tsinghua University, Beijing 100084, China, e-mail:

liuy5.13@sem.tsinghua.edu.cn; Hao Wang, School of Economics and Management, address: 318 WeilunBuilding, Tsinghua University, Beijing 100084, China, e-mail: wanghao@sem.tsinghua.edu.cn; tel: 86 10-62797482; Corresponding author: Lihong Zhang, School of Economics and Management, address: 322Weilun Building, Tsinghua University, Beijing 100084, China, e-mail: zhanglh2@sem.tsinghua.edu.cn; tel:86 10-62789963.

1

Electronic copy available at: http://ssrn.com/abstract=2383340

1 Introduction

The Efficiency Market Hypothesis tells that private information leads to trading profit in

semi-strong informationally efficient markets (Fama, 1970, 1998). Yet, investors have differ-

ent incentives and capability in gathering and processing information (Grossman and Stiglitz,

1980; Harris and Raviv, 1993), so it is a fact that investors possessing private information of

heterogeneous precision coexist in financial markets. How does private information asymme-

try affect trading behaviors? Moreover, information asymmetry induces learning (Veronesi,

1999, 2000; Pastor and Veronesi, 2003, 2009). Learning would improve subsequent trading

but ex ante discourage information revelation in the presence of profit competition. How do

trading and learning interact? How do they shape price formation and market efficiency?

What are the policy implications on information disclosure and investor protection? We

develop a dynamic model featuring investors with private information of heterogeneous pre-

cision and endogenous learning to address these intriguing but unexplored questions.

Our model considers two risk-neutral informed investors in a single-risky asset market

organized by a risk-neutral market maker. An insider receives a private signal precisely indi-

cating the asset fundamental value. A partially informed investor receives a private signal im-

precisely identifying the asset fundamental value. Information asymmetry between informed

investors is measured by the degree of imprecision of partially informed investor’s private in-

formation.1 Asset value variance reversely captures public information transparency. Insider

and partially informed investor exploit their informational advantage in competition. They

also learn about the other parties’ information conveyed by prices and incorporate learned in-

formation in subsequent trading. There are many noise traders trading for non-informational

reasons exogenous to the model. In each trading period, all traders simultaneously submit

anonymous market orders. Market maker takes on position and sets a uniform price to clear

1We in this paper use (private) information asymmetry or asymmetric (private) information to refer theinformation precision heterogeneity between informed investors, not the information discrepancy betweeninformed and uninformed investors. Although our model also incorporates the latter.

1

the market. Market making is assumed a perfectly competitive profession, so market maker’s

profit at the end of each trading period is expected to be zero. The model belongs to the

strategic category as informed trades have both transitory and permanent impact on prices

(Brunnermeier, 2001; Biais et al., 2005).

Private information asymmetry significantly shapes trading behaviors. In a unique linear

Bayesian Nash equilibrium, informed investors follow time-varying trading strategies. After

market opens, partially informed investor substantially increases his order flow to compete

for profit when private information asymmetry starts increasing from zero, while insider

reduces her order flow to avoid immediate information revelation. When information asym-

metry increases beyond a certain degree, insider alters her trading strategy to increase order

flow as information asymmetry further enlarges. Meanwhile, partially informed investor re-

duces his order flow, adding less competition but greater camouflage. Competition leads

to active trading after market opening, revealing a large amount of information and mov-

ing prices rapidly. Speculative trading reduces informed investors’ informational advantage,

trading volume gradually recedes and prices move less actively over time. However, before

market closes, insider is no longer concerned about preserving information for later trading.

Her order flow increases monotonically with information asymmetry increasing, outpacing

partially informed investor’s order flow reduction. Active trading of insider leads to rapid

price fluctuations before closure.

The results explain the U-shaped patterns of intraday stock trading volume (Jain and

Joh, 1988; Chan et al., 1996), mean return and volatility (Harris, 1986; Andersen and Boller-

slev, 1997). High trading volume, return and volatility at market opening and closure are,

however, explained by different strategic motives of informed trading. Stronger competition

leads to more active trading and price discovery at market opening. No or less need to pre-

serve information before market closure causes rapid increase in trading volume and price

movement. Since trading reveals information, trades before closure should have a weaker im-

pact on price formation than at opening, explaining the empirical finding that open-to-open

2

returns are more volatile than close-to-close returns (Amihud and Mendelson, 1987, 1991).2

Private information asymmetry time-varyingly affects market informational efficiency.

Within a trading period, information revelation decreases monotonically with information

asymmetry increasing. Market depth however rises after market opening but falls before mar-

ket closure. In the first trading period, when information asymmetry increases from zero,

insider withholds private information. Partially informed investor reveals noisy information

that hampers price discovery, leading to increase in market depth. When private information

asymmetry increases beyond a certain degree, insider reverts to trade aggressively, reveal-

ing more private information. Meanwhile, partially informed investor gradually loses his

influence on price formation with less trades. Market depth first increases above the level

of monopolistic insider as in Kyle (1985) and then falls as private information asymmetry

further increases. In the last trading period, market depth decreases convexly with private

information asymmetry increasing, but never falls below the level of monopolistic insider

as in Kyle (1985). The results highlight that involvement of partially informed investor in

a semi-strong efficient marketplace does not always lead to greater informational efficiency.

Its time-varying impact depends on the dominance of the competition effect or camouflage

effect of partially informed trades. The results further imply that per unit increase in stock

return and volatility at market opening is accompanied by greater increase in trading volume

than at market closure.

Trading profit of informed investors comes at the expense of uninformed investors. When

information asymmetry increases from zero, the total expected profit of insider and partially

informed investor first decreases from a level of two perfectly competitive insiders as in

Holden and Subrahmanyam (1992) and then reverts to increase but never exceeds the level

of monopolistic insider as in Kyle (1985). There exists a non-zero degree of private infor-

2Admati and Pfleiderer (1988, 1989) and Foster and Viswanathan (1990, 1993) theoretically show thatliquidity timing can lead to endogenous concentration of trades and price movements. Slezak (1994) developsa market closure model to explain the stylized patterns of time variation in trading and returns. Hong andWang (2000) propose a competitive model that incorporates both non-informational trades and privateinformation-motivated trades to explain these empirical patterns.

3

mation asymmetry at which informed investors extract the minimal amount of wealth from

uninformed investors. This information asymmetry, rather than zero asymmetry, minimizes

informational trading profit through balancing competition aggressiveness and information

revelation of informed investors, and thus maximizes market efficiency.

The finding has policy implications for information disclosure (Admati and Pfleiderer,

2000; Fishman and Hagerty, 2003) and investor protection. Private information asymme-

try is more influential in a market with more transparent public information, where private

information is more valuable. If information disclosure amplifies the influence of private in-

formation asymmetry by increasing public information transparency, market efficiency and

investor protection could suffer rather than benefit from it. Such consequence contradicts

the purpose of information disclosure. On the other side, disclosure could enhance efficiency

and investor protection by reducing private information asymmetry without changing public

information. Therefore, disclosure policy should consider its effects on both private infor-

mation asymmetry and public information transparency in maximizing efficiency and social

welfare.

Learning of informed investors naturally arises from private information asymmetry. Its

effects are, however, determined by private information quality and trading strategies. In-

sider learns from historical prices to analyze partially informed investor’s private signal and

trading strategies. On the other hand, she intentionally limits information revelation to

restrain partially informed investor learning that helps him to improve subsequent trading.

Learning depends on information revelation but discourages it ex ante. A higher portion

of partially informed investor’s information comes from learning as his information preci-

sion deteriorates, although the absolute amount of information acquired declines because his

learning capability decreases with his private information precision decreasing.

Our model is in the same spirit as those studying the profit-seeking motives of pri-

vately informed investors (see Kyle, 1985; Holden and Subrahmanyam, 1992; Foster and

Viswanathan, 1996; Back et al., 2000; Dridi and Germain, 2009, among others). A salient

4

feature of our model is that it considers not only information discrepancy between informed

and uninformed investors, but also information asymmetry between informed investors in

a dynamic framework. Kyle (1985) studies monopolistic insider trading and its effects on

market micro-structure. Holden and Subrahmanyam (1992) investigate perfectly compet-

itive insiders’ trading and market implications. Our model considers their models as two

special polar cases and bridges them. In such sense, our model explains the stylized empirical

patterns of trade clustering and price movements, while their models do not.

Foster and Viswanathan (1996) and Back et al. (2000) develop discrete- and continuous-

time models, respectively, to study the nature of competition among partially informed

investors receiving private signals of homogeneous imprecision. In comparison, the novelty of

our model resides on that heterogeneous information precision creates not only competition

but also camouflage. Their time-varying dominance over each other dynamically shapes

trading, prices, and market informational efficiency. Further, our model examines learning

that naturally arises from private information asymmetry and interacts with competitive

trading. More recently, Dridi and Germain (2009) develop a single-period trading model

featuring informed investors endowed with noisy signals of differential precision. Our model

extends theirs into a multiple-period framework to uncover the dynamic nature of trading

and learning, and their joint effects on market micro-structure. Ostrovsky (2012) provides a

theoretical foundation of information aggregation and revelation for strategic trading models,

based on which we explore their policy implications on disclosure and welfare in the presence

of private information asymmetry.

The rest of the paper is organized as follows. Section 2 describes the model. Section 3

discusses the equilibrium solution to the model. Section 4 analyzes the comparative statics

of simulated results and policy implications. Section 5 concludes.

5

2 The Model

We consider a security market where trading takes place in discrete time n = 0, 1, · · · , N−1.

The market is cleared in period n = N . There are two assets in the market: one riskfree

asset yields interest rate r = 0. One risky asset has a liquidation value of v ∼ N(v, σ2v),

where σv is a proxy of public information in-transparency. All uncertainty is supported on

a standard probability space (Ω,F , P ).

There are four types of risk-neutral agents: an insider, a partially informed investor, a

market maker, and many noise traders. The market marker trades the risky asset on his

own account and sets price competitively to clear the market. Market making is assumed

to be a competitive profession, so the market maker’s profit in each period n is expected

to be zero. Before trading starts, the insider receives a precise private signal ϕ on the

liquidation value v of the risky asset after period N , that is, ϕ = v. The partially informed

investor receives a imprecise private signal on the liquidation value of the asset θ = v + ε,

where ε ∼ N(0, σ2ε) and is independent of v. The insider and partially informed investor are

informed investors who exploit their private information in competition. They know that

their trading reveals information, exerting both transitory and permanent impact on prices.

Private information asymmetry refers to the information precision heterogeneity between

the insider and partially informed investor, and is measured by the publicly known σε. The

noise traders trade for non-informational reasons exogenous to the model. Their order flow,

denoted by un ∼ N(0, σ2u), is independent of ϕ and θ.

