Multi-period sentiment asset pricing model with information

International Review of Economics and Finance 34 (2014) 118–130

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Multi-period sentiment asset pricing model with information Jinfang Li ⁎ School of Business, Luoyang Normal University, Luoyang 410300, China School of Economics and Commerce, South China University of Technology, Guangzhou 510006, China

a r t i c l e

i n f o

Article history: Received 2 March 2014 Received in revised form 16 July 2014 Accepted 16 July 2014 Available online 24 July 2014 JEL classifications: G12 G14

a b s t r a c t This paper presents a multi-period trading sentiment asset pricing model under asymmetric information. In the model, the rational investor trades on information so that the information is gradually incorporated into prices, whereas the sentiment investor trades on sentiment as if it were information. Moreover, the speed of information incorporated into price becomes faster and faster, while the speed of sentiment factored into price gets slower and slower over time. We find that the existence of sentiment investor makes financial market possible, but decreases market efficiency. The model also offers a partial explanation to the financial anomalies of short-run momentum and long-term reversal. © 2014 Elsevier Inc. All rights reserved.

Keywords: Investor sentiment Multi-period trading equilibrium Momentum effect

1. Introduction Traditional asset pricing theory suggests that economists can safely ignore individual irrational trades, at least their impact on equilibrium at the aggregate level (Fama, 1965; Friedman, 1953). However, growing behavioral asset pricing theory argues that individual irrational trades tend to be correlated, and so irrational investors would create risk and their own space (De Long, Shleifer, Summers, & Waldmann, 1990, 1991). According to the irrational form, investors may be affected by noise (see, e.g., Grossman & Stiglitz, 1980; Kyle, 1985; Black, 1986; De Long et al., 1990; Mendel & Shleifer, 2012; Peri, Vandone, & Baldi, 2013), cognitive biases (e.g., Barberis, Shleifer, & Vishny, 1998; Daniel, Hirshleifer, & Subrahmanyam, 1998; Hong & Stein, 1999; Yan, 2010), or investor sentiment in the actual financial market. The shortcomings of both the noise pricing model and the psychological biases pricing model are that the noise and psychological biases are difficult to be measured, and thus can't be empirically testified in the realistic security market. Nevertheless, investor sentiment could be quantitatively measured; furthermore, the corresponding empirical tests can be carried out (Baker & Wurgler, 2006, 2007). In the recent years, many empirical studies have shown that investor sentiment has a systematic and significant impact on the financial asset prices, and that financial asset pricing is much higher with optimistic sentiment and vice versa (e.g., Baker & Wurgler, 2006, 2007; Baker, Wurgler, & Yuan, 2012; Brown & Cliff, 2004, 2005; Kim, Ryu, & Seo, 2014; Kumar & Lee, 2006; Lee, Jiang, & Indro, 2002; Seybert & Yang, 2012; Stambaugh, Yu, & Yuan, 2012; Yang & Zhang, 2014; Yu & Yuan, 2011; Zhu, 2013). Specifically, Corredor, Ferrer, and Santamaria (2013) analyze the investor sentiment effect in four key European stock markets, whose results show that sentiment has a significant influence on returns, varying in intensity across markets. Chan (2014) finds that the behavior of bullish retail investors can lead to overpricing of the U.S. IPO shares and price reversals in the long run. In addition, the related empirical results are supported by some financial experiments (see, e.g., Ganzach, 2000; Statman, Fisher, & Anginer, 2008; Kempf, ⁎ Tel.: +86 135 604 778 40. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.iref.2014.07.006 1059-0560/© 2014 Elsevier Inc. All rights reserved.

