Insider trading, representativeness heuristic insider, and market regulation

Insider trading, representativeness heuristic insider, and market regulation

Accepted Manuscript Insider trading, representativeness heuristic insider, and market regulation Hong Liu, Lina Qi, Zaili Li PII: DOI: Reference: S10...

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Accepted Manuscript Insider trading, representativeness heuristic insider, and market regulation Hong Liu, Lina Qi, Zaili Li PII: DOI: Reference:

S1062-9408(18)30216-X https://doi.org/10.1016/j.najef.2018.11.011 ECOFIN 882

To appear in:

North American Journal of Economics & Finance

Please cite this article as: H. Liu, L. Qi, Z. Li, Insider trading, representativeness heuristic insider, and market regulation, North American Journal of Economics & Finance (2018), doi: https://doi.org/10.1016/j.najef. 2018.11.011

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Insider trading, representativeness heuristic insider and market regulation Hong Liu



Lina Qi and Zaili Li

KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Abstract This article develops an insider-trading model with the presence of rational and representativeness heuristic insiders in the market with and without market regulation. We find that, regardless of whether the market regulation exist or not, the heuristic insider trades more aggressively on his information, than the rational insider does, with the result that there is a higher probability for him to earn greater profits. Further, a higher heuristic bias of the heuristic insider leads to lower expected profit for rational insider, but higher expected profits for the heuristic insider. Moreover, the heuristic insider and rational insider prefer to trade less aggressively on their information when there is market regulation, but, they are likely to get more ex-ante expected profit. Furthermore, we find that the price efficiency and market stability may all be sacrificed if insider trading is regulated. Keywords.

Rational insider; Representativeness heuristic insider; Market maker; Market

regulation JEL classifications: G14; D82; C72; D43



Corresponding author: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024,

P.R. China. Tel.: +86 13596039631. E-mail: [email protected] The first named author is grateful for financial support for National Natural Science Foundation of China ( No.11201060 and No. 11126107) and the Fundamental Research Funds for the Central Universities (No. 2412017FZ006).

1

Insider trading, representativeness heuristic insider, and market regulation

November 2, 2018

1

Introduction Insider trading has attracted a lot of public attention since its inception. As a response to

public criticism, the Insider Trading Sanction Act of 1984, and the Insider Trading and Securities Exchange Act of 1988, were enacted. The two acts provide for insider trading penalties if one were to surpass three times the profits gained from the trade. However, lots of scholars and economists have found that sometimes, insider trading benefits the market. For example, Jhinyoung Shin (1996) finds that tolerating some amount of insider trading can be an optimal regulatory policy. Seyhun (1989) and Arshadi and Eyssell(1991) have undertaken empirical studies, and find that insiders earn excess returns in spite of the regulation of the Securities Exchange Commission (SEC) on insider trading. Further, Chowdhry and Nanda (1991) consider the possibility of the insider’s profit being confiscated by the regulator, but the optimal regulatory policy they obtain in their model is either to choose unrestricted insider trading, or to completely ban it. In this paper, we consider the optimal regulation problem in the present of representativeness heuristic insider based on the model created by Kyle(1985). In his pioneering and insightful paper, Kyle (1985) introduced a dynamic model of insider trading where a risk-neutral insider receives only one signal, and the fundamental asset value does not change over time. Through trade, the insider progressively releases his private information to the market as he exploits his informational advantage1 . Using an extension of the framework by 1

Kyle(1985) has led to a large body of literature, for example: Holden and Subrahmanyam (1992) and Foster

and Viswanathan (1996) considered a market with multiple competing insiders and they showed that competition among insiders accelerates the release of their private information. Back (1992) formalized and extended the model by showing the existence of a unique equilibrium beyond the Gaussian-linear framework. Huddart, Hughes, and Levine (2001) presented an insider’s equilibrium trading strategy in a multi-period rational expectations framework based on Kyle (1985) and embodied the requirement that the insider must publicly disclose his stock trades after the fact. Zhang (2008) extended Huddart et al. (2001) by allowing for competition among identical informed agents

1

Kyle (1985), this paper characterizes the optimal trading behaviors of the rational and representativeness heuristic insiders and their influence on the market. Representativeness heuristic has been proposed by psychologists in their experiments as one type of psychological behavioral bias. People with representativeness heuristic put too much weight on their current information and too little on their prior knowledge. Many papers show that representativeness heuristic exists in financial market. For example, Chopra et al. (1992) suggest that the traders overreact to current information. Barberis et al. (1998) construct a model which includes representativeness heuristic to point out the reason for the asset price overreaction to new information. Ormos and Timotity (2016) implement a market microstructure model including informed, uninformed and heuristicdriven investors, by using the sample that covers every executed trade made through Budapest Stock Exchange between 2 January 2008 and 31 December 2008. There are many articles addressing the survival of various types of insiders. For example, DeLong et al.(1990) investigate whether or not noise traders would survive together with rational traders. Fischer and Verrecchia (1999) consider a trading model with two types of informed traders, heuristic (i.e., those who overreact) and Bayesian. Luo (2013) build a dynamic model of competitive securities market in which representativeness heuristic traders compete with rational traders, and get the result that heuristic traders can drive more expected profit than rational traders can. Liu and Du (2016) find that the representativeness heuristic insider can coexist with overconfident and rational insiders, and have a higher probability to earn more profits. In this paper, we address the following questions: what is the trading behavior of the rational insider and heuristic insider who is facing market regulation? What is the effect of market regulations on the equilibrium results such as insiders’ trading behaviors, the market depth, the price efficiency and insiders’ profits? We find that when the heuristic insider has a moderate heuristic bias, the linear equilibrium exists with and without the market regulation. We find that regardless of whether the market regulation exists or not, the heuristic insider trades more aggressively on his information, than the rational insider does, with the result that there is a higher probability for him to earn greater profits. Further, a higher heuristic bias of the heuristic insider leads to lower expected profit for the rational insider, but higher expected profits for the heuristic insider. Moreover, the heuristic insider and rational insider prefer to trade less aggressively on their information when there is market regulation, but, they are likely to get more ex-ante expected profit. Furthermore, we find that the price efficiency and market stability may all be sacrificed if insider trading is regulated. This means that tolerating some amount of insider trading can be an optimal regulatory policy, which coincides with the finding of Jhinyoung Shin (1996). This paper is structured as follows. In Section 2, we present the model. Section 3 explains insider trading without market regulation and Section 4 explains insider trading with market regulation. and the existence of outsiders, respectively. Kyle and Wang (1997), Wang (1998) and Zhou (2011) e.t.c. consider the insider trading model with overconfident traders. Moreover, there are lots of interesting results that are covered well in Vives(2010).

2

In Section 5, we compare the two models with and without regulation. Finally, Section 6 concludes the paper.

