Dynamic Asset Allocation

115 Pages·2003·0.71 MB·English

Dynamic Asset Allocation
(cid:3)
Claus Munk
y
This version: April 4, 2003
(cid:3)Informal lecture notes originally prepared for Ph.D. course on Continuous-time Finance, Danish Doctoral Re-
searchProgrammeinFinance,Spring2001. IappreciatecommentsandcorrectionsfromHeineJepsenandNicolai
Nielsen.
yDepartment of Accounting and Finance, University of Southern Denmark, Campusvej 55, DK-5230
Odense M, Denmark. Phone: [phone]. Fax: [phone]. E-mail: [email protected]. Internet homepage:
http://www.sam.sdu.dk/~cmu
Contents
1 Introduction 1
1.1 Investor classes and motives for investments . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Typical investment advice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 An overview of the theory of optimal investments . . . . . . . . . . . . . . . . . . . 3
1.4 The future of investment management and services . . . . . . . . . . . . . . . . . . 3
1.5 Outline of the rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Preferences 4
2.1 Expected utility representation of preferences . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Basic representation of preferences . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Expected utility of consumption plans . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Behavioral axioms and derived properties . . . . . . . . . . . . . . . . . . . 6
2.1.4 Expected utility representation of preferences . . . . . . . . . . . . . . . . . 6
2.1.5 Are the axioms reasonable? . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Comparison of risk aversion between individuals . . . . . . . . . . . . . . . 10
2.3 Frequently applied utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Preferences in multi-period settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 One-period models 15
3.1 The general one-period model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Mean-variance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Theoretical foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Mean-variance analysis with only risky assets . . . . . . . . . . . . . . . . . 17
3.2.3 Mean-variance analysis with both risky assets and a riskless asset . . . . . . 19
3.3 Critique of the one-period framework . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Introduction to multi-period models 21
4.1 A multi-period, discrete-time framework . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Wealth dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Dynamic programming in discrete-time models . . . . . . . . . . . . . . . . . . . . 23
4.4 The basic continuous-time setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
i
Contents ii
4.5 Dynamic programming in continuous-time models . . . . . . . . . . . . . . . . . . 28
4.6 The martingale approach to consumption-portfolio problems. . . . . . . . . . . . . 32
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Asset allocation with constant investment opportunities 37
5.1 General utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 CRRA utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Logarithmic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Discussion of the optimal investment strategy . . . . . . . . . . . . . . . . . . . . . 43
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Asset allocation with stochastic investment opportunities: the general case 47
6.1 One-dimensional state variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.1.1 General utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.1.2 Speci(cid:12)c utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Multi-dimensional state variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 What risks are to be hedged? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.4 Closed-form solution for CRRA utility: a(cid:14)ne models with one state variable . . . 56
6.4.1 Utility from terminal wealth only . . . . . . . . . . . . . . . . . . . . . . . . 57
6.4.2 Utility from consumption (and possibly terminal wealth). . . . . . . . . . . 59
6.5 Closed-form solution for CRRA utility: quadratic models with one state variable . 61
6.5.1 Utility from terminal wealth only . . . . . . . . . . . . . . . . . . . . . . . . 61
6.5.2 Utility from consumption (and possibly terminal wealth). . . . . . . . . . . 63
6.6 Closed-form solution for CRRA utility: multiple state variables . . . . . . . . . . . 64
6.7 Applying the martingale approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7 Asset allocation with stochastic investment opportunities: concrete cases 67
7.1 Stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.1.1 One-factor Vasicek interest rate dynamics . . . . . . . . . . . . . . . . . . . 68
7.1.2 One-factor CIR dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1.3 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.1.4 Other studies with stochastic interest rates . . . . . . . . . . . . . . . . . . 79
