Incomplete information and heterogeneous beliefs in continuous time finance

After a brief review of the literature, the author analyzes the consequences of incomplete informa-tion and heterogeneous beliefs for optimal portfolio and consumption choice and equilib

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Alexandre Ziegler

Incomplete

Information

and Heterogeneous Beliefs

in Continuous-time Finance

With 43 Figures

and 8 Tables

Ecole des HEC,

Universite de Lausanne

BFSH 1

CH -1 0 1 5 Lausanne-Dorigny, Switzerland

Mathematics Subject Classification (2003): 91 B28, 91 B70, 93 Ell, 93 E20

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Continuous-time finance was developed in the late sixties and early seventies

by R.C Merton Over the years, due to its elegance and analytical nience, the continuous-time paradigm has become the standard tool of anal-ysis in portfolio theory and asset pricing However, and probably because it was developed hand in hand with option pricing, in which investors' expecta-tions were thought not to matter, continuous-time finance has for a long time almost entirely neglected investors' beliefs More recently, the development

conve-of martingale pricing techniques, in which expectations playa dominant role, and the blurring boundary between those methods and the original methods

of continuous-time finance based on the Ito calculus, have allowed tions to regain their central role in finance

expecta-The habilitation thesis of Professor Alexandre Ziegler is entirely devoted

to the role of expectations in continuous-time finance After a brief review of the literature, the author analyzes the consequences of incomplete informa-tion and heterogeneous beliefs for optimal portfolio and consumption choice and equilibrium asset pricing Relaxing the assumption that investors can ob-serve expected dividend growth perfectly, the author shows that incomplete information affects stock prices and their dynamics, thus providing a potential explanation for the asset price bubble of the late 1990s He also demonstrates how the presence of heterogeneous beliefs among investors affects their opti-mal portfolios and their optimal consumption patterns This analysis, which nicely combines martingale methods and Ito calculus, provides the basis for

an investigation of the consequences of heterogeneous beliefs for equilibrium asset prices The author demonstrates that heterogeneous beliefs can have

a dramatic impact on equilibrium state-price densities, thus providing an explanation for the option volatility smile and the patterns of implied risk aversion recently documented in the literature Finally, the study considers costly information and issues of information aggregation It demonstrates that financial markets in general will not aggregate information efficiently, thus providing a plausible explanation for the equity premium puzzle It is truly exciting to observe the richness and diversity of the results obtained

by the author by simply relaxing the unrealistic assumptions of complete information and homogeneous beliefs It is my hope that this work stimu-

lates further research in the fascinating field of incomplete information and heterogeneous beliefs

Heinz Zimmermann

Professor of Economics and Finance

University of Basle

Any increase in wealth, no matter how insignificant, will always result in an increase in utility which is inversely proportionate to the quantity of goods already possessed

We obtain dy = b dx/x or y = blog(x/a)

Daniel Bernoulli, Exposition of a New Theory on the Measurement of Risk

It is with these words that in 1738, Daniel Bernoulli [6] first claimed that utility must be logarithmic Although logarithmic utility is no longer considered to describe investor preferences accurately today, it is nevertheless omnipresent in modern economics and finance The reason that this is so is not merely historical Indeed, logarithmic utility has a number of convenient properties

An alternate title for this study might be: Incomplete Information and Heterogeneous Beliefs with Non-Logarithmic Utility As will become clear be-low, logarithmic utility is in many situations a benchmark case in which things behave nicely, both analytically and in terms of results If agents have logarithmic utility, then, in most of the situations considered in this study, investors' information does not really matter for most economic variables As soon as one departs from the logarithmic utility assumption, however, infor-mation does matter, and can influence a whole range of economic variables, from agents' optimal portfolio behavior to asset prices and equilibrium in-terest rates This text deals with the implications of agents' information and beliefs for economic variables

This study is a revised version of my Habilitation thesis which was written while I was visiting Stanford University I would like to express my gratitude

to Professor Darrell Duffie, whose help was instrumental in making this work possible Not only did his class, Dynamic Asset Pricing Theory, provide me with the tools necessary for analyzing the problems addressed in this study

He also gave me some useful advice and references on some of the harder aspects of this work I would also like to thank Professors Heinz Zimmer-mann and Heinz Muller for their precious and devoted assistance, advice and encouragement while I was preparing this Habilitation Professors Sunil Ku-mar and George Papanicolaou advised me on the numerical methods used

in Chapter 3 In addition, Dr Stephanie Bilo and Professors Christian

Gol-lier and Louis Eeckhoudt provided me with some useful references on some aspects of this study

I am also deeply indebted to Dr Olivier Kern for his willingness to go through the formal arguments of this study and to Dr Alfonso Sousa-Poza for his invaluable help in correcting my English I would also thank Dr Hedwig Prey for her help with some Jb.'IE;X subtleties

Parts of this study have been previously published in academic journals Some aspects of Chapter 2 appeared in the Swiss Journal of Economics and Statistics [79], Chapter 3 in the European Finance Review [78], and parts

of Chapter 5 in the European Economic Review [80] My thanks go to the editors, Peter Kugler, Simon Benninga, and Harald Uhlig, as well as to the referees, for the many valuable suggestions they made, which greatly con-tributed to improving this text All errors remain mine

Last but by no means least, I would like to express my gratitude to the Swiss National Science Foundation and to my family for making my stay in Stanford possible, and to my colleagues and friends - both in Switzerland and Stanford - for providing the environment and encouragement required

to complete this Habilitation

Lausanne,

1 Incomplete Information: An Overview 1

1.1 Introduction 1

1.2 Portfolio Choice 1

1.2.1 Gennotte's Model 2

1.2.2 The Inference Process 4

1.2.3 Optimal Portfolio Choice 4

1.2.4 An Example 7

1.2.5 The Short Interest Rate 9

1.3 The Term Structure of Interest Rates 10

1.3.1 Dothan and Feldman's Models 10

1.3.2 A Characterization of the Term Structure 11

1.4 Equilibrium Asset Pricing 15

1.4.1 Honda's Model 16

1.4.2 The Equilibrium Price Process 17

1.5 Conclusion and Outlook 19

2 The Impact of Incomplete Information on Utility, Prices, and Interest Rates 23

2.1 Introduction 23

2.2 The Model 25

2.3 Equilibrium 27

2.3.1 The Equilibrium Expected Lifetime Utility 27

2.3.2 The Equilibrium Price 28

2.3.3 The Equilibrium Interest Rate 30

2.4 Logarithmic Utility 31

2.4.1 The Equilibrium Expected Lifetime Utility 31

2.4.2 The Equilibrium Price 32

2.4.3 The Equilibrium Interest Rate 33

2.5 Power Utility 35

2.5.1 The Equilibrium Expected Lifetime Utility 35

2.5.2 The Equilibrium Price 40

2.5.3 The Equilibrium Interest Rate 48

2.5.4 Hedging Demand and the Equilibrium Price of Estimation Risk 50 2.6 Information, Utility, Prices, and Interest Rates: A Synthesis 51

