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^^tA:'^^^-'ki^
i
ALFRED
P.
PAPR 241991
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH LOCAL SUBSTITUTION
by
Ayman Hindy
Scanford University
and
Chi-fu Huang
Massachusetts Institute of Technology
UP//3273-91-EFA
April 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
UITH LOCAL SUBSTITUTION
by
Ayman Hindy
Stanford University
and
Chi-fu Huang
Massachuseccs Inscituce of Technology
UP//3273-91-EFA
April 1991
lt^\ T.
APR
LIBi
2 4 19,
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH LOCAL SUBSTITUTION*
Ayman Hindy
Graduate School of Business
Stanford University
Stanford,
CA
94305
and
Chi-fu
Huang
Management
Sloan School of
Massachusetts Institute of Technology
Cambridge,
MA
02139
November 1989
Revised December 1990
Abstract
We
study the problem of optimal consumption and portfolio choice for a class of
utility
functions that capture the notion that consumptions at nearby dates are almost perfect substitutes.
The
class
utility functions
we consider excludes
used
in the literature.
all
time-additive and almost
We
all
the non-time-additive
provide necessary and sufficient conditions for a
consumption and portfolio policy to be optimal. Furthermore, we demonstrate our general
the-
ory by solving in a closed form the optimal consumption and portfoho policy for a particular
felicity
function
when
the prices of the assets follow a geometric
Brownian motion process. The
optimal consumption policy in our solution consists of a possible
"gulp" of consumption.
consumption with singular
sample paths. In almost all states of nature, the agent consumes periodically and invests more
in the risky assets than an agent with time-additive utility whose felicity function has the same
curvature and the same time-discount parameter. We also show that the ICAPM of Merton
holds and the CCAPM of Breeden does not. In addition, we compute the equilibrium riskless
rate in a representative investor economy with a single physical production technology whose
rate of return follows a Brownian motion.
initial
or a period of no consumption, followed by a process of accumulated
'We
are grateful to
Hua He
for pointing out
Jean-Luc Vila and Thaleia Zariphopoulou
an error in an earlier version, and to Darreil Duffie. John Heaton,
for helpful discussion.
1
Introduction and
Summary
Introduction and
1
1
Summary
study the problem of optimal consumption and portfolio choice
We
for
an agent whose prefer-
ences over consumption patterns are given by:
j
u(z(t),t)dt^V{W{T))\,
(1)
where
io-
and where
z{t)
is
the process of average past consumption derived from the consumption process
C{t) which denotes the ^o<a/ amount of consumption
factor, -(0~)
>
is
a constant, and \V(T)
is
the
till
time
t.
The parameter J
random wealth
are continuous and increasing functions. Furthermore.
V
is
left at
is
a weighting
T Both
time
concave and u
is
.
concave
u
and
V
in its first
argument.
Preferences for intertemporal consumption represented in (1)
ences analyzed by
come from
Huang and Kreps (1989) and Hindy and Huang
a class of prefer-
Such preferences
(1989a).
exhibit the notion that consumption at nearby dates are almost perfect substitutes. This notion
is
absent in the time-additive representations of preferences which are widely used in the
finance and economics literature.
ties; see. for
Many
researchers have recently used non-time-additive
utili-
example. Epstein (1987), Duffie and Epstein (1989), Epstein and Zin (1989), who
study generalizations of the
utility
form introduced by Koopmans (1960) and Uzawa (1968)
the context of continuous time equilibrium models, and Constantinides
(
in
Detemple and
1988).
Zapatero (1990). Heaton (1990), and Sundaresan (1989) who study habit formation models.
The
utility
representation in (1) differs from most specifications in the non-time-additive
preferences literature in one aspect which
stitution.
In
is
very crucial in capturing the notion of local sub-
most of the currently used non-time-additive models, the index of instantaneous
satisfaction, or felicity, at
Huang and Kreps
any time depends explicitly on the consumption rate
(1989) and Hindy and
Huang (1989a) show
depends explicitly on current consumption
substitution.
to the current
that
when the
rates, the resulting preferences
In essence, the marginal utility for
consumption
at
at that time.
felicity
function
do not exhibit
any moment
is
local
too sensitive
consumption rate and hence the agent would not tolerate delaying or advancing
consumption even
for a very small period of time.
The
utility function in
( 1 ),
on the other hand,
1
Introduction and
Summary
has the key feature that the
2
felicity
function at any time depends only
upon an exponentially
weighted average of past consumption - one derives satisfaction only from past consumption.
This feature
the
is
main
difference
between the
utility function
we study here and those studied
by other researchers, except for Heaton (1989, 1990).
The
specification of utility in
and also
1
)
leads to an economically interesting
to a technically challenging optimization problem.
for the agent
as
(
with preferences given by (I)
most models predict but only
from past consumption.
When
is
periodic.
consumption behavior
The optimal consumption behavior
The agent does
not consume
Most of the time the agent derives
periodically.
the marginal value of average past consumption,
all
the time
satisfaction
r, is
higher
than the marginal value of wealth, a situation that arises only periodically, the agent consumes.
Otherwise, the agent refrains from consumption.
From
the technical point of view, the standard dynamic
programming equations
of the
Bellman-Jacobi type do not apply since a heuristic derivation of these equations would suggest
a bang-bang control policy, but there
different
is
no natural upper bound on consumption.
We
follow a
approach to establish optimaUty.
In this paper,
we provide necessary and
policy to be optimal.
maximum
We
show that
it is
sufficient conditions for a
consumption-portfolio
necessary and sufficient that the value function, or the
attainable utility starting from any state, satisfies a differential inequality which can
be viewed as an application of the Bellman optimality principle. In particular, the conditions
require that at
all
times the marginal value of wealth
is
at least as large as
(f3
times) the
marginal value of the average past consumption. Furthermore, consumption should only occur
when
these marginal values are equal.
We also construct
class of felicity
a closed form solution of the optimal consumption problem for a particular
and bequeath functions
a geometric Brownian motion.
than a
critical
in the case
The idea
when
of the optimal solution
number, say k' and consume only when
,
asset returns, however, an interesting
the prices of the risky assets follow
—
—
k'.
phenomenon appears. The
is
to keep the ratio
—
less
Because of uncertainty about
ratio
—
is
subject to
random
shocks because of the randomness of the return on the risky assets. Furthermore, the sample
path properties of the Brownian motion lead to peculiar behavior of the quantity
the times
when
that whenever
it
it
is
very close to the critical
hits the value k'
it
number
k'
.
The quantity
bounces back and forth to
hit
it
^
—
during
fluctuates so fast
again uncountably infinite
1
Introduction and
number
Summary
3
consumption with nontrivial
of times. This results in a process of optimal cumulative
increasing sample paths in which consumption occurs only periodically.
provide a complete solution for the optimal consumption and portfolio rules in this
We
example and show how the solution changes with changes
whose
utility
We show
felicity
same curvature and the same time-discount parameter.
that the agent with local substitution preferences behaves in a /ess risk averse manner,
in the sense that
is
function has the
parameters of the problem.
of our agent to that of an agent with a time-additive
we compare the behavior
In particular,
in the
he invests higher proportions of his wealth in the risky assets. This behavior
reinforced by lower riskless rates, lower time-discount rate and higher effects of substitution
(lower 3).
It is
worthwhile to mention that one obtains periodic consumption behavior
models when the marginal
felicity of
an agent with such preferences
critical,
consumption evaluated
will refrain
possibly time varying, level; see
the optimal consumption
In addition, in
making
is
his
when the
(1989).
felicity
We
falls
below a
In contrast, in our analysis,
consumption choice, the agent considers both
his
wealth and his
effects of past
consumption on
utility.
also study the implication of the agent's behavior
and on the determination of the
riskless rate.
We
on the equilibrium
Consumption Capital Asset Pricing Model
no longer connected to the marginal
fails
On
markets
the other hand, the
to hold since the marginal utility of
utility of wealth.
Bergman (1985) and Grossman and Laroque
in asset
show that the agent's portfoho choice implies
two-fund separation and, hence, the Capital Asset Pricing Model holds.
is
wealth
function has infinite slope at zero.
consumption experience. Local substitution produces important
current
his
time additive
In such a case,
finite.
is
from consumption once
Cox and Huang
periodic, even
at zero
in
(1990).
consumption
Similar observations were
We
also
compute the
made by
riskless rate in
equilibrium in the presence of a production technology with constant stochastic returns to
scale.
than
We
in
show that the equilibrium
the case of time-additive
risk
premium
utility.
will
This result
be lower in the case of local substitution
is
consistent with the less risk averse
behavior of the agents with local substitution.
The
analysis of this paper suggests that
on equity.
