JOURNAL OF Economic Dynamics Journal of Economic Dynamics and Control Control ELSEVIER 22(1998)1027-1051 Dynamic portfolio choice and asset pricing with differential information Chunsheng Zhou* Federal Reserve Board,Mail Stop 91.Washington,DC 20551.USA Received 2 November 1996;accepted 24 July 1997 Abstract This paper presents a multi-asset intertemporal general equilibrium model of portfolio selection and asset pricing with differential information.A method of Sargent(1991)is used to resolve the 'infinite regress'problem in information extraction and to derive a rational expectations equilibrium.The model shows that rational investors trade stocks strategically according to their perceptions about economic states and provides a ration- ale for investors to hold less than perfectly diversified portfolios.The information distribution among investors has an important effect on stock prices,welfare,and the investment opportunities of investors.The model helps explain a number of interesting financial regularities such as imperfect portfolio diversification and home bias.Published by Elsevier Science B.V. JEL classification:G11;G12;G14 Keywords:Asset pricing:Multi-asset;Dynamic;Differential information 1.Introduction Portfolio choice and asset pricing under heterogeneous information in a multi-asset securities market are interesting and challenging issues in modern finance.Admati(1985)addresses this issue in a static setup.Zhou(1997)builds a dynamic model to study the issue under a special information structure: *Current address:Anderson Graduate School of Management,University of California, Riverside,CA 92521-0203;email:chunsheng.zhou@ucr.edu 0165-1889/Published by Elsevier Science B.V. PI1S0165-1889(97)00099-7
* Current address: Anderson Graduate School of Management, University of California, Riverside, CA 92521-0203; email: chunsheng.zhou@ucr.edu Journal of Economic Dynamics and Control 22 (1998) 1027—1051 Dynamic portfolio choice and asset pricing with differential information Chunsheng Zhou* Federal Reserve Board, Mail Stop 91, Washington, DC 20551, USA Received 2 November 1996; accepted 24 July 1997 Abstract This paper presents a multi-asset intertemporal general equilibrium model of portfolio selection and asset pricing with differential information. A method of Sargent (1991) is used to resolve the ‘infinite regress’ problem in information extraction and to derive a rational expectations equilibrium. The model shows that rational investors trade stocks strategically according to their perceptions about economic states and provides a rationale for investors to hold less than perfectly diversified portfolios. The information distribution among investors has an important effect on stock prices, welfare, and the investment opportunities of investors. The model helps explain a number of interesting financial regularities such as imperfect portfolio diversification and home bias. Published by Elsevier Science B.V. JEL classification: G11; G12; G14 Keywords: Asset pricing; Multi-asset; Dynamic; Differential information 1. Introduction Portfolio choice and asset pricing under heterogeneous information in a multi-asset securities market are interesting and challenging issues in modern finance. Admati (1985) addresses this issue in a static setup. Zhou (1997) builds a dynamic model to study the issue under a special information structure: 0165-1889/Published by Elsevier Science B.V. PII S0165-1889(97)00099-7
1028 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 information sets are completely ranked.However,the information structure in reality is more general and much richer.While some traders are better informed about certain aspects of the securities market,other traders may have better knowledge about some other aspects of the market.In other words,different investors may have different information which cannot be completely ranked.In line with He and Wang (1995),we use the term differential information to represent the information structure where heterogeneous information sets can- not be completely ranked. This paper considers an interesting example of a differential information story.In a noisy two-stock market,there are two classes of traders.