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Introduction to PortfolioIntroduction to Portfolio

Selection and Capital MarketSelection and Capital Market

Theory: Static AnalysisTheory: Static Analysis

BaoheWangBaoheWang

[email protected]@sina.com

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IntroductionIntroduction

The investment decision by households asThe investment decision by households ashaving two parts:having two parts:

(a) the(a) the consumptionconsumption--savingsaving choicechoice

(b) the(b) the portfolioportfolio--selectionselection choicechoice

In general the two decisions cannot beIn general the two decisions cannot bemade independently.made independently.

However, the consumptionHowever, the consumption--savingsavingallocation has little substantive impact onallocation has little substantive impact onportfolio theory.portfolio theory.

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OneOne--period Portfolio Selectionperiod Portfolio Selection

The solution to the general problem ofThe solution to the general problem ofchoosing the best investment mix is calledchoosing the best investment mix is called

portfolioportfolio--selection theoryselection theory.. There are n different investmentThere are n different investment

opportunities calledopportunities called securitiessecurities..

The random variable oneThe random variable one--period returnperiod returnper dollar on security j is denotedper dollar on security j is denoted jZ

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Any linear combination of these securitiesAny linear combination of these securitieswhich has a positive market value is calledwhich has a positive market value is called

aa portfolioportfolio.. denote the utility function.denote the utility function.

is the endis the end--ofof--period value of theperiod value of the

investorinvestors wealth measure in dollars.s wealth measure in dollars. is an increasing strictly concaveis an increasing strictly concave

function and twice continuouslyfunction and twice continuouslydifferentiable.differentiable.

So the investorSo the investors decision is relevant tos decision is relevant tothe subjective joint probability distributionthe subjective joint probability distributionfor .for .

( )U W

W

U

1 2( , , , )n Z Z Z L

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Assumption 1: Frictionless MarketsAssumption 1: Frictionless Markets

Assumption 2: PriceAssumption 2: Price--TakerTaker

Assumption 3: NoAssumption 3: No--Arbitrage OpportunitiesArbitrage Opportunities

Assumption 4: NoAssumption 4: No--Institutional RestrictionsInstitutional Restrictions

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Given these assumptions, the portfolioGiven these assumptions, the portfolio--selection problem can be formally statedselection problem can be formally statedasas

(2.1)(2.1)

WhereWhere EE is the expectation operator foris the expectation operator forthe subjective joint probability distribution.the subjective joint probability distribution.

1 20{ , , }

1

1

max { ( )}

. . 1

n

n

j jw w w

n

j

E U w Z W

S T w !

L

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If is a solution (2.1), then it willIf is a solution (2.1), then it will

satisfy the firstsatisfy the first--order conditions:order conditions:

Where is the random variableWhere is the random variablereturn per dollar on the optimal portfolio.return per dollar on the optimal portfolio.

With the concavity assumptions on U, ifWith the concavity assumptions on U, ifthe variancethe variance--covariance matrix of thecovariance matrix of thereturn is nonsingular and an interiorreturn is nonsingular and an interiorsolution exists, the the solution is unique.solution exists, the the solution is unique.

1 2( , , , )nw w w

L

0

0

{ ( )}jE U Z W ZW

Pd !

1

n

j j Z w Z

|

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Formula (2.1) rules out that any one ofFormula (2.1) rules out that any one of

the securities is a riskless security.the securities is a riskless security. If a riskless security is added to the menuIf a riskless security is added to the menu

of available securities then the portfolioof available securities then the portfolioselection problem can be stated as: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 w Z W w RW

E w Z R R W

L

L

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The firstThe first--order conditions can be writtenorder conditions can be written

as:as:

Where can be rewritten asWhere can be rewritten as

If it is assumed that the varianceIf it is assumed that the variance--covariance matrix of the returns on thecovariance matrix of the returns on therisky securities is nonsingular and anrisky securities is nonsingular and aninterior solution exits, then the solution isinterior solution exits, then the solution isunique.unique.

0{ ( )( )} 0 1, 2, ,jE U Z W Z R j nd ! ! L

Z

1( )

n

j jw Z R R

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

To rule out bankruptcy, the additionalTo rule out bankruptcy, the additionalconstraint that, with probability one,constraint that, with probability one,

could be imposed on .could be imposed on .

This constraint is too weak, because theThis constraint is too weak, because theprobability assessments on are subjective.probability assessments on are subjective.

An alternative treatment is to forbidAn alternative treatment is to forbidborrowing and shortborrowing and short--selling securities where,selling securities where,by law, .by law, .

0Z u*

1 2( , , , )nw w w

L

{ }jZ

0j

Z u

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The optimal demand functions for riskyThe optimal demand functions for riskysecurities, , and the resultingsecurities, , and the resultingprobability distribution for the optimalprobability distribution for the optimalportfolio will depend onportfolio will depend on

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

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

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

0{ }jw W

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The von NeumannThe von Neumann--Morgenstern utilityMorgenstern utilityfunction can only be determined up to afunction can only be determined up to apositive affine transformation.positive affine transformation.

The PrattThe Pratt--Arrow absolute riskArrow absolute risk--aversionaversionfunction is invariant to any positive affinefunction is invariant to any positive affinetransformation of .transformation of .( )U W

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The preference orderings of all choicesThe preference orderings of all choices

available to the investor are completelyavailable to the investor are completelyspecified byspecified by absolute riskabsolute riskaversionaversionfunctionfunction

The change in absolute risk aversion withThe change in absolute risk aversion withrespect to a change in wealth isrespect to a change in wealth is

( )( )

( )

U WA W

U W

dd

d

( )( ) ( )[ ( ) ]

( )

d A U W A W A W A W

dW U W

dddd! !

dd

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is positive, and such investor are callis positive, and such investor are call

risk averse.risk averse. An alternative, measure of risk aversion isAn alternative, measure of risk aversion is

thethe relative riskrelative risk--aversion functionaversion function defineddefined

byby

Its change with respect to a change inIts change with respect to a change in

wealth is given bywealth is given by

( )A W

( )( ) ( )

( )

U W WR W A W W

U W

dd| !

d

( ) ( ) ( )R W A W W A W d d!

