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Incomplete information and heterogeneous beliefs in continuous time finance

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Alexandre Ziegler

Incomplete
Information
and Heterogeneous
Beliefs
in Continuous-time
Finance
With 43 Figures
and 8 Tables

,

Springer


Professor Alexandre Ziegler


Ecole des HEC,
Universite de Lausanne
BFSH 1
CH -1 015 Lausanne-Dorigny, Switzerland

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

ISBN 978-3-642-05567-6
ISBN 978-3-540-24755-5 (eBook)
DOl 10.1007/978-3-540-24755-5

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Originally published by Springer-Verlag Berlin Heidelberg New York in 2003
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To my parents


Foreword

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 convenience, the continuous-time paradigm has become the standard tool of analysis in portfolio theory and asset pricing. However, and probably because it
was developed hand in hand with option pricing, in which investors' expectations were thought not to matter, continuous-time finance has for a long time
almost entirely neglected investors' beliefs. More recently, the development
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 expectations to regain their central role in finance.
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 information and heterogeneous beliefs for optimal portfolio and consumption choice
and equilibrium asset pricing. Relaxing the assumption that investors can observe 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 optimal 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-


VIII

Foreword

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

Heinz Zimmermann
Professor of Economics and Finance
University of Basle


Preface

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 below, 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, information does matter, and can influence a whole range of economic variables,
from agents' optimal portfolio behavior to asset prices and equilibrium interest 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 Zimmermann and Heinz Muller for their precious and devoted assistance, advice and
encouragement while I was preparing this Habilitation. Professors Sunil Kumar and George Papanicolaou advised me on the numerical methods used
in Chapter 3. In addition, Dr. Stephanie Bilo and Professors Christian Gol-


X

Preface

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 contributed 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,
October 2002

Alexandre Ziegler


Table of Contents

1

Incomplete Information: An Overview. . . . . . . . . . . . . . . . . . . .
1.1 Introduction...........................................
1.2 Portfolio Choice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Gennotte's Model. . . . . . . . . . . . .. .. . . . . . . .. . . .. . . . .
1.2.2 The Inference Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Optimal Portfolio Choice. . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 An Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 The Short Interest Rate. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . ..
1.3.1 Dothan and Feldman's Models. .. . . . . . . .. . . .. .. . . ..
1.3.2 A Characterization of the Term Structure ...........
1.4 Equilibrium Asset Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
1.4.1 Honda's Model. . . .. .. .. .. .. .. . . . . . . .. . . .. .. . . . . ..
1.4.2 The Equilibrium Price Process . . . . . . . . . . . . . . . . . . . ..
1.5 Conclusion and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

2

The Impact of Incomplete Information on Utility, Prices,
and Interest Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.1 Introduction...........................................
2.2 The Model ............................................
2.3 Equilibrium............................................
2.3.1 The Equilibrium Expected Lifetime Utility. . .. . .. . ..
2.3.2 The Equilibrium Price .............. . . . . . . . . . . . . ..
2.3.3 The Equilibrium Interest Rate ......... . . . . . . . . . . ..
2.4 Logarithmic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.4.1 The Equilibrium Expected Lifetime Utility ..........
2.4.2 The Equilibrium Price .......... . . . . . . . . . . . . . . . . ..
2.4.3 The Equilibrium Interest Rate. . . . . . . .. . . . . .. . . .. ..
2.5 Power Utility. . . . . .. . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. . . ..
2.5.1 The Equilibrium Expected Lifetime Utility.. . . .. . . ..
2.5.2 The Equilibrium Price ........ . . . . . . . . . . . . . . . . . . ..
2.5.3 The Equilibrium Interest Rate . . . . . . . . . . . . . . . . . . . ..
2.5.4 Hedging Demand and the Equilibrium Price of
Estimation Risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.6 Information, Utility, Prices, and Interest Rates: A Synthesis.

1
1
1
2
4
4
7
9
10
10
11
15
16
17
19
23
23
25
27
27
28
30
31
31
32
33
35
35
40
48
50
51


XII

Table of Contents

2.6.1 Expected Lifetime Utility. . . . . . . . . . . . . . . . . . . . . . . . ..
2.6.2 Share Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.6.3 Interest Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2.7 Time-Varying Parameters ...............................
2.7.1 The Equilibrium Expected Lifetime Utility ..........
2.7.2 The Equilibrium Price .. . . . . . . . . . . . . . . . . . . . . . . . . ..
2.7.3 The Equilibrium Interest Rate. .. . . .. . . . . . . . . . . . . ..
2.8 Conclusion............................................

