# A Theory of Market Equilibrium Under Conditions of Risks

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```					  Introduction to Portfolio
Selection and Capital Market
Theory: Static Analysis
BaoheWang
baohewang0592@sina.com
Introduction
   The investment decision by households as
having two parts:
(a) the “consumption-saving” choice
(b) the “portfolio-selection” choice
   In general the two decisions cannot be
   However, the consumption-saving
allocation has little substantive impact on
portfolio theory.
One-period Portfolio Selection
   The solution to the general problem of
choosing the best investment mix is called
portfolio-selection theory.
   There are n different investment
opportunities called securities.
   The random variable one-period return
per dollar on security j is denoted Z j
   Any linear combination of these securities
which has a positive market value is called
a portfolio.
   U (W ) denote the utility function.
    W is the end-of-period value of the
investor’s wealth measure in dollars.
    U is an increasing strictly concave
function and twice continuously
differentiable.
   So the investor’s decision is relevant to
the subjective joint probability distribution
for (Z1 , Z 2 , , Z n ).
   Assumption 1: Frictionless Markets

   Assumption 2: Price-Taker

   Assumption 3: No-Arbitrage Opportunities

   Assumption 4: No-Institutional Restrictions
   Given these assumptions, the portfolio-
selection problem can be formally stated
as
n
max E{U ( w j Z jW0 )}
{ w1 , w2 , wn }
1

(2.1)
n
S. T .          w
1
j   1

   Where E is the expectation operator for
the subjective joint probability
distribution.
   If (w1 , w2 , , wn ) is a solution (2.1), then it will
      

satisfy the first-order conditions:

E{U ( Z W0 Z j )} 


W0
   Where Z   1n wj Z j is the random variable
return per dollar on the optimal portfolio.
   With the concavity assumptions on U, if
the variance-covariance matrix of the
return is nonsingular and an interior
solution exists, the the solution is unique.
   Formula (2.1) rules out that any one of
the securities is a riskless security.
of available securities then the portfolio
selection problem can be stated as:
n
max E{U ( w j Z jW0  (1  1 w j ) RW0 )}
n

{ w1 , w2 , wn }
1
(2.4)
n
max E{U ([ w j ( Z j  R)  R]W0 )}
{ w1 , w2 , wn }
1
   The first-order conditions can be written
as:
E{U ( Z W0 )( Z j  R)}  0   j  1, 2,   ,n

Where Z can be rewritten as 1 wj (Z j  R)  R
n



   If it is assumed that the variance-
covariance matrix of the returns on the
risky securities is nonsingular and an
interior solution exits, then the solution is
unique.
   But neither (2.1) nor (2.3) reflect that end of
period wealth cannot be negative.
   To rule out bankruptcy, the additional
constraint that, with probability one, Z   0
   
could be imposed on (w1 , w2 , , wn ) .
*

   This constraint is too weak, because the
probability assessments on {Z j } are
subjective.
   An alternative treatment is to forbid
Z and
borrowingj  0 short-selling securities where,
by law,         .
   The optimal demand functions for risky
securities, {wjW0 } , and the resulting
probability distribution for the optimal
portfolio will depend on
(1) the risk preferences of the investor;
(2) his initial wealth;
(3) the join distribution for the securities’
returns.
   The von Neumann-Morgenstern utility
function can only be determined up to a
positive affine transformation.
   The Pratt-Arrow absolute risk-aversion
function is invariant to any positive affine
transformation of U (W ) .
   The preference orderings of all choices
available to the investor are completely
specified by absolute risk–aversion
function
U (W )
A(W ) 
U (W )
   The change in absolute risk aversion with
respect to a change in wealth is

dA                           U (W )
 A(W )  A(W )[ A(W )            ]
dW                           U (W )
   A(W ) ispositive, and such investor are call
risk averse.
   An alternative, measure of risk aversion is
the relative risk-aversion function defined
by
U (W )W
R(W )              A(W )W
U (W )
   Its change with respect to a change in
wealth is given by
R(W )  A(W )W  A(W )
   The certainty-equivalent end-of-period
wealth WC is defined to be such that
U (WC )  E{U (W )}
    WC is the amount of money such that the
investor is indifferent between having this
amount of money for certain or the
portfolio with random variable           W
outcome .
   We can proofU  follows directly by Jensen’s
U if is {U (W )}  U ( E{W
inequality:(W )  Estrictly concave})
C