7.2 Stochastic excess returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8 Non-(cid:12)nancial risks 85
8.1 Labor income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.1.1 Exogenous labor income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.1.2 Endogenous labor supply and income . . . . . . . . . . . . . . . . . . . . . 90
8.1.3 Further references on labor income in portfolio and consumption choice . . 92
8.2 In(cid:13)ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.3 Multiple and/or durable consumption goods . . . . . . . . . . . . . . . . . . . . . . 96
8.4 Uncertain time of death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Contents iii
9 Non-standard assumptions on investors 97
9.1 Preferences with habit formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
9.2 Recursive utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.3 Other objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.4 Consumption and Portfolio Choice for Non-price takers . . . . . . . . . . . . . . . 99
9.5 Non-Utility Based Portfolio Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.6 Allowing for Bankruptcy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10 Trading and information imperfections 101
10.1 Model/parameter uncertainty or incomplete information . . . . . . . . . . . . . . . 101
10.2 Trading constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.3 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11 Computational Methods 102
References 103
Chapter 1
Introduction
Financialmarketso(cid:11)eropportunitiestomovemoneybetweendi(cid:11)erentpointsintimeanddi(cid:11)erent
states of the world. Investors must decide how much to invest in the (cid:12)nancial markets and how to
allocate that amount between the many, many available (cid:12)nancial securities. Investors can change
their investments as time passes and they will typically want to do so for example when they
obtain new information about the prospective returns on the (cid:12)nancial securities. Hence, they
must (cid:12)gure out how to manage their portfolio over time. In other words, they must determine a
dynamic investment or asset allocation strategy. The term asset allocation is sometimes used for
the allocation of investments to major asset classes, e.g. stocks, bonds, and cash. In later chapters
we will often focus on this decision, but we will use the term asset allocation interchangeably with
the terms optimal investment or portfolio management.
Itisintuitivelyclearthatinordertodeterminetheoptimalinvestmentstrategyforaninvestor,
wemustmakesomeassumptionsabouttheobjectivesoftheinvestorandaboutthepossiblereturns
on the (cid:12)nancial markets. Di(cid:11)erent investors will have di(cid:11)erent motives for investments and hence
di(cid:11)erent objectives. In Section 1.1 we will discuss the motives and objectives of di(cid:11)erent types
of investors. We will focus on the asset allocation decisions of individual investors or households.
Individuals invest in the (cid:12)nancial markets to (cid:12)nance future consumption of which they obtain
some felicity or utility. We discuss how to model the preferences of individuals in Chapter 2.
1.1 Investor classes and motives for investments
We can split the investors into individual investors (households; sometimes called retail in-
vestors)andinstitutionalinvestors(includesboth(cid:12)nancialintermediaries{suchaspensionfunds,
insurance companies, mutual funds, and commercial banks { and manufacturing companies pro-
ducing goods or services). Di(cid:11)erent investors have di(cid:11)erent objectives. Manufacturing companies
probablyinvestmostlyinshort-termbondsanddepositsinordertotheirliquidityneedsandavoid
the deadweight costs of raising small amounts of capital very frequently. They will rarely set up
long-term strategies for investments in the (cid:12)nancial markets and their (cid:12)nancial investments is a
very small part of the total investments.
Individuals can use their money either for consumption or savings. Here we use the term
savings synonymously with (cid:12)nancial investments so that it includes both deposits in banks and
investments in stocks, bonds, and possibly other securities. Traditionally most individuals have
saved in form of bank deposits and maybe government bonds, but in recent years there has been
1
1.1 Investor classes and motives for investments 2
an increasing interest of individuals for investing in the stock market. Individuals typically save
when they are young by consuming less than the labor income they earn, primarily in order to
accumulate wealth they can use for consumption when they retire. Other motives for saving is to
beableto(cid:12)nancelargefutureexpenditures(e.g.purchaseofrealestate,supportofchildrenduring
theireducation,expensivecelebrationsorvacations)orsimplytobuildupabu(cid:11)erfor\hardtimes"
due to unemployment, disability, etc. The objective of an individual investor is to maximize the
utility of consumption throughout the life-time of the investor. We will discuss utility functions in
Chapter 2.