J. Li / International Review of Economics and Finance 34 (2014) 118–130

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Merkle, & Niessen, 2013). At the same time, some static sentiment asset pricing models have been proposed to emphasize the systematic role of investor sentiment in asset prices (Yang & Li, 2013; Yang, Xie, & Yan, 2012; Yang & Yan, 2011; Yang & Zhang, 2013a). Furthermore, Yang and Zhang (2013b) present a two-period trading asset pricing model with heterogeneous sentiments whose result shows that the equilibrium stock price is the wealth-share-weighted average of the stock prices. Yang and Li (2014) propose a twoperiod trading sentiment asset pricing model with information and find that sentiment trading quantity not only increases the market liquidity, but also causes the asset price overreaction if the intensity of sentiment demand is more than a constant value. Cen, Lu, and Yang (2013) consider a dynamic multiasset model and find that the breadth–return relationship can be either positive or negative depending on the relative offsetting forces of disagreement and sentiment. Much different from the behavior of the investors described in the static sentiment asset pricing model and the two-period trading model, investors often make multiple transactions in the realistic financial market. Furthermore, some rational investors who make the current trading strategy would take into account explicitly future trading opportunities, while some sentiment investors would change their sentiment over time. Ultimately, the equilibrium prices are multi-period game results among different types of investors. Therefore, multi-period trading equilibrium is closer to the realistic capital market and the setting of dynamic sentiment asset pricing model could better explain the characteristics of asset price movements. Moreover, the previous sentiment asset pricing models rarely involve some important factors such as fundamental information and the initial price. Where this paper differs from much of the previous literature is in analyzing the equilibrium prices in a dynamic setting. Based on the framework of Kyle (1985), we present a multi-period trading sentiment asset pricing model with information, in which there are both sophisticated rational investors and naive sentiment investors. In our model, sentiment is contrasted with information. The rational investor possesses valuable information and trades on information in the usual way, while the sentiment investor is vulnerable to sentiment shocks and trades on sentiment as if it were information. The features of our model, which distinguish itself from the previous sentiment asset pricing models, are as follows. First, there is a fundamental information release realized in the first period. Then the rational investor observes valuable information and trades in such a way that the fundamental information is incorporated into prices gradually. Second, the uninformed sentiment investor influenced by individual sentiment trades on his own sentiment so that the investor sentiment is also factored into prices, where the investor sentiment has a certain correlation with the fundamental information. The closer the linear correlation between sentiment and information is, the faster information gets factored into prices. Finally, we give an analytical solution to the sentiment equilibrium prices and then describe a dynamic price path anchoring to the initial price. Moreover, if the total trading number is just twice, then the multi-period trading model can degenerate to two-period trading equilibrium. So the multi-period trading model involves all the properties described in two-period trading equilibrium. The rest of the paper is organized as follows. In Section 2, we spell out the economy for formal model. In Section 3, we investigate the equilibrium of the economy. In Section 4, we employ the model to illustrate the impact of investor sentiment on equilibrium asset prices, market efficiency, and expected price path. Section 5 concludes the paper. 2. The economy We consider a simple economy with a single risky asset. There are N + 1 periods (N + 2 dates) with tn (n = 0, 1, …, N, N + 1), trading begins in period 1 and ends in period N, and then the asset pays its terminal value V at tN + 1. The terminal value is the sum of three terms. The first term is the unconditional expectation p0 which is the asset price at t0. The second term is a fundamental information release θ which is realized in period 1 and is normally distributed with mean zero and variance σθ2. Finally, there is a random disturbance term ε which is also normally distributed with mean zero and variance σ2 and is independent of θ. So the terminal value is given by V = p0 + θ + ε. There are three types of agents participating in the economy: a representative rational investor who possesses valuable information and trades completely rationally; a representative sentiment investor who is vulnerable to sentiment shocks and trades on sentiment; and market makers who make market clear and set prices efficiently conditional on the quantities traded by others. In our model, the rational investor, risk neutral and sophisticated, is assumed to maximize his expected profits and take into account his effect on prices in both the current trade and in future trades. While the sentiment investor, naive and vulnerable to sentiment, is assumed not to update his trading rule in the dynamic setting. Let us assume that the investor sentiment follows a random walk process, i.e. Sn = Sn − 1 + ξn, where the disturbance term ξn is normally distributed with mean ΔS and variance σ2S and the correlation coefficient between ξn and θ is ρn. In the economy, the sentiment investor would perceive the asset terminal value with his own sentiment. Generally, the sentiment investor overestimates the asset value with optimistic sentiment, and underestimates it with pessimistic sentiment (see, e.g., Ganzach, 2000; Statman et al., 2008; Kempf et al., 2013). Thus, the perceived terminal value is given by VS = p0 + θ + f(ΔS) + ε, where the sentiment function f(ΔS) is a monotonous increasing function of sentiment change and satisfies the properties as follows: (1) f(ΔS) N 0, if ΔSNΔS, i.e., VS N V; (2) f(ΔS) b 0, if ΔSbΔS, i.e., VS b V; and (3) f(ΔS) = 0, if ΔS ¼ ΔS, i.e., VS = V. To this end, we focus on evaluating the demand function for sentiment investor. In Appendix A, we prove the following lemma. Lemma 1. Given the linear pricing rule, the demand function for sentiment investor in the static setting can be written as
 S ΔX ¼ b ΔS−ΔS where b is a constant measuring the intensity of sentiment demand and ΔS is the level of sentiment change.

ð1Þ

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J. Li / International Review of Economics and Finance 34 (2014) 118–130

pN

p1

p0

R and sentiment investor submit quantities. There is an information release.

R and sentiment investor submit quantities.

The market makers observe the total quantities and set p1.

pN+1

Uncertainty resolves.

The market makers observe the total quantities and set pN.

Fig. 1. Timing of events.

Let XRn denote the aggregate position of the rational investor after the nth transaction, so ΔXRn = XRn − XRn − 1 denotes the quantity traded by the rational investor at the nth transaction. For n = 1, …, N, let πn denote the profits of the rational investor on positions acquired at transactions n, …, N. Explicitly, πn is given by πn ¼

N X R ðV−pk ÞΔX k :

ð2Þ

k¼n

Obviously, there are total N transactions, and each transaction takes place in two steps. In step one, the rational investor chooses
 the quantity ΔXRn that he will trade, and the sentiment investor simultaneously chooses the quantity1 ΔX Sn ¼ b ΔSn −ΔS that he will trade, where ΔSn is the level of sentiment change at the nth transaction. In step two, the market makers make market clear and set the price pn. Fig. 1 illustrates the timing of events. Definition of equilibrium. A sequential trading equilibrium is defined so that the three conditions hold as follows:
 (1) For n = 1, …, N, sentiment investor's trading rules are ΔX Sn ¼ b ΔSn −ΔS at each transaction, in which sentiment investor's demand is more with optimistic sentiment and vice versa. (2) The Rational investor's optimization demands {ΔXn⁎(⋅)} at each transaction satisfy expected profit maximization, i.e., h  i   ΔX n ¼ arg max E πn p1 ; …; pn−1 ; θ

ð3Þ

(3) The market clearing prices {pn⁎(⋅)} at each transaction satisfy
   R  S R S pn ¼ E V ΔX 1 þ ΔX 1 ; …; ΔX n þ ΔX n :

ð4Þ

3. The equilibrium of the economy In this section, we investigate the equilibrium of the economy in which a lot of trades take place sequentially. A recursive linear equilibrium is demonstrated as a sequential trading equilibrium where the component functions of the equilibrium price and the rational investor's demand are linear. The approach we obtain in the equilibrium of multi-period trades is similar in essence to that of Yang and Li (2014) in two-period trading model. In the rest of this section, we prove the existence of the equilibrium as we show in the following proposition. Proposition 1. For the economy described in Section 2, there exists a unique stationary equilibrium. The equilibrium is a recursive linear equilibrium and has the following linear forms:

 R pn ¼ pn−1 þ λn ΔX n þ b ΔSn −ΔS ; R

ð5Þ

ΔX n ¼ βn ðp0 þ θ−pn−1 Þ;

ð6Þ


  R 2 S R S σ θ;n ¼ Var θΔX 1 þ ΔX 1 ; …; ΔX n þ ΔX n ¼ Var ðp0 þ θ−pn Þ;

ð7Þ

h  i  2 E πn p1 ; …; pn−1 ; θ ¼ α n−1 ðp0 þ θ−pn−1 Þ þ δn−1 ðn ¼ 1; …; NÞ;

ð8Þ

1 In the static setting, we have verified that sentiment investor's demand function is just the proposed form of the text. Since the sentiment investor is naive (see, Ling, Naranjo, & Scheick, 2010; Stein, 2009; Victoravich, 2010), we assume that sentiment investor wouldn't update his trading rule in the dynamic setting.

J. Li / International Review of Economics and Finance 34 (2014) 118–130

121

where λn, βn, σ2θ,n, αn, δn are constants. And they are the solution to the following difference equation system: α n−1 ¼

1 ; 4λn ð1−α n λn Þ 2 2

ð9Þ

2

δn−1 ¼ α n λn b σ S þ δn ;

ð10Þ

βn ¼

1−2α n λn ; 2λn ð1−α n λn Þ

ð11Þ

λn ¼

βn σ 2θ;n−1 þ bρn σ S σ θ;n−1 2   σ θ;n ; 1−ρ2n b2 σ 2S σ 2θ;n−1

ð12Þ

2

σ θ;n ¼


 1−ρ2n b2 σ 2S σ 2θ;n−1 β2n σ 2θ;n−1 þ b2 σ 2S þ 2βn bρn σ S σ θ;n−1

;

ð13Þ

where the boundary conditions are σ2θ,0 = σ2θ and αN = δN = 0, and the second order condition is λn(1 − αnλn) N 0. Before we proceed with the proof, a few comments about the equilibrium are in order. Firstly, Eq. (5) shows that the equilibrium price could be decomposed to three terms. The first term is the previous price which is the anchoring value for the current price.
 The second term λnΔXRn is the rational investor's insufficient adjustment for the current price. The third term λn b ΔSn −ΔS is the sentiment investor's adjustment for the current price. In addition, the parameter 1/λn measures the depth of the market, i.e. the trading quantity necessary to induce prices to rise or fall by one unit (Kyle, 1985). Secondly, given the proposed price function, we solve the rational investor's optimization demand. The parameter βn measures the intensity of information demand for the rational investor. Thirdly, the parameter σ2θ,n measures the error variance of price after the nth trading, and reflects how much of the fundamental information isn't yet incorporated into prices. Finally, the parameters αn − 1 and δn − 1 define a quadratic expected profit function. Now we give the proof of Proposition 1. 3.1. The rational investor's optimization problem We employ a backward induction to prove the rational investor's profit maximization, and begin with the boundary condition αN = δN = 0 which indicates that no profit is made after trade is completed. Above all, we give the proposed price function in a stationary equilibrium, i.e.

 R pn ¼ pn−1 þ λn ΔX n þ b ΔSn −ΔS þ h

ð14Þ



 where h is a linear function of ΔX R1 þ b ΔS1 −ΔS ; …; ΔX Rn−1 þ b ΔSn−1 −ΔS . Now for constants αn and δn, we give the inductive hypothesis, i.e.  h i  2 E πnþ1 p1 ; …; pn ; θ ¼ α n ðp0 þ θ−pn Þ þ δn :

ð15Þ

From the definition of πn specified in Eq. (2), we obtain R

πn ¼ ðp0 þ θ þ ε−pn ÞΔX n þ πnþ1 :

ð16Þ

Thus the rational investor's optimization problem is i h   MaxΔX Rn E πn p1 ; …; pn−1 ; θ R

s:t: hπ n ¼ ðp0 þ θ þ ε−p i n ÞΔX n þ πnþ1 ;  2 E πnþ1 p1 ; …; pn ; θ ¼ α n ðp0 þ θ−pn Þ þ δn ;

 R pn ¼ pn−1 þ λn ΔX n þ b ΔSn −ΔS þ h: The solution is provided in the following Theorem.

ð17Þ

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J. Li / International Review of Economics and Finance 34 (2014) 118–130

Theorem 1. The solution to Eq. (17) is given by R

ΔX n ¼ βn ðp0 þ θ−pn−1 Þ

ð18Þ

n λn where βn ¼ 2λ1−2α , and the second order condition is λn(1 − αnλn) N 0. Furthermore, the constants αn − 1 and δn − 1 are given by n ð1−α n λn Þ

α n−1 ¼

1 2 2 2 ; δn−1 ¼ α n λn b σ s þ δn : 4λn ð1−α n λn Þ

ð19Þ

Proof. See Appendix B. 3.2. Market clearing The market makers observe the aggregate quantities traded by the rational investor and sentiment investor combined, and then they trade the quantity necessary to clear the market and set a clearing price. Given the total trading quantities in nth transaction and the market clearing condition, an application of the projection theorem for normally distributed random variable verifies that pn is just the form specified in Eq. (4), i.e.  h
i

  R R pn −pn−1 ¼ E V−pn−1 ΔX n þ b ΔSn −ΔS ¼ λn ΔX n þ b ΔSn −ΔS :