                   increase function, So
Because U is an WC  E{W }
   The certainty equivalent can be used to
compare the risk aversions of two
investor.
   If A is more risk averse than B and they
hold same portfolio, the certainty
equivalent end of period wealth for A is
less than or equal to the certainty
equivalent end of period wealth for B.
   Rothschild and Stiglitz define the meaning
of “increasing risk” for a security so we
can compare the riskiness of two
securities or portfolios.
   If E (W1 )  E (W2 ) , E{U (W1 )}  E{U (W2 )} for
all concave U with strict inequality holding
for some concave U , we said the first
portfolio is less risky than the second
portfolio.
   Its equivalence to the two following
definitions:
(1) W2 is equal in distribution to W1 plus some
“noise”.
(2) W2 has more “weight in its tails” than W1 .
 If there exists an increasing strictly
concave function V such that
E{V (Z )(Z j  R)}  0, j  1, 2, , n., we call this
portfolio is an efficient portfolio.
 All portfolios that are not efficient are
called inefficient portfolios.
   It follows immediately that every efficient
portfolio is a possible optimal portfolio, for
each efficient portfolio there exists an
increasing concave U such that the
efficient portfolio is a solution to (2.1) or
(2.3).
   Because all risk-averse investors have
different utility function, so they will be
indifferent between selecting their optimal
portfolios.
 Theorem 2.1: If Z denotes the random
variable return per dollar on any feasible
portfolio and if Ze  Ze is riskier than Z  Z
in the Rothschild and Stiglitz sense, then
Ze  Z ( Z e is an efficient portfolio)
Proof: By hypothesis
E{U[(Z  Z )W0 ]}  E{[(Ze  Ze )W0 ]}
If Z  Ze then           E{U ( ZW0 )}  E{U ( Z eW0 )}
trivially
Z                            .      Ze
But       is a feasible portfolio and Ze is Z     an
   Corollary 2.1: If there exists a riskless
security with return R, then Ze  R , with
equality holding only if Z e is a riskless
security.
   Proof: If Z e is riskless , then by
Assumption 3, Ze  R . If Z e is not riskless,
by Theorem 2.1, Ze  R .
   Theorem 2.2: The optimal portfolio for a
nonsatiated risk-averse investor will be the
riskless security if and only if Z j  R for
j=1,2,…..,n.
   Proof:  If Z   R is an optimal solution,
then we have U ( RW0 ) E{Z j  R}  0 By the
nonsatiation assumption, U ( RW0 )  0 so Z j  R
   If Z j  R j  1, 2 , n then Z   R will
satisfy U (Z W0 ) E{Z j  R}  0 because the
property of U, so this solution is unique.
   From Corollary 2.1 and Theorem 2.2, if a
risk-averse investor chooses a risky
portfolio, then the expected return on the
portfolio exceeds the riskless rate.
 Theorem 2.3: Let Z p denote the return on
any portfolio p that does not contain
security s. If there exists a portfolio p such
that, for security s, Zs  Z p   s , where
E{ s | Z j , j  1, 2, , n, j  s}  0 then the fraction
of every efficient portfolio allocated to
security s is the same and equal to zero.
Proof: Suppose Z e is the return on an
efficient portfolio with fraction  s  0
allocated to security s, Z be the return on
a portfolio with the same fractional
holding as Z e except that instead of
security s with portfolio P
Hence Ze  Z   s (Zs  Z p )  Z   s s
So Ze  Z
Therefore ,for  s  0 , Z e is riskier than
Z in the Rothschild-Stiglitz. This
contradicts that Z e is an efficient portfolio.
   Corollary 2.3: Let  denote the set of n
securities and   denote the same set of
securities except that Z s is replace with Z s.
If Z s  Z s   s and E{ s | Z}  0 , then all risk
averse investor would prefer to                        
choose           .
   Theorem 2.3 and its corollary demonstrate
that all risk averse investors would prefer
any “unnecessary” and “noise” to be
eliminated.
   The Rothschild-Stiglitz definition of
increasing risk is quite useful for studying
the properties of optimal portfolios.
   But this rule is not apply to individual
securities or inefficient portfolios.
2.3 Risk Measures for Securities and
Portfolios in The One-Period model
   In this section, a second definition of
increasing risk is introduced.
    Ze is the random variable return per dollar
k

on an efficient portfolio K.
    VK (ZeK ) denote an increasing strictly
concave function such that for VK      dVK
dZ Ke


E{VK (Z j  R)}  0   j  1, 2,   ,n   W0  1

VK  E{V }
Random variable         YK 
                                     
cov(VK , Z eK )
   Definition: The measure of risk bp of
K

portfolio P relative to efficient portfolio K
with random variable return ZeK is defined
by
b  cov(YK , Z P )
K
p
and portfolio P is said to be riskier than
portfolio P  relative to efficient portfolio K
if bp  bp .
K    K
 Theorem 2.4: If Z p is the return on a feasible
portfolio P and Ze is the return on efficient
K

portfolio K , then Z p  R  bp (ZeK  R) .
K

Proof: From the definition

E{VK (Z j  R)}  0               j  1, 2,   ,n
 j be the fraction of portfolio P allocated to
security j, then
n
Z P    j ( Z j  R)  R
and                    1

n

  E{V  (Z
1
j   K       j

 R)}  E{VK ( Z P  R)}  0

By a similar argument, E{VK (ZeK  R)}  0
Hence,
cov(VK                
 , Z eK )  E[VK ( Z eK  Z eK )]

