Overconfidence on public information

Economics Letters 112 (2011) 239–242

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Overconfidence on public information Deqing Zhou ∗ Department of Business Statistics and Econometrics, Guanghua School of Management, Peking University, 100871, China Center for Statistical Science, Peking University, China

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info

Article history: Received 5 January 2010 Received in revised form 28 March 2011 Accepted 20 April 2011 Available online 15 May 2011

abstract This work sets the market maker as overconfident and shows that this will lead to a higher informed trading intensity, a more efficient market, a larger informed profit and a lower adverse selection. © 2011 Elsevier B.V. All rights reserved.

JEL classification: C72 D82 G14 Keywords: Insider trading Public information Overconfidence

1. Introduction Theoretical models have predicted that traders will trade more when they are overconfident (Wang, 1998 and Glaser and Weber, 2007) while they trade less when they are rational but faced with an overconfident opponent due to the loss of the ‘‘first-mover advantage’’ (Benos, 1998 and Kyle and Wang, 1997). However, we find that when the traders’ overconfident opponent is the market maker, the rational insider would also like to trade more aggressively to take advantage of ‘‘mispricing’’ opportunities made by the market maker. Our model is based on the pioneering model proposed by Kyle (1985) investigating the problem of how private information is incorporated into public price in a semi-strong efficient speculative market. We extend it with the public information in a manner similar to that of Luo (2001), and with the heterogeneous beliefs setting. Strictly speaking, the market maker has a knowledge of a public signal concerning the liquidity value, with a precision overestimated by her. The insider also knows this public information, but unlike the market maker, he can estimate its precision correctly. The market maker sets the price conditional on information

∗ Corresponding address: Department of Business Statistics and Econometrics, Guanghua School of Management, Peking University, Beijing 100871, China. Tel.: +86 13811313349. E-mail address: [email protected]. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.04.009

available (both the order imbalance and the public signal) and conditional on her irrational belief that the public signal has a higher precision than it actually has. We also find that, to maximize his profit, the insider will trade strategically on private information and trade manipulatively on public information. More interestingly, unlike in models dealing with a monopolistic overconfident insider (or insiders with the same overconfidence degree and information) with intense trading motivated by his own optimistic belief, the rational insider in our model will benefit from his aggressive trading. Moreover, the market efficiency is improved and the adverse selection is indicated as lower compared to the case with a rational market maker. This work is structured as follows. In Section 2, the informed trading model with a monopolistic insider and an overconfident market maker is presented. In Section 3, the linear equilibrium is examined and the impacts of overconfidence are studied. Finally, Section 4 concludes. 2. The model Consider the natural extension of the Kyle (1985) one-period model in which there is one risky asset with a liquidation value  v , normally distributed with mean 0 and variance σv2 .  v is the private information and its realization is uniquely known by the insider.  s =  v + ϵ is the public information known by both the insider and the market maker, where  ϵ is normally distributed with mean 0 and variance σϵ2 and is independent with  v . σϵ−1 is the

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D. Zhou / Economics Letters 112 (2011) 239–242

precision of the public information, and the insider can estimate it correctly, while the overconfident market maker overestimates it as (kσϵ )−1 (k ≤ 1). In other words, the insider has the rational belief:  s = v + ϵ , while the market maker has the overconfident belief: s = v + k ϵ with (k ≤ 1). The single insider places an order to buy  x (sell | x| if it is negative) shares of risky asset at the start of auction. The market maker receives this order along with those of noise traders whose exogenously generated total demand  u is normally distributed with mean 0 and variance σu2 . We assume that  u is independent of  v and  ϵ . The total order  y =  x + u, observed by the market maker, together with the public information  s, is provided as the information available for determining the price. The insider’s trading strategy and the market maker’s pricing rule are assumed as the real-valued functions, such that, given the initial price normalized to zero,

 x = X ( s, v),

P = P ( s, y).

Use π to denote the insider’s profit, Ek to denote the expectation conditional on the belief  s =  v + k ϵ , and E (or E1 ) to denote the expectation conditional on the rational belief  s = v + ϵ . The concept of equilibrium is similar to that of Kyle (1985). Definition. An equilibrium consists of the insider’s trading strategy and the market price pair ( x, P ), such that the following two conditions hold: 1. Profit maximization: For any insider’s alternative trading strategy  x′ , E (π( x, P )| s, v) ≥ E (π( x′ , P )| s, v). 2. Market semi-strong efficiency under the market maker’s belief :

From Eqs. (5) and (6) we can see that β > 0, α < 0, which means that the insider puts a positive weight on private information but a negative weight on public information. This manipulative trading on public information provides the insider with more camouflage for trading on private information, and this result may offer a rationale for the trading contrary to public news. The nonzero expectation of order imbalance1 depending strongly on the heterogeneous prior beliefs is more likely to produce an unusually high trading volume widely documented by theoretical and empirical papers such as Glaser and Weber (2007) and Covrig and Ng (2004). As we turn to the overconfidence effects on pricing rule and trading strategy, we find from Proposition 1 the following results: Proposition 2. In the equilibrium, λ is an increasing function of k while γ is a decreasing function of k, which means that the overconfident market maker puts a larger weight on public signal and a smaller weight on order imbalance than she should rationally do.2 Moreover, β and |α| are decreasing functions of k, which means that the overconfidence of the market maker will motivate the aggressive trading of the insider on both the private and public information. As to the overconfidence effect on the market efficiency, we have: Proposition 3. The private information unrevealed after the auction satisfies

Σ = var( v|P , s) =

σv2 σϵ2 . σv2 /k + σv2 + 2σϵ2

(7)

Therefore, Σ is an increasing function of k, that is, an overconfident market maker will create a more efficient market. Proof. See Appendix. The overconfidence of the market maker implies an increased insider profit, stated as follows:

P = Ek ( v| s, y).