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TheThe certaintycertainty--equivalentendequivalentend--ofof--periodperiodwealthwealth is definedtobe such thatis definedtobe such that

is the amountofmoneysuch thattheis the amountofmoneysuch thattheinvestoris indifferentbetweenhaving thisinvestoris indifferentbetweenhaving this

amountofmoneyforcertainortheamountofmoneyforcertainortheportfoliowith random variable outcome .portfoliowith random variable outcome .

We canprooffollows directlyby JensenWe canprooffollows directlyby Jensenss

inequality: if is strictly concaveinequality: if is strictly concave

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

CW

( ) { ( )}C

U W E U W!

CW

W

U

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

{ }CW E W

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

IfA is more risk averse than B and theyIfA is more risk averse than B and they

hold same portfolio, the certaintyhold same portfolio, the certaintyequivalent end of period wealth for A isequivalent end of period wealth for A isless than or equal to the certaintyless than or equal to the certainty

equivalent end of period wealth for B.equivalent end of period wealth for B.

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Rothschild and Stiglitz define the meaningRothschild and Stiglitz define the meaning

ofofincreasing riskincreasing risk for a security so wefor a security so wecan compare the riskiness of twocan compare the riskiness of twosecurities or portfolios.securities or portfolios.

If forIf forall concave with strict inequality holdingall concave with strict inequality holdingfor some concave , we said the firstfor some concave , we said the firstportfolio is less risky than the secondportfolio is less risky than the second

portfolio.portfolio.

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

U

U

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If there exists an increasing strictlyIf there exists an increasing strictlyconcave function such thatconcave function such that

, we call this, we call this

portfolio is anportfolio is an efficient portfolioefficient portfolio.. All portfolios that are not efficient areAll portfolios that are not efficient are

calledcalled inefficient portfoliosinefficient portfolios..

V

{ ( )( )} 0, 1, 2, , .jE V Z Z R j nd ! ! L

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It follows immediately that every efficientIt follows immediately that every efficient

portfolio is a possible optimal portfolio, forportfolio is a possible optimal portfolio, foreach efficient portfolio there exists aneach efficient portfolio there exists anincreasing concave such that theincreasing concave such that theefficient portfolio is a solution to (2.1) orefficient portfolio is a solution to (2.1) or

(2.3).(2.3).

Because all riskBecause all risk--averse investors haveaverse investors havedifferent utility function, so they will bedifferent utility function, so they will be

indifferent between selecting their optimalindifferent between selecting their optimalportfolios.portfolios.

U

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Theorem 2.1: If denotes the randomTheorem 2.1: If denotes the random

variable return per dollar on any feasiblevariable return per dollar on any feasibleportfolio and if is riskier thanportfolio and if is riskier thanin the Rothschild and Stiglitz sense, thenin the Rothschild and Stiglitz sense, then

( is an efficient portfolio)( is an efficient portfolio)

Proof: By hypothesisProof: By hypothesis

If then trivially .If then trivially .But is a feasible portfolio and is anBut is a feasible portfolio and is anefficient 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 Zu

0 0

{ ( )} { ( )}e

E U ZW E U Z W"

Z eZ

eZ Z"

eZ

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Corollary 2.1: If there exists a risklessCorollary 2.1: If there exists a riskless

security with return R, then , withsecurity with return R, then , withequality holding only if is a risklessequality holding only if is a risklesssecurity.security.

Proof: If is riskless , then byProof: If is riskless , then byAssumption 3, . If is not riskless,Assumption 3, . If is not riskless,by Theorem 2.1, .by Theorem 2.1, .

eZ Rue

Z

eZ

eZ R! eZ

eZ R"

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Theorem 2.2: The optimal portfolio for aTheorem 2.2: The optimal portfolio for anonsatiated risknonsatiated risk--averse investor will be theaverse investor will be theriskless security if and only if forriskless security if and only if for

j=1,2,j=1,2,..,n...,n.

Proof: If is an optimal solution,Proof: If is an optimal solution,

then we have By thethen we have By thenonsatiation assumption, sononsatiation assumption, so

If then willIf then will

satisfy because thesatisfy because theproperty of U, so this solution is unique.property of U, so this solution is unique.

jZ R!

Z R !0( ) { } 0jU RW E Z Rd !

0( ) 0U RWd " jZ R!

1, 2 ,j Z R j n! ! L Z R !

0( ) { } 0jU Z W E Z Rd !

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From Corollary 2.1 and Theorem 2.2, if aFrom Corollary 2.1 and Theorem 2.2, if ariskrisk--averse investor chooses a riskyaverse investor chooses a riskyportfolio, then the expected return on theportfolio, then the expected return on theportfolio exceeds the riskless rate.portfolio exceeds the riskless rate.

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Theorem 2.3: Let denote the return onTheorem 2.3: Let denote the return onany portfolioany portfolio pp that does not containthat does not contain

securitysecurity ss. If there exists a portfolio p such. If there exists a portfolio p suchthat, for securitythat, for security ss, , where, , where

then the fractionthen the fractionof every efficient portfolio allocated toof every efficient portfolio allocated tosecuritysecurity ss is the same and equal to zero.is the same and equal to zero.

Proof: Suppose is the return on anProof: Suppose is the return on anefficient portfolio with fractionefficient portfolio with fraction

allocated to securityallocated to security s,s, be the return onbe the return ona portfolio with the same fractionala portfolio with the same fractionalholding as except that instead ofholding as except that instead ofsecuritysecurity ss with portfoliowith portfolio PP

pZ

s p sZ Z I!

