Issuu on Google+

THE CAPITAL ASSET PRICING MODEL WITH NON- HOMOGENEOUS EXPECTATIONS: THEORY AND EVIDENCE ON SYSTEMATIC RISKS TO THE BETA Cloduoldo

R. Francisco

April

1987

Harvard University. and Philippine Znstitute for DevelopmentStudies


ACKNOWLEDGEMENT

The

research

Environmental Government, Philippine Nations

and

office

Policy

facilities

Center,

Harvard University, Institute

Development

for

Program

John

Y.

at

the

Energy

Kennedy

and

School

of

and financial support from

the

Development Studies are gratefully

and

acknowledged.

the

United


ABSTRACT,

This

paper

investors assets

introduces

non-homogeneity

on the parameters of the probability

rates

of return and derives an

relationship which is non-linear.

systematic

investors

among.

distribution

equilibrium

of

return-risk

This relationshipshows

and additional form of riskcalled the

of expectations(NHE)

a new

theta risks I and II which are

biases to the beta risk arising from

NHE

among

on the mean and variance (covariance) respectively

of

the rates of return.

The beta is no longer a complete measure of

risk.

traditional

Under

assumption,

the

the

expectations(HE)

or if the theta risks vanish, the CAPM of Sharpe and

Lintner is a special case. to

homogeneous

An errors-in-variables model _sused

provide an indirect test and the results are explained within framework

anomalles

of

the model.

It appears

that

the

empirical

on the CAPM are due to attempts to fit a linear

on a fundamentally non-linear return-risk relationship.

model


TABLE

OF CONTENTS

A.

Acknowledgment

B.

ABSTRACT•

C.

List of Tables and Figures

I.

Introduction and Su.-._ry

II.

.

The CAPM with NHE: Explicit Non-Linear Return-Risk Relationships

I

' 6

_II.

Indirect Test of the CAPM.wi_h NHE

20

IV.

Explaining the Test Results

34

Concluding Remarks

45

Appendices

I

47

II

50

V. VI.

_II.

References

54


" LIST, OF TABLES

Table

I

:

Table 2

:

Table 3

:

Calculation Results for the 72 Securities for the Period 1976-1979

24

Bank-Commercial-lndustrial

28

Riskless Rate. andPortfolio. Rates of •Return. and Variances for the Market and. for Sectoral

Table 4

:

Table 5

:

Table 6

:

Figure 2

Figure 3

29

30

Results of Indirec_ Tests CAPM with NHE (OLS)

32

of the

Model Classification for the Indirect and Significant Test Results for the CAPM with Non Expectations

LIST

I

Groups of, Securities

Results of Tests of the CAPM (OLS) 72 Securities .... ,

Homogeneous

Figure

Securities

:

:

:

42

OF FIGURES

Plots of Significant Test Results for CAPM with Non-Homogeneity of Expectations Using Sectoral Groups of Securities

35

Monthly Traded

Total

40

Monthly

Ranges

Values

of Shares

of Price

Indices

of

40


The Capital Asset Pricing Model With Non- Homogeneous Expectations: Theory and Evidence on Systematic Risks to the Beta by Clodualdo

Z.

INTRODUCTION

R. Francisco*

AND SUMMARY

The Capital Asset Pricing Model (CAPM) of Sharpe (1964) Lintner

(1969)

determination

is

that

works

equilibrium

model

of

asset

price

based on the mean-variance portfolio selection

Markowitz (1952). recent

an

and

of

This model, which is the basis for most of the

in capital market theory and

under certain assumptions,

finance,

there is a linear

postulates relationship

i/ between the return of an asset and its non-diversifiable risk.

Tests

of

the

CAPM

indicate that

the postulated

linear

return-risk relationship is not consistent with the data and that

* Visiting Fellow, Harvard University and Research Fellow, Philippine Institute for Development Studies. This paper draws from the doctoral dissertation of the author submitted to the School of Economics, University of the Philippines.

For some works which provide a landscape of the literature on the theoretical and empirical development on the CAPM, see Fama (1968), Sharpe (1970), Jensen (1972), Friend (1977), Roll (1977), and Brealey and Myers (1981).


2

the

evidence

consistent

does not support with

the

a riskless

actual

market

measures

of

rate

of

return

riskless

rates

2/ of return.--

Aside

from the

deficiencies grounds,

of

the

theoretical

its assumptions,

One

of

homogeneous

the

empirical

attempts

to explain

CAPM on measurement extensions

and

expectations

assumptions (HE) among

assets.

This assumption

is observed attempted

of

the evidence.

investors

of the probabilitydistributi0nsof

of writers have

statistical

of the CAPM

variance

a number

ether

apparent

of the model, by relaxingsome 3_/

were also used to explain

crucial

the

is

onthe

mean

of and

of return

to

to be quite restrictive

and

to relax

rates

that

this assumption.

2--/ Friend and Blume (1970), Black_ Jensen and Scholes (1972_, Blume and Friend (1973), Fama and MacBeth (1973), and Petit and Westerfield (I974). For more recent tests taking into consideration Roll's (1977) critique, see Friend, Westerfield and Granito (1978), Cheng and Grauer (!980), and Best and Grauer (1985). _3/ Among the earliest writers who extended the CAPM by relaxing some of its assumptions were Black (1972) and Mayers (1972). Black dropped the assumption of the existence of unlimited borrowing and lending opportunities at a market risk-free rate and arrived at the well-known zero-beta portfolio. Mayer relaxed the assumption that all assets are marketable by introducing nonmarketable assets. Both model appear to provide explanation to the results documented by Black, Jensen and Scholes (1972). For a complete statement of all the assumptions instance, Black (1972).

of the CAPM,

see for


S Lintner

(1969) and Sharpe

expectations existence of

and

of diverse

equilibrium

equilibrium similar

arrived

pricesand

and

0wen (1978) derived

NHE

and

apply

CAPM

particular

concludes

assumes

that "...given

prices

can

average

and marginal

did

be considered

not derive

with

NEE.

costs

type

and NHE.

of

as determined

an explicit

Like Mayshar,

(p.

inside

of asset

return

a and

preferences. equilibrium

simultaneously Rebinovitch

provide

the

He assumes

relationship

they didnot

of

opinion,

127).

return-risk

under

(1983) extends

Of investors'

divergence

investors"

(NHE)

Rabincvitch

structures

distribution

a specific

derived

expectations

Mayshar

transaction

the

prices and portfolios

purposes.

of

structure

(1976)

(1970).

to different

type of probability

like Lintner, He

trading

by incorporating

Gonedes

(1972) and Sharpe

that

the linear

non-homogeneous

equilibrium

heterogeneity

conclusions

portfolios.

their model

for

similar

does notchange

under

to that of Black

information

at

opinion

conditions

(1970) considered

by

the

and Owen

for the CAPM

tests for

their

• models.

The

purpose

introducing explicit

Black,

this

NHE following equilibrium

given asset; attempt

of

provide

an alternative

is

Rabinovitch

non-linear

to;

explanation

extend

and Owen

return-risk

an indirect

Jensen and Scholes

on the CAPM.

paper

the

(1978);

relationship

CAPM

by

derive

an

for

a

test of the CAPM with NHE; to the evidence

(1972) and other

related

documented test

and by

results


4 The rest of the paper

is s,_mmarized as follows:

Section

the equilibrium

portfolios case.

under

An

asset

II

derives

NHE and shows

explicit

is

derived

additional

types of risks.

they

termed

are

observed from

theta

risks

are

vectors

that of the simple

Section NHE

consideration risks simple

are

of portfolio

an

estimates

given

a two-stage

security

arising

Approximations

for

relationships

CAPM.

corresponding

The

indirect

and related

_to

second among

test for the CAPM

model.

attempts

in the

using

market

stage postulates •

investors,

primary

tests

to overcome

tests in the literature ,

for the betas

The

is the fact that the

model,

and

of securities

in

the some

a

equal

stationary rates of

that _he theta risks investing i

theta

such as the use

the market

portfolio,

on the •probability distribution

security

are

covariance

The first stage

where the test procedure

The

name,

risks

and

are also derived

unobservable.

of indirect

return.

and

return-risk

errors-in-variables

of previous

assumption

new

CAPM.

basically

per

two

These

of means

to assets.

to the simple

limitations

weight

for each

risks" to the beta

for this test procedure

CAPM

of

and

special

relationship

I and II".

alternative

IIl provides

using

prices

Due to •lack of an appropriate

on the vector

yield

of

CAPM is a

the existence

of returns

comparable

equilibrium

with

shows

to be a form of "systematic

NHE among investors

which

return-risk

as "theta risks

matrix,., respectively, the

that •the simple

non-linear which

vector

for a

particular


5 subgroup of risks

securities,

among all investors

same

security

groups

of

expected

relationship return-risk

the simple

risks introduce

the

investing

becomes

the

sectoral

results.

non-linearity

theta on

CAPM using

and if the theta risk

relationship

IV

presents

of significant

explained

within

and explains

test results the framework

alternative

explanation

and Scholes

(1972) and other

V provides

better

than

This into

vanish,

is the

then the

linear and is given by

CAPM.

Section pattern

in the entire market

should yield

since the theta

resulting

smaller

such that tests of the simple

securities

return-risk

are

are relatively

some concluding

remarks.

test

is observed of the CAPM

of the empirical related

the

finding

works

results.

and these with

is provided.

results

NHE.

of Black,

A

An

Jensen Section


It

THE CAPM

WITH

NHE

:

EXPLICIT

NON-LINEAR

RE_URN-RISK

•RELATIONSHIPS A. ,-NHE Assumption,

Individual-Portfolio

and the Market

Clearing

Let V be an nxl vector, •random

one-period

to

be positive

j = k and_jk=

assumption V

exist

cov(Vj, Vk)

, j@k,

and

in the market,

the same

and

NHE,

1,,..,n.