 E[VK ( Z  R  R  Z )]
K
e                e
K


 ( Z eK  R)]  E[VK ( R  Z eK )]
 E[VK

 ( R  Z ) E[VK ]
e
K

and
                          
cov(VK , Z P )  ( R  Z P ) E{VK }
By Corollary 2.1 , Z  R . Therefore
K
e

Z p  R  b ( Z  R)  K
p   e
K
   Hence, the expected excess return on
portfolio P, Z P  R is in direct proportion
to its risk and the larger is its risk , the
larger is its expected return.
   Consider an investor with utility function U
and initial wealth W0 who solves the
portfolio-selection problem:
max E{U ([ wZ j  (1  w) Z ]W0 )}
w
   The first order condition:
E{U ([ w*Z j  (1  w* ) Z ]W0 )(Z j  Z )}
   If Z  Z * then the solution is W *  0 .
   However , an optimal portfolio is an
efficient portfolio. By Theorem 2.4
Z j  R  b ( Z  R)
*
j
*

   So w*W is similar to an excess demand
function . jb * Measures the contribution of
security j to the Rothsechild-Stiglitz risk of
the optimal portfolio.
   By the implicit function theorem, we have:

w *
w W0 E{U ( Z  Z j )}  E{U }
*


Z j       W0 E{U   ( Z  Z j ) 2 }

   Therefore , if Z j lies above the risk-return
line in the ( Z , b ) plane, then the investor
would prefer to increase his holding in
security j.
   bp is a natural measure of risk for
K

individual securities.
   The ordering of securities by their
systematic risk relative to a given efficient
portfolio will be identical with their
ordering relative to any other efficient
portfolio.
Lemma 2.1:

(i) E{Z P | VK }  E{Z P | Ze } for efficient
K

portfolio K.
(ii) If                                     
E{Z P | Z eK }  Z p then cov(Z p ,VK )  0

(iii) cov(Z p ,VK )  0 for efficient portfolio K if
and only if cov( Z PVL )  0 for every efficient
portfolio L.
Proof: (i) VK is a continuous monotonic
function of Ze and hence 
K
VK and ZeK are
in one to one correspondence.
(ii) cov(Z p ,VK )  E{VK (Z p  Z P )}  E{VK E{Z p  Z P | Z eK }}  0
(iii)Because bp  0  cov( Z p ,VK )  0
K

 if     bp  0 , then Z  R .
K
p

Property I: If L and K are efficient portfolios,
then for any portfolio p, bp  bL bp .
K    K L

Proof : From Theorem 2.4

Z R   L
Zp  R                 Zp  R
b 
K       e
b 
K
b 
L

Z R                       Z R                   Z eL  R
L        K             p        K             p
e                      e
   Property 2: If L and K are efficient
portfolios, then bK  1 and bK  0 .
K           L

   Hence, all efficient portfolios have positive
systematic risk, relative to any efficient
portfolio.
   Property 3: Z p  R if and only if bp  0 for
K

every efficient portfolio K.
   Property 4: Let p and q denote any two
feasible portfolios and let K and L denote
K  K
any two efficient portfolios. bp bq if
and only if bp  bqL
L                   

   Proof: From Property 1, we have
b b b
K
p
K
L
L
p    b b b
K
q
K L
L q

   Thus the b measure provides the same
K
p
orderings of risk for any reference efficient
portfolio.
   Property 5: For each efficient portfolio K
and any feasible portfolio p, Z p  R  bpK (ZeK  R)   p
where E{ p }  0 and  E{ pVL ( ZeL )}  0 for
every efficient portfolio L.
   Proof: From Theorem 2.4 E{ p }  0 . If
portfolio q is constructed by holding one
dollar p,     bp dollars riskless security, short
K

selling bp dollars portfolio K, then Zq  R   p
K

so bqL  0 for every efficient portfolio L.
But   bqL  0 implies 0  cov(Zq ,VL )  E{ p ,VL }
            
for every efficient portfolio L.