Alargepartofthesavingsofindividualsareindirectthroughpensionfundsandmutualfunds.
These funds are the major investors in today’s markets. Some of these funds are non-pro(cid:12)t funds
that are owned by the investors in the fund. The objective of such funds should represent the
objectives of the fund investors.
Let us look at pension funds. One could imagine a pension fund that determines the optimal
portfolio of each of the fund investors and aggregates over all investors to (cid:12)nd the portfolio of
the fund. Each fund investor is then allocated the returns on his optimal portfolio, probably
net of some servicing fee. The purpose of the fund is then simply to save transaction costs. A
practical implementation of this is to let each investor allocate his funds among some pre-selected
portfolios, for example a portfolio mimicking the overall stock market index, various portfolios of
stockindi(cid:11)erentindustries,oneormoreportfoliosofgovernmentbonds(e.g.oneinshort-termand
one in long-term bonds), portfolios of corporate bonds and mortgage-backed bonds, portfolios of
foreign stocks and bonds, and maybe also portfolios of derivative securities and even non-(cid:12)nancial
portfoliosofmetalsandrealestate. Somepensionfundsoperateinthiswayandthereseemstobe
a tendency for more and more pension funds to allow investor discretion with regards to the way
the deposits are invested.
However, in many pension funds some hired fund managers decide on the investment strategy.
Often all the deposits of di(cid:11)erent fund members are pooled together and then invested according
to a portfolio chosen by the fund managers (probably following some general guidelines set up by
the board of the fund). Once in a while the rate of return of the portfolio is determined and the
deposit of each investor is increased according to this rate of return less some servicing fee. In
manycasesthereturnsontheportfolioofthefundaredistributedtothefundmembersusingmore
complicated schemes. Rate of return guarantees, bonus accounts,.... The salary of the manager of
a fund is often linked to the return on the portfolio he chooses and some benchmark portfolio(s).
Arationalmanagerwillchooseaportfoliothatmaximizeshisutilityandthatportfoliochoicemay
be far from the optimal portfolio of the fund members....
Mutual funds...
This lecture note will focus on the decision problem of an individual investor and aims to
analyze and answer the following questions:
What are the utility maximizing dynamic consumption and investment strategies of an indi-
(cid:15)
vidual?
What is the relation between optimal consumption and optimal investment?
(cid:15)
How are (cid:12)nancial investments optimally allocated to di(cid:11)erent asset classes, e.g. stocks and
(cid:15)
bonds?
1.2 Typical investment advice 3
Howare(cid:12)nancialinvestmentsoptimallyallocatedtosinglesecuritieswithineachassetclass?
(cid:15)
How does the optimal consumption and investment strategies depend on, e.g., risk aversion,
(cid:15)
time horizon, initial wealth, income, and asset price dynamics?
Are the recommendations of investment advisors consistent with the theory of optimal in-
(cid:15)
vestments?
1.2 Typical investment advice
1.3 An overview of the theory of optimal investments
1.4 The future of investment management and services
See Bodie (2003), Merton (2003)
1.5 Outline of the rest
1.6 Notation
Since we are going to deal simultaneously with many (cid:12)nancial assets, it will often be mathe-
matically convenient to use vectors and matrices. All vectors are considered column vectors. The
superscript > on a vector or a matrix indicates that the vector or matrix is transposed. We will
use the notation 1 for a vector where all elements are equal to 1 { the dimension of the vector will
beclearfromthecontext. Wewillusethenotatione foravector(0;:::;0;1;0;:::;0)> wherethe
i
1 is entry number i. Note that for two vectors x = (x ;:::;x )> and y = (y ;:::;y )> we have
1 d 1 d
x>y = d x y . Also, x>1= d x and e>x=x .
