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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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Does Constant Relative Risk Aversion Imply Asset Demands that are Linear in ExpectedReturns?Author(s): Anthony S. CourakisSource: Oxford Economic Papers, New Series, Vol. 41, No. 3 (Jul., 1989), pp. 553-566Published by: Oxford University Press
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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Oxford
Economic
Papers
41
(1989),
553-566
DOES CONSTANT
RELATIVE RISK
AVERSION IMPLY
ASSET
DEMANDS
THAT
ARE LINEAR IN
EXPECTED RETURNS?
By
ANTHONY S. COURAKIS
Introduction
IT
is
often claimed'
that,
in
the
context of
discrete-time
analysis
of
portfolio
selection,
'
..
constant relative risk
aversion
and
joint normally
distributed
asset
return assessments
are
jointly
sufficient to
derive,
as
approximations,
asset demand functions [that exhibit] .
.
. wealth homogeneity and linearity
in
expected
returns'
(Friedman
and
Roley, 1987, p. 627).
The main
purpose
of this
paper
is
to
re-examine the characteristics
of the
asset
demands
that constant relative risk aversion
and
joint
normally
distributed
return assessments define.
In so
doing
it is
shown
that,
contrary
to the above
claim, given joint
normally distributed return assessments,
the
power
functions
typically employed
in
the relevant literature to
describe
investors'
preferences strictly
imply
asset demands that are
not
linear
in
expected
returns.
Furthermore,
no
additional
assumption
that delivers
linearity
as an
'approximation'
is admissible.
For
any
such
assumption
is
not
only
inconsistent with
portfolio
choices that are
based
on differences
between
expected
returns
on
the various
assets,
but
also implies
that
certain
distinctions between classes
of
utility
function that
we are
generally
prone
to
emphasize
are
then
completely
overlooked.
Like
the
asset demands
corresponding
to
the
quadratic utility
function,
and unlike the asset demands
corresponding
to the
negative
exponential
functions
(be
it in
wealth
or in
the portfolio rate
of
return),
contrary to the
consensus claims of previous studies, power functions deliver asset demands
where the
matrix
of
responses
to
expected
returns exhibits neither zero
row
sums
nor
symmetry.
On
the
other
hand,
unlike both the
quadratic
and
negative exponential functions,
power
functions
imply
that asset
demands
are
invariant
to
any multiplicative
scalar
change
in
the
vector of
perceived
returns
on the various assets.
A
corollary
of all this is that none
of
the
empirical
studies
purporting
to
model asset
demands in accordance with
power utility
functions
actually
relies on
a specification
that is
consistent
with such
behaviour
in
discrete
time.
I.
Mean-variance
models
of
asset
demands
Confining
our attention to
a
discrete-time
single period setting,
consider
an investor
whose wealth
at time
t,
the
point
of
deciding
the
composition
of
'
For example
Friedman (1980, 1982, 1985a,
and
1985b), Roley
(1981 and 1982), Frankel
(1985),
and Green
(1987).
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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554 A. S. COURAKIS
his
portfolio
as between k
assets,
is
readily
convertible into
any
feasible
bundle
of these assets at zero
costs of
transactions.
Suppose
that
his
preferences
over alternative
feasible
portfolios
are
defined
by
some con-
tinuous
and
twice
differentiable
utility
function
U(W,+f),
such that
U'(W,,f)
>
0 and
U (W,+f)
<
0,
where
W,,f
denotes
the value
of
his
portfolio2
at some
specific
time
in the
future,
t
+f,
that
defines his horizon.
Suppose
that
in
the interval
t to t
+f
decision costs
are infinite
(or
that
transactions
costs are infinite with
regard
to
all
trades)
so
that
no
intraperiod
revisions
in
the
composition
of
the
portfolio
can be entertained.
Let A,
be
a
k
X
1
vector
a
typical
element
of
which,
ai,
is the
amount
of
the
ith
asset
chosen
at
time
t.
Let
r,
be
a k x 1
vector of
holding period
subjective
returns
on
these
assets,
where
rj,,
the
t
+f period
return
per
unit
of the jth asset, is for all but at most one of these assets a random variable.