(Of course, one can extend it to any number of classes and any number of stocks in a straightforward way.)Class a traders are computer engineers who have better information and experience about the computer industry,especially IBM cor- poration;class b traders are communication experts who have better knowledge and insights about the communication industry,especially AT&T.In a friction- less market,should class a traders only hold IBM stock and class b traders only hold AT&T shares;or should they just hold the market portfolio?What is the difference between a and b's portfolios?What are the effects of interaction between traders on stock prices?The answers to these questions are useful for explaining a broad range of phenomena in the empirical literature,including mean reversion,excess volatility,and especially,home bias in international portfolio choices. In solving a dynamic rational-expectations asset pricing model with differen- tial information,one often faces the so-called'infinite regress'problem regarding rational information extraction of economic agents (i.e.,forecasting the fore- casts of forecasts ...of others).In this paper,we use an apparatus of Marcet and Sargent 1989a,b)and Sargent(1991)to handle it.Instead of modeling the beliefs of each class of economic agents as unobserved state variables,economic agents are modeled as forecasting the future by fitting finite-dimensional vector ARMA models for all information available to them,including endogenous variables such as prices. Other work which is closely related to this paper includes He and Wang (1995)and Hussman(1992).He and Wang present a differential information model with a finite horizon and an infinite number of investors,while Hussman gives a model with two classes of traders in which each class observes a compon- ent of stock dividends.Both models assume a single risky asset.The current work can be viewed as an extension of these previous papers in two ways.First, it presents a multi-asset model which can explore the cross-sectional properties See.e.g.Cooper and Kaplanis(1994),French and Poterba(1991),Stulz(1994),and Tesar and Werner(1993)for evidence and discussion
1 See, e.g., Cooper and Kaplanis (1994), French and Poterba (1991), Stulz (1994), and Tesar and Werner (1993) for evidence and discussion. information sets are completely ranked. However, the information structure in reality is more general and much richer. While some traders are better informed about certain aspects of the securities market, other traders may have better knowledge about some other aspects of the market. In other words, different investors may have different information which cannot be completely ranked. In line with He and Wang (1995), we use the term differential information to represent the information structure where heterogeneous information sets cannot be completely ranked. This paper considers an interesting example of a differential information story. In a noisy two-stock market, there are two classes of traders. (Of course, one can extend it to any number of classes and any number of stocks in a straightforward way.) Class a traders are computer engineers who have better information and experience about the computer industry, especially IBM corporation; class b traders are communication experts who have better knowledge and insights about the communication industry, especially AT&T. In a frictionless market, should class a traders only hold IBM stock and class b traders only hold AT&T shares; or should they just hold the market portfolio? What is the difference between a and b’s portfolios? What are the effects of interaction between traders on stock prices? The answers to these questions are useful for explaining a broad range of phenomena in the empirical literature, including mean reversion, excess volatility, and especially, home bias in international portfolio choices.1 In solving a dynamic rational-expectations asset pricing model with differential information, one often faces the so-called ‘infinite regress’ problem regarding rational information extraction of economic agents (i.e., ‘forecasting the forecasts of forecasts 2’ of others). In this paper, we use an apparatus of Marcet and Sargent 1989a,b) and Sargent (1991) to handle it. Instead of modeling the beliefs of each class of economic agents as unobserved state variables, economic agents are modeled as forecasting the future by fitting finite-dimensional vector ARMA models for all information available to them, including endogenous variables such as prices. Other work