i=1 i i i=1 i i i
If x = (x ;:::;x ) and f is a real-valued function of x, then the ((cid:12)rst-order) derivative of f
P 1 n P
with respect to x is the vector
@f @f >
f (x)= ;:::; :
0
@x @x
(cid:18) 1 n(cid:19)
This is also called the gradient of f. The second-order derivative of f is the n n Hessian matrix
(cid:2)
@2f @2f ::: @2f
@x21 @x1@x2 @x1@xn
0 @2f @2f ::: @2f 1
f00(x)=B@x2...@x1 @x...22 ... @x2...@xnC:
B C
BB @2f @2f ::: @2f CC
B@xn@x1 @xn@x2 @x2n C
@ A
If x and a are n-dimensional vectors, then
@
(a>x)=a:
@x
If x is an n-dimensional vector and A is a symmetric [i.e. A=A>] n n matrix, then
(cid:2)
@
(x>Ax)=2Ax:
@x
Chapter 2
Preferences
There is a large literature on how to model the preferences of individuals for uncertain outcomes.
The literature dates back at least to Daniel Bernoulli in 1738 (see English translation in Bernoulli
(1954)), but was put on a (cid:12)rm formal setting by von Neumann and Morgenstern (1944). Some
recenttextbookpresentationsaregivenbyHuangandLitzenberger(1988,Ch.1)andKreps(1990,
Ch. 3). The short presentation below mainly follows that of Huang and Litzenberger.
2.1 Expected utility representation of preferences
2.1.1 Basic representation of preferences
Assume a single consumption good and a one-period economy with uncertainty about the
(cid:15)
state ! (cid:10) at time 1 (end-of-period). The probabilities of the states of nature are given
2
[objective].
A consumption plan x is a speci(cid:12)cation of the number of units consumed in each state,
(cid:15)
x=(x ! (cid:10)). The set of possible consumption plans is denoted X.
!
j 2
Assume there are (cid:12)nitely many possible consumption levels, i.e. a (cid:12)nite set Z R exist such
(cid:15) (cid:18)
that x Z for all ! (cid:10) and all x X.
!
2 2 2
A preference relation on P is a binary relation satisfying
(cid:15) (cid:23)
(i) p q and q r p r [transitivity]
(cid:23) (cid:23) ) (cid:23)
(ii) p;q P: p q or q p [completeness]
8 2 (cid:23) (cid:23)
Derived relations , , and ...
(cid:15) (cid:24) 6(cid:23) (cid:31)
A probability distribution on Z is a function p : Z [0;1] such that p(z) 0; z Z and
(cid:15) ! (cid:21) 8 2
p(z)=1.
z Z
2
P
Assume that the preferences of an individual can represented by a preference relation
(cid:15) (cid:23)
on the set P P(Z) of probability distributions (aka. lotteries) de(cid:12)ned on Z. [...state-
(cid:17)
independence]
A utility function is a function U:P R such that
(cid:15) !
p q U(p) U(q):
(cid:23) , (cid:21)
4
2.1 Expected utility representation of preferences 5
state ! ! ! !
1 2 3
state prob. (cid:25) 0.2 0.3 0.5
!
cons. plan 1, x 3 2 4
1
cons. plan 2, x 3 1 5
2
cons. plan 3, x 4 4 1
3
cons. plan 4, x 1 1 4
4
Table 2.1: The possible state-contingent consumption plans in the example.
cons. level z 1 2 3 4 5
cons. plan 1, p 0 0.3 0.2 0.5 0
1
cons. plan 2, p 0.3 0 0.2 0 0.5
2
cons. plan 3, p 0.5 0 0 0.5 0
3
cons. plan 4, p 0.5 0 0 0.5 0
4
Table 2.2: The probability distributions corresponding to the state-contingent consumption plans
shown in Table 2.1. .
A preference relation can always be represented by a utility function (also if P is in(cid:12)nite).