Dropping
the
subscripts
t for notational
simplicity,
at
time
t
W
=
C
A
(1)
where
t
is
a k
x 1 unit
vector,
while
at
t
+f
the value
of
the
portfolio
is
Wf
=
(t
+
r)'A
=
(t
+ r^
0)'A
(2)
where ri
=
E(r)
and
0
=
(r
-
r),
E
denoting
the
expectation operator.
Suppose that the investor regards Wfto be normally distributed, or that
his
circumstances
are such
(see Tsiang, 1972)
as
to
entitle us
to
disregard
higher
than second order moments
in
the distribution
of
Wf.
Expected
utility
of
wealth may
then be
approximated
(Tsiang, 1972; Friedman
and
Roley,
1987) by
E[U(Wf)]
=
U(W
)
+
U
f
Wf
=
E
Wf
=
( t
+
r^)'A
(3)
V
=E[(W-
W)2] A'QA
where
U(Wf)
denotes
utility
as
a function of
expected wealth,
U (Wf)
is
the
second derivative
of
that function
with
respect
to
expected
wealth,
and
Vf
is
the variance
of
Wf,
with Q
=
E(00') being
the
variance-covariance
matrix
of
returns
on
the
various assets.
Maximizing (3)
in
terms
of
A, subject
to
(1), yields
the solution
A
=
(I/p)Q(t
+
r)
+
BW}
=
-U (Wf )/U'(Wf)
(4)
where:
(i)
in
the
absence
of
an asset
with a
known return
V
-'-tBB'l
t 'Q-'t
>0
(S)
2In
line
with the
analysis
of
my
precursors,
for most of what
follows
no distinctions shall
be
drawn
in the
analysis presented
between real and nominal
magnitudes.
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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CONSTANT RELATIVE RISK AVERSION
555
while
(ii)
in
the
presence of
an asset with a known
return:
Q[-l 'Q
-
,
t
]
0
t =Quil >0
(6)
where
i
denotes the variance-covariance matrix of
returns on the k-1
risky assets,
with the kth
asset carrying a known return.
Note
that,
whether with or without an
asset with
a
known
return,
t'B=
1
(7a)
*'Q=
Ot'
(7b)
Qt
=
Ot
(7c)
Q
=
Q
(7d)
z'Qz 0, where
z
is any non-zero vector
(7e)
The
first two of
these
conditions, (7a)
and
(7b), are sufficient to
ensure
that,
whatever the
exact form of
utility
function invoked to
describe the
investor's preferences,
the
system
of
demand
equations
described by (4)
conforms
to
the
Brainard and
Tobin
(1968) adding up restrictions
(which
follow, directly,
from
the
initial
wealth
constraint).
However (again,
whether
with or without
an
asset
with
a known
return) the
remaining three
conditions, (7c)
to
(7e)
are
not
sufficient to
ensure that the
(Jacobian)
matrix
of
responses
to
expected returns, [dA/dr]
=
J,
exhibits:
Zero row
sums,
Jt
=
Ot,
which is
to
say
that
an
equal
absolute
increase
in
expected
returns
on all
assets
will
leave asset
demands
unchanged;
Symmetry,
J =J',
i.e. that
(with regard
to all
i, j pairs
of
assets)
a unit
change
in the
expected
return
on
asset
j
causes
a
change
in
demand
for
asset
i which is equal in magnitude to the change in demand for asset j caused by
a unit
change
in the
expected
return on asset
i;
and
Concavity,
(daj/dri)
,
0,
i.e. the effect on demand for
some asset
of
a
change
in the
own
expected
return
is
non-negative.
For
given (4)
such
a
pattern
of
responses requires, (besides the features of
the
Q
matrix described
by (7c), (7d)
and
(7e)),
at least
(see Courakis 1987a
and
1988),
that
P
is invariant
to
changes
in
expected returns
on
the
various
assets.
With regard to the latter, and more generally to the variety of patterns of
behaviour
that
we
may entertain, Fig.