Many
mation leads to
researchers,
/ii'g/ierrisk
IocjlI
substitution alone leads to lower risk
premium
most notably Constantinides (1988), have shown that habit
premium on
equity. Habit formation,
which
is
for-
complementarity over
Formulation
2
4
longer periods, can be combined with local substitution to produce interesting and rich behavior
Eichenbaum and Hansen
and
asset price dynamics. Recent empirical evidences of
lant
and Tauchen (1989), and Heaton (1989, 1990) suggest that there
periods. In particular,
Heaton
(1990), Gal-
substitution over short
is
1990) showed that, after correcting for the problem of temporal
(
aggregation, the habit formation alone does not provide significant explanation power for asset
pricing over the time-additive models; while local substitution does.
substitution, habit formation in the longer horizon
becomes more
These imply that a model which combines
power.
worth pursuing. In
The
rest of this
significant in its explanatory
and habit formation
local substitution
is
next step in this research program.
fact, this is the
paper
In addition, given local
organized as follows. Section 2 sets up the consumption and portfoUo
is
problem under uncertainty
continuous time. Section 3 provides a heuristic derivation of the
in
necessary conditions for optimality, whereas Section 4 shows that these conditions are sufficient.
Section 5 provides a verification theorem for an infinite horizon program. In Section
6,
we
solve
the optimal consumption and portfolio problem in closed form for a particular felicity function
when
the prices of risky assets follow a process of geometrical Brownian motion.
Section 7
provides some comparative statics and equilibrium implications and computations of the riskless
rate are presented in section 8. Section 9 contains concluding remarks.
Formulation
2
Consider an agent
is
who
lives
from time
t
=
to
— T
t
m
a.
world of uncertainty where there
and T. The primitive source of
a single consumption good available at any time between
uncertainty in the agent's world
the agent's
life
modeled by a complete probability space (^,T,P). Over
is
span [0,T], there
is
an AZ-dimensional standard Brownian Motion defined on
the probability space and denoted by
5 =
{Bm{t);t £ [O.T],m
=
revealed to the agent via the realizations of this Brownian Motion.
information resolution by
F =
the filtration, generated by B.
the
is
P-nuU subsets and we note
resolved by time T, or
{J^u
We
t
[O'?"]), the
assume that
that
Tt =
S
T
.
F
is
On
use the notation
a.s. to
is
.
.
,.iVf }.
model
Information
is
this structure of
complete by assuming that Tq contains
the other hand,
=
We
.
family of increasing sub-sigma-flelds of/", or
right-continuous.
dimensional standard Brownian Motion, 5(0)
HVe
F
1,2,
a.s}
We
also
assume that
all
all
uncertainty
M-
Tq
is
AU
processes to be discussed will be
almost
trivial since for
denote statements which are true with probability one.
an
5
Formulation
2
adapted to F'.
moment, and can consume
the good at "gulps" at any
The agent can consume
over intervals. She can also refrain from consumption altogether for
sample path of her cumulative consumption at any time
t
Let
X+
be the space of
We
denoted by xl^.t~).
The
all
(.
processes x whose sample paths are positive, increasing and right
convention that x(^%0~)
will use the
sample paths of any x G X+, a jump of x{u,
stochastic process
C
X+
6
is
points of discontinuity of C(u^,
I)
are the
may have
Finally, C(u>,t)
a.s.
Aitw.
is
to time
moments when
Since
=
r)
left limits exist
x{u), r)
-
x{u;,
r~).
in state
t
uj.
For any
w'
G
t)
the
fi.
the agent consumes a "gulp". Moreover,
component over the
C(u;,t) has an absolutely continuous
at rates C'{uj,t),
at t
.)
=
6 (0,T]
t
a consumption pattern available to the agent with C(u^,
denoting the cumulative consumption from time
consuming
zero for almost
is
Recall that an increasing function x(u;,.) has a finite left-limit at any
continuous.
for the
all
time. Moreover, the
can have a singular component, that
a continuous nontrivial increasing function whose derivative
is
some
at finite rates
which the agent
intervals during
where C'{iJ,t) denotes the consumption rate at time
t
in state
is
uj.
a singular part. Singular components of consumption processes will
play an important role in our analysis of the optimal consumption policy.
The agent has
N+
1,
N
the opportunity to invest her wealth in a frictionless securities market with
< M, long
Security n, where n
at
time
t.
We
lived securities, continuously traded,
-
1,2,.....V,
sells for
riskless interest rate at
We
\i.
The
[5i(t), Si{t),
B{t)
time
= exp{/o
t
and
.
.
.
if
a
is
a matrix,
,
5a/(0]
as
^
Security
i.
is
locally riskless. does not
where
7-(5, i) is
/z
is
denote
a vector in J?^,
tr((TCT^),
where
let
tr is
|/x|
to
pay
the instantaneous
be the Euclidean norm of
the tract of a square matrix.
price process for the risky securities follows a diffusion process given by:
S(t)+ f p{S(s),s)ds = 5(0)+ [ niS{s),s)ds-\- f a{S{s),s)dBis)
Jo
^A
sells for Sn[t),
Borel measurable.
r(-,-) is
let \a\^
.V.
p„(5(0-0- where we have used S[t)
r{S{s)^s)ds) at time
will use the following notation: If
In addition,
pays dividends at rate PnlO^ a-^d
assume that Pn(0 can be written
denote the column vector
dividend, and
risky,
is
and indexed by n - 0.1,2
K
mapping Y.Q x[0,T]
Wte[0,T]
a.s.
(3
Jo
Jo
—
^
measurable with respect to .T® S([0, T]), the product
T]. For each u/ e n, Y{uj, .): [0, T]
sample
D? is a
path and for each t 6 [0, T], Vf, (): H
9? is a random variable. The process Y is said to be adapted to F if for
each ( € (0,7^, Y(t) is .Ft-measurable. This is a natural information constraint: the value of the process at time
t cannot depend on information yet to be revealed.
process
is
a
sigma-field generated by
F
—
The
superscript
that
is
and the Borel sigma-field of
denotes transpose.
[0,
—
Formulation
2
Assume
6
that this diffusion process
integer
m
we
sometimes use
will
>
is
with probability one and that
strictly positive
<
there exists a constant c^ so that £'[|5(0P'"]
0,
fi{t), p(t), a(t),
and
r(t) to
e'^'"'.'*
for
each
For brevity of notation
denote fi(S{t),t). p(S{t),t), a{S(t).t) and
r(S{t),t), respectively.
An
investment strategy
is
A =
an iV-dimensional process
{.4(0
= (Ai(0,
A,x{t))\t 6
[0,T]}, where An(t) denotes the proportion of wealth invested in the n-th risky security at time
t
before consumption and trading.
where
1 is
strategy
a vector of
.4.
I's.
A
The proportion invested
C
consumption plan
S
X+
in the riskless security
is
l-.4(i)^l,
said to be financed by an investment
is
if
W(t)
= WiO) + l^[wis)r{s) + Wis)A^{s)Is-As)iiLis) -
+
where W{t)
is
J^W(s)A^(s)[s-^{s)a{s)dB{s)
the wealth at time
t
V< e [0,T]
before consumption
r(s)S{s))) ds
-
C(r
)
a.s.,
and Is-i(t)
is
an
NxN
diagonal matrix
with the n-th element on the diagonal being equal to S{t)~^. Note that the wealth process has
left-continuous sample paths and that VF(f+)
The agent
felicity
=
^V{t)
derives satisfaction from her past
function at time
t
is
-
i\C{t).
consumption and from her
final
wealth.
The
defined over an exponentially weighted average of past consumption
z{t)
=
z(0-)€-^*
+
^3
f
e->^^'-'UC(s)
a.s.,
(5)
No-
where z{0~) >
is
a constant,
/3
is
a weighting factor, and the integral of (5)
is
defined path
by path in the Lebesgue-Stieltjes sense. Note that the lower limit of the integral in (5)
to account for the possible
jumps whenever
higher values of
C
jump
of
C
at
f
=
0,
and that
does. Moreover, z has a singular
z
is
It
has been shown by Hindy and
0~.
a right continuous process which
component whenever
imply higher emphasis on the recent past and
in the distant past.
is
less
Huang (1989)
C
does. Observe that
emphasis on consumption
that this kind of preferences
views consumption at nearby dates as close substitutes: delaying or advancing consumption
for a very small period of time,
without violating the condition that consumption at
only on information available up to
A
consumption plan
*The
latter
C
t,
t
depends
has a very small effect on the agent's total satisfaction.
and the investment strategy
can be ensured by a growth condition on
(/i
—
p)
A
that finances
and on
u; see
it
are said to be ad-
Friedman (1975, theorem
5.2.3)
7
Formulation
2
missible
if
(4)
is
and
well-defined*,
m
for all integers
>
0.
there exists
km
so that, for all
t,
E[\C{t)\'^'^]
<
e*-'
£'[|H^(OP'"]
<
e^-'.
(6)
Denote by C and
A
the space of admissible
consumption plans and trading
strategies, respec-
tively.
Formally, the agent manages her wealth dynamically to solve the following program:
supcec
s.t.