{ | , 1, 2, , , } 0s j E Z j n j sI ! { !L

eZ

0sH {

Z

eZ

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HenceHence

SoSoTherefore ,for , is riskier thanTherefore ,for , is riskier than

Z in the RothschildZ in the Rothschild--Stiglitz. ThisStiglitz. This

contradicts that is an efficient portfolio.contradicts that is an efficient portfolio. Corollary 2.3: Let denote the set of nCorollary 2.3: Let denote the set of n

securities and denote the same set ofsecurities and denote the same set of

securities except that is replace with .securities except that is replace with .If and , then all riskIf and , then all riskaverse investor would prefer to choose .averse investor would prefer to choose .

( )e s s p s s Z Z Z Z Z H H I! !

eZ Z!0sH { eZ

eZ

NNd

sZ

sZ

d

s s sZ Z Id! { | } 0sE ZI !

N

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Theorem 2.3 and its corollary demonstrateTheorem 2.3 and its corollary demonstrate

that all risk averse investors would preferthat all risk averse investors would preferanyany unnecessaryunnecessary andand noisenoise to beto beeliminated.eliminated.

The RothschildThe Rothschild--Stiglitz definition ofStiglitz definition ofincreasing risk is quite useful for studyingincreasing risk is quite useful for studyingthe properties of optimal portfolios.the properties of optimal portfolios.

But this rule is not apply to individualBut this rule is not apply to individualsecurities or inefficient portfolios.securities or inefficient portfolios.

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2.3 Risk Measures for Securities and2.3 Risk Measures for Securities and

Portfolios in The OnePortfolios in The One--Period modelPeriod model

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

is the random variable return per dollaris the random variable return per dollaron an efficient portfolioon an efficient portfolio K.K.

denote an increasing strictlydenote an increasing strictlyconcave function such that forconcave function such that for

Random variableRandom variable

k

eZ

( )K

K eV Z

{ ( )} 0 1, 2, ,K jE V Z R j nd ! ! L

KKK

e

dVV

dZ

d!

0 1W !

{ }

cov( , )

KK K

K e

V E VY

V Z

d d|

d

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Definition: The measure of risk ofDefinition: The measure of risk of

portfolioportfolio PP relative to efficient portfoliorelative to efficient portfolio KKwith random variable return is definedwith random variable return is definedbyby

and portfolioand portfolio PP is said to be riskier thanis said to be riskier thanportfolio relative to efficient portfolioportfolio relative to efficient portfolio KKif .if .

K

pb

K

eZ

cov( , )K p K P b Y Z|

PdK K

p pb b d"

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Theorem 2.4: If is the return on a feasibleTheorem 2.4: If is the return on a feasible

portfolio and is the return on efficientportfolio and is the return on efficientportfolioportfolio KK , then ., then .

Proof: From the definitionProof: From the definition

be the fraction of portfoliobe the fraction of portfolio PP allocated toallocated tosecurity j, thensecurity j, then

andand

pZ

PK

e

Z

( )K Kp p eZ R b Z R !

{ ( )} 0 1, 2, ,K jEV

Z R jn

d ! !L

jH

1( )

n

P j j Z Z R RH!

1

{ ( )} { ( )} 0n

j K j K PE V Z R E V Z RH d d ! !

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By a similar argument,By a similar argument,

Hence,Hence,

andand

By Corollary 2.1 , . ThereforeBy Corollary 2.1 , . Therefore

{ ( )} 0K

K eE V Z Rd !

cov( , ) [ ( )]

[ ( )]

[ ( )] [ ( )]( ) [ ]

K K K K e K e e

K K

K e e

K K

K e K e

K

e 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

d d!

d!

d d! d!

cov( , ) ( ) { }K P P KV

Z R Z E Vd d

! K

eZ R"

( )K K

p p eZ R b Z R !

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Hence, the expectedexcess returnonHence, the expectedexcess returnon

portfolioP, isindirectproportionportfolioP, isindirectproportiontoitsriskandthe largerisitsrisk , thetoitsriskandthe largerisitsrisk , thelargerisitsexpectedreturn.largerisitsexpectedreturn.

Consideraninvestorwithutility functionUConsideraninvestorwithutility functionUandinitialwealth whosolvestheandinitialwealth whosolvestheportfolioportfolio--selectionproblem:selectionproblem:

The firstordercondition:The firstordercondition:

PZ R

0W

0max { ([ (1 ) ] )}jw

E U wZ w Z W

* *

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

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If then the solution is .If then the solution is .

However , an optimal portfolio is anHowever , an optimal portfolio is anefficient portfolio. By Theorem 2.4efficient portfolio. By Theorem 2.4

So is similar to an excess demandSo is similar to an excess demandfunction . Measures the contribution offunction . Measures the contribution ofsecurity j to the Rothsechildsecurity j to the Rothsechild--Stiglitz risk ofStiglitz risk ofthe optimal portfolio.the optimal portfolio.

*Z Z!

* 0W !

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

w W *j

b

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

Therefore , if lies above the riskTherefore , if lies above the risk--returnreturnline in the plane, then the investorline in the plane, then the investorwould prefer to increase his holding inwould prefer to increase his holding in

security j.security j.

**

0

2

0

{ ( )} { }

{ ( ) }

j

j j

w W E U Z Z E Uw

Z W E U Z Z

dd d x!

ddx

jZ

( , )Z b

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Lemma 2.1:Lemma 2.1:

(i) for efficient(i) for efficientportfolioportfolio K.K.

(ii) If(ii) If thenthen

(iii) for efficient portfolio(iii) for efficient portfolio KK ififand only if for every efficientand only if for every efficientportfolioportfolio L.L.

Proof: (i) is a continuous monotonicProof: (i) is a continuous monotonicfunction of and hence and arefunction of and hence and arein one to one correspondence.in one to one correspondence.