1
outlines the
characteristics of
the
asset
demands
corresponding
to
four
popular types
of
utility
function.
Clearly,
q4
s
independent
of Wf, and
therefore of
ri,
for
the
two
negative
exponential utility
functions described
in
columns
(i)
and
(iv)
of this
figure.
However,
with both the
quadratic function, presented
in
column
(ii),
and
the
power function, presented
in column
(iii),
4
depends
on
Wtf.
Accordingly
the
asset demands
traced for these two
cases are
not
linear
in
expected
returns.
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556
A. S.
COURAKIS
- I
7
0
0
~~~~
_~~~~~~0
0
1-1~~~~~~~~~a
+ 0 I+
-
cl
0~~~~~~~~~~~~~
.0
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.
I
-
~ -'~' ~
-
41
2 T ~~~~~~~~~~~~~~~
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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CONSTANT
RELATIVE
RISK AVERSION
557
o
AV0
AV
AV
~~~~
A
.a,}?
AV
I
1
I
<
I
+
11 C
E | z
c
< + cc
,- t ccz
Q,
Ct Q
11
~ ~~~11
1 11
1
0
(
_
6,
S
Q -
-
N
~~~~~~~~~~
I
_
Q
u~~~~~~
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558
A.
S.
COURAKIS
Nor
do
they
exhibit
the zero row sum
property
that
the
asset
demands
corresponding
to the
negative
exponential
obey.
Given
the
adding up
restrictions
furthermore,
absence
of
zero row
sums
implies absence
of
symmetry.3 Moreover,
as with
the
asset demands corresponding to the
quadratic
so
too
for those
corresponding
to
the
power
function
we
cannot
generally presume
that the demand for
each
asset
is
positively
related
to
its
own
expected
return.4
11. Wealth
homogeneity
with and
without
linearity
in
expected returns
The
results presented
in
columns
(iii)
and
(iv)
of
Fig. 1, are at
variance
with the belief-expressed, for example, in Roley (1981, p. 1105; and 1982,
p.
651)
and
Friedman
and
Roley
(1987, p.
632),
and
encapsulated
in
all
econometric
specifications
of asset
demands
that
purport
to
model
pre-
ferences
in accordance with
power utility
functions-that
constant
relative
risk aversion
implies
asset demands of
the
very
same form as
those
traced
from a
negative
exponential utility
function
in
the
portfolio
rate
of return
(i.e. from
a
function
strictly analogous
to that
shown in
the last
column
of
Fig.
1).
What
these functions have
in
common is
that the asset
demands
that
they describe are
in both
cases
homogeneous
in
initial
wealth.
However,
for
the asset demands presented in column (iii) to exhibit zero row sums (and
symmetry)
of
response
to
expected
returns it is
necessary
to
assume
not
only
(as
we have
done)
that
Vf
is
quite
small
relative to
Wlf
but
also that
all
expected
returns are
equal.
It seems
unlikely
that
anyone
would wish
to
proceed on the
latter
premise.
For
(lack
of
realism
apart)
to invoke
such
a
premise is
clearly
inconsistent
with
a
paradigm
that
purports
to
explain portfolios
in
terms
of
differences
in
expected
returns
between assets.
Yet
scrutiny
of
the
literature
reveals that it is in fact willingness to invoke an extreme form of such an
equality
of
all
expected
returns-notably
that
Wf
=
W'is a
good approxima-
tion .
.
.'
(Friedman, 1985a, p. 339,
and
1985b
p.
200;
Friedman and
Roley,
1987, p. 631)-that
accounts
for
the
practice
of
deploying
the asset
demands
3That
absence of zero row sums
implies
absence of
symmetry,
in
the sense
that
symmetry
cannot hold for all
i, j pairs,
is, clearly,
true
irrespective
of
the
precise
form of
the
asset
demands and of whether
or
not the
portfolio includes an asset
with known return.