In each trading period n, the volumes of the market orders simultaneously submitted

by the insider, partially informed investor and noise traders are denoted by xn, yn, and un,

respectively. The market maker does not know asset liquidation value v, and observes only

the aggregate order flow Qn = xn + yn + un. He sets asset price Pn to clear the market.

Without loss of generality, we assume P0 = v = 0. Let Fn denote market information at

time n, i.e., Fn = σPi, i < n for n ≥ 1, and F0 = (Ω,Φ), where Φ denotes an empty set.

6

Since the orders of the noise traders are exogenous, we only need to consider the optimal

actions of the informed investors and market maker. Thus, a Bayesian Nash equilibrium

involves the trading strategies xn, n = 1, 2, · · · , N of the insider, the trading strate-

gies yn, n = 1, 2, · · · , N of the partially informed investor, and a pricing rule Pn, n =

1, 2, · · · , N of the market maker. The following conditions must be satisfied in forming an

equilibrium:

1. The insider submits order flows xn to maximize the sum of her expected profits after

periods n, n = 1, 2, · · · , N , conditional on her information in each period n, respec-

tively:

xn = Arg Max

[E

(N∑i=n

xi (v − Pi)∣∣∣ϕ,Fn)] ; (1)

2. The partially informed investor submits order flows yn to maximize the sum of his

expected profits after period n, n = 1, 2, · · · , N , conditional on his information in each

period n, respectively:

yn = Arg Max

[E

(N∑i=n

yi (v − Pi)∣∣∣θ,Fn)] ; (2)

3. In each period n, the market maker sets asset price Pn to clear the market, leading to

expected profit of zero. The asset price in each period n satisfies

Pn = E (v|Fn, Qn) . (3)

The above trading strategies and pricing rule can be expressed in functional forms as xn =

Xn(ϕ, P1, . . . , Pn−1), yn = Yn(θ, P1, . . . , Pn−1), and Pn = Pn(P1, . . . , Pn−1, Qn), respectively.

3 The Model Solution

This section solves for the market equilibrium. One salient feature of our model is that

informed investors are able to learn to improve information as trading proceeds. The insider

7

and partially informed investor consider not only their own private information, but also

information possessed by the other informed investor and the market maker. Without loss

of generality, we assume the following linear order flow function for the insider:

xn = αn1E(v|ϕ,Fn) + αn2E(θ|ϕ,Fn) + αn3ϕ+ αn4E(Pn|ϕ,Fn, xn) +n−1∑i=1

αn(i+4)Pi.

where E(v|ϕ,Fn) denotes the insider’s expected asset value based on her private and market

information. E(θ|ϕ,Fn) denotes her expectation of the partially informed investor’ infor-

mation. E(Pn|ϕ,Fn, xn) denotes her expected asset price based on her private information,

market information, and her order flow in period n. αn1, αn2, αn3, αn4 denote the coefficients

of these variables, respectively.∑n−1

i=1 αn(i+4)Pi denotes her application of information con-

veyed by historical prices. In the same logic, we assume the following linear order function

for the partially informed investor:

yn = βn1E(v|θ,Fn) + βn2E(ϕ|θ,Fn) + βn3θ + βn4E(Pn|θ,Fn, yn) +n−1∑i=1

βn(i+4)Pi,

where E(v|θ,Fn) denotes the partially informed investor’s expected asset value based on his

private and market information. E(ϕ|θ,Fn) denotes his expectation of the insider’ infor-

mation based on his private and market information. E(Pn|θ,Fn, yn) denotes his expected

asset price based on his private information, market information, and his order flow in period

n. βn1, βn2, βn3, βn4 denote the coefficients of these variables, respectively.∑n−1

i=1 βn(i+4)Pi

denotes his use of information conveyed by historical prices.

It is known that E(v|ϕ,Fn) ≡ ϕ and E(v|θ,Fn) ≡ E(ϕ|θ,Fn). Given that v, ε, and

un, n = 1, 2, ..., N are jointly normally distributed, E(Pn|ϕ,Fn, xn) and E(Pn|θ,Fn, yn) can

be expressed as the linear functions of ϕ, θ, Pi, (i < n), xn, and yn, respectively. Thus, the

above order flow functions of xn and yn can be rewritten as:

xn = ηn1ϕ+ ηn2E(θ|ϕ,Fn) +n−1∑i=1

ηn(i+2)Pi = τn1ϕ+n∑i=2

τniPi−1, (4)

8

yn = γn1θ + γn2E(ϕ|θ,Fn) +n−1∑i=1

γn(i+2)Pi = ρn1θ +n∑i=2

ρniPi−1. (5)

where τn1 denotes the coefficient of the insider’s endowed private information on her order

flow in period n; τni denotes the coefficient of asset price in an early trading period i on her

order flow in period n. ρn1 denotes the coefficient of the partially informed investor’s private

signal on his order flow in period n; ρni denotes the coefficient of asset price in an early

period i on his order flow in period n. Equations (4) and (5) capture the learning activities

of informed investors through gathering and processing market information.

Following the same logic, the market maker’s pricing rule can be expressed as:

Pn = λnQn +n−1∑i=1

ξniPi, (6)

where λn denotes the coefficient of aggregate order flow in shaping asset price in period

n. Market depth is defined as the reciprocal of λn, capturing the informational efficiency

of the market (Kyle, 1985; Biais et al., 2005; Pennacchi, 2007). ξni denotes the weight

assigned to asset price in an early period i in forming asset price in period n. Solving

for the market equilibrium is equivalent to solving for τn1, τn2, · · · , τnn, n = 1, · · · , N

for the insider, ρn1, ρn2, · · · , ρnn, n = 1, · · · , N for the partially informed investor, and

λn, ξn1, ξn2, · · · , ξnn−1, n = 1, · · · , N for the market maker by satisfying the equilibrium

conditions outlined in Equations (1), (2) and (3).

For illustration purpose, we study a two-period case with N = 2.3 We compute the values

of the following parameters: τ11, τ21, τ22, ρ11, ρ21, ρ22, , and λ1, λ2, ξ21 using backward

induction algorithm. (See Appendix for details.) The three parameters specifying the market

3We focus on examining the dynamic nature of strategic trading and endogenous learning of informedinvestors in the presence of private information asymmetry. The N = 2 case by and large enables us tocapture the time-varying nature of the variables of interest. In particular, it allows us to explain the stylizedpatterns of intraday trade clustering, return and volatility in stock market, based on reasonable predictionson the time-wise change patterns of the variables. Extending to N > 2 cases is technically plausible. Doingso however would lead to much tedious mathematical derivation, but add only marginal insights.

9

maker’s pricing rules in periods 1 and 2 are:

λ1 =(τ11 + ρ11)σ2

v

(τ11 + ρ11)2σ2v + ρ2

11σ2ε + σ2

u

, (7)

λ2 =λ1σ

2v

E

[λ1 (ρ21 + τ21)σ2

u + λ1ρ11 (τ21ρ11 − ρ21τ11)σ2ε

], (8)

ξ21 =λ1σ

2vF

E, (9)

where λ1 and λ2 capture how much price information is derived from the aggregate order

flows in periods 1 and 2, respectively. ξ21 denotes the coefficient of period 1 asset price P1 on

period 2 asset price. Since P1 is an outcome of market equilibrium, it incorporates private

information revealed to the market in period 1. E is given by:

E = λ21σ

4u + λ2

1

[((ρ11 + τ11)2 + (ρ21 + τ21)2

)σ2v + (ρ2

11 + ρ221)σ2

ε

]σ2u

+λ21(ρ21τ11 − ρ11τ21)2σ2

vσ2ε ,

and F is given by:

F = [ρ11 + τ11 − λ1(ρ21 + τ21)(ρ22 + τ22)]σ2u

+ (ρ21τ11 − ρ11τ21) [ρ21 + λ1ρ11 (ρ22 + τ22)]σ2ε .

For the insider’ order flow in period 2, we solve for

τ21 =1

2λ2

− ρ21 (σ2u − ρ11τ11σ

2ε)

2 (σ2u + ρ2

11σ2ε)

, (10)

τ22 = − ρ21ρ11σ2ε

2λ1(σ2u + ρ2

11σ2ε)− ρ22

2− ξ21

2λ2

, (11)

where τ21 is the coefficient of the insider’s private information on her order flow in period 2.

Parameter τ22 is the coefficient of period 1 asset price P1 on her order flow in period 2. For

10

the partially informed investor’ order flow in period 2, we solve for

ρ21 =σ2v(1− λ2τ21)(σ2

u − ρ11τ11σ2ε)

2λ2 [σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε ], (12)

ρ22 =τ11σ

2vσ

2ε (1− λ2τ21)

2λ2λ1 [σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε ]− λ2τ22 + ξ21

2λ2

, (13)

where ρ21 and ρ22 are the coefficients of the partially informed investor’s private information

and asset price in period 1, respectively. We also solve for τ11 and ρ11, respectively:

τ11 =1− λ1ρ11 − λ1τ21(λ2τ22 + λ2ρ22 + ξ21)

2λ1 [1 + λ1τ22 (λ2τ22 + λ2ρ22 + ξ21)]

+λ1τ22 [1− (λ2τ21 + λ2ρ21 + 2λ1ρ11 (λ2τ22 + λ2ρ22 + ξ21))]

2λ1 [1 + λ1τ22 (λ2τ22 + λ2ρ22 + ξ21)], (14)

ρ11 =[1− λ1τ11] σ2

v

σ2v+σ2

ε− λ1 (λ2τ22 + λ2ρ22 + ξ21)

(ρ21 + λ1τ11ρ22σ2

v

σ2v+σ2

ε

)2λ1[1 + λ1ρ22(λ2τ22 + λ2ρ22 + ξ21)]

+λ1ρ22

[[1− λ2τ22] σ2

v

σ2v+σ2

ε− λ2ρ21 − λ1τ11 (λ2τ22 + λ2ρ22 + ξ21) σ2

v

σ2v+σ2

ε

]2λ1[1 + λ1ρ22(λ2τ22 + λ2ρ22 + ξ21)]

, (15)

where τ11 is the coefficient of the insider’s private information on her order flow in period

1. ρ11 is the coefficient of the partially informed investor’s private information on his order

flow in period 1.

4 Result Analysis

This section first demonstrates the uniqueness of market equilibrium, followed by analyzing

informed investors’ trading in the presence of private information asymmetry. It shows

that insider’s trading strategies are different from those of monopolistic insider in Kyle

(1985) and perfectly competitive insiders in Holden and Subrahmanyam (1992). Information

acquisition through learning is subject to private information asymmetry, and discourages

information revelation ex ante. The dominance of the competition effect or the camouflage

11

effect of partially informed trading shapes market efficiency time-varyingly and situation-

dependently.