2.6.1 Expected Lifetime Utility 51

2.6.2 Share Prices 52

2.6.3 Interest Rates 53

2.7 Time-Varying Parameters 55

2.7.1 The Equilibrium Expected Lifetime Utility 56

2.7.2 The Equilibrium Price 58

2.7.3 The Equilibrium Interest Rate 58

2.8 Conclusion 61

3 Optimal Portfolio Choice Under Heterogeneous Beliefs 65

3.1 Introduction 65

3.2 The Model 67

3.3 The Deviant Agent's Problem 70

3.4 Optimal Portfolio Choice 71

3.5 An Example 74

3.6 Conclusion 78

4 Optimal Consumption Under Heterogeneous Beliefs 81

4.1 Introduction 81

4.2 The Cox-Huang Methodology 82

4.3 Heterogeneous Beliefs 84

4.4 An Example 86

4.4.1 The Model 86

4.4.2 Optimal Consumption Patterns Under Heterogeneous Beliefs 87

4.4.3 An Algebraic Solution 95

4.4.4 The Effect of the Time Horizon 102

4.5 Portfolios and Consumption: A Synthesis 105

4.6 Conclusion 107

5 Equilibrium Asset Pricing Under Heterogeneous Beliefs 109

5.1 Introduction 109

5.2 The Model 111

5.3 Equilibrium Consumption 113

5.4 Equilibrium Prices 116

5.4.1 The Equilibrium State-Price Density 117

5.4.2 The Equilibrium Short Rate 123

5.4.3 The Equilibrium Yield Curve 128

5.4.4 The Equilibrium Share Price 133

5.4.5 Equilibrium Option Prices and the "Smile Effect" 139

5.5 Implied Risk Aversion 145

5.6 Conclusion 146

6 Costly Information, Imperfect Learning, and Information

Aggregation '" 149

6.1 Introduction 149

6.2 The Model 151

6.2.1 The Economy 151

6.2.2 The Inference Process: Imperfect Learning 152

6.3 Portfolio Choice under Costly Information , 154

6.3.1 The Agent's Problem 154

6.3.2 The Agent's Optimal Investment and Research Policy 155 6.3.3 Determinants of the Demand for Information 157

6.3.4 Diversification and Information Costs , 160

6.4 Equilibrium Asset Pricing 161

6.5 Information Aggregation and the Equity Premium 165

6.6 Conclusion 170

7 Summary and Conclusion 173

A Conditional Mean and Variance of In(xs ) 179

B Conditional Mean and Variance of In(xs ) with Time-Varying Parameters 181

C The Short Rate Under Heterogeneous Beliefs 183

References 187

List of Figures 191

List of Tables 193

List of Symbols 195

1 Incomplete Information: An Overview

1.1 Introduction

Major classical portfolio choice and asset pricing theories to date assume that investors know the assets' expected return and volatility This assumption, however, is not fulfilled in practice In the real world, investors must estimate expected returns either from fundamentals, or from market data This is what

is meant when we speak of incomplete information

The literature on portfolio selection has analyzed many other capital ket imperfections, e.g transactions costs (Duffie and Sun [30], Dumas and Lu-ciano [32]) Very few papers in the literature analyze the problem of parameter estimation on asset markets and its consequences for optimal portfolio choice and asset pricing Williams [75], Detemple [23], Dothan and Feldman [27] and Gennotte [36] are notable exceptions More recently, Wang [73] and Honda [48, 49] have analyzed optimal portfolio choice and equilibrium asset pricing when mean returns or the rate of growth in dividends are unobservable This chapter provides a brief review of the existing literature on portfolio choice and asset pricing under incomplete information The discussion is cen-

mar-tered around three main aspects Section 1.2 analyzes portfolio choice Section 1.3 characterizes the term structure of interest rates Section 1.4 presents equi- librium asset pricing using state-price deflator techniques Finally, Sect 1.5 makes a few concluding remarks and sets the stage for the following chapters

1.2 Portfolio Choice

How do investors form optimal portfolios when they do not know the assets' expected returns? The answer to this question was provided by Williams [75]

in a continuous-time model for the special case of constant expected returns

He showed that an investor choosing an optimal portfolio under incomplete information does two things: the first is to replace the unknown expected returns with his current conditional expectations, i.e his best estimates of the unknown expected returns The second is to take a "hedging" position in the sense of Merton [61] to protect himself against unfavorable changes in his estimates of expected returns

A Ziegler, Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance

© Springer-Verlag Berlin Heidelberg 2003

The process of optimal portfolio choice under incomplete information in the more general case in which assets' true expected returns follow a diffusion process is surprisingly simple As was shown by Gennotte [36], when choosing their optimal portfolio, investors can proceed in two steps: first, they estimate expected returns from the history of security prices; second, they form an optimal portfolio of assets using estimated expected returns As is the case

in Williams [75], incomplete information has an effect on investors' optimal portfolios through the hedging component 1 Gennotte's model lies at the heart of the incomplete information literature and is therefore the focus of this section

