Can public information promote market stability?

Economics Letters 143 (2016) 103–106

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Can public information promote market stability? Binbin Chen a,∗ , Shancun Liu a , Qiang Zhang b a

School of Economics and Management, Beihang University, Beijing, 100191, China

b

School of Economics and Management, Beijing University of Chemical Technology, Beijing, 100029, China

highlights • • • •

With lower public information precision, multiple equilibria arise. Traders may reverse trading in multiple equilibria. There is a unique equilibrium if public information precision is sufficiently large. The reverse trading does not occur in unique equilibrium.

article

info

Article history: Received 3 December 2015 Received in revised form 29 March 2016 Accepted 8 April 2016 Available online 12 April 2016

abstract We study the effect of public information revealing part of underlying fundamentals on market stability. It shows that accurate public information reduces the uncertainty faced by informed traders and increases their responsiveness to private information and expected volume. The reverse trading and multiple equilibria arise under lower public information precision and they disappear when public information precision increases sufficiently. © 2016 Elsevier B.V. All rights reserved.

JEL classification: D40 D82 G10 Keywords: Information disclosure Public information Multiple equilibria Reverse trading

1. Introduction Does public information disclosure promote the market stability and thus enhance the social welfare? Morris and Shin (2002) find that public information can enhance social welfare when private information precision is very low. Cornand and Heinemann (2008) show that social welfare will be improved when part of traders make use of public information. By studying the monopolistic competition model with heterogeneous information, Hellwig (2005) finds that public information can reduce price deviation and improve social welfare. He points out that the state of traders abandoning private information is optimal. Colombo and Femminis (2014) extend the ‘‘beauty content model’’ by introducing the

∗ Correspondence to: Xueyuan Road 37#, Haidian distinct, Beijing 100191, China. Tel.: +86 13810162027. E-mail address: [email protected] (B. Chen). http://dx.doi.org/10.1016/j.econlet.2016.04.008 0165-1765/© 2016 Elsevier B.V. All rights reserved.

upper bound of public information precision and they also support the transparent market system. Scholars provide controversial evidence about the use of accurate public information. Amato et al. (2002) show that listed companies should be cautious in disclosing public information. Amador and Weill (2010) find that price informativeness and the welfare of traders decrease with public information precision. James and Lawler (2011) oppose the transparent market system because high public information precision can cause a consistent behavior of traders which in turn will reduce their utilities. Pancs (2014) evaluates the influence of public information based on the model of Glosten and Milgrom (1985) and finds that high public information precision deteriorates market quality. Under the setting of information asymmetry and the presence of short-term traders, researchers have conducted in-depth research on this topic. Chen et al. (2014) propose an information asymmetry model with short-term traders. They find that lower price informativeness exists in the market with low public

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information precision. Cespa and Vives (2015) also introduce short-term traders and they show that the retrospective inference is very strong and there exists an unstable equilibrium with high liquidity. Price informativeness increases when public information is overly precise compared to private information. We extend the recent literatures on public information issues by introducing asset liquidation value based on the assumption of Bernhardt and Taub (2008), who view the factors affecting asset liquidation value as many uncorrected multi-underlying fundamentals and assume that each informed trader has access to private information about part of them. However, in their research, the influence of other underlying fundamentals is not discussed in detail. We assume that asset liquidation value is a linear function, consisting of two underlying fundamentals, from one of which informed traders receive private information and with other listed company disclose public information. Then informed traders have incentives to reverse trading, and thus multiple equilibria occur. We find that the probabilities of reverse trading and multiple equilibria decrease with the increase of public information precision. 2. The model