{ } { }K

P K P eE Z V E Z Zd!

{ }K

P e p E Z Z Z !

cov( , ) 0p KZ

Vd!

cov( , ) 0p K

Z Vd!

cov( ) 0P L

Z Vd!

KVdK

eZ KVd

K

eZ

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(ii)(ii)

(iii)Because(iii)Becauseif , then .if , then .

Property I: IfProperty I: If LL andand KK are efficient portfolios,are efficient portfolios,then for any portfoliothen 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 d d d! ! !

0 cov( , ) 0K

p p Kb Z Vd! !0Kpb ! pZ R!

K K L

p L pb b b!

Lp p K K Le

L p p K K L

e e e

Z R Z RZ Rb b b

Z R Z R Z R

! ! !

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

Hence, all efficient portfolios have positiveHence, all efficient portfolios have positivesystematic risk, relative to any efficientsystematic risk, relative to any efficientportfolio.portfolio.

Property 3: if and only if forProperty 3: if and only if forevery efficient portfolioevery efficient portfolio K.K.

Property 4: LetProperty 4: Let pp andand qq denote any twodenote any two

feasible portfolios and letfeasible portfolios and let KK andand LL denotedenoteany two efficient portfolios. ifany two efficient portfolios. ifand only ifand only if

1KKb ! 0L

Kb "

pZ R! 0Kpb !

K K

p qb b

u

eL Lp q

b bu

e

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

Thus the measure provides the sameThus the measure provides the same

orderings of risk for any reference efficientorderings of risk for any reference efficientportfolio.portfolio.

Property 5: For each efficient portfolioProperty 5: For each efficient portfolio KKand any feasible portfolioand any feasible portfolio pp,,

where and forwhere and forevery efficient portfolioevery efficient portfolio L.L.

K K Lp L pb b b! K K Lq L qb b b!

K

pb

( )K K

p pe

p

Z R b Z R I!

{ } 0pE I ! { ( )} 0L

p L eE V ZI d !

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Hence , the systematic risk of a portfolio isHence , the systematic risk of a portfolio isthe weighted sum of the systematic risksthe weighted sum of the systematic risks

of its component securities.of its component securities. The Rothschild Stiglitz measure providesThe Rothschild Stiglitz measure provides

only for a partial ordering.only for a partial ordering.

measure provides a complete ordering.measure provides a complete ordering. They can give different rankings.They can give different rankings.

The Rothschild Stiglitz definition measureThe Rothschild Stiglitz definition measurethethe total risktotal risk of a security. It isof a security. It is

appropriate definition for identifyingappropriate definition for identifyingoptimal portfolios and determining theoptimal portfolios and determining theefficient portfolio set.efficient portfolio set.

K

pb

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The measure theThe measure the

systematic risksystematic risk

of aof asecurity.security.

To determine the , the efficientTo determine the , the efficientportfolio set must be determined.portfolio set must be determined.

The manifest behavioral characteristicThe manifest behavioral characteristicshared by all risk averse utilityshared by all risk averse utilitymaximization is to diversify.maximization is to diversify.

K

jb

K

jb

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The greatest benefits in risk reductionThe greatest benefits in risk reductioncome from adding a security to thecome from adding a security to theportfolio whose realized return tends to beportfolio whose realized return tends to behigher when the return on the rest of thehigher when the return on the rest of theportfolio is lower.portfolio is lower.

Next to suchNext to such countercyclicalcountercyclical investmentsinvestmentsin terms of benefit are the noncyclicin terms of benefit are the noncyclicsecurities whose returns are orthogonal tosecurities whose returns are orthogonal to

the return on the portfolio.the return on the portfolio.

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Theorem 2.5 : If and denote theTheorem 2.5 : If and denote the

returns on portfolioreturns on portfolio pp andand qq respectivelyrespectivelyand if, for each possible value of ,and if, for each possible value of ,

with strict inequalitywith strict inequalityholding over some finite probabilityholding over some finite probabilitymeasure of ,then portfoliomeasure of ,then portfolio pp is riskieris riskierthan portfolio q and .than portfolio q and .

Where , is theWhere , 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 Z

dZ dZu

eZ

p qZ Z"

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

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Proof:Proof:

is a strictly increasing function,is a strictly increasing function,

is a nondecreasing function, sois a nondecreasing function, so

From Theorem 2.4From 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

! !

!

!

!

( )e

Y Z ( ) ( )e p e q

G Z G Z

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

p qZ Z"

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Theorem 2.6: If and denote theTheorem 2.6: If and denote thereturns on portfolioreturns on portfolio pp andand qq respectivelyrespectivelyand if, for each possible value of ,and if, for each possible value of ,

, a constant, then, a constant, then

and .and .


pZ

qZ

eZ

( ) ( )p e q epq

e e

dG Z dG Za

dZ dZ !

p q pqb b a! ( ) p q pq

e Z Z a Z R!

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

cov[ ( ), ( ) ( )]

cov[ ( ), ]p q

e ep

eq

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! ! !

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Theorem 2.7: If, for all possible values ofTheorem 2.7: If, for all possible values 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 Z

dZ

" p eZ Z"

( )0 1p e

e

dG Z

dZ p e R Z Z

( )0p e

e

dG Z

dZ pR Z"

( )p ep

e

dG Za

dZ!

( )p p e Z R a Z R!

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Theorems 2.5, 2.6 and 2.7 demonstrate,Theorems 2.5, 2.6 and 2.7 demonstrate,the conditional expected return functionthe conditional expected return functionprovides considerable information about aprovides considerable information about asecuritysecuritys risk and equilibrium expecteds risk and equilibrium expectedreturn.return.

