Theoretical Economics Letters, 2013, 3, 306-316
Published Online December 2013 (http://www.scirp.org/journal/tel)
http://dx.doi.org/10.4236/tel.2013.36052
Open Access TEL
w-MPS Risk Aversion and the CAPM*
Phelim P. Boyle1, Chenghu Ma2#
1School of Business and Economics, Wilfrid Laurier University, Waterloo, Canada
2School of Management & Fudan Finance Center, Fudan University, Shanghai, China
Email: #machenghu@fudan.edu.cn
Received September 29, 2013; revised October 29, 2013; accepted November 6, 2013
Copyright © 2013 Phelim P. Boyle, Chenghu Ma. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the
intellectual property Phelim P. Boyle, Chenghu Ma. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
ABSTRACT
This paper establishes general conditions for the validity of mutual fund separation and the equilibrium CAPM. We use
partial preference orders that display weak form mean preserving spread (w-MPS) risk aversion in the sense of Ma
(2011). We derive this result without imposing any distributional assumptions on asset returns. The results hold even
when the market contains an infinite number of securities and a continuum number of traders, and when each investor is
permitted to hold some (arbitrary) finite portfolios. A proof of existence of equilibrium CAPM is provided for finite
economies by assuming that when preferences are constrained on the market subspace spanned by the risk free bond,
the market portfolios admit continuous utility representations.
Keywords: CAPM; w-MPS; Risk Aversion; Infinite & Incomplete Market; Non-Gaussian Returns
1. Introduction
This paper provides general conditions for the classical
CAPM as an equilibrium model in economies with a fric-
tionless market. First, we show that, if equilibrium exists,
then the asset returns must satisfy the CAPM if all in-
vestors are risk averse in a sense of mean-preserving
spread1 (Theorem 1). Second, we prove the existence of
equilibrium satisfying the CAPM for economies with
w-MPS risk averse investors, without assuming complete
markets (Theorem 2). Our results advance the literature
and lead to deeper understanding of the extent to which
the CAPM holds in equilibrium: 1) preferences satis-
fy w-MPS risk aversion, and such preferences can be
incomplete and non-transitive; 2) the population space,
or investors’ type space, can be finite or continuum infi-
nite; and 3) the market may contain a finite, or a count-
able infinite number of securities, while the market port-
folio is composed of a (fixed) finite number of risky as-
sets2.
It has been known for a long time that if all investors
have mean-variance preferences, then the CAPM holds
([5-7]). It is also known that mean-variance preferences
persist for general probabilistic sophisticated expected
and non-expected utility functions when asset returns are
elliptically distributed ([8,9]). It is, therefore, of particu-
lar interest to explore if the CAPM holds when prefer-
ences do not fall into the mean-variance class, particu-
larly when asset returns may follow arbitrary distribu-
tions3. Since the existence of equilibrium CAPM docu-
*This paper has benefitted from discussions with Jonathan Berk, Ber-
nard Cornet, Darrell Duffie, Editya Goenka, Chiaki Hara, Atsushi Kajii
Juan Pedro Gómez, Patrick Leoni, Ning Sun, Dong Chul Won, and
Zaifu Yang.
Ma acknowledges financial supports from the Economic and Social
Science Research Council, UK (R000223337), the Nature Science
Foundation of China (70871100, 71271058), and the Fudan Finance
Center (EZH4301102/020).
#Corresponding author.
1An investor exhibits w-MPS risk aversion if, for all random payoffs
and
XYX
, the investor would prefer to whenever
X Y
0Cov
and . The notion of mean-preserving-spead
used in this paper is weaker than the notion of MPS in [1,2], and is thus
denoted w-MPS. Our w-MPS risk averse preferences correspond to the
strict variance averse preferences used by [3].

, 0X
2The validity of CAPM, along with existence of equilibrium CAPM, for
the case of a finite number of securities and a finite number of investors
was documented in [4], Chapter 5. This paper can be thus regarded as a
generalization of [4] to infinite economies.
3In Duffie’s book [3] it contains a derivation of CAPM. Duffie’s deri-
vation rests on several explicit and implicit technical assumptions on
asset returns: first, the existence of a continuous linear pricing rule in
the market span; second, the market subspace is assumed to be a closed
subset in 2. Whilst the linearity of the pricing rule is necessarily im-
p
lied by the no-arbitrage condition as part of the equilibrium conditions,
the closedness assumption on the market space is somewhat arbitrary.
P. P. BOYLE, C. H. MA 307
mented in literature can be largely built up on the finite
specification with respect to the number of tradable secu-
rities and the number of investors, it is thus also desirable
to consider infinite economies.
In comparison with the above cited work, this paper
makes no distributional assumption on asset returns, and
purely focuses on investor’s risk preferences over the
random payoffs. The assumption of w-MPS risk aversion
is appealing because 1) it captures investor’s psycho-
logical aversion towards “increase in risk” in a natural
way, 2) w-MPS risk averse preferences are not restricted
to certain classes of expected or non-expected utility
functions. Therefore, they are not subject to criticisms
such as the well-known Allais Paradox and other defi-
ciencies associated with expected utility functions. It is
noted that expected utility functions may violate w-MPS
risk aversion—and it could even be argued that this con-
stitutes another drawback of expected utility functions. In
fact, we shall show that CAPM holds even when inves-
tors’ preferences are not complete, nor transitive.
The relationship between mean-variance utility func-
tions and the w-MPS risk averse preferences has been
analyzed. It is shown in [10] that, when the market con-
tains a finite number of risky assets (but no less than
three) and when the market span forms a convex cone of
2 if the w-MPS risk averse preference is represented
by continuously Frechét-differentiable (in the 2-norm)
utility functions, then the preference must admit a mean-
variance utility representation. Nevertheless, one may
still want to treat the mean-variance preferences as a sub-
set of the w-MPS risk averse preferences. In fact, a bi-
nary relationship satisfying the w-MPS risk aversion
property constitutes a partial order, which may not admit
an utility representation. As an illustrative example, con-
sider the following Lexicographic binary relation
defined on : For a given positive integer ,
2n
X
Y if
X
dominates by the first order stochas- Y
tic dominance (i.e.,
dist
XY
, where 0
and
0
); or, if there exists such that (a) 2mn
ii
Y
 
 