V and

k = 1,...,m,

the

_

is

_jk

the

be an assumed

= var(Vj),

Given

that there

expectations

(HE)

that all investors• perceive

However,

have non-homogeneous

Given

k =

if

an individual

_ differently,

_i such that they are•not

_krespectively,

investors

returns which

the homogeneous

manner.

represents

each element

implies

to perceive

of which

j, 3 = 1,...,n;

total

and where

assumed Vi

matrix•of

of the simpleCAPM

and _ in

instead

each element

definite

are m investors

Problem

Condition

total return Of asset

nxn variance-covariance

Selection

i

i.e.,

necessarily

there••

equal

k # I, then this means

is

to V k

that the

expectations.

portfolio

selection

problem

of investor

i

is to maximize

Ui( E i, @i )

subject_,to: W.l = P'xi

where •"

(I)

_ di

(2)

E. 1 = V!X. i i - 8d"i

(3)

__ , _i

and

Wi

beginning

(_)

Xi_IXi

is the total marketable of the period,

which represents

assets

X i is an

the fraction

of

nxl

of investor vector,

the total market

i

at

each element value

the of

of asset


7

j

held by investor

beginning of

i,

d i is the net debt of investor

of the period and

which

represents

of the period,

risk-free

rate of return.

such

is an

nxl

the total market

beginning

define• _ and Q

P

=

and 8

is

rj an

is

nTn

the

one-period

random

variance-covariance

such

_jk

j = k and

number

= var(rj),

S = ($I,82,.'._)'

of shares

be the

nxl

investor held

i

by•investor

shares

i.

of all assets

being

j

at

the

the one-period

transposition.

Also

the

(5)

return

of

each element

m

Q

which

Ojk. is

rk) , j _ k..

i.e.,

Sj is the total

let si = (s_i s2i,...,Sni)' of risky assets of

total number of shares is cleared

are held by t_._ m

j,

vector• of the _umber of

the portfolio

The market

asset

of return

= cov(rj,

nxl

Further,

representing

in terms of

Ojk

in the market,

of asset j.

vector

of asset

of rates

and where

be the

shares of•assets

of

matrix

definite

outstanding

each element

= I +"r.. 3

rate

to be positive

Let

RF

at the

that

is assumed that

,

denotes

pj = (V./Pj)3 = ((Pi + Pjrj)/Pj)

where

value

I + RF

A prime

vector,

i

if

all

investors

of asset

j

outstanding

so that

m

El s ji = Sj , j = 1 '"" ,n or i=l E s. i_ = S.

By definition, Xji .is the fraction held by investor i, that is,

of the total

Xj i = sjiPj/Sjp j '

(6)

market

Value

of asset

(7)

j


8

where Pj is the prevailing

price per share

of asset

j, so that

z xjl = Z -z sji -= I. i i sJiPj/SjpJ pjsji.,._ pjSj. _. Therefore, written

•the

market-clearing

condition

given by (6)

can

be

as E Xi = G i

where G is an

B.

(8)

The

nxl

(9)

vector with all elements

Equilibrium

Vector

of Prices

and

equal

to I.

Portfolio

Under

NHE

problem,

the

and HE

Solving portfolio

the

vector

individual

Portfolio

of investor

i

selection

is

xi o hlnT1(vl - eP)., i : i,....,,_,

(1o)

where _U.

I

_U.

i

h._: - .f.1 / 2_771 , a

measure

introducing

of

risk

(ll.)

.aversion,

Slightly

the market-clearingcondltion

all investors

rearranging

(I0),

(9) and summing

up over

result in

O'E h,[l-iP = Z hl_-lv i - )_X i i i i i or

e P _ ( _ i h,a _ -I z, )-i (Zi This

gives

vector

the equilibrium

of portfolio

hi_.Ivi

vector

is arrived

-

C ).

of prices.

(12)

The

at by substituting

equilibrium

(12) in. (I0),

so that

Xi -_ hiOi

1 _ Vi

_ (._. hiRi i

I

)-1

( Z h.0-. 1 V, i J- i z

C)).

(13)


9 _:

The traditional (13). _

More

.

or simple

specifically,

CAPM is a special

.

if there is HE, i.e., V i = V

f_r.,all i, then the equilibrium

.

case of (12) and

price vector

and

_i =

is

l

ST= V-EnG

(14)

where h-=F. h i i For each asset j,

Vj

+iE m . . _ k jk

=e_

=SPj

where VM =j?.V.3" PM-- Z Pj

oov ( _, vM ),

+I

But coy ( Vj, V M ) = PjPM-C°V

, so that dividing

j

(15) by

(15)

( Pj , OM),

P. results

]

where

in

1 PH "coy • ( Pj, OH )

p. = e +_ J Taking

expectation

and subtracting

(16)

I from both sides give

PM-

(17)

E(rj> --_.+ v _ov_( _,oM). Appendix •I- shows that

2 ..... PM o

rM

•••

where ] ••

E( rM••

the variance that

where,

(18)

h_ R.(rM)-

'

(17)

.

) is the expected

market

of

rM

.

Also,

coy (%,

2

rate of return and _rM pM ) -- coy(

is

rj , rM.) so

becomes.

.E(_rj.:) =_.+ coy

Bj =

( E(r M) -_)

( r__

Sj

(19)

, rM )

oa rM

"

(20).


10 Equation

(19) is the simple

CAPM due to Sharpe. (1964) and Lintner

(1965).

Again, of investor

if there is HE, the equilibrium i given by(13)

= hi_-I

Xi or

An

vector

of portfolio

becomes

( V-

( V -_ 1 _ C ))

hi

(21)

x ov-G. investor

proportion relative

i

holds all assets

and it is a function to

the

sum

in his portfolio

of his degree

of the degree

of

in

the

of aversion

risk

same

to risk

aversion

of

all

4/ investors.

The equilibrium results. define •i

To

simplify,

.Vi and

is

an

component

_isuch

nxl by

respectively. is an identity matrix

solutions

following

that Vi = V

vector

matrix -i

the

+ _

and

Bi =

for all

and 0wen

and _i nxn

bias of

conveniently,

( I - A i) exists

Rebinovitch

and Bi is an

component More

given by (12) and (13) are general !1978),

= _ + Bi ' where

matrix

Vi and _i

representin_ from V

and

let _i = ( I - Ai ) _where -Ai_ i

.

Further,

assume

I

that the

and can be approximated

such

4_/ This is an important limitation of the CAPM which has the object of criticism and investigations. See for example (I978) and Green (I986).

been Levy


11

that

(I-

Aiej

Ai )-l-_l + _

0 ,

differences

i,

j = 1,...,m.

_or

differences

approximate

ep

where

AIAj=O

for

These assumptions

vi ann

in

_i in _12), uslng

perceptions

price vector under

1 =V-_nC+r

i

8i = hi/h.

A i over all

If

i

6

ei

and.

that

the

Similarly,

simplifying

yield

•

of an

+ I _ A._CI )

(22)

and A i are zero for all

6i as weights,

i,

that is,

sums of

ei

the eolut_Qn

and

reverts

CAPM.

tNe equilibrium

can be derived

t_e assumption

NHE,_i.e.,

(e.

i

are zero with

back to the simple

"

imply

5/

thece is HE, or if there is NHE but the weighted

i

J ,

I

all

_in perce_ptions are "small".

Substituting small

and

vector of portfolia

for investor

as

X.x = 6. i ( C •+ hfi-l_i + fi I A i _ C )

(23)

.where

ei = e. - _6ie i and Ai = Ai - _ SiAi" and

A_

reverts

=

0

for alli,

If there

the equilibrium

lvec-tor ..... Of

back to (19) which is that of the simple

there is NHE,

as

long as e

and A

is HE, i.e:,

are equal

CAPM.

to zero,

ei =

portfolio Even

if

(19) still

holds.

For added details Owen (Iq78). p. 579.

on this assumption,

see Rabinovitch

and


12 C.

Non-Linear

Return-Risk

Rabinovitch

and

Relationshi_

0wen

(1978) did not provide

return-risk _elationship. is

given here.

It

new and additional

Appendix

1

explicit•

relationship

will be seen that this relationship

components•of

(22) can be re-written

0 P = Vh.. -_

Risks

an

An explicitreturn-risk

risks,

the beta risk, which are referred

Equation

and the Theta

_ G +_

i

the "systematic

has

two

risks"

to

to as "theta risks I and II".

so 'that

I " hi fl z ei +h2 Ez. A.z G

E i

(24)

i shows that 2

2

'"v M

PN YoN.

( E( vM) - e _M) Substituting

2

the value of

h

(25)

PM ( E( DM) - e ) in (24) gives

2 0 P = V -

pM. (' E(pbl) -e 2 2

) (fig -Z

hie i ) +

P2M( E(OM)-0 " 4 4

i

PM °pM

) ( E h.A._CI i

PM o PM

i

For each asset j, •

iZ coy( vj, V.z) -

V. = OP.+ 3 3

( E(OM) - O ) ( -

2 PM °p M

_i h._e..3_ 2 ) PM o PM

Z h.A.._G i 3l _ ( • m(pM) •

_ e )2

( i 2

4

(27) ).

PM °p M

...

The term Z i cov( Vj, V i ) = coy( Vj, VM ) = PjPM coy( pj, pM ). _call that = l,...,n.

V i --V + ei so that V.31 = Vj + eji , i = l,...,m; For asset

j,

.eji = Vji - Vj

(28)

kz_)


13

Dividing

both

sides

of

(28)

by

PJ

Pj yields

P3

eJ

" Pjl - Pj = ( I + E(r_il) - ( I.+ E(r_) = E(

rji

).-

E( rjl ) is the expected

forecast.of

J by.*nvestor

_Ji - E(rji)

i.

Define

E( rj

)

).

(29)

the rate of. return and pj - E(.rj

of asset ).