Proposition 4. The ex ante expected profit of insider satisfies: 3. The linear equilibrium We focus on the linear Nash equilibrium to avoid the technical inconvenience due to the problems associated with higher-order expectation (forecast the forecast of others and so on), and we find the equilibrium satisfying the following:

 x = β v + α s,

(1) (2)

in which

λ=

kσv σϵ  , 2σu σv2 + k2 σϵ2

σv , σv2 + k2 σϵ2  σu σv2 + k2 σϵ2 β= , kσv σϵ σv σu  α=− . kσϵ σv2 + k2 σϵ2 γ =

σv σϵ σu (k4 σϵ2 + σv2 ) , 2k(σv2 + k2 σϵ2 )3/2

(8)

where π (k) is a decreasing function of k when k ≤ 1. This means that the overconfidence of the market maker will enhance the profit of the insider. Proof. See Appendix.

Proposition 1. The unique linear Nash equilibrium is given by P = λ y + r s,

π (k) = E [( v − P ) x] =

On the other hand, noise traders gain an ex ante expectation of profit E [( v − P ) u] = −λσu2 , decreasing with k. This means that noise traders will lose less money when trading at a price set by the overconfident market maker. Therefore, in the Bertrand game with competitive rational and overconfident market makers, the overconfident one will survive.

(3) 4. Conclusion

2

(4)

(5) (6)

The expectation of an insider trading quantity conditional on the public information is zero if and only if k = 1. Proof. See Appendix.

We investigate Kyle’s (1985) extended model with the setting of private and public information and heterogeneous prior beliefs. We find that the overconfidence of the market maker intensifies the aggressiveness of informed trading on both the private and public information, leading to an increased insider profit, and a more efficient and more stable market.

1 Note that the expectation of the liquidity trading quantity is always zero. 2 When the market maker is overconfident, the decreased λ also implies: (i) a more stable market since λ−1 can be explained as the market depth; (ii) a lower adverse selection which is measured by λ.

D. Zhou / Economics Letters 112 (2011) 239–242

Appendix

and

Firstly, a well-known regression result (Lemma 1) and another result (Lemma 2) implied by it are prepared for later use. Lemma 1. Suppose that X1 and X2 are two normal random vectors, satisfying

 µ1 , ∼ N (µ, Σ ) with µ = µ2

 



X1 X2

241

Σ11 Σ= Σ21



 Σ12 . Σ22

= Ek ( v − ck s) 2 = E ( v − ck ( v + k ϵ))2 2 2 = (1 − ck ) σv + k2 ck2 σϵ2 =

Then the random vector X1 conditional on X2 , denoted as X1 |X2 , is also a random vector, satisfying X1 |X2 ∼ N (µ1 + Σ12 Σ22 (X2 − µ2 ), Σ11 − Σ12 Σ22 Σ21 ). −1

H = Ek ( v − Ek ( v| s))2

−1

k2 σv2 σϵ2

σv2 + k2 σϵ2

.

(14)

Comparing the price expression Eq. (12) with Eq. (1), we have

λ=

βH , β 2 H + σu2

(15)

γ = λ(−α − β ck ) + ck .

In particular,

(16)

Substituting Eqs. (10) and (11) into Eq. (16) yields

−1 E (X1 |X2 ) = µ1 + Σ12 Σ22 (X2 − µ2 ).

γ = ck .

(17)

Applying Lemma 1 in a special case, we have:

Substituting Eq. (17) into Eq. (11), we have

Lemma 2. If (x, y, z ) is a normal random vector, satisfying cov(x, z ) = 0, cov(y, z ) = 0, then

α=−

E (x|y, z ) = E (x|y).

From Eq. (15), we can see that

Proof of Proposition 1. Suppose that for constants λ, γ , β, α , the linear rule P and the Gaussian strategy x are given by

H =

P = λ y + γ s,

The insider’s profit conditional on his information available and his rational belief is E [( v − P ) x| v, s] = E [( v − λ y − γ s) x| v, s] (9)

For maximizing this profit expression over  x, the first-order condition yields

 x = β v + α s,

2λ

,

γ α=− . 2λ

(10) (11)

The second-order condition yields λ > 0, satisfied by Eq. (3). By the semi-strong efficiency and lemmas, noting the facts that  v − Ek ( v| s) is independent with  s under the belief  s = v + k ϵ and that σ {β v + α s + u, s} = σ {β( v − Ek ( v| s)) +  u, s}, we have Ek [ v − Ek ( v| s) + Ek ( v| s)|β v + α s + u, s]

kσv σϵ  . 2σu σv2 + k2 σϵ2

(21)

Substituting Eq. (21) into Eq. (10), the intensity trading on private information satisfies

 σu σv2 + k2 σϵ2 β= . kσv σϵ

(22)

α=−

ck 2λ

=−

σv σu kσϵ



σv2 + k2 σϵ2

.