2

The Model A single risky asset is traded in the competitive security market. The ex-post liquidation value of

the risky asset, denoted as v˜, is normally distributed with mean 0 and variance σv2 . There are four kinds of traders in the market: a rational insider, a heuristic insider, noise traders, and competitive risk-neutral market makers. The quantity traded by noise traders is a random variable, denoted as u ˜, which has mean zero and variance σu2 . Before the trade takes place, no trader knows the payoff of the risky asset, but each insider (including the rational insider and the heuristic insider) observes an information signal with respect to the payoff of the risky asset. This information signal is modeled as s˜ = v˜ + ˜, in which the residual error ˜ is normally distributed with mean zero and variance σ2 , we assume that v˜, u ˜, and ˜ are mutually independent. The rational insider can perceive the distribution of ˜ correctly after receiving the information signal. He updates his belief about the mean and variance of the risky asset as follows: Er (˜ v |˜ s) = V arr (˜ v |˜ s) = σv2 −

σv2 s˜, σv2 + σ2

(2.1)

σv4 σv2 σ2 = , σv2 + σ2 σv2 + σ2

(2.2)

where subscript r indicates rational insider. The derivation of Eqs.(2.1) and (2.2) follow from the Lemma 1 of the Appendix. The trader with representativeness heuristic (heuristic insider), as mentioned in literature on psychology, places too much weight on his current information signal, but too little on his prior knowledge. Following the way of Fischer and Verrecchia (1999) and Luo (2013) to model the heuristic traders’ updated mean and variance, we have the heuristic insider’s conditional mean and variance for the risky asset’s payoff, which are as follows: Eh (˜ v |˜ s) = m[Er (˜ v |˜ s)] =

mσv2 s˜, σv2 + σ2

and V arh (˜ v |˜ s) = σv2 + m[V arr (˜ v |˜ s) − σv2 ] = σv2 −

(2.3)

mσv4 , σv2 + σ2

(2.4)

respectively, where subscript h indicates heuristic insider and m is a heuristic bias parameter and m > 1. For parameter m, the higher above one it is, the more the heuristic bias is. Moreover, to ensure that V arh (˜ v |˜ s) > 0, Eq.(2.4) implies that m<1+

3

σ2 . σv2

(2.5)

We conform to the trading process mentioned by Luo (2001) and Liu and Zhang (2011). There are two periods, period 0 and period 1, in the economy. At period 0, the information signal is released and trading takes place. After observing the information signal, the rational insider and the heuristic insider choose their trading quantities as x ˜ = X(˜ s), y˜ = Y (˜ s), respectively, in which X, Y, are measurable functions representing trading strategies of the rational insider and the heuristic insider, respectively. Then market makers determined the price p˜ = P (˜ x + y˜ + u ˜) with measurable function P after received these orders along with noise traders’ u ˜, (but not x ˜, y˜, u ˜ separately). At period 1, the uncertainty is resolved and the risky asset payoff is realized. Let π(X, P ) = (˜ v − p˜)x and π(Y, P ) = (˜ v − p˜)y denote the profit of the rational insider and the heuristic insider, respectively. Let Er , Eh , denote the expectations of the rational insider and the heuristic insider, conditional on their information, respectively. Next, we will list out the market equilibriums for insider trading with market regulation and insider trading without market regulation, respectively.

3

Insider trading without market regulation Equilibrium without market regulation is defined by the following:

Definition 1. Equilibrium consists of the rational insider’s and the heuristic insider’s trading strategies, and the market makers’ pricing rule (X, Y, P ), such that the following two conditions hold: (1) Profit maximization: For any alternate trading strategy X 0 of the rational insider, Er0 [π(X, P )|˜ s] ≥ Er0 [π(X 0 , P )|˜ s]. For any alternate trading strategy Y 0 of the heuristic insider, Eh0 [π(Y, P )|˜ s] ≥ Eh0 [π(Y 0 , P )|˜ s]. (2) Market efficiency: P = P (˜ x + y˜ + u ˜) = E(˜ v |˜ x + y˜ + u ˜).

3.1

The unique linear equilibrium

In this section, we say there exists an equilibrium in which the rules X, Y and P are simple linear functions, as shown in the following: Theorem 3.1. For 1 < m < min{1 +

σ2 , 2} σv2

, there exists a unique linear Nash equilibrium, in 0

which X, Y , P are linear functions, with the constants α00 , α10 , β00 , β10 , γ 0 and λ such that x ˜ = X(˜ s) = α00 + α10 s˜, 4

(3.1)

y˜ = Y (˜ s) = β00 + β10 s˜,

(3.2)

p˜ = P (˜ x + y˜ + u ˜) = γ 0 + λ0 (˜ x + y˜ + u ˜),

(3.3)

α00 = β00 = γ 0 = 0, s r 2−m σu2 0 α1 = , m + 1 σv2 + σ2 s 2m − 1 σu2 β10 = p , (m + 1)(2 − m) σv2 + σ2 p (m + 1)(2 − m) σv2 0 p λ = . 3 (σv2 + σ2 )σu2

(3.4)

in which

(3.5)

(3.6)

(3.7)

Proof. See the appendix. The inequality condition for the belief parameter m, that is, 1 < m < min{1 +

σ2 , 2} σv2

results

from Eq.(2.5) and the existence of a positive root of the liquidity parameter, that is λ > 0, and consequently the existence of the unique linear equilibrium (See Eq.(3.7)). If the precision of the private information is high enough, that is σ2 < σv2 , we can get 1 < m < 1 +

σ2 σv2

< 2, and if the

precision is not very high that is σ2 ≥ σv2 , we have 1 < m < 2. This means that high precision of private information narrows down the interval of heuristic bias. The economics intuition is that the heuristic insider should moderately overreact to the new or current information for the equilibrium to exist.

3.2

Properties of the linear equilibrium

In this section, we discuss some properties of linear equilibrium pertaining to trading intensity and market depth, the ex-ante profit of the rational insider and the heuristic insider, and the information revealed by the price. 3.2.1

Trading intensity and market depth

The constants α10 and β10 measure the trading intensity of rational insider and heuristic insider, respectively. From Eqs.(3.5) and (3.6), we have the following proposition. Proposition 3.1. α10 < β10 , which indicates that the rational insider trades less than the heuristic insider does. The quantity

1 λ0

measures the “depth” of the market, that is, the order flow necessary to induce

the price to rise or fall by one unit. From the expression of α10 , β10 and λ0 in Theorem 3.1, it is easy to see the following propositions.

