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
nonhomogeneity
on the parameters of the probability
rates
of return and derives an
relationship which is nonlinear.
systematic
investors
among.
distribution
equilibrium
of
returnrisk
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 errorsinvariables 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 nonlinear returnrisk relationship.
model
TABLE
OF CONTENTS
A.
Acknowledgment
B.
ABSTRACTâ€˘
C.
List of Tables and Figures
I.
Introduction and Su.._ry
II.
.
The CAPM with NHE: Explicit NonLinear ReturnRisk 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 19761979
24
BankCommerciallndustrial
28
Riskless Rate. andPortfolio. Rates of â€˘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 NonHomogeneity 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 meanvariance 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 nondiversifiable risk.
Tests
of
the
CAPM
indicate that
the postulated
linear
returnrisk 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 riskfree rate and arrived at the wellknown zerobeta 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).
returnrisk
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
nonhomogeneous
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
â€˘ models.
The
purpose
introducing explicit
Black,
this
NHE following equilibrium
given asset; attempt
of
provide
an alternative
is
Rabinovitch
nonlinear
to;
explanation
extend
and Owen
returnrisk
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 twostage
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
returnrisk
errorsinvariables
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
returnrisk
as "theta risks
matrix,., respectively, the
that •the simple
nonlinear which
vector
for a
particular
5 subgroup of risks
securities,
among all investors
same
security
groups
of
expected
relationship returnrisk
the simple
risks introduce
the
investing
becomes
the
sectoral
results.
nonlinearity
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
returnrisk
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
NONLINEAR
RE_URNRISK
•RELATIONSHIPS A. ,NHE Assumption,
IndividualPortfolio
and the Market
Clearing
Let V be an nxl vector, •random
oneperiod
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 nonhomogeneous
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 variancecovariance
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,
riskfree
rate of return.
such
is an
nxl
the total market
beginning
define• _ and Q
P
=
and 8
is
rj an
is
nTn
the
oneperiod
random
variancecovariance
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 oneperiod
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
â€˘the
marketclearing
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 marketclearingcondltion
all investors
rearranging
(I0),
(9) and summing
up over
result in
O'E h,[liP = 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= VEnG
(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 ) = PjPMC°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 ovG. 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 â€˘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
â€˘
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 â€˘+ hfil_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
lvector ..... 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.
NonLinear
ReturnRisk
Rabinovitch
and
Relationshi_
0wen
(1978) did not provide
returnrisk _elationship. is
given here.
It
new and additional
Appendix
1
explicit•
relationship
will be seen that this relationship
components•of
(22) can be rewritten
0 P = Vh.. _
Risks
an
An explicitreturnrisk
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
nonllnear
re'turnrlsk •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 nonhomogeneity J'
eli and eiij
k ffiI,...,n. are
bias,
from the nonhomogeneity
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
ReturnRisk
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
returnrisk
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 "marginalopinion 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. 581585), Mayshar and Palmon (1985, p. 85).
•(1983, pp. 114115),
and Givoly
17
The resulting theta risk Even
returnrisk
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. â€˘ 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
returnrisk
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
returnrisk
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 nonlinear.
The Corresponding
In (23), define C,
j,
•each
E_uilibriumVector
C H Gland where
k = 1,...,n. j
returnrisk
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
Â°.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,
returnrlsk
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 â€˘ 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 twopart 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
,
Â˘. and A
J
j..
â€˘
(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
nonzero, 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
returnrlsk relationship, and that the estimate
o
i.e.,
j
are and
may not be expected. F
This is Part I of the twopart 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
twopart 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 49day
The matrix
offdiagonal 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 â€˘ F
variancecovariance 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
newlylisted.
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.