1
1
m
for , and (b) 0im
1
m
m

1
m
X
Y


 . In this binary relation,
in order to rank random payoffs
X
and Y, which may
not dominate each other by first order stochastic domi-
nance, one may need to compare their moments up to the
th order. The binary relation displays w-MPS risk
aversion property, yet it does not admit a mean-variance
representation! Indeed, most of the analyses to be carried
out in this paper do not rely on the assumption on the
existence of a mean-variance utility representation.
n
Equilibrium with continuum of traders was thoroughly
studied in literature ([11-13]). [13] also contains an ab-
stract treatment of equilibrium with incomplete orders.
Existence of equilibrium with incomplete and non-transi-
tive preferences (with finite number of traders) traces to
[14,15]. The w-MPS risk averse preferences assumed in
this paper is neither complete, nor transitive, thus falls
into the category studied in [14]. Also, in this paper we
adopt general population type space with arbitrary po-
pulation mass distribution functions which may accom-
modate the case of finite, countable infinite, as well as a
continuum range of investors.
It is noted that the existing proofs of equilibrium pro-
vided in these earlier studies cited above are all with re-
spect to the standard Warasian equilibrium in a context
of certainty. In contrast, the existing proof provided in
this paper is for a particular stochastic finance economy.
Specifically, we assume that the financial market can be
incomplete, and that the economy consists of an arbitrary
(finite, countable infinite or continuum) number of inves-
tors and each investor has (incomplete and non-transitive)
a preference order displaying w-MPS risk aversion. The
key assumption underlying the existing proof in our pa-
per is that when investor’s preference is restricted to the
market subspace of efficient portfolios, it admits a con-
tinuous utility representation, even though an utility rep-
resentation over the entire market span may not exist.
From technical point of view, our existing proofs are
very much in line with those in the literature ([16,17]).
[16] assumes a finite number of mean-variance investors
and a complete market, while [17] assumes a finite num-
ber of investors and that each investor has an well-de-
fined utility function defined on the market subspace
spanned by the risk free bond and a common risky asset
—the market portfolio. The literature contains several
other existing proofs for the equilibrium CAPM, all re-
lying crucially on the assumption of mean-variance pref-
erences (See, for instance, [18,19]).
The derivation of CAPM provided in this paper fol-
lows the heritage of the traditional approach ([5,6]). It is
based on the relevance of mean-variance efficient fron-
tier to investor’s optimal portfolio holdings. Following
the standard treatment in literature ([18,20]) we restrict
investors to hold portfolios involving in only a finite
number of securities even though the market may contain
an infinite number of tradable securities. The finiteness
assumption on the market portfolio is not restrictive since,
in practice, the market portfolio is an index of a finite
number of stocks. Derivative securities written on the
stocks and indeces of stocks, which represent a large or
even an infinite number of traded financial securities, are
not in the composite of the market portfolio. Our model
thus provides a useful mechanism to price, which pro-
vides not only those primitive securities in the composite
of the market portfolio, but also securities out of the
composite of the market portfolio.
The remainder of the paper is organized as follows:
Section 2 describes the model and summarizes the main
Open Access TEL
P. P. BOYLE, C. H. MA
308
results. Section 3 introduces the notion of generalized
efficient frontiers. Section 3 also discusses w-MPS risk
averse investor’s optimal portfolio choice problem and
its relevance to the generalized efficient frontier. Section
4 includes a formal derivation of the CAPM, along with
the two-fund separation theorem. The proofs of the exis-
tence of equilibrium are outlined in Section 5. Section 6
concludes the paper.
2. Outline of the Model and Main Results
This section describes the basic framework and summa-
rizes the main results of the paper. We start with several
useful notations. Let and
f
g
be two real-valued
functions (vectors). We write if
fg

f
tgt for
all ; if
tfg
 
f
tgt for all t and gf
; and
f
g if
 
f
tgt for all t.
We consider a two-period exchange economy with a
frictionless capital market and heterogenous agents. The
uncertainty is summarized by a probability state space
with probability measure . The topological
properties of the state space are otherwise not specified.
There exists a countable set of non-redundant risky secu-
rities available for trade4. Let
,
1,, ,Jj #J
j
be the number of tradable (risky) securities. Security
is associated with a state-contingent random payoff
. Security 0 is a risk free bond with unit
payoff in all future states (i.e., ). Let

,
2
j
0
1
j
p be the
price for security . The (total) return on security is
j j
thus given by
2,
j
j
j
Rp
 
(if ). The 0
j
p
return for the risk free bond is denoted
f
R.
Let #
J
 be a set of admissible portfolios on
J
5.
We restrict to consist of portfolios that involve only
a finite number of risky assets; that is,


#:sup: 0.
J
j
jJ

  (1)
Here,
j
represents the number of shares in risky
security . The market span consists of
all possible portfolio payoffs obtainable by trading:
j
2, 
.

2,: s.t.dd

  (2)
Following the standard literature, we may refer the
composite of a portfolio by its weight allocated to each
of the risky assets, as well as the number of shares in-
vested in the assets. Precisely, let be a copy of
.
Let be the initial wealth. Let
0
W
 be a portfolio
with
j
representing the proportion of initial wealth
invested in risky security . The number of shares in
j
security held is thus given by
j0j
j
W
p
, and the amount
invested in the risk free bond is
011W

, where
#
1
J
is the vector of unit elements. For each
,
the portfolio return is
j
f
j
jJ
RR R
 
f
R, with
expected portfolio return and standard deviation respec-
tively denoted by
and
.
There is a bunch of investors, indexed by tT
.
can be taken as a closed interval, say
T
0,1 , ohe Eucli-
dean space . Theopulation distribution is summa-
rized by a positive measure

 on such t for
each Borel set BT, population mass
belonging to B. Typt investr is endowed with an
vector of shares
n
p
t
hat,
T
e

B is th
o
e
et
. Investors express preferences
over . The typet investor’spreference is summa-
rized by a partial binary relation t
on
.
With these, the exchange economy is summarized by
 

,,;,,,
t
tT
Tet
  . (3)
Economy is finite if the state space contains
only finite elements; otherwise, it is infinite. For finite
economies, it is sensible to assume the number of non-
redundant securities is finite and no more than the num-
ber of states. Throughout this paper, we make the fol-
lowing assumptions for :
A1 0
0
and the payoffs for the risky securities
have a positive definite (infinite-dimensional) covariance
matrix
; that is, for all , , and x 00xx
00xx x

.
A2
satisfies Fellers property: Let be an open
set in , and
O
O be the closure of . For any real-val-
ued integrable function on ,
O
fTO
tyy ,f to be
continuous on ,
O
-a.s., implies

,d
T
yfyt
t
to be continuous on ; particularly, for all
O0
yO
,

d
00
lim,lim ,
yy yy
TT
d
f
ytf yt

t t
whenever the limit on the right hand side converges6.
A3 is :eT
-measurable with
0et
.
Moreover,

Tet
 
0dt
m

and
0
mm

.
A4 is :T
-measurable. And, for each ,
is strictly monotonic and homothetic, and displays
w-MPS risk aversion in the sense that
t
t
t
X
Y when-
ever
dist
YX
, where
0
and
Cov ,0X
,
and that the preference strictly holds if 0
.
4By non-redundancy, we mean that for each security , its payoff jj
cannot be duplicated by holding a finite portfolio
 of other
tradable securities.
5Much of the analysis below applies for a general admissible set such as
defined in Section 3, which may contain portfolios with an
infinite number of securities.