4.1 Unique Linear Market Equilibrium

In a discrete framework, the existence of a unique linear market equilibrium can be demon-

strated numerically. The equilibrium convergence conditions are that the difference between

two adjacent numerical solutions is less than 10−5, and that the coefficients of aggregate

orders on asset price in periods 1 and 2 must satisfy λ1 > 0 and λ2 > 0.

We illustrate the existence and uniqueness of market equilibrium in two steps. In the

first step, the exogenous variable values are fixed at σv = 1, σε = 1, and σu = 1. We adapt

our model with different initial values of the nine parameters λ1, λ2, ξ21, τ11, τ21, τ22, ρ11, ρ21,

and ρ22 in an extensive range of -100 to 100, respectively. We report the numerical results of

λ1 in Table 1. It shows that all converged solutions yield the same result with widely ranged

initial values of λ1, suggesting that the interaction of the agents establishes a unique linear

market equilibrium.4

In the second step, we examine whether market equilibrium uniqueness is general and

robust for different exogenous variable values. Calculations are carried out with an exten-

sive selection of initial values of σ2v , σ

2ε , and σ2

u ranging between 0.1 and 100, respectively.

Table 2 shows that the numerical calculations generate unique market equilibria for an ex-

tensive combination of exogenous parameter values, confirming that the market equilibrium

is unique.

4The last column in Table 1 reports 1 when the numerical solution converges, and 0 otherwise. Sincethe range of the initial values of λ1 is large, some calculations produce counter-intuitively negative λ1 or λ2during the numerical process. In such situation, the numerical process would be terminated and reportedas non-convergence. The numerical results of the other parameters are consistent. For brevity, we do notreport the results in the paper but make them available upon request.

12

4.2 Competitive Trading

Before examining the dynamic perspective of trading strategies adopted by informed in-

vestors, we first set out the background with an analysis of the expected profits of insider

and partially informed investor in competition. In particular, we investigate how their profit

share-out changes with respect to information asymmetry between insider and partially in-

formed investor and asset value variance, respectively.5

As reported in Table 3, the insider’s expected profit in periods 1 and 2 increases with

private information asymmetry σ2ε increasing and asset value variance σ2

v increasing, respec-

tively. Parameters σ2ε and σ2

v capture her informational advantage over the partially informed

investor and the market maker, respectively. The insider’s expected profit is higher than the

expected profit of one of the perfectly competitive insiders (σε = 0) as in Holden and Subrah-

manyam (1992), but lower compared to the expected profit of monopolistic insider (σε →∞)

as in Kyle (1985). The partially informed investor’s expected profit in periods 1 and 2 in-

creases with asset value variance increasing, because his informational advantage over the

market maker grows. But his expected profit decreases as private information asymmetry

increases, that is, his private information becomes more inferior to the insider’s. The re-

sults are consistent with the semi-strong market efficiency feature that private information

generates extra trading profits.

Table 4 reports the profit share-out between insider and partially informed investor with

respect to private information asymmetry and asset value variance, respectively. Profit

share-out is measured by the ratio of the partially informed investor’s total expected profit

to the insider’s total expected profit. It is intuitive that the insider extracts higher portion of

profit as she possesses greater informational advantage over the partially informed investors.

The insider’s share of profit decreases as asset value variance increases. Asset value variance

has two opposite effects on the insider’s expected profit. On the one hand, it increases the

5We find that the magnitude of noise trading, measured by σu, does not affect the nature of competitivetrading between informed investors. We discuss this finding in detail in Section 4.5.

13

value of private information and, consequently, the expected profit of informed investors. On

the other hand, it dilutes the insider’s informational advantage over the partially informed

investor eroding her share of profit. Only the second effect plays a role in profit share-out.

We find that profit share-out between insider and partially informed investor is fixed

when the ratio of σ2ε to σ2

v is fixed. Given that profit share-out is a direct outcome of com-

petitive trading, private information asymmetry and asset value variance tend to jointly

shape informed investors’ trading strategies. Asset value variance reversely captures public

information transparency. The result implies that the impact of private information asym-

metry is subject to public information transparency. We therefore mainly use the ratio of σ2ε

to σ2v as public information transparency adjusted private information asymmetry between

insider and partially informed investor in our subsequent investigation.

Order flows and expected profits of the informed investors in different trading periods cap-

ture the dynamic nature of their strategic trading and asymmetric competition. It shows in

Figure 1 that the partially informed investor’s expected profit in period 1 decreases convexly

with private information asymmetry increasing, although his order flow does not exhibit a

monotonically decreasing pattern. The partially informed investor increases his order flow

when private information asymmetry starts increasing from zero—he becomes slightly less

competitive than the insider. In this situation, it is optimal for him to trade aggressively

as he possesses only marginally inferior private information than the insider. Otherwise, he

may quickly lose his informational advantage over the market maker as the insider trades

and reveals information. However, when private information asymmetry rises above a certain

degree, the partially informed investor switches to decrease his order flow as the precision of

his private information further deteriorates.

The insider first reduces her order flow when information asymmetry starts increasing

from zero. Her action may look counter-intuitive at the first glance, but indeed reflects the

optimal strategy that trades off profit gain due to active trading versus profit loss due to

information revelation in the current period, and profiting in the current period versus profit-

14

ing in the later period. When the partially informed investor is informationally competitive,

the insider intentionally reduces her order flow because the benefit of preserving information

outweighs the profit derived from head-to-head competition in an early trading period. Her

expected profit increases with less order flow nevertheless, confirming the rationality and

optimality of the insider’s trading strategy.

However, when private information asymmetry increases beyond a certain degree, the

insider alters her strategy to trade more aggressively as information asymmetry further in-

creases. In this situation, partially informed trading provides rising camouflage accompanied

by fading competition. The increase in the insider’s order flow gradually slows down as pri-

vate information asymmetry approaches infinity. The partially informed investor reduces

his order flow and is eventually driven out of market. The insider’s trading strategy and

expected profit then converge to those of monopolistic insider in Kyle (1985), where no

partially informed investor is considered.

When private information asymmetry rises slightly above zero, the partially informed in-

vestor has more order flow than the insider, but generates less expected profit than the latter.

The partially informed investor’s expected profit in period 1 is computed as y1 (v − P1). Per

order profit v − P1 is negatively correlated to order flow y1 because more informed buying

order drives up asset price and in turn reduces profit. The partially informed investor has

lower expected profit than the insider since the latter could more accurately calculate her

order flow. As shown in Figure 2, the insider’s order flow increases monotonically in period

2 when she possesses stronger informational advantage over the partially informed investor.

Since trading ends after period 2, the insider is no longer concerned over that information

revelation affects subsequent trading. Her trading strategy becomes completely driven by

the current profit.

Consistently, the partially informed investor’s order flow and expected profit in period 2

decrease more dramatically than in period 1 when private information asymmetry enlarges.

As his private information becomes less precise, the partially informed investor relies more

15

on market information derived from historical asset price. So his expected asset value based

on information conveyed by historical price falls closer to the latter, reducing expected profit

and trading incentive. When the partially informed investor’s information is significantly

imprecise, he almost stops trading. His expected profit in period 2 displays a convex-shape

decreasing pattern with respect to private information asymmetry.

The finding explains the U-shaped patterns of intraday stock trading volume (Jain and

Joh, 1988; Chan et al., 1996), mean return and volatility (Harris, 1986; Andersen and Boller-

slev, 1997) from the private information asymmetry-induced strategic trading angle. Differ-

ent strategic motives of speculative trading explain the high trading volume, return and

volatility at market opening and closure. Stronger competition leads to more active trading

and price discovery at market opening. No or less need to preserve information before mar-

ket closure causes rapid movement in volume and prices. Since speculative trading reveals

information, trades should have a weaker impact on price formation at market closure than

opening, explaining the empirical finding that open-to-open returns are more volatile than

close-to-close returns (Amihud and Mendelson, 1987, 1991).

If the informed investors design their optimal trading strategies based on their asym-

metric private information, how do they allocate expected profit between trading periods?

Graph A in Figure 3 shows that the ratio of the insider’s expected profit in period 1 to her

expected profit in period 2 changes non-monotonically with respect to private information

asymmetry. The insider extracts the highest proportion of profit in period 1 when she pos-

sesses information of identical precision as the partially informed investor, consistent with

Foster and Viswanathan (1996) and Back et al. (2000) in that competition between investors

endowed with similar signals is more intense in early stages. The insider sharply reduces

her proportion of profit in period 1 as the partially informed investor’s information becomes

slightly less precise. She rationally sacrifices some period 1 profit in exchange for greater

period 2 profit when the partially informed investor imposes stiff competition. The insider

shifts to increase the proportion of early profit after the partially informed investor’s infor-

16

mation precision declines below a certain degree. It becomes optimal to increase order flow

to profit more in period 1 than to preserve information. Her profit allocation across trad-

ing periods converges to that of monopolistic insider in Kyle (1985) as private information

asymmetry enlarges (σε →∞).

Graph B in Figure 3 shows that the ratio of the partially informed investor’s expected

profit in period 1 to his expected profit in period 2 increases concavely as private information

asymmetry increases. He allocates a greater portion of profit to the early period when his

private information becomes less precise, although the absolute level of his expected profit in

period 1 decreases with private information asymmetry increasing. In period 2, the partially

informed investor’s informational advantage over the market maker will decline since the

latter acquires asset value information from trading in period 1. Waiting leads to less profit,

so it is optimal to allocate a greater portion of profit to the early trading period.

4.3 Endogenous Learning

A salient feature of our model is that informed investors learn about the other parties’ infor-

mation conveyed by prices and incorporate learned information in subsequent trading. In-

teresting questions arise: how does private information asymmetry affect informed investors’

learning incentives? How does learning interact with trading and information revelation?

This section addresses these questions.

Table 2 shows that learning does not help the insider to improve her information. In

period 2, the insider assigns the same absolute weights to her private information and the

market information in deciding her order flow. τ21 and τ22 have identical values but oppo-

site signs, suggesting that they perfectly substitute each other. The market is semi-strong

informationally efficient as the market maker sets asset price equal to the expected asset

value based on the market information. The result confirms Kyle (1985)’s intuition of mod-

eling an insider’s trading strategy as a linear function of the difference between her private

information and information conveyed by current price. The insider uses less of her private

17

information in period 1 than in period 2, that is, τ21 is greater than τ11. The result suggests

that the insider strategically delays information revelation to restrict learning of the partially

informed investor, beside to hamper price discovery.

