Introduction to Portfolio Introduction to Portfolio Selection and Capital Market Selection and Capital Market

Theory: Static AnalysisTheory: Static Analysis

BaoheWangBaoheWangbaohewang0592@sina.combaohewang0592@sina.com

IntroductionIntroduction

The investment decision by The investment decision by households as having two parts:households as having two parts:

(a) the “consumption-saving” choice(a) the “consumption-saving” choice (b) the “portfolio-selection” choice(b) the “portfolio-selection” choice In general the two decisions cannot In general the two decisions cannot

be made independently.be made independently. However, the consumption-saving However, the consumption-saving

allocation has little substantive allocation has little substantive impact on portfolio theory.impact on portfolio theory.

One-period Portfolio SelectionOne-period Portfolio Selection

The solution to the general problem The solution to the general problem of choosing the best investment mix of choosing the best investment mix is called is called portfolio-selection theoryportfolio-selection theory..

There are n different investment There are n different investment opportunities called opportunities called securitiessecurities..

The random variable one-period The random variable one-period return per dollar on security j is return per dollar on security j is denoteddenoted

jZ

Any linear combination of these Any linear combination of these securities which has a positive securities which has a positive market value is called a market value is called a portfolioportfolio..

denote the utility function. denote the utility function. is the end-of-period value of the is the end-of-period value of the

investor’s wealth measure in dollars.investor’s wealth measure in dollars. is an increasing strictly concave is an increasing strictly concave

function and twice continuously function and twice continuously differentiable.differentiable.

So the investor’s decision is relevant So the investor’s decision is relevant to the subjective joint probability to the subjective joint probability distribution for .distribution for .

( )U W

W

U

1 2( , , , )nZ Z Z

Assumption 1: Frictionless MarketsAssumption 1: Frictionless Markets

Assumption 2: Price-TakerAssumption 2: Price-Taker

Assumption 3: No-Arbitrage Assumption 3: No-Arbitrage OpportunitiesOpportunities

Assumption 4: No-Institutional Assumption 4: No-Institutional RestrictionsRestrictions

Given these assumptions, the Given these assumptions, the portfolio-selection problem can be portfolio-selection problem can be formally stated asformally stated as

(2.1)(2.1)

Where Where E E is the expectation operator is the expectation operator for the subjective joint probability for the subjective joint probability distribution.distribution.

1 20

{ , , }1

1

max { ( )}

. . 1

n

n

j jw w w

n

j

E U w Z W

S T w

If is a solution (2.1), then it will If is a solution (2.1), then it will satisfy the first-order conditions:satisfy the first-order conditions:

Where is the random variable Where is the random variable return per dollar on the optimal return per dollar on the optimal portfolio.portfolio.

With the concavity assumptions on U, if With the concavity assumptions on U, if the variance-covariance matrix of the the variance-covariance matrix of the return is nonsingular and an interior return is nonsingular and an interior solution exists, the the solution is solution exists, the the solution is unique.unique.

1 2( , , , )nw w w

00

{ ( )}jE U Z W ZW

1

n

j jZ w Z

Formula (2.1) rules out that any one of tFormula (2.1) rules out that any one of the securities is a riskless security.he securities is a riskless security.

If a riskless security is added to the menIf a riskless security is added to the menu of available securities then the portfoliu of available securities then the portfolio selection problem can be stated as:o selection problem can be stated as:

(2.4)(2.4)1 2

1 2

0 01{ , , }1

0{ , , }

1

max { ( (1 ) )}

max { ([ ( ) ] )}

n

n

nn

j j jw w w

n

j jw w w

E U w Z W w RW

E U w Z R R W

The first-order conditions can be The first-order conditions can be written as:written as:

Where can be rewritten as Where can be rewritten as If it is assumed that the variance-If it is assumed that the variance-

covariance matrix of the returns on covariance matrix of the returns on the risky securities is nonsingular the risky securities is nonsingular and an interior solution exits, then and an interior solution exits, then the solution is unique. the solution is unique.

0{ ( )( )} 0 1,2, ,jE U Z W Z R j n

Z 1

( )n

j jw Z R R

But neither (2.1) nor (2.3) reflect that But neither (2.1) nor (2.3) reflect that end of period wealth cannot be negative.end of period wealth cannot be negative.

To rule out bankruptcy, the additional To rule out bankruptcy, the additional constraint that, with probability one,constraint that, with probability one,

could be imposed on .could be imposed on . This constraint is too weak, because the This constraint is too weak, because the

probability assessments on are probability assessments on are subjective.subjective.

An alternative treatment is to forbid An alternative treatment is to forbid borrowing and short-selling securities borrowing and short-selling securities where, by law, .where, by law, .

0Z *

1 2( , , , )nw w w

{ }jZ

0jZ

The optimal demand functions for The optimal demand functions for risky securities, , and the risky securities, , and the resulting probability distribution for resulting probability distribution for the optimal portfolio will depend on the optimal portfolio will depend on

(1) the risk preferences of the (1) the risk preferences of the investor;investor;

(2) his initial wealth; (2) his initial wealth;

(3) the join distribution for the (3) the join distribution for the securities’ returns.securities’ returns.

0{ }jwW

The von Neumann-Morgenstern The von Neumann-Morgenstern utility function can only be utility function can only be determined up to a positive affine determined up to a positive affine transformation.transformation.

The Pratt-Arrow absolute risk-The Pratt-Arrow absolute risk-aversion function is invariant to any aversion function is invariant to any positive affine transformation of positive affine transformation of . .

( )U W

The preference orderings of all The preference orderings of all choices available to the investor are choices available to the investor are completely specified by completely specified by absolute absolute risk–aversion functionrisk–aversion function

The change in absolute risk aversion The change in absolute risk aversion with respect to a change in wealth iswith respect to a change in wealth is

( )( )

( )

U WA W

U W

( )( ) ( )[ ( ) ]

( )

dA U WA W A W A W

dW U W

is positive, and such investor are call is positive, and such investor are call risk averse.risk averse.

An alternative, measure of risk aversion iAn alternative, measure of risk aversion is the s the relative risk-aversion functionrelative risk-aversion function defi defined byned by

Its change with respect to a change in wIts change with respect to a change in wealth is given byealth is given by

( )A W

( )( ) ( )

( )

U W WR W A W W

U W

( ) ( ) ( )R W A W W A W

The The certainty-equivalent end-of-period certainty-equivalent end-of-period wealth wealth is defined to be such thatis defined to be such that

is the amount of money such that is the amount of money such that the investor is indifferent between the investor is indifferent between having this amount of money for certain having this amount of money for certain or the portfolio with random variable or the portfolio with random variable outcome .outcome .