+ V{W{T^))
by Ae A with W{0) - Wq,
E
[jj u(z(t),t}dt
C
is
financed
and ir(<) - AC(i) >
where \Vq
is
the initial wealth of the agent. u(-,t)
is
(7)
V^ G [0,r],
the felicity function at time
the felicity function for final wealth. Note that the second constraint of (7)
is
t.
and
V'(-) is
a positive wealth
constraint - wealth after consumption at any time must be positive.
One
pattern
objective of this paper
C
is
E C financed by some
to derive necessary
A
^
G
and
to be optimal.
To achieve
the utility maximization problem of the agent at time
the exponentially weighted past consumption
is z,
sufficient conditions for a
t
>
we
this end,
consumption
will also consider
W
and
at that time.
We
It^,
(8)
given that her wealth
both before consumption
is
define^
J(iy,z,5,0
=
E, [// u{z{s),s)ds
supcec.
C
s.t.
is
financed by
.4
+ V{WiT^))\
e ^( with H''(0
and VF(5) - AC(5) >
where
from
t
Ct
t
V5e[<.r],
and At are the spaces of admissible consumption plans and trading strategies starting
with the convention that C(-
replaced by
s, for all 5
=
for all
C
€
Ct, the admissibility
ze"*^'*-''
Note that J{W{t),z{t ),S{t),t)
is
the agent's optimal expected
her consumption purchcises at that time
mean both
+13
life
and J{W(t'^),z{t),S{t),t)
time
is
utility at
the Lebesgue integral and the Ito integral are well-defined.
W
to the stochastic differential equation
Et denotes expectation conditional on /*!.
(4).
time
t
before
her optimal expected
control depending on {\V{i), z{t~), S{t),t) and C{t) depends on the history of (VV'.S),
^
defined by (6) with
r e-'^^''-'UC{v).
=
For this we
is
G [<,T], and
z{s)
solution
=
When
A(t)
we mean
is
life
a feedback
there exists a
3
Heuristic Derivation of Necessary Conditions
consumption purchases
utility after the
time
8
at time
Often,
t.
we
will refer to
J
as the value
function.
Note that
if
C
a solution to (7) financed by
is
solution to (8) financed by
then Ct
.4,
=
{C{s) - C(t~};s G [^T]}
is
a
restricted to [t,T].
.4
Heuristic Derivation of Necessary Conditions
3
We
will use
that
it
Bellman's optimality principle to derive necessary conditions about J assuming
has certain smoothness properties.
One
difficulty that arises in this
usual Bellman equation in dynamic programming
is
derived
and may have singular
components,
it
parts.
cannot be expressed
in
Moreover,
if
that the
is
the admissible consumption
Here, cumulative consumption can be
plans are purely at rates and are feedbaclc controls.
in gulps
when
endeavor
the cumulative consumption has singular
Thus, our derivation of the necessary
feedback form.
conditions will be heuristic in nature.
First,
—
z{t~)
we observe that
and S{t) =
z,
S,
if
a consumption gulp of size
is
{2:(5);5
size of the
J{W-
A,z +
(3A,S,t).
(9)
Jw
u(z{s),s)ds
+ V{W(T+))],
£ [^T]} and W{T'^) are defined along the optimal path on [t,T]. Moreover, the
gulp should be chosen to maximize J
Jw{W where
when W{t) — W,
so because both quantities are equal to
Et[J
where
t
we must have
J{W,z,S,t) =
This
A is prescribed at time
.
Thus, we must have
^,z + i3A,Sj) ^ I3J,{W -
and Jz denote the
first
partial derivatives of
X- + I3A,SJ),
J with respect
(10)
to its first
and second
arguments, respectively.
We now
show that
(9)
and (10) imply that J\v must be equal to
where a gulp of consumption
that
A
is
is
=
at
prescribed. Differentiating (9) with respect to
an implicit function of VV and
Jw{W,z,S,t)
fSJ,
z defined
[-JwiW-A,z +
any (\V,z,S,t)
W while noticing
through (10) gives
(3A,S,t)
+ i3J,{W-A,z + l3A,Sj)]
dA
dw
9
Heuristic Derivation of Necessary Conditions
3
+7vv(ir- A.z + /3A,5,i)
=
Jw{\y - ^.z + ^3A.SJ),
= J;(ir -
where we have used (10) to get the second equality. Similarly, we have y.(U^;.5.
A,
2
+ a3A,5,0- Given
(10),
we then have
Jw{W,z,Sj)^3J,iW,z,S,t)
at
{W,z,S,t) where a consumption gulp
Second, assume that J
any time
t
< T,
lemma imply
=
is
sufficiently
(11)
prescribed.
is
smooth
for the generalized Ito's
lemma
to apply'
.
For
the principle of optimality in dynamic programming and the generalized Ito's
that
maxjc.A Et[jl^^ u(z(s).s)ds
+
//+^[2?^ J(5)
+
+ /;+^[/3J,(3) - Jw(s)]dCis) + E,[J{r,^) + J,(t,)Az{t,)]^
J,{s)]ds
J{r,)]
- Z.[Jw{t^)AW{t,)
(12)
a.s..
lemma we will use throughout can be found in Krylov (1980, theorem 2.10.1) and
Meyer (1982, VIII. 27): Fix a consumption policy financed by a trading strategy and define the
proportion invested in risky securities A{t) accordingly. Let S{t),W{t), z(i} follow the dynamics in (3), (4) and
» be once continuously differentiable on 3?''^+' x [0, T] in
(5), respectively. Let /(W^, 2, 5, t): R'^'*'' x [O.T]
''"^
x [0, T] in its first N + 2 arguments, except
all of its arguments and twice continuously differentiable on 3?
The
generaJized Ito
Dellacherie and
—
possibly on a
smooth manifold
«^+' X [O,T]:0,(W,z,S.t) =
Z-
<
A'
+
3, is
M.
0,
/
(A smooth manifold
=
l\. where
1,2
continuously differentiable in
f{W(t*),z{t),S(t),t)
=
M
all its
in 'R^'*'^
0,(^^'
x [0,7^
-. ^S, <):
arguments.) Then,
»'^+^ x
for all
[0,
7]
—
»
optioned limes
f{W(0),z{0-),S(0).0)+ f [V^f{s)
M
given by:
is
+
,
/
t
=
<
=
1,
(W,z,S,t) €
<
2
T,
we
L,
with
have:
f,{s)]ds
Jo
+
/
Jo
[fw(s)W{s)A(sf Is-^i^Ms) + fs(sf a(3)] dB(s)
+ I (0Us)-fw(s))dC(s)
Jo+ ^A/(r.)-^[/w(r,)AVV(r,)+Mr.)Jiz(r.)]a.3.,
I
where ri.ri,..., are the jump points of
{W{t), z(t~),S(t),i), where V* f is the
C
on
I
[O.t],
where
aJl
the partial derivatives are evaluated at the points
differential generator of
/ associated with the trading strategy and
given by:
+2/^5(^^<^'^''/5-"-1)
and where A/(r,)
=
+
'M/s5<T(t"')
/{VV(r+),2(r.),5(r.),r.)-/(VV(r,),r(r.-),S(r.),r.).
.
is
Heuristic Derivation of Necessary Conditions
3
where
dC
denotes the consumption plan on the time interval
prescribed by
dC on
+ A),
[t,t
J(s) and
its
7.
and we have assumed without
lemma
is
a martingale and thus vanishes
Since no consumption
is
always feasible, putting
is
appearing in
Ito's
[t, (
+ A ),
r, is
derivatives are evaluated at
defined in footnote
V'^J
is
10
(
the i-th
jump
VV(s),z(5~
),
point
5(5),5),
loss of generality that the Ito integral
when
the conditional expectation
taken.
dC =
in (12)
and
letting
A
decrease to
zero gives
>
This relation must hold
Suppose
t
is
u(:{t),t)
for all values of
+
W.
A
:,
S, and
continuity of C, there will not be any gulps on
A,
is
[t,t
+
a consumption gulp at
only occur at time
t
f,
(13)
+
for small
enough A. by the
i3Jz(t). This,
for
is
continuous on
[t,t
(13) holds as an equality at
t
+
maximize the right-hand
when
A
In
at
t,
side of (12) for small
at
it
given that
t
[i, i
+
A). For
must be the case that
dC =
(13) holds as an equality.
sum, we have derived the following necessary conditions
consumption gulp
form, can
decreases to zero.
enough A,
< Jw(W{t],z(t-),S{t)J). Moreover,
i3J^{W{t),z(t-),Sit),t)
consumption plan
its
there
A), standard arguments in dynamic
Next suppose that optimal consumption plan prescribes zero consumption on
to
when
when (5Jz{W{t),z{r),S(t).t) = JwiW{t),z(t-),Sit),t). Moreover, when
programming show that
dC
small enough
together with earlier discussion
implies that nontrivial consumption, independent of
optimal cumulative consumption
zero
right-
For a nontrivial consumption dC,
A).
A), to maximize the right-hand side of (12)
must be the case that Jw{t) =
it
[t,t
Jt[t).
t.