Learning however helps the partially informed investor to acquire useful market informa-

tion as trading proceeds. The partially informed investor’s application of total information

in trading is measured by the sum of the absolute coefficients of his private information and

the market information. Figure 4 shows that his informational trading decreases convexly as

private information asymmetry enlarges. In period 1, his application of private information

decreases sharply as information asymmetry increases. The coefficient of market information

first increases as learning substantially improves the partially informed investor’s informa-

tion. It then decreases as his capability to infer market information decreases with the

precision of his private information decreasing. When private information asymmetry is

high, the partially informed investor almost completely relies on the market information.

The same pattern is also observed in period 2. His market information usage is relatively

higher in period 2 because he acquires price information at the end of period 1. The result

suggests that investors improve subsequent trading with technical analysis of historical price

and trading information (Treynor and Ferguson, 1985; Grundy and McNichols, 1989; Brown

and Jennings, 1989; He and Wang, 1995). The result is consistent with the Efficient Market

Hypothesis in that only privately informed investors can earn extra profits in a semi-strong

informationally efficient market. Technical analysis leads to greater profit. The magnitude

of increase in profit depends on the investor’s private information quality.

Learning affects trading and market efficiency from the dynamic perspective. On the one

hand, it improves market efficiency by encouraging informational trading. On the other hand,

it discourages ex ante information revelation, which negatively affects market efficiency. The

net learning effect is endogenously determined by private information asymmetry induced

strategic trading.

18

4.4 Market Depth

Table 2 shows that the market maker assigns greater weight on aggregate order flow in pricing

when asset value variance increases—private information revealed in trading becomes more

valuable. He assigns relatively a lower weight on aggregate order flow in period 2 because he

partially discovers the asset fundamental value in period 1. Figure 5 shows that the amount

of information revealed to the market maker decreases as private information asymmetry

increases. Previous studies also find that information is revealed more quickly in the pres-

ence of stronger competition between informed investors (Holden and Subrahmanyam, 1992;

Foster and Viswanathan, 1996; Back et al., 2000). The dynamic is however more complex

here. In the current model, partially informed trading not only exerts weaker competition,

but also produces more camouflage when private information asymmetry increases.

Market depth reflects the sensitiveness of change in asset price in response to change

in order flow. The market depth is deeper when informed trading has a weaker impact on

price movement. Since the market is semi-strong informationally efficient, the market depth

reflects both the amount and quality of private information revealed in trading. Figure 6

shows that in period 1, the market depth first increases and then switches to decrease with

private information asymmetry increasing. When information asymmetry starts increas-

ing from zero, the insider withholds private information as the partially informed investor

trades aggressively in competing for profit. The negative effect of noisy information pro-

duced by partially informed trading on price discovery outweighs the positive effect of clean

information revealed by insider trading. The market depth rises. When private informa-

tion asymmetry increases beyond a certain degree, the insider reverts to trade aggressively,

revealing more private information. The partially informed investor gradually loses his in-

fluence on price formation with less trades. The market depth gradually reverts its course

to decrease. So the market depth in period 1 will first rise above the level of monopolistic

insider in Kyle (1985) and then decrease as private information asymmetry further increases,

driven by the trade-off between the competition effect and the camouflage effect of partially

19

informed trading.

In period 2, the market depth decreases monotonically with private information asym-

metry increasing. The insider trades to take full advantage of her informational superiority,

increasing the amount of clean information revealed to the market. Price formation becomes

more sensitive to order flow. The influence of private information asymmetry on market

depth changes in trading stages, depending on the amount and quality of private informa-

tion revealed in the asymmetric competition between the informed investors. The results

suggest that partially informed investor’s trading in a semi-strong informationally efficient

market does not necessarily improve market informational efficiency for certain. The re-

sults also imply that per unit increase in stock return and volatility at market opening is

accompanied by greater increase in trading volume than at market closure.

4.5 Noise Trading

The noise traders trade for non-informational reasons exogenous to the model. This section

examines how noise trading affects the informed trading and learning and market micro-

structure. Table 5 shows that the sum of order flows and expected profits of the insider

and partially informed investor increase in proportion to the magnitude of noise trading σu

in both periods 1 and 2. The magnitude of noise trading negatively affects price discovery,

creating more profitable opportunities for informational trading. However, the magnitude

of noise trading does no affect the allocation of the order flows and expected profits of

the informed investors among trading periods, neither the profit share-out between them.

The informed investors adjust their trading strategies according to changes in noise trading.

Thus, noise trading does not alter the nature of competition between the informed investors.

Table 6 shows that the market depth increases with the magnitude of noise trading in

proportion to σu in both periods, ceteris paribus. Noise trading hampers price discovery,

reducing the sensitivity of price adjustment to order flow. However, price discovery of the

market maker will not be affected by noise trading in equilibrium, because the informed

20

investors adjust their order flows according to change in the magnitude of noise trading in

order to maintain optimal information revelation in equilibrium. As a result, learning of

the partially informed investor does not change with the magnitude of noise trading either.

The results imply that noise trading does not affect the nature of information revelation and

learning.

4.6 Policy Implications

Profit of the informed investors comes at the expense of the noise traders. Private information

asymmetry affects non-monotonically their profit. Figure 7 shows that the total expected

profit of the informed investors first decreases with private information asymmetry when it

starts increasing from zero. In this situation, the partially informed investor’s profitability

decreases as his information precision deteriorates, while the insider does not realize full

profit as she preserves information. Their total expected profit falls below a level of perfectly

competitive insiders in Holden and Subrahmanyam (1992). The total expected profit of the

informed investors reverts to increase as private information asymmetry further increases.

The insider’s expected profit rapidly increases as the partially informed investor becomes

less informationally competitive and his trading provides camouflage. However, their total

expected profit never exceeds the expected profit of monopolistic insider in Kyle (1985).

There exists a non-zero degree of private information asymmetry between informed investors

that leads to the least amount of value extraction by privately informed investors from

uninformed investors.

The results have important policy implications for information disclosure (Admati and

Pfleiderer, 2000; Fishman and Hagerty, 2003) and investor protection. The impact of pri-

vate information asymmetry is subject to asset return variance, which is a proxy for public

information transparency. If disclosure only reduces asset return variance, but does not

change information asymmetry between informed investors, the influence of private informa-

tion asymmetry increases rather than decreases with greater disclosure. That could cause

21

unintended deterioration in market efficiency and protection to uninformed investors, contra-

dicting the purpose of disclosure. On the other side, disclosure could enhance efficiency and

protection by reducing private information asymmetry without changing public information.

Therefore, disclosure policy should be simultaneously targeted to both public information

transparency and private information asymmetry in optimizing efficiency and welfare.

5 Conclusions

It is known that investors possessing private information of heterogeneous precision coexist

in financial markets. They also rationally learn to enhance their information as trading

proceeds. Therefore, we in this paper investigate how such private information asymmetry

affects trading and learning behaviors in the presence of profit competition, and illustrate

their externalities to market micro-structure and implications on disclosure policy.

We find that information asymmetry between informed investors significantly affects price

formation, information revelation, and market efficiency. Insider displays a time-varying and

situation-dependent trading pattern that is more complicated than those of monopolistic in-

sider in Kyle (1985) and perfectly competitive insiders in Holden and Subrahmanyam (1992).

Our model explains the stylized asymmetric U-shape patterns of intraday stock trading vol-

ume, return, and volatility from a strategic trading perspective. It implies that market

depth rises at market opening but decreases at market closure with private information

asymmetry increasing. A non-zero degree of private information asymmetry leads to maxi-

mal market efficiency by forcing informed investors to trade off competition aggressiveness

versus information preservation. The effects of private information asymmetry are subject to

public information transparency. Therefore, it is important for disclosure policy to consider

its effects on both private information asymmetry and public information transparency in

optimizing efficiency and investor protection.

Our work constitutes one of the first efforts to introduce asymmetrically informed in-

vestors and endogenous learning into a dynamic trading framework. Our discrete model by

22

and large illustrates the nature of time-varying impact of private information asymmetry

on trading, learning, and market efficiency. Its caveat is not able to identify the continuous

patterns of these effects. Such limitation, however, leads to an interesting avenue in which

to extend the current work by going continuous-time. A continuous-time model would also

allow us to better quantify the effects of private information asymmetry between informed

investors.

23

References

Admati, Anat R., and Paul Pfleiderer, 1988, A theory of intraday patterns: volume andprice variability, Review of Financial Studies 1, 3–40.

Admati, Anat R., and Paul Pfleiderer, 1989, Divide and conquer: A theory of intraday andday-of-the-week mean effects, Review of Financial Studies 2, 189–223.

Admati, Anat R., and Paul Pfleiderer, 2000, Forcing firms to talk: Financial disclosureregulation and externalities, Review of Financial Studies 13.

Amihud, Yakov, and Haim Mendelson, 1987, Trading mechanisms and stock returns: Anempirical investigation, Journal of Finance 42, 533–53.

Amihud, Yakov, and Haim Mendelson, 1991, Volatility, efficiency, and trading: Evidencefrom the japanese stock market, The Journal of Finance 46, pp. 1765–1789.

Andersen, Torben G., and Tim Bollerslev, 1997, Intraday periodicity and volatility per-sistence in financial markets, Journal of Empirical Finance 4, 115 – 158, ¡ce:title¿HighFrequency Data in Finance, Part 1¡/ce:title¿.

Back, Kerry, Henry Cao, and Gregory Willard, 2000, Imperfect competition among informedtraders, Journal of Finance 55, 2117–2155.

Biais, Bruno, Larry Glosten, and Chester Spatt, 2005, Market microstructure: A survey ofmicrofoundations, empirical results, and policy implications, Journal of Financial Markets8, 217–264.

Brown, David, and RH Jennings, 1989, On technical analysis, Review of Financial Studies2, 527–551.

Brunnermeier, Markus, 2001, Asset Pricing under Asymmetric Information: Bubbles,Crashes, Technical Analysis, and Herding (Oxford University Press).

Chan, K.C., Wai-Ming Fong, Bong-Chan Kho, and RenM. Stulz, 1996, Information, tradingand stock returns: Lessons from dually-listed securities, Journal of Banking Finance 20,1161 – 1187.

Dridi, Ramdan, and Laurent Germain, 2009, Noise and competition in strategic oligopoly,Journal of Financial Intermediation 18, 311–327.

Fama, Eugene, 1970, Efficient capital markets: A review of theory and empirical work,Journal of Finance 25, 383–417.

Fama, Eugene, 1998, Market efficiency, long-term returns, and behavioral finance, Journalof Financical Economics 49, 283–306.

Fishman, Michael J., and Kathleen M. Hagerty, 2003, Mandatory vs. voluntary disclosurein markets with informed and uninformed customers, Journal of Law, Economics andOrganization 19, 45–63.