1.2.1 Gennotte's Model

Gennotte [36] considers a continuous-time economy with a single physical good, the numeraire, which may be allocated to consumption or investment There is one instantaneously riskless asset paying a rate of return of rt and

n risky assets (production technologies) whose values follow

dSt = ISlLtdt + Is:EtdBt , (1.1) where

as a hedge against changes in mean returns He demonstrates that the hedging demand is large if there is significant uncertainty about the drift, or if the variance

of returns is small

is the n x n matrix of the process' instantaneous standard deviation and dBt

is a n-dimensional Brownian motion vector The variance-covariance matrix

of rates of returns, EtE~, is positive definite for all t and known to all agents.2

Gennotte assumes that the drift (1.3) is not constant, but evolves domly through time according to the stochastic differential equation

ran-(1.5)

where ItIl,O is a n x 1 vector known to all agents, It/J,l and Ell,S are known

n x n matrices, and Ell is a known n x m matrix The formulation in (1.5) is

sufficiently general to capture such phenomena as mean-reverting expected returns.3 It is assumed that the m-dimensional Brownian motion vector dB/J,t

is independent of dBt Equation (1.5) thus captures the fact that part of the change in the assets' drift is correlated with the unexpected change in asset value

There is a fixed number K of agents in the economy, identical in their

preferences and endowments Agents are characterized by their initial wealth

Wt and their preferences, u Although agents know the deterministic functions

of time E t, It/J,o, Itll,l, Ell,S and Ell and observe instantaneous returns on

the n assets, dSt , they do not know the true expected return Itt That is,

they only have the filtration FS = {Fl}, where Fr = a(Su : u ~ t) They therefore seek to maximize their expected lifetime time-additive utility of

consumption conditional on all available information as of date t,

(1.6)

where u is increasing and concave in current consumption Cs The investors' decision variables in this maximization problem are their portfolio holdings

w and their current consumption c

2 The assumption that Et be known may, at first, seem somewhat restrictive However, as shown by Williams [751 for the case of constant parameters, an unknown volatility parameter can be estimated to any desired degree of accuracy from the history of returns by increasing the sampling frequency Such is not the case, however, for unknown drift parameters It is therefore natural to focus

on unknown expected returns when analyzing portfolio choice under incomplete information

3 In order to capture mean-reverting expected returns, it suffices to set P/J,1 <

O To see why, suppose that n = 1 and let li = Et(p.) Then, li satisfies dli = (P/J,o + P/J,1li.)ds, with lit = Pt The solution to this ordinary differential equation is

li = _P/J,o + (Pt + P/J'o) exp(p/J,1(S - t)) ,

P/J,1 P/J,1 which is mean-reverting whenever P/J,1 < o

1.2.2 The Inference Process

Because they cannot observe the true drift /-Lt, agents seek to extract mation on future expected returns from their observation of past returns

infor-At initial time 0, agents view the distribution of /-Lo as Gaussian with mean vector mo and variance-covariance matrix Yo As they observe new returns, agents update their estimate mt of expected returns according to

where

(1.8) denotes the unexpected component of the asset prices' change from the agents' viewpoint, i.e conditional on their information Thus, B is a mar-tingale with respeCt to the filtration FS

The variance-covariance matrix of the agents' estimate, denoted by V t ==

E ((mt - /-Lt)(mt - /-Lt),I.rf) , follows

dVt = (E/L,SE~,S + E/LE~ + /-L/L,l V t + Vt/-L~,l

- (E/L,sE~ + Vt) (EtED-1 (EtE~,s + Vt) )dt , (1.9) and is thus a deterministic function of time Equations (1 7) and (1.9) describe the agents' optimal update of the estimated drift and the change in the conditional variance of the estimated drift when they use all available (return) information F?

1.2.3 Optimal Portfolio Choice

Since dBt is Brownian motion with respect to {F t S }, it contains no tion on the future variations of St and mt Moreover, Vt is a deterministic function of time, so the distribution of dmt is characterized by the state vector mt As a result, the system [St, mt] is Markov, i.e St and mt de-termine the probability distribution of S and m over the next infinitesimal time interval [t, t+dt] Thus, the vector mt fully characterizes the investment opportunity set perceived by investors at any time t

informa-This leads to the following separation result: agents solve the investment decision problem in two stages:

- derivation of the vector of (conditional) expected returns mt, and

- choice of an optimal portfolio of assets using estimated expected returns mt·

Each agent's problem is to choose consumption c and an optimal portfolio w

so as to maximize his expected lifetime utility of consumption conditional on his information at time t, .'Ft

subject to the budget constraint

dWt = Wt (w'Is1dSt + (1 - w'l)rtdt) - Ctdt (1.11) Substituting (1.1) into (1.11) yields

dWt = Wt (w'(J.ttdt + I:tdBt) + (1 - w'l)rtdt) - Ctdt (1.12) The problem at this point is that the agent does not know the true parameter

J.l.t, but only its estimated value mt Using dBt = dBt + I:;-1 (J.l.t - mt)dt,

which is the unexpected component of the asset prices' change from the agent's viewpoint, i.e conditional on his information, (1.12) can be rewritten

as

dWt = Wt (w' (mtdt + I:tdBt) + (1 - w'l) rtdt) - ctdt (1.13)

In this Markovian setting, the methodology developed by Merton [61] to solve the dynamic investment-consumption problem can be applied directly Defining

the necessary optimality condition for (1.10) is

and tr(·) denotes the trace, subject to the boundary condition J(W,m,T) =

O Differentiating (1.15) partially with respect to the decision variables yields the following first-order conditions:

- Second, the n hedging portfolios

I:tI: t "'-'t"'-'/l,S + t JwwWt (1.20)

are constructed so as to hedge against changes in the estimated investment opportunity set That is, in addition to the correlation between prices and the true investment opportunity set, I:/l,sI:L investors take the cumulated estimation risk V t into account when forming their portfolios

When there is no parameter uncertainty, mt = J.Lt and V t = 0, and Merton's [61] optimal portfolio strategy,

uncer The first is the term (I:tI:~,s + Vd, i.e., the sum of the covariance between the true investment opportunity set and asset prices I:tI:~,s and of the degree of parameter uncertainty V t