Focussing
on
the two cases
where
zero
row
sums
and
symmetry
do
not
hold,
however,
one
may note
that,
for
portfolios
comprising
more than one
risky asset, with regard
to the pattern of
responses
to
expected
returns
a
distinction can be drawn
in
terms of the
presence
or
absence
of
an asset
with
known return. Specifically, for the asset demands corresponding to the quadratic and power
functions it can be
easily
established
that
the
presence
of
an
asset
with known
return
ensures
some
partial
symmetry,
in
that the subset of
responses
of
demands for the k
-
1
risky
assets
to
changes
in
expected
returns
on
these
assets
is
then
symmetric.
See
Appendix
A.
4
Notice that the
responses
to
changes
in
expected
returns
described
in
Fig.
1
qualify Arrow's
(1971, p. 108)
claim
that
'It is ...
obvious
that an
increase
in
the
rate of return on
the secure
asset
.
.
.
[leads]
to a decrease in demand
for
the
risky
asset
and an increase in
that
for
the
secure asset'.
For
in the
case of
the asset demands
corresponding
to the
power functions the
results reveal that
we
cannot dismiss the
possibility
that
an
increase
in
return on
the
secure
asset
causes the demand
for
that asset to fall.
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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CONSTANT RELATIVE
RISK
AVERSION 559
shown
in
the last column of
Fig.
1
as
descriptions of behaviour in
accordance
with
preferences
that
reflect constant
relative risk
aversion.
In fairness to those that choose to
proceed
in
this
fashion,
one
should
note
that such exercises are often
preambled (or
postscripted) by aphorisms
of the form:
'If the
time unit is
sufficiently
mall
to
render W a
good
approximationWf or the
purposes of the underlyingexpansionthen the
(scalar) term
[(aW)-'Wf,
in the
last
row of
column
(iii)
of
Fig.
1]
is
simply the
reciprocal
of
the constant
coefficientof relative
risk
aversion'
(Friedman,1985a, p. 339, emphasisadded);
and
'Whenthe argument[in the utility function]is ... the portfoliorate of return,
with wealth
homogeneity
.
., symmetry mpliesconstant relative risk aversion
if
the time
unit is
sufficiently
mall to render
W,
a
good approximation
or
Wf',
(Friedman
and
Roley, 1987, p. 633, emphasis
added).5
Yet to
presume
that the time unit is
'small',
or
'sufficiently small', strictly
does not
dispense
with
the issue
of
consistency
raised above.
Conversely,
though,
if
we are
disposed
to overlook this
dilemma,
then
we
should
at least
be
prepared
to
recognise
that the
same
presumption
will
suffice
to
render
linear in expected returns also the asset demands corresponding to the
quadratic utility
function.6
Again,
one
suspects
that
the latter
will not
be
palatable
to
many.
For
though,
at
a
stretch,
we
may
still
regard
the asset demands described in
the
first
two columns of
Fig.
1
as
referring
to
competing hypotheses
of
behaviour vis
a
vis
each
other
and
vis
a
vis
the
pattern
of
behaviour
implied
by (iii)
or
(iv),
no
assessment
of their relative
validity
can rest on
whether
zero
row
sums and
symmetry
can be shown to hold. In
other
words,
if
we
concede
to the
approximation described,
it can no
more be
argued
that
(ibid,
pp. 632-633)
'As is true
in
the
standard consumer demand
paradigm
the coefficient matrix
applicable
to the vector
of
expected
asset
returns consists of a
combination of
symmetricSlutsky
substitutioneffects
and
(in
general) asymmetricSlutsky
wealth
effects' emphasis
added).
5
Others
are
more laconic. For
example,
in
using the asset
demands
shown
in
column (iv)
Green (1977 p. 211) invokes
'. . . the second order approximation to
the Pratt-Arrow
coefficient of relative risk aversion . . .' (emphasis added); while Frankel (1985, pp. 1052-1053)
invokes
a
power utility
function
as
the antecedent
of
asset demands that are linear
in expected
returns, having
cited
an
earlier
version of Friedman and
Roley (1987),
but thereafter
commenting only (p.
1053,
footnote
13)
on the
intertemporal
maximization
implications
of
the
model.