Foster, Douglas, and S. Viswanathan, 1990, A theory of the interday variations in volume,variance, and trading costs in securities markets, Review of Financial Studies 3, 593–624.

24

Foster, Douglas, and S. Viswanathan, 1993, The effect of public information and competitionon trading volume and price volatility, Review of Financial Studies 6, 23–56.

Foster, Douglas, and S. Viswanathan, 1996, Strategic trading when agents forecast the fore-casts of others, Journal of Finance 51, 1437–1478.

Grossman, Sanford J., and Joseph E. Stiglitz, 1980, On the impossibility of informationallyefficient markets, American Economic Review 70, 393–408.

Grundy, Bruce, and Maureen McNichols, 1989, Trade and the revelation of informationthrough prices and direct disclosure, Review of Financial Studies 2, 495–526.

Harris, Lawrence, 1986, A transaction data study of weekly and intradaily patterns in stockreturns, Journal of Financial Economics 16, 99 – 117.

Harris, M, and A Raviv, 1993, Differences of opinion make a horse race, Review of FinancialStudies 6, 473–506.

He, Hua, and Jiang Wang, 1995, Differential informational and dynamic behavior of stocktrading volume, Review of Financial Studies 8, 919–972.

Holden, Craig W., and Avanidhar Subrahmanyam, 1992, Long-lived private information andimperfect competition, Journal of Finance 47, 247–270.

Hong, Harrison, and Jiang Wang, 2000, Trading and returns under periodic market closures,The Journal of Finance 55, 297–354.

Jain, Prem, and Gun-Ho Joh, 1988, The dependence between hourly prices under transac-tions and return uncertainty, Journal of Financial and Quantitative Analysis 23, 269 –283.

Kyle, Albert S., 1985, Continous auctions and insider trading, Econometrica 53, 1315–1335.

Ostrovsky, Michael, 2012, Information aggregation in dynamic markets with strategictraders, Econometrica 80, 2595–2647.

Pastor, Lubos, and Pietro Veronesi, 2003, Stock valuation and learning about profitability.,Journal of Finance 58, 1749–1790.

Pastor, Lubos, and Pietro Veronesi, 2009, Technological revolutions and stock prices, Amer-ican Economic Review 99, 1451–1483.

Pennacchi, George, 2007, Theory of Asset Pricing (Prentice Hall).

Slezak, Steve L., 1994, A theory of the dynamics of security returns around market closures,The Journal of Finance 49, pp. 1163–1211.

Treynor, Jack, and Robert Ferguson, 1985, In defense of technical analysis, The Journal ofFinance 40, 757–773.

Veronesi, Pietro, 1999, Stock market overreactions to bad news in good times: A rationalexpectations equilibrium model, Review of Financial Studies 12.

Veronesi, Pietro, 2000, How does information quality affect stock returns?, Journal of Fi-nance 55, 807–837.

25

A Appendix

A.1

We illustrate how to calculate the order flows of the insider and partially informed investor,

and the market price set by the market maker. We first give the following lemma:

Lemma: If X1 and X2 have joint normal distribution N

[(µ1

µ2

),

(Σ11 Σ12

Σ21 Σ22

)], then

the conditional expectation is

E(X2|X1) = µ2 + Σ12Σ−111 (X1 − µ1),

and the conditional variance is

D(X2|X1) = Σ22 − Σ12Σ−111 Σ21.

It is assumed that ϕ = v, θ = v + ε and that v and ε are mutually independent with means

of zero and variances of σ2v and σ2

ε , respectively. Hence, we know that the joint distribution

of ϕ and θ is

N

[(00

),

(σ2v σ2

v

σ2v σ2

v + σ2ε

)].

We have the following propositions:

Proposition A1. For the insider in period 1:

E(θ|ϕ) = ϕ, D(θ|ϕ) = σ2ε . (A.1)

For the partially informed investor in period 1:

E(ϕ|θ) =σ2v

σ2v + σ2

ε

θ, D(ϕ|θ) =σ2ε

σ2v + σ2

ε

σ2v . (A.2)

For the market maker, the information observed in period 1 is Q1 = τ11ϕ + ρ11θ + u1 =

(τ11 + ρ11)v + ρ11ε+ u1. Hence, the joint distribution of v and Q1 can be expressed as

N

[(00

),

(σ2v (τ11 + ρ11)σ2

v

(τ11 + ρ11)σ2v (τ11 + ρ11)2σ2

v + ρ211σ

2ε + σ2

u

)].

Proposition A2: For the market maker,

E(v|Q1) =(τ11 + ρ11)σ2

v

(τ11 + ρ11)2σ2v + ρ2

11σ2ε + σ2

u

Q1, D(v|Q1) =(ρ2

11σ2ε + σ2

u)σ2v

(τ11 + ρ11)2σ2v + ρ2

11σ2ε + σ2

u

. (A.3)

26

According to the semi-strong form market efficiency conditions in Eq.s (3) and (6), we have

λ1 =(τ11 + ρ11)σ2

v

(τ11 + ρ11)2σ2v + ρ2

11σ2ε + σ2

u

. (A.4)

In period 2, by previous assumptions, we have asset price in period 1 as

P1 = λ1 (τ11ϕ+ ρ11θ + u1) = λ1 [(τ11 + ρ11) v + ρ11ε+ u1] .

The joint distribution of

(θϕP1

)is

N

( 000

),

σ2v + σ2

ε σ2v λ1ρ11(σ2

v + σ2ε) + λ1τ11σ

2v

σ2v σ2

v λ1ρ11σ2v + λ1τ11σ

2v

λ1ρ11(σ2v + σ2

ε) + λ1τ11σ2v λ1ρ11σ

2v + λ1τ11σ

2v λ2

1 [σ2u + (ρ11 + τ11)2σ2

v + ρ211σ

2ε ]

.We then have(

σ2v λ1 (ρ11 + τ11)σ2

v

λ1 (ρ11 + τ11)σ2v λ2

1 [σ2u + (ρ11 + τ11)2σ2

v + ρ211σ

2ε ]

)−1

=1

λ21σ

2v(σ

2u + ρ2

11σ2ε)

(λ2

1 [σ2u + (ρ11 + τ11)2σ2

v + ρ211σ

2ε ] −λ1(ρ11 + τ11)σ2

v

−λ1(ρ11 + τ11)σ2v σ2

v

),

and (σ2v + σ2

ε λ1 [ρ11(σ2v + σ2

ε) + τ11σ2v ]

λ1 [ρ11(σ2v + σ2

ε) + τ11σ2v ] λ2

1 [σ2u + (ρ11 + τ11)2σ2

v + ρ211σ

2ε ]

)−1

=1

λ21 [σ2

u(σ2v + σ2

ε) + τ11σ2vσ

2ε ]

·(λ2

1[σ2u + (ρ11 + τ11)2σ2

v + ρ211σ

2ε ] −λ1[(ρ11 + τ11)σ2

v + ρ11σ2ε ]

−λ1[(ρ11 + τ11)σ2v + ρ11σ

2ε ] σ2

v + σ2ε

).

For the insider, we have

E(θ|ϕ, P1)

=σ2u − ρ11τ11σ

2ε

σ2u + ρ2

11σ2ε

ϕ+ρ11σ

2ε

λ1(σ2u + ρ2

11σ2ε)P1, (A.5)

27

and for the partially informed investor, we have

E(ϕ|θ, P1)

=σ2v(σ

2u − ρ11τ11σ

2ε)

σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε

θ +τ11σ

2vσ

2ε

λ1 [σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε ]P1. (A.6)

The market maker’s information comes from aggregate order flow, so he knows P1 and

Q2 = x2 + y2 + u2, where

Q2 = τ21ϕ+ ρ21θ + (τ22 + ρ22)P1 + u2

= [τ21 + λ1τ11(τ22 + ρ22)]ϕ+ [ρ21 + λ1ρ11(τ22 + ρ22)] θ + λ1(τ22 + ρ22)u1 + u2.

The joint distribution of

(ϕP1

Q2

)can be expressed as

N

[(000

),

(σ2v λ1 (τ11 + ρ11)σ2

v Cλ1 (τ11 + ρ11)σ2

v λ21[σ2

u + ρ211σ

2ε + (ρ11 + τ11)2σ2

v ] BC B A

)],

where

A =[1 + λ2

1(ρ22 + τ22)2]σ2u + [ρ21 + λ1ρ11(ρ22 + τ22)]2 σ2

ε +Dσ2v ,

B = λ1(τ11 + ρ11) [τ21 + ρ21 + λ1 (τ11 + ρ11) (τ22 + ρ22)]σ2v

+λ1ρ11 [ρ21 + λ1ρ11 (τ22 + ρ22)]σ2ε + λ2

1 (ρ22 + τ22)σ2u,

C = [ρ21 + τ21 + λ1(τ11 + ρ11)(τ22 + ρ22)]σ2v ,

D = [ρ21 + τ21 + λ1(ρ11 + τ11)(ρ22 + τ22)]2 .

Further, we have (λ2

1[σ2u + ρ2

11σ2ε + (ρ11 + τ11)2σ2

v ] BB A

)−1

=1

E

(A −B−B λ2

1[σ2u + ρ2

11σ2ε + σ2

v(ρ11 + τ11)2]

),

28

where

E = λ21σ

4u + λ2

1

[((ρ11 + τ11)2 + (ρ21 + τ21)2

)σ2v + (ρ2

11 + ρ221)σ2

ε

]σ2u

+λ21(ρ21τ11 − ρ11τ21)2σ2

vσ2ε ,

and

E(v|Q1, Q2)

=λ1σ

2v

E

(F λ1 (ρ21 + τ21)σ2

u + λ1ρ11 (τ21ρ11 − ρ21τ11)σ2ε

)( P1

Q2

), (A.7)

where

F = [ρ11 + τ11 − λ1(ρ21 + τ21)(ρ22 + τ22)]σ2u

+ (ρ21τ11 − ρ11τ21) [ρ21 + λ1ρ11 (ρ22 + τ22)]σ2ε .

According to the semi-strong form market efficiency conditions in Eq.s (3) and (6), we

compute

λ2 =λ1σ

2v

E

[λ1 (ρ21 + τ21)σ2

u + λ1ρ11 (τ21ρ11 − ρ21τ11)σ2ε

], (A.8)

ξ21 =λ1σ

2vF

E. (A.9)

A.2

We present solutions to market equilibrium. Based on the market equilibrium conditions

in Eq.s (1) and (2), we obtain equilibrium solution using back-ward induction algorithm.