Then

eJi " PJ ( PJi - _ ) and therefore

the term

_. t hie.1i

(30)

in (27) becomes

ir. hi pj ( _J.t " _J ) ° PJ _ h_ ( PJi " _J )" Substituting

values

in

• vj =oPj + (z(p M)-e.)

(27)

gives

(

PjPM

cov(pj,

PM )

PM 2 .0 M

••

P" _ hi

,--- J i

:,

( 2

_J±-

_J ) )

PM _0 M

- ( E(PM). - 8 )2 ( _ hiAji_

G

) .

(31)

PM _p.

Dividing

through

by Pj and knowing

that pj = I + _j , 8 = 1 + _ 2

coy( PJ' PM ) _ coy( rj, rM ) and oOM

2

OrM,

(31) becomes

,


Z hi( _ji - Uj ) r_ , rM ) i 2 2

coy( E( rj ) = RF + (E(rM)

- b

)(

ar M

)

PM °rM

E hl A.•.•n C - ( E(r M) - RF )2( i

3i

p. 24

),

(32)

3 PM arM

Equation

(32) provides

an ex_llclt

ship for the CAPM with

Now, define

non-llnear

re'turn-rlsk •relation-

NHE.

81j

and 811j so that

Z hi( _ji - #j ) = i

eli -

(33)

2

PM arM

O

Z hiAji _ i

and 8IIj

Let RF = s 0

and

(E(rM)

Substituting

values in

=

_G (34)

2 4

Pj PM o rM

- RF ) = _i , the market" risk premium. (32) gives 2

E( rj ) = so + _i ( SS - eli ) - _i eiij providing

a simplification

for (32).

(35)


is One

may

systematic forecast bias"

elj as

risk Bj that arises

the "systematic

that

arises _k

'

, and 81ij

from non-homogeneity J'

eli and eiij

k ffiI,...,n. are

bias,

from the non-homogeneity

of the rate of return Wj

covariances name,

interpret

referred

of

the

in

the

as the "systematic

in the forecast

For lack of an to as "theta

of

the

appropriate

risks I and

II"

respectively.

D.

Approximations Alternative

Appendix2 II.

for the Theta

Return-Risk

provides

Risks

and the Corresponding

,

Relationships

approximations

for the theta

risks I and

More specifically,

(36)

= --I-

eli --e_j

a1

aria

i. o-

where

m

(37)

z i (u31-

_j )

In

and

.

m

n

Z ( Z Ajk i / n )

.._. ffi i 3.-

(38)

k m


16 m

In brief, for in

ej

•the average the

A_i.... represent•some

crude approximations

differences •in perceptions •by the , • •, • •

market

parameters

and

from the first

of the •probability

two moments

distribution

m

investors

respectively

• 0f

of the rate Of

the

return

6/ of asset

j.

•Substituting •

E( rj

(36) and (37)•in

) = _0 +al

If •there is HE, then (40)

(35) gives _" 3 )

( 8j -

&l.

2

(

- al

Aj..

al

••

)

(40)

becomes

E( rj )_ a0 + aI 8j •which three

is

the simple • CAPM. •

alternative

the approximations

Suppose

Under

return-risk

varying

relationships

assumptions

on

NHE,

can be derived

from

given by (40).

NHE only exists on

for some k and i but

_

= _

E( rj ) = aO +al

V

but not on _ , i.e.,

for all i,

( 8j __i

Then,

Vk # Vi

(40) becomes

)'

or

E(

r. j ) "

( a0 -

e.j ) + a i 8 j .

(41)

6/ -Alternatively, e. and A_ could arise from the differences in perceptions on th_ probability distributions of assets rates of return by the "marginal-opinion investors" as exemplified by "insiders" who possess or believe that they possess information not available to the rest of the investors. See Rabinovitch and 0wen (1978, pp. 581-585), Mayshar and Palmon (1985, p. 85).

•(1983, pp. 114-115),

and Givoly


17

The resulting theta risk Even

return-risk

I

is only to

if NHE exists

holds.

However,

and the effect

Suppose all i

relationship shift

for both

V

the

intercept

and

_ but

if (36) does not hold,

oo

_1

i$ _ k

_

The effect

_ 0,

(41) still

does not cancel Out a bias on

but not on V, i.e.,

for some k and i.

of

by the amount c. • 3

of tbeta risk I is to introduce

NHE only exists on

but _

is linear.

Then,

_j.

Vi = V for

(40)becomes

A.

or

E(

r. 3 ) = a 0 + a I ( B. 3 -Aj

The resulting

return-risk

relationship

Of theta risk II introduces exist for both

V

if

not

(37)

does

and

_

hold,

relationshi p is no longer

Finally, and A.

= 0.

suppose

o- ).

(42)

is linear and the presence

a bias on the beta risk.

but

e. = O, (42)still 3

ml

does

not

holds.

cancel

out

However, and

the

linear.

NHE exists

for both V and_

and _hat

The (40) becomes

_I.-

E.

3 ) = _0 + al ( Bj ____ E( r. _I )

or

Even if_NHE

2

-,a

1 (

Aj..

at

)

ej _ 0


18 Under

NHE

in

both

V

and

D

,

the

relationship

is linear but the intercept

is biased•by

the magnitude

A.•

.

resulting

return-risk

is shifted

However,

by

cj

and

8j

if (36) and (37) do

9""

not

hold,

_I

relationship

E.

does not cancel out in (43) and the

in non-linear.

The Corresponding

In (23), define C,

j,

•each

E_uilibriumVector

C H G-land where

k = 1,...,n. j

return-risk

Using

of Portfolio

cjk

(30), the term

are the elements h_-le i•

of

in (23), for

_ecomes

• h _ Cjk( eji- Z i _i eji) or

h z k ejk ej ((

Using

Uji

(38), (44) becomes

h gZCjk pj

Now,

consider

- uj )

- _l _ hi

approximately

( cj

to zero,

term inside

the values

(45)

the parenthesis

of Ai

results

of (23),

in

C A.l _ G - C iZ 6i_G Without

considering

P_k AJ kiPkPM

C in (46), for each asset

COy( rk, rM..) _ 1 ( hl ( EkAjkiPkPM +

(44)

i.e.,

lh .eJ li h i ) -- O.

the third

i.e., CA i _G. Substituting

equal

( Uji - Uj )).

(46) J, this becomes

coy(

rk, rM )) + ...

( Z PM Coy( ))) h m k AJ kiPk rk' rM '

(47)


19

By

(A2.6)

and

(A2.7)

it

is

assumed

that

Aj.1

and

°.M

exist

such

that

E k

Z k akM

AJk:l. ffi

11

.i

Substituting

2 PM

;.j

.i

these values

_ ,M-

Thus,

_1 PM2

based

approximations the

alternative

.i -a.M z i

on

the

of

hl=

risks

I and IS,

return-rlsk

the equilibrium

(21) which

in

O.

assumptions

(23) are zero,

approximate

(43),

is given by

;.j

ffi O.M

rt

in (47) results

for there

parenthesis

(42) and

and

(48)-

which

yield

the

crude

the last two terms

This means

that

relationships

given by

vector • of portfolio

is that of the simple

under

in the

(41),

of investor

i

CAPM.

46 it able

might be worthwhile to

provide

segmentation oocurenoes of

to suggest

insights

on

as well as a possible _nexplained

large

that the there studies

on

alternative

fluctuations

risks might

be

capital

market

explanation

of the

in stock

prices.


20 •Ill. INDIRECT TEST OF THE CAPM WITH NHE

A.

Design and Test Procedure

Designing a

precise •empirical procedure to test••(32) is not

easy since estimating the theta risks may not be possible. is due to two reasons.

First,

hi

This

which is a measure•of the ith

investor's risk aversion may not be measurable; second, _Ji Aji

which reflect the perceptions of investor

i

on _'3

and

and O.k3

respectively are not observable.

However, might

be

(43)

of

an approximate and two-part indirect set of

tests

provided for (32) by using the approximation given which (41),

(42) and (19)

are

special

cases.

by More

specifically, the following model can be estimated.

rj

= _0+

a 1 _j

+ e.3

(49)

where r.,=

*

r.

ej

ej

3

and

ej

assumed. • and A

is

the

+

3 "

(50)

E.

3

(51/

regression

The variables

error andB. 3*

rj*

term with

the usual

properties

are not measurable•due to E. 3

which are not observable. 3..

Substituting values in (49) and simplifying give ( r. 3 + _. 3 ) = _0 . + al ( 8j - A. 3"-

) + e. J

or

r] = _0 + _l B.3+ e.3

(52)

ej = ( e.3- E.3- CtlA.3..)"

(53)

where


21 Since

rj

estimated

and_ and

are the ones measurable, it_ can

be

taken as

a

(52) can

model

with

instead

be

errors

in

w

variables, the errors being

Except

for

e.

J

,

¢. and A

J

j..

•

(52) is similar to

(19).

If _. and

J

are ih fact equal to zero or negligible in magnitude, theta risks vanish, then the correct model isgiven model

should be equal to R

non-zero, the

then

F

.

linear

of the intercept

On the other hand,if c. and A

j

j..

the errors will yield insignificant results

hypothesized equality between u and 0

R

the

by a testable

for (19) and test of (52) should yield significant

return-rlsk relationship, and that the estimate

o

i.e.,

j

are and

may not be expected. F

This is Part I of the two-part indirect test of the CAPM with NHE given by (32).

Looking

at (35) which is a simplification of

(32),

it

is

seen that if %lj and el!j approach zero, then (35) reverts back to the simple CAPM given by (19). The lesser the degree of NHE among investors on the parameters of the probability distribution of the rate of return of an asset,

the smaller the values of the

theta risks.

Given mining,

a

the

probability investing the

..

security within a

sectoral

group,

for example,

differences in perceptions on the parameters of the distribution

of

this

security

among

investors

on this group could be relatively smaller compared

differences

in

perceptions: of

the parameters

of

to the


22 distribution

of this particular security among all investors

the market.