51
52
53
55
56
58
58
61

3

Optimal Portfolio Choice Under Heterogeneous Beliefs ...
3.1 Introduction...........................................
3.2 The Model ............................................
3.3 The Deviant Agent's Problem. . . . .. .. . . .. .. .. . . . . .. . ... ..
3.4 Optimal Portfolio Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
3.5 An Example....................................... ....
3.6 Conclusion............................................

65
65
67
70
71
74
78

4

Optimal Consumption Under Heterogeneous Beliefs ......
4.1 Introduction...........................................
4.2 The Cox-Huang Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.3 Heterogeneous Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.4.1 The Model.. .. ... . . . .. .. .. .. .. .. .. .. . . . . .. . . . . ..
4.4.2 Optimal Consumption Patterns Under Heterogeneous
Beliefs. . . . .. .. .. .. . . .. . .. . . . .. .. . . .. .. . . . . .. .. ..
4.4.3 An Algebraic Solution ............................
4.4.4 The Effect of the Time Horizon ....................
4.5 Portfolios and Consumption: A Synthesis . . . . . . . . . . . . . . . . ..
4.6 Conclusion............................................

81
81
82
84
86
86
87
95
102
105
107

Equilibrium Asset Pricing Under Heterogeneous Beliefs ..
5.1 Introduction ...........................................
5.2 The Model ............................................
5.3 Equilibrium Consumption ...............................
5.4 Equilibrium Prices ......................................
5.4.1 The Equilibrium State-Price Density ................
5.4.2 The Equilibrium Short Rate .......................
5.4.3 The Equilibrium Yield Curve .. . . . . . . . . . . . . . . . . . . ..
5.4.4 The Equilibrium Share Price. . . . . . . . . . . . . . . . . . . . . ..
5.4.5 Equilibrium Option Prices and the "Smile Effect" .....
5.5 Implied Risk Aversion ...................................
5.6 Conclusion ............................................

109
109
111
113
116
117
123
128
133
139
145
146

5


Table of Contents

6

Costly Information, Imperfect Learning, and Information
Aggregation ....... '" ....................................
6.1 Introduction ...........................................
6.2 The Model ............................................
6.2.1 The Economy ....................................
6.2.2 The Inference Process: Imperfect Learning ...........
6.3 Portfolio Choice under Costly Information ............. , ...
6.3.1 The Agent's Problem .............................
6.3.2 The Agent's Optimal Investment and Research Policy.
6.3.3 Determinants of the Demand for Information ........
6.3.4 Diversification and Information Costs .............. ,
6.4 Equilibrium Asset Pricing ...............................
6.5 Information Aggregation and the Equity Premium ..........
6.6 Conclusion ............................................

XIII

149
149
151
151
152
154
154
155
157
160
161
165
170

7

Summary and Conclusion ................................. 173

A

Conditional Mean and Variance of In(x s )

B

Conditional Mean and Variance of In(x s ) with
Time-Varying Parameters ................................ 181

C

The Short Rate Under Heterogeneous Beliefs ............. 183

.................

179

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 market imperfections, e.g. transactions costs (Duffie and Sun [30], Dumas and Luciano [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 centered around three main aspects. Section 1.2 analyzes portfolio choice. Section
1.3 characterizes the term structure of interest rates. Section 1.4 presents equilibrium 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


2

1 Incomplete Information: An Overview

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
dS t = ISlLtdt + Is:EtdBt ,

(1.1)

where
81 0

Is =

0

0 82

(1.2)
0

0

o 8n

is the diagonal matrix of current asset prices,

(1.3)
is the n x 1 vector of the process' drift,

(1.4)

1

More recently, Honda [48] analyzed the question of optimal portfolio choice for
unobservable and regime-switching mean returns. Assuming that the unknown
expected return coefficient is an unobserved two-state continuous-time Markov
chain, Honda [48] shows that the uncertainty about the true expected return
affects the investors' portfolio policy by adding a portfolio that can be viewed
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.