- The second factor driving investors' hedging demand and therefore the direction and magnitude of the effect of incomplete information on portfolio demands is the investor's wealth-state risk aversion J Wm, and thus depends

on investor preferences

Under additional assumptions about the two factors Vt and JWm, one can

say a little more about how optimal portfolio demands under incomplete information compare to those under complete information Suppose that Vt

is nonnegative.4 Then, when JWm > 0, -JWm/(JwwWd > 0 and the demand for risky assets is higher under incomplete information than under complete information When JWm < 0, then -JWm/(JWW Wt} < 0 and incomplete information leads to a reduction in the demand for assets whose expected returns are uncertain.5

It is instructive to compare these results with those obtained in static models As shown by Klein and Bawa [53}, estimation risk will lead to a decrease in investors' optimal portfolio demand This is not necessarily the case in a continuous-time model, where price risk and estimation risk are, to some extent, distinct Whereas price risk influences the investors' tangency portfolio, estimation risk influences their hedging portfolio Because the sign

of hedging demand depends on investor preferences (and risk aversion is not sufficient to induce hedging behavior), the influence of estimation risk on optimal portfolio choice is in general ambiguous

1.2.4 An Example

To illustrate that the effect of incomplete information on investors' portfolio demands depends on investor preferences, consider an investor in the follow-ing situation, analyzed by Brennan [9] Suppose that there is a single risky asset available for investment, with price dynamics

(1.22) Suppose that p, is a constant unknown to the investor As a result, the investor estimates it from past price data Suppose that at initial time 0, the investor views the distribution of p, as normal with mean mo and variance Vo Using

(1 7), as new price information becomes available, the investor updates his estimate mt according to

5 From the analysis in Benveniste and Scheinkman [5], the value function J will

be concave in wealth whenever u is concave in current consumption c Thus,

-Jwm/(Jww W t ) has the same sign as JWm

where dBt = dBt + ((J.t - mt)la)dt The mean square error of this estimate,

Vt, follows

v,2 dVt = t-dt

Consider now the investor's investment decision, and suppose for ity that he derives utility exclusively from final consumption,

where WT denotes the investor's terminal wealth Suppose further that his

terminal utility of wealth is of the isoelastic class,

WoO<

B(WT) = .:r , a < 1

(The limiting case in which a ~ 0 corresponds to logarithmic utility.) The

investor chooses the share of his wealth invested in the risky asset, w, so

as to maximize his expected utility from terminal wealth, conditional on his information at time t,

(1.27) subject to the budget constraint

Defining

J(Wt,mt,t) == maxE(B(WT» = maxE ( ; ) , (1.29)

J must satisfy the Bellman equation

(1.30)

with the boundary condition J(W, m, T) = WO< la

Under the assumed investor preferences, J can be rewritten as the product

of two functions,

(1.31)

Then, computing the partial derivatives of J as a function of I, substituting into the Bellman equation (1.30) and simplifying yields

1 V'?)

subject to the boundary condition l(m, T) = l

Differentiating this expression partially with respect to w yields the order condition

first-(1.33) Solving for w gives the optimal percentage investment in the risky asset

mt - Tt Vam/ I

w -- (1 - +

To characterize the investor's portfolio demand more precisely, note that due

to non-satiation, expected lifetime utility must rise as investment ties improve, i.e Jm > O Since 1m = Jm/(Wo /a), this implies that 1m > 0 for a > 0 and 1m < ° for a < 0 Thus, the investor's demand for the risky asset is higher under incomplete information than under complete informa-tion whenever a > 0, i.e whenever the investor is less risk-averse than the log-utility investor Conversely, when a < 0, so that the investor is more risk-averse than the log-utility investor, 1m < ° and his demand for the risky asset

opportuni-is lower under incomplete information than under complete information In the special case of logarithmic utility (a = 0), the investor's hedging demand

is zero in both cases and his demand for the risky asset under incomplete information is the same as it would be under complete information

1.2.5 The Short Interest Rate

'I\uning back to the general case of Gennotte's model, under the assumption

of homogeneous beliefs, one can compute the equilibrium interest rate from investors' optimal portfolio demands (1.19) Premultiplying this expression

by 1', remembering that l'w = 1 in equilibrium since the net supply of the risk-free asset is zero and solving for Tt yields the following expression for the equilibrium short rate:

(J~:W + I' (EtED-1 mt

+ I , (~ ~')-l (~~' ,ut,ut ,ut,u/l-,S + V) Jwm) t Jw ( 1.35 )

Under some additional assumptions regarding investor preferences, it is sible to characterize the term structure of interest rates implied by (1.35), a question to which we now turn

pos-1.3 The Term Structure of Interest Rates

How does incomplete information influence the term structure of interest rates? Although they have not been able to produce a definite answer, a number of models have investigated this question

Stulz [70] analyzes interest rate behavior in a model in which optimizing households with logarithmic utility functions are uncertain about monetary policy Agents learn the true dynamics of the money supply as they acquire more data about changes in the money supply He shows that the variance of interest rates increases with the households' uncertainty about the monetary authority's policy However, he does not solve for the full term structure of interest rates

Dothan and Feldman [27] and Feldman [34] apply the methodology of Cox, Ingersoll and Ross [18, 19] in the context of an incomplete information economy and analyze multiperiod bonds and the term structure of interest rates This section presents the results of their models

1.3.1 Dothan and Feldman's Models

Dothan and Feldman [27] and Feldman [34] analyze an economy very ilar to that of Gennotte [36] Their model is that of a production economy with a single unobservable productivity factor lit in which returns in the n

sim-production technologies evolve according to

(1.36) where Is is the diagonal matrix of current asset prices, Ao and Ai are known

n x 1 vectors of constants, :E is a constant n x n matrix, and dB t is a n-dimensional Brownian motion vector The scalar li is assumed to have dynamics

(1.37) where J.LI-',O, J.LI-',i and (J'I-' are known scalar constants and :E I-',S is a known 1 x n

vector of constants Agents are unable to observe the true parameter lit At initial time 0, they view the distribution of flo as Gaussian with mean rno and variance Vo Agents seek to maximize their expected lifetime time-additive

utility of consumption conditional on all available information as of date t,

(1.38)

where u is increasing and concave in current consumption Ca Their decision variables in this maximization problem are their portfolio holdings w and their current consumption c

