Information and asset prices in complete markets exchange economies

Economics Letters 65 (1999) 75–83

Information and asset prices in complete markets exchangeeconomies

*,1Tom Krebs

Department of Economics, Brown University (and University of Illinois), Providence, RI 02912, USA

Received 27 April 1998; received in revised form 22 April 1999

Abstract

This note analyzes the effect of public information, called news, on asset price behavior in Markovian(recursive) exchange economies with dynamically complete markets and risk-averse agents. A strongrelationship between asset prices in two economies with the same fundamental structure but a differentinformation structure is derived: asset prices in the economy with more information are a weighted average ofasset prices in the corresponding economy with less information. This representation theorem implies that theunconditional mean of asset prices is independent of the information structure (news process) and that anincrease in agents’ information (a decrease in agents’ forecast error of future dividends and/or discount factors)increases the volatility of asset prices (the unconditional variance of asset prices). 1999 Elsevier ScienceS.A. All rights reserved.

Keywords: Dynamic general equilibrium; Asset pricing; Information

JEL classification: D80; G12

1. Introduction

This note analyzes the effect of public information, called news, on asset price behavior in a Lucasasset pricing model (Lucas, 1978) with heterogeneous agents and sequentially complete markets. Theunderlying state process is a joint Markov process of economic fundamentals and news. News isuseful in forecasting future economic fundamentals in the sense that current news affects theconditional probability of future fundamentals. Agents’ preferences are time- and state-additive withstrictly concave one-period utility functions (risk-aversion) and common discount factors. The set of

*Tel.: 11-401-863-2097; fax: 11-401-863-1970.E-mail address: [email protected] (T. Krebs)1On leave from the University of Illinois. Current mailing address: Department of Economics, Box B, Brown University,Providence, RI 02912, USA.

0021-9673/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0021-9673( 97 )00065-4

76 T. Krebs / Economics Letters 65 (1999) 75 –83

assumptions adopted here implies that the equilibrium consumption allocations are independent of thenews realization. This news-independence of equilibrium consumption is used to derive a strongrelationship between asset prices in two economies with the same fundamental structure but adifferent information structure (news process): asset prices in the economy with more information area weighted average of asset prices in the corresponding economy with less information. Thisrepresentation theorem implies that the unconditional mean of asset prices is independent of theinformation structure and that the unconditional variance of asset prices is increasing in agents’information. Put differently, a reduction in agents’ forecast error of future dividends and/or discountfactors (conditional variance of future dividends and/or discount factors) does not change averageasset prices but increases asset price volatility.

The positive relationship between agents’ information and asset price volatility has a simpleeconomic rationale. An increase in the quality of news received by economic agents induces them toreact more strongly to variations in news. For an unchanged fundamental structure of the economy,the increase in agents’ willingness to trade will lead to stronger asset price movements. As a simpleexample, consider the case of serially uncorrelated dividends and constant discounting. If agents onlyobserve current and past realizations of the dividend process, their forecast of future dividends isconstant over time. Hence, agents never change their asset demand and equilibrium asset prices areconstant – the case of zero asset price volatility. If, however, agents observe in addition the realizationof a news variable which is correlated with future dividends, then their expectations of futuredividends will vary with the realization of the news variable. Thus, asset demand and thereforeequilibrium asset prices will randomly vary over time – the case of strictly positive asset pricevolatility.

This note assumes a Markovian structure since a large part of the macroeconomic literature dealingwith stochastic dynamic general equilibrium models has been formulated within that framework(Stokey and Lucas, 1989). This note also adds an additional stationarity assumption in order to beable to interpret the unconditional mathematical expectations as averages over one particular sample

2path (time series). Clearly, similar mathematical results could be proved in a non-Markovian settingwithout the stationarity assumption.

In this note the idea of information quality, or informational content of news, is given a precisemeaning by extending Blackwell’s analysis (Blackwell, 1951) to Markovian models. The formalcomparison of different information structures employed here is applicable to a broader class ofstochastic processes than the class of linear–stochastic processes often considered in the empiricalmacroeconomics literature (Sargent, 1987), but is consistent with that approach in the sense that moreinformation still means a smaller conditional variance (forecast error). The definition of moreinformation employed in this note might also be of interest to macroeconomic modeling not using alinear (or log–linear) approximation (Stokey and Lucas, 1989; Cooley, 1995).

The result that an increase in agents’ information increases asset price volatility links this note tothe excess-volatility studies (LeRoy and Porter, 1981 and Shiller, 1981), which use variance upperbounds to test the present value model of asset pricing: since more information increases asset pricevolatility the full information equilibrium establishes an upper bound on theoretically possible assetprice variances. This note shows that the assumptions usually made in the excess-volatility literature,namely constant discounting and/or linear stochastic processes, are not essential for deriving this

2Assuming ergodicity, an assumption stronger than stationarity.