6Clearly,
for
Wf
=
W,
we
have for
the
quadratic
A
=
((1
-
aW)/a)Qi
+
BW,
which is to
say
that on the 'time unit
being
small'
reasoning
the quadratic
too
implies
asset demands
that are
linear
in
expected
returns
and
exhibit
zero
row sums
and
symmetry.
The fact that no
approximation
is needed
for
the
quadratic utility
function to translate
into
mean-variance is, of
course, quite
irrelevant in this
context.
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560
A.
S. COURAKIS
Or that
Only '...
in some
specific
ases he relevant wealth
terms do exhibit
symmetry.
The
... asset demands
derived
.. . under constant
relative
risk aversion and
joint
normalasset return distributionsprovidea clear example. ... By contrast, the
symmetry property
does
not follow
from (for example) the quadratic utility
function
. .'
(emphasis
added).
Or
that
Since 'the
symmetry
property
.. does not
necessarily
hold for
any
reasonablebut
arbitrarily
hosen
form
of
expected utility maximizing
behaviour,
.
..
evidence
indicating
whether investors' behaviour
does or
does not exhibit symmetry
providespotentially
useful
information
about
investors'
preferences]'.
Conversely, if we insist on distinguishing, as we generally do, between the
quadratic
and the
negative exponential
in
terms of
Slutsky
conditions-
noting
that is that zero
row sums and
symmetry
warrant
that the
Royama
and
Hamada
(1967)
'expected
wealth effects' of
changes
in
expected
returns
are
zero,
i.e. such
as the
quadratic
unlike
the
negative
exponential
will not
in general
exhibit-we
are
hardly
entitled to
abstract
from the
analogous
distinctions
that
comparisons
of
the asset demands shown
in column
(iii)
to
those shown
in column
(iv)
of
Fig.
1
immediately
brings
to
mind.
Notice
that the condition
for
symmetry (and
zero
row
sums)
is that
expected
wealth effects
be
zero.
For
given
the second
of the above
quotations
it must be stressed
that
strictly
non-zero
expected
wealth effects
can never
be
all
symmetric.
To
drive the
point home,
let
J=S+H (8)
where
S
=
[Sin]
denotes
the matrix of
Slutsky (equivalent)
substitution
effects
(i.e.
effects
of
changes
in
r when the
investor is
compensated
for
each such
change so as to anticipate the same expected wealth with the same degree of
risk),
and
H
=
[hin]
denotes
the matrix
of
expected
wealth
effects (i.e.
of
effects
due to the
change
in
the
marginal utility
of
wealth
that,
for
any given
initial
portfolio,
changes
in
expected returns,
and hence in
Wf,
imply).
With
a view to
distinguishing
between
these two
effects
of
changes
in
expected
returns, imagine
first a case
where a
lump
sum
tax, Tf
>
0,
payable
at t
+f,
is
imposed
on the investor.
The net of tax
value
of the
portfolio
at
t
+f
is then
Wf =(t
+ r+
0)'A-Tf
(9)
whence
Wf
=
(l
+
r^)'A
Tf
=
Wgf
-a
Tf(10)
where
the
subscript g
denotes the
gross
of tax
expected
value of
the
portfolio.
Fig. 2,
first
row, presents
the
corresponding
asset demands derived as
in
7A
point
stressed
by Roley
(1983 p.
126),
and also
by
Courakis
(1974
pp. 180-181).
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
10/15
CONSTANT
RELATIVE RISK
AVERSION
561
+~~~~3
14~~~~~~~+
E
~
II
E
-i
~
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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562
A. S. COURAKIS
Section
I
above,
while the second row shows the effect
of
changes
in
Tf
on
demand
for the various
assets. The
lump
sum tax effects
the
asset demands
drawn
from the quadratic
and
power functions,
but
is
of no
consequence
when
preferences
conform
to
either
of
the two
negative exponential
functions.
Moreover,
from the asset demands
shown in
columns
(ii)
and
(iii)
it is
clear
that
[dA/dWgI
hcfnst
=
-[dA
/d
Tf]
(11)
Granted
this consider
again
the
expressions
for
J,
i.e. the matrices
of
responses
to
changes
in
expected
returns shown in
columns (ii) and (iii)
of
Fig.