Then
not a point for consumption gulps.
with no discontinuities on
+
m^\[V-'^J(t)]
for optimality:
Is
the optimal
4
11
Sufficiency
In additions to the above necessary conditions,
it is
J must
clear that
satisfy the following
boundary conditions:
The
first
JiW,z,S,T)
=
J{0,z,Sj)
-
boundary condition
involve no gulps at time
T
ViW),
/
(15)
u{ze-^^'-'\s)ds
+
V{0).
(16)
implied by the fact that any optimal consumption plan must
is
since such a gulp cannot contribute to the agent's satisfaction from
increases in z and. furthermore,
reduces the terminal wealth.
it
The second condition
is
implied
by the constraint that wealth at any time cannot become negative. Thus, whenever wealth
zero, the only feasible policy afterwards
is
is
no consumption.
Sufficiency
4
we provide a
In this section
two steps.
First,
we show
theorem
verification
that
if
for
optimal controls.
J{W,z,Sj) > J(W.z,Sj)
proceed with
will
there exists a solution J to the differential inequality (14)
with the two boundary conditions (15) and (16), and J
then
We
for all
(W.z.SJ).
satisfies
some
regularity conditions,
Second, we then give conditions so that
attained by a candidate feasible investment and consumption policy.
J{W,z,S,t)
is
follows that
J(W,z,S,t) = J(W,z,S,t) and the candidate investment and consumption policy
is
then
the optimal policy.
Proposition
cave in
of
It
its
1
its first
Let there be a differentiable function
two arguments, which
is
J(K 0:
arguments, except possibly on a smooth manifold
M,^
K\ >
and
Hx,t)\
<
\J{Y,t)\
<K,{\ +
A"i(l
+
A'2
Vx G
\Y\)^'^
w
5?
U {-oo}, con-
J?'^"*"^
x [0,T]
x [0,T]
in its first
(16). In addition, let u
>
|x|)^''
For the definition of a smooth manifold see footnote
JJ^."^^
—
in all
N+
2
satisfying the differential inequality (14)
boundary conditions (15) and
the growth conditions: there exist
x [0,T]
continuously differentiable over
arguments, and twice continuously differentiable over
in the interior o/R^"*"^ with
??+"'"^
3?+,
(
G [0,r],
e R'^^\
t
e [0,T],
and J
satisfy
.
12
Sufficiency
4
Then, for
all
t
G [O.T],
rT
for
all
+ V(W{T+)) < j{W(t).z{r),Sit)j) <
u(z{s),s)ds
f
E,
C
some A £ At, where
€ Ct financed by
consumption and wealth,
Proof.
Let
C
z(s)
and W{s),
C
respectively, associated with
A
£ Ct financed hy
£ At he
Sl
.7(17(0, -(r),
s
5(0.0.
(18)
6 [t,T], are the average past
and A.
feasible policy for (8). Let \V{s)
and z{s~
)
be
the wealth and average past consumption at time s 6 [t,T] associated with (C, A). Since wealth
after
consumption must be nonnegative, whenever the wealth reaches
policy
to
is
consume and
Recall that
W
invest zero afterwards.
has left-continuous and that z has right-continuous sample paths. For sim-
we
plicity of notation,
time, evaluated at
zero, the only feasible
(
denote J and
will
VF(s), z(5~
),
5(3),.s),
its
partial derivatives with respect to
W ,z,S
and
by J(s), Jw(s), Jzis), Js{s), Js{s), respectively.
Note that, given the admissibility condition
(6)
and the growth condition on u given
in (17),
we know
Et
Define g
= mi{s >
t
1^(5)
:
does not exist, we set
it
stochastic interval
be
[t,
Generalized Ito's
u{z(s),s) ds
J'
g)
=
•
Also,
ti(uj),t2{u)),
00.
(19)
let
the hypothesis that
J
is
..
.
S{g), g)
J(Wit),z(t-),S(t),t)
concave in
J{tt)-J{U)
'
g
is
an optional time in that,
the infimum
the points of discontinuity of C{uj,s) on the
+ J\v^J{s) +
Js{s)]ds
+ j"[Jwis)W{s)Aisf Is-i{s)a(s) + Js(sfais)]dB(s) + j[
By
if
implies that
+ J{W{g),z(g-),
= J\{z(s),s)ds +
<
where we have used the convention that
0},
to be T.^
lemma
!«(-(•«), s)\dt
JIt
(u)
its first
Jw{t,)AW{t,) + Mt,)Az{ti)
=
-7w(f,)AC(i.)
g Q:
q(uj)
t}
- Jw{s)]dC{s)
two arguments, we know that,
<
<
[fij,{s)
+
£ Tt ^t £
7,(i.)/?AC(i.)
[0, 7"].
V^
a.s.
13
Sufficiency
4
Thus.
lfu{z{s),s)ds
+ J{\V{g),z(g-),S(g),Q)
(21)
<J{W{t),zit-),S{t),t)
where we have used the
Now
J
fact that
satisfies
the differential inequality and
C
is
increasing.
define
=
Pn
From
+ lf[Jw(s)W{s)A(s}^Is-i(sMs) + Js(sfa(s)]dB{s)
inf{5
<
g
:
f' [Jw{s)W{s)Aisf Is-i(s)ais)
the application of the generalized Ito's
lemma
in (20),
+ Jsisf <j(s)]'^ds =
we know that
/ {Jwis)W{s)A{sf Is-i{s)a{s) + JsS{sf a{s)]'^ds < oo
and. hence, we conclude that gn
]
On
9 a-s.
n}.
a.s.,
the stochastic interval [i,On],
J'[Jwiv)Wiv)A{vf Is-i{v)aiv) + Js{vf (T(v}]dB{v)
is
a martingale; see, for example, Liptser and Shiryayev (1977, chapter
4).
Replacing g of (21) by gn and taking expectation gives
Et[jf"u(z(s),s)ds + J{\V{gr,),z(g-),S{gn),g.)
(22)
<JiW(t),z(t-},S(t),t).
Note that (19) and the
fact that gn
^
Q a.s. imply
/en
re
u(z(s),s)ds
Et
—^ Et
by Lebesgue dominated convergence theorem.
Et[J(W{gn),z{g-),S(gn),gn)]
as n
—
We
-
/
u{z(s),s)ds
want to show that
Et[J(W{g),z{g~),S{g),g)]
oo. In our current context, this conclusion follows
from Lebesgue theorem by the
continuity of the sample paths of (W(5),z(5~),5(5)), the growth condition on
the
moment
conditions on
W
and on
C
in (6); see, for
J
in (17)
left-
and
example, Fleming and Rishel (1975,
theorem V.5.1).
The
assertion then follows from the
fact that
V
is
increasing and
boundary conditions of 7
W(T-^) < WiT).
I
at
T
and
at
W=
0.
and the
4
14
Sufficiency
Now,
there exists, for any time
if
tion plan
C£
Ct financed
t
< T and
given {W{t),z(t~),Sit)J), a feasible consump-
by A' 6 At so that
rT
J{W{t),z(r),S{t]J) = Et
where
z'
and
W
u(z'{s),s)ds^V(\V'(T^
I
denote the weighted average past consumption and the wealth associated with
(C*,.4'). respectively, then
it
must be the case that J(W[t),z(r ),S{t),t) = J(W(t),z{r).S[t).t),
(CA*) is the optimal consumption and investment policy starting from the (W{t), z[t~),S(t),t]
and
The
following
is
the
main theorem
consumption and investment policy
Theorem
1
to be optimal.
Let J satisfy the conditions of Proposition
Assume furthermore
I.
S [0,T], with \V{t),z(t~),S(t), there exists a consumption policy
at
any time
by
A' £ At with
t
of this section, which provides conditions for a candidate
the associated state variables
W
and
z'
,
such
that,
that, starting
CG
putting g
—
Ct financed
inf{5
>
t
:
P^*(5)=0},
J,
=
V a.e.5 e
\jw{W\z\S,v)-iih{\y\z\S,v)\dC'{v)
=
V5G(f,£»]a.5.,
u{z' ,s)-\-V'^' J
i
^
(«,p) a.5..
(23)
(24)
and
J{Wit,),z(t-),S{t,),t,)
where
(, 's
- J{W(t,) - AC'M,z(t-) +
are the times of gulps prescribed by
J{W{t),z{t~),S{t),t) and {C',A')
is
C
on
f3^C'{tr),S{t,).t,)
[t,Q).
=
generalized Ito's
E,
an optimal consumption and investment policy starting
=
Et
lemma
u{z'is), s) ds
[J'
allows us to write
+ J(W'{e), z'iQ-), S{q), g)
ru(z'{s),s)ds + JiW'{t+),z'(t),S(t),t)
+ j [Jw{s)W'{s)A'{sf Is-i{s)a{s) +
=
(25)
Then J{W{t),z{t~),S{t),t) =
att.
Proof. The
a.s..