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Proposition 3.2. The trading intensities and the market depth satisfy: ∂α10 < 0, ∂m

(3.8)

∂β10 > 0, ∂m ∂λ0 < 0. ∂m

(3.9) (3.10)

Proof. See the appendix. The above proposition states that the rational insider’s trading intensity varies negatively with the heuristic bias, that is to say, the greater the heuristic bias, the lesser the rational insider trades on his information. However, the heuristic insider acts in the opposite way, that is, the greater the heuristic bias, the more the heuristic insider trades on his information. The higher heuristic bias leads to a deeper market depth. 3.2.2

The profits of insiders

Proposition 3.3. The expected profit of rational insider conditional on his available information is as below: 2−m Er0 [π(X, P )|˜ s] = Er0 [(˜ v − p˜)x|˜ s] = 3

r

2 − m σv2 m + 1 σv2 + σ2

s

σu2 s˜2 . σv2 + σ2

(3.11)

The expected profit of a heuristic insider conditional on his available information is as below: s (2m − 1)2 σv2 σu2 0 0 (3.12) Eh [π(Y, P )|˜ v + ˜] = Eh [(˜ v − p˜)y|˜ s] = p s˜2 . 2 + σ2 2 + σ2 σ σ 3 (m + 1)(2 − m) v  v  Hence, their ex-ante expected profit satisfies the following: 2−m Er0 [π] = Er0 {Er0 [(˜ v − p˜)x|˜ s]} = 3

r

2−m 2 σ m+1 v

s

and (2m − 1)2

σv2 Eh0 [π] = Eh0 {Eh0 [(˜ v − p˜)y|˜ s]} = p 3 (m + 1)(2 − m)

σv2 s

σu2 , + σ2

(3.13)

σu2 , σv2 + σ2

(3.14)

Proof. See the appendix. The expected profit for all types of traders are functions of the variance of the risky asset’s payoff (σv2 ), the variance of the signal perceived by the rational insider (σv2 + σ2 ), the standard variance of noise traders (σu ) and the heuristic bias parameter (m). From the equations (3.13) and (3.14), we have the following corollary:

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Corollary 3.1.

∂Eh0 [π] > 0, ∂m

∂Er0 [π] < 0, ∂m

(3.15)

and Er0 [π] < Eh0 [π]

(3.16)

Proof. See the appendix. From the above corollary, the profit of the rational insider is decreasing the functions of the heuristic bias, which means that the higher the heuristic bias, the lesser the profits for a rational insider, while the heuristic bias will enhance the profit of the heuristic trader. Moreover, the heuristic insider can get more expected profit than the rational insider can. 3.2.3

Information revelation

To obtain a measure of the informativeness of price, we define Σ01 = V ar(˜ v |˜ p),

(3.17)

which is a measure of the residual information after the information is incorporated into the price. Let I 0 = V ar(˜ v ) − Σ01 ,

(3.18)

denote how much information has been incorporated into the equilibrium price. Then, we offer the following proposition. Proposition 3.4. In the equilibrium I0 = and it is easy to have

∂I 0 ∂m

m + 1 σv4 , 3 σv2 + σ2

(3.19)

> 0.

The information revealed by the price, denoted by I 0 , is an increasing function of m, which means that the higher the heuristic bias is, the more efficient the market is. The intuition is that the heuristic insider with higher heuristic bias trades more aggressively on his information, with a result that the heuristic insider can get more expected profit from the misvaluations than can rational traders (See Eq.3.16). This result is consistent with the finding of Luo (2013). Also, heuristic insider’s aggressive trading on his information leads to more information revealed, hence more efficient (See Pro.3.4) and more stable(See Eq.3.10) market.

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4

Inside trading with the market regulation In this section, we follow Jhinyoung Shin(1996) to model the market regulation. The regulatory

policy is composed of two parts, namely, the probability of convicting an insider denoted by q, and the penalty imposed on the convicted insider, if convicted, where the fine is t times the square of his trading order. We assume that the probability and the penalty can be arbitrarily set by the regulator without cost, and the regulator set this regulation policy to minimize the trading losses of noise traders. As shown by Jhinyoung Shin (1996), the parameter R =

tq 1−q

is the unique measure

of how strictly insider trading is regulated. Definition 2. Equilibrium consists of the trading strategies of the rational insider and the heuristic insider, the market makers’ pricing rule, and the optimal regulation (X, Y, P, R), such that the following three conditions hold: (1) Profit maximization: For any alternate trading strategy X 0 of the rational insider, Er [π(X, P )|˜ s] ≥ Er [π(X 0 , P )|˜ s]. For any alternate trading strategy Y 0 of the heuristic insider, Eh [π(Y, P )|˜ s] ≥ Eh [π(Y 0 , P )|˜ s]. (2) Market efficiency: P (˜ x + y˜ + u ˜) = E(˜ v |˜ x + y˜ + u ˜). (3) Optimal regulation (Loss minimization of noise traders): R = argminE[(˜ p − v˜)˜ u]

4.1

The unique linear equilibrium

In this section, we say there exists an equilibrium in which the rules X, Y , P and R are simple linear functions, as shown in the following: Theorem 4.1. For 1 < m < 1 +

σ2 , σv2

there exists a unique linear Nash equilibrium, in which X,

Y , P are linear functions, with the constants α0 , α1 , β0 , β1 , γ and λ such that x ˜ = X(˜ s) = α0 + α1 s˜,

(4.1)

y˜ = Y (˜ s) = β0 + β1 s˜,

(4.2)

p˜ = P (˜ x + y˜ + u ˜) = γ + λ(˜ x + y˜ + u ˜),

(4.3)

α0 = β0 = γ = 0,

(4.4)

in which

8

s

σu2 , σv2 + σ2  s 1 σu2 β1 = 1 − , 2m σv2 + σ2 1 α1 = 2m

(4.5)

(4.6)

σv2 λ= p . 2 (σv2 + σ2 )σu2

(4.7)

And the optimal regulation R satisfies R=

2m − 1 σv2 p . 4 (σv2 + σ2 )σu2

(4.8)

Proof. See the appendix. The inequality condition for the belief parameter m, that is, 1 < m < 1 +

σ2 σv2

results from

Eq.(2.5).

4.2

Properties of the linear equilibrium

In this section we discuss some properties of the linear equilibrium pertaining to the trading intensity and the market depth, the ex-ante profit of the rational insider and the heuristic insider, and the information revealed in price. 4.2.1

Trading intensity and market depth

The constants α1 and β1 measure the trading intensity of a rational insider and a heuristic insider, respectively. From Eqs. (4.5), (4.6) and (4.7) it is easy to see the following proposition. Proposition 4.1. (1) α1 < β1 , which indicates that the rational insider trades less than the heuristic insider does. (2) α1 is a decreasing function of m and β1 is an increasing function of m. (3) The market depth, measured by

1 λ,

is a constant.