In effect, within a sectoral group, the theta risks

could

be relatively smaller compared to the theta risks for

in

the

entire marke_

Since/_i

and _j.. are directly proportional to theta risks

I and l_/respectively, the_

relatively smaller theta risks imply that

"error" terms are also relatively smalland

/%he

tests for (52) using sectoral betasshould

the results of

show

improvement

over the test results using betas for the entire market. results of the tests in fact show such improvement, an

indirect

confirmation of the existence of the

If the

then this is theta

risks.

Test using sectoral groups of securities comprise Part II of

the

two-part indirect test of the CAPM with NHE.

B.

Data

Most

tests oflthe simple CAPM use indirect estimates of the

betas using Sharpe's (1963) diagonal or market model.

This model

relates

index

the rate of return of a security to a market

and

the regression coefficient estimate provides the estimate for the beta weight

risk of that security.

Also,

these tests assume an equal

for each security in computing for the

market

portfolio

rate of return r . -m

?_/i Friend and Blume (1970), Black, Jensen and Scholes (1972) and Fama and MacBeth (1973). Brown mad Barry (1984, p. 814) following Roll's (1977) conjecture cOnClude that "...we have found evidence that observed anomalies in excess returns are associated with misspecification in the market model used to estimate systematic risk."


23 Available limitations small.

Philippine

could overcome

since the %otal number

Monthly

available

data

data published

1976

extended included

by the Manila

to December

1979.

to

the

the number

is

reduced

past,

since some

are also delisted

the total number

of securities.

maximum

number

sectoral

These

groups:

19 Mining;

and 35 Small

yield of the 49-day

The matrix

off-diagonal computed

deviations

Also,

zero

are

(mostly

of the

be accounted so

using

that

and

for

from

further can

for

the outstanding

the

Industrial

the 48-

three (BCI);

The average R • F

variance-covariance entirely

by

diagonal

model

value

random iS

of each security

of each security

shows that the equal weight

be

reducing

into

oil) issues.

72x72

are

newly-listed.

categorized

Sharpe's

basis for ,computing the weight

portfolio

period is

Bill is used to estimate

terms

cannot

from

inadequate. as

Treasury

Exchange

further

can be included

Commercial

Board

time,

relatively

which

are

test

The total of 72 is therefore

securities

18 Bank,

of securities

through

of securitieswhich

period.

Stock

for the entire

securities

two

is

If the time period

Other securities

month

of securities

f6r a total of 72 securities

January

these

in the market

per security

assumption

is

of the betas

were computed.

inadmissible.

Based on (20), precise The

results

the

72 securities

estimates

of the calculations

for the period

are shown in Table

I.

1976-1979

Similar

for all

calculations


24 TABLE

I

Calculation Results for the 72 Securities for the Period 1976-1979

PERIOD

1976-1979

MARKET •"

SEQ

SEC NR

" rM"

WEIGHT

0.1229 " MEAN, RETURN

VARIANCE VARIANCE

OF THE MARKETo 2 r COV(RJ.RM)

M

0.7259 BETA-J

_u

i 2 3 4 5 6 7 8 9 IO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

44 64 71 I 2 8 7 8 28 27 33 34 35 22 43 40 45 49 31 54 72 73 85 62 69 81 79 5 9 11 12 13 14 15 16 18 21 29

0.0049 0.0019 0.0001 0.0256 0.O233 0.0575 0.0294 0.0829 O. 1423 0;0082 0.001 3 0.0436 O. 0792 0.0324 0.001 3 0.0072 0.OO21 0.0007 0.0025 0.0022 0.0094 0.0048 0.0007 0.0043 0.0007 O.0015 0.0019 0.0460 0..0756 0.0206 0.0032 O.0144 0.01 97 0.0353 0.0151 0.00i.8 0.01 69 0.O109

O.4621 7.8975 0.1554 3.4681 -0.0445 O. 3284 0.2329 0.8787 O. 2662 I .0295 -0.0996 2.7013 -0.0621 I.6608 0.0898 2.8762 0.0305 0.8703 O.4921 I .8909 O.0704 2.9305 -0.1805 I .7363 0.0622 i. 6045 0.4878 IO. 1840 -0.3ae6 I .9412 -0.3504 2.1983 -0.1276 8.5314 -0.2080 6.3441 O.1098 .... I .0520 -O.4872 4.1243 0.1460 2.7127 O.1108 10.9461 -0.1770 4.8468: 0.1513 2.1449 0.2700 3.5703 -0.2284 6.5677 0.0079 4.2631 0.2189 0.7723 0.4543 5.6961 0.3274 I.2653 -O.2150 I .0701 0.1378 I .3201 -O. 1043 3.5443 '0.1204 3.1173 -0.1047 I .0810 O.1232 0.2053 O. 1752 4.9767 .0.0695 0.6986

I .1763 0.6192 -0.0720 0.2256 -0.1158 i .0872 O. 2747 0.9866 O. 5768 0.0990 0.3443 0.7769 O.8517 0:5836 0.5639 0.8507 0.9118 I .0878 0.2788 O. 9445 O.6168 i .4074 0.7246 0.4799 0.4508 I.4572 O.8143 0.3803 0.7522 0.5579 0.0277 O.2610 O.8729 I.0166 0.3130 0.0887 0.7003 0.4214

I .6204 0.8529 -0.0992 0.3108 ,0.1.595 I -4977 O. _784 I .3591 0.7945 O. 1363 0.4743 I .0702 I •1732 0.8040 0.7768 I .1719 I .2560 I.4985 0.3841 I.3011 0.8497 I .9388 0.9982 0.6612 O.6210 2.0074 I .1218 0.5239 I .0362 0.7685 0.0382 0.3595 I •2024 I .4005 0.4311 0.1222 0.9647 0.5805


25

Table

SEQ

SEC NR

WEIGHT

1 (cont'd)

MEAN RETURN

VARIANCE

COV(RJ.RM)

r

39

46

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

75 78 85 84 3 4 28 32 47 17 68 36 37 38 51 70 95 91 92 93 94 90 89 39 59 58 41 87 56 52 61 65. 76

o.0003 0.0001 0.0247 0.0042 0.0029 0.0283 0.0041 0.0003 O. 000_ 0.0023 0.0017 0.0009 0.0256 0.0172 0.0009 0.0004 0.0005 0.0005 0.0014 O. 0005 0.0005 0.0003 0.0016 0.0023 0.0010 0.0042 0.0070 0.0104 0.001 2 0.0118 0.000.9 0.0005 0.0050 0.0039

-0.3151 -0.5397 0.6770 0.0900 0.0642 0.1 915 0.1610 -0.1621 O. 0773 -0.1 325 O. 1281 0.0051 0.3606 0.2450 -0.4604 0.0675 0.0386 -0.0000 -0.1450 -0.2708 -0.2339 -0.1338 -0.2140 0.1244 -0.3'785 -0.1 396 -0.0775 0.0071 -0.1.689 0.3293 -0.2268 -0.1067 0.3574 0.2532

BETA-J v,

4.7235 6.4785 11.0981 3.2487 5.9394 O. 3347 0.3234 4.6335 0.8091 3-5621 O. 1392 2.1815 3.8089 3.7678 8.4052 I .0351 16.5145 4.5677 9.0980 4.5856 9.71 99 2.6321 4.9195 4.4605 I .4236 5.0084 I .5486 3.4429 6.3086 8.0484 2.5053 0.9525 6.0202 5.8979

.-

-0.3463

-0.477o

O. 5283 I .73Q3 0.9474 0.9882 O. 0471 0.1966 0.1115 0.2850 I .0686 0.0697 0.7142 I .3691 I .3830 -0.1189 0.2241 -0.0822 0.9327 I. 2788 O. 9974 I .4656 0.5592 0.9517 0.8115 O. 2440 I•_384 0.6658 0.8854 I .4521 I .1023 0.6495 0.2297 0.9140 0.8428

0.7278 2.3Q60 I .3051 I .3613 O.0648 0.2708 0.1538 0.3928 I .4721 0.0960 0.9838 I .8859 I .9051 -0.1638 0.3087 -0.I 132 I .2848 I .7616 I.3740 2.0189 0.7703 I.3110 I .1179 O. 3361 I .8437 0.9171 1.21 97 J.0004 I.5184 0.8947 0.3164 I .2591 I. 1610


26 were

done

done of

for the annualperiods.

for each of the three the calculations

the

entire

securities riskless

C.

sub-groups

of securities.

for BCI for 1976-1979

Table 3 shows the market for

The same calculations

portfolio

market

The results

are shown

in Table

rates of return and

and for the three

sec_oral

for the annual as well as the 1976-1979

rates of return are also shown

were

2.

variances groups

periods.

of _The

for each period.

Test Results

Initially, period

(52) was tested using

for the entire market

result of the regression

using

analysis

the data over the 48-month

all the 72

using

securities.

ordinary

least

The

squares

is

insignificant.

The

observed

possibility time such the

volatility

of significant

of

the

differences

that the parameters

market

the

behavior

over

in market

of the probability

rates of return of securities

indicates

distributions

may not be stationary

over

of the

_8/ 48-month able

period.

The betas

to take into account

using annual

The

data.

results

improvement

could also

these possibilities,

The results

of

the

change over time.

are shown

tests using

over the results

To be

tests were

made

in Table 4.

annual

of the test using

data

show

the data for

some the

s_/ For

-evidence

which

shows

that the beta could

change

time, see Jacob (1971), Blume (1975), Fabozzi and _rancis Olson and Rosenberg (i982) and Bos and Newbald (1984).

over.