We can proof follows directly by Jensen’s We can proof follows directly by Jensen’s inequality: if is strictly concaveinequality: if is strictly concave

Because U is an increase function, So Because U is an increase function, So

CW

( ) { ( )}CU W E U W

CW

W

U

( ) { ( )} ( { })CU W E U W U E W

{ }CW E W

The certainty equivalent can be used The certainty equivalent can be used to compare the risk aversions of two to compare the risk aversions of two investor.investor.

If A is more risk averse than B and If A is more risk averse than B and they hold same portfolio, the certainty they hold same portfolio, the certainty equivalent end of period wealth for A equivalent end of period wealth for A is less than or equal to the certainty is less than or equal to the certainty equivalent end of period wealth for B. equivalent end of period wealth for B.

Rothschild and Stiglitz define the meaniRothschild and Stiglitz define the meaning of “increasing risk” for a security so ng of “increasing risk” for a security so we can compare the riskiness of two secwe can compare the riskiness of two securities or portfolios.urities or portfolios.

If for all concavIf for all concave with strict inequality holding for some with strict inequality holding for some concave , we said the first portfolio is e concave , we said the first portfolio is less risky than the second portfolio.less risky than the second portfolio.

1 2 1 2( ) ( ) , { ( )} { ( )}E W E W E U W E U W U

U

Its equivalence to the two following Its equivalence to the two following definitions:definitions:

(1) is equal in distribution to plus (1) is equal in distribution to plus some “noise”. some “noise”.

(2) has more “weight in its tails” (2) has more “weight in its tails” than . than .

2W 1W

2W 1W

If there exists an increasing strictly If there exists an increasing strictly concave function such that concave function such that

, we call this , we call this portfolio is an portfolio is an efficient portfolioefficient portfolio..

All portfolios that are not efficient are All portfolios that are not efficient are called called inefficient portfoliosinefficient portfolios..

V

{ ( )( )} 0, 1,2, , .jE V Z Z R j n

It follows immediately that every It follows immediately that every efficient portfolio is a possible efficient portfolio is a possible optimal portfolio, for each efficient optimal portfolio, for each efficient portfolio there exists an increasing portfolio there exists an increasing concave such that the efficient concave such that the efficient portfolio is a solution to (2.1) or (2.3).portfolio is a solution to (2.1) or (2.3).

Because all risk-averse investors Because all risk-averse investors have different utility function, so they have different utility function, so they will be indifferent between selecting will be indifferent between selecting their optimal portfolios.their optimal portfolios.

U

Theorem 2.1: If denotes the random varTheorem 2.1: If denotes the random variable return per dollar on any feasible poiable return per dollar on any feasible portfolio and if is riskier than in the rtfolio and if is riskier than in the Rothschild and Stiglitz sense, thenRothschild and Stiglitz sense, then

( is an efficient portfolio) ( is an efficient portfolio) Proof: By hypothesis Proof: By hypothesis

If then trivially .If then trivially . But is a feasible portfolio and is an But is a feasible portfolio and is an

efficient portfolio. By contradiction, efficient portfolio. By contradiction,

Z

e eZ Z Z Z

eZ Z

0 0{ [( ) ]} {[( ) ]}e eE U Z Z W E Z Z W

eZ Z 0 0{ ( )} { ( )}eE U ZW E U Z W

Z eZ

eZ Z

eZ

Corollary 2.1: If there exists a riskless secCorollary 2.1: If there exists a riskless security with return R, then , with equaurity with return R, then , with equality holding only if is a riskless security. lity holding only if is a riskless security.

Proof: If is riskless , then by AssumptioProof: If is riskless , then by Assumption 3, . If is not riskless, by Theorem n 3, . If is not riskless, by Theorem 2.1, .2.1, .

eZ R

eZ

eZ

eZ R eZ

eZ R

Theorem 2.2: The optimal portfolio for a Theorem 2.2: The optimal portfolio for a nonsatiated risk-averse investor will be tnonsatiated risk-averse investor will be the riskless security if and only if for jhe riskless security if and only if for j=1,2,…..,n.=1,2,…..,n.

Proof: If is an optimal solution, thProof: If is an optimal solution, then we have By the nonsatien we have By the nonsatiation assumption, soation assumption, so

If then will satisfy If then will satisfy because the property of U, so t because the property of U, so this solution is unique. his solution is unique.

jZ R

Z R

0( ) { } 0jU RW E Z R

0( ) 0U RW jZ R

1,2 ,jZ R j n Z R

0( ) { } 0jU Z W E Z R

From Corollary 2.1 and Theorem 2.2, if a From Corollary 2.1 and Theorem 2.2, if a risk-averse investor chooses a risky portfrisk-averse investor chooses a risky portfolio, then the expected return on the porolio, then the expected return on the portfolio exceeds the riskless rate.tfolio exceeds the riskless rate.

Theorem 2.3: Let denote the return on Theorem 2.3: Let denote the return on any portfolio any portfolio pp that does not contain that does not contain security security ss. If there exists a portfolio p . If there exists a portfolio p such that, for security such that, for security ss, , where , , where

then the fraction then the fraction of every efficient portfolio allocated to of every efficient portfolio allocated to security security ss is the same and equal to zero. is the same and equal to zero.

Proof: Suppose is the return on an Proof: Suppose is the return on an efficient portfolio with fraction efficient portfolio with fraction allocated to security allocated to security s, s, be the return be the return on a portfolio with the same fractional on a portfolio with the same fractional holding as except that instead of holding as except that instead of security security s s with portfolio with portfolio PP

pZ

s p sZ Z { | , 1,2, , , } 0s jE Z j n j s

eZ0s

Z

eZ

HenceHence So So Therefore ,for , is riskier than Z in Therefore ,for , is riskier than Z in

the Rothschild-Stiglitz. This contradicts tthe Rothschild-Stiglitz. This contradicts that is an efficient portfolio.hat is an efficient portfolio.

Corollary 2.3: Let denote the set of n seCorollary 2.3: Let denote the set of n securities and denote the same set of securities and denote the same set of securities except that is replace with . If curities except that is replace with . If and , then all risk averse inv and , then all risk averse investor would prefer to choose . estor would prefer to choose .

( )e s s p s sZ Z Z Z Z

eZ Z0s

eZ

eZ

sZ sZ

s s sZ Z { | } 0sE Z

Theorem 2.3 and its corollary demonstrTheorem 2.3 and its corollary demonstrate that all risk averse investors would pate that all risk averse investors would prefer any “unnecessary” and “noisrefer any “unnecessary” and “noise” to be eliminated.e” to be eliminated.

The Rothschild-Stiglitz definition of incrThe Rothschild-Stiglitz definition of increasing risk is quite useful for studying theasing risk is quite useful for studying the properties of optimal portfolios. e properties of optimal portfolios.

But this rule is not apply to individual seBut this rule is not apply to individual securities or inefficient portfolios.curities or inefficient portfolios.