1.
Suppose
that
simultaneously
with
a
change
in
r
a
lump
sum
tax
dTf
is
imposed, such
as to
imply
that
were the investor to continue to hold the
same portfolio as before these changes occurred, he will enjoy the same
combination
of risk and return as before.
This
implies
dTf
=
A'[drF],
whence
-[(dA/dTf ][dTf
dr]
=
-[(dA/dTf
]A'
=
H (12)
Given
the
expressions
for
[dA/dTf]
shown
in
columns
(ii)
and
(iii)
of
Fig.
2,
H
is, therefore,
identical
to
the
last
components
of
the
expressions
for J
shown
in columns
(ii)
and
(iii)
of
Fig.
1.
Correspondingly,
the remainder
of
the.expressions
for J shown
in
columns
(ii)
and
(iii)
of
Fig. 1,
denote
the
S
matrices
of substitution
effects.
From the
properties
of
Q,
see
(7)
above, it follows that the substitution
matrices,
S,
shown in
Fig. 2,
are
symmetric
with zero row sums.
Moreover,
though
the substitution
effects
differ
in
magnitude
across
the
four classes
of
function, remembering
that
U'(Wf)
>
0
implies
for the quadratic (1-
aWf)
>
0,
it is also clear
that across the
four
classes of function
they
exhibit
the same
signs.
As for
H,
the results
reveal
that
in
terms
of
expected wealth
effects
the real difference
between
the
quadratic
and
power
functions does
not lie
in
these effects
being asymmetric
for
the
former
and
symmetric
for
the latter;8 for they are clearly asymmetric in both cases.' Rather, the
difference
is
that the
expected
wealth effects
of
a
change
in
any expected
return
will
carry
opposite signs
if
preferences
conform
to
a
power function
to those
which
these
effects
will
carry
if
preferences
conform
to the
quadratic.
On the
other
hand,
from
the same
viewpoint
the difference
between
the
power
functions and
the
allegedly
'isomorphic negative
8
Nor, evidently,
are
they
zero in
the case of the asset
demands corresponding
to the power
functions,
contrary to Flemming's
(1974, pp. 145-146) claim drawn, albeit, in the context of
continuous time.
'As shown in Appendix A, in the presence of an asset with known return the responses of
demands
for the k -
1
risky
assets to
changes
in
expected
returns on these assets do indeed
imply that
the
corresponding expected
wealth effects are
symmetric.
However, this does not
imply that the full (k
x
k)
matrix
of
expected wealth effects is symmetric. Moreover, the partial
symmetry
of
expected
wealth effects that in the presence
of an
asset
with known
return is a
feature
of the asset demands
corresponding
to the
power
function is
also
a feature of the asset
demands
corresponding
to
the
quadratic
function.
(In passing,
I
may also add
that
at the
empirical
level the studies cited in footnote 1
above
proceed
on
the premise that decisions
relate to
the real value
of the
portfolio
and
due
to
stochastic inflation
all assets are risky; see
also Courakis (1987a)).
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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CONSTANT
RELATIVE
RISK
AVERSION
563
exponential
with the coefficient of
absolute risk
aversion
inversely
depend-
ent
on initial wealth'
(Friedman
and
Roley,
1987, p. 632),
is that the
latter,
unlike
the
former, implies
zero
expected
wealth
effects.
III.
Homogeneity
in
initial
wealth
with and without
homogeneity
in
final
wealth
With regard
to the effects
of
changes
in
the determinants of
asset
demands the discussion
so far
has focussed on
changes
in
expected
returns
and in initial wealth.
But other
thought
experiments
confirm
that it
would
be
unwise
to assume
that
the behaviour
of
investors whose
preferences
conform
to
power
functions
will
replicate
that of
investors whose
pre-
ferences
conform
to a
negative exponential
in
the rate of
return
on
the
portfolio.