2
 
When trading takes place at time 0, after observing
security prices #
0,,,
J
j
pp p



investor
t
6The Feller’s condition is purely technical. It obviously holds true when
the economy consists of a finite number of investors. In this case,
corresponds to a measure of finite supports.
Open Access TEL
P. P. BOYLE, C. H. MA 309
may assess his initial wealth and revise
his portfolio holding accordingly.
 
0
Wt pet
 is feasible (for
) if . A feasible allocation
t

0
pWt

is
optimal (for ) if t
t

for all
 that is
feasible for . By strict monotonicity of the preference
relation, at the optimum, the budget constraint must hold
with equality; that is, .
t
0

0
Wtp

If security prices are all non-zero as must be the case
when
, the optimality condition can be expressed
in terms of portfolio weights


00Wt
,
allocated to each
risky securities. For instance, for , we have
 is optimal if, for all
t
RR
m
:T

 
d
Ttt
.
The market equilibrium for consists of a price
vector and an asset allocation that is
p
-measurable such that (i)

p
 
:T

; and (ii)
for all , is optimal w.r.t. for type t inves-
tors.
t

t
Again, for the case when equilibrium security prices
are all non-zero, and when investors’ initial wealth at
are all positive, the equilibrium condition can be equiva-
lently stated in terms of portfolio allocation
p
with condition (i) replaced by

d
0jm
j
Tj
tW
p
tt

for all , in addition to the optimality condition (in term
of
j
) for all type investors.
Let be the
market portfolio with
m

,
m
jj
m
pjJ
p

m
j
. Let be the return of the market
m
R
portfolio, and let
mm
R
. We state the first main
result of the paper:
Theorem 1 Suppose conditions A1-A4 hold. If the
economy achieves its equilibrium at (with p0
j
p
for all ) with equilibrium portfolio allocation j
, then
(1) there exists :T
d1t
, that is -measurable and
satisfies , such that
 
Tt

tm
t
for all ; tT
(2) the capital asset pricing model (CAPM) holds

,
f
R
fm
R
 



  (4)
where Cov
Var
,m
m
RR
R.
Theorem 1-(1) is about mutual fund separation of
equilibrium allocation—investors of all type optimally
hold a proportion of the market portfolio, in addition to a
position in the risk free bond. It extends the well known
Black's two-fund separation theorem [21] to this infinite
economy. The proof of the statement is based on insight
on the mathematical properties of the efficient frontier
covered in Section 3. The notion of efficient frontier,
direct extension to that of Markowitz ([22,23]) by con-
sidering the possibility of an infinite number of tradable
securities. Proof of the equilibrium CAPM outlined in
Theorem 1-(2) is summarized in Section 4.
To find an equilibrium price vector, we
which is formally introduced below in Definition 3.1, is a
work with a
set of “normalized prices” satisfying the CAPM, and the
term “normalized prices” will be made precise in Section
5. Let #
J
 be the set of normalized price vector
satisfyingAPM. We shall show that the set the C
forms a straight line on the vector space #
J
(Re: Lem-
mas 4 and 5), and that, for each p, iestor’s opti-
nv
expre
the market subspace spanned by
th
mal portfolio, if it exists, must bessed as a combi-
nation of the risk free bond and the market portfolio (Re:
Lemma 8).
Denote by0,m
nd and the risk free boe market portfolio. We write
:, .
m
xy xy
 
0,m(5)
Recall that, by assumption A3, m
is finite and
,
m
 is bounded. To prove e existence of
assume further that investor’s preference
restricted on 0,m
admits an utility representation7. Put
this formally
A5 For eact
equilibrium, we
th
h, the restriction of on ad-
m
t
0,m
nd,its a utility representation: 0,
:
tm
U. A the
utility function
,t
xyU x
m
y
 tinuous and
strictly quasi-co
Assumption A5 is logi
is con
nve. ca
cally consistent with the w-MPS
ris
ite and satisfies A1-A5.
Th
heorem 2 is contained in Section 5. For
th
e restriction ofon the market subspace
sp
we see that A5’ implies A5 when
se
3. MV Analysis and Portfolio Choice
-variance
k averse behavioral assumption (A4) since there exists
no w-MPS dominating relations for each bundles in
0,m
. The second main result of this paper concerns the
ence of equilibrium CAPM.
Theorem 2 Suppose is fin
exist
en, the equilibrium exists and, at equilibrium, the
CAPM holds.
The proof of T
e validity of Theorem 2, we may replace A5 with A5’
below:
A5’ Th t
anned by those efficient portfolios (if exists) admits a
utility representation.
In fact, by Lemma 8,
curity prices are governed by the CAPM.
This section starts with a discussion of mean
efficiency and generalized mean-variance efficiency as
an infinite dimensional extension to Markowitz mean-
variance efficiency. It follows with an in-depth discus-
sion on the relevance of the mean-variance efficiency to
7Since payoffs within the market subspace do not display mean pre-
serving spead to each other, assumption A5 is thus logically consistent
with assumption A4.
Open Access TEL
P. P. BOYLE, C. H. MA
310
optimal portfolio choice by w-MPS risk averse investors.
3.1. Efficient Frontier
price vector Taking as given an arbitrary#
J
p with
0
j
p for all j so that the asset returns are well-de-
r all tradle securities. Let #
fined foab
J
 be an ad-
missible portfolio space. For each
, the portfolio
return has its mean return and standaviation respec-
tively given by

rd de



1
2
1, ,
ff
RR
  
 
where
and represent the vector of expected re-
arturns a the viance-covariance matrix of the risky
returns. Similarly, for two arbitrary risky portfolios
nd
and
, the covariance of the two portfolio returns is
,
 

. For pure risky portfolios
we have
11.