which is closely related to this paper includes He and Wang (1995) and Hussman (1992). He and Wang present a differential information model with a finite horizon and an infinite number of investors, while Hussman gives a model with two classes of traders in which each class observes a component of stock dividends. Both models assume a single risky asset. The current work can be viewed as an extension of these previous papers in two ways. First, it presents a multi-asset model which can explore the cross-sectional properties 1028 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1029 of asset prices.Second,it has implications for a number of real world financial issues such as imperfect portfolio diversification and the home-bias puzzle' which the previous papers do not have. The rest of this paper is structured as follows.Section 2 describes the eco- nomic model.Section 3 considers the benchmark case of a perfect information discrete-time model.Section 4 shows rational information extraction in a noisy market with differential information sets which cannot be completely ranked. Section 5 solves for a differential information equilibrium of the market. Section 6 uses a couple of numerical examples to show economic implications of the current models and to explain some important findings cited in the empirical literature.Section 7 concludes. 2.The economic model In this paper,we will consider a hypothetical exchange economy where one riskless asset and more than one risky asset are traded.Economic agents are differently informed,but no one informationally dominates all other agents. Formally,we have the following assumptions: Assumption 1 (Physical good).There is only a single physical good in the economy,which can be allocated either to consumption or to investment.All values are expressed in the units of this good. Assumption 2(Equity).This is a multi-asset economy.For simplicity and without loss of generality,we assume that there are two risky assets(stock 1:IBM stock and stock 2:AT&T stock)available in the economy.The dividend process for each stock is driven by a(partially)persistent component and a(purely) transitory component Dit Fit+ViD. Ft=aiFFia-1+tF,n（-1≤aF≤1 (2) where Di is the dividend payment of stock i in period t,Fir is the persistent component of Dir and viD.is the transitory component of Dit.Noise terms iand vi.are i.i.d.Gaussian processes with means zero and variances and ir,respectively. Assumption 3(Bond).There is one risk-free asset(bond)which generates a fixed rate of dividend r(r>0)per unit time.The bond supply is perfectly elastic,so the price of bond will not be affected by the bond demand
of asset prices. Second, it has implications for a number of real world financial issues such as imperfect portfolio diversification and the ‘home-bias puzzle’ which the previous papers do not have. The rest of this paper is structured as follows. Section 2 describes the economic model. Section 3 considers the benchmark case of a perfect information discrete-time model. Section 4 shows rational information extraction in a noisy market with differential information sets which cannot be completely ranked. Section 5 solves for a differential information equilibrium of the market. Section 6 uses a couple of numerical examples to show economic implications of the current models and to explain some important findings cited in the empirical literature. Section 7 concludes. 2. The economic model In this paper, we will consider a hypothetical exchange economy where one riskless asset and more than one risky asset are traded. Economic agents are differently informed, but no one informationally dominates all other agents. Formally, we have the following assumptions: Assumption 1 (Physical good). There is only a single physical good in the economy, which can be allocated either to consumption or to investment. All values are expressed in the units of this good. Assumption 2 (Equity). This is a multi-asset economy. For simplicity and without loss of generality, we assume that there are two risky assets (stock 1: IBM stock and stock 2: AT&T stock) available in the economy. The dividend process for each stock is driven by a (partially) persistent component and a (purely) transitory component Dit"Fit#v iD,t , (1) Fit"a iFFi,t~1#v iF,t , (!14a iF41), (2) where Dit is the dividend payment of stock i in period t, Fit is the persistent component of Dit and v iD,t is the transitory component of Dit. Noise terms v iD,t and v iF,t are i.i.d. Gaussian processes with means zero and variances p2 iD and p2 iF, respectively. Assumption 3 (Bond). There is one risk-free asset (bond) which generates a fixed rate of dividend r(r'0) per unit time. The bond supply is perfectly elastic, so the price of bond will not be affected by the bond demand. C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1029