The above proposition states that the rational insider’s trading intensity varies negatively with the heuristic bias, that is to say, the more the heuristic bias is, the less the rational insider trades on his information. However, the heuristic insider behaves in the opposite way, that is, the more the heuristic bias is, the more the heuristic insider trades on his information. The higher heuristic bias has no effect on market depth. 4.2.2

The profits of insiders

Proposition 4.2. The expected profit of the rational insider conditional on his available information is as below:

 σv2 Er [(˜ v − p˜)x|˜ s] = s˜ − λ(α1 + β1 )˜ s α1 s˜. σv2 + σ2 

9

(4.9)

The expected profit of the heuristic insider conditional on his available information is as below:   mσv2 Eh [(˜ v − p˜)y|˜ s] = s˜ − λ(α1 + β1 )˜ s β1 s˜. (4.10) σv2 + σ2 Hence, their ex-ante expected profit satisfies: σ2 Er [π] = Er {Er [(˜ v − p˜)x|˜ s]} = v 4m and

s σv2

σu2 , + σ2

   s 1 1 σu2 Eh [π] = Eh {Eh [(˜ v − p˜)y|˜ s]} = m − 1− σv2 . 2 2m σv2 + σ2

(4.11)

(4.12)

The expected loss of noise traders is as follows: σ2 En [L] = λσµ2 = v 2

s σv2

σu2 . + σ2

(4.13)

Proof. See the appendix. The expected profit for all types of traders are functions of the variance of risky asset’s payoff (σv2 ),

the variance of the signal perceived by the rational insider (σv2 + σ2 ), the standard variance

of noise traders (σu ) and the heuristic bias parameter (m). From the equations (4.11), (4.12) and (4.13), we have the following corollary: Corollary 4.1. ∂Er [π] < 0, ∂m

∂Eh [π] > 0, ∂m

∂En [L] = 0. ∂m

(4.14)

Proof. See the appendix. From the above corollary, the profit of the rational insider is a decreasing function of heuristic bias, which means the higher heuristic bias leads to lesser profits for the rational insider, while the heuristic bias will enhance the profit of the heuristic trader. From Eqs. (4.11) and (4.12), we can get s    s σv2 σu2 1 1 σu2 2 Er [π] − Eh [π] = − m − 1 − σ v 4m σv2 + σ2 2 2m σv2 + σ2 s σu2 =(1 − m)σv2 < 0, σv2 + σ2

(4.15)

Eq.(4.15) implies that the heuristic insider can get more expected profits than the rational insider can.

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4.2.3

Information revelation

To obtain a measure of the informativeness of price, we define (4.16)

Σ1 = V ar(˜ v |˜ p),

which is a measure of the residual information after the information is incorporated into the price. Let (4.17)

I = V ar(˜ v ) − Σ1 ,

denote how much information has been incorporated into the equilibrium price. Then, we make the following proposition. Proposition 4.3. In the equilibrium 1 I= 2



σv4 σv2 + σ2

 .

(4.18)

Proof. See the appendix. The information revealed by the price denoted by I is not affected by the heuristic bias. The intuition is that the heuristic insider trades less aggressively on his information (See Pro. 5.1) when there is market regulation. The heuristic insider is afraid of being convicted and releases his information at a constant speed(See Pro. 4.3) whatever the heuristic bias is, with a result that the market depth is constant (See Pro.4.1).

5

The comparison of two models with and without regulation In this section, we analyze the impacts of the regulation on the market equilibrium by comparing

the two equilibrium results of the two models. We discuss some properties of insider trading with and without market regulation pertaining to the trading intensity and the market depth, the ex-ante profit of the rational insider and the heuristic insider, and the information revealed in price.

5.1

Trading intensity and market depth

Proposition 5.1. (1) The trading intensity of a rational insider satisfies: when m is big enough 1.7808 < m < 2, we have α1 > α10 ,

(5.1)

α1 < α10 ,

(5.2)

and when 1 < m < 1.7808, we have

(2) For all 1 < m < 2, the trading intensity of heuristic insider satisfies: β1 < β10 . 11

(5.3)

Proof. See the appendix. Since m satisfies 1 < m < min{1 +

σ2 , 2}, σv2

the above proposition shows that most of the time,

the rational insider and heuristic insider trade less aggressively on their information when there is market regulation. Proposition 5.2. The quantity

1 λ

measures the “depth” of the market, when 1 < m < 2 , we have 1 1 < 0. λ λ

(5.4)

Proof. See the appendix. When market regulation exists, the “depth” of the market becomes smaller. This means that market stability is worse.

5.2

The profits of the two kinds of insiders

Proposition 5.3. (1) The ex-ante expected profit of rational insider satisfies: Er [π] > Er0 [π].

(5.5)

(2) The ex-ante expected profit of heuristic insider satisfies: when 1 < m < 1.04, we have Eh [π] < Eh0 [π].

(5.6)

Eh [π] > Eh0 [π].

(5.7)

and when 1.04 < m < 2, we have

Proof. See the appendix. When the market is regulated by the regulator, the rational insider can get more ex-ante expected profits and most of the time the heuristic insider can also get more ex-ante expected profits. The intuition is that the market maker can implement the insider trading regulation without cost, while the insider has a risk of paying a fine proportional to the number of shares he traders. The insider has a higher risk and meanwhile has a higher ex-ante expected profits.

5.3

Information revelation

Proposition 5.4. I < I 0. Proof. See the appendix. 12

(5.8)

The above proposition indicates that the price reveals less information when market regulations exist, which means that market regulation leads to less efficiency in the market. This means that tolerating some insider trading can be the optimal regulatory policy, this is consistent with the finding of Shin (1996).

6

Conclusion We investigate Kyle’s (1985) extended model with the assumption that there are two kinds of

insiders in the market: rational and heuristic insiders. We first analyze the trading behavior of rational and heuristic insiders with the assumption that there is no market regulation in the market. Next, we examine the insider trading model with market regulation. Finally, we compare the two models with and without market regulation. We find that, regardless of whether market regulation exists or not, the heuristic insider trades more aggressively on his information than the rational insider does, with the result that there is a higher probability for him to earn more profits. The more the heuristic bias is, the lesser the rational insider trades on his information, while the heuristic insider behaves in the opposite way. When there is no market regulation, the higher heuristic bias leads to greater market depth, which means that the existence of a heuristic insider leads to a more stable market. But, when there are market regulations, the market depth is a constant, that is to say, the existence of a heuristic insider has no effect on the market depth. Moreover, whether the market regulation exists or not, the heuristic insider can get more expected profit than the rational insider can, and the higher heuristic bias leads to lesser profits for a rational insider, and increased profits for a heuristic insider. Furthermore, when there is no market regulation, the information revealed by the price is affected by the heuristic bias. The higher the heuristic bias, the more efficient the market is. When there are market regulations, market efficiency is not affected by the heuristic bias. By comparing the two models with and without market regulations, we find that most of the time the rational insider and heuristic insider trade less aggressively on their information when there is market regulation. Further, when the market is regulated, the rational insider can get more ex-ante expected profits and most of the time the heuristic insider can also get more ex-ante expected profits. Furthermore, we find that the price efficiency and market stability may all be sacrificed if insider trading is regulated. Thus, tolerating some amount of insider trading can be an optimal regulatory policy.