Consider
a
multiplicative
shift in the vector
of
perceived
returns on
the
various assets. Such
a shift can
be due
to a
change
in
the tax
rate
on
final
wealth,
or
profit
(Wf
-
W)
with
complete
loss
offset
provisions,
as
ex-
amined,
for
example, by
Tobin
(1958, p. 41)
and
Atkinson and
Stiglitz
(1980, Ch.
4). Alternatively,
following
Courakis
(1987
a &
b;
and
1988
pp.
626-8),
when
U(Wf)
relates
to
the
real
value of
wealth,
such
a
shift can
be
due
to
a
change
in
the
anticipated
rate
of
inflation in
circumstances where
the investor's perceptions of nominal returns on the various assets are not
conditional
upon
the
anticipated
rate of inflation.
Let
p
denote
the shift
parameter-so
that if
r
is the tax
rate
on
final
wealth
then
i
=
(1
-
v),
while
if 6 is
the
perfectly
anticipated
rate of
inflation then
i
= (1
+
s)-f.
In
place
of
equation
(2)
we now
have
Wf
=M(t
+
r
+
0)'A
(13)
whence
W
=
(t
+
r)'A
(14)
V-=
2A'QA
(
and
hence,
by
the same
manipulations
as in
Section
I,
the
solution
A
=
(p41Q(t
+
r)
+ BW
(15)
=
-U (Wf)/U'(W)
I
where
Q
and
B have
exactly
as
in
(5)
or
(6)
above,
depending
on
the
absence
or
presence
of an asset with known
return.
The explicit forms of the asset demands corresponding to the four utility
functions
are shown
in
Fig.
3. Notice that the
assets demands
drawn from
the
power
function are
independent
of
M.
On the other
hand,
as for
the
negative
exponential
in
wealth
and
the
quadratic,
the asset
demands
drawn
from the
negative exponential
in
the
rate
of return
on
the
portfolio
depend
on
y.
In
other
words, changes
in
the
proportionate
tax
rate
on
wealth, r,
or
in the
perfectly
anticipated
rate
of inflation
6
(in
the
circumstances
described
above),
have no
effect
on
asset
demands
when
preferences
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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564
A. S.
COURAKIS
FIG. 3. Taxation, inflation, and asset
demands
U(Wf)
Asset demands
Responses to changes
in ,u
(i) A
=
(ay)-iQP'+ BW
Z =-(ay-1)-1QP
(ii)
A
=
[I
+
QWF'-f1[(ay)
fQP + BW]
Z
=
+
Qpp'-fQp
(iii)
A
=
c[al
-QPP']-'[a-'Qr
+
B]W
z =
Ot
(iv)
A
=
(q)-1WQP
+
BW
Z
=-(ay-1)-1WQP
where z
=
[dA/dy]
conform
to
power
functions;
but this is not what one would surmise were
he
to think that power functions imply the asset demands that correspond to
the negative exponential
in the rate
of
return on
the
portfolio.
Concluding
remarks
All
in all
then,
short of
'throwing away
the
baby
with
the
bathwater', we
must
acknowledge
that10
power
functions
imply
asset demands
that are
not
linear
in
expected
returns
and exhibit neither zero
row sums nor symmetry.
For
those
prone
to think a
priori
that
preferences
conform to
power
functions,
however,
there
is some comfort
to be
found
in
this verdict.
For
it
follows directly
from the
analysis presented
here
that
rejection
of
zero
row
sums
and
symmetry by
econometric
studies11
that
rely
on
specifications
of
asset
demands
of
the
form shown
in
column
(iv)
of
Fig. 1, (though
such
rejection
can also be due
to factors
quite
unrelated to
the underlying choice
of
utility
function), 2
is
not inconsistent with
choices made
in
accordance
with
preferences
that conform
to
power
functions.
Indeed,
other
things
being equal,
if
preferences
do
in
fact conform to
power
functions this is
precisely what one must expect.
Brasenose
College, Oxford
APPENDIX:
CHOOSING
AMONG
RISKY ASSETS IN THE
PRESENCE
OF A SAFE
ASSET: ON
'PARTIAL SYMMETRY'
As
in
(6)
above,
let
Q denote the variance-covariance matrix
of
returns on the k
-
1
risky
assets.