The fowing definition of efficient portfolio and effi-
ci
For
oll
ent frontier for applies whether or not there is a
risk free asset.
Definition 10
, 0
is said to be effi-
cient at 0
if

0
:
0
arg min

 
. The
curve

 

,:is efficient
 

(6)
is referred to as the mean-variance efficient fronti
roy of
ef
t
er, or
simply the efficient frontier”, with respect to .
The next proposition describes a general ppert
ficient portfolios.
Proposition 1 Le 0
be an efficient portfolio
with mean 0
. For all
with
0,

we
have: RR
0
 with
0 and
Cov
0
0, .R
Proof. Consider the set of portfolios
nvex combina-

1:

  formed by co

0
tions of 0
and
. The
rn
se portfolios all have the same
mean retu given by 0
Since 0
is efficient at 0
with standard deviation0
,

0
1




m
achieve its minimum at 0;
us t
that is,

22
0argmin



22
0
21, 1
0



The first order condition leads to
2
00
,
 
. Let
0
RR
. We have:
0
and Co

0
v ,0R
.
Notice thf does nuire assumption
th
ite portfolios,
w
e that is relevant to choices made
by
at this prooot req on
e finiteness of the number of securities, nor requires the
covariance matrix to be non-singular.
Since investors are restricted to hold fin
e shall naturally pay special attention to the efficient
frontier

with respect to . In contrast to
the finitensional case orig considered by
Markowitz in [22,23], we do encounter one technical dif-
ficulty; that is, when the number of tradable securities is
infinite the efficient frontier

could be empty and
thus not well defined.
There is another cas
e diminally
w-MPS risk averse investors, and is of particular in-
terest. This refers to the efficient frontier for
2
.
Here,
#
2:
J

 
is an Hilbert space with inner product ,


and norm ,

. The portfolio space
forms
t not closed su
a dense bubset of

2. Since elements
in
2
are with possibly infimber of securi-
tiesficient frontier w.r.t.

2, denoted
nite nu
, the ef
g
, is
called the generalized efficient frg.e.f.).
Unlike
ontier (
, we shall show that
g
is well de-
fined under general conditions (see Proposition 3).
Similar to the finite dimensional case, an analytic ex-
pression for the g.e.f. can be obtained. As it turns out,
fairly
g
forms a hyperbola on the μ-σ plane in absence of
free asset. And, in presence of risk free asset, risk
g
is
well defined when the mean return for the minimuisk
portfolio differs from the risk free interest rate. For this
latter case,
m r
g
is composed of the generalized tangent
ray and the reflection of the generalized tangent ray. Ac-
cordingly, the classical Black-Tobin mutual fund separa-
tion theorems (see [21] and [24]) extend to this infinite
dimensional setting for
2.
3.2. Portfolio Choice
mber of a risky asset, it is well When there is a finite nu
known that the optimal portfolio for mean-variance in-
vestors, if it exists, must be located on the Markowitz
mean-variance efficient frontier. The existing literature
tells us little about the relevance of efficient frontier for
portfolio choices made by investors whose preferences
are not in the mean-variance class8. The difficulties in
establishing such relevancy are well known: First, an in-
vestor’s optimal portfolio may not exist even though the
efficient frontier is well defined. This occurs, for exam-
ple, when the security prices violate the no-arbitrage con-
ditions which are necessary for the existence of an opti-
mal portfolio for all investors with increasing and con-
tinuous utility functions. Second, even if optimal portfo-
lio exists, the efficient portfolio with mean return corre-
sponding to that of the optimal portfolio may not exist.
This occurs, for instance, when the mean return of the
minimum variance portfolio is equal to the risk free rate.
Finally, when the optimal portfolio and efficient portfolio
both exist, it is still not obvious if the investor would
choose to optimally hold the efficient portfolio because
8See [25,26] for results on two-fund separation for investors with risk
averse expected utility functions. They prove that, the separating port-
folios, if they exist, must be on the efficient frontier. See also [27] fo
r
conditions on asset returns and the expected utility functions that are
sufficient for both two-fund separation and the CAPM.
Open Access TEL
P. P. BOYLE, C. H. MA 311
the investor may care about higher moments beyond the
first two. In this section, we study the optimal choice
behavior for risk averse investors and explore the rele-
vance of the efficient portfolios in their optimal choices.
Here, we restrict our attention to investors who are risk
averse in the sense of w-MPS risk aversion.
Recalling first the definition of w-MPS risk aversion.
An investor is said to display w-MPS risk aversion if
X
Y whenever
dist
YX
, where
0
and
Cov X
, 0
. The rence hold0strict prefes if
. In
this case, we say that Y is identical in distribo a
w-MPS of
ution t
X
.
For w-MPS risk averse investors, our next result fol-
lows as a corollary to Proposition 1.
Proposition 2 Let
be an optimal portfolio holding
for a w-MPS risk ave investor. Let rse
be the portfo-
lio mean return for the optimal portfolio
. Then, if the
efficient portfolio at
exists, the optimal portfolio
must be efficient.
It is well known that, with a finite number of securities,
efficient frontiers are well-defined and all efficient port-
folios can be expressed as a convex combination of two
efficient portfolios. The optimal portfolio holdings for
w-MPS risk averse investors can be easily characterized
because they would have to be located on the efficient
frontier. When investors face an infinite number of in-
vestment opportunities

#J
, it is not clear if mu-
tual fund separation holdt, as to be illustrated
below, in presence of infinite number of risky assets, the
efficient frontier for is generically not defined (non-
existence), and the timal portfolio correspondence
(valued in ) for the w-MPS risk averse investors is
generically epty.
We first conside
s. In fac
po
m
r the case when the market contains
pu
d
rely risky portfolios. We impose the following two
conditions on the coefficients9:
C1 is positive definite, an
11
2
,1

;
C2 Nn-degeneracy: o
is not proportional to 1.
nd
e following im-
Suppose conditions C1 and C2 hold.
Fo
Under conditions C1 a C2, the g.e.f


 is well defined, and th
2g
portant risk decomposition result holds for the general-
ized portfolios:
Proposition 3
r all

2
, there exists a unique 0
g
such
that R
be expressed as a w-MPS of R0
. Moreover,
for a0
ll
, the generalized efficient portfolio at 0
is
given by


111
00
,,1eA
 

 2
 

(7)
where
 
1
,,
A
ee


is a positive definite
matrix.
22
Proof. The first statement follows the same argument
as in Proposition 1. Condition C2 implies is not
proportional to
1e
1
, which in turn implies
1111
,ee
 

 , and matrix
A
has a
well-defined inverse 1
A
. The quadratic optimization
problem

2
211
0
min:,1and,e



  
can be readily solved with the Lagrangean method, and
the optimal solution is given by Equation (7).
Notice that, the risk-decomposition theorem holds for
all generalized risky portfolios in , thus in par-
ticular for risky portfolios in . That is, re-
turn for each risky portfolio