   Property 6: If a feasible portfolio p has
portfolio weight (1 , ,  n ) ,then bp  1  j b jK
K    n
   Hence , the systematic risk of a portfolio is
the weighted sum of the systematic risks
of its component securities.
   The Rothschild Stiglitz measure provides
only for a partial ordering.
    bp measure provides a complete
K

ordering.
   They can give different rankings.
   The Rothschild Stiglitz definition measure
the “total risk” of a security. It is
appropriate definition for identifying
optimal portfolios and determining the
efficient portfolio set.
   The b measure the “ systematic risk” of a
K
j
security.
   To determine the b j , the efficient
K

portfolio set must be determined.
   The manifest behavioral characteristic
shared by all risk averse utility
maximization is to diversify.
   The greatest benefits in risk reduction
come from adding a security to the
portfolio whose realized return tends to be
higher when the return on the rest of the
portfolio is lower.
   Next to such “ countercyclical” investments
in terms of benefit are the noncyclic
securities whose returns are orthogonal to
the return on the portfolio.
   Theorem 2.5 : If Z p and Z q denote the
returns on portfolio p and q respectively
and if, for each possible value of Z e ,
dGp ( Ze )
dZe

dGq ( Ze )
dZe
with strict inequality
holding over some finite probability
measure of Z e ,then portfolio p is riskier
than portfolio q and Z p  Z q .
Where Gp (Ze )  E{Z p | Ze } , Z e is the
realized return on an efficient portfolio.
   Proof:

bp  bq  cov[Y ( Z e ), Z p  Z q ]  E[Y ( Z e )( Z p  Z q )]
 E[Y ( Z e )( E{Z p | Z e }  E{Z q | Z e })]
 E[Y ( Z e )(Ge ( Z p )  Ge ( Z q ))
 cov[Y ( Z e ), Ge ( Z p )  Ge ( Z q )]

is a strictly increasing function, Ge (Z p )  Ge (Zq )
Y (Ze )
is a nondecreasing function, so
bp  bq  cov[Y (Ze ), Ge (Z p )  Ge (Zq )]  0
From Theorem 2.4 Z p  Z q
 Theorem 2.6: If Z p and Z q denote the
returns on portfolio p and q respectively
and if, for each possible value of Z e ,
 a pq , a constant, then
dGp ( Z e )       dGq (Z e )

dZe              dZ e
bp  bq  a pq and Z  Z  a ( Z  R) .
p      q    pq e
Proof: By hypothesis
Ge (Z p )  Ge (Zq )  a pq  h
bp  bq  cov[Y ( Z e ), Ge ( Z p )  Ge ( Z q )]
 cov[Y ( Z e ), a pq Z e  h]  a pq
Z p  R  bp ( Z e  R)  R  bq (Z e  R)  a pq (Z e  R)  Z q  a pq (Z e  R)
  Theorem 2.7: If, for all possible values of Z e
(i)dG (Z )  1 , then Z p  Z e
p     e
dZe

(II)   0
dGp ( Ze )
dZe
1    , then R  Z p  Z e

(III)      dG p ( Z e )
dZe
0     , then    R  Zp

dG p ( Z e )
(IV)                      dZe
 ap    , a constant, then
Z p  R  a p ( Z e  R)
   Theorems 2.5, 2.6 and 2.7 demonstrate,
the conditional expected return function
security’s risk and equilibrium expected
return.
2.4 Spanning, Separation, and
Mutual-Fund Theorems
   Definition: A set of M feasible portfolios
with random variable returns ( X 1 , X M )
is said to span the space of portfolios
contained in the set  if and only if for
any portfolio in  with return denoted by Z p
there exist numbers (1 ,  M ) , 1  i  1
M

such that Z p  1M  j X j
   A mutual fund is a financial intermediary
that holds as its assets a portfolio of
securities and issues as liabilities shares
against this collection of assets.
   Theorem 2.8 If there exist M mutual funds
whose portfolio span the portfolio set  ,
then all investors will be indifferent
between selecting their optimal portfolios
from  and selecting from portfolio
combination of just the M mutual funds.
   Therefore the smallest number of such

funds M is a particularly important
spanning set.
   When such spanning obtain, the investor’s
portfolio-selection problem can be
separated into two steps.
   However, if the smallest funds can be
constructed only if the fund managers
know the preferences, endowments, and
probability beliefs of each investor.
Theorem 2.9: Necessary conditions for the
M feasible portfolios with return ( X1 , , X M )
to span the portfolio set  are (a) that
f

the rank of   M and (b) that there exist
numbers   (1 , ,  M ), 1  j  1 such that the
M

random variable 1  j X j has zero variance.
M

Proposition 2.1: If Z p  1 a j Z j  b is the
n

return on some security or portfolio and if
there are no “ arbitrage opportunities”
then
(a) b  (1  1 a j ) R and (b) Z p  R  1 a j ( Z j  R)
n                               n
   Proof: Let Z  be the return on a portfolio
with fraction  j allocated to security j, j  1,
                                 , n;

 p allocated to the security with                 Zp
return  p  1  j
1  ; and               allocated to the
n 

j 
riskless security with return R, if is
chosen          a j
 j suchpthat   Z  R   p,then  1 a j )]