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AA mutualfundmutualfund is afinancial intermediaryis afinancial intermediarythatholds as its assets aportfolio ofthatholds as its assets aportfolio ofsecurities and issues as liabilities sharessecurities and issues as liabilities sharesagainstthis collectionofassets.againstthis collectionofassets.

Theorem 2.8 Ifthere existTheorem 2.8 Ifthere existMM mutualfundsmutualfunds

whose portfolio spanthe portfolio set ,whose portfolio spanthe portfolio set ,thenall investors willbe indifferentthenall investors willbe indifferentbetweenselecting their optimalportfoliosbetweenselecting their optimalportfolios

from andselecting from portfoliofrom andselecting from portfoliocombinationofjustthecombinationofjustthe MM mutualfunds.mutualfunds.

]

]

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Therefore the smallest number of suchTherefore the smallest number of suchfunds is a particularly importantfunds is a particularly importantspanning set.spanning set.

When such spanning obtain, the investorWhen such spanning obtain, the investorssportfolioportfolio--selection problem can beselection problem can be

separated into two steps.separated into two steps.

However, if the smallest funds can beHowever, if the smallest funds can beconstructed only if the fund managersconstructed only if the fund managers

know the preferences, endowments, andknow the preferences, endowments, andprobability beliefs of each investor.probability beliefs of each investor.

M

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Theorem 2.9: Necessary conditions for theTheorem 2.9: Necessary conditions for the

M feasible portfolios with returnM feasible portfolios with returnto span the portfolio set are (a) thatto span the portfolio set are (a) thatthe rank of and (b) that there existthe rank of and (b) that there existnumbers such that thenumbers such that the

random variable has zero variance.random variable has zero variance.

Proposition 2.1: If is theProposition 2.1: If is thereturn on some security or portfolio and ifreturn on some security or portfolio and if

there are nothere are no

arbitrage opportunitiesarbitrage opportunities

thenthen

1( , , )MX XL

f]

M; e

1 1( , , ), 1

M

M jH H H !L

1

M

j jXH

1

n

p j j Z a Z b!

1 1( ) (1 ) ( ) ( )

n n

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

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Proof: Let be the return on a portfolioProof: Let be the return on a portfoliowith 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 risklessand allocated to the risklesssecurity with return R, if is chosen suchsecurity with return R, if is chosen such

that ,then isthat ,then isriskless security and therefore butriskless security and therefore butcan be chosen arbitrarily. So we get thecan be chosen arbitrarily. So we get theresult.result.

Z

jH 1, , ;j n! L

pH pZ

11

n

p jH H

jH

j p jaH H

! 1[ (1 )]n

p jZ R b R aH

! Z R

!

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Hence, as long as there are no arbitrageHence, as long as there are no arbitrageopportunities, it can be assumed withoutopportunities, it can be assumed withoutloss of generality that one of the portfoliosloss of generality that one of the portfoliosin any candidate spanning set is thein any candidate spanning set is theriskless security.riskless security.

Theorem 2.10: A necessary and sufficientTheorem 2.10: A necessary and sufficientcondition for to span is thatcondition for to span is thatthere exist number such thatthere exist number such that

1( , , , )mX X RLf]

{ }ija

1 ( ) 1, 2, , .

m

j ij i Z R a X R j n! ! L

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Proof: If span , thenProof: If span , then

such that . Becausesuch that . Because

and substituting , weand substituting , wehavehave

we pick the portfolio weightswe pick the portfolio weights

for and , fromfor and , fromwhich it follows that .But everywhich it follows that .But everyportfolio in can be written as a portfolioportfolio in can be written as a portfolio

combination of and R.combination of and R.

1( , , , )mX X RL

11

M

ijH !

f]

1

M

j ij iZ XH!

MX R! 11m

Mj ijH H!

1( ) 1, 2, , .

m


ij ijaH !

1, ,i m! L1

1m

Mj ijH H! 1

M

j ij iZ XH!

f]

1( , , )nZ ZL

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Corollary 2.10: A necessary and sufficientCorollary 2.10: A necessary and sufficientcondition for to be the smallestcondition for to be the smallest

number of feasible portfolio that span isnumber of feasible portfolio that span isthat 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. Moreoverare linearly independent. Moreover

hence, if the rank of then therehence, if the rank of then thereexist number such thatexist number such that

for . Thereforefor . Thereforewhere by Theorem 2.10where by Theorem 2.10spanspan

1( , , , )mX X RL

; X m; !

X m; !

m; !

{ }ij

a1

( )m

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

1

m

j j ij iZ b a X!

1

m

j j ij ib Z a X! f]

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It follows from Corollary 2.10 that aIt follows from Corollary 2.10 that anecessary and sufficient condition fornecessary and sufficient condition for

nontrivial spanning of is that some ofnontrivial spanning of is that some ofthe risky securities are redundantthe risky securities are redundantsecurities.securities.

By Theorem 2.10, if investors agree on aBy Theorem 2.10, if investors agree on aset of portfolios such thatset of portfolios such that

and if theyand if they

agree on the number ,thenagree on the number ,thenspan even if investors do not agree onspan even if investors do not agree onthe joint distribution ofthe joint distribution of

f]

1( , , , )mX X RL

1( ) 1, 2, , .

m


{ }ija

1( , , , )mX X RLf]

1( , , , )mX X RL

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Proof: LetProof: Let

because , thusbecause , thus

bybyconstruction , and henceconstruction , and hence

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

thereforetherefore

1 1(1 )

m m

p j i i iZ Z X RH H H H !

1( )

m

j j ij i i j Z Z a X X I!

1

[ ( )]m

p jij

ij

Z R Z R a X RH HI!

{ } 0jE I ! cov( , ) 0p KZ Vd!

K

pb

pZ R!

1( )

m

j ij i Z R a X R!