2

2
in must admit as a
w-MPS to that of a generalized efficient portfolio
02
on
g
. This implies that w-MPS risk
averse investors would tend to hold generalized efficient
portfolios if they were allowed to hold an infinite number
of securities. In other words, investors would, in general,
not be satiated with holding any arbitrary finite number
of securities10. This results in (generic) non-existence of
optimal portfolio holdings for w-MPS risk averse inves-
tors who are restricted to hold finite portfolios in
.
Secondly, we consider the case when the market con-
tains a risk free bond, and maintain conditions C1 and C2
for the risky assets. Let
11
2
1
1,
1


11. If
f
R
,
then the generalized tangent portfolio would be well-
defined and be given by
11
2
11 1
1.
1, 1
f
m
f
R
R

 


 (8)
When m
contains an infinite number of non-zero
elements (that is, m
), all generalized efficient port-
folios (except the risk free bond) would not be admissible.
This causes the generic non-existence of optimal portfo-
lios for w-MPS risk averse investors who are restricted to
hold finite portfolios. For the extreme case when the
generalized tangent portfolio is finite (that is, m
),
10However, there are two exceptions to this statement: (a) when
#J
, the g.e.f. g
reduces to the Markowitz efficient frontier,
and all generalized efficient portfolios become efficient; (b) when
#J
and the (normalized) risky assets contain just a finite number
distinct expected returns.
11Here,
is the mean return for the generalized minimum variance
p
ortfolio
1
2
1
1
1
.
9We consider normalized securities that forms an orthonormal basis o
f
the market span . Each normalized security is a finite portfolio in
J
. The set of normalized securities are uncorrelated to each other, and
they generate the same market span from the original set of securi-
ties
J
. For normalized securities,
is diagonal with positive di-
agonal elements, its inverse is also well defined and diagonal.
1
Open Access TEL
P. P. BOYLE, C. H. MA
312
the tangent ray and its reflection constitute the efficient
frontier w.r.t. in presence of an risk free bond.
Based on this discussion, the optimal portfolio hold-
ings for w-MPS risk averse investors can be character-
ized.
Proposition 4 Suppose conditions C1 and C2 hold.
Consider an investor with monotonic and w-MPS risk
averse preference.
a) The investors optimal portfolio, if it exists, must
have an expected return no less than the risk free interest
rate,
f
R
.
b) When
f
R
, the optimal portfolio, if it exists,
must be on the generalized efficient rays and be ex-
pressed as a combination of the risk free bond and the
generalized tangent portfolio m
. In this case, the gen-
eralized tangent portfolio must be finite; that is, m
.
c) When
f
R
, the optimal portfolio, if not risk free,
does not exist.
If the investor were allowed to hold an infinite number
of securities, say , the optimal portfolio
would exist and be expressed as a combination of the risk
free bond and the generalized tangent portfolio. Since the
generalized tangent portfolio may not be finite, hence
does not belong to , the optimal demand correspon-
dence in for an w-MPS risk averse investor can be
empty.

2

4. Proof of Theorem 1
This section builds upon our earlier results to derive the
CAPM with w-MPS risk averse investors. Recall that
investors have homogeneous beliefs and so they will
perceive the same generalized efficient frontier as de-
scribed in the previous section. Further to the two-fund
separation property (Proposition 4-(b)), to prove the va-
lidity of the CAPM we shall first show that, in equilib-
rium, (a) the generalized tangent portfolio exists and be-
longs to , and (b) the generalized tangent portfolio
coincides with the market portfolio.
Lemma 1 Existence of equilibrium implies
f
R
.
Proof. Suppose to the contrary that the equilibrium
exists with
f
R
. By Proposition 4-(c), the optimal
portfolio would be either given by the risk free bond, or
not exist. Since all investors investing in the risk free
bond will necessarily violate the market clearing condi-
tions for the risky assets, we thus conclude that, the opti-
mal portfolios do not exist for at least one investor. The
latter contradicts the assumption on the existence of equili-
brium. Therefore, in equilibrium, we must have
f
R
.
Since
f
R
, in equilibrium the generalized tangent
portfolio m
is well defined. We can further identify the
generalized tangent portfolio to coincide with the market
portfolio (in equilibrium) following the standard separat-
ing portfolio arguments. So, we may state without proof
the following claim.
Lemma 2 Suppose economy has an equilibrium
that is supported by non-zero equilibrium security prices
(
0
j
p
for all jJ
). The equilibrium generalized
tangent portfolio must be finite ; and, in par-
ticular, it must be given by the market portfolio; that is,

m
m
m
12.
The following risk decomposition theorem is an infi-
nite-dimensional generalization to that of Huang and
Litzenberger [26], Chapter 3.18 & 3.19.
Lemma 3 Suppose
f
R
, the generalized tangent
portfolio m
is well defined; in particular, for all
2
it holds true that13

2
,m
m
ff
m
RRR R


(9)
where
has a zero mean and is uncorrelated with
; in particular,
m
R


2
,.
m
fm
m
RR

 
 
f
(10)
Lemmas 1, 2 and 3 together lead to the equilibrium
CAPM that is valid for all generalized portfolios, par-
ticularly for those admissible finite portfolios in
as a
subset of
2
. This concludes the validity to Theo-
rem 1-(2).
Since, in equilibrium, the generalized tangent portfolio
is given by the market portfolio which, by assumption
A3, is finite, the equilibrium efficient frontier for
with #J
(and with a risk less asset) is well-defined
and is given by the tangent ray and its reflection ray. This,
in turn, implies (by Proposition 4-(b)) the validity of two-
fund separation; that is, in equilibrium, each investors
optimally holds a portfolio that involves a combination
of the risk free bond and the market portfolio. This con-
firms the validity of Theorem 1-(1).
Notice further that the CAPM remains valid as an
equilibrium model if w-MPS risk averse investors are
allowed to hold an infinite number of securities, say port-
folios belonging to
2
. This is because, under con-
ditions C1 and C2, the efficient frontier w.r.t.
2
is
well-defined, and because w-MPS risk averse investors
would all invest in the risk-free asset and the efficient
tangent portfolio. Following the same argument as in
Lemma 3, the tangent portfolio must coincide with the
12We can further show that, if equilibrium exists with for all
00
i
W
i, then we must have mf
R
 and the market portfolio m
must be on the efficient ray.
13Notice that, when 0
j
p
for all , the risky returns have a non-
singular variance-covariance matrix as long as the covariance matrix
for the risky payoffs
j
is non-singular. This last assumption is to
ensure that none of the tradable securities are redundant and that
0
m
.
Open Access TEL
P. P. BOYLE, C. H. MA 313
market portfolio14. This would let us conclude the valid-
ity of the CAPM along with the two-fund separation for
.