[b  R(1
n

is riskless security and therefore   R
Z
but       can be chosen arbitrarily. So we get
the result.
   Hence, as long as there are no arbitrage
opportunities, it can be assumed without
loss of generality that one of the portfolios
in any candidate spanning set is the
riskless security.
   Theorem 2.10: A necessary and sufficient
condition for ( X1 , , X m , R) to span is that
f

there exist number {aij } such that
Z j  R  1 aij ( X i  R) j  1, 2,
m
, n.
   Proof:  If ( X1 ,   , X m , R) span  f , then
1  ij  1 such that Z j  1 ij X i . Because
M                           M

X M  R and substituting  Mj  1  1  ij , we
m

have Z j  R  1 aij ( X i  R) j  1, 2, , n.
m

 we pick the portfolio weights ij  aij
for  i  1, , m and  Mj  1  1m  ij , from
which it follows that Z j  1  ij X i .But every
M

portfolio in can be written as a portfolio
f

combination of ( Z1 , , Z n ) and R.
   Corollary 2.10: A necessary and sufficient
condition for ( X1 , , X m , R) to be the smallest
number of feasible portfolio that span is
that the rank of  equals the rank of  X  m
   Proof:  If the rank of  X  m , then X
are linearly independent. Moreover
hence, if the rank of   m then there
exist number {aij }such that Z j  Z j  1 aij ( X i  X i )
m

for j  1, , n . Therefore Z j  b j   aij X i
m
1

where b j  Z j  1 aij X i by Theorem 2.10
m

span  f
   It follows from Corollary 2.10 that a
necessary and sufficient condition for
nontrivial spanning of         f is that some of
the risky securities are redundant
securities.
   By Theorem 2.10, if investors agree on a
set of portfolios ( X1 , , X m , R) such that
Z j  R  1 aij ( X i  R) j  1, 2, , n. and if they
m

agree on the number {a } ,then ( X1 , , X m , R)
ij

span  f even if investors do not agree on
the joint distribution of ( X1 , , X m , R)
   Proposition 2.2: If Z e is the return on a
portfolio contained in  e , then any
portfolio that combines positive amount of Z e
with the riskless security is also contained
in   e , where  e is the set of all efficient
portfolios contained in     f .
   Proof: Let Z   Z e  (1   ) R , because Z e is
an efficient portfolio, so E{V (Ze )(Z j  R)}  0
Define U (W )  V (aW  b) where a  1 and
b  (  1)R

, Hence E{U (Z )(Z j  R)}  0 ,
thus Z is an efficient portfolio.
   It follows from Proposition 2.2 that, for
every number Z such that Z  R , there
exists at least one efficient portfolio with
expected return equal to Z .
   Theorem 2.11: Let ( X 1 , , X m ) denote the
return on m feasible portfolios. If, for
security j, there exist number {aij } such that
Z j  Z j  1 aij ( X i  X i )   j where E{ jVK (ZeK )}  0
m


for some efficient portfolio K, then
Z j  R  1 aij ( X i  R)
m
Proof: Let Z p   Z j  1  i X i  (1    1  i )R
m                m


because Z j  Z j  1 aij ( X i  X i )   j , thus
m

Z p  R   [ Z j  R  1 aij ( X i  R)]   j by
m

construction , E{ j }  0 and hence cov(Z p ,VK )  0
Therefore the systematic risk of portfolio
p, bp is zero. From Theorem 2.4 Z p  R
K

therefore Z j  R   m aij ( X i  R)
1
   Hence, if the return on a security can be
written in this linear form relative to the
portfolios ( X1 , , X m ) , then its expected
excess return is completely determined by
the expected excess returns on these
portfolios and the weights {aij } .
   Theorem 1.12: If, for every security j,
there exist numbers {aij } such that
Z j  R  1 aij ( X i  R)   j
m

where E{ j | X1, , X m}  0 , then ( X1 , , X m , R)
span the set of efficient portfolios      e .
   Proof:
Z  1 w j Z j  1 w j [ R  1 aij ( X i  R )   j ]
K       n        K               n       K           m
e

 1 w j R  1 1 w j aij ( X i  R )  1 w K  j
n   K            n       m       K                         m
j

 R  1  iK ( X i  R )   K
m

Where   1 wK aij                               K  1 wK  j
K           n                           m
j i                                            j