Starting in period 2, we know P2 = λ2(x2 + y2 + u2) + ξ21P1. The insider’s expected profit

can be expressed as

E(πx2

∣∣∣ϕ, P1, x2

)= E

[x2 (v − P2)

∣∣∣ϕ, P1, x2

]= x2

[E (v|ϕ, P1)− E

(P2

∣∣∣ϕ, P1, x2

)]= x2

[ϕ− λ2

(x2 + ρ21E

(θ∣∣∣ϕ, P1

)+ ρ22P1

)− ξ21P1

].

To maximize her expected profit, we solve the first order condition as∂E(πx

2 |ϕ,P1,x2)

∂x2= 0, i.e.,

ϕ− λ2

(x2 + ρ21E

(θ∣∣∣ϕ, P1

)+ ρ22P1

)− ξ21P1 − λ2x2 = 0.

29

We solve for the insider’s order flow in period 2:

x2 =ϕ

2λ2

− ρ21E(θ|ϕ, P1) + ρ22P1

2− ξ21P1

2λ2

.

Plugging (A.5) into the above equation and using Eq. (4) , we have

τ21 =1

2λ2

− ρ21 (σ2u − ρ11τ11σ

2ε)

2 (σ2u + ρ2

11σ2ε)

, (A.10)

τ22 = − ρ21ρ11σ2ε

2λ1(σ2u + ρ2

11σ2ε)− ρ22

2− ξ21

2λ2

. (A.11)

Similarly, for the partially informed investor in period 2, we have

E(πy2

∣∣∣θ, P1, y2

)= y2

[E(v∣∣∣θ, P1

)− E

(P2

∣∣∣θ, P1, y2

)]= y2

[E(ϕ∣∣∣θ, P1

)− λ2

(τ21E

(ϕ∣∣∣θ, P1

)+ τ22P1 + y2

)− ξ21P1

].

The first order condition is:

E(ϕ∣∣∣θ, P1

)− λ2

(τ21E

(ϕ∣∣∣θ, P1

)+ τ22P1 + y2

)− ξ21P1 − λ2y2 = 0.

Solving it for the order flow of the partially informed investor in period 2:

y2 =1− λ2τ21

2λ2

E(ϕ∣∣∣θ, P1

)− λ2τ22 + ξ21

2λ2

P1.

Plugging (A.6) into the above equation and using equation (5), we have

ρ21 =σ2v(1− λ2τ21)(σ2

u − ρ11τ11σ2ε)

2λ2 [σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε ], (A.12)

ρ22 =τ11σ

2vσ

2ε (1− λ2τ21)

2λ2λ1 [σ2u(σ

2v + σ2

ε) + τ 211σ

2vσ

2ε ]− λ2τ22 + ξ21

2λ2

. (A.13)

After obtaining period 2 solutions, we solve for the period 1 variable values . The insider’s

30

expected profit in period 1 is

E(πx1

∣∣∣ϕ, x1

)= E

[x1 (v − P1) + πx2

∣∣∣ϕ, x1

]= x1

[ϕ− E

[P1

∣∣∣ϕ, x1

]]+(τ21ϕ+ τ22E

[P1

∣∣∣ϕ, x1

]) [ϕ−

(λ2τ21ϕ+ λ2ρ21E [θ|ϕ] + (λ2τ22 + λ2ρ22 + ξ21)E

[P1

∣∣∣ϕ, x1

])]−λ1τ22ρ11[λ2ρ21 + λ1ρ11(λ2τ22 + λ2ρ22 + ξ21)]V ar(θ|ϕ, x1)

−λ21τ22(λ2τ22 + λ2ρ22 + ξ21)σ2

u

where

E[P1

∣∣∣ϕ, x1

]= λ1

(x1 + ρ11E

(θ∣∣∣ϕ)) .

The first order condition is∂E(πx

1 |ϕ,x1)

∂x1= 0, i.e.,

ϕ− E[P1

∣∣∣ϕ, x1

]− λ1x1 − λ1

(τ21ϕ+ τ22E

[P1

∣∣∣ϕ, x1

])(λ2τ22 + λ2ρ22 + ξ21)

+λ1τ22

[ϕ−

(λ2τ21ϕ+ λ2ρ21E [θ|ϕ] + (λ2τ22 + λ2ρ22 + ξ21)E

[P1

∣∣∣ϕ, x1

])]= 0,

which can be written as

2λ1 [1 + λ1τ22 (λ2τ22 + λ2ρ22 + ξ21)]x1

= [1− λ1ρ11 + λ1τ22 [1− (λ2τ21 + λ2ρ21 + 2λ1ρ11 (λ2τ22 + λ2ρ22 + ξ21))]]ϕ

− [λ1τ21(λ2τ22 + λ1ρ22 + ξ21)]ϕ.

Solving the equation yields

τ11 =1− λ1ρ11 + λ1τ22 [1− (λ2τ21 + λ2ρ21 + 2λ1ρ11 (λ2τ22 + λ2ρ22 + ξ21))]

2λ1 [1 + λ1τ22 (λ2τ22 + λ2ρ22 + ξ21)]

− λ1τ21(λ2τ22 + λ1ρ22 + ξ21)

2λ1 [1 + λ1τ22 (λ2τ22 + λ2ρ22 + ξ21)]. (A.14)

31

Similarly, the partially informed investor’s expected profit function in period 1 is

E(πy1

∣∣∣θ, y1

)= E

[y1 (v − P1) + πy2

∣∣∣θ, y1

]= y1

[E(v|θ)− E

[P1

∣∣∣θ, y1

]]+(ρ21θ + ρ22E

[P1

∣∣∣θ, y1

])·[E(v|θ)−

(λ2τ21E(v|θ) + λ2ρ21θ + (λ2τ22 + λ2ρ22 + ξ21)E

[P1

∣∣∣θ, y1

])]+λ1ρ22τ11[1− λ1τ11(λ2τ22 + λ2ρ22 + ξ21)− λ2τ21]V ar(ϕ|θ)−λ2

1ρ22(λ2τ22 + λ2ρ22 + ξ21)σ2u,

where

E[P1

∣∣∣θ, y1

]= λ1 (τ11E [ϕ|θ] + y1) .

The first order condition is∂E(πy

1 |θ,y1)

∂y1= 0, i.e.,

E[ϕ|θ]− E[P1

∣∣∣θ, y1

]− λ1y1 − λ1

(ρ21θ + ρ22E

[P1

∣∣∣θ, y1

])(λ2τ22 + λ2ρ22 + ξ21)

+λ1ρ22

[E[ϕ|θ]−

(λ2τ21E[ϕ|θ] + λ2ρ21θ + (λ2τ22 + λ2ρ22 + ξ21)E

[P1

∣∣∣θ, y1

])]= 0,

which can be written as

2λ1[1 + λ1ρ22(λ2τ22 + λ2ρ22 + ξ21)]y1

=

[[1− λ1τ11]

σ2v

σ2v + σ2

ε

− λ1 (λ2τ22 + λ2ρ22 + ξ21) (ρ21 +λ1τ11ρ22σ

2v

σ2v + σ2

ε

)

]θ

+λ1ρ22

[[1− λ2τ21]

σ2v

σ2v + σ2

ε

− λ2ρ21 − λ1τ11 (λ2τ22 + λ2ρ22 + ξ21)σ2v

σ2v + σ2

ε

]θ.

Solving the equation yields

ρ11 =[1− λ1τ11] σ2

v

σ2v+σ2

ε− λ1 (λ2τ22 + λ2ρ22 + ξ21)

(ρ21 + λ1τ11ρ22σ2

v

σ2v+σ2

ε

)2λ1[1 + λ1ρ22(λ2τ22 + λ2ρ22 + ξ21)]

+λ1ρ22

[[1− λ2τ21] σ2

v

σ2v+σ2

ε− λ2ρ21 − λ1τ11 (λ2τ22 + λ2ρ22 + ξ21) σ2

v

σ2v+σ2

ε

]2λ1[1 + λ1ρ22(λ2τ22 + λ2ρ22 + ξ21)]

. (A.15)

32

Table

1U

niq

ue

Mark

et

Equil

ibri

um

Th

ista

ble

show

sth

eex

iste

nce

of

un

iqu

em

arke

teq

uil

ibri

um

wh

enth

ein

itia

lva

lue

ofλ

1ch

an

ges

ina

ran

geof

-100

to100.

Th

eeq

uil

ibri

um

conve

rgen

ceco

nd

itio

nre

qu

ires

that

the

diff

eren

ceb

etw

een

two

adja

cent

solu

tion

sis

less

than

10−

5.

Som

eca

lcu

lati

on

sd

on

ot

conve

rge

wh

enλ

1orλ

2fa

lls

out

of

reaso

nable

ran

ges

du

rin

gth

enu

mer

ical

pro

cess

.T

he

Diff

eren

ceco

lum

nre

port

sth

ed

iffer

ence

bet

wee

ntw

o

adja

cent

solu

tion

sw

hen

calc

ula

tion

conve

rges

.T

he

Indic

ato

rco

lum

nre

por

ts1

for

conve

rgen

ce,

0ot

her

wis

e.λ

1d

enote

sth

eco

effici

ent

of

agg

rega

teor

der

flow

on

mark

etp

rice

inp

erio

d1.λ

2an

dξ 2

1d

enot

eth

eco

effici

ents

ofag

greg

ate

ord

erfl

owin

per

iod

2an

dass

etp

rice

inp

erio

d1,

resp

ecti

vely

,on

ass

etp

rice

inp

erio

d2.τ 1

1d

enot

esth

eco

effici

ent

ofin

sid

er’s

pri

vate

info

rmat

ion

on

her

ord

erfl

owin

per

iod

1.τ 2

1an

dτ 2

2d

enote

the

coeffi

cien

tsof

insi

der

’sp

riva

tein

form

atio

nan

dth

emar

ket

info

rmati

on,

resp

ecti

vely

,on

her

ord

erfl

owin

per

iod

2.ρ

11

den

ote

the

coeffi

cien

tof

par

tiall

yin

form

edin

vest

or’s

pri

vate

info

rmat

ion

onh

isor

der

flow

inp

erio

d1.ρ

21

an

dρ

22

den

ote

the

coeffi

cien

tsof

par

tiall

yin

form

edin

ves

tor’

sp

riva

tein

form

atio

nan

dth

em

arket

info

rmat

ion

,re

spec

tive

ly,

onh

isord

erfl

owin

per

iod

2.