Correspondingly
let A'
=
[A':
aJ
and
P'
=
[rJ
denote the
partitioned
vectors of
quantities
and
expected
returns on the various
assets,
where
Ax
and
FP
are the
quantities
of
and
expected
returns
on the
k
-
1
risky
assets,
while
a,
and
r,
denote
the
quantity
of and return on
the
safe asset.
'
Certainly
with joint normally
distributed
asset return assessments
and/or
when the
risk on
the portfolio
is small relative
to
wealth.
Symmetry
is
rejected
in all three
studies of the
US that report tests of this property
in the
context
of asset
demands of
the form shown in column
(iv)
of
Fig.
1
(viz.
Roley 1983;
Friedman,
1985b;
and Friedman and Roley 1987).
12
See,
for
instance,
Courakis
(1980, 1987a,
and
1988).
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8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns
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CONSTANT
RELATIVE
RISK AVERSION
565
FIG. A.I. On 'partial
symmetry'
U(Wf)
(ii)
(iii)
Ax [+QPP]((x+
- (1 +
r)W)Q'lp [cxl
-
Qpp']
'(
+ r,)WQlp
Jxx
[1 +
Q'pp'['Q'[(a
-
1f)l
-
pAj1
=J . [cl-
Q'pp'] Q'[Wfl
+
pAxj
=
J
Hxx
-
[Il
+
Qpp']
'Q
'pAX
=
Hx
[I1l-
Q'pp' ]'Q'pA
=
HX
[I
+
Q'pp' ]Q'(c
-1i)l
=
SX
[cl
-
Q'pp']'Q
WI
=
S5
JXc -11
+
Q'pp]
'Q [(aK
-
Wf)t
+
alpJ
/J
- [cl-
pp']
'Q
[Wf
-
acpJ ]
a,:
W
-
t'Ax
W-
t'A
J~~~~~x
t~~~~LJx+c
4LJxx
+Axc
J'c
t'Jxc
O
t
'Jxc
2 0
where p
=x
- rt)
From
the
definitions of
Q
and
B shown in
(6)
it follows
that,
as
shown in Fig.
A. 1, in the
presence
of
an
asset
with
known return, both
for the
quadratic and for the
power
functions
(a)
the sub-matrix
of
responses
of
demands for
the k
-
1
risky assets to
changes in expected
returns on these
assets,
denoted
by
Jxx,
is
symmetric;
correspondingly.
(b)
symmetry
holds
not only
for
the
analogous sub-matrix of
substitution effects between the
k
-
1
risky
assets,
denoted
by
Sxx,
but also
for the
sub-matrix of
expected wealth
effects
between the
k
-
1
risky assets, denoted
by
Hxx
On the other
hand,
as indicated also
by
the
results
pertaining
to
the
general case
described in
Fig. 1, (a)
and
(b)
above should
not be
misconstrued
to
imply
that
the
presence of an asset with
known
return ensures
symmetry
and
zero row
sums
of
the
full matrix,
J,
of
responses
of
asset
demands to
changes
in
expected
returns
on
all these
assets. In
other
words, both
for
the
quadratic and
the
power
functions,
(c) responses
of
demands for
the k
-
1
risky
assets to
changes
in
the known
return
of
the
safe
asset,
denoted
by
the column
vector
Jx,
are not
symmetric
to
the responses of
demand for the safe asset
to
changes
in the
expected
returns
on
the
k
-
1
risky assets,
denoted
by
the row vector
J'
;
the
corollary
of
course
being that the zero row
sums
condition
does not hold
either.
Comments
by
Ben Friedman
and
Vance
Roley,
on
an
earlier
draft,
are
gratefully
acknowledged.
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K.
J.
(1971), Essays
in
the
Theory of
Risk-Bearing,
(Chicago:
Markham).
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A.
B.
and
STIGLITZ,
J.E.
(1980), Lectures
on Public
Economics,
(Maidenhead:
McGraw
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W.
C. and
TOBIN,
J.
(1968),
Pitfalls
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A. S.
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http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp
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S.
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Price
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