Construct portfolio Z  1m  iK X i  (1  1m  iK ) R
Thus Ze  Z   where E{ | Z }  0
K        K           K

Hence, for   0 , Ze is riskier than Z,
K        K

which contradicts that ZeK is and efficient
portfolio. So  K  0 . We get the result.
                    w K denote the fraction
Theorem 2.13: Let j
of efficient portfolio K allocation to
security j, j  1, , n. ( X1 , , X m , R) span      e if
and only if there exist number {aij } for every
security j such that Z j  R  1 aij ( X i  R)   j
m

where E{ j | 1  i X i }  0,  i  1 w j aij for
m K              K     n K

every efficient portfolio K.
   Corollary 2.13: (X,R) span  e if and only if
there exist a number a j for each security j,
j  1, , n, such that Z j  R  a j ( X  R)   j
where E{ j | X }  0
   Proof: By hypothesis,                        for
ZeK   K ( X  R)  R
every efficient portfolio K. If X  R , then
from Corollary 2.1   0 for every
K

efficient portfolio K and R span         e.
Otherwise, from Theorem 2.2,  K  0 for
every efficient portfolio. By Theorem 2.13,
E{ j |  K X }  0 so E{ j | X }  0
   Since   e is contained in  f , any properties
proved for portfolios that span  e must be
properties of portfolio that span  f .
   From Theorem 2.10, 2.12, 2.13, the
essential difference is that to span the
efficient portfolio set it is not necessary
that linear combinations of the spanning
portfolios exactly replicate the return on
each available security.
   All the models that do not restrict the
class of admissible utility function, the
distribution of individual security returns
must be such that
Z j  R  1 aij ( X i  R)   j
m
   Proposition 2.3: If, for every security j,
E{ j | X1 , , X m}  0 with ( X 1 , , X m ) linearly
independent with finite variances and if
the return on security j, Z j has a finite
variance, then the {aij } i  1, , m, in
Theorems 2.12 and 2.13 are given by
aij  1 vik cov( X K , Z j ) where vik is the ikth
m

element of X       1 .
   Hence given some knowledge of the joint
distribution of a set of portfolio that span 
e

with Z j  Z j , we can determining the aijand Z j
   Proposition 2.4: If (Z1 , , Z n ) contain no
redundant securities,  j denotes the
fraction of portfolio X allocated to security

j, and w j denotes the fraction of any risk-
averse investor’s optimal portfolio
allocated to security j, j  1, , n, then for
every such risk-averse investor
w
j
          j, k  1, 2,
j
,n
w k
*
k
   Because every optimal portfolio is an
efficient portfolio and the holding of risky
securities in every efficient portfolio are
proportional to the holding in X.
If there exist numbers  j where   , j, k  1,
            
*


j
 *
j
,n
k       k

and 1  j ,then the portfolio with
n *

proportions (1 ,  n ) is called the Optimal
*    *

Combination of Risky Assets.
   Proposition 2.5: If ( X , R) span  e , then  e
is a convex set.
   Proof: Let     Ze  1 ( X  R)  R
1
Ze2   2 ( X  R)  R
and     1   2 , Z   Ze  (1   )Ze2 . By
1

substitution, the expression for Z can be
rewritten as         Z   (Ze  R)  R , where
1

    ( 2  )(1   ) .Therefore by Proposition
2.2, Z is an efficient portfolio. It follow by
1

induction that for any integer k and
number i such that 0  i  1, i  1, , k and
1  i  1, Z  1 i Z ei is the return on an
k            k       k

efficient portfolio. Hence ,            e is a convex
set.
   Definition: A market portfolio is defined as
a portfolio that holds all available
securities in proportion to their market
values.
   The equilibrium market value of a security
for this purpose is defined to be the
equilibrium value of the aggregate
demand by individuals for the security.
   The market value of a security equals the
equilibrium value of the aggregate amount
of that security issued by business firms.
   We use V j denote the market value of
security j and VR denote the value of the
riskless security, then  jM is the fraction of
security j held in a market portfolio.
Vj
   M

V
j        n
1    j    VR

   Theorem 2.14: If  e is a convex set, and if
the securities’ market is in equilibrium,
then a market portfolio is an efficient
portfolio.
   Proof: Let there be K risk averse investor
in the economy.Define Z  R  1 w j (Z j  R)
K      n k

to be the return on investor k’s optimal
portfolio. In equilibrium, 1 wkj W0k  V j ,
K

where    W0k is the initial wealth of investor
K, and 1 W0  W0  1 V j  VR . Define
K   K         n