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Hence, if the return on a security can beHence, if the return on a security can bewritten in this linear form relative to thewritten in this linear form relative to the

portfolios , then its expectedportfolios , then its expectedexcess return is completely determined byexcess return is completely determined bythe expected excess returns on thesethe expected excess returns on these

portfolios and the weights .portfolios and the 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 , thenwhere , thenspan the set of efficient portfolios .span the set of efficient portfolios .

{ }ija

1( , , )mX XL

{ }ij

a

1 ( )

m

j ij i jZ Ra X

R I! 1{ , , } 0j mE X XI !L 1( , , , )mX X RL

e]

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Proof:Proof:

WhereWhere

Construct portfolioConstruct portfolio

Thus whereThus whereHence, for , is riskier than Z,Hence, for , is riskier than Z,which contradicts that is and efficientwhich contradicts that is and efficientportfolio. So . We get the result.portfolio. So . We get the result.

1 1 1

1 1 1 1

1

[ ( ) ]

( )

( )

j j

j j j

n n m K K K

ej

ij

ij

n n m m K K K

ij i j

m K K

i i

Z w Z w R a X R

w R w a X R w

R X R

I

I

H I

! !

!

!

1

nK K

i j ijw aH ! 1 j

mK K

jwI I!

1 1(1 )

m mK K

i i iZ X RH H!

K K

e

Z ZI

!

{ } 0

KE Z

I!

0KI {K

eZ

K

eZ

0KI !

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Theorem 2.13: Let denote the fractionTheorem 2.13: Let denote the fractionof efficient portfolioof efficient portfolio KK allocation toallocation to

securitysecurity j j, spa, span ifand only if there exist number for everyand only if there exist number for everysecurity j such thatsecurity j such that

where forwhere forevery efficient portfolioevery efficient portfolio K.K.

Corollary 2.13: (X,R) span if and only ifCorollary 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 thatsuch that

wherewhere

K

jw

11, , . ( , , , )mj n X X R! L Le]

{ }ij

a

1( )

m

j ij i j Z R a X R I!

1 1{ } 0,m n K K K

j i i i j ijE X w aI H H! !

e]

ja

1, , ,j n! L ( ) j j j Z R a X R I!

{ } 0jE XI !

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Proof: By hypothesis, forProof: By hypothesis, forevery efficient portfolio K. If , thenevery efficient portfolio K. If , then

from Corollary 2.1 for everyfrom Corollary 2.1 for everyefficient portfolio K and R span .efficient portfolio K and R span .Otherwise, from Theorem 2.2, forOtherwise, from Theorem 2.2, for

every efficient portfolio. By Theorem 2.13,every efficient portfolio. By Theorem 2.13,soso

Since is contained in , any propertiesSince is contained in , any properties

proved for portfolios that span must beproved for portfolios that span must beproperties of portfolio that span .properties of portfolio that span .

( )K Ke

Z X R RH!

X R!

0KH !e

]

0KH {

{ } 0K

jE XI H ! { } 0jE XI !

f]

f]e

]

e

]

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From Theorem 2.10, 2.12, 2.13, theFrom Theorem 2.10, 2.12, 2.13, theessential difference is that to span theessential difference is that to span the

efficient portfolio set it is not necessaryefficient portfolio set it is not necessarythat linear combinations of the spanningthat linear combinations of the spanningportfolios exactly replicate the return onportfolios exactly replicate the return on

each available security.each available security. All the models that do not restrict theAll the models that do not restrict the

class of admissible utility function, theclass of admissible utility function, the

distribution of individual security returnsdistribution of individual security returnsmust be such thatmust be such that

1( )

m

j ij i j Z R a X R I!

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Proposition 2.3: If, for every security j,Proposition 2.3: If, for every security j,

with linearlywith linearlyindependent with finite variances and ifindependent with finite variances and ifthe return on security j, has a finitethe return on security j, has a finitevariance, then the invariance, then the in

Theorems 2.12 and 2.13 are given byTheorems 2.12 and 2.13 are given bywhere is the ikthwhere is the ikth

element of .element of .

Hence given some knowledge of the jointHence given some knowledge of the jointdistribution of a set of portfolio that spandistribution of a set of portfolio that span

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

1{ , , } 0j mE X XI !L 1( , , )mX XL

jZ

{ } 1, , ,ija i m! L

1cov( , )

m

ij ik K ja v X Z! ikv1

X

;

e]

j jZ Z ija jZ

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Proposition 2.4: If contain noProposition 2.4: If contain noredundant securities, denotes theredundant securities, denotes the

fraction of portfoliofraction of portfolio XX allocated to securityallocated to security j, and denotes the fraction of any risk j, and denotes the fraction of any risk--averse investoraverse investors optimal portfolios optimal portfolioallocated to security j, then forallocated to security j, then forevery such riskevery such risk--averse investoraverse investor

1( , , )nZ ZL

jH

jw

1, , ,j n! L

*, 1, 2, ,

j j

k k

wj k n

w

H

H

! ! L

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Because every optimal portfolio is anBecause every optimal portfolio is anefficient portfolio and the holding of riskyefficient portfolio and the holding of risky

securities in every efficient portfolio aresecurities in every efficient portfolio areproportional to the holding in X.proportional to the holding in X.

If there exist numbers whereIf there exist numbers where

and ,then the portfolio withand ,then the portfolio withproportions is called theproportions is called the OptimalOptimalCombination of Risky Assets.Combination of Risky Assets.

Proposition 2.5: If span , thenProposition 2.5: If span , thenis a convex set.is a convex set.

jH

*

* , , 1, ,j j

k k

j k nH H

H H! ! L

*

1

n

jH* *

1( , )nH HL

( , )X Re

]e

]

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Proof: LetProof: Let

and , . Byand , . By

substitution, the expression for Z can besubstitution, the expression for Z can berewritten as , whererewritten as , where

.Therefore by Proposition.Therefore by Proposition

2.2, Z is an efficient portfolio. It follow by2.2, Z is an efficient portfolio. It follow byinduction that for any integer k andinduction that for any integer k andnumber such that andnumber such that and

is the return on anis the return on anefficient portfolio. Hence , is a convexefficient portfolio. Hence , is a convexset.set.

