Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns

8/18/2019 Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns

1/15

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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2/15

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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3/15

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.




4/15

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.




5/15

556

A. S.

COURAKIS

- I

7

0

0

~~~~

_~~~~~~0

0

1-1~~~~~~~~~a

+ 0 I+

-

cl

0~~~~~~~~~~~~~

.0

I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

I

-

~ -'~' ~

-

41

2 T ~~~~~~~~~~~~~~~




6/15

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~~~~~~




7/15

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.




8/15

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.




9/15

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).




10/15

CONSTANT

RELATIVE RISK

AVERSION

561

+~~~~3

14~~~~~~~+

E

~

II

E

-i

~




11/15

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


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)).




12/15

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


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


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




13/15

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).




14/15

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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J.

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Documents

Does Constant Relative Risk Aversion Imply Asset Demands That Are Linear in Expected Returns