ð20Þ

Moreover, we simultaneously obtain the following explicit expression for λn:

 Cov V−pn−1 ; ΔX Rn þ b ΔSn −ΔS

 λn ¼ Var ΔX Rn þ b ΔSn −ΔS ¼

βn σ 2θ;n−1 þ bρn σ S σ θ;n−1 β2n σ 2θ;n−1 þ b2 σ 2S þ 2βn bρn σ S σ θ;n−1

ð21Þ :

Here, the parameter σ2θ,n − 1, which measures the information variance of price after the n − 1th transaction, is a unknown constant. Using Bayesian updating rule, we obtain the information variance after the n − 1th transaction, i.e.,
 2 R S R S σ θ;n−1 ¼ Var θjΔX 1 þ ΔX 1 ; …; ΔX n−1 þ ΔX n−1
 R S R S ¼ Var θjΔX 1 þ ΔX 1 ; …; ΔX n−2 þ ΔX n−2

 2 R R S R S Cov θ; ΔX n−1 þ b ΔSn−1 −ΔS jΔX 1 þ ΔX 1 ; …; ΔX n−2 þ ΔX n−2

 − Var ΔX Rn−1 þ b ΔSn−1 −ΔS jΔX R1 þ ΔX S1 ; …; ΔX Rn−2 þ ΔX Sn−2
2 βn−1 σ 2θ;n−2 þ bρn−1 σ S σ θ;n−2 2 ¼ σ θ;n−2 − 2 βn−1 σ 2θ;n−2 þ b2 σ 2S þ 2βn−1 bρn−1 σ S σ θ;n−2
 1−ρ2n−1 b2 σ 2S σ 2θ;n−2 ¼ 2 : βn−1 σ 2θ;n−2 þ b2 σ 2S þ 2βn−1 bρn−1 σ S σ θ;n−2

ð22Þ

Obviously, these are equivalent to Eqs. (12) and (13). The results are summarized in the following theorem. Theorem 2. For the economy described in Section 2, the market clearing price at the nth transaction satisfies

 R pn ¼ pn−1 þ λn ΔX n þ b ΔSn −ΔS :

ð23Þ

Additionally, the pricing rule λn at the nth transaction and the remained information variance σ2θ,n after the nth transaction are respectively λn ¼

βn σ 2θ;n−1 þ bρn σ S σ θ;n−1 2   σ θ;n 1−ρ2n b2 σ 2S σ 2θ;n−1

ð24Þ

J. Li / International Review of Economics and Finance 34 (2014) 118–130

2 σ θ;n

¼


 2 2 2 2 1−ρn b σ S σ θ;n−1 β2n σ 2θ;n−1 þ b2 σ 2S þ 2βn bρn σ S σ θ;n−1

:

123

ð25Þ

Combining Theorem 1 with Theorem 2 obtains Proposition 1. This completes the proof of Proposition 1. 4. Price variability, market efficiency and expected price path In this section, we illustrate the properties of the equilibrium in more detail. In particular, we analyze how the investor sentiment affects the equilibrium prices, market efficiency and asset price movements. 4.1. The equilibrium asset prices To quantitatively illustrate the impact of sentiment change on risky asset equilibrium prices, we give a three-period trading equilibrium numerical simulation. The parameters are chosen as follows: p0 = 10, σ2S = 1, σ2θ,3 = 0.2, S ¼ 0, b = 1.5, ρn = 0.1, θ = 2, and ΔS1, ΔS2, ΔS2 ∈ [−6, 6]. In the third period transaction, we begin with the boundary condition α3 = 0. Therefore, plugging α3 = 0 into Eq. (11) gives β3 ¼

1−2α 3 λ3 1 ¼ : 2λ3 ð1−α 3 λ3 Þ 2λ3

ð26Þ

Thus, from Eq. (13) we have λ3 ¼

σ θ;2 : 2bσ S

ð27Þ

Combining Eq. (26), Eq. (27) with Eq. (13) and solving for σ2θ,2 gives 2

σ θ;2 ¼ 0:44:

ð28Þ

Furthermore, solving for λ3 gives λ3 ¼ 0:22

ð29Þ

where this root satisfies the second order condition. In the second period transaction, from Eq. (9) we obtain α2 ¼

1 : 4λ3

ð30Þ

And then combining Eq. (30), Eq. (11) with Eq. (12) and solving for λ2 gives λ2 ¼ 0:25:

ð31Þ

Moreover, solving for β2 and σ2θ,1 yields β2 ¼ 1:24 2

σ θ;1 ¼ 0:74:

ð32Þ ð33Þ

In the first transaction, we similarly obtain λ1 ¼ 0:26

ð34Þ

β 1 ¼ 0:79:

ð35Þ

Ultimately, the explicit expressions for p1, p2 and p3 are given by p1 ¼ p0 þ 0:21θ þ 0:39ΔS1

ð36Þ

p2 ¼ p0 þ 0:45θ þ 0:27ΔS1 þ 0:37ΔS2

ð37Þ

124

J. Li / International Review of Economics and Finance 34 (2014) 118–130 3 2.5 2 1.5

Δp

1 0.5 0 -0.5 -1

Δ p1 Δ p2 Δ p3

-1.5 -2 -6

-4

-2

0

2

4

6

ΔS Fig. 2. The changes of sentiment equilibrium prices.

p3 ¼ p0 þ 0:73θ þ 0:14ΔS1 þ 0:19ΔS2 þ 0:33ΔS3 :