(1978)• "


27 48-month

period.

generally poor. positive

On

the

whole,

however,

the •results are

For the year 1976, where there is a significant

return-beta

relationship,

only

11

percent

of

the

variations are explained by the explanatory variable beta and the intercept is at -0.2937, significantly less than R • F 1979,

a

In fact for

perverse result is obtained giving a negative

estimate

for the coefficient of the beta.

These magnitudes that

the

results

could

be

interpreted

to

mean

that

the

of the errors in (52) are large which could also mean theta

insignificant

risks

results.

are

relatively

large

thereby

giving

Due to the presence of the theta risks,

Part I of the indirect test of the CAPM with NHE attempted to fit a linear model on a basically non-linear relationship.

Thus, the

9/ generally insignificant test results may not be unexpected.

To preclude the possibility that the generally poor test results is due to some other factors, investigations were made on: I) the possible non-stationarity of the probability diatributione of securities rates of return; 2) the possible autocorrelation of monthly rates of return following the test procedure of Brown (1979); 3) tests by groups of securities using ranked beta_ to reduce the possible effects of measurement errors similar to the procedure used by Black, Jensen and Scholes 1972); 4) possible non-stationarity of the betas; and, ) tests of the CAPM using the variance as risk surrogate _ollowing Levy's (1978) constraints onthe number of securities in the investor's portfolio. All these did not provide adequate explanation for the generally poor test results. The probability distributions of securities rates of return were found to exhibit departures from normality and some degree of liptokurtosis and asymmetry. For further details, see Francisco (1983). These findings on liptoku_tosis are in consonance with those of Mandelbrot (1963) and Fama (1965, 1976). For evidence on asymmetric distributions, see Fieletz and Smith (1972)and Leitch and Paulson (1975). The existence of liptokurtosis and asymmetry could be interpreted as possible indications of the presence of NHE among investors.


TABLE

2

Benk-Commercial-Industrial

PERIOD

SEQ

1976-I 979

SEC NR

MARKET rM : O.2578

Securities

VARIANCE

OF THE MARKET e 2r M

0.6229

WEIGHT

MEAN RETURN

VARIANCE

COV(RJ.RM)

BETA-J

O. 1207 -0.O584

O. 1938 -O.O937

I 2

I 2

O .0744 O.O677

O.2529 O. 2662

O.8787 I .0295

3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17 18

7 27 22 31 5 9 11 12 13 16 18 29 3 4 32 !7

0.0855 0.0238 0 .O941 O.O073 O. 1338 O. 2197 O. 0600 O. 0092 O.0419 O. 0440 0.0053 0.O316 0.0822 O.O120 0,0025 0.0050

-O .O621 O.4921 O.4878 O. 1098 O. 2189 O.4543 O.3274 -O. 2150 0.1378 -O. 1047 O. 1232 0,0695 O. 1915 O. 1610 0.0773 O. i281

_I.6608 I .8909 IO. 1840 I .O520 0.7723 5.6981 I.2653 I .O701 I .3201 I.08i0 0.2053 0.6966 O.3347 0.3234 O.8091 O. 1392

O .6267 O. 101.O I .0562 O ,2024 0.3865 I.5802 0.4209 O. 1260 O,3744 O. 1508 0.1215 O.1 930 0.0583 0.1036 O. 1427 0.0389

I .0060 O. 1621 I .6956 O. 3249 0,.6204 2.5368 O. 6757 O.2023 O.6010 O. 2421 O.1951 0.3098 0.O937 O. 1663 O.2291 O.0624


TABLE Risklese for

3

Rate and Portfolio Rates the Market and for 8eotoral (Manila Stock Exchange,

Market

Riskless

of Return and Variances Groups of Securities Philippines)

BCI

Mining

Small Board

o_/ Rate Period

_

72 Securities

18 Securities

19 Securities

rM

O2

rM

rM

o2

0_

35 Securities rM

OM 2 8.8508

1976

0.1022

-0.0033

I .3641

0.1573

0.3636

-0.0980

I .6386

0.1518

1977

O. 1074

0.0434

0.3703

0.2717

0.0601

-0.1075

0.8945

O. 1108

I.0750

1978

O. 1040

O.4910

O.7244

O.6442

I .4656

0.3339

I .3747

0.7080

I .2202

1979

O. 1194

-O,O565

O.4585

-0 .O648

O.2398

-O.O704

I .0803

O.O269

O. 7647

1976-79

O.1083

O.1229

0.7259

0.2578

0.6229

O.0141

I .1916

0.2685

3.2510

a/ The portfolio

rates of return and variances

are computed

using

_b/ The average

cJ

yield of the 49-day

2 Notationally,

oM

Treasury

Bill, Philippines.

2 and

OrM

are equivalent.

value weights.


3O

•TABLE

4.

Results of Tests of the CAPM (OLS) 72 Securities (Manila Stock Exchange)

( rj--_o+ _I Bj+ ej ) Period Covered

Jan.-Dec. 1976

1977

Intercept

Beta Coefficient

-0.2937"_ (0.0780)"

0.1550* (0.0522)

-0.0418

(0.055_) 1978

1979

an. 1976 _c. 1979

0.2600

0.0575

R2

D.W. Statistic

0.1104

2.25

0.0289

2.07

(0.0396) 0.0628

_ _.0107

2.18

-0.2272" (0.0670)

0.1393

1.56

0.0246 (0.0464)

0.0039

1.92

(o.o8o9)

(o.0717)

0.0226 (0.0693)

-0.0045 (0.0515)

See equation (52) in the text. The results here comprise Part I of the 2-part indirect test of the CAP?{with non-homogeneous expectations.

Values in parenthesis are the standard errors. *Significant at least at the .005 level.


31

Table

5 shows the results of .PartII of the_ indirect

test

for the CAPH with NHE using sectoral groups of .securities for the annual and the 1976-1979 periods for BCI, Mining and Small Board.

For BCI, the coefficient of the sectoral beta is significant for

1977

and the 1976-1979 periods and

_0 = RF

at .05

This means that the simple CAPM explains the data.

For

level. Mining,

the coefficient of the sectoral beta is significant only for 1977 but

m

the

data.

0

< R

.

F

This means that the simple CAPM cannot explain

Finally,

the

coefficient of the sectoral beta " for

Small Board is significant for 1977 and the 1976-1979 periods but _0

_10_FI " data.

Again,

the Simple

CAPM

is

unable to

explain the

Of the total 15 regression estimates made, " only five showed significant

relationship at the .05 level.

1977, the explanatory power ofthe coefficient test

Except for

BCI

sectoral beta, in terms of the

of determination for the significant portion of

results,

is

generally

in

low,

with

at

most

25%

of

the the

variations in the dependent variable explained.

These different results by sectoral groups might also be interpreted in the light of the segmentation hypothesis. For a study on capital market segmentation using the CAPM framework, see Errunza and Losq _1985).


ITAB Results

of Indirect

LE

Tests

( rj = e0 +

,5 of the CAPH with NHE (OLS) o_ 1 Bj + e_ )a. 2

Period

Covered

Jan.'Dec.

Group

Intercept

Beta Coefficient

1976

:

Ib 25

0.0666 -0.4286 -0.2480

(0.0629) c ,(0.1712) (0.1597 )

0.0246 O. 1962 O. 2409

i977

:

I 2 3

0.1358" -0.4068* '0.2214"

(0.0417) (0.1577) (0.0839)

0.0742" 0.3102" 0.1739"

I•978 :

I 2 5

0.5225 O.1706 0.2312

(0.1530) (0.2057) (0.0915)

1979

:

I 2 3

0.1129 -0.0744 -0.1051

(0.1569) (0.2439) (0.I_40)

:

I 2 3

0.1056" -0.2805 -0.1786"

(0.0521) (0.1 484) (0.0658)

Jan. 1976 , Dec. 1979

D.W.

Statistic

0.0070 0.0869 0.0393

2.70 • I .43, . 2.24

(0.0114) (0.1504) (0.0752)

0.7145 0.1912 0.1358

2.27• 1.58 1.80

0.1790 0.0800 0.1686

(0.1468) (0.1762) (0.0944)

0.0805 0.0113 0.0859

2.02 I .71 I.93

-0.1627 -0.0464 -0.1448

(0.159!) (0.2584) (0.1357)

0.0579 0.0018 0.0324

2.42 2.12 I .87

(0.0630) (0.1442) (0.0917)

0.2052 O. 1i72 0.2530

2.42 I .56 2.08

0.1319" 0.2229 0.3113"

(0.0713) (0.1498.) (0.2044 )

R

aJ see equation (52) in the text. of the CAPM with non-homogeneous

The results here comprise expectations.

Part II of the 2-part

indirect

test

h_/ Numbers I, 2 and 5 refer to tests of the CAPM using 18 securities of Bank-CommercialIndustrial(BCI), 19 securities of Mining, and 35 of Small Board (mostly oil) respectively. (M

_o/ Values

in parenthesis

*Significant

at least

are the standard at .05 level.

errors.


33 11__/ Figure

I shows the graph

These

results

significant market. c

using

improvement

sectoral over

This improvement

and

A•

J

embedded

in

could

group

of

groups

results.

of

securities

the test results

could

j..

which

of the significant

e*

for

be interpr@ted

the

that

show entire

the errors

•of (52) are relatively

smaller

j

also mean

securities

that the theta risks

could be smaller

within

than those

a

sectoral

for the

entire

market.

Two First, and

observations

the results

Small

Board.

can

Second,

relationship.

relationship

is noted only

all

The

next

on

for BCI are better

return-risk

for

be made

there

foregoing• results.

than the results

is a pattern

More specifically,

for Mining

of

significant

the

significant

for 1977 and for the 1976-1979

the three groups of securities section

the

•provides

explanation

except

for Small

for these

two

period Board. sets

of

observations.

The graph for Small Board for 1976-1979 is included although it is not significant at the .05 level. The t ratios of the beta coefficient and the intercept are 1.55 and -1.89 respectively. It is postulated below that the nature of the data for 1977 which yields significant results for all the three sectoral groups is the primary reason for the significant result for the 1976-1979 period since tests for this period include the 1977 data.