Js{s)^ais)]dBis)]
J{W'{t+),z'{t).Sit)J) = J{W'{t).z'it-),Sit)J),
A
5
Theorem
Verification
where the
first
for the Infinite
Horizon Program
equality follows from (23). (24) and (25), the second equality follows from the
arguments used to prove Proposition
identical
15
1,
and the third equality follows from
the boundary conditions of (15) and (16) allows us to conclude that J
an optimal policy.
is
5
The necessary and
for
presented in the above two sections are
sufficient conditions for optimality
C
replace the admissibility conditions of (6) for
for all
t
cissociated with
is
C, and
financed by
T =
let
.4
oc
and
1'
=
in (7)
for all integers
m
>
and
and
by:
\u(z{s),s)\ds]<^,
E[
z
of course satisfies
Horizon Program
for the Infinite
the finite horizon program of (7). In this section, wo will
where
1
3.
Theorem
\ erification
Theorem
portfolio policy characterized in
the necessary conditions of Section
.\
thus (C.-l")
I
The optimal consumption and
all
- J and
(25). Using
(26)
T
G 3?+, there exists
kj^ so that,
< r,
^[|C(OP'"]
<
e*-',
(27)
Similarly
let
A
C and
policies starting at
spaces starting at
Consider the
=
<
denote the space of admissible consumption policies and investment
0, respectively. In addition, Ct
and At are the corresponding admissible
t.
infinite
horizon program:
J(W,z,S,t)
=
supcec.
s.t.
£'[/r u(z{s),s)ds\J't]
is financed hy A e At with \V{t) -
C
and Wit) - AC(s) >
We
V5 >
\V,
(28)
<.
record below a verification theorem for this infinite horizon program:
Theorem
2 Let J
:
differentiable over 3?^
in its first
N+2
inequality of
(
^^'^
"*"
—
3?+ be positive, concave in
in all of its arguments,
and
its first
two arguments, continuously
twice continuously differentiable over
arguments, except possibly on a smooth manifold
M
I4) with the boundary condition
7(0,2,5,0=
/
u(ze-'^^'-'\s)ds< 00.
,
3?^"*"
satisfying the differential
.
5
A
Verification
In addition,
KJ >
J
Theorem
satisfies the
growth condition: for every
T
16
KJ
£ ^+, there exists
and
>
so that
\J{Y,t]\
Assume furthermore
consumption policy
such
z',
Horizon Program
for the Infinite
that,
putting g
any time
R^+\
Vy
e
<
oc
t
t
6 [0,T].
(29)
with \V{t), :(t~
),
there exists a
S{t),
€ Ct financed by A' 6 At, with the associated state variables
=
inf{s
>
t
ir'(5)
=
0},
u{z\s)
+
'D'^'
:
W
and
J,
=
"^
a.e.s £ {t,Q) a.s.
(30)
r[Jw(W\:',S,v)- 3j,{W\z',S,v)]dC'{v)
=
'i
se(t,g]a.s.
(31)
lim £,[J(VF"(<
5
starting at
that,
C
< A'f(l + \Y\f?
—'OO
+
5),z"(«
+ 5-),5(« +
J+
5),i
+
5)]
=
0,
a.s..
(32)
and
J{W{t,),z{t-),S{t,)J,)- J(W{t,) - ^C'{t,),z(t-)
where
t,'s
are the times of gulps prescribed by
C
on
optimal consumption and investment policy starting at
Proof. For any
finite
+
pAC'{t,),S(t,},t,)
=
a.s.,
(33)
Then J = J and {C',.A')
[t,g).
is
an
t.
T, using identical arguments as in proving
Theorem
1
we
get.
rT
J{W{t),z{t-),S{t)J)> Et
Since
and
/
still
is
positive,
we can drop
u{z(s),s)ds
J
+ J(W{T),z(T-),S(T),T]
the second term on the right-hand side of the above relation
maintain the inequality. Since the inequality holds
for all
T >
t.
We
can
let
T
—
oo
and conclude that J > J
Then the
assertion follows from arguments identical to those used to prove
addition to using condition (32).
more conditions are added.
First,
the expected value of J{t) along the optimal path must converge to zero as
latter condition ensures that the agent exhibits
wealth indefinitely without consumption
technical convenience and
1
in
I
In the infinite horizon program, two
The
Theorem
is
we can replace
holds not just for the optimal policy but for
t
is
positive. Second,
increases to infinity.
enough impatience so that accumulating
not optimal.
it
J
We
imposed the former condition
for
by a stronger condition which requires that (32)
all
feasible plans.
Closed Form Solution for an Infinite Horizon Program
A
6
Closed Form Solution for an Infinite Horizon Program
A
6
17
In this section
we provide a
closed form solution for a particular
sumption and portfolio problem formulated
analysis,
we use a
felicity
function u{zj)
example of the optimal con-
which the horizon
in section 5, in
= ^e"*';", where
< a <
1,
is infinite.
In this
and where the discount
factor 6 expresses the agent's impatience.
In addition,
we assume
that the prices of the risky securities follow a geometric Brownian
motion given by:
S{t)+ [ p(s)ds = S(0)+ f [s(s)^lds+ f Is(s)adB{s),
Jo
Jo
where /s(0
where
/x is
is
N
a diagonal ;V x
matrix whose diagonal elements are Sn{t)-'n =
an iV-dimensional vector of constants, and a
instantaneous interest rate
Note that since the
a constant.
r is
We
an
is
assume that
and the
asset price parameters
N
M
x
aa''' is
=
e"^*
constant impatience factor.
J{W,z,0), since the
We
Q
A
The boundary condition when Vr =
function
is
r,
and
t.
Also,
The
differential inequality of (14)
6J,i3J,
- Jw} =
it is
easy
time separable and with a
our attention on the function J{W,
henceforth denote this function simply by J{\V,z).
max{— + max[P^ J] -
matrix of constants. The
interest rate are constant, the value
felicity
will therefore focus
-V,
1
nonsingular.
function J depends only on wealth \V, the average past consumption
to see that J{\V,z,t)
(34)
Jo
and
z. 0)
becomes
0.
(35)
is
^(0,-^)=—4tTT-'"-
We
will
show that the optimal solution
barrier policy.
The optimal investment
the iV risky assets at
ratio of wealth
all
W
states of nature.
all
times.
to
is
consumption
problem takes the form of a
'
_'
.
is
strictly less
consume nothing and wait while
Rec^l that we use weak
relations
ratio
to invest constant fractions of wealth in
is
the
amount required
z less than^° a critical
the agent starts at time zero with wealth
experience z{0~) such that
is
decision
The optimal consumption
to average past
If
to the agent's
(36)
than k'
.
number
W(0) and
initial
to keep the
k' in almost
consumption
then the optimal consumption policy
W increases on average and z declines until the (random)
and hence r
less
than y means that i
is strictly less
than or equal to
y.
A
6
Closed Form Solution
-4^
time r when the ratio
—
when
only
On
=
k' the
the ratio
^
if
see, the erratic
equal to k'
<
Starting from that
.
—
to keep
< k'
moment, the agent consumes
forever in
states of nature.
all
,^J.
>
k'
.
then the optimal
W and increasing
consumption, reducing
-,
to bring
Following this gulp, the optimal amount of consumption
k'.
k' forever,
and consumption occurs only when -^ = k'
.
As we
is
shall
behavior of the price of the risky assets implies that the optimal consumption
pattern will have "singular" sample paths. In almost
a non-trivial
18
the agent starts at time zero with
immediately to
—
Horizon Program
Infinite
to take a "gulp" of
is
that required to keep
is
an
amount required
the other hand,
consumption policy
for
amount
at infinitely
many
all
states of nature, the agent
points of time. However, the times
consumes
when consumption
occurs are a set of Lebesgue measure zero.
In essence,
when
into consumption.
the wealth in relation to z
Conversely,
when
is
too high, the agent will convert wealth
the wealth relative to ;
from consumption altogether to accumulate her wealth and
in the
low, the agent will refrain
mean
time, will enjoy the
from her past consumption purchases. The singularity of the sample paths
satisfaction derived
of the optimal consumption pattern
This
is
is
caused by the erratic behavior of a Brownian motion.
a feature of the modeling choice in which the returns on risky securities follow a Brownian
is
motion. The economic content of the solution
the periodicity of consumption, which
is
is
in
part due to the fact that the agent derives satisfaction from past consumption.
We
will
employ probabilistic methods
and portfolio
tion
policies.
We
to
compute the function J and the optimal consump-
present the logic behind the optimal solution in two steps.
First,
we analyze general consumption and portfolio
policies of the k-ratio barrier form.
any
>
the policy of investing constant fractions
A;
0,
the corresponding k-ratio 6arner policy
of wealth in the
less
A''
risky assets together with a
than or equal to k forever in
policy
,J_|
is
>
all
is
consumption policy of keeping the ratio
states of nature
and consuming only when
followed after an initial "gulp" of consumption
k, or after
For
if
a (random) period of no consumption
—
=
k.
—
This
the initial conditions are such that
if
^.J_(
<
k.