Appendix: Proofs At the beginning of appendix, we state a famous result that will be used later. ! X1 Lemma 6.1. Let X1 and X2 have joint normal distribution, satisfying ∼ N (µ, Σ) with X2 13

µ=

µ1

!

,Σ= µ2 distribution, and

Σ11 Σ12 Σ21 Σ22

! . Then the random variable X1 conditional on X2 has a normal

E[X1 |X2 ] = µ1 + Σ12 Σ−1 22 (X2 − µ2 ), V ar(X1 |X2 ) = Σ11 − Σ12 Σ−1 22 Σ21 . Proof of Theorem 3.1. We conjecture that the linear equilibrium is given by: x ˜ = X(˜ s) = α00 + α10 s˜,

(A1)

y˜ = Y (˜ s) = β00 + β10 s˜,

(A2)

p˜ = P (˜ x + y˜ + u ˜) = γ 0 + λ0 (˜ x + y˜ + u ˜),

(A3)

0

where the parameters α00 , α10 , β00 , β10 , γ 0 and λ are constants that need to be determined. We will verify the conjecture and identify the parameters. Firstly, the rational insider’s profit conditional on his information available and his rational belief is: Er [(˜ v − p˜)x|˜ s] =Er {[˜ v − γ 0 − λ0 (x + β00 + β10 s˜ + u ˜)]x|˜ s} =[Er (˜ v |˜ s) − γ 0 − λ0 x − λ0 β00 − λ0 β10 s˜]x   σv2 0 0 0 0 0 0 = s˜ − γ − λ x − λ β0 − λ β1 s˜ x. σv2 + σ2 By the first-order condition, we have   1 σv2 0 0 0 0 0 x= 0 s˜ − γ − λ β0 − λ β1 s˜ , 2λ σv2 + σ2

(A4)

(A5)

and the second-order condition implies λ0 > 0. Comparing Eqs.(A5) and (A1), we have α00 = − α10 =

1 2λ0



γ 0 + λ0 β00 , 2λ0

 σv2 0 0 − λ β 1 . σv2 + σ2

(A6) (A7)

Secondly, the heuristic insider’s profit conditional on his information available and his rational belief is: Eh [(˜ v − p˜)y|˜ s] =Eh {[˜ v − γ 0 − λ0 (α00 + α10 s˜ + y + u ˜)]y|˜ s} =[Eh (˜ v |˜ s) − γ 0 − λ0 α00 − λ0 α10 s˜ − λ0 y]y   mσv2 0 0 0 0 0 0 = s˜ − γ − λ α0 − λ α1 s˜ − λ y y. σv2 + σ2 14

(A8)

By the first-order condition, the maximizing y is easily shown to be given by   1 mσv2 0 0 0 0 0 y= 0 s˜ − γ − λ α0 − λ α1 s˜ . 2λ σv2 + σ2

(A9)

Comparing Eqs. (A9) and (A2), we have β00 = − β10

1 = 0 2λ



γ 0 + λ0 α00 , 2λ0

 mσv2 0 0 − λ α1 . σv2 + σ2

(A10) (A11)

By the semi-strong efficient condition of the market and using the lamma 6.1, we have p˜ = E(˜ v |˜ z ) = E(˜ v) +

Cov(˜ v , z˜) [˜ z − E(˜ z )] V ar(˜ z)

(A12)

in which z˜ = x ˜ + y˜ + u ˜ = (α00 + β00 ) + (α10 + β10 )(˜ v + ˜) + u ˜,

(A13)

E(˜ z ) = α00 + β00 ,

(A14)

Cov(˜ v , z˜) = (α10 + β10 )σv2 ,

(A15)

V ar(˜ z ) = (α10 + β10 )2 (σv2 + σ2 ) + σu2 ,

(A16)

we get p˜ = E(˜ v |˜ z) = −

(α00 + β00 )(α10 + β10 )σv2 (α10 + β10 )σv2 + z˜, (α10 + β10 )2 (σv2 + σ2 ) + σu2 (α10 + β10 )2 (σv2 + σ2 ) + σu2

(A17)

Comparing Eqs.(A17) and (A3), we can know γ0 = − λ0 =

(α00 + β00 )(α10 + β10 )σv2 , (α10 + β10 )2 (σv2 + σ2 ) + σu2

(α10 + β10 )σv2 , (α10 + β10 )2 (σv2 + σ2 ) + σu2 γ 0 = −(α00 + β00 )λ0 ,

(A18)

(A19)

(A20)

Now we calculate α00 , β00 , γ 0 . From the first equations of Eqs.(A6) and (A10), we get 1 [2γ 0 + (α00 + β00 )λ0 ], 2λ0

(A21)

−2λ0 (α00 + β00 ) = 2γ 0 + (α00 + β00 )λ0 ,

(A22)

α00 + β00 = − i.e.

that is

2γ 0 . 3λ0 Plug Eq.(A23) into the Eq.(A20), we immediately have α00 + β00 = −

γ 0 = 0. 15

(A23)

(A24)

Substituting γ 0 = 0 into the equation of Eqs.(A6) and (A10) yields α00 = β00 = 0.

(A25)

then we get Eq.(3.4). Next, we consider the parameters α10 , β10 , λ0 . Combining the second equations of (A7) and (A11) we have α10 + β10 =

(m + 1)σv2 1 − (α0 + β10 ) 2λ0 (σv2 + σ2 ) 2 1

(A26)

(m + 1)σv2 3λ0 (σv2 + σ2 )

(A27)

that is α10 + β10 =

Then plug Eq.(A27) into the equation of (A19), we have λ0 =

3λ0 (m + 1)σv4 , (m + 1)2 σv4 + 9λ02 (σv2 + σ2 )σu2

(A28)

it equals to

hence

3(m + 1)σv4 = (m + 1)2 σv4 + 9λ02 (σv2 + σ2 )σu2 ,

(A29)

p (m + 1)(2 − m) σv2 p λ = . 3 (σv2 + σ2 )σu2

(A30)

0

Also from Eqs.(A7), (A11) and (A27) we have 2 − m σv2 , 3λ0 σv2 + σ2

(A31)

2m − 1 σv2 , 3λ0 σv2 + σ2

(A32)

α10 = β10 =

Then plug Eq.(A30) into Eqs.(A31) and (A32), we get s r 2 − m σu2 α10 = , 2 m + 1 σv + σ2 s

2m − 1

β10 = p (m + 1)(2 − m)

σu2 , σv2 + σ2

(A33)