2
#J
0
π0
5. Existence of Equilibrium
We now proceed with the proof of the existence of equi-
librium (Theorem 2). We assume finite economies with a
finite number of states and a finite number of assets
.

To find the equilibrium, we shall restrict security
prices to satisfy the CAPM.
Lemma 4 If equilibrium exists, then there exists
and such that, for all
1
π0x

01
ˆ
πwith πππ,
x
xc  (11)
where ˆ
c
mm
m





,
0
π10

and


1
π
10
mm
m








.
The CAPM pricing rule in Lemma 4 is well known
(see, for instance, [3]). Here, is a discount factor.
Therefore, given the random payoff for the market port-
folio, and given the expressions for 0 and 1, the
equilibrium pricing rule is fully determined by
π
π π
1
and m


, which are the prices of the risk free bond
and that of the market portfolio.
Lemma 5 If is an equilibrium discount factor,
then for all positive constant , is also an
equilibrium discount factor.
π
0kπk
By Lemmas 4 and 5, we can normalize the equilibrium
discount factor in (11) to be such that 01 and
write for some
ππ1

ππr
0,1r
,
, where

1r
ˆ
rrc
. Denote by the pricing rule
resulting from . Let
rx

rπ
,r
be the positive price
vector for the
J
risky assets.
To prove the existence of equilibrium, we need to
show that there exists an such that, given prices de-
termined by (11) with
r
rππ, the optimal portfolio
exists for each investor and satisfies the market clearing
conditions.
To ensure existence of an optimal portfolio, we shall
restrict
r
so that the pricing rule (11) satisfies the no-
arbitrage condition. By the fundamental theorem of fi-
nance (see [28]), the no-arbitrage condition is equivalent
to the existence of

2,vr

vr
such that
 
π0r
rv and is orthogonal to the market
span 15. For the case when , we can choose

π0r
0vr
.
Let be the subset of for which no-arbitrage
condition holds. We have
O

0,1
Lemma 6 The set
0O is an open convex subset
of
0,1 ; in particular, we can write
0,Or
for
some 01rO
 .
Proof. The proof proceeds in five steps. First, 0O
with
01π
and 0v
. Second, 1. This is be-
cause, with
O
ˆ
1cππ
 , the price for the market
portfolio is negative:

1, 0.
m

m


0,1

r
This
violates the no-arbitrage condition since, by assumption
A3, . Third, there exists s.t.
0
m
rO
.
Since is a finite,
m
takes finite possible values,
and has a finite
-norm, denoted m
. Since m
is risky, we have mm


. Let

0,1
m

m
r

mm m

 


 

 
0, m
rr . For all
we have

11
11 0;
mm
m
mm
m
rr
r








 








 




π
that is,
0rπ with
0vr
rO
. Fourth, we show that
is connected: Let
O
with and
 
π0rvr
vr
. For any
0,1a we have
av r
and

πa ar


0vrπar
ar O
1avr . So,
. Finally,
0O is open. For any 0
0rO
with
0r
00 and . For
sufficiently small, set
πrv

0
vr
0

0
vrvr for all
,r
00
rr
. We have
  
000 00
πrvr
ˆˆ
ππ.rvrrrcrvrc


Since
0
πrvr
0
is strictly positive and takes fi-
nite values, when is sufficiently small, the right hand
side, namely
00
ˆ
πrvrc
, must be strictly posi-
tive. This yields
π
0,Or
0rvr. Consequently, we can
write
for some
00
,1r.
We understand that, the no-arbitrage condition is vio-
lated at r
. For all rO
, let and

mr
mr
be
respectively the generalized tangent portfolio and the
market portfolio. For all #
J
x, let
diag
x
be the
diagonal matrix with jth diagonal element given by
j
x
.
Lemma 7 For all 0rO
, the generalized tangent
portfolio
mr
is well-defined and is given by







11
0
11
0
diag,,1,.
,,1
m
rr
rrr

 


 

 
14Here, of course, we must assume

2
m
.
,
r
r
(12)
15We see that is orthogonal to
2,v
2,i
f
0vd
for all . We write whenever is ortho-
gonal to . The set is the orthogonal complement of .
d
v
v
Open Access TEL
P. P. BOYLE, C. H. MA
314
Proof. Since all risky securities are with non-negative
payoffs, the no-arbitrage condition implies the price vec-
tor
,r
to be strictly positive. By assumption A1,
0 is positive definite, so the co-variance matrix
for
asset returns is well defined, and is positive definite. We
can further verify that, for all
0,rO
 
1,1
f
rrR
 
. This in turn implies, by Proposi-
tion 4, the existence of the efficient rays with a well de-
fined generalized tangent portfolio given by


1
2
11 1
1
1, 1
f
m
f
R
r
R
 


 in which
2
11 1
1,10
f
R
 

 for 0. The de- rO
sired expression for the generalized tangent portfolio is
valid because
 


 

 
11
0
diag ,diag ,
,diag ,1
,1
,1diag,.
f
rr
rrR
r
rr




 

 

 


We can further show that
Lemma 8 For all , the generalized tangent
portfolio is given by the market portfolio:
0rO
 
m
mrr

.
Proof. The market portfolio is
 

diag ,
,
m
m
m
r
rr



. Since both the generalized
tangent portfolio and the market portfolio have unit
length, and since diag

,r
is non-singular, it suf-
fices to show that m
is proportional to



1
0
,
,1
r
r





. In fact, with




0
,1
ˆ
,1 11
m
m
rrr
c
rr r

 
 


 we ob-
tain



1
0
,
1
,1
mm r
r
rr
 



as desired.
In the light of this observation, by the two-fund sepa-
ration theorem, all w-MPS risk averse investors will
choose to hold a combination of the market portfolio and
the risk free bond. By assumption A3, the market portfo-
lio is finite, hence the efficient frontier exists. Let
be the efficient frontier that is
composed of combinations of the risk free bond and the
market portfolio. Therefore, for all , the portfolio pay-
offs generated by efficient portfolios must coincide with
the market subspace 0,m
that is spanned by the risk
free bond and the market portfolio.
 