We
will
show that
the expected life-time utility associated with any /c-ratio barrier policy and initial state {W,z),
denoted j''{W,z),
satisfies
max {— + D-^7*-(57*} =
if
W<kz
J^ =
if
W>kz.
A
J^ -
13
and
A
6
Closed Form Solution
an
for
Infinite
Second, we will show that there
together with
and hence
its
Horizon Program
>
a unique vjilue k'
is
19
such that the
associated value function. J'(W,z), satisfy
all
pattern with
make two
initial state variables
observations.
consumption pattern
j''(W,z)
0.
=
for the
same
{W,z)
foT
A:-
a.
ratio policy.
fc-ration policy
with
Thus j''{a\V,az) = a'^j''(W,z). That
z'^j''{W/z,l)-
Second,
must be time independent and
is
is
it
-
{aa'^)(fi
rl) and
a function of
1 is
be the optimal consumption
By
the linearity of the dynamics
and
(5), respectively,
J^
homogeneous of degree q
is
the
is
{a\V,az) for any
initial state variables
is.
aC
in that
easily seen that the optimal investment strategy
W
and
IV
where F =
2
C
First, let
of wealth and the average past consumption, given in (4)
>
Theorem
the conditions of
the optimal policy.
is
Before proceeding, we
scalar a
A:*-ratio policy
an .V-vector of
J
z.
A'
Direct computation shows that
WW
The investment
I's.
policy
is
always propor-
tional to the vector F.'^
The
object of control under a A:- ratio policy
keep this ratio less than
and suppose that she
by
.4*^,
A;.
Suppose that the agent
>
us
r.
and consumes nothing
The wealth dynamics
compute the expected
^From the
until the
starts
—
is
to
from a state {W,z) such that -r <
k,
of this policy
utility
=
e\
is
=
e[
1).
k.
-^^ <
k for cdl
j''{W,z) obtained from such a policy.
we can
write, for
(W,z) such that
f^.J
l
-
e-(<^'3+5)^]|
We
thus
^
<
^'•
'
+ E\e-'^j'{W(T),ze->'n]
for a point
•
on the boundary {W{r),z{T)),
know that
j'{wir),z(T)) =a,z{Tr,
"Here we note
Also suppose that
well-defined by Lions and Sznitman (1984). Let
homogeneous of degree q,
J{W{t),z{t)) = z(TfJ{k,
=
r e-^'—e-''^'ds
+ e-^^j''{W{T),ze-'^^)
a
,
is
.^V
required to keep the ratio
'Jo
Given that J*
and the purpose of control
stopping time r when
minimum amount
definition of the utility,
j''{W,z)
the ratio
invests constant proportions of her wealth in the iV risky assets, denoted
from T on. the agent consumes the
t
is
the two-fund separation property
imphed by
(37); see
(38)
Merton (1971).
6
A
for
some constant
Closed Form Solution
an
for
we can rewrite J^{W,z)
a^. Hence,
=
J^'i^V.z)
f+ 6)
,
,^
Q(Q/i
Assume
^
motion with
drift
=
[//
-
+
-
[flfc
<
k, the
=
^i/t
+
r
-
rl]^(i7<r^)~^[/i
+
/j
b^f
—
bJ.-f/2
From Harrison
rl].
^
,
logarithm of the ratio
^,
20
as:
a(a3I +
'
=
that the constant investment policy A''
from a state
7
Horizon Program
Infinite
J z°E[e-'"^+^'^]
S)'
i
some
bt-T for
before
^
(39)
.
^
J
scalar
reaching
/c,
and a standard deviation
(1985, proposition 3.23),
6*;.
Then
follows a
ctj.
'
starting
Brownian
= bi^^. where
we conclude
that
where
ak
=
—[{^il
+
2al(af3
^'(^-') =
Now, consider the
A=
^
to a point
~^^f
situation
J^ =
/JJj
when
—
f"^
^"^
'
>
-
The
k.
S))2-fi^].
<
Jw -
when
«.
^
QJz
=
A;.
=
(a/3
+
for all
From
^
>
A:.
^^]-"(E)'
a(al3 +
6)'''
We
^
+
utility
^''^
•
^•-barrier policy prescribes a
jump
of size
is
then choose the value of the constant a^
we
this condition,
(5)[ajt(l
(41)
A;,
on the boundary and the corresponding
Notice that (43) implies
such that
+
t(af3 + S)
m£tT)
^
when
Substituting from (40) into (39), we get,
+
r
/3A;)-Q/3it]
+
get
^^^.
+
a(Q/J
<5)
m
To summarize, the value function associated with a constant proportions investment strategy
A =
bkT and a
A;-ratio
consumption policy
+
*
*
where a^
is
is
given by
zn^
[^l>-.(^]
l«4l^)"
given in (44) and a^
is
'f^<^'
ifT>^-
given by (41).
Note that in deriving (45) we started by assuming a constant proportions investment
egy
A'^
=
bkT. This investment strategy together with the fc-ratio
strat-
consumption policy implies
,
A
6
Closed Form Solution
the functional form
J''
If
.
for
an Infinite Horizon Program
J^ were the optimal value function, the optimal investment strategy
would be determined through
In particular, since the fc-ratio policy
(37).
the optimal investment strategy would be equal to r/(l
composed
to the system of equations
1
make
to
j''
of (41)
a concave function, then
investment behavior of
21
and
=
bk
-
1/(1
-
Qfc)
and
.
We
this solution
is
Assumption
> ar
6
Given these
1/(1
-
1
Qjt)
'''
=
need to make enough
as-
exist.
We
whose proof uses
ele-
problem satisfy
and
(\
-
>
a)l3
6
-
r
.
easily establish the following result,
There exist a unique q^ <
1
that solves the
system of equations of (41) and b^ =
and
[7/2
+
(r
+
3)
As a consequence of
will
than
less
for the reader.
is left
Moreover, a^ > a and a^
We
the
Q7
2(1 - Q)
-^
we can
restrictions,
mentary algebra and
Lemma
k,
this section that:
The parameters of
1
<
that the implied
J*^ in
sumptions on the parameters of the problem to ensure that a desired solution of a^
assume throughout
y
there exists a solution
if
we have produced a consistent
generates the functional form of J^
,/''
Thus,
Q/t).
would keep
+ {a3 +
is
6)]
- ^[7/2 +
+ (3) +
^i^TW)
independent of
[r
in the definition of a^ of (44).
The
-
q*).
k, A''
= r/(l -
and
+ J)(ad +
S)
It is
independent of
Q,t) is also
Lemma
1
and by
also understood that the q^
following proposition gives properties of
Proposition 2 The function J^ of (45) with a^ replaced by a'
strictly concave,
4( r
•
henceforth denote by a' the solution characterized in
proportions investment strategy r/(l
+ 6)^ -
k.
being independent of
Q/t
(al3
is
.4'
the constant
replaced by a"
is
j''
k.
defined in (45).
increasing,
continuous,
twice differentiable. has continuous first derivatives, has continuous second
derivatives except possibly on the smooth manifold
Mk
=
{{^^^^)
'•
^
=
^^}>
''"'^ satisfies,
together with A'
^^
Q
+ D^V*^ -SJ'= =
J^v- ^Jz
=0
if
if
W < kz
W > kz,
and
(46)
(47)
A
6
and
Closed Form Solution
the
for
an Infinite Horizon Program
22
boundary conditions
J{0,z)
1
=
a(Q/3
+
and
(48)
6]
\imE\e-'' j'={Wk{t),z,(t))\
=0,
(49)
t\oo
where
W^ and
and
z^ are the wealth
consumption policy and
the average past
the investment strategy A'.
consumption associated with
the k-ratio
In addition, e~^^J^ satisfies the growth
condition of (29).
Proof.
All the assertions except for the last can be verified by direct computations using
the fact that
t
>
0,
q < q' <
1.
For the last boundary condition note tb
and hence J^{\Vk{t),Zk{t))
is less
t
Wk{t) <
than akz'^(t), where we recall that a^
is
kzt;{t) for all
defined in (44).
Note that
Zk{t)
=
z[Q-)e-^'^i3
f e-^^'-'UC,{s)
Jo
<
where Ch denotes the
see that Ch{t)
<
consumption
A:-ratio
lV{t)
a.s.,
portfolio rule .4' with no
z(O-]e-^'
where W{t)
+
[0Ck{t)],
policy.
is
^From the budget
constraint of (4), we can
the wealth realized at time
consumption withdrawal before
t.
when
t
following the
Using the dynamics of
W.
one can
easily verify that
limE
e-''j(Wk{t),Zk(t))\
(Tco
J
< limE[a,e-^'r?(0] < rimE[a,,e-%{0-)e-^' + (3W{t)Y
tfoo
(loo
But
z(0)e-^'
+
l3W{t)\
-^ \tSW{t)\
uniformly in
w
as
t
]
oo
hence
limEle-^^ iz{0)e-'^' +l3W(t))°-]
=
limE[e-*'/3'*Pr'^(T)
\imE\e-'"J{Wk(t},Zk(t))\
<
-St,
limEle-"
ff^Wit
f\oo
J
1-
(Too
I-
But
E Vr(0
=
W{0)e {4^-^T^hi^f^y
and thus
A
6
Closed Form Solution
where the
an Infinite Horizon Program
for
q < q", and the
inequality follows from the property that
first
qV Hence E[e-*ny°'(o] <
the definition of
23
W{0)e>^^''-''"i'
last inequality
Noting that q - q' <
.
from
0.
the
required result follows.