Para

met

erIn

itia

lV

alu

eD

iffer

ence

λ1

λ2

ξ 21

τ 21

τ 22

ρ21

ρ22

τ 11

ρ11

Ind

icato

rλ1

100

1.7

11E

-12

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

50

2.8

34E

-12

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

20

4.8

93E

-09

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

10

2.4

98E

-10

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

53.8

32E

-13

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

42.5

81E

-08

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

3.0

12.7

65E

-08

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

22.1

15E

-09

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

1.0

51.1

98E

-07

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

-1.0

51.0

61E

+00

-1.0

584

1.0

592

0.4

284

0.7

973

0.3

157

2.4

519

0.6

627

-6.2

493

164.7

142

0λ1

-21.6

76E

+00

-1.6

472

0.9

915

0.5

739

1.4

910

0.3

170

1.4

123

0.4

537

5.8

736

16.7

901

0λ1

-3.0

16.1

59E

-10

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

-41.8

21E

-10

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

-51.5

82E

-13

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

-10

1.7

59E

-10

-0.4

508

0.2

792

1.0

000

1.7

366

-1.7

366

0.1

743

-0.2

055

-1.2

373

-0.2

689

0λ1

-20

6.7

08E

-09

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1λ1

-50

7.3

43E

-01

-0.4

754

-0.1

106

1.2

875

-0.9

605

4.1

660

2.5

098

3.6

334

198.8

155

88.3

278

0λ1

-100

2.7

10E

-07

0.4

676

0.3

654

1.0

000

1.2

800

-1.2

800

0.2

489

-0.3

432

0.5

768

0.3

516

1

33

Table 2 Unique Market Equilibrium (Cont.)This table shows the existence of unique market equilibrium with initial values of asset value

variance σ2v , private information asymmetry σ2

ε , and magnitude of noise trading σ2u changing in

a range of 0.10 to 100, respectively. The equilibrium convergence condition requires that the

difference between two adjacent solutions is less than 10−5. Some calculations do not converge

when λ1 or λ2 falls out of reasonable ranges during the numerical process. The Difference column

reports the difference between two adjacent solutions when calculation converges. The Indicator

column reports 1 for convergence, 0 otherwise. λ1 denotes the coefficient of aggregate order flow

on market price in period 1. λ2 and ξ21 denote the coefficients of aggregate order flow in period 2

and asset price in period 1, respectively, on asset price in period 2. τ11 denotes the coefficient of

insider’s private information on her order flow in period 1. τ21 and τ22 denote the coefficients of

insider’s private information and the market information, respectively, on her order flow in period

2. ρ11 denote the coefficient of partially informed investor’s private information on his order flow

in period 1. ρ21 and ρ22 denote the coefficients of partially informed investor’s private information

and the market information, respectively, on his order flow in period 2.

Panel A: With Different Values of σ2v

σ2v σ2

ε σ2u Deviation λ1 λ2 ξ21 τ21 τ22 ρ21 ρ22 τ11 ρ11

0.1 1 1 5.31E-09 0.1458 0.1291 1.0000 3.8526 -3.8526 0.0747 -0.1411 2.0623 0.20130.4 1 1 7.03E-11 0.2922 0.2463 1.0000 1.9796 -1.9796 0.1606 -0.2598 0.9756 0.32010.7 1 1 2.28E-13 0.3887 0.3144 1.0000 1.5186 -1.5186 0.2135 -0.3140 0.7090 0.34831 1 1 9.58E-08 0.4676 0.3654 1.0000 1.2800 -1.2800 0.2489 -0.3432 0.5768 0.35162 1 1 3.36E-12 0.6725 0.4874 1.0000 0.9056 -0.9056 0.3036 -0.3723 0.3892 0.32183 1 1 2.00E-09 0.8322 0.5787 1.0000 0.7308 -0.7308 0.3169 -0.3675 0.3130 0.28794 1 1 1.14E-06 0.9674 0.6562 1.0000 0.6244 -0.6243 0.3161 -0.3548 0.2701 0.26055 1 1 1.29E-07 1.0864 0.7252 1.0000 0.5513 -0.5513 0.3099 -0.3407 0.2416 0.238710 1 1 4.34E-10 1.5519 1.0018 1.0000 0.3730 -0.3730 0.2683 -0.2818 0.1728 0.175920 1 1 4.96E-12 2.2064 1.4023 1.0000 0.2538 -0.2538 0.2122 -0.2176 0.1240 0.126250 1 1 3.05E-08 3.4996 2.2066 1.0000 0.1554 -0.1554 0.1441 -0.1456 0.0795 0.0802100 1 1 1.57E-11 4.9543 3.1168 1.0000 0.1085 -0.1085 0.1044 -0.1050 0.0565 0.0568

Panel B: With Different Values of σ2ε

σ2v σ2

ε σ2u Deviation λ1 λ2 ξ21 τ21 τ22 ρ21 ρ22 τ11 ρ11

1 0.1 1 2.44E-15 0.4908 0.3168 1.0000 1.1794 -1.1794 0.8484 -0.8912 0.5465 0.55631 0.4 1 5.68E-08 0.4783 0.3386 1.0000 1.2740 -1.2740 0.4949 -0.5877 0.5451 0.48091 0.7 1 6.54E-08 0.4714 0.3545 1.0000 1.2841 -1.2841 0.3357 -0.4346 0.5617 0.40851 1 1 9.58E-08 0.4676 0.3654 1.0000 1.2800 -1.2800 0.2489 -0.3432 0.5768 0.35161 2 1 1.72E-08 0.4628 0.3845 1.0000 1.2595 -1.2595 0.1278 -0.1994 0.6080 0.23631 3 1 4.08E-10 0.4615 0.3932 1.0000 1.2461 -1.2461 0.0840 -0.1397 0.6237 0.17701 4 1 2.50E-11 0.4611 0.3981 1.0000 1.2376 -1.2376 0.0620 -0.1073 0.6329 0.14121 5 1 2.67E-12 0.4609 0.4013 1.0000 1.2317 -1.2317 0.0490 -0.0870 0.6389 0.11741 10 1 1.99E-08 0.4609 0.4082 1.0000 1.2183 -1.2183 0.0236 -0.0446 0.6522 0.06371 20 1 4.74E-09 0.4612 0.4119 1.0000 1.2106 -1.2106 0.0115 -0.0226 0.6594 0.03321 50 1 6.26E-09 0.4614 0.4143 1.0000 1.2056 -1.2056 0.0045 -0.0091 0.6640 0.01361 100 1 2.63E-11 0.4615 0.4151 1.0000 1.2039 -1.2039 0.0022 -0.0045 0.6655 0.0069

Panel C: With Different Values of σ2u

σ2v σ2

ε σ2u Deviation λ1 λ2 ξ21 τ21 τ22 ρ21 ρ22 τ11 ρ11

1 1 0.1 6.47E-10 1.4786 1.1555 1.0000 0.4048 -0.4048 0.0787 -0.1085 0.1824 0.11121 1 0.4 5.09E-07 0.7393 0.5778 1.0000 0.8095 -0.8095 0.1574 -0.2170 0.3648 0.22231 1 0.7 6.44E-08 0.5589 0.4368 1.0000 1.0709 -1.0709 0.2083 -0.2871 0.4826 0.29411 1 1 9.58E-08 0.4676 0.3654 1.0000 1.2800 -1.2800 0.2489 -0.3432 0.5768 0.35161 1 2 7.47E-10 0.3306 0.2584 1.0000 1.8102 -1.8102 0.3520 -0.4853 0.8157 0.49721 1 3 1.37E-12 0.2700 0.2110 1.0000 2.2170 -2.2170 0.4311 -0.5944 0.9990 0.60891 1 4 2.11E-14 0.2338 0.1827 1.0000 2.5600 -2.5600 0.4978 -0.6863 1.1536 0.70311 1 5 2.22E-08 0.2091 0.1634 1.0000 2.8622 -2.8622 0.5566 -0.7673 1.2897 0.78611 1 10 1.23E-08 0.1479 0.1156 1.0000 4.0477 -4.0477 0.7871 -1.0852 1.8239 1.11171 1 20 4.41E-13 0.1046 0.0817 1.0000 5.7243 -5.7243 1.1132 -1.5347 2.5794 1.57221 1 50 5.00E-12 0.0661 0.0517 1.0000 9.0509 -9.0509 1.7601 -2.4266 4.0785 2.48591 1 100 9.77E-15 0.0468 0.0365 1.0000 12.8000 -12.8000 2.4891 -3.4317 5.7678 3.5156

34

Table 3 Expected Profit of Informed InvestorsThis table reports the expected profits of insider and partially informed investor in periods 1 and 2.

Asset value variance σ2v changes column-wise in a range of 1 to 9. Private information asymmetry

σ2ε changes row-wise in a range of 0 to 9. Magnitude of noise trading σ2

u is normalized to be 1. The

numbers in parentheses are the expected profits of partially informed investor.

Insider/(Partially Informed Investor)PPPPPPσ2

ε

σ2v 1 2 3 4 5 6 7 8 9

0 0.4036 0.5708 0.6991 0.8073 0.9026 0.9887 1.0679 1.1417 1.2109

(0.4036) (0.5708) (0.6991) (0.8073) (0.9026) (0.9887) (1.0679) (1.1417) (1.2109)

1 0.6652 0.8234 0.9330 1.0239 1.1046 1.1785 1.2472 1.3120 1.3734

(0.1678) (0.3364) (0.4780) (0.5997) (0.7070) (0.8036) (0.8919) (0.9736) (1.0499)

2 0.7421 0.9408 1.0672 1.1645 1.2466 1.3194 1.3860 1.4481 1.5065

(0.1052) (0.2373) (0.3618) (0.4757) (0.5800) (0.6760) (0.7650) (0.8481) (0.9261)

3 0.7782 1.0070 1.1522 1.2608 1.3496 1.4262 1.4947 1.5575 1.6160

(0.0765) (0.1829) (0.2906) (0.3937) (0.4910) (0.5827) (0.6690) (0.7506) (0.8279)

4 0.7991 1.0494 1.2106 1.3305 1.4271 1.5093 1.5815 1.6468 1.7069

(0.0601) (0.1487) (0.2426) (0.3355) (0.4254) (0.5117) (0.5941) (0.6728) (0.7481)

5 0.8127 1.0789 1.2530 1.3831 1.4875 1.5755 1.6522 1.7208 1.7834

(0.0494) (0.1253) (0.2081) (0.2922) (0.3751) (0.4559) (0.5340) (0.6094) (0.6821)

6 0.8224 1.1005 1.2853 1.4242 1.5357 1.6295 1.7108 1.7830 1.8485

(0.0420) (0.1082) (0.1822) (0.2587) (0.3354) (0.4109) (0.4848) (0.5568) (0.6267)

7 0.8295 1.1170 1.3106 1.4571 1.5752 1.6743 1.7600 1.8360 1.9044

(0.0365) (0.0951) (0.1619) (0.2320) (0.3031) (0.3740) (0.4439) (0.5124) (0.5794)

8 0.8350 1.1301 1.3310 1.4841 1.6079 1.7120 1.8020 1.8816 1.9531

(0.0323) (0.0849) (0.1457) (0.2103) (0.2765) (0.3431) (0.4092) (0.4745) (0.5387)

9 0.8394 1.1407 1.3478 1.5067 1.6356 1.7442 1.8382 1.9212 1.9957

(0.0289) (0.0767) (0.1325) (0.1923) (0.2542) (0.3168) (0.3795) (0.4418) (0.5033)

35

Table

4P

rofit

Share

-out

betw

een

Info

rmed

Invest

ors

Th

ista

ble

rep

orts

pro

fit

share

-ou

tb

etw

een

the

insi

der

and

par

tial

lyin

form

edin

vest

orw

ith

resp

ect

top

riva

tein

form

ati

on

asy

mm

etry

and

asse

tva

lue

vari

an

ce,

resp

ecti

vely

.P

rofi

tsh

are-

out

ism

easu

red

by

the

rati

oof

the

exp

ecte

dp

rofi

tof

part

iall

yin

form

edin

vest

or

in

per

iods

1an

d2

toth

eex

pec

ted

pro

fit

ofin

sid

erin

per

iod

s1

and

2in

per

centa

gep

oint.