1.2 Portfolio Choice

3

is the n x n matrix of the process' instantaneous standard deviation and dB t
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 randomly through time according to the stochastic differential equation
(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 dB t . 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, dS t , they do not know the true expected return Itt. That is,
they only have the filtration F S = {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

3

The assumption that E t 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.
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.


4

1 Incomplete Information: An Overview

1.2.2 The Inference Process

Because they cannot observe the true drift /-Lt, agents seek to extract information on future expected returns from their observation of past returns.
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 martingale with respeCt to the filtration F S .
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 dB t is Brownian motion with respect to {FtS }, it contains no information on the future variations of St and mt. Moreover, V t 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 determine 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.
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·


1.2 Portfolio Choice

5

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

n:,~E (J U(c,,')d'iJ1)

,

(1.10)

w'l)rtdt) - Ctdt .

(1.11)

subject to the budget constraint

dWt

= Wt (w'Is1dS t + (1 -

Substituting (1.1) into (1.11) yields

dWt

= W t (w'(J.ttdt + I:tdB t ) + (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 dB t = dB t + 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 = W t (w' (mtdt + I:tdB t) + (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
0= max (u(c, t)
c,w

+ VJ) ,

(1.15)

where

VJ

= Jt + J~(J.I.~,o + J.I.~,1mt) + Jw (Wt(w'(mt - rtl) + Tt) +~JwwWlw'I:tI:~W + WtW'(I:tI:~,s + Vt)JWm

c)
(1.16)

+~tr ((I:~,s + VtI:~-1) (I:~,s + I:;-1Vt ) J mm)
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:


6

1 Incomplete Information: An Overview

0= uc(c,t) - J w ,

(1.17)

0= JwWt(mt - Ttl) + JwwW?I:tI:~w
+Wt (I:tI:~,s + Vt) JWm .

(1.18)

Equation (1.17) is the usual consumption optimality condition derived by
Merton [61]. Equation (1.18), which describes the optimal portfolio decision,
can be rewritten as
1)-1 (
) -JW
w = (I:tI: t
mt - Ttl J
W
WW

1)-1 (~I
+ ( I:tI:t
I:t"'-'/l

t

V) -JWm
S + t J
W
'
WW t

(1.19)

In essence, (1.19) is similar to the expression derived by Merton [61]. Two
important differences deserve to be mentioned, however.
- First, the expected rate of return used by agents when forming the tangency
portfolio (I: t I:D- 1 (mt - Ttl)(-Jw/JwwWt) is the estimated drift mt,
not the true drift J.Lt.
- Second, the n hedging portfolios
(

1)-1 (~ ~I
I:tI: t
"'-'t"'-'/l,S

V) -JWm
+ 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
Merton's [61] optimal portfolio strategy,
w

= ( I:tI:t1)-1 (J.Lt -

) -Jw

Ttl J

W
ww t

= 0,

1 -JWm
+ (~/)-1
I:t"'-'t
I:tI:/l S J
W'
' ww t

(

and

1.21

)

results.
Note that it is through the hedging demand (1.20) that parameter uncertainty V t influences agents' portfolio demands. However, one cannot tell in
general whether the demand for risky assets will be higher under complete
information than under incomplete information. The reason is that hedging
demand depends on two factors:
- 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.


1.2 Portfolio Choice

7

- 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 V t and JWm, one can
say a little more about how optimal portfolio demands under incomplete
information compare to those under complete information. Suppose that V t
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 following 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
dmt
4

5

Vi = -dBt
(J

,

(1.23)

This assumption is required because, although V t is positive definite, it may still
contain negative elements, which may lead to ambiguous effects of incomplete
information on optimal portfolio demands.
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.


8

1 Incomplete Information: An Overview

where dB t
Vt, follows

= dBt + ((J.t -

mt)la)dt. The mean square error of this estimate,

dVt

v,2

= --t-dt
.
a2

(1.24)

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

U

= E(B(WT»

,

(1.25)

where WT denotes the investor's terminal wealth. Suppose further that his
terminal utility of wealth is of the isoelastic class,

B(WT)

WoO<

= --.:r..,
a

a <1.

(1.26)

(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)


1.2 Portfolio Choice

9

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

o~ "':' (It + (Q (Tt + w(m, +lm aw vt

V'?)