(23)

Therefore, Eqs. (1)–(6) have been proved. Conditional on the public information  s, the insider trading quantity is expected to be E ( x| s) = E (β v + α s| s) = β c1 s + α s

σv σu = kσϵ



 σv2 + k2 σϵ2 1  s. − σv2 + σϵ2 σv2 + σϵ2

Proof of Proposition 3. By the definition of Σ and the lemmas, we have

v| s)|β( v − Ek ( v| s)) +  u, s] + Ek ( v| s) Ek [ v − Ek (

Ek [ v − Ek ( v| s)|β( v − Ek ( v| s)) +  u] + Ek ( v| s) βH = 2 [β( v − Ek ( v| s)) +  u] + ck s β H + σu2

(12)

in which

σ2 ck = 2 v 2 2 , σv + k σϵ

(20)

Therefore, E ( x| s) = 0 if and only if k = 1.

Ek [ v − Ek ( v| s)|β v + α s + u, s] + Ek ( v| s)

[ ] βH βH  = 2 y + (−α − β c ) + c s, k k  β H + σu2 β 2 H + σu2

(19)

Combining Eqs. (14) and (20) yields the liquidity parameter as

P = Ek [ v| y, s]

= = = =

λσu2 . β − λβ 2

Substituting Eqs. (17) and (21) into Eq. (11), the intensity trading on public information satisfies

where 1

(18)

H = 4λ2 σu2 .

λ=

= E [( v − λ( x + u) − γ s) x| v, s]

β=

.

Substituting Eq. (10) into Eq. (19), we have

 x = β v + α s.

= ( v − γ s) x − λ x2 .

ck 2λ

Σ = = = = =

= (13)

var( v|P , s) var ( v − E ( v| s)|λ y + r s, s)

var ( v − E ( v| s)|β v + α s + u, s)

var ( v − E ( v| s)|β( v − E ( v| s)) +  u, s) var ( v − E ( v| s)|β( v − E ( v| s)) +  u)

β

E ( v − E ( v| s))2 σu2 2E

in which

( v − E ( v| s))2 + σu2

,

(24)

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D. Zhou / Economics Letters 112 (2011) 239–242

E ( v − E ( v| s))2 = E ( v − c1 s)2 =

σv2 σϵ2 . σv2 + σϵ2

(25)

Substituting Eqs. (25) and (5) into Eq. (24), we have

σv σϵ . σv /k + σv2 + 2σϵ2 2

Σ=

2

2

E [( v − P ) x]

= EE [( v − P ) x| v, s] = E [( v − γ s)(β v + α s) − λ(β v + α s)2 ] (by Eq. (9)) 2 2 = (β + α − γ β − γ α − λβ − λα − 2λβα)σv2 + (−γ α − λα 2 )σϵ2   1 1 1 2 1 1 2 1 1 − ck − ck + ck − − ck + ck σv2 = 2λ 2λ 2λ 2λ 4λ 4λ 2λ   1 2 1 2 ck − ck σϵ2 + 2λ 4λ 1 4λ

(1 − ck )2 σv2 +

1 4λ

σv σϵ σu (k4 σϵ2 + σv2 ) . 2k(σv2 + k2 σϵ2 )3/2

Hence, Eq. (8) has been proved. Further, a complex calculation concerning the derivative of insider’s profit indicates that

π ′ (k) =

Proof of Proposition 4. A calculation concerning the expectation of insider’s profit shows that

=

=

ck2 σϵ2

σv3 σϵ σu [(3k4 − 4k2 )σϵ2 − σv2 ] 5

2k2 (σv2 + k2 σϵ2 ) 2

.

(26)

We find from Eq. (26) that π ′ (k) < 0 when k ≤ 1. References Benos, A., 1998. Aggressiveness and survival of overconfident traders. Journal of Financial Markets 1, 353–383. Covrig, V., Ng, L., 2004. Volume autocorrelation, information, and investor trading. Journal of Banking & Finance 28, 2155–2174. Glaser, M., Weber, M., 2007. Overconfidence and trading volume. Geneva Risk and Insurance Review 32, 1–36. Kyle, A., 1985. Continuous auctions and insider trading. Econometrica 53, 1315–1336. Kyle, A., Wang, F., 1997. Speculation duopoly with agreement to disagree: can overconfidence survive the market test? Journal of Finance 52, 2073–2090. Luo, S., 2001. The impact of public information on insider trading. Economics Letters 70, 59–81. Wang, F., 1998. Strategic trading, asymmetric information and heterogeneous prior beliefs. Journal of Financial Markets 1, 321–352.