is a linear function of β1 vI + u1 and p2 is a linear function of 1β2 vI + u2 and sP . Then β1 vI + u1 is a sufficient statistic of p1 and {β1 vI + u1 , 1β2 vI + u2 , γ sP } is sufficient statistic of {p1 , p2 } in the estimation of v , where 1β2 = β2 − β1 . Thus G1 and G2 are observationally equivalent to {si1 , β1 vI + u1 } and {˜si2 , β1 vI + u1 , 1β2 vI + u2 , sP } respectively. 2.2. Model solution In our paper, vO (vI ) is the uncertainty faced by informed traders from the view of private (public) information. The uncertainty disappears when the precision of vO (vI ) is infinity. Cespa and Vives (2012) define the uncertainty of risk asset liquidation value as ‘‘residual uncertainty’’. And let vO as ‘‘residual uncertainty’’ which informed traders faced in this paper. Eqs. (2) and (3) yield the equilibrium shown in Proposition 1. Proposition 1. There is a unique linear equilibrium in period 1. The demand schedule of informed traders is: xi1 = β1 si1 −

2.1. Model assumptions

p1

λ1

,

(4)

and equilibrium price is: Consider a two-period market with a risky asset whose liquidation value is v . v consists of underlying fundamentals vI and vO (v = vI + vO ), where vI ∼ N (0, τI−1 ) and vO ∼ N (0, τO−1 ). pt refers to asset price in period t (t = 1, 2). And the asset is liquidated periodically. There are noise traders and a continuum of informed traders indexed in the interval [0, 1] in market. In period t, the net demand −1 of noise traders is ut , where ut ∼ N (0, τut ). One informed trader i receives private information sit = vI + εit about the underlying 1 fundamental vI , where εit ∼ N (0, τε−t 1 ) and 0 εit di = 0. It can be proved that s˜i2 = (τε1 + τε2 )−1 (τε1 si1 + τε2 si2 ) is a sufficient statistic of {si1 , si2 } in the estimation of v . Furthermore, informed traders also receive public information sP = vO + η disclosed by listed companies in period 2, where η ∼ N (0, τP−1 ). Their demand schedule is xi1 = X (si1 , p1 ) in period 1 and xi2 = X (si1 , si2 , sP , p2 ) in period 2, respectively. All variables in set {vI , vO , εit , εP , ut } are independent. Informed traders follow CARA utility function, U (πi ) = − exp(−ρ −1 πi ), where ρ is common risk-tolerance coefficient and πit = (v − pt )xit is return. Maximization condition expected of utility function is equal to max E [(v − pt )|Gt ]xit − xit

1 2

ρ Var[(v − pt )|Gt ]x2it .

(1)

p1 = λ1 (β1 vI + u1 ), where β1 =

(5)

ρ var[vI |G1 ]τε1 , var[v|G1 ]

λ1 = ρ −1 var[v|G1 ] + β1 τu1 var[vI |G1 ], var[vI |G1 ] = (τI +β τ +τε1 )−1 and var[v|G1 ] = var[vI |G1 ]+τO−1 . 2 1 u1

Informed traders’ responsiveness to private information is β1 which is affected by ρ , τO , τε1 and τI . Taking the partial derivative of β1 with respect to other variables yields ∂β1 /∂ρ > 0, ∂β1 /∂τO > 0, ∂β1 /∂τε1 > 0 and ∂β1 /∂τI < 0. In other words, informed traders’ responsiveness to private information increases with common risk-tolerance coefficient, the precision of vO and private information and it decrease with the precision of vI . Meanwhile, it can be proved that β1 ≤ min{ρτO , ρτε1 }. We  denote market depth as  λ1 , consisting of two parts: ρ −1 (τI + β12 τu1 + τε1 )−1 + τO−1 which captures inventory risk premium due to traders’ risk aversion, and (τI +β12 τu1 +τε1 )−1 β1 τu1 which captures adverse selection risk premium faced by informed traders. Inventory risk premium equals zero when ρ approaches infinite, since traders are risk-neutral and thus require no compensation for their inventories. Adverse selection risk premium derives from the presence of informed traders in the market. In period 2, we obtain the equilibrium shown in Proposition 2.