2.3 Risk Measures for Securities and 2.3 Risk Measures for Securities and Portfolios in The One-Period modelPortfolios in The One-Period model

In this section, a second definition of In this section, a second definition of increasing risk is introduced.increasing risk is introduced.

is the random variable return per is the random variable return per dollar on an efficient portfolio dollar on an efficient portfolio K.K.

denote an increasing strictly denote an increasing strictly concave function such that for concave function such that for

Random variable Random variable

keZ

( )KK eV Z

{ ( )} 0 1,2, ,K jE V Z R j n

KKKe

dVVdZ

0 1W

{ }

cov( , )K

K KK e

V E VY

V Z

Definition: The measure of risk of Definition: The measure of risk of portfolio portfolio PP relative to efficient relative to efficient portfolio portfolio K K with random variable with random variable return is defined by return is defined by

and portfolio and portfolio PP is said to be riskier is said to be riskier than portfolio relative to efficient than portfolio relative to efficient portfolio portfolio K K if .if .

Kpb

KeZ

cov( , )Kp K Pb Y Z

PK Kp pb b

Theorem 2.4: If is the return on a Theorem 2.4: If is the return on a feasible portfolio and is the return feasible portfolio and is the return on efficient portfolio on efficient portfolio K K , then , then . .

Proof: From the definitionProof: From the definition

be the fraction of portfolio be the fraction of portfolio P P allocated allocated to security j, then to security j, then

and and

pZ

P KeZ

( )K Kp p eZ R b Z R

{ ( )} 0 1,2, ,K jE V Z R j n

j

1

( )n

P j jZ Z R R

1

{ ( )} { ( )} 0n

j K j K PE V Z R E V Z R

By a similar argument,By a similar argument,

Hence,Hence,

andand

By Corollary 2.1 , . ThereforeBy Corollary 2.1 , . Therefore

{ ( )} 0KK eE V Z R

cov( , ) [ ( )]

[ ( )]

[ ( )] [ ( )]

( ) [ ]

K K KK e K e e

K KK e e

K KK e K e

Ke K

V Z E V Z Z

E V Z R R Z

E V Z R E V R Z

R Z E V

cov( , ) ( ) { }K P P KV Z R Z E V KeZ R

( )K Kp p eZ R b Z R

Hence, the expected excess return Hence, the expected excess return on portfolio P, is in direct on portfolio P, is in direct proportion to its risk and the larger is proportion to its risk and the larger is its risk , the larger is its expected its risk , the larger is its expected return.return.

Consider an investor with utility Consider an investor with utility function U and initial wealth who function U and initial wealth who solves the portfolio-selection solves the portfolio-selection problem: problem:

The first order condition:The first order condition:

PZ R

0W

0max { ([ (1 ) ] )}jwE U wZ w Z W

* *0{ ([ (1 ) ] )( )}j jE U w Z w Z W Z Z

If then the solution is . If then the solution is . However , an optimal portfolio is an efficHowever , an optimal portfolio is an effic

ient portfolio. By Theorem 2.4ient portfolio. By Theorem 2.4

So is similar to an excess demand funSo is similar to an excess demand function . Measures the contribution of section . Measures the contribution of security j to the Rothsechild-Stiglitz risk of curity j to the Rothsechild-Stiglitz risk of the optimal portfolio.the optimal portfolio.

*Z Z * 0W

* *( )j jZ R b Z R *wW

*jb

By the implicit function theorem, we By the implicit function theorem, we have:have:

Therefore , if lies above the risk-Therefore , if lies above the risk-return line in the plane, then return line in the plane, then the investor would prefer to increase the investor would prefer to increase his holding in security j.his holding in security j.

**0

20

{ ( )} { }

{ ( ) }j

j j

wW E U Z Z E Uw

Z W E U Z Z

jZ( , )Z b

is a natural measure of risk for is a natural measure of risk for individual securities.individual securities.

The ordering of securities by their The ordering of securities by their systematic risk relative to a given systematic risk relative to a given efficient portfolio will be identical efficient portfolio will be identical with their ordering relative to any with their ordering relative to any other efficient portfolio.other efficient portfolio.

Kpb

Lemma 2.1: Lemma 2.1: (i) for efficient portfolio (i) for efficient portfolio K. K.

(ii) If (ii) If then then (iii) for efficient portfolio (iii) for efficient portfolio KK if an if an

d only if for every efficient portfd only if for every efficient portfolio olio L.L.

Proof: (i) is a continuous monotonic funProof: (i) is a continuous monotonic function of and hence and are in onction of and hence and are in one to one correspondence.e to one correspondence.

{ | } { | }KP K P eE Z V E Z Z

{ | }KP e pE Z Z Z

cov( , ) 0p KZ V

cov( , ) 0p KZ V

cov( ) 0P LZ V

KV KeZ KV K

eZ

(ii)(ii)

(iii)Because (iii)Because

if , then .if , then .

Property I: If Property I: If LL and and KK are efficient are efficient portfolios, then for any portfolio portfolios, then for any portfolio pp, , . .

Proof : From Theorem 2.4Proof : From Theorem 2.4

cov( , ) { ( )} { { | }} 0Kp K K p P K p P eZ V E V Z Z E V E Z Z Z

0 cov( , ) 0Kp p Kb Z V

0Kpb

pZ R

K K Lp L pb b b

Lp pK K Le

L p pK K Le e e

Z R Z RZ Rb b b

Z R Z R Z R

Property 2: If L and K are efficient Property 2: If L and K are efficient portfolios, then and .portfolios, then and .

Hence, all efficient portfolios have Hence, all efficient portfolios have positive systematic risk, relative to positive systematic risk, relative to any efficient portfolio.any efficient portfolio.

Property 3: if and only if Property 3: if and only if for every efficient portfolio for every efficient portfolio K.K.

Property 4: Let Property 4: Let pp and and qq denote any denote any two feasible portfolios and let two feasible portfolios and let KK and and LL denote any two efficient portfolios. denote any two efficient portfolios. if and only if if and only if

1KKb 0L

Kb

pZ R 0Kpb

K Kp qb bL L

p qb b

Proof: From Property 1, we haveProof: From Property 1, we have

Thus the measure provides the Thus the measure provides the same orderings of risk for any same orderings of risk for any reference efficient portfolio.reference efficient portfolio.

Property 5: For each efficient portfolio Property 5: For each efficient portfolio KK and any feasible portfolio and any feasible portfolio pp,,

where and for where and for every efficient portfolio every efficient portfolio L.L.

K K Lp L pb b b K K L

q L qb b b

Kpb

( )K Kp p e pZ R b Z R

{ } 0pE { ( )} 0Lp L eE V Z

Proof: From Theorem 2.4 . If portfProof: From Theorem 2.4 . If portfolio olio qq is constructed by holding one doll is constructed by holding one dollar ar pp, dollars riskless security, short sel, dollars riskless security, short selling dollars portfolio ling dollars portfolio KK, then, then

so for every efficient portfolio so for every efficient portfolio LL.. But implies But implies for every efficient portfolio for every efficient portfolio L.L.