1030 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 Assumption 4 (Equity supply).The total supply of each stock i (i=1,2)is normalized to 1+Ni,where Ni is the noisy supply of stock i.Ni follows an AR(1)process: Na=awNa-1+tw.t(-1≤aw≤1， (3) where viN.is an ii.d Gaussian process with mean zero and variance i.We will call the total supply of a stock with noise the noisy supply of the stock and the total supply of a stock excluding noise the pure supply of the stock and will call the market portfolio with noise components the noisy market portfolio and the market portfolio excluding noise components the pure market portfolio. Assumption 5(Information structure).There are two classes of rational economic agents,indexed by j=a,b.The total population is normalized to 1,with a proportion k in class a and 1-k in class b.Class a has perfect information about F but does not observe F2.Symmetrically,class b has perfect informa- tion about F2 but does not observe F of stock 1.Nobody observes noisy asset supplies.Mathematically,their information sets can be represented by 乎日={P1P2oD1,D2e,F1t≤t, (4) 乎={P1eP2oD1D2F2lt≤t (5) For expositional convenience,we sometimes simply call a representative agent of class a agent a and a representative agent of class b agent b. Assumption 6 (Common knowledge).The structure of the economy is common knowledge. Assumption 7 (Preferences).All economic agents have the same constant abso- lute risk aversion (CARA)preference.At any time t,agents maximize their expected utilities of next period wealth W,+by solving maxE,[u(W+i门=maxE,[-exp(-φWr+i】中>0. (6) Assumption 8(Trading mechanism).Trading in assets takes place once each period t at equilibrium prices P and P2 after dividends for that period D and D2,have been paid out.No trading takes place at non-equilibrium prices. For simplicity,we assume the following covariance relations:Cov(Up,vr) Cov(UD,UN)=Cov(UF,UN)=0,Cov(v1D,D2D)=ID,Cov(U1F,U2F)=F,and
Assumption 4 (Equity supply). The total supply of each stock i (i"1,2) is normalized to 1#Ni , where Ni is the noisy supply of stock i. Ni follows an AR(1) process: Nit"a iNNi,t~1#v iN,t (!14a iN41), (3) where v iN,t is an i.i.d Gaussian process with mean zero and variance p2 iN. We will call the total supply of a stock with noise the noisy supply of the stock and the total supply of a stock excluding noise the pure supply of the stock and will call the market portfolio with noise components the noisy market portfolio and the market portfolio excluding noise components the pure market portfolio. Assumption 5 (Information structure). There are two classes of rational economic agents, indexed by j"a,b. The total population is normalized to 1, with a proportion k in class a and 1!k in class b. Class a has perfect information about F1 but does not observe F2 . Symmetrically, class b has perfect information about F2 but does not observe F1 of stock 1. Nobody observes noisy asset supplies. Mathematically, their information sets can be represented by Fa t "MP1q , P2q , D1q , D2q , F1q Dq4tN, (4) Fb t "MP1q , P2q , D1q , D2q , F2q Dq4tN. (5) For expositional convenience, we sometimes simply call a representative agent of class a agent a and a representative agent of class b agent b. Assumption 6 (Common knowledge). The structure of the economy is common knowledge. Assumption 7 (Preferences). All economic agents have the same constant absolute risk aversion (CARA) preference. At any time t, agents maximize their expected utilities of next period wealth ¼t`1 by solving max Et [u(¼t`1 )]"max Et [!exp(!