(A34)

then we get Eqs.(3.5) , (3.6) and (3.7). Proof of Proposition 3.2. From the Eqs.(3.5), (3.6) and (3.7) we know that q 2  q σu 2−m ∂ 0 m+1 ∂α1 σv2 +σ2 = ∂m  ∂m  s 2 σu  1 1 −3  q = σv2 + σ2 2 2−m (m + 1)2 m+1   s 2 σu  3 1 1  < 0, = − q 2 2 σv + σ 2 2−m (m + 1)2 m+1

16

(A35)

and 

√ 2m−1 (m+1)(2−m)

q

2 σu σv2 +σ2



∂ ∂β10 = ∂m ∂m  p  s 2 (m + 1)(2 − m) − 12 (2m − 1) √ 1−2m 2 σu  (m+1)(2−m)  = σv2 + σ2 (m + 1)(2 − m)  p  2 s 2 (m + 1)(2 − m) + 12 √ (2m−1) 2 σu  (m+1)(2−m)  =   > 0. 2 2 σv + σ (m + 1)(2 − m)

(A36)

and √

(m+1)(2−m) 3



σv2 2 (σv2 +σ2 )σu



∂ ∂λ0 = ∂m ∂m " # 2 σv 1 1 1 =p × p [(2 − m) − (m + 1)] (σv2 + σ2 )σu2 3 2 (m + 1)(2 − m) " # σv2 1 1 p =p (1 − 2m) < 0. (σv2 + σ2 )σu2 6 (m + 1)(2 − m)

(A37)

Then we can have equations in Proposition (3.2). Proof of Proposition 3.3. From the Theorem (3.1) and the information the rational insider observed, the profit of rational insider is: Er0 [π(X, P )|˜ v + ˜] = Er0 [(˜ v − p˜)x|˜ s] =Er0 {[˜ v − λ0 (α10 + β10 )˜ s − λ0 u ˜]x|˜ s} =[Er0 (˜ v |˜ s) − λ0 (α10 + β10 )˜ s]α10 s˜   σv2 0 0 0 − λ (α1 + β1 ) α10 s˜2 = 2 σv + σ2 s r 2 − m 2 − m σv2 σu2 = s˜2 . 3 m + 1 σv2 + σ2 σv2 + σ2

(A38)

the ex-ante expected profit of rational insider shows that: Er0 [π] = Er0 {Er0 [(˜ v − p˜)x|˜ s]} s " # r 2 2 2 − m 2 − m σ σ v u =Er0 s˜2 3 m + 1 σv2 + σ2 σv2 + σ2 s r 2−m 2−m 2 σu2 = σv . 2 3 m+1 σv + σ2

17

(A39)

And the profit of heuristic insider is: Eh0 [π(Y, P )|˜ v + ˜] = Eh0 [(˜ v − p˜)y|˜ s] =Eh0 {[˜ v − λ0 (α10 + β10 )˜ s − λ0 u ˜]y|˜ s} =[Eh (˜ v |˜ s) − λ0 (α10 + β10 )˜ s]β10 s˜   mσv2 0 0 0 = 2 − λ (α1 + β1 ) β10 s˜2 σv + σ2 s (2m − 1)2 σv2 σu2 = p s˜2 . 2 + σ2 2 + σ2 σ σ 3 (m + 1)(2 − m) v  v 

(A40)

the ex-ante expected profit of heuristic insider shows that: Eh0 [π] = Eh0 {Eh0 [(˜ v − p˜)y|˜ s]} s " # (2m − 1)2 σv2 σu2 0 2 =Eh p s˜ 3 (m + 1)(2 − m) σv2 + σ2 σv2 + σ2 s (2m − 1)2 σu2 = p σv2 . σv2 + σ2 3 (m + 1)(2 − m) Proof of Corollary 3.1. For the first Eq. in Corollary (3.1), q 2  q  σu 2−m 2−m 2 ∂ σ 0 3 m+1 v ∂Er [π] σv2 +σ2 = ∂m ∂m s √ " 3√ # 3 1 2 2 − m(−1) − 12 √m+1 2−m 1 2 σu 2 = σv 3 σv2 + σ2 m+1 s   1 2 σu2 1 √ 2−m = σv (− ) 2 − m 3 + √ 3 σv2 + σ2 2 m−1 s   1 2 σu2 √ 2−m √ =(− )σv 2−m 3+ < 0. 6 σv2 + σ2 m−1

(A41)

(A42)

and  ∂

√ (2m−1)

2

σv2

q

2 σu σv2 +σ2



3 (m+1)(2−m) ∂Eh0 [π] = ∂m ∂m   p s 1 2 [(2−m)−(m+1)] √ 4(2m − 1) (m + 1)(2 − m) − (2m − 1) 2 1 σu2  (m+1)(2−m)  = σv2   3 σv2 + σ2 (m + 1)(2 − m)

1 = σv2 3



s

σu2 σv2

+

σ2

 

p 4(2m − 1) (m + 1)(2 − m) + 12 √ (m + 1)(2 − m)

18

(2m−1)3 (m+1)(2−m) 

  > 0.

(A43)

From Eqs.(A39) and (A41), we have Er0 [π] − Eh0 [π] " # s r 2−m 2−m (2m − 1)2 σu2 = − p σv2 3 m + 1 3 (m + 1)(2 − m) σv2 + σ2 s 3(1 − m2 ) σu2 = p σv2 <0 σv2 + σ2 3 (m + 1)(2 − m)

(A44)

Proof of Proposition 3.4. Σ01 = V ar(˜ v |˜ p) = V ar(˜ v) −

[Cov(˜ v , p˜)]2 . V ar(˜ p)

(A45)

in which Cov(˜ v , p˜) = Cov(˜ v , λ0 (˜ x + y˜ + u ˜))   =Cov v˜, λ0 [(α00 + β00 ) + (α10 + β10 )(˜ v + ˜) + u ˜]

(A46)

=λ0 (α10 + β10 )σv2 ,   V ar(˜ p) = V ar λ0 (˜ x + y˜ + u ˜)   =V ar λ0 [(α00 + β00 ) + (α10 + β10 )(˜ v + ˜) + u ˜]

(A47)

=λ02 (α10 + β10 )2 (σv2 + σ2 ) + σu2 . Then, easy to have Σ01 = V ar(˜ v |˜ p) λ02 (α10 + β10 )2 σv4 λ02 (α10 + β10 )2 (σv2 + σ2 ) + σu2 λ02 (α0 + β 0 )2 σ 2 σ 2 + σ 2 σ 2 = 02 0 1 0 12 2v  2 v u 2 λ (α1 + β1 ) (σv + σ ) + σu   1 (m + 1)σv2 σ2 2 = + (2 − m)σv . 3 σv2 + σ2

=σv2 −

so 0

I =

σv2

  1 (m + 1)σv2 σ2 m + 1 σv4 2 + (2 − m)σ = − v 3 σv2 + σ2 3 σv2 + σ2

(A48)

(A49)

Proof of Theorem 4.1. We conjecture that the linear equilibrium is given by: x ˜ = X(˜ s) = α0 + α1 s˜,

(B1)

y˜ = Y (˜ s) = β0 + β1 s˜,

(B2)

P = P (˜ x + y˜ + u ˜) = γ + λ(˜ x + y˜ + u ˜),

(B3)

R = argminE[(˜ p − v˜)˜ u].