Property 6: If a feasible portfolio p has pProperty 6: If a feasible portfolio p has portfolio weight ,then ortfolio weight ,then

{ } 0pE

Kpb

Kpb q pZ R

0Lqb

0Lqb 0 cov( , ) { , }q L p LZ V E V

1( , , )n 1

nK Kp j jb b

Hence , the systematic risk of a portfolio Hence , the systematic risk of a portfolio is the weighted sum of the systematic risis the weighted sum of the systematic risks of its component securities.ks of its component securities.

The Rothschild Stiglitz measure provideThe Rothschild Stiglitz measure provides only for a partial ordering.s only for a partial ordering.

measure provides a complete orderinmeasure provides a complete ordering.g.

They can give different rankings.They can give different rankings. The Rothschild Stiglitz definition measurThe Rothschild Stiglitz definition measur

e the “total risk” of a security. It is appe the “total risk” of a security. It is appropriate definition for identifying optimaropriate definition for identifying optimal portfolios and determining the efficient l portfolios and determining the efficient portfolio set.portfolio set.

Kpb

The measure the “ systematic The measure the “ systematic risk” of a security.risk” of a security.

To determine the , the efficient To determine the , the efficient portfolio set must be determined.portfolio set must be determined.

The manifest behavioral The manifest behavioral characteristic shared by all risk characteristic shared by all risk averse utility maximization is to averse utility maximization is to diversify.diversify.

Kjb

Kjb

The greatest benefits in risk reduction cThe greatest benefits in risk reduction come from adding a security to the portfoome from adding a security to the portfolio whose realized return tends to be higlio whose realized return tends to be higher when the return on the rest of the poher when the return on the rest of the portfolio is lower.rtfolio is lower.

Next to such “ countercyclical” investNext to such “ countercyclical” investments in terms of benefit are the noncycments in terms of benefit are the noncyclic securities whose returns are orthogolic securities whose returns are orthogonal to the return on the portfolio.nal to the return on the portfolio.

Theorem 2.5 : If and denote the Theorem 2.5 : If and denote the returns on portfolio returns on portfolio pp and and qq respectively and if, for each possible respectively and if, for each possible value of ,value of ,

with strict inequality with strict inequality holding over some finite probability holding over some finite probability measure of ,then portfolio measure of ,then portfolio pp is is riskier than portfolio q and . riskier than portfolio q and .

Where , is the Where , is the realized return on an efficient portfolio.realized return on an efficient portfolio.

pZ qZ

eZ( ) ( )p e q e

e e

dG Z dG ZdZ dZ

eZ

p qZ Z

( ) { | }p e p eG Z E Z Z eZ

Proof: Proof:

is a strictly increasing function, is a strictly increasing function, is a nondecreasing function, sois a nondecreasing function, so

From Theorem 2.4 From Theorem 2.4

cov[ ( ), ] [ ( )( )]

[ ( )( { | } { | })]

[ ( )( ( ) ( ))

cov[ ( ), ( ) ( )]

p q e p q e p q

e p e q e

e e p e q

e e p e q

b b Y Z Z Z E Y Z Z Z

E Y Z E Z Z E Z Z

E Y Z G Z G Z

Y Z G Z G Z

( )eY Z ( ) ( )e p e qG Z G Z

cov[ ( ), ( ) ( )] 0p q e e p e qb b Y Z G Z G Z

p qZ Z

Theorem 2.6: If and denote the Theorem 2.6: If and denote the returns on portfolio returns on portfolio pp and and qq respectively and if, for each possible respectively and if, for each possible value of ,value of ,

, a constant, then , a constant, then

and .and .

Proof: By hypothesis Proof: By hypothesis

pZ qZ

eZ( ) ( )p e q e

pqe e

dG Z dG ZadZ dZ

p q pqb b a ( )p q pq eZ Z a Z R

( ) ( )e p e q pqG Z G Z a h

cov[ ( ), ( ) ( )]

cov[ ( ), ]

p q e e p e q

e pq e pq

b b Y Z G Z G Z

Y Z a Z h a

( ) ( ) ( ) ( )p p e q e pq e q pq eZ R b Z R R b Z R a Z R Z a Z R

Theorem 2.7: If, for all possible Theorem 2.7: If, for all possible values ofvalues of

(i) , then (i) , then

(II) , then (II) , then

(III) , then (III) , then

(IV) , a constant, then(IV) , a constant, then

eZ( )

1p e

e

dG ZdZ p eZ Z

( )0 1p e

e

dG ZdZ p eR Z Z

( )0p e

e

dG ZdZ pR Z

( )p ep

e

dG ZadZ

( )p p eZ R a Z R

Theorems 2.5, 2.6 and 2.7 Theorems 2.5, 2.6 and 2.7 demonstrate, the conditional demonstrate, the conditional expected return function provides expected return function provides considerable information about a considerable information about a security’s risk and equilibrium security’s risk and equilibrium expected return.expected return.

2.4 Spanning, Separation, and 2.4 Spanning, Separation, and Mutual-Fund TheoremsMutual-Fund Theorems

Definition: A set of M feasible Definition: A set of M feasible portfolios with random variable portfolios with random variable returns is said to span the returns is said to span the space of portfolios contained in the space of portfolios contained in the set if and only if for any portfolio set if and only if for any portfolio in with return denoted by there in with return denoted by there exist numbers , such exist numbers , such thatthat

1( , )MX X

pZ

1( , )M 1

1M

i 1

M

p j jZ X

A A mutual fund mutual fund is a financial is a financial intermediary that holds as its assets a intermediary that holds as its assets a portfolio of securities and issues as portfolio of securities and issues as liabilities shares against this collection liabilities shares against this collection of assets.of assets.

Theorem 2.8 If there exist Theorem 2.8 If there exist MM mutual mutual funds whose portfolio span the funds whose portfolio span the portfolio set , then all investors will portfolio set , then all investors will be indifferent between selecting their be indifferent between selecting their optimal portfolios from and optimal portfolios from and selecting from portfolio combination of selecting from portfolio combination of just the just the MM mutual funds. mutual funds.

Therefore the smallest number of Therefore the smallest number of such funds is a particularly such funds is a particularly important spanning set.important spanning set.

When such spanning obtain, the When such spanning obtain, the investor’s portfolio-selection problem investor’s portfolio-selection problem can be separated into two steps.can be separated into two steps.

However, if the smallest funds can be However, if the smallest funds can be constructed only if the fund constructed only if the fund managers know the preferences, managers know the preferences, endowments, and probability beliefs endowments, and probability beliefs of each investor.of each investor.

M

Theorem 2.9: Necessary conditions Theorem 2.9: Necessary conditions for the M feasible portfolios with for the M feasible portfolios with return to span the portfolio return to span the portfolio set are (a) that the rank of set are (a) that the rank of and (b) that there exist numbers and (b) that there exist numbers such that the random such that the random variable has zero variance.variable has zero variance.