T. Krebs / Economics Letters 65 (1999) 75 –83 77

result. Recent empirical studies by Ederington and Lee (1993), Harvey and Huang (1991), and Joneset al. (1994) provide support for the view that information arrival and price volatility are positivelycorrelated.

2. The model

Time is divided into a countable number of time periods t 5 1,2, . . . . The exogenous state space isa finite set X 5 1, . . . , X . Finiteness of the state space is only assumed to avoid unnecessaryh j

3mathematical technicalities. A generic element of the set X is denoted by x. Uncertainty andìnformation in this economy is described by a stationary Markov process (chain) hX j on X. Thet t51

`stationary Markov process hX j is constructed in the canonical way from the state space X, at t514stationary transition function, p :X 3 X → [0,1] , and one of the invariant distributions, p : X → [0,1],s

5of the transition function as initial distribution. A typical value of the transition function p is denotedby p(x9ux) and a typical value of the stationary distribution p is denoted by p (x). p(x9ux) stands fors s

the probability that the state in the next period is x9given that the current state is x and p (x) stands fors

the unconditional probability of state x.The state space X is assumed to be the Cartesian product of two sets, X 5 Y 3 Z, with Y 5 h1, . . . ,

Yj and Z 5 h1, . . . , Zj. Correspondingly, the current state of the economy decomposes: x 5 ( y, z). Thecomponent z represents an exogenous shock to economic fundamentals, whereas the component yrepresents public information, called news, about economic fundamentals. The joint Markov process

`X in conjunction with the above decomposition of the state space uniquely defines a marginalh jt t51` ` ˜process Y and a marginal process Z . The transition function p :Z 3 Z → [0,1] of the marginalh j h jt t51 t t51` ˜process Z and the corresponding stationary distribution p :Z → [0,1] can be calculated from theh jt t51 s

joint probabilities using

˜;z: p (z) 5Op ( y, z)s sy

˜;z, z9: p(z9uz) 5O O p( y9, z9uy, z)p ( yuz) (2.1)sy[Yy9[Y

p ( y, z)s]]]with p ( yuz) 8 ,s p̃ (z)s

˜where the formula for the marginal transition probabilities is valid for all z with p (z) . 0. Clearly,s

3The convention followed here is to denote sets by uppercase boldface letters, the cardinality of sets by the correspondinguppercase letters, and elements of the set by the corresponding lower case letters. Similarly, random variables are denoted byupper case letters and their realizations by the corresponding lower case letters.

4Strictly speaking, a transition function is a mapping p :X 3 3(X) → [0,1], where 3(X) denotes the power set of the set X.5Since the state space is finite, an invariant distribution always exists, even though it may not be unique. If there are

multiple invariant distributions, we (arbitrarily) pick one as initial distributions and thereby define a stationary Markovprocess in an unambiguous way.


`there is a continuum of joint processes X giving rise to the same marginal process ofh jt t51`fundamentals Z , each of which corresponds to a different information structure (news process).h jt t51

There is a finite number of (types of) agents i [ I ; h1, . . . , Ij. Agents’ preferences are time- andstate-additive. The one-period utility function and endowment of an agent are time- and news-

i˜invariant, that is, they are defined by (possibly fundamental-dependent) functions u : Z 3 R → R1i iand v : Z → R . The discount factor is common across agents: b 5 b, i [ I.11

3. Results

All proofs in this note are based on a present value formula for asset prices in which the discountfactor is the intertemporal marginal rate of substitution of any agent (the representative agent) andnews only affects conditional probabilities. In order to save space and since the proofs arewell-known, we only outline the argument leading to the particular present value representation ofasset prices used in this note. For a detailed treatment of this issue, see Krebs (1995).

The argument proceeds in two steps. First, Bewley (1972) ensures the existence of a contingentmarket equilibrium (CME) for the static Arrow–Debreu economy with an infinite-dimensionalcommodity space (assuming strictly positive endowment vectors). The CME consumption allocationsare evidently Pareto optimal. When agents are risk-averse and have a common discount factor, thePareto optimal allocations are news-invariant. The proof is by contradiction using Jensen’s inequalityand resembles the Cass and Shell sunspot result (Cass and Shell, 1983). In the second step, the CME(the Pareto optimal allocation) is implemented by repeated trading of a set of dynamically completesecurities. The resulting sequential (financial) market equilibrium (Radner, 1972) is by constructionPareto optimal and therefore also displays a news-independent consumption allocation. To summarize,we have the following:

3.1. Fact

Assume that agents’ preferences are time- and state-additive (expected utility) and that endowmentsand one-period utility functions are news-invariant. Suppose further that endowment vectors arestrictly positive and discount factors are common across agents. If in each period agents have theopportunity to trade X one-period (short-lived) securities with linearly independent asset payoffvectors (full rank of payoff matrix), then there exists a sequential market equilibrium with news-independent consumption allocation.