1
+ '21mm
a2

Tt))+

~Q(Q - 1)W'a') I
(1.32)

'

subject to the boundary condition l(m, T) = l.
Differentiating this expression partially with respect to w yields the firstorder condition
(1.33)
Solving for w gives the optimal percentage investment in the risky asset

w - mt - Tt
- (1 - a)a 2

Vam/ I
+ -,----':--:2
(1 - a)a



(1.34)

To characterize the investor's portfolio demand more precisely, note that due
to non-satiation, expected lifetime utility must rise as investment opportunities 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 information 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 riskaverse than the log-utility investor, 1m < and his demand for the risky asset
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
~')-l (~~'
+ V)t Jwm)
+ I , (~
,ut,ut
,ut,u/l-,S
Jw
. (1.35)


10

1 Incomplete Information: An Overview

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

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 similar 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
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)


1.3 The Term Structure of Interest Rates

11

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
J.Lt

= Ao + Allit

(1.39)

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

(1.40)
where

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

dVi

= (EI',sE~,s + a; + 2J.L1',1 lit
- (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 function is separable in wealth Wt and in the state variables mt and t,

(1.44)
Therefore, one has JWm
as

rt

=

= 0, Wdww / Jw = -1, and (1.35) can be rewritten
1

1

l' (EtED- 1

Using the fact that mt = Ao

( 1, (,)-1
EtE t
mt - 1)

+ Almt

then yields

(1.45)


12

1 Incomplete Information: An Overview

Thus, under incomplete information, the equilibrium instantaneous spot interest rate Tt is a linear function of mt, the conditional mean of the unobservable factor. Since Tt is a one-to-one function of mt, it represents the best
estimate of the unknown productivity factor lit.
Using the dynamics of the estimated unknown productivity parameter,
mt, the dynamics of the short rate can be computed as
dTt =

I' (EE') -1 A
I dmt
I' (EE') -1 1

}' (EE,)-1 Al (
= I' (EE') I}
J.LJ.t,O
-

-

(

}' (EE,)-1 A
1
}' (EE') - I } J.LJ.t,O

+ }' (EE') -1 Al
}' (EE') -1 }

(E

_ )d

+ J.LJ.t,1 mt

+

J.t,s

t

+

(~

""'J.t,sE

,

' ) ( ,)-1 -)
E
dB t

+ Al lit

1- }' (EE,)-1 A
0
}' (EE') - I } J.LJ.t,1

)

+ J.LJ.t,1 T t dt

(147)


E' + A' lit) (E,)-1 dB
1
t

== (J.Lr,o + J.Lr,I Tt) dt + !7r dB t

,

where the third equality follows from (1.46). As can be seen in (1.47), the
variance of the spot rate depends on two factors. The first,
I' (EE') -1 Al (:E s:E') (:E,)-1
}' (:E:E') I}
J.t,
,

(1.48)

is driven by :EJ.t,s:E', which is the covariance between the true investment
opportunity set and prices. The second component,
}' (EE,)-1 Al A' V; (E,)-l
}' (EE') -1 }
1 t

,

(1.49)

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 investment opportunity set are observable, and equilibrium interest rates are set
to perfectly imitate these changes (see Cox, Ingersoll and Ross [18]). As a result, low variability of the spot rate implies a low volatility of the investment


1.3 The Term Structure of Interest Rates

13

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 correspond anymore. The variance of the spot rate now reflects the variability
in the estimated investment opportunity set, which is driven by the weight
that consumers put on the new information contained in the observed realized outputs. Low volatility of the spot rate might now mean low learning
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
-rA + At ,

(1.50)

1, where the subscripts of A
subject to the boundary condition A(t, t)
denote partial derivatives.
Observe from (1.50) that incomplete information influences the term
structure of interest rates through two channels. The first is the variability 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.
The solution to (1.50) is

A(t,s)=exp (J.Lr,o(S_t)
J.Lr,l

+ (J.Lr,o +r) l-exp(J.Lr,l(S-t))
J.Lr,l

J.Lr,l

8

-~ la~ (1- exp(J.Lr,l(U J.Lr,l

t)))du

(1.51)

t

~ 1ara~ (1 - exp(J.Lr,l (u - t)))2 dU)
J.Lr,l

+2

8

t

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


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