This model can be analyzed by setting

in Gennotte's model Given the new information structure, agents update their estimate of lit, mt, following

where

The conditional variance of mt, Vi (now also a scalar), follows

dVi = (EI',sE~,s + a; + 2J.L1',1 lit

(1.40)

- (EI',sE' + A~ Vi) (EE,)-l (EE~,s + Al Vi) )dt , (1.42) and is again a deterministic function of time

1.3.2 A Characterization of the Term Structure

After solving the investor's consumption and portfolio problem and obtaining expressions similar to those presented in Sect 1.2.3 above for suitable changes

in parameters, Dothan and Feldman [27] characterize the term structure of interest rates In order to obtain explicit solutions, they assume that investors have logarithmic preferences,

(1.43)

The analysis in Merton [62] shows that, in this case, the indirect utility tion is separable in wealth Wt and in the state variables mt and t,

func-(1.44) Therefore, one has JWm = 0, Wdww / Jw = -1, and (1.35) can be rewritten

Thus, under incomplete information, the equilibrium instantaneous spot terest rate Tt is a linear function of mt, the conditional mean of the unob-servable factor Since Tt is a one-to-one function of mt, it represents the best estimate of the unknown productivity factor lit

in-Using the dynamics of the estimated unknown productivity parameter,

mt, the dynamics of the short rate can be computed as

is driven by lit, the degree of parameter uncertainty The observed variability

of the spot rate is driven by the sum of these two effects, EJ.t,sE' + Ai lit

As a result, one cannot tell in general if the instantaneous variance of the estimated investment opportunities will be lower than, equal to, or higher than the instantaneous variance of the true investment opportunities It is therefore also impossible to tell whether the volatility of the spot rate will be higher, identical or lower under incomplete information than under complete information

In a complete information economy, the changes of the stochastic ment opportunity set are observable, and equilibrium interest rates are set

invest-to perfectly imitate these changes (see Cox, Ingersoll and Ross [18]) As a sult, low variability of the spot rate implies a low volatility of the investment

re-opportunity set Under incomplete information, however, it is impossible for investors to perfectly duplicate changes they cannot observe The variability

of the spot rate and that of the true investment opportunity set do not respond anymore The variance of the spot rate now reflects the variability

cor-in the estimated cor-investment opportunity set, which is driven by the weight

that consumers put on the new information contained in the observed ized outputs Low volatility of the spot rate might now mean low learning

real-ability about the changes of the investment opportunity set rather than low

volatility of it Conversely, high volatility of the spot rate might imply high

estimation error of the changes in the investment opportunity set rather than high volatility of it

To solve for the term structure of interest rates, note that the price at time t of a default-free zero-coupon bond maturing at time s, A(t, s), must

satisfy the partial differential equation

o = ~ara~Arr + (J.Lr,o + J.Lr,lr - (A~ + A~mt - rl/rE-la~) Ar

subject to the boundary condition A(t, t) 1, where the subscripts of A

denote partial derivatives

Observe from (1.50) that incomplete information influences the term structure of interest rates through two channels The first is the variabil-

ity of the spot rate a r , which changes as a function of individuals' quality

of information The second is the covariance between returns on optimally invested wealth and the perceived changes in the investment opportunities, i.e., the market risk premium (Ah +Ai mt -rl/)~-la~ These effects combine and influence the dynamics of the spot rate, thus leading to a term structure

of interest rates under incomplete information that differs from that under complete information

In a complete information economy, the term structure of interest rates is

determined by the expectations regarding future spot rates, the market risk premium, and the Jensen's inequality bias In an incomplete information

economy, an additional factor is involved: the price of the bond A( t, s) is a

function of the path of the estimation error through the life of the bond This factor influences both the market risk premium bias and the Jensen's inequality bias

in general if the market risk premium will be lower, equal, or higher under incomplete information than under complete information

Using (1.51), the yield curve can be computed as

infor-The third and fourth terms, the market risk premium bias

6 To see this, let f = Et(r.) Then, fa satisfies df = (I-'r,O + I-'r,lf.)ds with ft = rt The solution to this ordinary differential equation is given by

Ts = _.!:!: Q + #-£7',1 (rt +.!:!: Q.) P, ,l exp(l-'r,l(S - t» Using Fubini's theorem then yields

E (_1_ f· r dU) = _1_ f· T du = _.!:!: Q _ (.!:!: Q + r) 1-eXP(Pr,t(S-t»

t s-t Jt U s-t Jt U Pr,t Pr,t Pr,t(s-t)

in-1.4 Equilibrium Asset Pricing

The models presented so far took an unknown but given asset price process and analyzed the consequences for portfolio choice and the term structure of interest rates Another strand of the literature assumes that firms' dividends

follow a given but unknown process and derive equilibrium prices and price dynamics using state-price deflator techniques

Wang [73] analyzes intertemporal asset pricing under asymmetric tion He assumes two categories of investors with exponential utility Whereas the informed investors have perfect private information about a state vari-able determining the expected growth in dividends, the uninformed do not Since the growth rate of dividends determines the rate of appreciation of stock prices, changes in prices provide signals about the future growth of div-idends Uninformed investors rationally extract information about the state

informa-of the economy from the behavior informa-of prices as well as dividends However, prices and dividends are not sufficient to reveal the value of the state to the uninformed investors As a result, information asymmetry persists in equi-librium Wang demonstrates that information asymmetry can increase price volatility and negative autocorrelation in returns

Honda [49] analyzes equilibrium asset pricing under unobservable switching mean earnings growth Although much simpler than Wang's model, Honda's model focuses on incomplete information and abstracts from issues

regime-of strategic trading This section presents the main results regime-of Honda's model

1.4.1 Honda's Model

Honda [49] considers a continuous-time representative-agent economy with one perishable consumption good, which is produced by a single firm The firm is completely financed by equity and has one share outstanding The uncertainty in the economy is the dividend process Xt paid by the firm This dividend follows

(1.57) where yt E {O, 1} denotes the state of the economy and a is constant It is assumed that state 1 is associated with higher expected growth in dividends than state 0, that is, J.I.(yt = 1) == J.l.l > J.I.(yt = 0) == J.l.o At initial time 0, the state is 1 with probability p and 0 with probability 1 - p The process Y,

starting at i, remains there for an exponentially distributed length of time, and then jumps to state j =f i The exponential density has parameter >