  W W k  1, K . By definition of a market
k
0
k

portfolio 1 wkj k   jM j  1, , n. Multiplying
0     K

by Z j  R and summing over j, it follows
that
   w ( Z  R)                         K ( Z  R)
K    n k               K   K
1       k        1       j   j       1

   ( Z  R)  Z  R
n        M
1       i        j           M
because 1      k  1, Z M  1 K Z k
. Hence, Z M is
K                   K

a convex combination of the returns on K
efficient portfolios. Therefore , if e is
convex, then the market portfolio is
contained in  .
e

 The efficiency of the market portfolio
provides a rigorous microeconomic
justification for the use of a
“ representative man” to derive equilibrium
prices in aggregated economic models.
   Proposition 2.6: In all portfolio models
with homogeneous beliefs and risk-averse
investors the equilibrium expected return
on the market portfolio exceeds the return
on the riskless security.
   Proof: From the proof of Theorem 2.14
and Corollary 2.1.   Z M  R  1 k ( Z k  R) ,
K

because Z k  R , k  0 . Hence ZM  R
   The market portfolio is the only risky
portfolio where the sign of its equilibrium
expected excess return can always be
predicted.
   Returning to the special case where  e is
spanned by a single risky portfolio and the
riskless security, the market portfolio is
efficient. So the risky spanning portfolio
can always be chosen to be the market
portfolio.
   Theorem 2.15: If      ( Z M , R) span  e , then the
equilibrium expected return on security j
can be written as Z j  R   j ( Z M  R)
where           cov( Z j , Z M )
j 
var( Z M )
   This relation, called the Security Market
Line, was first derived by Sharpe.
   In the special case of Theorem 2.15,  j
measure the systematic risk of security j
relative to the efficient portfolio Z M .
    j can be computed from a simple
covariance between Z j and Z M . But the
sign of b j can not be determined by the
k

sign of the correlation coefficient between
Z j and Zek
   Theorem 2.16: If (Z1 , , Z n ) contain no
redundant securities, then (a) for each

value  ,  j , j  1, , n, are unique, (b)
there exists a portfolio contained in
with return X such that ( X , R) span  min ,
and (c) Z j  R  a j ( X j  R) where,
cov( Z j , X )
aj                                , j  1,   , n.
var( X )
   Where  min denote the set of portfolios
contained in  such that there exists no
f

other portfolio in  with the same
f

expected return and a smaller variance.
   Proof: Let  ij denote the ijth element of 
and  vij denote the ijth element of 1 . So
all portfolios in  min with expect return u,
we need solutions the problem
min 1 1  i j ij
n    n

S .T Z (  )  
If   R then Z ( R)  R and  jR  0, j  1, 2 , n
Consider the case when   R . The n
first-order conditions are
0  1  j ij  u ( Zi  R) i  1, 2,
n
,n
Multiplying by             and summing, we get
1 1  i  j  ij   i  i  ( Zi  R)  0
   
n    n               n

u  var[ Z (  )] (   R)

By definition of  min ,  must be the same
for all Z ( ) . Because  is nonsingular,
the linear equation has unique solution
 j  u 1 vij ( Zi  R) j  1, , n
        n

This prove (a). From this solution we have
 j  k are the same for every value  .
Hence all portfolios in  min are perfectly
correlated. Hence we can pick any
portfolio in  min with   R and call its
return X. Then we have
Z ( )    ( X  R)  R
Hence ( X , R) span  min which proves (b).
and from Corollary 2.13 and Proposition
2.3 (c) follows directly.
   From Theorem 2.16, ak will be equivalent
to bkK as a measure of a security’s
systematic risk provided that the
chosen for X is such that   R .
   Theorem 2.17: If ( X , R) span    e and if X
has a finite variance, then  e is contained
in  min .
   Proof: Let Ze  R  ae ( X  R) . Let Z p be
the return on any portfolio in     f such
that Z e  Z p . By Corollary 2.13 Z  R  a ( X  R)  
p      p            p

where E{ p }  E{ p | X }  0
Therefore a p  ae
Thus
var( Z p )  a 2 var( X )  var( p )  a p var( X )  var( Z e )
p

Hence, Z e is contained in  min .
    Theorem 2.18: If ( Z1 , , Z n ) have a joint
normal probability distribution, then there
exists a portfolio with return X such that
( X , R) span  .
e
   Proof: construct a risky portfolio contained
in  min , and call its return X. Define
 k  Z k  R  ak ( X  R), k  1, , n by
Theorem 2.16 part (c) E{ k }  0 and by
construction cov( k , X )  0 . Because Z1 Z n
are normally distributed, X will be
normally distributed. Hence  k is normal
distributed , and because cov( X ,  k )  0 , so
they are independent. Therefore
E{ k }  E{ k | X }  0 , From Corollary 2.13
it follows that ( X , R ) span      e
   Theorem 2.19: If p(Z1 , , Z n ) is a symmetric
function with respect to all its arguments,
then there exists a portfolio with return X
such that ( X , R )span  .
e