1

1( )eZ X R RH! 2

2( )

eZ X R RH!

1 2H H{1 2(1 )e e

Z Z Z P P!

1( )e

Z Z R RH!

2

1

( )(1 )H

H P PH

!

iP 0 1, 1, ,i i kPe e ! L

1 11,

k kk i

i i eZ Z

P P! !

e]

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We use denote the market value ofWe use denote the market value ofsecurity j and denote the value of thesecurity j and denote the value of the

riskless security, then is the fraction ofriskless security, then is the fraction ofsecurity j held in a market portfolio.security j held in a market portfolio.

Theorem 2.14: If is a convex set, and ifTheorem 2.14: If is a convex set, and ifthe securitiesthe securities market is in equilibrium,market is in equilibrium,

then a market portfolio is an efficientthen a market portfolio is an efficientportfolio.portfolio.

jV

RV

M

jH

1

jM

j n

j R

V

V V

H !

e

]

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Proof: Let there be K risk averse investorProof: Let there be K risk averse investorin the economy.Definein the economy.Define

to be the return on investor kto be the return on investor ks optimals optimalportfolio. In equilibrium, ,portfolio. In equilibrium, ,where is the initial wealth of investorwhere is the initial wealth of investor

K, and . DefineK, and . Define. By definition of a market. By definition of a market

portfolio Multiplyingportfolio Multiplying

by and summing over j, it followsby and summing over j, it followsthatthat

1( )

nK k

j j Z R w Z R!

01

K k k

j jw W V!0

kW

0 01 1

K nK

j RW W V V | ! 0

0

1,k

k

Wk K

WP ! ! L

11, , .

K k M

j k jw j nP H! ! LjZ R

1 1 1

1

( ) ( )

( )

K n Kk K

k j j K

n M

i j M

w Z R Z R

Z R Z R

P P

H

!

! !

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Proposition 2.6: In all portfolio modelsProposition 2.6: In all portfolio modelswith homogeneous beliefs and riskwith homogeneous beliefs and risk--averseaverse

investors the equilibrium expected returninvestors the equilibrium expected returnon the market portfolio exceeds the returnon the market portfolio exceeds the returnon the riskless security.on the riskless security.

Proof: From the proof ofTheorem 2.14Proof: From the proof ofTheorem 2.14and Corollary 2.1. ,and Corollary 2.1. ,because , . Hencebecause , . Hence

The market portfolio is the only riskyThe market portfolio is the only riskyportfolio where the sign of its equilibriumportfolio where the sign of its equilibriumexpected excess return can always beexpected excess return can always bepredicted.predicted.

1( )

K k

M k Z R Z RP ! k

Z Ru 0kP " MZ R"

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Returning to the special case where isReturning to the special case where isspanned by a single risky portfolio and thespanned by a single risky portfolio and the

riskless security, the market portfolio isriskless security, the market portfolio isefficient. So the risky spanning portfolioefficient. So the risky spanning portfoliocan always be chosen to be the marketcan always be chosen to be the market

portfolio.portfolio. Theorem 2.15: If span , then theTheorem 2.15: If span , then the

equilibrium expected return on security jequilibrium expected return on security jcan be written ascan be written as

wherewhere

e]

( , )M

Z Re]

( )j j M Z R Z RF!

cov( , )

var( )

j M

j

M

Z Z

ZF !

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

In the special case ofTheorem 2.15,In the special case ofTheorem 2.15,measure the systematic risk of security jmeasure the systematic risk of security jrelative to the efficient portfolio .relative to the efficient portfolio .

can be computed from a simplecan be computed from a simplecovariance between and . But thecovariance between and . But thesign of can not be determined by thesign of can not be determined by the

sign of the correlation coefficient betweensign of the correlation coefficient betweenandand

jF

MZ

jF

jZ MZk

jb

jZk

eZ

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Theorem 2.16: If contain noTheorem 2.16: If contain noredundant securities, then (a) for eachredundant securities, then (a) for eachvalue are unique, (b)value are unique, (b)there exists a portfolio contained inthere exists a portfolio contained inwith returnwith return XX such that span ,such that span ,

and (c) where,and (c) where,

Where denote the set of portfoliosWhere denote the set of portfolios

contained in such that there exists nocontained in such that there exists noother portfolio in with the sameother portfolio in with the sameexpected return and a smaller variance.expected return and a smaller variance.

1( , , )nZ ZL

, , 1, , ,j j nQQ H ! L

( , )X R min]

( ) j j j Z R a X R ! cov( , )

, 1, , .var( )

jj

Z Xa j n

X! ! L

min]f

] f]

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Proof: Let denote theProof: Let denote the ijthijth element ofelement ofand denote theand denote the ijthijth element of . Soelement of . So

all portfolios in with expect return u,all portfolios in with expect return u,we need solutions the problemwe need solutions the problem

If then andIf then and

Consider the case when . The n firstConsider the case when . The n first--

order conditions areorder conditions are

ijW

ijv

;1;

min=

1 1min

. ( )

n n

i j ij

S Z

H H W

!

RQ ! ( ) Z R R! 0, 1, 2 ,R

j j nH ! ! L

RQ {

10 ( ) 1, 2, ,

n

j ij u iZ R i n

QH W P! ! L

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portfolio in with and call itsportfolio in with and call itsreturn X. Then we havereturn X. Then we have

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

and from Corollary 2.13 and Propositionand from Corollary 2.13 and Proposition2.3 (c) follows directly.2.3 (c) follows directly.

min] RQ {

( ) ( )Z X R RQQ H!