34

IV.

EXPLAINING

A.

THE

SectoraI

Given the

a security in

differences

probability

group,

Risks

within

_f

in

a sectoral on

this

could

distribution

the

security

results. indirect

This

of this particular

is

results

investors compared

security

market.

Tests

the

among

all

sectoral

comparedto

the

of (52) using sectoral

are therefore

expected

to

yield

underlying

hypothesis

of

the

in Part

to

of

within a

small

the

The overall

better two-part

improvement

II over those in Part I

of

appears

to

investors

are

this.

Moreover, expected

the

among small

In effect,

test of the CAPM with NHE.

tests

confirm

securities

of

on the same parameters

in the entire market.

of

say Mining,

parameters

the theta risks could be relatively

groups

group,

be relatively

perceptions

theta risks for the entire

the

"

perceptions

on this subgroup

investors

RESULTS

distribution

investing the

Theta

differences

probability

TEST

the relatively

to invest more

and Small Board

(mostly

on the more oil).

risk-averse

investors

less

risky assets

such as BCI.

the differences

risky assets

On the other

more

risky assets,

less risk-averse

are expected

such as

hand,

to invest

the relatively more

For those investing

in opinions

among

Mining

on

the

on the more

them are expected


FIGURE I Plots

of

Signiflcant

Test Results for CAPM Uslng Sectoral Groups

with Non-Homogenelty of Securities

of

Expectations

-.3 rj â&#x20AC;˘

e

(BM_

o

'

R2.o.n

o'._ i!o_.!_ _'.o "_

3

._

( BM,rM) 0

o N,;_)_/..,

5

_.i

-.2 / -.3

_J _I

1.52.0 BCI: 1976.-79

_

Mining: 1976'79

i -.2 r-

Small Board: 1976-79


36 to

be

more

diverse,

aorrespondingly,

the

theta

risks

are

12/ expected assets, less

to be larger.-the differences

diverse.

For those investing

in opinions

"blue

available more

chips",

than information

significant

and

Small Board.

test

are

results

about

test result

Mining

consists

reliable

and Small

is expected

This provides

observed

more

Also,

mostly

and

Board.

of

readily Thus,

a

for BCI than for Mining

an explanation

for BCI compared

to be

to be smaller.

on the BCi, which

generally

'Tisky

them are expected

The theta risks are expected

â&#x20AC;˘it is known that information the

among

on the less

for the

to that of

better

Mining

and

Small Board.

12/

â&#x20AC;˘

In other words, the departure from the simple CAPM is larger if the overall level of risk is larger. This is in line with the conclusion of Friend and Blume (1970 p. 574) that "...our analysis raises some questions about the usefulness of the theory in its present form to explain market behavior. The Sharpe, Treynor and Jensen one-parameter measure of portfolio performance based on this theory seem to yield seriously biasedestimate of performance, with the magnitudes of the bias related to portfolio risk.

Equality of risk premia among BCI, Mining and Small Board for 1977 and 1976-1979 is rejected _ least at .01 level of significance using two-tailed t test, with BCI having the lowest risk premium. While the estimates may have been biased By the presence of the theta risks which is an added form o_ :risks, it may not be unreasonable to expect that the theta risks directly affect the risk premium. The risk premium could therefore change over time as investors' expectation change. See for example Corhay, Hawawlni and Michael (1987) for seasonal variations of risk premium.


37

, B.

Thotaâ&#x20AC;˘R_ske

The

and

general

perception

General

market

of

investments.

the

Harket

conditions

investors

on

could

the

Correspondingly,

Conditions

affect

prospects

the

values

the

of

overall

their

of the

stock

theta

risks

could change.

For changes the

an

assumed

value

in the values

of the market

of the theta

values

of

c

denominator

is

fixed so that

hi

and

and A

j

( _ji - _j )

"

J..

risks

. In (33), 8

Suppose

premium

_i'

correspondingly

affect

for any security

depends

ljâ&#x20AC;˘

â&#x20AC;˘risk

that hi,

j,

the

only on the

value

of

i = 1,...,m

is fixed

14/ at

least

magnitude the

for

the

perceived

by

expected individual

of the ending

differences

in

i ffi1,...,m

rate

investigation.---

price

Since

among the

and iZ ( _ji - _J )

relatively

when

of

i,nvestors magnitude

by _ji which security

the

on of

the

j

the ( _

as is

grea_er ending

ji

is

a the

price,

- _. ) ,

j

may "not be zero.

the market

more investors

Then

this rate of return

of the security,

the greater

Specifically,

determined

of return

i.

expectations

of the security,

are

under

of theta _risk I is entirely

one-period

function

period

is bullish

in the market

or active,

who may tend

there to

be

This assumption may not be unreasonable especially if one considers 'tests over a one-year perlod, noting that h is a measure of rick aversion.


38

optimistic

and

of portfolio security risks

hence may accept

return.

difference return

This implies

or subgroup

as measured

overestimate, of investors

an

such that ( _jl-

with

( _ji - _J ) > 0

_i > O,

l,.Slj

> 0

i.e_,

the market

a relativelysubdued minimize

investment

the relatively to

be

more

-

by

choose a higher is the

from the true rate investor

_J r) > O. means

_. )

unit

may

tend

to

The preponderance

that i_' ( _ji-

(33) and

of

(36)_j

_J ) >

• 0

> 0... given

_._.

On the other hand, during or dull,

i

optimistic

_j

risk per

relatively

Since ( _

of individual

j,

Since hi > 0 for.all that

which• have

by their betas.

security

higher

that these investors

of securities

in perception

of

relatively

periods

is relatively

trading,

more risk-averse

be close to or lesser

pull out of the investors.

and conservative

ending •price and hence their than _j

inactive

estimate

is bearish

as indicated

the less risk-averse

or entirely

sautious

when the market

individual

market,

These

such that

may

leaving

investorstend

in their forecasts

of _j •

by

Thus,

_ji

i_ ( _j i -

_j)

of

the

could < O° --

There are some indications which show that investors may shift investments •from one group of stocks to another or may enter and pull out of the market. For the period under study, percentage changes in the total monthly values of shares traded ranged from 372 to 984% for the entire market. By sector, the corresponding percentage changes are 282 to 625% for BCI, 199 t_ 1,209% for Mining and 387 to 7,074% for Small Board.


$9

Consequently,

8

< 0 and by

(36) E

lJ

In (34),

if

over the period depends or

the

A

matrix

the_rates

,

i = 1,...,m

and P

of return,

prices

perceptions

on

in

expectations

Pj

â&#x20AC;˘

_n

of

on the ending

differences

investors

are small,

or active;

market

risk II implies

The compared total

year

that

price high(H)

indices

Also

Thus,

_

the

differences

_.

differences

of security

embedded

to be relatively

j

differences

it is small when

large

in these

risk I,

when the

to be relatively

A non-zero

in

market

small

when

value

ofthe

theta

to be a relatively

dull

market

@ O.

shares

Figure 2 shows

traded are relatively

that the much

by

and low (L) price

3 shows that the sectoral the

difference

indices

between

are relatively

monthly

lower

for each of the three sectoral

Figure

given

in

are due to the

as in the case of theta

or dull.

those of the other years securities.

by

are large;

1977 is observed

of

whiah are in turn determined

price

to _he other periods.

values

of

it is expected

is bearish

811j

are functions

Aji

Similarly

theta risk II is expected is bullish

of

theta risk II is large when

among

of

G, P 2 and _4 are the same M rM Since _ is the variance-

elements

on

at least

,

securities.

_ reflected

effect,

perceptions

the

the

fixed,

then the magnitude

since_

of return of securities

ending

is assumed

ji J all securities.

to

covariance

hi

under investigation,

only on

common

! 0.

j

than

groups ranges

the

smaller

of of

monthly in 1977


4O FIGURE 2 Ithly Total Values of Shares Traded (Million Pesos(t='))

A

8O •

"P'M

"

'

//

20"

_r

40

. "

o

Commer©lol-

,

/

I

1976

/ \

"°°lit,_ ,001 ,, o...-

1976

I

"

".

Induslrlal

_i

•1977

"'''

"

.....

I

,976

I

1978

1979

Months

,.76

.o,<,,,

...... _19. _-''--_-'i_

"P-M

1°°olr .,J

_ 1976

19"/"/'

Table 3' Monthly Ranges of Price indices of Seoforol Groupsof Securities January 1976 to December 1979

(N-LI

'

1 • 80

.

.- -.

..

Id_th.ly..IndexPrice IMng_l (H-Li:.Commer¢lol-Znduitrlel

'

*__

O"

•-

w

1976

'

"

/

/_ _/

A "_

|

" I

I9?7

1978

(X-l.I K)O0 .

1979

• •

5OO ioo I00

o-

_t

(x-i.i

1976 ..

,_

......

19TT

I

,,

1976

1979

..

I.Iil I.e hO 0,11 0,6 0,4 0.2 O.

-1976

I _

. 1977

'

I 1976

..

1979

m

"

'


41 z6/. compared â&#x20AC;˘ to the

other

periods:--

, These

could

be interpreted

to

mean that the theta risks for i977 are generally smaller than the

theta

risks

for the other periods.--

If the

theta

risks

are

generally small, E. and A. are correspondingly small such that J J.. tests

based

on

(52) could yield

relatively

more

significant

pattern of

significant

results.

This results

provides an explanation Ofthe for 1977

for

each of the three

sectoral

groups

securities.

The

1979 period

could be attributed to the effects of the 1977

which

of

significant relationship observed for the 1976' data

form part of the data used for the tests for the 1976-1979

period. relatively

For

the large

regression model.

other so

periods,

that

(52)

the is

an

theta

risks

entirely

could

be

inadequate

Hence, the generally insignificant results for

these periods.