/¼t`1 )], /'0. (6) Assumption 8 (¹rading mechanism). Trading in assets takes place once each period t at equilibrium prices P1t and P2t after dividends for that period D1t and D2t have been paid out. No trading takes place at non-equilibrium prices. For simplicity, we assume the following covariance relations: Cov(v D , v F ) "Cov(v D , v N )"Cov(v F , v N )"0, Cov(v 1D , v 2D )"g D , Cov(v 1F , v 2F )"g F , and 1030 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1031 Cov(vIN,)=nN.That is,we assume that the shocks in different categories are uncorrelated but that the shocks in the same categories can be correlated.Every random shock is assumed to be i.i.d over time. A comment on our notation is in order here.We use letters with subscript i(i=1,2)to denote coefficients or variables associated with stock i,e.g,air and Fi.After dropping subscript i,those letters in boldface will represent the corresponding diagonal matrices (for coefficients)or column vectors (for vari- ables),e.g, and F (7) 0 d2F F2 We use the capital Greek letter >(with subscripts)to represent variance- covariance matrices,e.g.,r Var(F)=E[rF]. Generally,variables used in this paper have a time subscript while constants do not have a time subscript.When no confusion exists,subscripts may be suppressed. 3.Benchmark case:perfect information equilibrium Before proceeding to study the differential information model,we will first consider the perfect information equilibrium in which rational economic agents observe the current and the past values of all of the underlying economic variables described earlier.This relatively simple perfect informa- tion setup provides useful intuition and will serve as a benchmark for evaluating the differential information equilibrium considered in subsequent sections. 3.1.Stock fundamentals and investment opportunities We define the fundamental value of a stock as the expected present value ofits dividend flows discounted at the riskless interest rate r.It is easy to see that Theorem 1.The fundamental value Vit)of stock i is given by Vt)=⊙：*Ft),i=1,2 (8) where⊙：*=ar/1+r-ae
Cov(v 1N ,v 2N )"g N . That is, we assume that the shocks in different categories are uncorrelated but that the shocks in the same categories can be correlated. Every random shock is assumed to be i.i.d over time. A comment on our notation is in order here. We use letters with subscript i (i"1,2) to denote coefficients or variables associated with stock i, e.g., a iF and Fi . After dropping subscript i, those letters in boldface will represent the corresponding diagonal matrices (for coefficients) or column vectors (for variables), e.g., a F "C a 1F 0 0 a 2F D and F"C F1 F2 D . (7) We use the capital Greek letter R (with subscripts) to represent variancecovariance matrices, e.g., RF "Var(F)"E[ F @ F ]. Generally, variables used in this paper have a time subscript while constants do not have a time subscript. When no confusion exists, subscripts may be suppressed. 3. Benchmark case: perfect information equilibrium Before proceeding to study the differential information model, we will first consider the perfect information equilibrium in which rational economic agents observe the current and the past values of all of the underlying economic variables described earlier. This relatively simple perfect information setup provides useful intuition and will serve as a benchmark for evaluating the differential information equilibrium considered in subsequent sections. 3.1. Stock fundamentals and investment opportunities We define the fundamental value of a stock as the expected present value of its dividend flows discounted at the riskless interest rate r. It is easy to see that ¹heorem 1. ¹he fundamental value »i (t) of stock i is given by »i (t)"Hi *Fi (t), i"1,2, (8) where Hi *"a iF/(1#r!a iF). C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1031
1032 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 Proof.By Assumption 2 Y.ELEtDo (9) (10) aF-Fa□ =T+r-diF (11) To obtain the market equilibrium,we need to describe the investment oppor- tunities first.Let I7i denote the undiscounted cumulative cash flow from a zero-wealth portfolio long one share of stock i financed by selling the risk-free bond.We have Ii+1(Pis+1+Dit+1)-(1 +r)Pir (12) ein vin, (13) where Pi is the price of stock i,ein =E,[i+1]is the one-period-ahead expecta- tion of excess return and vin=Ii-ein is the corresponding expectation error. 3.2.The equilibrium According to Assumption 7,a representative economic agent's optimization problem can be written as maxE[-exp(-Wt+i门， (14) subject to W:+1 =(1 r)W:Qen.Qien.+1, (15) where W is the agent's wealth and o is the vector of his or her stock holdings. Let us conjecture that Un is Gaussian.2 With the conjecture,we immediately have that Q=En'en EnE[P+1+D+1-(1+r)P ] (16) where En is the variance-covariance matrix of the innovations Un. 2We will see shortly that the conjecture is true since Eq.(18)implies that P,and therefore I,are linear functions of D.F:and N