We assume that in addition to the X short-lived securities there are j [ J 5 1, . . . , J long-livedh jassets. For asset pricing purposes, an Inada-type condition on marginal utilities ensuring interiority ofagents’ optimal consumption choices is useful. With this additional assumption it follows that for eachlong-lived asset j [ J with asset payoffs (dividends) defined by a function d : Z → R the equilibriumj 1

asset price function q : X → R solvesj 1

;( y, z): q ( y, z)u9(z) 5 bO d (z9) 1 q ( y9, z9) u9(z9)p( y9, z9uy, z) (3.1)s dj j jx9

˜where u9(z) 5 u 9(c(z), z)is the equilibrium marginal utility of any agent and b his pure time discount


factor (for simplicity we skipped the index for agent i). Notice that the marginal utilities do notdepend on y.

For each asset j, Eq. (3.1) is a system of X linear equations in the X unknowns q ( y, z). Exploitingj

the transversality condition, the recursive asset pricing formula (3.1) can be solved by repeated use ofthe law of iterated expectations to yield the fundamental solution for the price of any long-lived assetj [ J

T u9(z )tt (t )]];( y, z): q ( y, z) 5 lim O O b d (z )p (z uy, z), (3.2)j j t tu9(z)T →` t51 z [Zt

(t )where p (z uy, z) stands for the probability of moving from state ( y, z) to any state with fundamentalt

shock z in t steps (periods).t

In order to analyze the effects of news on the first and second unconditional moment of asset prices,we compare two economies with different information structures (news processes). We begin with the

`simplest case, namely the comparison of an economy with fundamental process Z and no newsh jt t51` `(state process Z ) to an economy with the same fundamental process Z and news processh j h jt t51 t t51

``Y (state process (Y , Z ) ). Evidently, in the economy with news, agents have more informationh j h jt t51 t t t51

than in the corresponding news-free economy, even though the ‘fundamental structure’ is the same forJ˜both economies. Asset prices in the news-free economy are defined by a function q : Z → R with1

T u9(z )tt (t )˜ ˜]];z: q (z) 5 lim O O b d (z )p (z uz) (3.3)j j t tu9(z)T →` t51 z [Zt

Jand asset prices in the economy with news are described by a function q:X → R with1

T u9(z )tt (t )]];( y, z): q ( y, z) 5 lim O O b d (z ) p (z uy, z). (3.4)j j t tu9(z)T →` t51 z [Zt

p ( y, z)s]]]Define the conditional probabilities p ( yuz)8 . The following proposition is a simples p̃ (z)s

implication of the rules for conditional probabilities applied to (3.3) and (3.4):

Proposition 1. Asset prices in an economy without news are a weighted average of asset prices in acorresponding economy with news, that is, for all assets j [ J we have:

˜;z: q (z) 5O q ( y, z)p ( yuz).j j sy[Y

Proof. Consider first the set of news Y* for which p ( yuz) . 0. Multiplying the present value relations

(3.4) by p ( yuz), y [ Y*, and summing over all news y [ Y* yieldss

T u9(z )tt (t )]];z: O q ( y, z)p ( yuz) 5 lim O O b d (z ) O p (z uy, z)p ( yuz) (3.5)j s j t t su9(z)T →`y[Y* t51 z [Z y[Y*t

where the summation over y can be interchanged with the limit operation T → ` since only one limitoperation is involved. We also have


(t ) (t )Õ p (z uy, z)p ( yuz) 5 p (z uz). (3.6)t s ty[Y*

Combining (3.5) and (3.6) we have

T u9(z )tt (t )˜ ˜]];z: O q ( y, z)p ( yuz) 5 lim O O b d (z )p (z uz) 5 q (z). (3.7)j s j t t ju9(z)T →`y[Y t51 z [Z* t

Adding o q ( y, z)p ( yuz) 5 0 to the LHS of Eq. (3.7) proves Proposition 2. hy[Y\Y* j s

˜Define for each asset j [ J the price difference function Dq : X → R , Dq ( y, z) 8 q ( y, z) 2 q (z).j 1 j j j