ð38Þ

Trivially, p4 = p0 + θ = 12. Then we give a numerical example with ΔS1 = ΔS2 = ΔS3 in Fig. 2. Fig. 2 shows that the sentiment equilibrium price changes Δpi (i = 1, 2, 3) increase substantially with respect to investor sentiment change. When ΔS = 0, the equilibrium price changes Δpi (i = 1, 2, 3) gradually increase over time. Thus, it shows that the speed of information incorporated into price becomes faster and faster, while the sensitivity of price to sentiment ∂Δp/∂ΔS gets smaller and smaller over time, which shows that the speed of sentiment incorporated into price gets slower and slower. Moreover, we consider the case of time-varying sentiment so as to deeply analyze the relationship between investor sentiment and the equilibrium price. A numerical example is given in Fig. 3. Fig. 3 shows that if the investor sentiment in period 1 is higher, then the equilibrium price in period 2 is much bigger. From Fig. 3 we also know that the ∂p2/∂ΔS2 is smaller than ∂p1/∂ΔS1, so the speed of sentiment incorporated into price in period 2 is slower than that in period 1. And the high sentiment results in the overreaction of the asset prices eventually. Fig. 4 illustrates the sentiment equilibrium price in period 3 with different sentiment combinations in period 1 and period 2. Obviously, the sentiment equilibrium price is much higher with optimistic investor sentiment. And the speed of sentiment incorporated into price in period 3 is also slower than that in period 1.

14

13

12

P

11

10

p1

9

p2(ΔS1=-3) p2(ΔS1=3)

8

p4 7 -6

-4

-2

0

2

4

ΔS Fig. 3. The equilibrium price with changing sentiment in period 2.

6

J. Li / International Review of Economics and Finance 34 (2014) 118–130

125

15

14

13

P

12

11

p1 p3(ΔS1=-3,ΔS2=-3)

10

p3(ΔS1=-3,ΔS2=3) p3,ΔS1=3,ΔS2=-3)

9

p3(ΔS1=3,ΔS2=3)

8 -6

p4 -4

-2

0

2

4

6

ΔS Fig. 4. The equilibrium price with changing sentiment in period 3.

4.2. The market stability and efficiency We consider a metric for stability and efficiency of the market, as defined by Yang and Zhang (2013a). This is a ratio of informationto-sentiment in prices. The market efficiency metric E in this case can be written as E¼

FR : FR þ jFSj

ð39Þ

The equilibrium price is naturally split into two pieces: the efficient market price, FR = p0 + θ, and the sentiment price |FS| caused by sentiment shocks. A numerical example, which quantitatively illustrates the effect of sentiment change on the market efficiencies Ei (i = 1, 2, 3) in each period, is given in Fig. 5. Fig. 5 shows that moderate expansion of investor sentiment drives the risky asset prices to the rational expected value and thus increases the market efficiency when there is a positive information release. Nevertheless, dramatic expansion of investor sentiment forces prices of the risky asset to move away from the rational expected value and results in the asset price overreaction eventually. Therefore, the excessive reaction to fundamental information decreases the market efficiency. 4.3. Expected price path The ultimate object of interest is how the risky asset price moves in every period. From the market clearing condition, we know that the equilibrium price depends on the total demands for the informed rational investor and uninformed sentiment investor. 1

0.95

E1 E2 E3

E

0.9

0.85

0.8

0.75

0.7 -6

-4

-2

0

2

ΔS Fig. 5. The efficiency of the market.

4

6

126

J. Li / International Review of Economics and Finance 34 (2014) 118–130

Expected Price

Favorable Signal Rational Expected Value

2

1

3

4

Rational Expected Value

Unfavorable Signal

Fig. 6. The average price path.

Given the quantity traded by the rational investor and the quantity traded by the sentiment investor, we can describe an expected price path. Let us assume that there is a positive fundamental information release in period 1. Then the rational investor buys the risky asset based on the positive information, whereas the sentiment investor buys the risky asset based on his own sentiment as if it were information. If the intensity of sentiment demand is greater than a certain level, then the total demand would lead to asset price overreaction. This implies that, on average, positive information can increase investor sentiment, intensifying overreaction. The continuing overreactions result in short-run momentum during the initial overreaction phase. However, the rational investor draws the asset prices back toward the rational expected value of the risky asset, and the initial overreaction is corrected in the long run. The process described above yields an impulse-response function for positive fundamental information, as illustrated by the upper curve in Fig. 6. Fig. 6. shows the average price path as a function of time for the three-period trading equilibrium. The upper solid curve exhibits expected price path with continuing overreactions, the middle dashed curve shows expected price path conditional on period 3 correction, and the thin horizontal shows the rational expected price level. Because the period 1, period 2 and period 3 overreactions must be reversed by the rational investor in the long run, the overreaction and reversal imply the unconditional covariance between the date 4 price change relative to the date 1 price and the date 1 price change is negative, i.e.