Proposition 2.1: If is the Proposition 2.1: If is the return on some security or portfolio return on some security or portfolio and if there are no “ arbitrage and if there are no “ arbitrage opportunities” thenopportunities” then

1( , , )MX Xf

M 1 1

( , , ), 1M

M j

1

M

j jX1

n

p j jZ a Z b

1 1( ) (1 ) ( ) ( )

n n

j p j ja b a R and b Z R a Z R

Proof: Let be the return on a portfolio Proof: Let be the return on a portfolio with fraction allocated to security j, with fraction allocated to security j,

allocated to the security with return ; allocated to the security with return ; and allocated to the riskless secand allocated to the riskless security with return R, if is chosen such thurity with return R, if is chosen such that ,then is riskless sat ,then is riskless security and therefore but can be ecurity and therefore but can be chosen arbitrarily. So we get the result.chosen arbitrarily. So we get the result.

Z

j 1, , ;j n

p pZ1

1n

p j j

j p ja 1

[ (1 )]n

p jZ R b R a Z R

Hence, as long as there are no arbitrage Hence, as long as there are no arbitrage opportunities, it can be assumed withouopportunities, it can be assumed without loss of generality that one of the portfot loss of generality that one of the portfolios in any candidate spanning set is the lios in any candidate spanning set is the riskless security.riskless security.

Theorem 2.10: A necessary and sufficienTheorem 2.10: A necessary and sufficient condition for to span is that tt condition for to span is that there exist number such that here exist number such that

1( , , , )mX X R f{ }ija

1( ) 1,2, , .

m

j ij iZ R a X R j n

Proof: If span , thenProof: If span , then such that . Becausesuch that . Because and substituting , we have and substituting , we have we pick the portfolio weights for we pick the portfolio weights for

and , from which it follow and , from which it follows that .But every portfolio in cas that .But every portfolio in can be written as a portfolio combination n be written as a portfolio combination of and R.of and R.

1( , , , )mX X R

11

M

ij

f

1

M

j ij iZ XMX R

11

m

Mj ij 1

( ) 1,2, , .m

j ij iZ R a X R j n

ij ija 1, ,i m

11

m

Mj ij 1

M

j ij iZ Xf

1( , , )nZ Z

Corollary 2.10: A necessary and sufficienCorollary 2.10: A necessary and sufficient condition for to be the smallest t condition for to be the smallest number of feasible portfolio that span is number of feasible portfolio that span is that the rank of equals the rank ofthat the rank of equals the rank of

Proof: If the rank of , then XProof: If the rank of , then X are linearly independent. Moreover are linearly independent. Moreover hence, if the rank of then there exihence, if the rank of then there exi

st number such that st number such that for . Therefore where for . Therefore where

by Theorem 2.10 span by Theorem 2.10 span

1( , , , )mX X R

X m

X m

m { }ija 1

( )m

j j ij i iZ Z a X X 1, ,j n

1

m

j j ij iZ b a X 1

m

j j ij ib Z a X f

It follows from Corollary 2.10 that a It follows from Corollary 2.10 that a necessary and sufficient condition for necessary and sufficient condition for nontrivial spanning of is that some nontrivial spanning of is that some of the risky securities are redundant of the risky securities are redundant securities.securities.

By Theorem 2.10, if investors agree on By Theorem 2.10, if investors agree on a set of portfolios such that a set of portfolios such that

and if they and if they agree on the number ,then agree on the number ,then span even if investors do not agree span even if investors do not agree on the joint distribution of on the joint distribution of

f

1( , , , )mX X R

1( ) 1,2, , .

m

j ij iZ R a X R j n { }ija 1( , , , )mX X R

f1( , , , )mX X R

Proposition 2.2: If is the return on a poProposition 2.2: If is the return on a portfolio contained in , then any portfolirtfolio contained in , then any portfolio that combines positive amount of wito that combines positive amount of with the riskless security is also contained ih the riskless security is also contained in , where is the set of all efficient pn , where is the set of all efficient portfolios contained in .ortfolios contained in .

Proof: Let , because is an eProof: Let , because is an efficient portfolio, sofficient portfolio, so

Define where andDefine where and , Hence , thus Z is a, Hence , thus Z is a

n efficient portfolio.n efficient portfolio.

eZe

eZ

e ef

(1 )eZ Z R eZ

{ ( )( )} 0e jE V Z Z R

( ) ( )U W V aW b 1a ( 1)Rb

{ ( )( )} 0jE U Z Z R

It follows from Proposition 2.2 that, for It follows from Proposition 2.2 that, for every number such that , there every number such that , there exists at least one efficient portfolio exists at least one efficient portfolio with expected return equal to .with expected return equal to .

Theorem 2.11: Let denote Theorem 2.11: Let denote the return on the return on mm feasible portfolios. If, feasible portfolios. If, for security j, there exist number for security j, there exist number such that such that

where where

for some efficient portfolio K, thenfor some efficient portfolio K, then

Z Z R

Z

{ }ija

1( , , )mX X

1( )

m

j j ij i i jZ Z a X X { ( )} 0Kj K eE V Z

1( )

m

j ij iZ R a X R

Proof: LetProof: Let

because , thusbecause , thus

by by construction , and hence construction , and hence

Therefore the systematic risk of Therefore the systematic risk of portfolio p, is zero. From Theorem 2.4 portfolio p, is zero. From Theorem 2.4

therefore therefore

1 1(1 )

m m

p j i i iZ Z X R

1( )

m

j j ij i i jZ Z a X X

1[ ( )]

m

p j ij i jZ R Z R a X R { } 0jE cov( , ) 0p KZ V

Kpb pZ R

1( )

m

j ij iZ R a X R

Hence, if the return on a security can Hence, if the return on a security can be written in this linear form relative to be written in this linear form relative to the portfolios , then its the portfolios , then its expected excess return is completely expected excess return is completely determined by the expected excess determined by the expected excess returns on these portfolios and the returns on these portfolios and the weights .weights .

Theorem 1.12Theorem 1.12: If, for every security j, : If, for every security j, there exist numbers such thatthere exist numbers such that

where , then where , then span the set of efficient portfolios .span the set of efficient portfolios .

{ }ija

1( , , )mX X

{ }ija

1( )

m

j ij i jZ R a X R 1{ | , , } 0j mE X X

1( , , , )mX X Re

Proof:Proof:

Where Where

Construct portfolio Construct portfolio

Thus whereThus where

Hence, for , is riskier than Z, Hence, for , is riskier than Z, which contradicts that is and which contradicts that is and efficient portfolio. So . We get efficient portfolio. So . We get the result.the result.