Using the probability space X, 3(X), p , with p : X → [0,1] being the stationary distribution, thes ds s6˜˜ ˜ ˜ ˜functions q , q , Dq define random variables Q , Q , DQ . Define the sets Z(q )8 z [ Zuq (z) 5 qh jj j j j j j j j j

˜ ˜for any q in the image of the function q :Z → R and define the following conditional probabilitiesj j 1

p ( y, z)s˜ ˜ ]]]];y [ Y, ;z [ Z(q ): p ( y, zuq )8 .j s j Õ p (z)s˜z[Z( q )j

˜ ˜ ˜The expectation of the random variable DQ conditional on the event Q 5 q is given byj j j

˜ ˜E[DQ uq ]8 O O Dq ( y, z)p ( y, zuq ).j j j s jy[y˜z[Z( q )j

An immediate consequence of Proposition 1 is

˜ ˜;q : E[Q uq ] 5 0. (3.8)j j j

˜˜Moreover, by definition we have ;y, z: q ( y, z) 5 q (z) 1 Dq ( y, z), or, equivalently: Q 5 Q 1 DQ .j j j j j j

To summarize, in random variable notation Proposition 1 reads:

Proposition 19. Asset prices in an economy with news are equal to asset prices in the correspondingnews-free economy plus noise, that is, for all assets j [ J we have:

˜Q 5 Q 1 DQj j j

˜ ˜;q : E[DQ uq ] 5 0.j j j

Remark 1. The term noise for a random variable DQ with zero conditional mean is taken fromj

Rothschild and Stiglitz (1970) who also show that proposition 19 is equivalent to the statement that˜ ˜the random variable Q is a mean-preserving spread of Q . Note also that E[DQ uq ] 5 0 implies (butj j j j

˜is not necessarily implied by) the lack of correlation between the random variables DQ and Q . Thej j

random variables DQ and Q , however, are in general correlated.j j

6 ˜To be pedantic, the function q : Z → R should in this case be thought of as a functionj 1

˜ ˜ ¯ ˜ ˆ ¯ ˆq : X → R with q (y, z) 5 q (y, z) ;y, y, z.j 1 j j


Denote the unconditional mean of random variables by E[.] and the unconditional variance by˜Var[.]. Since Q is a mean-preserving spread of Q , we have:j j

Corollary 1. Average asset prices do not depend on the news process (information structure)

˜; j: E[Q ] 5 E[Q ].j j

Corollary 2. Asset prices in an economy with news are more volatile than asset prices in thecorresponding economy without news:

˜; j: Var[Q ] $Var[Q ].j j

¯ ˆRemark 2. The inequality in corollary 2 is strict whenever q (y,z) ± q (y,z) for at least onej j

¯ ˆfundamental shock z [ Z and at least one pair of news realizations y,y [ Y, which is typically thecase (see Krebs, 1995 for a formal genericity argument).

Remark 3. Because of corollary 1, an inequality analogous to corollary 2 also holds for thecoefficient of variation, that is, the standard deviation divided by the mean.

Finally, we compare two economies with different news processes, that is, we compare an economy`` `˜(Y , Z ) to an economy (Y , Z ) . Denote the signal space of news process Y by Y and theh jh j h jt t t51 t t t51 t t51

`˜ ˜signal space of news process Y by Y. Further, denote the transition functions corresponding to theh jt t51`` ˜ ˜˜Markov processes (Y , Z ) and (Y , Z ) by p : (Y 3 Z) 3 (Y 3 Z) → [0,1] and p : (Y 3 Z) 3h jh jt t t51 t t t51

˜(Y 3 Z) → [0,1], respectively. The following definition partially orders the set of news processes withrespect to their informational content:

`Definition 1. The news process Y is said to be more useful in forecasting the fundamental processh jt t51`` ˜ ˜Z than the news process Y , if Y , Y and for all current shocks z [ Z and for all newsh jh jt t51 t t51

˜˜ ˜realizations y [ Y there exists a set of news realizations A(y, z) , Y such that

˜ ˜;z9: p(z9uy, z) 5 O p(z9uy, z)p ( yuz). (3.9)s˜y[A( y,z)

Remark 4. The following example might be useful in illustrating the main idea behind the definition.˜Suppose Y 5 1, 2, 3 and Y 5 1, 2 and fix z, z9 [ Z. One arrangement that satisfies the definition ish j h j

p̃(z9u1, z) 5 p(z9u1, z)

p̃(z9u2, z) 5 p(z9u2, z)p(2uz) 1 p(z9u3, z)p(3uz).