Gt represents information set of informed traders where G1 = {si1 , p1 } and G2 = {si1 , si2 , p1 , p2 , sP }. The optimal solution of Eq. (1) is

Proposition 2. There is always the equilibrium in period 2. The demand schedule of informed traders is:

ρ E [(v − pt )|Gt ] . xit = Var[(v − pt )|Gt ]

xi2 = β2 s˜i2 + γ sP −

(2)

According to market clearing mechanism, the total position between informed and noise traders is equal to zero. Then

λ2

+

p1 λ∗2 − λ2

λ1

xit di + θt = 0.

(3)

0

Considering normal distribution theory, we prove that xi1 and xi2 are linear functions of private information, asset price and public information, i.e., xi1 = β1 si1 + f (p1 ) and xi2 = β2 s˜i2 + γ sP + h(p1 , p2 ), where f (p1 ) and h(p1 , p2 ) are the linear functions of p1 and {p1 , p2 }, respectively. According to Eqs. (2) and (3), p1

λ2

,

(6)

and equilibrium price is: p2 = λ2 (1β2 vI + u2 + γ sP ) +

1



p2

p1 λ∗2

λ1

,

ρ var[vI |G2 ](τε1 +τε2 ) var[v|G2 ] τ , γ = (τ +τP )λ , λ2 = var[v|G2 ] ρ O P 2 G2 ] 1β2 τu2 var[vI |G2 ], λ∗2 = var[v| + β τ var [v | G ] , var [v | G2 ] 1 u1 I 2 I  −ρ1 2 τI + t =1 (τεt + 1βt2 τut ) , 1β2 = β2 − β1 , var[v|G2 ]

where β2 =

var[vI |G2 ] + (τO + τP )

−1

.

(7)

+ = =

B. Chen et al. / Economics Letters 143 (2016) 103–106

105

Compared with the counterpart in period 1, the responsiveness to private information in period 2 includes public information, resulting in a decrease in ‘‘residual uncertainty’’ faced by informed traders and adjustment of their responsiveness to private information conversely. We can also obtain β2 ≤ min{ρ(τε1 + τε2 ), ρ(τO + τP )}. The reduction of ‘‘residual uncertainty’’ affects market liquidity as well. In the expression of λ2 , inventory risk premium is ρ −1 var[v|G2 ] and it can be arranged as follow:

−1  . (8) ρ −1 (τO + τP )−1 + ρ −1 τI + τε1 + τε2 + β12 τu1 + 1β22 τu2 The first (second) term in Eq. (8) represents risk premium which comes from uncertainty of vO (vI ). The first term tends to 0 as the precision of public information increases, while the second term is indirectly influenced by public information through the net change in responsiveness to private information. The effect for adverse selection risk premium will be discussed in Section 3.

Fig. 1. Market liquidity with public information precision for τε2 = 1, 5, 9.

3. Impact of public information on market equilibrium 3.1. Unique equilibrium or multiple equilibria? Cespa and Vives (2012) show that multiple equilibria arise under the existence of ‘‘residual uncertainty’’ in period 2. In our paper, β2 is the solution of

β23 τu2 − 2β1 τu2 β22 + (β12 τu2 + β1−1 ρτO τε1 + τP + τε2 )β2 − ρ(τO + τP )(τε1 + τε2 ) = 0.

relationship between market liquidity and public information precision in the equilibrium of β2max in this section. It can be easily obtained that inventory risk premium and public information precision are negatively correlated. However, the effect of public information on adverse selection risk premium is ambiguous in period 2. Let ASR = 1β2 τu2 var[vI |G2 ] represent adverse selection risk premium and rearrange ASR as

(9)

And triple equilibria arise in period 2 if there are three real solutions to this cubic equation. Then we can obtain the necessary and sufficient condition for triple equilibria as follows:

√

−2τu2 ∆3 < 2β13 τu2 + 18β1 (τP + τε2 )