A utility function is unique up to a strictly positive transformation.
A von Neumann-Morgenstern utility function is a function u:Z R such that
(cid:15) !
p q p(z)u(z) q(z)u(z):
(cid:23) , (cid:21)
z Z z Z
X2 X2
E[u(z~p)] E[u(z~q)]
Given a von Neumann-Morgenstern|utili{tzy fun}ctio|n u,{az util}ity function U is de(cid:12)ned by
U(p)=E[u(z~ )].
p
Example 2.1 Consider an economy with three possible states and four possible state-contingent
consumption plans as illustrated in Table 2.1. Note that Z = 1;2;3;4;5 . The probability
f g
distributions corresponding to these consumption plans are then as shown in Table 2.2. Note that
p and p are indistinguishable in this representation. 2
3 4
2.1 Expected utility representation of preferences 6
2.1.2 Expected utility of consumption plans
Expected utility of consumption plan x corresponding to a probability distribution p:
E[u(x)]= (cid:25) u(x )
! !
! (cid:10)
X2
= (cid:25) u(x )
! !
zX2Z!:Xx!=z
= u(z) (cid:25)
!
zX2Z !:Xx!=z
= u(z)p(z):
z Z
X2
2.1.3 Behavioral axioms and derived properties
Axiom 2.1 is a preference relation.
(cid:23)
Axiom 2.2 (Substitution/independence) For all p;q;r P and all a (0;1]:
2 2
p q ap+(1 a)r aq+(1 a)r:
(cid:31) ) (cid:0) (cid:31) (cid:0)
Axiom 2.3 (Archimedean) For all p;q;r P with p q r there exist constants a;b (0;1)
2 (cid:31) (cid:31) 2
such that
ap+(1 a)r q bp+(1 b)r:
(cid:0) (cid:31) (cid:31) (cid:0)
From these basic axioms the following properties can be derived:
Theorem 2.1 (1) p q and 0 a<b 1 bp+(1 b)q ap+(1 a)q.
(cid:31) (cid:20) (cid:20) ) (cid:0) (cid:31) (cid:0)
(2) p q r and p r !a [0;1]:q a p+(1 a )r.
(cid:3) (cid:3) (cid:3)
(cid:23) (cid:23) (cid:31) ) 9 2 (cid:24) (cid:0)
(3) p q, r s, and a [0;1] ap+(1 a)r aq+(1 a)s.
(cid:31) (cid:31) 2 ) (cid:0) (cid:31) (cid:0)
(4) p q and a [0;1] p ap+(1 a)q.
(cid:24) 2 ) (cid:24) (cid:0)
(5) p q and a [0;1] ap+(1 a)r aq+(1 a)r for all r P.
(cid:24) 2 ) (cid:0) (cid:24) (cid:0) 2
(6) z0;z Z such that P p P for all p P.
9 0 2 z0 (cid:23) (cid:23) z0 2
2.1.4 Expected utility representation of preferences
Theorem 2.2 Assume Z (cid:12)nite. A binary relation has an expected utility representation if and
(cid:23)
only if satis(cid:12)es Axioms 1-3.
(cid:23)
Proof: Firstweprovetheimplication ‘ ’: Letz0;z beasinProperty6. IfP P thenp q
( 0 z0 (cid:24) z0 (cid:24)
for all p;q P and consequently any u(z)=k; z Z, will do.
2 8 2
Assume now P P . By Property 2, there exists for any p a unique a [0;1] such that
z0 (cid:31) z0 p 2
a P +(1 a )P p. De(cid:12)ne the function U :X R by U(p)=a . Then
p z0 (cid:0) p z0 (cid:24) ! p
U(p) U(q) U(p)P +(1 U(p))P U(q)P +(1 U(q))P p q:
(cid:21) , z0 (cid:0) z0 (cid:23) z0 (cid:0) z0 , (cid:23)