   Proof: By hypothesis
p( Z1 , Zi , Z n )  p( Zi , Z1 , Z n ) for each set
of given values. Therefore every risk
averse investor will choose 1  i . But this
is true for all i. Hence , all investor will
hold all risky securities in the same
relative proportions. Then ( X , R) span  e
 The APT model developed by Ross
provides an important class of linear-factor
models that generate spanning without
assuming joint normal probability
distributions.
 If we can construct a set of m portfolios
with returns ( X1 , , X M ) such that X i and Yi
are perfectly correlated, i  1, , m, then
( X 1 , , X M , R) will span  e
   The APT model is attractive because the
equilibrium structure of expected returns
and risks of securities can be derived
without explicit knowledge of investors’
preferences or endowments.
   For the study of equilibrium pricing, the
usual format is to derive equilibrium V j 0
given the distribution of V j .
   Theorem 2.20: If ( X1 , , X m ) denote a set of
linearly independent portfolios that satisfy
the hypothesis of Theorem 2.12, and all
securities have finite variances, then a
necessary condition for equilibrium in the
securities’ market is that
V j  1   
m       m
vik cov( X k ,V j )( X j  R)
Vj0               1

R

where   vik is the ikth          element of X
1
   Proof: By linear independence V j  Z jV j 0
by Theorem 2.12 V j  V j 0 [ R   m aij ( X i  R)   j ]
where E{ j | X1, , X m}  0 . Take
1

expectations, we have
V j  V j 0 [ R  1 aij ( X i  R)]
m

Noting that cov( X k ,V j )  V j 0 cov( X k , Z j )
From Proposition 2.3 aij  1 vik cov( X K , Z j )
m

Thus V j 0 aij  1 vik cov( X K ,V j )
m

We can get
V j  1   
m       m
vik cov( X k ,V j )( X j  R)
Vj0                1

R
   Hence, from Theorem 2.20, a sufficient
set of information to determine the
equilibrium value of security j is the first
and second moments for the join
distribution of ( X1, , X m ,V j ) .
   Corollary 2.20a: If the hypothesized
conditions of Theorem 2.20 hold and if the
end-of-period value a security is given by
V  1  jV j then in equilibrium
n

V0  1  jV j 0
n

   This property of formula is called “ value
   Corollary 2.20b: If the hypothesized
conditions of Theorem 2.20 hold and if the
end-of-period value of a security is given
by V  qV j  u , where E{u}  E{u | X1 , , X m }  u
and E{q}  E{q | X 1 , , X m }  q then in
equilibrium V0  qV j 0  u R
   Hence, to value two securities whose end
of period values differ only by
multiplicative or additive “noise”, we can
simply substitute the expected values of
the noise terms.
   Theorem 2.20 and its corollaries are
central to the theory of optimal
   Although the optimal investment and
financing decisions by a form generally
require simultaneous determination, under
certain conditions the optimal investment
decision can be made independently of
the method of financing.
   Theorem 2.21: If firm j is financed by q
different claims defined by the function
fk (V j ) k  1, , q, and if there exists an
equilibrium such that the return
distribution of the efficient portfolio set
remains unchanged from the equilibrium
in which firm j was all equity financed,
then

q
1
fk 0  V j 0 (I j )

where f k 0 is the equilibrium initial value of
financial claim k.
   Hence, for a given investment policy, the
way in which the firm finances its
investments changes the return
distribution of the efficient portfolio set.
   Clearly, a sufficient condition for Theorem
2.21 to obtain is that each of the financial
claims issued by the firm are “ redundant
securities”.
   An alternative approach to the
development of nontrivial spanning
theorems is to derive a class of utility
functions for investors .
   Such that even with arbitrary joint
probability distributions for the available
securities,investors within the class can
generate their optimal portfolios from the
spanning portfolios.
   Let  u denote the set of optimal portfolios
selected from  f by investors with strictly
concave von Neumann-Morgenstern utility
functions.
   Theorem 2.22 There exists a portfolio with
return X such that         ( X , R) span  u if and
only if Ai (W )  1 (ai  bW )  0 , where Ai is the
absolute risk-aversion function for investor
i in  u .
   Because the b in the statement of
Theorem 2.22 does not have a             i
u
subscript , therefore all investors in
must have virtually the same utility
function.
   Cass and Stiglitz (1970) conclude: it is
requirement that there be any mutual
funds, and not the limitation on the
number of mutual funds.
   This is a negative report on the approach
to developing spanning theorems.
The End

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