√ (10) + 18ρτO τε1 − 27ρ (τO + τP ) (τε1 + τε2 ) < 2τu2 ∆3   τ + τ ρτ τ P ε2 O ε1 ∆ = β12 − 3 + > 0. (11) β1 τu2 τu2 Let β2min , β2mid and β2max denote the real solution of Eq. (9) in

order and we can prove that √ they are respectively in the inter√ √ 2β − ∆ ), ( 2β1 −3 ∆ , 2β1 +3 ∆ ) and (β1 , +∞). The net responval (0, 1 3 siveness to private information of informed traders is less than zero in the equilibrium of β2min and β2mid . In other words, informed traders adopt reverse trading strategy and the equilibrium price is negatively correlated with vI . In the equilibrium of β2max , the net responsiveness to private information is large than zero and reverse trading disappears. We can conclude following Corollary 1 from the fact that β2min and β2mid disappear when the precision of public information increases sufficiently.

Corollary 1. Informed traders have incentives to reverse trading in the multiple equilibria. But reverse trading and multiple equilibria disappear when the precision of public information increases sufficiently. Along with its increasing, there is at least a unique equilibrium in market. The real solution of Eq. (9) is only β in unique equilibrium according to Corollary 1. By analyzing the derivation of β2max with respect to τP , we can obtain Corollary 2 as follows. max 2

Corollary 2. In the equilibrium of β2max , as τP increases public information influences the informed traders through two aspects: (1) enhancing their responsiveness to private information, (2) enhancing the expectation of their positions. 3.2. Market liquidity Since the equilibrium of β2min and β2mid disappear when public information is larger than the cutoff value, we will analyze the

 ASR =

τI + τε1 + τε2 + β12 τu1 + 1β2 1β2 τu2

 −1

.

(12)

According to Eq. (12) and combining with Corollary 2, adverse selection risk premium will increasewith the disclosure of  public



information when 1β2 is located in

0,

τI +τε1 +τε2 +β12 τu1 τu2

 it will decrease when 1β2 is located in

, while

τI +τε1 +τε2 +β12 τu1 , +∞ τu2

 .

Then we can depict the total impact of public information on market liquidity by numerical simulations. Fig. 1 shows the change of market liquidity along with public information precision with parameter values ρ = 10, β1 = 8, τε1 = 5, τO = 5 and τu2 = 5 in respective. Specifically, the bold, dashed and dotted are associated with τε2 = 1, τε2 = 5 and τε2 = 9. The nexus between τP and market liquidity is negatively correlated when public as well as private information precision is very low, since the increment of adverse selection risk premium is greater than the decrement of inventory risk premium along with the increase of public information precision (see the decline part in dotted line). As the growth of adverse selection risk premium marginally decreases, market liquidity increases with public information precession (see the raise part in dashed line). When private information is extremely accurate, with   market liquidity increases



τP because 1β2 is always in

τI +τε1 +τε2 +β12 τu1 , +∞ τu2

and both

inventory and adverse selection risk premium are negatively correlated with public information (the bold line). 4. Conclusion Our paper examines the impact of public information on the market quality and equilibrium when liquidation value of risky asset includes multiple underlying fundamentals. The uncertainty of underlying fundamentals from which informed traders cannot get any information makes their behavior become more cautious. It also strengthens their motivation to adopt reverse trading and

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thus multiple equilibria arise. But reverse trading and multiple equilibria disappear when public information precision is accurate because it reduces the ‘‘residual uncertainty’’ faced by informed traders. In perfectly competitive market, equilibrium price is only decided by the average of private information, noise trading and public information and it is unaffected by private information of a single trader. Higher public information precision will increase the weight of public information in determining equilibrium price, the responsiveness to private information and expected volume. There are two effects on market liquidity with the disclosure of public information: (1) strengthening adverse selection risk premium; (2) weakening inventory risk premium. A higher (lower) private information precision leads to the adverse selection risk premium lower (higher) than inventory risk premium and consequently market liquidity will increase (decrease) with the accuracy of public information. Therefore, we conclude that accurate public information always reduces uncertainty faced by informed traders, strengthens market stability and enhances market liquidity. Acknowledgment This work is supported in part by National Natural Science Foundation of China Grant No. 71371023, No. 71371024 and No. 71171146.

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