:
m
rara
r
By assumption A5, there exists a utility representation
on 0,m that is given by

,m
t
xyU xy
. With
initial wealth


,ret
0
Wr,t
, the shares in-
vested in the bond and the market portfolio can be ex-
pressed as

0,1
,1
Wrta
xr
and


0,
,m
Wrta
yr
,
where is the proportion (of the wealth) invested in
the market portfolio. Since, by assumption,
a
0>
te

,0t
,
the positivity of the pricing rule implies 0 for
all
Wr
rO
. By the homotheticity of the utility function
(Re: assumption A4), the optimal choice problem re-
duces to
max ,
at
Var, where
 

1
,,1 ,
m
tt m
aa
Var Urr





.
For any , assumptions A4 and A5 together imply
that
r
t
taV ar,, is
-measurable in and con-
tinuous in
t
a
. By the strict-quasi-concavity of
r,
t
aVa
O
r
, the optimal solution (if it exists) must be
unique. Since no arbitrage condition is satisfied for
(Re: Lemma 6), by the Fundamental Theorem of
Finance, the optimal solution exists (for rO
). Let
,r
0at be the optimal solution for
, where
,amaxrt V
O
rg
r
a
t
. Here, we restrict the number of shares
invested in the market portfolio to be non-negative.
By the measurable maximum theorem ([29], Theorem
14.91), we have
,tr
t to be -measurable.
Lemma 9 below shows that is continuous
on .
rt
,r
O
Lemma 9 For all rO
and , we have: tT
1.
0, 0.t
2. If converges to , then

1
nn
r
OrO
lim,.
nn
rt


3.
,:tO
 is continuous.
Proof. At 0r
the price of the market portfolio is
given by m
. The expected return of the market
portfolio is thus given by the risk free interest rate
10,1 1

. This implies that, for all , the portfolio
return
a


1
10,1 0,
m
m
aa

is a w-MPS of the
risk free return

1
0,1. Therefore, all risk averse
investors would optimally invest in the risk free bond
with zero position in the market portfolio. That is,
0, 0t
for all .t
To prove the second statement, let con-
verge to

1
nn
rO
rO
. Consider

1
,
nn
rt
. To show
lim ,
nn
rt

 , suppose, to the contrary, that


1
,
nn
rt
has a finite limit point given by . 0
Let

1
kk
n
be a convergent subsequence that con-
Open Access TEL
P. P. BOYLE, C. H. MA 315
verges to
,V
. We have: for all ,a
 

,
tntn n
Varrt r,
for all n, particularly holds
true for the subsequence

k
n1
k
k. Let , by con-
tinuity of , we have:

t
V

,
t
r V
r
,
t
r
Va , which
holds true for all . Therefore,
a
arg ,
at
Var

Gr
max

. This, however, contradicts
the emptiness of at . Therefore, we must
have .

lim ,
nn
rt


,:tO
To show that is continuous, it is suf-
ficient to show that, for all
1
nn
r

,
nn
rt
O converging to
, the resulting sequence converges to
rO
1
,rt
. Firstly, we show to constitute a


1
,
nn
rt
1
nn
a
,
bounded sequence. Suppose, without loss of generality,
that . Let

be an ar-

lim ,
nn
rt

0
bitrary sequence that converges to . For any ar-
rt
bitrary let
a
,
n
n
a
xrt
for all . For suf- n n
ficiently large, we have
0,1
n
x
. The sequence
 

1
n nn
xa xrt
1
nn
 , converges to
,rta
. By
the quasi-concavity of , and by the optimality of
we have
t
V
,
n
rt

,
tn
Va

,
n
1,
tnnnnn
axrr r
Vx t . Let , n
it yields


,, ,
t
Vrtar r


,
t
Vrt
,rt
. This contra-
dicts the unique optimality of for the given
.
rO
Now, let 0
be any finite limit point of

,
n
rt
1
n
, and let be the convergent subse-

1
kk
n
quence. We have, for all a
and for all ,k
 

,,
k
tntnn
VarVrtr

,
k
k
k
. Let , by continuity
of , we have
t
V
r,VraV tt ,
rOsup
, which holds for all
a. Also, for , achieves its maxi-

,Var

,
n
rt
t
a
mum uniquely at . Therefore, we conclude that

,rt
,rt

,rt

. This implies that has a uni-
1
n
que limit point ; or, equivalently,

,rt
,rt

limnn . This ends the proof of the third
statement.
As a necessary condition for the existence of a market
equilibrium, the market clearing condition for the risky
assets implies


dt
rO
,, ,
m
Tret r

 
;
rt (13)
that is, the aggregate investment in all risky assets equals
to the value of the market portfolio. We have,
Lemma 10 There exists 0
that solves Equa-
tion (13).
Proof. Let



,,d
,
,
mT
m
rt rett
rr
r

 O
. By
assumption A2 (the Feller’s property) and by Lemma 9,
we have :
mO
to be continuous and to satisfy (i)
0
m
0
and (ii) for all

m
nn
r
 
r
lim
O

r
1
nn converging to . By the Intermediate Va-
lue Theorem, there exists an such that
0,rr
1
mr
; or, equivalently,
,,d
Tret t

,
m
rrt


.
The validity of the main existence theorem, namely
Theorem 2, can be concluded with the proof of Lemma
11:
Lemma 11 There exists an such that

0,1r
πr
constitutes an equilibrium discount factor.
Proof. Let 0rO
be a solution to Equation (13).
Lemma 8, together with Proposition 4, implies that ’s
optimal portfolio
t
t
, for the given , is proportional
to the market portfolio. We write
r
 
,m
trt

r.
With
0,Wr t,t re

we have

 


0
1, ,d
,,,d
T
m
T
rt W rtt
rrtrett
 

0;
 
that is, the net borrowing in the risk free bond is zero.
Moreover, for each risky asset , we have
j

 