The agent is unable to observe the current state of the economy, the cess yt He only has the filtration:Ft = a(xs,s ~ t), that is, he only observes

pro-the path of dividends and uses this information, as well as his knowledge of

a, p, >., J.l.l and J.I.o to update his beliefs about the state

This incomplete information economy can be reduced to a complete formation economy by using the filtered probability

in-tPt = Prob(yt = 11:Ft) , tPo = p (1.58) Using Theorem 9.1 in Liptser and Shiryayev [56], Honda [49] shows that the agent updates his beliefs about the state according to

(1.59) where

(1.60)

Note that dEt denotes the "surprise" component of the change in dividends,

in a fashion similar to Gennotte's [36] and Dothan and Feldman's [27, 34] models The Brownian motion Et is a martingale with respect to the filtration

:Ft·

The filtered probability tPt follows a mean-reverting stochastic process with mean 1f = 1/2 It is worth noting that tPt summarizes both the estimated

expected return p and the precision of this estimate When tPt is close to 0

or 1, the investor is fairly confident about the current regime of the economy

As a result, the variability of tPt is low, as can be seen in (1.59) On the other hand, when tPt is close to 1/2, the investor is not confident about the current regime, and the variability of tPt is high

The representative agent has a utility function defined by

(1.61)

where p is nonnegative and u is concave in current consumption C s • Honda [49] considers the case of power utility, u(c) = cOt la

1.4.2 The Equilibrium Price Process

Let St denote the equilibrium price of the firm's share at time t In librium, the agent holds one share Since he has no other source of income,

equi-he is restricted to consume his current dividend income, Ct = Xt Hence, the state-price deflator 1ft is given by

1ft = e-ptu'(Xt) = e-Ptxr- l

and the equilibrium asset price process St by

(1.62)

s, = ;, E (! ~.'.dsl.r:) = .;-1 E (! e-p('-')'~dslr.) (1.63) Honda [49J shows that the equilibrium price process St satisfies

St = J('l/Jt, t)Xt ,

where f('l/J, t) solves the partial differential equation

(1.64)

1 2 2 ( JLl - JLo ) 2 0= ('x(1 - 2'l/J) + a'l/J(1- 'l/J)(JLl - JLO))!1/J + 2'l/J (1- 'l/J) a !1/J1/J

Since exp (its (o:p,(tPu) + 0:(0: - 1)112/2 + p) du) > 0 and 8tP~(w)/8x ~ 0, one has f", > 0 if 0: > 0 and f", < 0 if 0: < 0.7 Thus, when the investor is less risk averse than the log-utility investor (0: > 0), the share price increases with the conditional probability tPt of being in the high-dividend growth state The reverse is true when the investor is more risk-averse than the log-utility investor (0: < 0)

Using Ito's formula, the dynamics of the stock price can be computed as

dSt = (p,(tPt) + 7-(f",(-\(l - 2tPt) + tPt(l - tPt)(J1.1 - J1.o))

+'if"''''tPt (1 - tPd 11 + ft Stdt (1.68) + (11 + f; tPt(1- tPt)J1.1 ~ J1.0) StdBt

== J1.sStdt + I1sStdBt

From (1.68), the following properties of the equilibrium share price process can be noted:

- the share price displays stochastic volatility,

- when tPt is close to 0 or 1, Le., when the investor is fairly confident about the current regime, the volatility of asset returns I1s is close to the underlying uncertainty, Le., the volatility of dividends 11,

- when the difference between the expected growth rate in dividends between the two regimes PI -Po is large, then the difference between the asset return volatility and the dividend volatility is also large,

- when 11 is high, so that dividends are very volatile and the volatility of tPt

is low, then I1s is close to 11,

- if the investor is less risk-averse than the log-utility investor (0: > 0), then

f1/J > 0 and I1S > 11 Conversely, when the investor is more risk-averse than the log-utility investor (0: < 0), then f1/J > 0 and I1S < 11

Note that whereas most models assume stochastic volatility at the tal level to explain stochastic price volatility, Honda's model demonstrates that under incomplete information, stochastic volatility may arise even though

fundamen-the volatility of fundamen-the underlying dividend process is constant This illustrates that by leading to richer price dynamics than under complete information,

7 In order to show that ot/JHw)/ox ~ 0, let Zt == ot/J;(w)/ox Then, from (1.59)

and the theory of stochastic Hows, Zt satisfies

incomplete information models can help explain some of the phenomena served in financial markets

ob-1.5 Conclusion and Outlook

This brief literature review demonstrates that there is a very close spondence between a complete-information and an incomplete-information economy In general, an incomplete-information economy can be reduced to

corre-a complete-informcorre-ation economy through the use of corre-appropricorre-ate stcorre-ate vcorre-ari-ables Depending on the information structure at hand, two cases can be distinguished

vari For Gaussian information structures, Detemple [23] shows that the

con-sumer's optimization problem under incomplete information can be formed into an equivalent program with a completely observed state: the conditional expectation of the underlying unobservable state variable This

trans-is the essence of the separation principle, and trans-is prectrans-isely what occurred in the context of Gennotte's [36] and Dothan and Feldman's [27, 34] models

of Sects 1.2 and 1.3 First, the conditional expectation of the true pected returns vector ILt, mt = Et(ILt), was estimated In a second step, portfolio optimization was performed, with ILt replaced with its conditional expectation mt

ex For non-Gaussian information structures, additional state variables

be-sides the conditional mean will in general be necessary to characterize investors' posterior beliefs (i.e the conditional distribution of the cur-rent state given the past observations) For example, in the case of non-Gaussian prior beliefs, Detemple [24] shows that the investor's posterior

beliefs are characterized by two sets of sufficient statistics: (i) the vector

of conditional means and (ii) a set of sufficient statistics for the tional variance-covariance matrix Thus, with non Gaussian priors, the conditional variance-covariance matrix becomes stochastic Similarly, in the context of Honda's model of Sect 1.4, which did not have a Gaus-sian information structure, the state variable was the filtered probability

condi-.,pt, which summarized both the agent's beliefs about the state and the

confidence of his beliefs As a result, both the conditional mean and the

conditional variance were stochastic

A consequence of the use of state variables is that classic results in finance remain valid under an appropriate reinterpretation of the state variables On the basis ofthis result, it may at first glance seem that incomplete information can be dealt with quite trivially Such is not the case, however, as can be illustrated by the following open issues:

- The analysis in Sect 1.3 showed that it is not possible to say in general how interest rates under incomplete information compare to those under

complete information In the context of Honda's model of Sect 1.4, no statement was made as to how asset prices under incomplete information compare to those under complete information Thus, whereas the method-ology to be used for analyzing incomplete information is clear, the economic effects of incomplete information are ambiguous Chapter 2 presents a sim-ple continuous-time, representative-agent economy and analyzes the effect

of incomplete information on the representative agent's expected lifetime utility, equilibrium asset prices and interest rates Contrary to what conven-tional wisdom would suggest, the analysis demonstrates that asset prices can be higher under incomplete information than under complete infor-mation Sufficient conditions under which utility, prices and interest rates will be higher or lower under incomplete information than under complete information are derived

- What are the consequences of heterogeneous beliefs for agents' optimal trading strategies, consumption patterns, and equilibrium asset pricing? Using the price dynamics from the homogeneous-beliefs economy of Chap

2, Chap 3 analyzes the consequences of heterogeneous beliefs for an vidual price-taking agent's portfolio demand It thus addresses the practi-cally very relevant question of how someone convinced that he can "beat the market" should behave The analysis distinguishes two types of hetero-geneous beliefs: relative optimism/pessimism and confidence It is shown that whereas the agent's relative optimism/pessimism influences the tan-gency component of his optimal asset demand, his confidence influences his hedging demand Numerical computations show that the effect of hetero-geneous beliefs on optimal portfolio allocations can be significant Chapter

indi-4 takes a closer look at the effects of heterogeneous beliefs in terms of sumption Again, two types of heterogeneous beliefs can be distinguished, each leading to some specific consumption patterns Building upon the re-sults from Chaps 3 and 4, Chap 5 then analyzes equilibrium asset pricing

con-under heterogeneous beliefs It shows that consumption can be considered

as the "bridge" between heterogeneous beliefs and the equilibrium lent martingale measure, which is common to all agents Using state-price deflator techniques, the analysis demonstrates that under heterogeneous beliefs, the state-price density function may become multi-modal Whereas the effect of the introduction of a mean-preserving spread in beliefs on the yield curve is ambiguous, Chap 5 demonstrates that the effect of mean-preserving spreads on stock prices depends on the agents' degree of risk aversion An analysis of option prices shows that heterogeneous beliefs cause a "smile effect" in implied option volatility Furthermore, taking het-erogeneous beliefs into account leads to a sizable improvement in option price forecasting accuracy

equiva What is the effect of costly information on asset prices? Which factors drive the demand for information? Chapter 6 analyzes agents' trading strategies and their demand for information under costly information When infor-

mation is costly, agents can be expected to perform the inference process needed to estimate unknown parameters imperfectly As a result, their uncertainty about mean returns will generally differ from its value when information processing is costless Agents' demand for information is shown

to depend on two factors: their ''pure'' state risk aversion and the extent

to which they dislike uncertainty about future expected returns by itself

An analysis of agents' portfolio behavior shows that each investor uses his own estimate of unknown expected returns and his own private parameter uncertainty when forming his optimal portfolio Whereas the estimate of the expected return influences the agent's tangency portfolio, his param-eter uncertainty influences his hedging portfolio When agents are more risk-averse than the log-utility investor, they reduce their demand for as-sets whose expected returns are uncertain In this case, investors' demand for assets falls as the costs of acquiring information about these assets rise Information costs can therefore help explain such phenomena as the home bias observed in international finance Investors' hedging behavior

is also shown to have implications for aggregate asset demands Because each agent uses his private parameter uncertainty when forming his op-

timal portfolio, the equilibrium risk premium will generally be distorted away from its equilibrium value under information pooling, implying that the market does not aggregate information efficiently Costly information and imperfect learning can therefore explain the equity premium puzzle

Utility, Prices, and Interest Rates

- How does the quality of agents' information influence their utility? Does better information always mean higher expected lifetime utility?

- How does the quality of information influence share prices and interest rates? Is it the case that better information leads to higher asset prices? The model used to answer these questions is that of a simple representative-agent economy with a single risky asset available for investment The agent does not know the expected growth rate of dividends, and must therefore estimate it from the historical path of dividends The representative agent's expected lifetime utility in equilibrium as well as equilibrium share prices and interest rates conditional on his information are derived Computing utility, share prices and interest rates conditional on the quality of the agent's infor-mation allows analyzing the impact of changes in the quality of information

on these variables The analysis below shows that monotonicity results that could be expected intuitively - such as "better information increases expected utility because people are risk averse" and "better information increases share prices because people face less uncertainty" - do not hold

In static models, the effect of information on expected utility is driven by economic agents' risk aversion (see, for example, Willinger [76]) The analysis

in this chapter demonstrates that in continuous time, risk aversion as such is not sufficient for more information to lead to an increase in expected lifetime utility Rather, the impact of information on expected lifetime utility is driven

by the agent's state risk aversion, i.e by those same factors that induce agents

to hedge against unfavorable changes in the investment opportunity set in Merton's [61] original model of continuous-time portfolio selection