16__/ F-statistics comparing the variances of price ranges (highlow) between _riods for each sectoral group of securities show that in general the ranges in prices in1977 are significantly smaller than those for the other years.

One possible implication of this is that the theta risks could influence stock price movements. Thus, the theta risks might provide an alternative explanations for unexplained large stock price fluctuations. For more recent development in this area of study, see Poterba and Summer (1986).

,_


42 The significant test results can be classified according the

three

alteraative

approximate

relationships given by (41), (42)and in Table 6 below.

linear

(43)-

to

return-risk

These are s_mmarized

The graphs for these are previously shown

in

Figure I.

TABLE

6

Model Classificationfor the Indirect and SignificantTest Results for the CAPM with Non-Homogeneous Expectations (NHE)

Year/Period

BCI . ..

Mining

.

.

Small Board

.

. .

.

1977

(41) or (42)

!41) or (43)

(41) or (43)

1976-1977

.(41)or (42)

(41) or (43)

(41) or (43)

.

m

m

Am Alternative Explanation of U.S. Results on the CAPM

C.

Equation (41), explanation works

of

(1972),

to

(42) and (43) might provide an

the test results of the CAPM in- the

Friend and Blume (1970), Blume

and

Friend (1973),

Black,

securities have

greater than

RF

and

The

Scholes

One result which has

extensive investigation is

have intercept less than _

intercept

1972).

to

Jemsen

U.S.

Fama and MacBeth (197.3) and

others do not support the CAPM hypothesis. been subjected

alternative

that

and low beta

high

(Black,

Jensen

beta

and

securities Scholes,


43 In the U.S.,

in

V

exists

is

non-zero

but with negligible NHE but

that individual portfolio

it may not be unreasonable

ellj i

return

relatively

may accept and

higher

less risk-averse

C

>

J

thus,

there

relatively

therefore • a as measured

investor

believes

O.

in_.

is a downward

of

is

( _

o

eli

implies of

•-chosen• has A

relatively

_" J ) > O,

In effect

in (41)

hi

J

( _Ji-

that

risk •per unit

by its beta.

on security• j.

The intercept

higher

security

that

that NHE

This means

A higher value

risk

wise he may not invest (36),

vanishes.

to suppose

8

other-

> 0

zj

and by

- _ )

and

j

bias on the •intercept for

high

beta

securities.

On

the

individual portfolio a

i

may

return,

security

measured

other hand,

j

a lower value

accept

relatively

i.e., a more chosen

by its beta.

In this case,

or

an

ands, _ by

i implies

an

lesser •risk per• unit

of

investor.

could have a relatively

for the rate of return •of security

ezj <_ 0

h

that

risk-averse

i

under-estimate

of

such

lower

the•perception j

( PJ i - _j)

< O. (36•) Ej _

Therefore,

could be

<

0

The _ intercept

risk

as

of investor conservative

In in

effect, (41)

is

ISl The presence of insiders confirms this. On insiders, Jaffe (1974), Finnerty (1976) and Givoly and Palmon (1985).

see


44 19/ ( _

- c ) giving

0

NHE

j

in

there• perhaps which

_ is

also exists a

on low beta

and do not vanish,

bias o_ both

explain made

an upward bias

the intercept

then (43)

and

the generally •poor explanatory

investigators

conclude

securities._

beta.

holds This

power

If

of the

that the CAPM of Sharpe

and could beta

(1964)

2o__/ and Lintner

(1965) is not consistent

withthe

data.

19/ Since the h s for low beta securities are relatively smaller than the h s for high beta securities, it is not unreasonable to expect a preponderance of downward biases on the intercept, assuming that the number of investors in the low beta and the high beta • securities are approximatelF equal. This is observed for the results shown in Figure I. Otherwise•, if there are relatively more risk-averse investors, then there could be an upward bias on the intercept as in the case of BCI for 1977.

20__/ The alternative explanation here for the empirical anomalies of the CAPM might be more appealing in the light of more recent doubts raised on conclusions drawn from zero-beta based_tests for the CAPM (Best and Grauer, 1986, p. 98).


45 V.

CONCLUDING REMARKS When

investors'

distributions resulting This

of

diverse

assets

equilibrium

rates of

called

theta

biases

to the beta arising

among

investors

on

asset.

If

relationship Lintner

considered, is

risks are

mean

and

the

non-linear. form of risk

the of

systematic expectations

variance

(covariauce)

of an asset.

the beta is not a complete

the theta risks vanish,

NHE

market

empirical attempt

to

There indirect

is

than

fit

the

the simple

more

measure

of risk of an

resulti_

return-risk

CAPM of Sharpe

appropriate

the apparent

(1964)

and

characterization

anomalies

on the CAPM couldperhaps

a linear model

on

a

in

the

existing

be explained

fundamentally

of

by the

non-linear

relationship.

are

methodological

and

data

limitations

test of the CAPM with NHE undertaken

risks are basically used.

a HE,

evidence

return-risk

is

probability

(I965).

Since the

are

from non-homogeneity

the

to

the

relationship

These

of the rate of return

reverts

on

of a new and additional

risks I and II.

Under NHE,

return

return-risk

is due to the existence

respectively

expectations

Also,

capital

market,

further

verify

unobservable tests using such

The

and an errors-in-variables

the theta model

data from a larger and more mature

as that of the

the theoretical

here.

on

U.S.,

predictions

are

necessary

of the model.

to

Also,


46 it able

might be worthwhile to

segmentation occurences

provide

to suggest

insights

on

that the theta studies

on

as well as a possibl e alternative of unexplained

large fluctuations

risks might capital

explanation in stock

be

market of the

prices.


47 APPENDIX I

To show

that: 2 PM 0 r M

m

h E i lhi= .Proof:

R F

;' The problem

ation

:E(r M)

problem

with

given

by (I)

and (2)

the Lagrangian

L _U i ( E,,_ i ) + _i

and substituting

values

_Ei

v +_

De fine

By assumption, fl > 0,

equation

( Wi - P'Xi

Diffe_entiatingwith, respect to

is

given

by

+di

)" i

and

Xi, equating to

di

yield

a constraiued.maxi_niz-

= 1,...,m.

(AI.1) zero

-.

2_ xi e-_:P_ o. . i •

_AI.2>

fi = _Ol / _El

(AI.3)

gi = _ui / _i

(A1.4)

hi -- -fi / 2gi"

(Ai.5)

gl < 0 such that hi > 0

•Dividing A(I.2) through by _Oi / _Ei

for

all

and substituting

i.

the

defined variables y_p_d

v+_2_x f:i

i- er-o.

_,_s>

Or •.---I n Xi ffi V -.,8_. hi ,

(&,l..7)


48 For each security

J,

that

(A1.7)means n

E h. = k _jk Xki l V. - 8P. 3 3

The condition his marginal

for optimality

of individual

rate of substitution

all securities.

between

i, i = l,...,m _i

summing up the numerator

will not change

is that

and Ei are equal

for

That is,

-2"_E')1=I".... :( 2_E')J=I"'" = ( Thus,

(AI,8)

' j = l,...,n.

the value

2_Ei)n= h.1

and denominator

over

all j in

(A1.9) (AI.8)

of h i , i.e.,

zz

i k hl ffi _.(v.3

. eP.

]

Since by definition,

(At lO)

)

3

h _ iE hi , Z ÂŁ Wjk_i jk

h=Z(

) VM -

i

OPM

or

â&#x20AC;˘ j

k

i

j

h_

V M - 0 PM

By

k _jk

_

(AI.I1) ....

@

VM - 0 PH

(9), Xk ffi1 for all k, k ffil,...,n.

By definition

2 Jl kE _jk ffiJr coy( Vj, VM ) ffiw VM

On this point,

see for instance,

o

Mayers

(1972, p. 227).


49

Thtls

2 (_M

2 02 PM PM

h:vM__2 _M("M-e_ OE

2 •

h =

PM _M E( rM ) -_F

*


50

APPENDIX Approximations

A.

for

II

Thera

Rlsks

I and

II

Theta P_sk I By (33),

:£ hi( _j i - _j ) i 81

J• = PM o

Estimating

h i and

,

2

( _Jl-

_Jl are not observable.

i

= l,...,m;

j

= l,...,n.

rH

_J ) may not be possible

since

To have •an idea of the magnitude

hi and of 8Tj,

suppose

= e. J

m where

E. is defined 3

as the average

from pj by all investors imatlon

(A2. I)

deviation

in the market.*

of the forecasts

Then a crude approx-

for eli is /; hl i

zj

PM orM

From Appendix

_.3 i l: h.i

¢..h. j

PM °r M

PM. °r M

1, PM 02 rM h=

E ( rM ) -_

* In effect, (A2.1) enables us to substitute as a very crude approximation.

ej for (

Pji- Wj )


51 • so

that e:j £'M o 2 r_

¢.3

4 ¢.3

(E(rM)- 5, )p_o2 E(,%) -_ N B.

Theta

Risk

._

IT

By (34), i hi Aj i n O 8TI

=

2 Pj

J

It Is readily

4

, i = l,...,m;

j = l,...,n.

PM °rM

seen that

t_c= • (co,,(vl_ , ffi ( PiPHcov(rl

V

M)

cov(V 2,

VH)

.....

, rM) P2PHCOV(r2,

cov(V n , rM )

,..

vH) .)' pnpMcoV(rn,

rH)

)'

(A2.4)

Thus, n Aji _ G ffik Z Ajk i Pk PM UkM

Again,

to

have

an

idea

of

the

(A2.5)

approx_nate

magnitude

of

QIIj'

suppose

n and

_,6)

"Aj.i

let tZ o10 I n

:_(_i_.6)i'/means ;.e_n . ._xtx ,

that Ai

-

(_.7

= O.M

J_'4 i

is

t/_e average

and . _(A2.7)_ :!.... , me,us .

of rate of return

of a security

that with

value

-O. M

is

of the

the average

Jth

row

of

covari-,

the market .rate of return.