0,d
,
,, d;
,
j
T
j
mm
T
j
j
j
tW rtt
r
rt rettr
r



or, the total number of shares invested in equals to
the number of shares outstanding for the security. There-
fore, the pricing rule resulting from the discount factor
j
πr constitutes a market equilibrium. This concludes
the proof.
6. Concluding Remarks
In this paper, we prove that the CAPM holds for econo-
mies with w-MPS risk averse investors. The CAPM
model is shown to be valid without imposing any distri-
butional restriction on asset returns and the number of
tradable securities, and to be valid for economies with a
continuum type of investors. This approach is compared
to multi-factor models in the literature based on assump-
tions on the existence of some exogenous factors’ struc-
ture in modeling asset returns. Our results suggest that,
so long as investors exhibit w-MPS risk aversion, the
relevance of all those factors that affect asset returns
Open Access TEL
P. P. BOYLE, C. H. MA
Open Access TEL
316
vestors
en
REFERENCES
[1] M. Rothschildng Risk. I: A Defi-
/10.1016/0022-0531(70)90038-4
would all be summarized through the return of the mar-
ket portfolio. This is true, at least, in equilibrium.
It is implicitly assumed in our analysis that in
dowments lie in the market span. For cases with non-
spannable endowments as discussed in [17], the results
on the validity of the CAPM and the existing proofs are
still valid so long as the part of endowment, which is not
market spannable is orthogonal to the market span.
and J. Stiglitz, “Increasi
nition,” Journal of Economic Theory, Vol. 2, No. 3, 1970,
pp. 225-243.
http://dx.doi.org
. II: Its
022-0531(71)90034-2
[2] M. Rothschild and J. Stiglitz, “Increasing Risk
Economic Consequences,” Journal of Economic Theory,
Vol. 3, 1971, pp. 66-84.
http://dx.doi.org/10.1016/0
,”
Asset Pricing Theory,” Imperial Col-
[3] D. Duffie, “Security Markets: Stochastic Models Aca-
demic Press, 1988.
[4] C. Ma, “Advanced
lege Press, 2011. http://dx.doi.org/10.1142/p745
[5] W. Sharpe, “Capital Asset Prices: A Theory of Capita
ts and the Selec-
l
Market Equilibrium under Conditions of Risk,” Journal
of Finance, Vol. 19, 1964, pp. 425-442.
[6] J. Lintner, “The Valuation of Risky Asse
tion of Risky Investment in Stock Portfolios and Capital
Budgets,” Review of Economics and Statistics, Vol. 47,
No. 1, 1965, pp. 13-37.
http://dx.doi.org/10.2307/1924119
[7] J. Mossin, “Equilibrium in a Capital Asset Market,” Eco-
nometrica, Vol. 34, No. 4, 1966, pp. 768-783.
http://dx.doi.org/10.2307/1910098
[8] G. Chamberlain, “A Characterization of the Distribution
that Imply Mean-Variance Utility Functions,” Journal of
Economic Theory, Vol. 29, No. 1, 1983, pp. 185-201.
http://dx.doi.org/10.1016/0022-0531(83)90129-1
[9] J. Owen and R. Rabinovitch, “On the Class of Elliptical
Distributions and Their Applications to the Theory of
Portfolio Choice,” Journal of Finance, Vol. 38, No. 3,
1983, pp. 745-752.
http://dx.doi.org/10.1111/j.1540-6261.1983.tb02499.x
[10] A. Löffler, “Variance Aversion Implies μ-σ2-Criterion,”
Journal of Economic Theory, Vol. 69, No. 2, 1996, pp.
532-539. http://dx.doi.org/10.1006/jeth.1996.0067
[11] R. J. Aumann, “Markets with a Continuum of Traders,”
Econometrica, Vol. 32, No. 1-2, 1964, pp. 39-50.
http://dx.doi.org/10.2307/1913732
[12] R. J. Aumann, “Existence of Competitive Equilibria
4
in
Markets with a Continuum of Traders,” Econometrica,
Vol. 34, No. 1, 1966, pp. 1-17.
http://dx.doi.org/10.2307/190985
bria in Markets with a[13] D. Schmeidler, “Competitive Equili
Continuum of Traders and Incomplete Preferences,” Eco-
nometrica, Vol. 37, No. 4, 1969, pp. 578-585.
http://dx.doi.org/10.2307/1910435
[14] A. Mas-Colell, “An Equilibrium Existence Theorem with-
out Complete or Transitive Preferences,” Journal of Ma-
thematical Economics, Vol. 1, No. 3, 1974, pp. 237-246.
http://dx.doi.org/10.1016/0304-4068(74)90015-9
[15] D. Gale and A. Mas-Colell, “An Equilibrium Existence
016/0304-4068(75)90009-9
Theorem for a General Model without Ordered Prefer-
ences,” Journal of Mathematical Economics, Vol. 2, No.
1, 1975, pp. 9-15.
http://dx.doi.org/10.1
of [16] R. A. Dana, “Existence, Uniqueness and Determinacy
Equilibrium in CAPM with a Riskless Asset,” Journal of
Mathematical Economics, Vol. 32, No. 2, 1999, pp. 167-
175. http://dx.doi.org/10.1016/S0304-4068(98)00050-0
[17] C. Hara, “Equilibrium Prices of the Market Portfolio in
Jour-
the CAPM with Incomplete Financial Markets,” Working
Paper, University of Cambridge, 2001.
[18] L. Nielsen, “Existence of Equilibrium in CAPM,”
nal of Economic Theory, Vol. 52, No. 1, 1990, pp. 223-
231. http://dx.doi.org/10.1016/0022-0531(90)90076-V
[19] N. Sun and Z. Yang, “Existence of Equilibrium and
rbitrage, Factor
Zero-Beta Pricing Formula in the Capital Asset Pricing
Model with Heterogeneous Beliefs,” Annals of Econom-
ics and Finance, Vol. 4, 2003, pp. 51-71.
[20] G. Chamberlain and M. Rothschild, “A
Structure, and Mean-Variance Analysis on Large Asset
Markets,” Econometrica, Vol. 50, 1983, pp. 1281-1304.
http://dx.doi.org/10.2307/1912275
[21] F. Black, “Capital Market Equilibrium with Restricted
Borrowing,” Journal of Business, Vol. 45, No. 3, 1972,
pp. 444-455. http://dx.doi.org/10.1086/295472
[22] H. Markowitz, “Portfolio Selection,” Journal of Finance,
on,” John Wiley and Sons,
reference and Behavior towards
Vol. 7, 1952, pp. 77-99.
[23] H. Markowitz, “Portfolio Selecti
Inc., New York, 1959.
[24] J. Tobin, “Liquidity P
Risk,” Review of Economic Studies, Vol. 25, 1958, pp.
65-86. http://dx.doi.org/10.2307/2296205
[25] S. Ross, “Mutual Fund Separation in Financial Theory—
The Separating Distributions,” Journal of Economic The-
ory, Vol. 17, 1978, pp. 254-286.
http://dx.doi.org/10.1016/0022-0531(78)90073-X
s for the
[26] C. F. Huang and R. Litzenberger, “Foundations for Fi-
nancial Economics,” Prentice Hall, Inc., 1988.
[27] J. Berk, “Necessary and Sufficient Condition
CAPM,” Journal of Economic Theory, Vol. 73, 1997, pp.
245-257. http://dx.doi.org/10.1006/jeth.1996.2218
[28] P. Dybvig and S. Ross, “Arbitrage,” In: The New Pal-
nd K. Border, “Infinite Dimensional
04-2
grave: A Dictionary of Economics, The MacMillan Press
Limited, 1987.
[29] C. Aliprantis a
Analysis,” Springer-Verlag, 1994.
http://dx.doi.org/10.1007/978-3-662-030