The
last assertion
is
Now. we show that
the function
y*"'
is
obvious.
I
a ^--ratio policy financed by
indeed the expected
.4' is
an admissible policy in (28). Moreover,
agent gets from following this /c-ratio
utility that the
policy.
Proposition 3 The wealth process and
the
consumption pattern associated a k-ratio barrier
consumption policy financed by the investment strategy A' satisfy
e~^'j''{W,z) gives the expected
Proof.
Let
W^, Ck, and
utility
of following these policies at time
consumption policy financed by A'. Recall from the
^--ratio
proof of Proposition 2 that Ck{t) < W{t)
at
t.
time
t
when
Similarly,
t.
denote the wealth, the consumption, and the average past
Zk
consumption associated with the
(27). In addition, the function
a.s.,
where we
recall that
Wit)
is
the wealth realized
following the investment strategy .4" with no consumption withdrawal before
Wk{t) < W{t)
a.s.
easily verified that
It is
Brownian motion and thus W'(i)
is
W
satisfies (27) since it is a
log-normally distributed.
The
first
geometric
assertion follows.
For the second assertion, identical arguments used to prove Theorem 2 show that
^-S'
^^^ dv + e-"'
J'{\Vk{s),Zk{s))
Sv^kiv)
SvthM>
^1/' e-'^
J,.
where we have used Proposition
by letting
5
—
00.
2.
The
assertion follows from
discussions,
we know
satisfies
fail
to satisfy the differential inequality (35).
is
monotone convergence theorem
that any Ar-ratio consumption policy together with
A'
(35)
-St jk.
e'^' r{Wk{t),Zk{t)),
J
I
iFrom the above
many
=
,
Q
of the sufficient conditions for optimality. All policies, except one, however,
the optimal solution, which
is
The unique
barrier policy that satisfies inequality
recorded in the following proposition.
6
A
Closed Form Solution
an Infinite Horizon Program
for
24
Proposition 4 Let
f3{a'
and
>
-a)
A' and
the k' -ratio
Proof.
First k'
Next
the value function associated with the A;'-ratio barrier policy be J'.
In view of
Then
k'
the solution to (28) consists of the investment strategy
harrier consumption policy.
let
Proposition
We
2,
>
lemma
by
we only need
to
show that J'
W<
satisfies:
Jw - JJ: <0
if
W < k'z
—a + V'^'J' -SJ' <0
if
VV>k'z.
and
Using the definition of J', we can write J^y — 3J'
start with the first inequality.
region
1.
in the
k' z as:
=
7^-/37;
where
az^-'gi'Q-}
VV
=
g(y)
where a
is
/3(a--Q)t/<^-
a constant. Note that ^(1)
Computing the
we
derivative of g,
—
= a
dy
j3{a
=
+ ^y(-'-i'-—
k'
ap +
^,
and that since q' >
0,
1/
>
1.
It
+
-a)y
Now, we
verify the second inequality.
y
0, for all
^T ~
which one would jump
if
f
cxd.
.
one starts at
x.
for
1/
=
1,
and that j^ >
W < k'z.
such that W
points {W,z) such that
= {W,z)
W = k'z and the straight
^^ other words, a
~h'
=
Consider any point x
Let a be the point on the intersection of the line
the point x with slope
oo as y
k'
J^ - pJ' >
then follows that
f
get
Substituting k' in the expression for j^, we conclude that j^
for
0, 5(1/)
is
line passing
the point on the boundary
By construction, J'{x) =
>
k'z.
through
W = k'z,
to
J'{a).
Let
fiW,
z)
and
-a + J'^rW - -^7 - J'J^
'Jww
and note that /(a) - SJ'{a) =
straight line connecting a
=
x,
0.
,
Applying the fundamental theorem of calculus along the
we obtain
/(£)-/(£)-
r fwdW + f^dz.
Ja
.
A
6
Closed Form Solution
an
for
and
N'oting that along the line connecting a
direction from a to x,
then follows that
it
l- +
f\y
The
at
=
(
0.
J/. <
is
_^y
= -3d\V. and
—>
if
in the region
fw -
<0
3fz
W
in the
k'
^
>
=
for k'
d\V >
conclude that
sufficient to
<
that
and using the properties of q', the reader can
k':.
jffrr^^-
following proposition records the optimal consumption and investment policy starting
The optimal
Proposition
5
policy from
any
>
(
let the
can be constructed similarly.
Let
——
^-
A'(t)=
and
/» -
:
25
2A*ivv
- J/j
easily verify that
we have d
x,
Jl_^ _y;^,
j-^.r\V
a
Computing
Horizon Program
Infinite
budget feasible consumption process
r
for
C
all
t.
which has continuous sample paths almost
surely, be given by:
C-{t)
= AC-{0) +
P-a.s.
J^^^^dl{s)
(50)
where
AC-,0,
=
l{t)
=
Wit)
=
(vVo-AC-(0))e(''"^^^^''"'^^'^'''^""
z(t)
=
[zo
ma.(o. "-'°|;^;f-'
}.
r
sup
and where W'is) and z'(s)
Wis)
^-
log—
log
r
+ 3AC'(0))e-^'
1
+
(51,
p-a.s.,
(52)
P-a.s..
(53)
P-a.s..
(54)
are the state variables associated with
C
.
The strategy
.\'
and
C
defined above are the optimal solution for the agent's problem.
Proof. To prove
this proposition, all
ratio barrier strategy associated with k'
if
>
wo=)
^'
The
size of the
from which we compute
Let us proceed
policy
is
now
A
to
jump
A
we need
The
.
is
A:'
to
show
is
that the above strategy
ratio barrier policy calls for a
jump
at
calculated to achieve the condition .(q-i Tj^
'
is
(
=
the
=
0,
^''
as in (51).
compute the consumption process
after
t
—
0.
The
/:'-ratio barrier
equivalent to the condition that
W'[t)
z-[t)
<
k'
'it.P-a.s.
or
(55)
6
A
Closed
Form
Solution for an Infinite Horizon
loglI-1^
<
Program
26
^t.P-a.s.
\ogk-
(56)
z'{t)
Now
consider the "unregulated" process {W,z), which gives the wealth of the agent and her
past average consumption under the assumption that no consumption takes place after the
initial
jump
at
i
=
0.
The
values of \V{t) and z{t) are given in (53) and (54).
will fail to satisfy condition (55) for
the solution
is
to "regulate" this ratio using the
optimal solution condition (55)
To achieve
let
is
for
•
is
some sample paths, and the idea
C
consumption process
=
[log——--
sup
P-
log/:']
u).
to ensure that at the
a.s.
F-a.s.
(07,
the sample path /(^,.) has the following properties:
=
increasing and continuous with l{^,0)
log^p^ <
logfc" for all
• l{u>,.) increases only
t
when
>
0. for
almost
all cj.
0. P-a.s.
log
^"^'V =
This "regulated" process (log 7777,
)
is
log/:'.
a candidate for the logarithm of the ratio of the state
variables associated with the optimal consumption plan, since from the above construction,
can easily conclude that condition (55)
exists a feasible
is
satisfied.
The question now becomes whether
consumption process C" that could enforce the relationship
both sides of (57) using
,
of
be given by:
-;t-
log^ = .og|li>-,(„
• /(u;,.)
-~^
satisfied.
the "regulated" process log
For each state
ratio
this regulation, define the process
l(t)
and
some periods and
The
W{t)
''^1^
W{t)
Ito's
lemma, we
get
H^(0)-AC-(0)
p.
+
JAC-(0)+yot'"
+
,
,
=
^°^.(o-)
W{0)-AC'{0)
log
,
f\
,
,
,
,,
,
i(0
+ f
Jo
-T^crdBis)
j^- —
a-)
(1
7(1
-2a') ,,
^^^(T^^^'^
P-a.s.
7(1
-2a') ,,
in (57).
we
there
Expanding
A
6
Closed Form Solution for an Infinite Horizon Program
27
Thus, we conclude that the consumption process given by:
U'(5)
7o
/
increases for
condition (31) of
Now
let
sample paths.
all
Theorem
2
is
=
C
Therefore,
=
y{0)l{0)
pw'{t)+z'{t)
/(O)
=
.
P-
a. 5.,
, .
^'
=
k'
when
and hence
=
lemma, we
Ito's
P-
get:
a..5.