Covðp4 −p1 ; p1 −p0 Þ b 0; s:t: b N

λ3 λ1 ρ3 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ23 λ21 ρ23 þ λ2 ð4λ3 −λ2 Þ½λ2 ð4λ3 −λ2 Þ−2λ3 λ1  σ  θ: σS λ1 ½2λ2 ð4λ3 −λ2 Þ−2λ3 λ1 

ð40Þ

Similarly, we also have Covðp4 −p2 ; p2 −p1 Þ b 0; s:t: b N

Covðp4 −p3 ; p3 −p2 Þ b 0; s:t: b N

λ2 ρ2 þ

σθ : 2λ3 σ S

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ22 ρ22 þ 8λ3 ð2λ3 −λ2 Þ σ  θ; σS 2λ2 ð4λ3 −λ2 Þ

ð41Þ

ð42Þ

The explicit calculation and proof for the covariance are in Appendix C. The results are summarized in the following proposition. Proposition 2. If there is a fundamental information release in period 1 and sentiment investor's response is sufficiently intense, then continuing overreactions lead to unconditional short-term momentum and long-term reversal, but the asset price momentum effect may also come from continuing underreactions to information in the model. 5. Conclusion Recently, a lot of empirical tests and financial experiments on behavioral finance have found significant relationship between asset price and investor sentiment. However, the sentiment-based asset pricing model is still in the exploratory stage. The existing sentiment asset pricing models investigate either single trading equilibrium or two-period trading equilibrium. Moreover, they rarely involve some important factors such as fundamental information and the initial price and so on. Based on bounded rationality and limited arbitrage, we presents a multi-period trading sentiment asset pricing model with information which extends the noisy rational expectation model of Kyle (1985). The model involves one class of uninformed sentiment

J. Li / International Review of Economics and Finance 34 (2014) 118–130

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investor whose sentiment would change over time. The sentiment investor trades on sentiment as if it were information, and his influence on the equilibrium price becomes apparent. Consequently, the risky asset price reflects both the information that rational investor trades on and the sentiment that sentiment traders trades on. Summaries and conclusions for the characteristics of our model are as follows: Firstly, the information that rational investor put into asset prices is cumulative, and the speed of information incorporated into price is much faster across periods when the market is closed than across periods when the market is open. Nevertheless, the speed of sentiment factored into price gets slower and slower over time. Thus the asset prices tend to move back toward rational expected value of the risky asset over time. Secondly, the more sentiment trading there is, the more liquid the market is. The existence of sentiment investor makes financial market possible, but also decreases the market efficiency. Finally, on average, positive information can increase investor sentiment, intensifying overreaction. The continuing overreactions lead to short-run momentum during the initial overreaction phase, while the initial overreaction is corrected in the long run. Our findings could raise some significant issues for future research. For example, we need to build a continuous sentiment asset pricing model with information as trading takes place frequently. Acknowledgments I would like to thank the two anonymous referees and the editor Beladi H. for their helpful and constructive comments and suggestions. All remaining errors and omissions are mine. This work was supported by the Fundamental Research Funds for the Central Universities (no. 20148DXMPY03), the Doctoral Fund of Ministry of Education of China (no. 20120172110040), the National Natural Science Foundation of China (no. 70871042) and the Major Program of National Social Science Foundation of China (no. 11 & ZD156). Appendix A. Proof of Eq. (1) Let us consider the special case in which there is a representative sentiment investor so as to obtain the demand function for sentiment investor in the static setting. This is similar in essence to the case considered by Yang and Li (2014). Above all, the equilibrium price function is given by a simple linear form, i.e. p = p0 + λXS, where λ is a constant. Then expected profit for sentiment investor can be written as h i S E ðV S −pÞX jΔS
 h
i S S X jΔS ¼ E p0 þ θ þ f ðΔSÞ þ ε− p0 þ λX
 S S ¼ f ðΔSÞ−λX X :

ðA:1Þ

Maximizing this quadratic objective with respect to XS yields the demand function for sentiment investor S

X ¼

f ðΔSÞ : 2λ

ðA:2Þ


 In addition, let us assume that the sentiment function f(ΔS) is given by f ðΔSÞ ¼ ϕ ΔS−ΔS which satisfies the proposed properties in the text. So the demand function for sentiment investor can be written as
 S X ¼ b ΔS−ΔS

ðA:3Þ

where ϕ and b = ϕ/2λ are constants. Appendix B. Solution to optimization problem of the rational investor Plugging Eq. (16), Eq. (15) and Eq. (14) into the expected profit equation of the rational investor yields h i E π n jp1 ; …; pn−1 ; θ h i R ¼ E ðp0 þ θ þ ε−pn ÞΔX n þ πnþ1 jp1 ; …; pn−1 ; θ
 R R ¼ p0 þ θ−pn−1 −λn ΔX n −h ΔX n
2 R 2 2 2 þ α n p0 þ θ−pn−1 −λn ΔX n −h þ α n λn b σ S þ δn :

ðB:1Þ

Taking the first order condition with respect to ΔXRn we obtain
 R R ðp0 þ θ−pn−1 −hÞ−2λn ΔX n −2α n λn p0 þ θ−pn−1 −h−λn ΔX n ¼ 0:

ðB:2Þ

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Solving for ΔXRn gives R

ΔX n ¼

ð1−2α n λn Þðp0 þ θ−pn−1 −hÞ 2λn ð1−α n λn Þ

ðB:3Þ

subject to the second order condition λn ð1−α n λn ÞN0:

ðB:4Þ



 λn Þðp0 þθ−pn−1 −hÞ , we have Given pn ¼ pn−1 þ λn ΔX Rn þ b ΔSn −ΔS þ h and ΔX Rn ¼ ð1−2αn2λ n ð1−α n λn Þ  
 R R E Δpn jΔX 1 þ b ΔS1 −ΔS ; …; ΔX n−1 þ b ΔSn−1 −ΔS 



 R R R ¼ E λn ΔX n þ bΔSn þ hjΔX 1 þ b ΔS1 −ΔS ; …; ΔX n−1 þ b ΔSn−1 −ΔS

ðB:5Þ

h : ¼ 2ð1−α n λn Þ From the market clearing condition, we have 

   R R E Δpn ΔX 1 þ b ΔS1 −ΔS ; …; ΔX n−1 þ bðΔSn−1 −ΔS ¼ 0:

ðB:6Þ

Thus, we prove that h = 0. And then we obtain R

ΔX n ¼ βn ðp0 þ θ−pn−1 Þ

ðB:7Þ

n λn . where β n ¼ 2λ1−2α n ð1−α n λn Þ

Furthermore, plugging ΔXRn = βn(p0 + θ − pn − 1) into the expected profit equation of the rational investor yields h i E π n jp1 ; …; pn−1 ; θ

 R R R 2 2 2 2 ¼ p0 þ θ−pn−1 −λn ΔX n ΔX n þ α n p0 þ θ−pn−1 −λn ΔX n þ α n λn b σ S þ δn

ðB:8Þ

ðp þ θ−pn−1 Þ2 2 2 2 þ α n λn b σ S þ δn : ¼ 0 4λn ð1−α n λn Þ Therefore, the constants αn − 1 and δn − 1 are given by an−1 ¼

1 2 2 2 ; δn−1 ¼ α n λn b σ s þ δn : 4λn ð1−α n λn Þ

ðB:9Þ

Appendix C. Proof of Eqs. (40)–(42) From the aforementioned equilibrium price function, the period 1 price change is p1 −p0 ¼


 λ2 ð4λ3 −λ2 Þ−2λ3 λ1 θ þ λ1 b ΔS1 −ΔS : 2λ2 ð4λ3 −λ2 Þ−2λ3 λ1

ðC:1Þ

And the date 4 price change relative to the date 1 price is p4 −p1 ¼


 λ2 ð4λ3 −λ2 Þ θ−λ1 b ΔS1 −ΔS : 2λ2 ð4λ3 −λ2 Þ−2λ3 λ1

ðC:2Þ

Thus, the unconditional covariance between the date 4 price change and the period 1 price change is given by Covðp4 −p1 ; p1 −p0 Þ 
 λ ð4λ −λ Þ−2λ λ
 λ2 ð4λ3 −λ2 Þ 3 2 3 1 θ−λ1 b ΔS1 −ΔS ; 2 θ þ λ1 b ΔS1 −ΔS ¼ Cov 2λ2 ð4λ3 −λ2 Þ−2λ3 λ1 2λ2 ð4λ3 −λ2 Þ−2λ3 λ1 ¼

2 2 2 −λ1 σ S b

λ3 λ21 ρ3 σ θ σ S λ ð4λ3 −λ2 Þ½λ2 ð4λ3 −λ2 Þ−2λ3 λ1  2 þ bþ 2 σθ: λ2 ð4λ3 −λ2 Þ−λ3 λ1 ½2λ2 ð4λ3 −λ2 Þ−2λ3 λ1 2

ðC:3Þ

J. Li / International Review of Economics and Finance 34 (2014) 118–130

Therefore, we obtain Covðp4 −p1 ; p1 −p0 Þ b 0 subject to b N Similarly, the period 2 price change is given by p2 −p1 ¼

λ3 λ1 ρ3 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

λ3 λ1 ρ23 þλ2 ð4λ3 −λ2 Þ½λ2 ð4λ3 −λ2 Þ−2λ3 λ1  λ1 ½2λ2 ð4λ3 −λ2 Þ−2λ3 λ1 


 ð2λ3 −λ2 Þðp0 þ θ−p1 Þ þ λ2 b ΔS2 −ΔS : 4λ3 −λ2

129

 σσ θS .

ðC:4Þ

And the date 4 price change relative to the date 2 price is given by p4 −p2 ¼


 2λ3 ðp0 þ θ−p1 Þ −λ2 b ΔS2 −ΔS : 4λ3 −λ2

ðC:5Þ

We thus have Covðp4 −p2 ; p2 −p1 Þ 
 ð2λ −λ Þðp þ θ−p Þ
 2λ3 ðp0 þ θ−p1 Þ 3 2 0 1 −λ2 b ΔS2 −ΔS ; þ λ2 b ΔS2 −ΔS ¼ Cov 4λ3 −λ2 4λ3 −λ2 2

2 2

¼ −λ2 σ S b þ

λ22 ρ2 σ θ σ S 4λ3 −λ2

bþ

2λ3 ð2λ3 −λ2 Þ 2 σθ: ð4λ3 −λ2 Þ2

Then we obtain Covðp4 −p2 ; p2 −p1 Þ b 0 subject to b N Similarly, the period 3 price change is given by p3 −p2 ¼

ðC:6Þ

λ2 ρ2 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 λ2 ρ22 þ8λ3 ð2λ3 −λ2 Þ 2λ2 ð4λ3 −λ2 Þ


 p0 þ θ−p2 þ λ3 b ΔS3 −ΔS : 2

 σσ θS .

ðC:7Þ

And the period 4 price change is given by p4 −p3 ¼


 p0 þ θ−p2 −λ3 b ΔS3 −ΔS : 2

ðC:8Þ

Therefore, we have Covðp4 −p3 ; p3 −p2 Þ 
 p þ θ−p
 p þ θ−p2 2 −λ3 b ΔS3 −ΔS ; 0 þ λ3 b ΔS3 −ΔS ¼ Cov 0 2 2 ¼

σ 2θ 4

ðC:9Þ

2 2 2

−σ S λ3 b :