52 Then

Aji _ C_"eM;'j.i _._ k _ ek "

C_..S_

But kE Pk " PM so

that

z h

2

_.

-

=

suppose

there

m

,

= 2 4 PJ PM °r M

eIlj

Further,

-

exists

Pj

a value

"t..

4 a

"

(A2.9)

rM

Aj..such

Chat

m

i

k_Aj kl

(A2. I0) m

mn

Then

(A2.9) becomes

_.MAj"" E i h_ 8IlJ

:itutlug

the

=

value

P. o 4 3 rM of

_.M_" 3.. h =

P. o 4 3 rM

"

(A2. II)

h yields

_..MA. ].. PM Ur2N

A" 3.-

--

O.M

) (7-) E(r•M) - RF

urM

PM

c--) P.3

(A2.,2)


53

-

By definition, -O.M

-a

- E,o_/ ,,, _ut_:,_- ,,2 k rM so-,:_t

ON

2rM /n.

Thus,

2

0

-_.M 2 =

°rM 2

U 0

1 " -- _"

rM

(A2.13)

n

rM

For the n securitles,

let

Pj be the average

total

security J at the beg£nnln8 of the period.

_j Thus,

a very

crude

approximation

e_z:_" ezzj " ( -

is

Since PM "_

_ Pj I n .. PM I n.

@O

It

market

e(,%)- as

noted,: that

for

for

PJ"

851j is given

by.

) ( - ) n = _1

.

J and for

"

given

(_.15)

•values • of Aj _.

and (z I , (A2.15) means that •

w

.if Pj > _j => e_zj ,: ezzj

(A2.16)

and tf

of

(A2.14)

],

any security

value

Pj < Pj =:_ eli j > 811J

'

(A2.17)


54 REFEREN CES

Best, Michael J. and Robert R. Grauer. _'CapitalAsset Pricing Compatible with Observed Market Value Weights." Journal of Finance, Vol. 40, No. I (March 1985), 85-I03. _ Black, Fischer. • "Capital Market Eqhilibrium with Restricted BorroWing." Journal of Business, Vol. •45,• No. 3 (July 1972), 444-455. Black, Fischer, Michael C. Jensen, and Myron Scholes. "The Capital Asset Pricing Model: Some Empi_£cal •Tests." In Michael C_ Jensen (ed.), Studies in the Theor_ of Capital Markets, New York: Praeger Publishers, 1972. •

Blume, Marshall E. and Erwin Friend. "A New Look at Capital Asset Pricing Model." Journal of Finance, Vol. No. I (March 1973), 19-34.•

the 28,

Blume, Marshall E. "Betas•and their Regression Tendencies." •Journal•of Finance, Vol. 30, No. 3'(June 1975), 785-795. Bos,T. and P. Newbald. "An Empirical Investigation of the possibility of Stochastic Systematic ••Risk in the Market Model." Journal of Business, Vol. 57, No. I (January 1984), 35-41 • Brealey, Richard and Stewart Myers. PrincAples_of Cor_orate_ Finance, New York: McGraw Hill, Inc., 1981. Brown, Stewart. "Autocorrelation, Market Imperfections, and the CAPM." Journal of Financial and Quantitative Analysis, Vol. 14, No. 5 (December1979), IO27-IO33. Brown, Stephen J. and Christopher B. Barry. "Anomalies in •Security Returns and the Specification of the Market Model." Journal of Finance, Vol. 39, No. 3 (July 1984), 807-817. Cheng, Pao L. and Robert R. Grauer. "An Alternative Test of the Capital Asset Pricing Model." American Economic Review, Vol. 70, No. 4 (September 1980), 660-671. Corhay, Albert, Gabriel Hswawini and Pierre Michael. "Seasonality in the Risk-Return _Relationship: Some International Evldence." Journal of Finance, Vol. 42, No. I (March•1987), 49-68.


Er_unza, Vihang ,and .Etienne Losq, "International :Under Mild Segmentation: Theory and Test." Finance, Vol. 40, No. ,I (March 1985), 105-124.

Asset Pricin_ Journal of

Fabozzi, F.J. and J.C. Francis. "Beta as Coefficient." Journal of Financial and Quantitative Vol. 13, No. I (March 1978), 101-116.

Yama, Eugene F. "The Behavior of Stock Market of Business, VOI. 37, No. I (January 1965),

Comments." i

a Random Analysis,

Prices." 34-I05.

JOurnal

Risk, of Return and Equilibrium: Clarifying Journal Finance, Vol, 23 (MarchSome 1968),29-40.

,

and James D. MacBeth. "Risk, Return, and Equilibrium: Empirlcs/ Tests." Journal of Political Economy, Vol. 81, No. 3 (May/June 1973), 607-6364 Fama, Eugene F. Inc., 1976.

Foundations =

of Finance,

New York:

Basic

Books,

Fieletz, B.D. and E.W. Smith. "Asymmetric Stable Distibutions of Stock Price Changes." Journal of the American Statistical Association, Vol. 67, No. 340 _December 1972), " 813_.,814, ' _ .. ,

â&#x20AC;˘

!

,_

,

.,

Finnerty, : Joseph E. ,. "Insiders and Market Efficiency/' Journal of Finance, Vol. 31, No. 4 (September 1976), 1141-1148. Francisco, Clodualdo R. "Portfolio Theory and Capital Market Phenomena: Tests and Extensions cf the Capital Asset Pricing Model." Ph. D. Dissertation, School of Economics, University of the Philippines, Nay 1983. Friend, Errin, Randolph Nesterfield and Michael Granito. "New Evidence cn the _ CapitalAsset Pricing Model." Journal of Finance, Vol. 33, No. 3 (June 1978), 903-920, Friend, Irwin and Marshall Blume. "Measurement of Portfolio Performance .Under Uncemtainty." American Economic Review, Vol. 60, No.4 (September 1970), 561-575. ,!

Friend _ Irwin. "Recent Bankin_ and Finance,

Development_ in Finance, Journal of Vol. I, No. 2 (October 1977), 103-117.

Givoly, Dan and Dan Palmon.."Insider Trading and the Exploitation of Inside Information: Some Empirical Evidence." Journal of Business, Vol. 58. No. I (january 1985), 69-87.


56 Gonedes, Nicolas J. "Capital Market Equilibrium for a Class Heterogeneous Expectations in a Two-Parameter World." Journal of Finance, Vol. 31, No. 1 9March 1976), 1-15. Green, Richard C. "Positively Weighted Portfolios on Minimum-Variance Frontier." Journal of Finance, Vol. 41, No. 5 (December 1986), 1051'1068.

the

Jacob, N.L. "The Measurement of Systematic Risk for Securities and Portfolios:Some Empirical Results." Journal of Financial and Quantitative AnalEsis , Vol. 6, no. 2 (March 1971),

815-834.

Jaffe, Jeffrey F. "Special Information and Insider Trading." Journal of Business, Vol. 47, N0.3 (July 1974), 410-428. Jensen, Michael C. 'LOapita!Markets: Theory and Evidence." Bell Journal of Economics and Management Science, Vol.3, no. 2 (Autumn 1972), 357-398. Leitch, R.A. and A.S. Paulson. "Estimation of :Stable Law Parameters: Stock Price Behavior Application." Journal of American Statistical Association, Vol. 11, No. 7 (May 1965), 681-689. Levy, Haim. "Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio." American Economic Review, Vol. 68, No. 4 (September 1978), 648-658. Lintner, John. "The Valuation of Risk Assets and the Selection of Risky Investment in Stock Portfolios and Capital Budgets." Review of Econom_icsand S_tatistics,Vol. 47, No. I (February 1965_, .13-37. • "The Aggregation of Investors' Diverse Judgement and Preferences in Purely Competitive Securities Markets." Journal of Financial an Quantitative Analysis, Vol. 4, No. 4 _December 1969), 347-400. _ndelbrot, Prices." 394-429.

Benoit. "The Variation Journal of Business,• Vol.

of Certain :Speculative 36, No.2 (October 1963),

arkowitz, Harry M. "Portfolio Selection." Voi. 7 (March 1952), 77-91.

Journal of Finance,

_yers, -David. "Nonmarketable Assets and Capital Marke¢ Equilibrium, Under Uncertainty." In Michael C.,Jensen (ed.), Studies in the Theory of Capital Market, New York: Praeger Publishers, 1972, 223-248.


57 Ma_shar, Joram. "On Divergence of opinion and Imperfections in Capital Markets." American Economic Review, Vol. 73, No. I (March 1983), 114-128. Ohlson, James and Barr Rosenberg. "Systematic Risk of the CRSP Equal-Weighted Common Stock Index: A History Estimated by Stochastic-Parameter Regression." Journal of Business, Vol. 55, No. I (January 1982),121-145. Petit, R. and R. Westerfield. "Using the Market Model and Capital Asset Pricing Model to Predict •Security Returns." Journal of Financial and Quantitativ _ Ana!ysis, Vol. 9, No. 4 (September 1 974 ) ,_579-605 • Pcterba, James M. and Lawrence H. Summers. "The,Persistence of Volatility and Stock Market Fluctuations." American Economic R eview, Vol. 76, No. 5 (December 1986), 1142-1151 ". Rabinovitch, Ramon and joel owen. "Nonhomogeneous Expectations and Information in the Capital Market." Journal of Finance, Vol. 33, No. 2 (May 1978), 575-587. Roll, Richard. "A Critique of the Asset Pricing Theory's Tests." Journal of Financial Economics, Yol. 4, No.2 (march 1977), 129-176. Sharpe, William F. "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk." Journal of Finance, Vol. 19, No.3 (September 1964), 425-442. • Portfolio Theor_ And Capital McGraw-Hill, Inc., 1970.

Markets.

New York:


/pidssp8704