2We will see shortly that the conjecture is true since Eq. (18) implies that Pt and therefore Pt are linear functions of Dt , Ft and Nt . Proof. By Assumption 2 »it,EtC = + s/1 1 (1#r)s Di,t`sD (9) " = + s/1 A a iF 1#rB s Fit (10) " a iF 1#r!a iF Fit h (11) To obtain the market equilibrium, we need to describe the investment opportunities first. Let Pit denote the undiscounted cumulative cash flow from a zero-wealth portfolio long one share of stock i financed by selling the risk-free bond. We have Pi,t`1 ,(Pi,t`1 #Di,t`1 )!(1#r)Pit (12) "e i P#v i P, (13) where Pi is the price of stock i, e i P"Et [Pi,t`1 ] is the one-period-ahead expectation of excess return and v i P"Pi !e i P is the corresponding expectation error. 3.2. The equilibrium According to Assumption 7, a representative economic agent’s optimization problem can be written as max Qt Et [!exp(!¼t`1 )], (14) subject to ¼t`1 "(1#r)¼t #Q@ t eP,t #Q@ t P,t`1 , (15) where ¼ is the agent’s wealth and Q is the vector of his or her stock holdings. Let us conjecture that P is Gaussian.2 With the conjecture, we immediately have that Q"R~1 P eP"RP~1Et [Pt`1 #Dt`1 !(1#r)Pt ], (16) where RP is the variance—covariance matrix of the innovations P. 1032 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1033 The total supply of stocks to rational economic agents is 1+N,where 1 is a two-dimensional vector of ones.Market clearing condition =1 +N then implies P=(1+r)-1EP+1+D+)-(1+)-1E1+N) (17) which may be solved forward to yield P,=V,-(1/r)En1-ΣnΦW, (18) whereΦisa2×2 diagonal matrix 1 0 1+r-aiN Φ= (19) 0 1+r-a2N」 Theorem 2.The equilibrium conditions of the model imply that en=En(1 +N) (20) and that En satisfies the following matrix equation: En-∑nΦEvΦ'En=ED+P∑FP', (21) where里=I+⊙*isa2×2 natrix. Proof.The first part of the theorem about en is pretty straightforward since market clearing implies o=1 N. Note that D+1=F+1+p.+1 as specified in Section 2.From price equa- tion,Eq.(18),we have P+1+D+1=V+1+D+1-(1/r)En1-EnΦN+1 =0*F+1+F+1+D.+1-(1/r)En1-EnN+1 =ΨF+1+p.+1-(1/r)Enl-∑nΦN,+1, (22) where平=I+⊙*. On the other hand,the definition of I implies that Σn=Var(Pr+i+D+i (23) As a result,we have Σn=∑D+平ErΨ'+EnΦEwΦΣm口 (24) Eq.(21)has a real-valued solution if and only if the matrix is not too large in magnitude.Therefore,if the market is too noisy and/or the noise is too
The total supply of stocks to rational economic agents is 1#N, where 1 is a two-dimensional vector of ones. Market clearing condition Q"1#N then implies Pt "(1#r)~1Et (Pt`1 #Dt`1 )!(1#r)~1RP(1#Nt ) (17) which may be solved forward to yield Pt "Vt !(1/r)RP1!RPUNt , (18) where U is a 2]2 diagonal matrix U" C 1 1#r!a 1N 0 0 1 1#r!a 2N D . (19) ¹heorem 2. ¹he equilibrium conditions of the model imply that eP"RP(1#N) (20) and that RP satisfies the following matrix equation: RP!RPURN U@RP"RD #WRF W@, (21) where W"I#H* is a 2]2 matrix. Proof. The first part of the theorem about eP is pretty straightforward since market clearing implies Q"1#N. Note that Dt`1 "Ft`1 #D,t`1 as specified in Section 2. From price equation, Eq. (18), we have Pt`1 #Dt`1 "Vt`1 #Dt`1 !(1/r)RP1!RPUNt`1 "H*Ft`1 #Ft`1 #D,t`1 !(1/r)RP1!RPUNt`1 "WFt`1 #D,t`1 !(1/r)RP1!RPUNt`1 , (22) where W"I#H*. On the other hand, the definition of P implies that RP"Vart (Pt`1 #Dt`1 ). (23) As a result, we have RP"RD #WRF W@#RPURN U@RP. h (24) Eq. (21) has a real-valued solution if and only if the matrix URN U@ is not too large in magnitude. Therefore, if the market is too noisy and/or the noise is too C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1033
