competitive trading and endogenous learning of ... · the market. market making is assumed a...
TRANSCRIPT
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:
[email protected]; Hao Wang, School of Economics and Management, address: 318 WeilunBuilding, Tsinghua University, Beijing 100084, China, e-mail: [email protected]; tel: 86 10-62797482; Corresponding author: Lihong Zhang, School of Economics and Management, address: 322Weilun Building, Tsinghua University, Beijing 100084, China, e-mail: [email protected]; tel:86 10-62789963.
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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
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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
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etw
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der
and
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tial
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ith
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ect
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tein
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etry
and
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out
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red
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toth
eex
pec
ted
pro
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ofin
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erin
per
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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