(B4)

where the parameters α0 , α1 , β0 , β1 , γ and λ are constants that need to be determined. We will verify the conjecture and identify the parameters. 19

Firstly, the rational insider’s profit conditional on his information available and his rational belief is: Eγ [(1 − q)(˜ v − p˜)x − qtx2 |˜ s] =Eγ [(1 − q)[˜ v − γ − λ(x + β0 + β1 s˜ + u ˜)]x − qtx2 |˜ s] =(1 − q)[Eγ (˜ v |˜ s) − γ − λx − λβ0 − λβ1 s˜]x − qtx2   σv2 =(1 − q) 2 s˜ − γ − λx − λβ0 − λβ1 s˜ x − qtx2 , σv + σ2 By the first-order condition, we have   σv2 (1 − q) 2 s˜ − γ − 2λx − λβ0 − λβ1 s˜ − 2qtx = 0, σv + σ2 so

  1 σv2 x= s˜ − γ − λβ0 − λβ1 s˜ . 2(R + λ) σv2 + σε2

(B5)

(B6)

(B7)

and the second-order condition implies λ + R > 0. Comparing Eqs.(B1) and (B7), we have α0 = −

α1 =

γ + λβ0 , 2(R + λ)

1 σ2 ( 2 v 2 − λβ1 ). 2(R + λ) σv + σ

(B8)

(B9)

Secondly, the heuristic insider’s profit conditional on his information available and his heuristic belief is: Eh [(1 − q)(˜ v − p˜)y − qty 2 |˜ s] =Eh {(1 − q)[˜ v − γ − λ(α0 + α1 s˜ + y + u ˜)]˜ y − qty 2 |˜ s] =(1 − q)[Eh (˜ v |˜ s) − γ − λα0 − λα1 s˜ − λy]y − qty 2   mσv2 =(1 − q) 2 s˜ − γ − λα0 − λα1 s˜ − λy y − qty 2 . σv + σ2

(B10)

The first-order condition implies (1 − q)[

mσv2 s˜ − γ − λα0 − λα1 s˜ − 2λy] − 2qty = 0, σv2 + σ2

i.e. y=

  1 mσv2 s ˜ − γ − λα − λα s ˜ . 0 1 2(R + λ) σv2 + σ2

(B11)

(B12)

and comparing Eqs.(B2) and (B12), we have β0 = −

β1 =

γ + λα0 , 2(R + λ)

1 mσ 2 ( 2 v 2 − λα1 ). 2(R + λ) σv + σ

20

(B13)

(B14)

By the semi-strong efficient condition of the market and using the lamma 6.1, we have p˜ = E[˜ v |˜ z ] = E(˜ v) +

Cov(˜ v , z˜) [˜ z − E[˜ z ]]. V ar(˜ z)

(B15)

where z˜ = x ˜ + y˜ + u ˜ = (α0 + β0 ) + (α1 + β1 )(˜ v + ˜) + u ˜, E(˜ z ) = α0 + β0 , Cov(˜ v , z˜) = (α1 + β1 )σv2 , V ar(˜ z ) = (α1 + β1 )2 (σv2 + σ2 ) + σu2 . so

(α1 + β1 )σv2 z˜ (α1 + β1 )2 (σv2 + σ2 ) + σu2

P = E[˜ v |˜ z ] = −(α0 + β0 )λ +

(B16)

and comparing Eqs.(B3) and (B16), we can know γ = −(α0 + β0 )λ. λ=

(α1 + β1 )σv2 , (α1 + β1 )2 (σv2 + σ2 ) + σu2

(B17) (B18)

Now we calculate α0 and β0 . From the Eqs.(B8) and (B13), we get α0 + β0 = −

1 [2γ + λ(α0 + β0 )]. 2(R + λ)

i.e.

(B19)

−2γ . 2R + 3λ

(B20)

(m + 1)σv2 . (2R + 3λ)(σv2 + σ2 )

(B21)

α0 + β0 = Eqs.(B9) and (B14) imply α1 + β1 =

Plug Eq.(B20) into the Eq.(B17), we immediately have (λ + 2R)γ = 0. 2R + 3λ

(B22)

Since λ + R > 0 and R ≥ 0, so we have λ + 2R > 0, and γ = 0. Plug Eq.(B21) into the Eq.(B18), we can have λ=

(m + 1)(2R + 3λ)σv4 . (m + 1)2 σv4 + (2R + 3λ)2 (σv2 + σ2 )σu2

(B23)

Plug γ = 0 into Eqs.(B8) and (B13), we get α0 = β0 = 0. From Eq.(B21), we have α1 =

(m + 1)σv2 − β1 . (2R + 3λ)(σv2 + σ2 ) 21

(B24)

Combing with Eq.(B9), we have β1 =

2mR + (2m − 1)λ σv2 , (2R + 3λ)(2R + λ) σv2 + σ2

(B25)

α1 =

2R + (2 − m)λ σv2 . (2R + 3λ)(2R + λ) σv2 + σ2

(B26)

Next, we consider the loss minimization of the noise traders. Since En [(˜ p − v˜)u] = En [(λ˜ z − v˜)˜ u] (B27)

=En {[λ(˜ x + y˜ + u ˜) − v˜]˜ u} =λσu2 , the optimization problem transfer to find the maximization of −λ. From Eq.(B23), we have λ2R + 3λ2 − m + 1[(2 − m)λ + 2R]

σv4 = 0. σu2 (σv2 + σ2 )

(B28)

Therefore, the Lagrange function is constructed as follows: σv4 . σu2 (σv2 + σ2 )

(B29)

∂L σ4 = λ(2R + 3λ)2 − (m + 1)[(2 − m)λ + 2R] 2 2 v 2 = 0, ∂η σu (σv + σ )

(B30)

∂L σ4 = η[4λ(2R + 3λ) − 2(m + 1) 2 2 v 2 ] = 0. ∂R σu (σv + σ )

(B31)

L = −λ + η[λ(2R + 3λ)2 − m + 1[(2 − m)λ + 2R] Using the Lagrange method, we have:

Eq.(B31) implies 2R + 3λ =

m+1 σv4 . 2λ σu2 (σv2 + σ2 )

(B32)