19761979
Similar
for all
calculations
24 TABLE
I
Calculation Results for the 72 Securities for the Period 19761979
PERIOD
19761979
MARKET •"
SEQ
SEC NR
" rM"
WEIGHT
0.1229 " MEAN, RETURN
VARIANCE VARIANCE
OF THE MARKETo 2 r COV(RJ.RM)
M
0.7259 BETAJ
_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
BETAJ v,
4.7235 6.4785 11.0981 3.2487 5.9394 O. 3347 0.3234 4.6335 0.8091 35621 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.
subgroups
of securities.
for BCI for 19761979
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 19761979
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 48month
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/ 48month 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)â€˘ "
27 48month
period.
generally poor. positive
On
the
whole,
however,
the â€˘results are
For the year 1976, where there is a significant
returnbeta
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 â€˘ 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 nonlinear 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 nonstationarity 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 nonstationarity 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
BenkCommercialIndustrial
PERIOD
SEQ
1976I 979
SEC NR
MARKET rM : O.2578
Securities
VARIANCE
OF THE MARKET e 2r M
0.6229
WEIGHT
MEAN RETURN
VARIANCE
COV(RJ.RM)
BETAJ
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
197679
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 49day
2 Notationally,
oM
Treasury
Bill, Philippines.
2 and
OrM
are equivalent.
value weights.
3O
â€˘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 2part indirect test of the CAP?{with nonhomogeneous 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 19761979 periods for BCI, Mining and Small Board.
For BCI, the coefficient of the sectoral beta is significant for
1977
and the 19761979 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 19761979 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 nonhomogeneous
The results here comprise expectations.
Part II of the 2part
indirect
test
h_/ Numbers I, 2 and 5 refer to tests of the CAPM using 18 securities of BankCommercialIndustrial(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
returnrisk
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 19761979
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 19761979 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 19761979 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 twopart
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).
riskaverse
investors
less
risky assets
such as BCI.
the differences
risky assets
On the other
more
risky assets,
less riskaverse
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 NonHomogenelty of Securities
of
Expectations
.3 rj â€˘
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: 197679
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€˘it is known that information the
among
on the less
for the
to that of
better
Mining
and
Small Board.
12/
â€˘
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 oneparameter 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 19761979 is rejected _ least at .01 level of significance using twotailed 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€˘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€˘
â€˘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
oneperiod
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 oneyear 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 riskaverse
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 riskaverse
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€˘
_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 nonzero
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_
"PM
1°°olr .,J
_ 1976
19"/"/'
Table 3' Monthly Ranges of Price indices of Seoforol Groupsof Securities January 1976 to December 1979
(NLI
'
1 • 80
.
. .
..
Id_th.ly..IndexPrice IMng_l (HLi:.Commer¢lolZnduitrlel
'
*__
O"
•
w
1976
'
"
/
/_ _/
A "_

" I
I9?7
1978
(Xl.I K)O0 .
1979
• •
5OO ioo I00
o
_t
(xi.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€˘ 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 19761979
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__/ Fstatistics 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
returnrisk
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 NonHomogeneous Expectations (NHE)
Year/Period
BCI . ..
Mining
.
.
Small Board
.
. .
.
1977
(41) or (42)
!41) or (43)
(41) or (43)
19761977
.(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
nonzero
but with negligible NHE but
that individual portfolio
it may not be unreasonable
ellj i
return
relatively
may accept and
higher
less riskaverse
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
riskaverse
i
underestimate
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 riskaverse 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 zerobeta 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
nonlinear. 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_
returnrisk
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
nonlinear
relationship.
are
methodological
and
data
limitations
test of the CAPM with NHE undertaken
risks are basically used.
a HE,
evidence
returnrisk
is
probability
(I965).
Since the
are
from nonhomogeneity
the
to
the
relationship
These
of the rate of return
reverts
on
of a new and additional
risks I and II.
Under NHE,
return
returnrisk
is due to the existence
respectively
expectations
Also,
capital
market,
further
verify
unobservable tests using such
The
and an errorsinvariables
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 ero.
_,_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€˘ 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("Me_ 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)
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