Accordingly, if bN 2λσ2θσ S , then Cov(p4 − p3, p3 − p2) b 0. References Baker, M., & Wurgler, J. (2006). Investor sentiment and the cross-section of stock returns. The Journal of Finance, 61(4), 1645–1680. Baker, M., & Wurgler, J. (2007). Investor sentiment in the stock market. Journal of Economic Perspective, 21(2), 129–151. Baker, M., Wurgler, J., & Yuan, Y. (2012). Global, local, and contagious investor sentiment. Journal of Financial Economics, 104, 272–287. Barberis, N., Shleifer, A., & Vishny, R. (1998). A model of investor sentiment. Journal of Financial Economics, 49(3), 307–343. Black, F. (1986). Noise. The Journal of Finance, 41(3), 529–543. Brown, G., & Cliff, M. (2004). Investor sentiment and the near-term stock market. Journal of Empirical Finance, 11(1), 1–27. Brown, G., & Cliff, M. (2005). Investor sentiment and asset valuation. Journal of Business, 78, 405–440. Cen, L., Lu, H., & Yang, L. (2013). Investor sentiment, disagreement, and the breadth–return relationship. Management Science, 59, 1076–1091. Chan, Y. (2014). How does retail sentiment affect IPO returns? Evidence from the internet bubble period. International Review of Economics and Finance, 29, 235–248. Corredor, P., Ferrer, E., & Santamaria, R. (2013). Investor sentiment effect in stock markets: Stock characteristics or country-specific factors? International Review of Economics and Finance, 27, 572–591. Daniel, K., Hirshleifer, D., & Subrahmanyam, A. (1998). Investor psychology and security market under- and overreactions. The Journal of Finance, 53(6), 1839–1885. De Long, J., Shleifer, A., Summers, L., & Waldmann, R. (1990). Noise trader risk in financial markets. Journal of Political Economy, 98(4), 703–738. De Long, J., Shleifer, A., Summers, L., & Waldmann, R. (1991). The survival of noise traders in financial markets. Journal of Business, 64, 1–19. Fama, E. (1965). The behavior of stock market prices. Journal of Business, 38, 34–106. Friedman, M. (1953). The Case for Flexible Exchange Rates. Essays in Positive Economics. Chicago: University of Chicago Press. Ganzach, Y. (2000). Judging risk and return of financial assets. Organizational Behavior and Human Decision Processes, 83(2), 353–370. Grossman, S., & Stiglitz, J. (1980). On the impossibility of informationally efficient markets. American Economic Review, 70, 393–408. Hong, H., & Stein, J. (1999). A unified theory of underreaction, momentum trading, and overreaction in asset markets. The Journal of Finance, 54(6), 2143–2184. Kempf, A., Merkle, C., & Niessen, A. (2013). Low risk and high return-affective attitudes and stock market expectations. European Financial Management, 00, 1–36. Kim, J. S., Ryu, D., & Seo, S. W. (2014). Investor sentiment and return predictability of disagreement. Journal of Banking & Finance, 42, 166–178. Kumar, A., & Lee, C. M. (2006). Retail investor sentiment and return comovements. The Journal of Finance, 61, 2451–2486. Kyle, A. (1985). Continuous auctions and insider trading. Econometrica, 53(6), 1315–1335.

130

J. Li / International Review of Economics and Finance 34 (2014) 118–130

Lee, W., Jiang, C., & Indro, D. (2002). Stock market volatility, excess returns, and the role of investor sentiment. Journal of Banking & Finance, 26, 2277–2299. Ling, D. C., Naranjo, A., & Scheick, B. (2010). Investor Sentiment and Asset Pricing in Public and Private Markets. 46th Annual AREUEA Conference Paper. Mendel, B., & Shleifer, A. (2012). Chasing noise. Journal of Financial Economics, 104, 303–320. Peri, M., Vandone, D., & Baldi, L. (2013). Internet, noise trading and commodity futures prices. International Review of Economics and Finance, 33, 82–89. Seybert, N., & Yang, H. (2012). The role of earning guidance in resolving sentiment-driven overvaluation. Management Science, 58(2), 308–319. Stambaugh, R., Yu, J., & Yuan, Y. (2012). Investor sentiment and anomalies. Journal of Financial Economics, 104, 288–302. Statman, M., Fisher, K., & Anginer, D. (2008). Affect in a behavioral asset-pricing model. Financial Analysts Journal, 64(2), 20–29. Stein, J. (2009). Presidential address: Sophisticated investors and market efficiency. The Journal of Finance, 64(4), 1517–1548. Victoravich, L. M. (2010). Overly optimistic? Investor sophistication and the role of affective reactions to financial information in investors' stock price judgments. Journal of Behavioral Finance, 11, 1–10. Yan, H. (2010). Is noise trading cancelled out by aggregation. Management Science, 57(7), 1047–1059. Yang, C., Xie, J., & Yan, W. (2012). Sentiment capital asset pricing model. International Journal of Digital Content Technology and its Applications, 6(3), 254–261. Yang, C., & Li, J. (2013). Investor sentiment, information and asset pricing model. Economic Modelling, 35, 436–442. Yang, C., & Li, J. (2014). Two-period trading sentiment asset pricing model with information. Economic Modelling, 36, 1–7. Yang, C., & Yan, W. (2011). Does high sentiment cause negative excess return? International Journal of Digital Content Technology and Its Applications, 5(12), 211–217. Yang, C., & Zhang, R. (2013a). Sentiment asset pricing model with consumption. Economic Modelling, 30, 462–467. Yang, C., & Zhang, R. (2013b). Dynamic asset pricing model with heterogeneous sentiments. Economic Modelling, 33, 248–253. Yang, C., & Zhang, R. (2014). Does mixed-frequency investor sentiment impact stock returns? Based on the empirical study of MIDAS regression model. Applied Economics, 46(9), 966–972. Yu, J., & Yuan, Y. (2011). Investor sentiment and the mean–variance relation. Journal of Financial Economics, 100(2), 367–381. Zhu, M. (2013). Return distribution predictability and its implications for portfolio selection. International Review of Economics and Finance, 27, 209–233.