A Ziegler, Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance

© Springer-Verlag Berlin Heidelberg 2003

The actual sufficient conditions that must be satisfied by the tative agent's (static) utility function for better information to lead to an increase in expected lifetime utility, share prices and interest rates are de-rived It is shown that risk aversion by itself is not sufficient for this to be the case

represen-In using a representative agent model, the analysis of this chapter sumes homogeneous beliefs among agents and therefore abstracts from the value of information arising from advantages in trading under asymmetric information This rationale for information acquisition and the way in which agents' information is incorporated into prices have been analyzed in a rich literature and shall not be addressed here.1 Rather, this chapter focuses on the aggregate effects of better information on expected lifetime utility As will become clear below, the effect of better information on economic agents' expected lifetime utility need not be positive in this setting The reason is that the representative agent is unable to change his behavior in response to new information because he is constrained to hold the asset in the amount available.2 As a result, better information in such a setting may even lead to lower expected lifetime utility 3

as-The chapter is organized as follows Section 2.2 presents the model tion 2.3 describes the agent's expected lifetime utility and computes the equi-librium share price and interest rates using state-price deflator techniques Sections 2.4 and 2.5 analyze equilibrium in the special cases of logarithmic and power utility, respectively Section 2.6 provides a theoretical discussion

Sec-of the effect Sec-of the quality Sec-of information on utility, asset prices, and interest rates, and derives sufficient conditions on the representative agent's utility function for these values to be higher and lower under complete information than under incomplete information Section 2.7 extends the basic results to the case of a time-varying unknown dividend growth Section 2.8 concludes

1 See, for example, Grossman [38, 391 and Kyle [541

2 Hirshleifer [461 provides an early analysis of the difference between the private and social value of information He shows that in a pure exchange economy, private information may lead to large private profits, but is of no social value The reason is that in a pure exchange setting, information does not lead to improvements in productive arrangements It merely has redistributive effects, and one agent's gain is essentially an other agent's loss

3 As shown in Gollier [371, the value of information is extracted from the fact that the observation of a signal allows the agents to better adapt their decisions to the risky environment that they face and is thus closely related to the "value

of flexibility" developed in the real options literature This correspondence tween the value of information and options is a direct consequence of the result established by Kihlstrom [521 that the expectation of any convex function is an increasing function of the informativeness of a signal in the Blackwell [71 sense

be-2.2 The Model

Consider a continuous-time representative-agent economy with a single firm.4 The firm produces a single good It is completely financed by equity and has one share outstanding The firm pays a dividend to its shareholder at a rate

Xt at time t Suppose that the dividend process follows

(2.1) where J.t is the constant instantaneous increase in expected dividends, a de-

notes the dividend process' constant instantaneous volatility, and B t denotes

a standard Brownian motion Suppose, however, that the agent does not know the true mean J.t and must estimate it from past data We assume that at ini-

tial time t = 0, the agent has the following prior information on J.t: he views J.t

as normally distributed with mean mo and variance Vo = E((mo - J.t)2) The

investor has no additional prior information on J.t He only has the filtration

:Ft = a(xs, s ~ t) The parameter a is assumed to be known

From filtering theory, as new dividend information arrives, the agent dates his estimate mt of the mean growth in dividends J.t using the relation-

up-ship5

where

Vi dmt = -dBt , a

which is a measure of his relative uncertainty about J.L From the agent's

viewpoint, this partially observed economy with constant mean expected growth in dividends J.t is equivalent to a perfectly observed economy with

stochastic, time-varying mean expected growth in dividends mt Note that

the instantaneous change in dividends dxt/ Xt and the investor's estimate of

4 Rubinstein [681 provides an overview of the conditions under which a tative agent can be used

represen-5 See Theorem 11.2 of Liptser and Shiryayev [561

J.1, are perfectly correlated Thus, the investor's expectations formation is trapolative" (see Brennan [9]): when dxt/xt > mtdt, the investor revises his estimate of J.1, upwards When dxt/ Xt < mtdt, he revises his estimate of J.1,

"ex-downwards

In (2.2) and (2.4), Vi = Et{{mt - J.1,}2} denotes the mean square error of

mt and has dynamics

of information When Vi = 0, then J.1, is perfectly known and the economy is a complete-information economy When Vi is high, then the agent's uncertainty about J.1, is high and his revisions of mt as new dividend information becomes available are large

To gain some intuition for the agent's updating behavior as presented in (2.2) and (2.5), consider the case in which the agent only has diffuse prior information on J.1" i.e Vo + 00 Then, given the fact that

or

t

In(xt} = In(xo} + (J.1, - ~2) t + (1 / dEs,

o his best estimate for J.1, at time t will be

In(xt) -In(xo) (12

The mean square error of this estimate, Vi, is given by

where the first equality on the last line results from the Ito Isometry.6 To termine how the agent updates his estimate mt of the instantaneous increase

de-in expected dividends /-l, one can apply Ito's formula to (2.9), yielding the following expression for the dynamics of mt:

2.3 Equilibrium

This section characterizes prices, utility and interest rates in this economy

In equilibrium, the representative agent holds one share and consumes the

dividend stream Xt The results presented here are fairly general; examples for particular utility functions will be provided in Sects 2.4 and 2.5

2.3.1 The Equilibrium Expected Lifetime Utility

Using the equilibrium condition Ct = Xt, the representative agent's expected lifetime utility of consumption is given by

J(Xt, mt, t) = E (! e -P'u(x,)d.!n ) (2.14)

6 See 0ksendal [65], page 26

By Ito's Lemma, one has

(2.19)

As can easily be seen by looking at the expression for 'OJ in (2.16), expected lifetime utility will depend on the quality of information lit through two chan-nels: the cross-partial derivative Jxm and the ''pure state" risk aversion Jmm

Thus, how expected lifetime utility under incomplete information compares

to that under complete information will depend closely on Jxm and Jmm

2.3.2 The Equilibrium Price

Let St denote the equilibrium price of the firm's share at time t In rium, the representative agent holds one share Since he has no other source

equilib-of income, his consumption is equal to the current dividend, Ct = Xt Hence, the state-price deflator trt is given by

(2.20)

and the equilibrium asset price process St by 7

To analyze the properties of equilibrium prices in more detail, define

Therefore,

E, (I x.d') + I VJ(x"m",)ds ~ 0, and for all t, J satisfies the partial differential equation

of information lit only if at least one of these derivatives is nonzero It is worth noting that the valuation equation (2.26) is very similar (although not identical in this case) to the HJB equation (2.19) for the lifetime expected utility of consumption J at the optimal portfolio-consumption process derived

by Merton

7 Honda [491 (see Sect 1.4) provides an example of equilibrium asset pricing under incomplete information using state-price deflator techniques