Remark 5. The condition in the definition can be rewritten as

˜ ˜ ˜;z,z9,y : p(y uz,z9) 5 O p( yuz,z9)p( yuz). (3.10)˜y[A( y,z)

˜Eq. (3.10) says that for fixed realization z, z9 [ Z the random variable Y is a garbling of the randomvariable Y and that therefore, for given current shock z, every expected utility maximizer whose


˜utility is a function of the future shock z9 prefers observing Y to observing Y (Blackwell, 1951). Inthis sense, our definition of more useful information is an extension of the Blackwell definition to theMarkovian case in which the current fundamental shock might also be useful in forecasting futurefundamentals.

˜˜ ˜Remark 6. If ;z [ Z,y [ Y we have E[Z uy,z] 5 E[Z uy,z], then it follows immediately thatt11 t11

˜˜ ˜ ˜;z [ Z, ;y [ Y,;y [ A(y,z): Var[Z uy,z] $Var[Z uy,z]. (3.11)t11 t11

Hence, Definition 1 also covers the case usually considered in the macroeconometric literature inwhich the conditional variance is used as a measure of the quality of the forecast of economic agentsand the econometrician (Sargent, 1987).

` `Remark 7. If a news process Y can be decomposed into two component processes, Y 5 (Y ,h j h j ht t51 t t51 1t` `Y ) , then according to Definition 1 the joint process Y is more useful in forecasting thej h j2t t51 t t51

` ` `fundamental process Z than any of its component processes Y and Y . Hence,h j h j h jt t51 1t t51 2t t51

Definition 1 encompasses the case considered by LeRoy and Porter (1981) and others.

Remark 8. In order to avoid ‘fake’ refinements of the information structure, one could add the˜requirement that at least one of the sets A(y, z) has more than one element and at least one of the

corresponding conditional probabilities satisfies p ( yuz) ± 0, 1 /2.s

Using Definition 1, we have the following analog to Proposition 1, from which analogous versionsof Corollary 1 and Corollary 2 immediately follow:

` `Proposition 2. If the news process Y is more useful in forecasting the fundamental process Zh j h jt t51 t t51` `˜ ˜than the news process Y , then asset prices in the economy with news process Y are ah j h jt t51 t t51

`weighted average of asset prices in the corresponding economy with news process Y , that is, forh jt t51

all assets j [ J we have:

˜˜ ˜ ˜;y [ Y, ;z [ Z: q (y, z) 5 O q ( y, z)p ( yuz).j j s˜y[A( y,z)

Proof. Repeat, mutatis mutandis, the proof of Proposition 1.

Acknowledgements

I would like to thank John Donaldson, Prajit Dutta, Duncan Foley, Richard Ericson, and PaoloSiconolfi as well as seminar participants at Carnegie-Mellon, Columbia University, University ofIllinois, McMaster University, and UC-Riverside for helpful comments. All remaining errors aremine.

References

Bewley, T., 1972. Existence of equilibria in economies with infinitely many commodities. Journal of Economic Theory 4,514–540.


Blackwell, D., 1951. Comparison of experiments. In: Neymann, J. (Ed.), Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability, University of California Press.

Cass, D., Shell, K., 1983. Do sunspots matter. Journal of Political Economy 91, 193–227.Cooley, T. (Ed.), 1995. Frontiers of Business Cycle Research, Princeton University Press, Princeton, New Jersey.Ederington, L., Lee, J., 1993. How markets process information: news releases and volatility. Journal of Finance 1161–1191.Harvey, H., Huang, R., 1991. Volatility in foreign currency future market. Review of Financial Studies 4, 543–570.Jones, C., Kaul, K., Lipson, M., 1994. Information, trading, and volatility. Journal of Financial Economics 36, 127–154.Krebs, T., 1995. Expectational equilibrium in sequential market economies, Columbia University, Ph.D. thesis.LeRoy, S., Porter, R., 1981. The present-value relation: test based on implied variance bounds. Econometrica 49, 555–574.Lucas, R., 1978. Asset prices in an exchange economy. Econometrica 66, 1429–1445.Radner, R., 1972. Existence of equilibrium of plans, prices and price expectations in a sequence of markets. Econometrica

40, 289–303.Rothschild, M., Stiglitz, J., 1970. Increasing risk: a definition. Journal of Economic Theory 2, 225–243.Sargent, T., 1987. Macroeconomic Theory, 2nd edn, Academic Press, Inc.Shiller, R., 1981. Do stock prices move too much to be justified by subsequent changes in dividends? American Economic

Review 71, 421–435.Stokey, N., Lucas, R., 1989. Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge MA.

Documents

Information and asset prices in complete markets exchange economies