1 1 1

1 1 1 1

1

[ ( ) ]

( )

( )

j j

j j j

n n mK K Ke j ij i j

n n m mK K Kij i j

m K Ki i

Z w Z w R a X R

w R w a X R w

R X R

1

nK Ki j ijw a 1 j

mK Kjw

1 1(1 )

m mK Ki i iZ X R

K KeZ Z { | } 0KE Z

0K KeZ

KeZ

0K

Theorem 2.13: Let denote the fraction Theorem 2.13: Let denote the fraction of efficient portfolio of efficient portfolio KK allocation to allocation to security security jj, span if , span if and only if there exist number for and only if there exist number for every security j such that every security j such that

where for every where for every efficient portfolio efficient portfolio K.K.

Corollary 2.13: (X,R) span if and only if Corollary 2.13: (X,R) span if and only if there exist a number for each security j,there exist a number for each security j,

such that such that

wherewhere

Kjw

11, , . ( , , , )mj n X X R e{ }ija

1( )

m

j ij i jZ R a X R 1 1

{ | } 0,m nK K K

j i i i j ijE X w a

eja

1, , ,j n ( )j j jZ R a X R

{ | } 0jE X

Proof: By hypothesis, for everProof: By hypothesis, for every efficient portfolio K. If , then from Cy efficient portfolio K. If , then from Corollary 2.1 for every efficient portforollary 2.1 for every efficient portfolio K and R span . Otherwise, from Tholio K and R span . Otherwise, from Theorem 2.2, for every efficient portfoeorem 2.2, for every efficient portfolio. By Theorem 2.13,lio. By Theorem 2.13,

so so Since is contained in , any properties Since is contained in , any properties

proved for portfolios that span must bproved for portfolios that span must be properties of portfolio that span .e properties of portfolio that span .

( )K KeZ X R R

X R0K

e0K

{ | } 0KjE X { | } 0jE X

f

fee

From Theorem 2.10, 2.12, 2.13, the From Theorem 2.10, 2.12, 2.13, the essential difference is that to span essential difference is that to span the efficient portfolio set it is not the efficient portfolio set it is not necessary that linear combinations of necessary that linear combinations of the spanning portfolios exactly the spanning portfolios exactly replicate the return on each available replicate the return on each available security.security.

All the models that do not restrict the All the models that do not restrict the class of admissible utility function, class of admissible utility function, the distribution of individual security the distribution of individual security returns must be such thatreturns must be such that

1( )

m

j ij i jZ R a X R

Proposition 2.3: If, for every security j,Proposition 2.3: If, for every security j, with linearly indepenwith linearly indepen

dent with finite variances and if the returdent with finite variances and if the return on security j, has a finite variance, thn on security j, has a finite variance, then the in Theorems 2.12 and 2.en the in Theorems 2.12 and 2.13 are given by13 are given by

where is the ikth element where is the ikth element of . of .

Hence given some knowledge of the joinHence given some knowledge of the joint distribution of a set of portfolio that spt distribution of a set of portfolio that span an

with , we can determining the andwith , we can determining the and

1{ | , , } 0j mE X X 1( , , )mX X

jZ

{ } 1, , ,ija i m

1cov( , )

m

ij ik K ja v X Z ikv1X

ej jZ Z ija jZ

Proposition 2.4: If contain no Proposition 2.4: If contain no redundant securities, denotes the redundant securities, denotes the fraction of portfolio fraction of portfolio XX allocated to allocated to security j, and denotes the fraction security j, and denotes the fraction of any risk-averse investor’s optimal of any risk-averse investor’s optimal portfolio allocated to security j, portfolio allocated to security j, then for every such risk-averse then for every such risk-averse investorinvestor

1( , , )nZ Z

j

jw

1, , ,j n

*, 1, 2, ,j j

k k

wj k n

w

Because every optimal portfolio is an effBecause every optimal portfolio is an efficient portfolio and the holding of risky sicient portfolio and the holding of risky securities in every efficient portfolio are pecurities in every efficient portfolio are proportional to the holding in X.roportional to the holding in X.

If there exist numbers where If there exist numbers where and ,then the portfolio with proportiand ,then the portfolio with proporti

ons is called the ons is called the Optimal CombinaOptimal Combination of Risky Assets.tion of Risky Assets.

Proposition 2.5: If span , then is Proposition 2.5: If span , then is a convex set.a convex set.

j *

* , , 1, ,j j

k k

j k n

*

1

n

j* *1( , )n

( , )X R e e

Proof: LetProof: Let

and , . By and , . By substitution, the expression for Z can substitution, the expression for Z can be rewritten as , where be rewritten as , where

.Therefore by .Therefore by Proposition 2.2, Z is an efficient Proposition 2.2, Z is an efficient portfolio. It follow by induction that for portfolio. It follow by induction that for any integer k and number such that any integer k and number such that and and

is the return on an is the return on an efficient portfolio. Hence , is a efficient portfolio. Hence , is a convex set.convex set.

11( )eZ X R R 2

2 ( )eZ X R R

1 2 1 2(1 )e eZ Z Z

1( )eZ Z R R

2

1( )(1 )

i 0 1, 1, ,i i k

1 11,

k kk ii i eZ Z

e

Definition: A Definition: A market portfoliomarket portfolio is defined is defined as a portfolio that holds all available as a portfolio that holds all available securities in proportion to their market securities in proportion to their market values.values.

The equilibrium market value of a The equilibrium market value of a security for this purpose is defined to be security for this purpose is defined to be the equilibrium value of the aggregate the equilibrium value of the aggregate demand by individuals for the security.demand by individuals for the security.

The market value of a security equals The market value of a security equals the equilibrium value of the aggregate the equilibrium value of the aggregate amount of that security issued by amount of that security issued by business firms.business firms.

We use denote the market value of seWe use denote the market value of security j and denote the value of the riscurity j and denote the value of the riskless security, then is the fraction of sekless security, then is the fraction of security j held in a market portfolio.curity j held in a market portfolio.

Theorem 2.14: If is a convex set, and if Theorem 2.14: If is a convex set, and if the securities’ market is in equilibrium, the securities’ market is in equilibrium, then a market portfolio is an efficient pothen a market portfolio is an efficient portfolio.rtfolio.

jVRV

Mj

1

jMj n

j R

V

V V

e

Proof: Let there be K risk averse investor Proof: Let there be K risk averse investor in the economy.Definein the economy.Define

to be the return on investor k’s optimal to be the return on investor k’s optimal portfolio. In equilibrium, , where portfolio. In equilibrium, , where is the initial wealth of investor K, and is the initial wealth of investor K, and . Define . Define

. By definition of a market portf. By definition of a market portfolio Multiplying by anolio Multiplying by and summing over j, it follows thatd summing over j, it follows that

1( )

nK kj jZ R w Z R

01

K k kj jw W V

0kW

0 01 1

K nKj RW W V V

0

01,

k

kW k KW

11, , .