Ass

etva

lue

vari

an

ceσ

2 vch

an

ges

colu

mn

-wis

ein

a

ran

ge

of

1to

10.

Pri

vate

info

rmati

onas

ym

met

ryσ

2 εch

ange

sro

w-w

ise

ina

ran

geof

0to

9.

Mag

nit

ud

eof

nois

etr

ad

ingσ

2 uis

norm

ali

zed

tob

e1.

PP

PP

PP

σ2 ε

σ2 v

12

34

56

78

910

0100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

100.0

0%

125.2

2%

40.8

5%

51.2

3%

58.5

7%

64.0

0%

68.1

9%

71.5

1%

74.2

1%

76.4

4%

78.3

2%

214.1

7%

25.2

2%

33.9

0%

40.8

5%

46.5

3%

51.2

3%

55.1

9%

58.5

7%

61.4

7%

64.0

0%

39.8

3%

18.1

6%

25.2

2%

31.2

3%

36.3

9%

40.8

5%

44.7

6%

48.1

9%

51.2

3%

53.9

4%

47.5

2%

14.1

7%

20.0

4%

25.2

2%

29.8

1%

33.9

0%

37.5

6%

40.8

5%

43.8

3%

46.5

3%

56.0

8%

11.6

1%

16.6

1%

21.1

3%

25.2

2%

28.9

4%

32.3

2%

35.4

1%

38.2

5%

40.8

5%

65.1

1%

9.8

3%

14.1

7%

18.1

6%

21.8

4%

25.2

2%

28.3

4%

31.2

3%

33.9

0%

36.3

9%

74.4

0%

8.5

2%

12.3

5%

15.9

2%

19.2

5%

22.3

4%

25.2

2%

27.9

1%

30.4

2%

32.7

8%

83.8

7%

7.5

2%

10.9

5%

14.1

7%

17.2

0%

20.0

4%

22.7

1%

25.2

2%

27.5

8%

29.8

1%

93.4

5%

6.7

2%

9.8

3%

12.7

6%

15.5

4%

18.1

6%

20.6

5%

22.9

9%

25.2

2%

27.3

3%

36

Table 5 Role of Noise TradingThis table reports the relationships between expected profits and order flows of insider and partially

informed investor and magnitude of noise trading. Panel A reports expected profits of informed

investors. Panel B reports order flows of informed investors.

Panel A: Profits

Insider Partially Informed Investor

σu Period 1 Period 2 Period 1 Period 2

1 0.3264 0.3388 0.1412 0.0266

2 0.6528 0.6776 0.2823 0.0532

3 0.9792 1.0165 0.4235 0.0798

4 1.3057 1.3553 0.5647 0.1064

5 1.6321 1.6941 0.7058 0.1330

6 1.9585 2.0329 0.8470 0.1596

7 2.2849 2.3717 0.9882 0.1862

8 2.6113 2.7105 1.1293 0.2128

9 2.9377 3.0494 1.2705 0.2394

10 3.2642 3.3882 1.4117 0.2660

Panel B: Order Flows

Insider Partially Informed Investor

σu Period 1 Period 2 Period 1 Period 2

1 0.5768 0.9629 0.4972 0.2698

2 1.1536 1.9258 0.9943 0.5396

3 1.7303 2.8887 1.4915 0.8094

4 2.3071 3.8517 1.9887 1.0792

5 2.8839 4.8146 2.4859 1.3490

6 3.4607 5.7775 2.9830 1.6188

7 4.0375 6.7404 3.4802 1.8886

8 4.6143 7.7033 3.9774 2.1584

9 5.1910 8.6662 4.4746 2.4282

10 5.7678 9.6292 4.9717 2.6980

37

Table 6 Information Revelation, Market Depth and Noise TradingThis table reports the relationships between information revelation and market depth and mag-

nitude of noise trading. Information Revelation (P) denotes the amount of information revealed

to partially informed investor. Information Revelation (M) denotes the amount of information

revealed to market maker.

Market Depth Information Revelation (P) Information Revelation (M)

σ2u period 1 period 2 period 1 period 2 period 1 period 2

1 2.1387 2.7366 0.5000 0.5713 0.4341 0.7353

2 4.2773 5.4732 0.5000 0.5713 0.4341 0.7353

3 6.4160 8.2098 0.5000 0.5713 0.4341 0.7353

4 8.5547 10.9464 0.5000 0.5713 0.4341 0.7353

5 10.6933 13.6830 0.5000 0.5713 0.4341 0.7353

6 12.8320 16.4196 0.5000 0.5713 0.4341 0.7353

7 14.9706 19.1562 0.5000 0.5713 0.4341 0.7353

8 17.1093 21.8928 0.5000 0.5713 0.4341 0.7353

9 19.2480 24.6294 0.5000 0.5713 0.4341 0.7353

10 21.3866 27.3660 0.5000 0.5713 0.4341 0.7353

38

Figure 1 Informed Investor Trading in Period 1This figure depicts order flows and expected profits of insider and partially informed investor with

respect to private information asymmetry σ2ε in trading period 1. Asset value variance σ2

v and

magnitude of noise trading σ2u are normalized to be 1.

0 5 10 150

0.2

0.4

0.6

0.8

Private Information Asymmetry σε2

Graph A: Order Flows in Period 1

InsiderPartially Informed Investor

0 5 10 150

0.1

0.2

0.3

0.4

0.5

Private Information Asymmetry σε2

Graph B: Profit in Period 1

InsiderPartially Informed Investor

39

Figure 2 Informed Investor Trading in Period 2This figure depicts order flows and expected profits of insider and partially informed investor with

respect to private information asymmetry σ2ε in trading period 2. Asset value variance σ2

v and

magnitude of noise trading σ2u are normalized to be 1.

0 5 10 150

0.2

0.4

0.6

0.8

1

Private Information Asymmetry σε2

Graph A: Order Flows in Period 2

InsiderPartially Informed Investor

0 5 10 150

0.1

0.2

0.3

0.4

0.5

Private Information Asymmetry σε2

Graph B: Profit in Period 2

InsiderPartially Informed Investor

40

Figure 3 Informed Investor Profit Allocation between Periods 1 and 2This figure depicts profit allocation of insider and partially informed investor between periods 1

and 2 with respect to private information asymmetry σ2ε . Profit allocation is measured by the ratio

of investor’s expected profit in period 1 to expected profit in period 2. Asset value variance σ2v and

magnitude of noise trading σ2u are normalized to be 1.

0 5 10 150.8

1

1.2

1.4

1.6

Private Information Asymmetry σε2

Graph A: Insider

0 5 10 150

5

10

15

Private Information Asymmetry σε2

Graph B: Partially Informed Investor

41

Figure 4 Information of Partially Informed InvestorThis figure depicts information composition of partially informed investor with respect to private

information asymmetry between informed investors σ2ε . Public Information denotes the information

partially informed investor learns from historical asset prices and other public information. Pri-

vate Information denotes the private information possessed by partially informed investor. Total

Information denotes the sum of Public Information and Private Information. Asset value variance

σ2v and magnitude of noise trading σ2

u are normalized to be 1.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Private Information Asymmetry σε2

Graph A: Information of Partially Informed Investor in Period 1

Total InformationPrivate InformationPublic Information

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Private Information Asymmetry σε2

Graph B: Information of Partially Informed Investor in Period 2

Total InformationPrivate InformationPublic Information

42

Figure 5 Informed Revelation to Market MakerThis figure depicts information revelation to market maker in periods 1 and 2 with respect to private

information asymmetry between informed investors σ2ε , respectively. The Kyle Model represents

the Kyle (1985) model. The HS Model represents the Holden and Subrahmanyam (1992) model.

Asset value variance σ2v and magnitude of noise trading σ2

u are normalized to be 1.

0 5 10 15

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Private Information Asymmetry σε2

Graph A: Amount of Information Revealed to Market Maker in Period 1

0 5 10 150.65

0.7

0.75

0.8

0.85

0.9

Private Information Asymmetry σε2

Graph B: Amount of Information Revealed to Market Maker in Period 2

Our ModelThe HS ModelThe Kyle Model

Our ModelThe HS ModelThe Kyle Model

43

Figure 6 Market DepthThis figure depicts market depth in periods 1 and 2 with respect to private information asymmetry

between informed investors σ2ε , respectively. The Kyle Model represents the Kyle (1985) model.

The HS Model represents the Holden and Subrahmanyam (1992) model. Asset value variance σ2v

and magnitude of noise trading σ2u are normalized to be 1.

0 5 10 152

2.05

2.1

2.15

2.2

2.25

Private Information Asymmetry σε2

Graph A: Marmet Depth in Period 1

Our ModelThe HS ModelThe Kyle Model

0 5 10 15

2.6

2.8

3

3.2

3.4

Private Information Asymmetry σε2

Graph B: Market Depth in Period 2

Our ModelThe HS ModelThe Kyle Model

44

Figure 7 Expected Profits of Informed InvestorsThis figure depicts the sum of total expected profits of insider and partially informed

investor in periods 1 and 2, respectively, with respect to private information asymme-

try σ2ε . The HS Model represents the Holden and Subrahmanyam (1992) model. As-

set value variance σ2v and magnitude of noise trading σ2

u are normalized to be 1.

0 0.05 0.1 0.15 0.20.8065

0.807

0.8075

0.808

0.8085

0.809

0.8095

0.81

0.8105

0.811

Private Information Asymmetry σε2

Total Profits of the Informed Investors

Our ModelThe HS Model

45