Plug Eq.(B32) into the Eq.(B30), we can have 2R =

M +1 σ4 · 2 2 v 2 − (2 − M )λ. 4λ σu (σv + σ )

(B33)

σv4 . 4[σu2 (σv2 + σ2 )]

(B34)

Eqs.(B32) and (B33) imply λ2 = so we have

σv2 λ=± p . 2 σu2 (σv2 + σ2 )

Next we discuss the value of λ. 2 σv2 √ 2 σv2 2 . , it is easy to have R = 2m−1 4 2 (σ 2 +σ 2 ) σu σ (σ v  u v +σ ) 2 −σv2 1−2m √ 2 2 2 , it is easy to have R = 4 √ 2 σv2 2 . σu (σv +σ ) 2 σu (σv +σ )

(1) If λ = √ 2

(2) If λ =

22

(B35)

Since m > 1 and R > 0, we have σv2 λ= p . 2 (σv2 + σ2 )σu2 R=

2m − 1 σv2 p . 4 σu2 (σv2 + σ2 )

(B36)

(B37)

Then we prove the theorem. Proof of Proposition 4.2. From the Theorem (4.1) and the information the rational insider observed, the profit of rational insider is: Er [(˜ v − p˜)x|˜ s] = Er {[˜ v − λ(α1 + β1 )˜ s − λ˜ u]x|˜ s} =[Er (˜ v |˜ s) − λ(α1 + β1 )˜ s]α1 s˜   σ2 = 2 v 2 s˜ − λ(α1 + β1 )˜ s α1 s˜. σv + σ

(B38)

so the ex-ante expected profit of rational insider is Er [π] = Er {Er [(˜ v − p˜)x|˜ s]} σ2 =Er {[ 2 v 2 s˜ − λ(α1 + β1 )˜ s]α1 s˜} σv + σ   σv2 = 2 − λ(α1 + β1 ) αE[˜ s2 ] σv + σ2   σv2 = 2 − λ(α1 + β1 ) ασs2 . σv + σ2 Substitute the expression of λ, α1 and α2 into the above equation, we have s σv2 σu2 Er [π] = Er {Er [(˜ v − p˜)x|˜ s]} = , 4m σv2 + σ2

(B39)

(B40)

And the profit of heuristic insider is: Eh [(˜ v − p˜)y|˜ s] =Eh {[˜ v − λ(α1 + β1 )˜ s − λ˜ u]y|˜ s} =[Eh (˜ v |˜ s) − λ(α1 + β1 )˜ s]β1 s˜   mσv2 = 2 s˜ − λ(α1 + β1 )˜ s β1 s˜. σv + σ2

(B41)

the ex-ante expected profit of heuristic insider shows that: Eh [π] = Eh {Eh [(˜ v − p˜)y|˜ s]} mσ 2 =Eh {[ 2 v 2 s˜ − λ(α1 + β1 )˜ s]β1 s˜} σv + σ   mσ 2 = 2 v 2 − λ(α1 + β1 ) βE[˜ s2 ] σv + σ   mσv2 = 2 − λ(α1 + β1 ) βσs2 . σv + σ2 23

(B42)

Substitute the expression of λ, α1 and α2 into the above equation, we have    s 1 1 σu2 Eh [π] = Eh {Eh [(˜ v − p˜)y|˜ s]} = m − 1− σv2 . 2 2m σv2 + σ2

(B43)

Proof of Proposition 4.3. [Cov(˜ v , p˜)]2 V ar(˜ p) (α1 + β1 )2 σv4 =σv2 − (α1 + β1 )2 (σv2 + σ2 ) + σu2 (α1 + β1 )2 σv2 σ2 + σv2 σu2 = . (α1 + β1 )2 (σv2 + σ2 ) + σu2

Σ1 = V ar(˜ v |˜ p) =V ar(˜ v) −

(B44)

Substitute the expressions of α1 and α2 into the above equation, we have σv2 (σ2 + σv2 + σ2 ) 2(σv2 + σ2 ) 1 σ2σ2 = ( 2 v  2 + σv2 ). 2 σv + σ

Σ1 = V ar(˜ v |˜ p) =

So I=

1 σv4 . 2 σv2 + σ2

Proof of Proposition 5.1. From Eqs.(4.5) and (3.5), we have √ α1 m+1 =p , 0 α1 4m2 (2 − m)

(B45)

(B46)

(C1)

Let f (m) = m+1−4m2 (2−m). It is easy to know that f (0) = 1 > 0, f (1) = −2 < 0, f (2) = 3 > 0, so this is only one root in (1, 2) for f (m) = 0, and the root about 1.7808. So we can have when 1 < m < 1.7808,

α1 α01

< 1, and when 1.7808 < m < 2,

α1 α01

> 1.

Similarly from Eqs.(4.6) and (3.6), we get p (m + 1)(2 − m) β1 = , 0 β1 2m

(C2)

Let g(m) = (m + 1)(2 − m) − 2m. we can get when 1 < m < 2, g(m) < 0 is permanent. Hence, we get the conclusion: when 1 < m < 2,

β1 β10

< 1.

Proof of Proposition 5.2. From Eqs.(4.7) and (3.7), we get λ 3 = p , 0 λ 2 (m + 1)(2 − m)

(C3)

The proof method is similar to the proposition 5.1, so we can easy to know when 1 < m < 2, λ λ0

> 1. Proof of Proposition 5.3. Comparing the ex-ante expected profit of rational insider, we can

get Er [π] > Er0 [π], 24

(C4)

because from Eqs.(4.11) and (3.13), we have

Let

√ Er [π] 3 m+1 =p , Er0 [π] 16m2 (2 − m)3

(C5)

√ 3 m+1 p h(m) = , 16m2 (2 − m)3

(C6)

when 1 < m < 2, h(m) is an increasing function of m, so we can easy to get √ √ 3 m+1 3 2 h(m)min = p = > 1, 4 16m2 (2 − m)3 hence, when 1 < m < 2,

Er [π] Er0 [π]

(C7)

> 1.

Comparing the ex-ante expected profit of heuristic insider, from Eqs.(4.12) and (3.14), we have p 3 (m + 1)(2 − m) Eh [π] (C8) = , Eh0 [π] 4m Let p 3 (m + 1)(2 − m) (C9) p(m) = = 1, 4m It is easy to know that one root in (1, 2) for p(m) = 1, and the root about 1.04. So we can have when 1 < m < 1.04,

Eh [π] Eh0 [π]

< 1, and when 1.04 < m < 2,

Eh [π] Eh0 [π]

> 1.

Proof of Proposition 5.4. From Eqs. (4.18) and (3.19), we have I 3 = , I0 2(m + 1) let q(m) =

(C10)

3 = 1, 2(m + 1)

we can get m = 12 , so when 1 < m < 2, it is easy to know that

(C11) I I0

< 1.

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