Jo
and that
yit)lit)
From
-
+ fyis)dl{s)+ fl{s)dy{s)
Jo
Noting that
when
increases only
increases only
satisfied.
the process y be given by y{t)
y{t)l{t)
C
Note from the above equation that
the condition in (57).
satisfies
:-(s)
dl{s)
+
C'it)
= ^—f-, we
find that
P-a.i
f l(s)dyis)
Jo
Integrating the second term in the right-hand side by parts, we have that
ii\V'{s)
Jo
But from the properties of
/,
we know that
/
+
z'[s)
increases only
when
../^y
=
k'
.
We
thus conclude
that
W{3)
"
'^"*" =
The
solution
we constructed has
agent consumes the
equal to
(/3
minimum
possible
If
sample paths.
amount required
z.
to keep the marginal ratio of wealth
Consumption takes place only when these marginal
the marginal value of wealth drops below
of 2, the agent stops consuming.
times
the following features. After the initial (possible) gulp, the
times) the marginal value of
values are equalized.
all
iniF'"w '-''
These
Consumption occurs
at
rules result in a
uncountably
times) the marginal value
consumption process with singular
infinite
when consumption occurs has Lebesgue measure
(/j
number
of times, but the set of
zero, for almost all
sample paths.
This result occurs because of the unbounded variation property of the Brownian motion sample
paths.
7
Comparative Statics
Comparative Statics
7
It
28
is
compare the behavior described
interesting to
an agent with a time-additive
in the previous section to the behavior of
on the subspace of absolutely continuous
utility function defined
consumption patterns and given by:
a
Jo
where
c{t) is the
consumption rate
at
time
t.
J
Note that the
of the utility function analyzed in section 6 as
/3 t
utility function in (59) is the limit
oo.
Consider the consumption-investment problem of an agent whose preferences are given by
(59) and
who
by (34) and a constant
faces the investment opportunities given
Merton 1971) shows that the agent's optimal policy
(
and
to a constant fraction of total wealth,
is
to
consume continuously
to invest constant proportions
riskless rate.
at a rate equal
of his wealth
yt^^
in the risky assets.
Now compare
the investment policies of the agent
we studied
Merton's time- additive agent, assuming that the corresponding
in this
felicity
lemma
1,
Hence the agent whose preferences exhibit
risky assets
and appears
we know that q' >
6,
and face the same
a. Furthermore, \\mpi~^ a'
local intertemporal substitution invests
Furthermore, the weaker the
to be less risk averse.
same
functions have the
curvature q, and that the agents have the same time preference parameter
investment opportunities. From
paper with those of
more
=
q.
in the
effect of local
substitution (higher 3), the less that the agent invests in risky assets, ceteris paribus.
One
could explain such a behavior by arguing that the fact that past consumption affects current
and future
utility leads the
agent to tolerate higher probabilities of reduced future level of
wealth and hence take more risky positions.
The reader can
easily verify that
^^
<
0,
and
^^
<
0.
In other words, higher values
of riskless rate and higher discount factors lead to less investment in the risky assets, ceteris
paribus.
Another interesting property of the solution
state distribution
for the
and the expected value of the the
non-time-additive utility
ratio -p-.
z{t)
-
the steady
Following the computations in
Harrison (1985, §5.6), we conclude that, from any starting point (Wo,zq), as
^
is
t
—
'
oo:
(60)
/
1
1
ifi>fc-
8
29
Equilibrium Riskless Rate
and
"
'^""
z(t)
/i-
i
+a-^/2
k-.
(61)
where Pq and Eq are the probability distribution and expectation, respectively, conditional on
starting at (lV(0),z(0))
and where
7
and where 7
Denote
=
[;i
-
rl]'''(CTi7"'"
lim(_,3o Eo[t77)
d
One can
]
)
(1-a-)'
^[fJ.
-
d
W
d
W
d
VV
understand these relationships as follows. The ratio of wealth to average
past consumption eventually settles
which the agent holds
rl).
by [Vloo- The reader can easily verify that:
W
intuitively
v/7
=
(T
7
for
down
to
[— ]oo- Wealth
is
the investment in risky assets
purposes of future consumption, whereas z
is
an indication of
past consumption experience, which also contributes to his future satisfaction.
his
The agent
decides on an optimal division between these two sources of future satisfaction. This division
is
naturally dependent on the strength of the local substitution effect and the outside investment
opportunities.
relatively
more
When
the local substitution effect
satisfaction from ;.
and hence
[
—
investment opportunities are more attractive (high
is
]oo
high (low
is
r or
low.
t3),
On
the agent tends to derive
the other hand,
low a), [y-Joo
is
when the
high, reflecting
more
emphasis on deriving future satisfaction from wealth.
8
Equilibrium Riskless Rate
In this section,
on asset prices
we
briefly state
some
of the implications of the form of utility studied in section 6
We
adopt the representative agent equilibrium framework of
in equilibrium.
Cox, Ingersoll and Ross (1985a).
In the general
framework of section
4,
we note
that the
optimal investment policy implies the mutual fund separation results of Breeden (1979) and
Merton (1971). Thus the Intertempor2Ll Capital Asset Pricing Model of Merton (1973) holds
in equilibrium.
In the special case of constant
parameter production processes, the constant
Concluding Remarks
9
30
proportion optimal policy given
in (37) implies
two-fund separation.
Therefore, the Capital
Asset Pricing Model holds. However, the Consumption Capital Asset Pricing Model of Breeden
(
1979)
to the
fails
to hold because the marginal utility of instantaneous
marginal
utility of wealth.
consumption
no longer equal
is
Bergman (1985) and Grossman and
For similar results, see
Laroque (1990).
Next, we consider the determination of the
riskless rate.
Suppose,
for simplicity^^. that there
one risky production technology whose rates of return follow a (/i,<T)-Brownian.''' Consider
is
an agent with preferences as given
Using arguments similar to those of Cox,
in section 6.
IngersoU and Ross (1985b), and using the computations of Harrison (1985, §3.2), we conclude
that the equilibrium riskless rate,
In contrast,
r. is
r
=
/i-(l-d)cr^
a
=
-^ yifi
if
given by:
where
+ ,3-
(tV2)2
+
2a^{at3
a/i
<
6)
-
[p
+
^3
-
a^l'l)
.
the agent has a time additive utility function whose felicity has the
curvature a and the same time discount parameter
Given that
^
6,
b,
the interest rate,
the reader can easily verify that a
case of local substitution preferences
words, the risk premium, which
is
is
given by
is
> a and hence
/x
—
(
1
same
— a)a^.
the interest rate in the
/izg/ierthan that for time-additive preferences. In other
the excess of
of preferences with local substitution. This
is
/i
over the riskless rate,
is
lower
in the case
consistent with our earlier observations that the
agent whose preferences exhibit local substitution behaves in a /ess risk averse manner compared
to
an agent with time-additive
the risk
premium
utility. In
addition, as local substitution decreases
(/3
increases),
increases.
Concluding Remarks
9
In this paper,
we have provided necessary and
sufficient conditions for a
consumption and
portfolio policy to be optimal for a class of time-nonseparable preferences that consider con-
sumptions at nearby dates to be almost perfect substitutes.
We
demonstrated our general
This result can be easily generalized to the case of many production technologies.
t starting from one unit continuously reinvested,
evolves according to the following equation: dV(t) = nV{i) dt-\-<TV(i) dB(t), where /i >
and u > are constants
'
^''More specifically, V(t), the level of risky capital at time
such that an < S, and 5 is a one-dimensional standard Brownian motion. Note that we use n and
meaning from that in section 6, and 6 is the parameter of time discount.
<t
in a different
31
References
10
theory by explicitly solving in closed form the oplimaJ consumption and portfolio policy for
a particular felicity function
when
the prices of the risky assets follow a geometric
Brownian
motion process.
We
have also explicitly solved
(1985) type
economy with a
for the
equilibrium interest rate in a Cox, IngersoU, Ross
physical production technology whose rate of return follows a
Brownian motion.
avenues of future research deserve attention. First, we can incorporate the notion of
Two
habit formation into our model while preserving the feature that consumptions at nearby dates
are almost perfect substitutes. This can be accomplished by defining two weighted averages of
past consumption, zi and
than does
Consider
that ui2
Z2
and
felicity
>
0,
^2
Z2,
where
::i
has heavier weights on more recent past consumption
has heavier weights on
functions defined on
zi
and
more
tt,
distant past
unit of more recent past
consumption
is,
will be.
;i.
denoted by u{zi,Z2,t), having the property
where ui2 denotes the cross partial derivative of the
the higher the distant past consumption
consumption than does
first
two arguments of
u.
Then
the higher the marginal felicity for an additional
And
Z2
has the interpretation of a living standard.
This formulation treats consumption at nearby dates as close substitutes; while incorporates
the notion that "habit"
is
formed not over a short period of time but over a longer horizon.
Second, the interest rate derived in Section 8
curve.
One needs
explicit
and interesting term structure of
10
1.
is
of limited interest as
it
implies a
flat
yield
to explore possible specifications of the production technology to derive
interest rates.
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Date Due
^21
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