1034 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 persistent,no stable equilibrium can be established.We will exclude this possi- bility in the subsequent analysis. 4.Information filtration and perceived investment opportunities 4.1.Perceived laws of motion Now we consider the differential information model.To solve for an equilib- rium with non-completely-ranked information sets,we need a tractable method to deal with the information extraction problem. According to Assumption 5,agents in class a observe a record of current and past values Sat [Pit P2n Die D2e FitT, (25) where Pi=Pir-p1 and P2=P2-P2 are 'demeaned'stock prices.Pi and p2 reflect the unconditional expected risk premia,which will be discussed later. Define Xar =Sar-E[S]as the period-ahead conditional expectation error in Sr.Following Sargent(1991),we assume that the filtration rule of agent a,or equivalently,the agent's perceived law of motion for Sat,is a first-order ARMA process of the form Sat AaSa.t-1 BaXa.t-1 Xat (26) We will solve for matrices A and B and show that this assumption is appropri- ate to establish a rational expectations equilibrium. The above perceived law of motion can also be written as []-]+] (27) or Yat gaYa.t-1 +UYa. (28) where (29) Se Xor Yor and go can be defined and analyzed symmetrically for agents in class b.For example,Spr is defined as Sot [Pi P2e Di D20 F2t] (30) and then Xor =Sor-E[Sh1]is defined straightforwardly
persistent, no stable equilibrium can be established. We will exclude this possibility in the subsequent analysis. 4. Information filtration and perceived investment opportunities 4.1. Perceived laws of motion Now we consider the differential information model. To solve for an equilibrium with non-completely-ranked information sets, we need a tractable method to deal with the information extraction problem. According to Assumption 5, agents in class a observe a record of current and past values S at"[PI 1t , PI 2t , D1t , D2t , F1t ]@, (25) where PI 1t "P1t !p 1 and PI 2t "P2t !p 2 are ‘demeaned’ stock prices. p 1 and p 2 reflect the unconditional expected risk premia, which will be discussed later. Define Xat"S at!E[S atDFa t~1] as the period-ahead conditional expectation error in S at. Following Sargent (1991), we assume that the filtration rule of agent a, or equivalently, the agent’s perceived law of motion for S at, is a first-order ARMA process of the form S at"Aa S a,t~1#Ba Xa,t~1#Xat. (26) We will solve for matrices Aa and Ba and show that this assumption is appropriate to establish a rational expectations equilibrium. The above perceived law of motion can also be written as C S at XatD "C Aa Ba 0 0 DCSa,t~1 Xa,t~1D #C Xat XatD, (27) or Yat"ua Ya,t~1#Ya,t , (28) where Yat"C S at XatD, Ya,t "C Xat XatD, and u a "C Aa Ba 0 0 D . (29) S bt, Xbt, Ybt and u b can be defined and analyzed symmetrically for agents in class b. For example, S bt is defined as S bt"[PI 1t , PI 2t , D1t , D2t , F2t ] (30) and then Xbt"S bt!E[S btDFb t~1] is defined straightforwardly. 1034 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1035 Given the perceptions outlined above,agents form period-ahead forecasts according to E[Yatl Ya.t-1]=gaYa.t-1, (310 E[Yhal Yb.-1]=goYb.t-1. (32) The actual law of motion for prices results from the dynamic market equilib- rium that equates asset supplies and asset demands arising from these expecta- tions.The rational expectations assumption requires that agents'perceptions be consistent with the actual law of motion. Let denote the state vector of the economy which contains Yt,Yor and some other state variables.With a proper choice of elements,will evolve according to =T(g-1+H(g)we (33) where w,is a vector of innovations.For a given set of perceptionsg=(ga g),the actual law of motion,Eq.(33),can be used to obtain the projections of Yir on Yja-1 for j=a,b. E[Yal Ya.t-1]=Ia(g)Ya.t-1, (34) E[YbrlYb.t-1]=Th(g)Yb.t-1, (35) where rj(g)(j=a,b)are obtained using the linear least squares projection formula. For the current asset pricing model,state vector z can be expressed as =(Pie P20 Die D2e Fi F2e Nie N2 Xat Xit' (36) and the vector of innovations in w,can be written as W:=[U1D U2D,UiF,U2F,UIN,U2N]' (37) The equilibrium of the market can be formally defined as: Definition.A(limited-information)rational expectations equilibrium(REE)with heterogeneous information is the fixed point (gg)=(r(g),T(g))such that the market clears in equilibrium. This kind of equilibrium concept was previously used by Sargent(1991)in investigating optimal investment in a production economy,and was then used by Hussman(1992)in an asset pricing model similar to ours.3 Both authors have 3 Hussman(1992)assumes that there is a single risky asset in the market.As we mentioned in the introduction,this single-asset setup is not appropriate to address the effects of private information on portfolio choices and related issues