K k Mj k jw j n

jZ R

1 1 1

1

( ) ( )

( )

K n Kk Kk j j K

n Mi j M

w Z R Z R

Z R Z R

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

The efficiency of the market portfolio The efficiency of the market portfolio provides a rigorous microeconomic provides a rigorous microeconomic justification for the use of a “ justification for the use of a “ representative man” to derive representative man” to derive equilibrium prices in aggregated equilibrium prices in aggregated economic models. economic models.

1 11,

K K kk M KZ Z

e

e

MZ

Proposition 2.6: In all portfolio models wProposition 2.6: In all portfolio models with homogeneous beliefs and risk-averse ith homogeneous beliefs and risk-averse investors the equilibrium expected returinvestors the equilibrium expected return on the market portfolio exceeds the ren on the market portfolio exceeds the return on the riskless security.turn on the riskless security.

Proof: From the proof of Theorem 2.14 aProof: From the proof of Theorem 2.14 and Corollary 2.1. , because nd Corollary 2.1. , because , . Hence , . Hence

The market portfolio is the only risky porThe market portfolio is the only risky portfolio where the sign of its equilibrium etfolio where the sign of its equilibrium expected excess return can always be prexpected excess return can always be predicted.dicted.

1( )

K kM kZ R Z R

kZ R 0k MZ R

Returning to the special case where is Returning to the special case where is spanned by a single risky portfolio and tspanned by a single risky portfolio and the riskless security, the market portfolio he riskless security, the market portfolio is efficient. So the risky spanning portfoliis efficient. So the risky spanning portfolio can always be chosen to be the market o can always be chosen to be the market portfolio.portfolio.

Theorem 2.15: If span , then the eTheorem 2.15: If span , then the equilibrium expected return on security j quilibrium expected return on security j can be written as can be written as

where where

e

( , )MZ R e

( )j j MZ R Z R cov( , )

var( )j M

jM

Z Z

Z

This relation, called the Security Market This relation, called the Security Market Line, was first derived by Sharpe.Line, was first derived by Sharpe.

In the special case of Theorem 2.15, In the special case of Theorem 2.15, measure the systematic risk of security measure the systematic risk of security j relative to the efficient portfolio .j relative to the efficient portfolio .

can be computed from a simple can be computed from a simple covariance between and . But covariance between and . But the sign of can not be determined by the sign of can not be determined by the sign of the correlation coefficient the sign of the correlation coefficient between between

and and

j

MZ

j

jZ MZkjb

jZkeZ

Theorem 2.16: If contain no Theorem 2.16: If contain no redundant securities, then (a) for each redundant securities, then (a) for each value are unique, (b) value are unique, (b) there exists a portfolio contained in there exists a portfolio contained in with return with return XX such that span , such that span , and (c) where,and (c) where,

Where denote the set of portfolios Where denote the set of portfolios contained in such that there exists contained in such that there exists no other portfolio in with the same no other portfolio in with the same expected return and a smaller expected return and a smaller variance.variance.

1( , , )nZ Z

, , 1, , ,j j n

( , )X R min( )j j jZ R a X R

cov( , ), 1, , .var( )

jj

Z Xa j nX

minf

f

Proof: Let denote the Proof: Let denote the ijthijth element of element of and denote the and denote the ijthijth element of . So a element of . So all portfolios in with expect return u, wll portfolios in with expect return u, we need solutions the problem e need solutions the problem

If then and If then and Consider the case when . The n first-Consider the case when . The n first-

order conditions areorder conditions are

ijijv

1

min

1 1min

. ( )

n n

i j ij

S T Z

R ( )Z R R 0, 1,2 ,Rj j n

R

10 ( ) 1,2, ,

n

j ij u iZ R i n

Multiplying by and summing, we getMultiplying by and summing, we get

By definition of , must be the same By definition of , must be the same for all . Because is nonsingular, the for all . Because is nonsingular, the linear equation has unique solutionlinear equation has unique solution

This prove (a). From this solution we haveThis prove (a). From this solution we have

are the same for every value .are the same for every value .

Hence all portfolios in are perfectly Hence all portfolios in are perfectly correlated. Hence we can pick any correlated. Hence we can pick any

1 1( ) 0

var[ ( )] ( )

n n n

i j ij i ii

u

Z R

Z R

min ( )Z

1( ) 1, ,

n

j u ij iv Z R j n

j k

min

portfolio in with and call its portfolio in with and call its return X. Then we havereturn X. Then we have

Hence span which proves Hence span which proves (b).(b).

and from Corollary 2.13 and and from Corollary 2.13 and Proposition 2.3 (c) follows directly.Proposition 2.3 (c) follows directly.

min R

( ) ( )Z X R R

( , )X Rmin

From Theorem 2.16, will be From Theorem 2.16, will be equivalent to as a measure of a equivalent to as a measure of a security’s systematic risk provided security’s systematic risk provided that the chosen for that the chosen for XX is such that is such that . .

Theorem 2.17: If span and if X Theorem 2.17: If span and if X has a finite variance, then is has a finite variance, then is contained in .contained in .

Proof: Let . Let be Proof: Let . Let be the return on any portfolio in such the return on any portfolio in such that . By Corollary 2.13 that . By Corollary 2.13

where where

kaKkb

R ( , )X R e

emin

( )e eZ R a X R pZf

e pZ Z ( )p p pZ R a X R

{ } { | } 0p pE E X

ThereforeTherefore

Thus Thus

Hence, is contained in .Hence, is contained in . Theorem 2.18: If have a Theorem 2.18: If have a

joint normal probability distribution, joint normal probability distribution, then there exists a portfolio with then there exists a portfolio with return return X X such that such that

span . span .

p ea a

2var( ) var( ) var( ) var( ) var( )p p p p eZ a X a X Z

mineZ

1( , , )nZ Z

( , )X R e

Proof: construct a risky portfolio Proof: construct a risky portfolio contained in , and call its return X. contained in , and call its return X. DefineDefine

by by Theorem 2.16 part (c) and by Theorem 2.16 part (c) and by construction . Becauseconstruction . Because

are normally distributed, X will be are normally distributed, X will be normally distributed. Hence is normal normally distributed. Hence is normal distributed , and because , so distributed , and because , so they are independent. Therefore they are independent. Therefore

, From Corollary 2.13 , From Corollary 2.13 it follows that span it follows that span

min

( ), 1, ,k k kZ R a X R k n { } 0kE

cov( , ) 0k X 1 nZ Z

kcov( , ) 0kX

{ } { | } 0k kE E X ( , )X R e

Theorem 2.19: If is a symmetric Theorem 2.19: If is a symmetric function with respect to all its function with respect to all its arguments, then there exists a portfolio arguments, then there exists a portfolio with return X such that span .with return X such that span .