3 Hussman (1992) assumes that there is a single risky asset in the market. As we mentioned in the introduction, this single-asset setup is not appropriate to address the effects of private information on portfolio choices and related issues. Given the perceptions outlined above, agents form period-ahead forecasts according to E[YatDYa,t~1]"ua Ya,t~1, (31) E[YbtDYb,t~1]"ub Yb,t~1. (32) The actual law of motion for prices results from the dynamic market equilibrium that equates asset supplies and asset demands arising from these expectations. The rational expectations assumption requires that agents’ perceptions be consistent with the actual law of motion. Let z t denote the state vector of the economy which contains Yat, Ybt and some other state variables. With a proper choice of elements, z t will evolve according to z t "T (u)z t~1#H (u)wt , (33) where wt is a vector of innovations. For a given set of perceptions u"(u a , u b ), the actual law of motion, Eq. (33), can be used to obtain the projections of Yjt on Yj,t~1 for j"a,b. E[YatDYa,t~1]"Ca (u)Ya,t~1, (34) E[YbtDYb,t~1]"Cb (u)Yb,t~1, (35) where Cj (u)( j"a, b) are obtained using the linear least squares projection formula. For the current asset pricing model, state vector z t can be expressed as z t "MPI 1t , PI 2t , D1t , D2t , F1t , F2t , N1t , N2t , Xat, XbtN@ (36) and the vector of innovations in z t , wt , can be written as wt "[v 1D , v 2D , v 1F , v 2F , v 1N , v 2N ]@ (37) The equilibrium of the market can be formally defined as: Definition. A (limited-information) rational expectations equilibrium (REE) with heterogeneous information is the fixed point (u a , u b )"(Ca (u), Cb (u)) such that the market clears in equilibrium. This kind of equilibrium concept was previously used by Sargent (1991) in investigating optimal investment in a production economy, and was then used by Hussman (1992) in an asset pricing model similar to ours.3 Both authors have C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1035
1036 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 discussed the properties of this equilibrium concept in detail,so we will not discuss them further.Below we use this equilibrium concept to investigate various implications of our multi-asset differential information asset pricing model. 4.2.Investment opportunities Now we consider the optimization problems faced by rational economic agents given the perceived laws of motion.Because of the symmetry between classes a and b,we will only consider a's optimization behavior. Investment opportunities characterize the distributions of stock returns.De- note I,+1=P+1+D+1-(1+r)P,as the excess returns earned by each share of stock.Then based on agent a's information sets,II can be expressed as Ⅱ，+1=-Tp+hgaYat-(1+rP,+ha+1 =-rp hgaYat-(1 +r)Yat heya+1 =ena.t Una.t+1, (38) where (39) (40) enat -rp hgaYat -(1 +r)P =-rp hgaYat -(1 +r)hYat (410 Una hUya. (42) h and h are selector matrices. Given investment opportunities,for a portfolio agent a receives a total excess payoff on.His wealth therefore evolves according to Wat+1=(1+r)Wam+aⅡ =(1+r）Wa+ena+a'nam (43) where Wa.+i is agent a's wealth at time t +1
discussed the properties of this equilibrium concept in detail, so we will not discuss them further. Below we use this equilibrium concept to investigate various implications of our multi-asset differential information asset pricing model. 4.2. Investment opportunities Now we consider the optimization problems faced by rational economic agents given the perceived laws of motion. Because of the symmetry between classes a and b, we will only consider a’s optimization behavior. Investment opportunities characterize the distributions of stock returns. Denote Pt`1 "Pt`1 #Dt`1 !(1#r)Pt as the excess returns earned by each share of stock. Then based on agent a’s information sets, P can be expressed as Pt`1 "!rp#hu a Yat!(1#r)PI t #h Ya,t`1 "!rp#hu a Yat!(1#r)hIYat#h Ya,t`1 , "ePa,t #Pa,t`1 , (38) where h"C 10100 2 0 01010 2 0D, (39) hI"C 100 2 0 010 2 0D, (40) ePat"!rp#hu a Yat!(1#r) PI t "!rp#hu a Yat!(1#r) hIYat, (41) Pa "h Ya. (42) h and hI are selector matrices. Given investment opportunities, for a portfolio Qa , agent a receives a total excess payoff Q@ a P. His wealth therefore evolves according to ¼a,t`1 "(1#r)¼at#Q@ a P "(1#r)¼at#Q@ a ePa #Q@ a Pa , (43) where ¼a,t`1 is agent a’s wealth at time t#1. 1036 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051