Proof: By hypothesis Proof: By hypothesis

for each set for each set of given values. Therefore every risk of given values. Therefore every risk averse investor will choose . But averse investor will choose . But this is true for all this is true for all i. i. Hence , all investor Hence , all investor will hold all risky securities in the same will hold all risky securities in the same relative proportions. Then spanrelative proportions. Then span

( , )X R e

1( , , )np Z Z

1 1( , , ) ( , , )i n i np Z Z Z p Z Z Z

1 i

( , )X R e

The APT model developed by Ross The APT model developed by Ross provides an important class of linear-provides an important class of linear-factor models that generate factor models that generate spanning without assuming joint spanning without assuming joint normal probability distributions.normal probability distributions.

If we can construct a set of m If we can construct a set of m portfolios with returns such portfolios with returns such that and are perfectly that and are perfectly correlated, thencorrelated, then

will span will span

1( , , )MX X iX iY1, , ,i m

1( , , , )MX X R e

The APT model is attractive because The APT model is attractive because the equilibrium structure of expected the equilibrium structure of expected returns and risks of securities can be returns and risks of securities can be derived without explicit knowledge of derived without explicit knowledge of investors’ preferences or investors’ preferences or endowments.endowments.

For the study of equilibrium pricing, the For the study of equilibrium pricing, the usual format is to derive equilibrium giusual format is to derive equilibrium given the distribution of .ven the distribution of .

Theorem 2.20: If denote a set of liTheorem 2.20: If denote a set of linearly independent portfolios that satisfnearly independent portfolios that satisfy y the hypothesis of Theorem 2.12the hypothesis of Theorem 2.12, and al, and all securities have finite variances, then a l securities have finite variances, then a necessary condition for equilibrium in thnecessary condition for equilibrium in the securities’ market is thate securities’ market is that

where is the where is the ikthikth element of element of

0jV

jV

1 10

cov( , )( )m m

j ik k j jj

V v X V X RV

R

ikv1X

1( , , )mX X

Proof: By linear independence Proof: By linear independence by Theorem 2.12 by Theorem 2.12 where . Take where . Take expectations, we have expectations, we have

Noting thatNoting that

From Proposition 2.3From Proposition 2.3

Thus Thus

We can get We can get

0j j jV Z V

0 1[ ( ) ]

m

j j ij i jV V R a X R 1{ | , , } 0j mE X X

0 1[ ( )]

m

j j ij iV V R a X R 0cov( , ) cov( , )k j j k jX V V X Z

1cov( , )

m

ij ik K ja v X Z0 1

cov( , )m

j ij ik K jV a v X V

1 10

cov( , )( )m m

j ik k j jj

V v X V X RV

R

Hence, from Theorem 2.20, a sufficient sHence, from Theorem 2.20, a sufficient set of information to determine the equiliet of information to determine the equilibrium value of security j is the first and sbrium value of security j is the first and second moments for the join distribution econd moments for the join distribution of .of .

Corollary 2.20a: If the hypothesized conCorollary 2.20a: If the hypothesized conditions of ditions of Theorem 2.20 Theorem 2.20 hold and if the ehold and if the end-of-period value a security is given bynd-of-period value a security is given by

then in equilibriumthen in equilibrium

This property of formula is called “ valuThis property of formula is called “ value additivity”.e additivity”.

1( , , , )m jX X V

1

n

j jV V0 01

n

j jV V

Corollary 2.20b: If the hypothesized Corollary 2.20b: If the hypothesized conditions of conditions of Theorem 2.20 Theorem 2.20 hold and hold and if the end-of-period value of a if the end-of-period value of a security is given by , security is given by , where and where and then in equilibrium then in equilibrium

Hence, to value two securities whose Hence, to value two securities whose end of period values differ only by end of period values differ only by multiplicative or additive “noise”, we multiplicative or additive “noise”, we can simply substitute the expected can simply substitute the expected values of the noise terms.values of the noise terms.

jV qV u 1{ } { | , , }mE u E u X X u

1{ } { | , , }mE q E q X X q 0 0jV qV u R

Theorem 2.20 and its corollaries are Theorem 2.20 and its corollaries are central to the theory of optimal central to the theory of optimal investment decisions by business investment decisions by business firms.firms.

Although the optimal investment and Although the optimal investment and financing decisions by a form financing decisions by a form generally require simultaneous generally require simultaneous determination, under certain determination, under certain conditions the optimal investment conditions the optimal investment decision can be made independently decision can be made independently of the method of financing.of the method of financing.

Theorem 2.21: If firm j is financed by q Theorem 2.21: If firm j is financed by q different claims defined by the functiondifferent claims defined by the function

and if there exists an and if there exists an equilibrium such that the return equilibrium such that the return distribution of the efficient portfolio set distribution of the efficient portfolio set remains unchanged from the remains unchanged from the equilibrium in which firm j was all equilibrium in which firm j was all equity financed, thenequity financed, then

where is the equilibrium initial value where is the equilibrium initial value of financial claim of financial claim kk..

( ) 1, , ,jkf V k q

0 01( )

q

k j jf V I0kf

Hence, for a given investment policy, Hence, for a given investment policy, the way in which the firm finances its the way in which the firm finances its investments changes the return investments changes the return distribution of the efficient portfolio distribution of the efficient portfolio set.set.

Clearly, a sufficient condition for Clearly, a sufficient condition for Theorem 2.21 to obtain is that each Theorem 2.21 to obtain is that each of the financial claims issued by the of the financial claims issued by the firm are “ redundant securities”.firm are “ redundant securities”.

An alternative approach to the An alternative approach to the development of nontrivial spanning development of nontrivial spanning theorems is to derive a class of utility theorems is to derive a class of utility functions for investors .functions for investors .

Such that even with arbitrary joint Such that even with arbitrary joint probability distributions for the probability distributions for the available securities,investors within available securities,investors within the class can generate their optimal the class can generate their optimal portfolios from the spanning portfolios from the spanning portfolios.portfolios.

Let denote the set of optimal portfolioLet denote the set of optimal portfolios selected from by investors with strics selected from by investors with strictly concave von Neumann-Morgenstern tly concave von Neumann-Morgenstern utility functions.utility functions.

Theorem 2.22 There exists a portfolio wiTheorem 2.22 There exists a portfolio with return X such that span if and oth return X such that span if and only if , where is the absolutnly if , where is the absolute risk-aversion function for investore risk-aversion function for investor

in . in .

uf

u

u( , )X R

( ) 1 ( ) 0i iA W a bW iA

i

Because the b in the statement of TheorBecause the b in the statement of Theorem 2.22 does not have a subscript , therem 2.22 does not have a subscript , therefore all investors in must have virtuallefore all investors in must have virtually the same utility function.y the same utility function.

Cass and Stiglitz (1970) conclude: it is reCass and Stiglitz (1970) conclude: it is requirement that there be any mutual funquirement that there be any mutual funds, and not the limitation on the number ds, and not the limitation on the number of mutual funds.of mutual funds.

This is a negative report on the approacThis is a negative report on the approach to developing spanning theorems.h to developing spanning theorems.

iu

The EndThe End

thanksthanks