( , )X Rmin]

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From Theorem 2.16, will be equivalentFrom Theorem 2.16, will be equivalentto as a measure of a securityto as a measure of a securityss

systematic risk provided that thesystematic risk provided that thechosen forchosen for XX is such that .is such that .

Theorem 2.17: If span and if XTheorem 2.17: If span and if X

has a finite variance, then is containedhas a finite variance, then is containedin .in .

Proof: Let . Let beProof: Let . Let be

the return on any portfolio in suchthe return on any portfolio in suchthat . By Corollary 2.13that . By Corollary 2.13

wherewhere

ka

K

kb

RQ "

( , )X Re]

e

]min]

( )e e

Z R a X R! pZf

]e p

Z Z! ( ) p p p Z R a X R I!

{ } { } 0p pE E XI I! !

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ThereforeTherefore

ThusThus

Hence, is contained in .Hence, is contained in .

Theorem 2.18: If have a jointTheorem 2.18: If have a jointnormal probability distribution, then therenormal probability distribution, then thereexists a portfolio with returnexists a portfolio with return XX such thatsuch that

span .span .

p ea a!

2var( ) var( ) var( ) var( ) var( ) p p p p eZ a X a X ZI! u !

min

]eZ

1( , , )nZ ZL

( , )X Re]

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Th 2 19 If i t iTh 2 19 If i t i( )

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Theorem 2.19: If is a symmetricTheorem 2.19: If is a symmetricfunction with respect to all its arguments,function with respect to all its arguments,

then there exists a portfolio with return Xthen there exists a portfolio with return Xsuch that span .such that span .


for each setfor each setof given values. Therefore every riskof given values. Therefore every riskaverse investor will choose . But thisaverse investor will choose . But thisis true for allis true for all i.i. Hence , all investor willHence , all investor willhold all risky securities in the samehold all risky securities in the samerelative proportions. Then spanrelative proportions. Then span

( , )X R e]

1( , , )np Z ZL

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

1 iH H !

( , )X R e]

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The APT model developed by RossThe APT model developed by Rossprovides an important class of linearprovides an important class of linear--factorfactor

models that generate spanning withoutmodels that generate spanning withoutassuming joint normal probabilityassuming joint normal probabilitydistributions.distributions.

If we can construct a set of m portfoliosIf we can construct a set of m portfolioswith returns such that andwith returns such that andare perfectly correlated, thenare perfectly correlated, then

will spanwill span

1( , , )MX XL iX iY

1, , ,i m! L

1( , , , )

M

X X RLe]

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F th t d f ilib i i i thF th t d f ilib i i i th

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For the study of equilibrium pricing, theFor the study of equilibrium pricing, theusual format is to derive equilibriumusual format is to derive equilibrium

given the distribution of .given the distribution of . Theorem 2.20: If denote a set ofTheorem 2.20: If denote a set of

linearly independent portfolios that satisfylinearly independent portfolios that satisfy

the hypothesis ofTheorem 2.12the hypothesis ofTheorem 2.12, and all, and allsecurities have finite variances, then asecurities have finite variances, then anecessary condition for equilibrium in thenecessary condition for equilibrium in thesecuritiessecurities market is thatmarket is that

where is thewhere is the ikthikth element ofelement of

0jV

jV

1 10

cov( , )( )m m

j ik k j j

j

V v X V X RV

R

!

ikv

1

X

;

1( , , )mX XL

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Hence from Theorem 2 20 a sufficientHence from Theorem 2 20 a sufficient

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Hence, from Theorem 2.20, a sufficientHence, from Theorem 2.20, a sufficientset of information to determine theset of information to determine the

equilibrium value of security j is the firstequilibrium value of security j is the firstand second moments for the joinand second moments for the joindistribution of .distribution of .

Corollary 2.20a: If the hypothesizedCorollary 2.20a: If the hypothesizedconditions ofconditions ofTheorem 2.20Theorem 2.20 hold and if thehold and if theendend--ofof--period value a security is given byperiod value a security is given by

then in equilibriumthen in equilibrium

This property of formula is calledThis property of formula is called valuevalueadditivityadditivity..

1( , , , )

m jX X VL

1

n

j jV VP!

0 01

n

j jV VP!

C ll 2 20b If h h h i dC ll 2 20b If h h h i d

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Corollary 2.20b: If the hypothesizedCorollary 2.20b: If the hypothesizedconditions ofconditions ofTheorem 2.20Theorem 2.20 hold and if thehold and if the

endend--ofof--period value of a security is givenperiod value of a security is givenby , whereby , whereand then inand then in

equilibriumequilibrium Hence, to value two securities whose endHence, to value two securities whose end

of period values differ only byof period values differ only bymultiplicative or additivemultiplicative or additive noisenoise, we can, we can

simply substitute the expected values ofsimply substitute the expected values ofthe noise terms.the noise terms.

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

1{ } { , , }

mE q E q X X q! !L

0 0jV qV uR!

Th 2 20 d it ll iTh 2 20 d it ll i

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Theorem 2.20 and its corollaries areTheorem 2.20 and its corollaries arecentral to the theory of optimalcentral to the theory of optimal

investment decisions by business firms.investment decisions by business firms. Although the optimal investment andAlthough the optimal investment and

financing decisions by a form generallyfinancing decisions by a form generally

require simultaneous determination, underrequire simultaneous determination, undercertain conditions the optimal investmentcertain conditions the optimal investmentdecision can be made independently ofdecision can be made independently ofthe method of financing.the method of financing.

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Hence for a given investment policy theHence for a given investment policy the

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Hence, for a given investment policy, theHence, for a given investment policy, theway in which the firm finances itsway in which the firm finances its

investments changes the returninvestments changes the returndistribution of the efficient portfolio set.distribution of the efficient portfolio set.

Clearly, a sufficient condition for TheoremClearly, a sufficient condition for Theorem

2.21 to obtain is that each of the financial2.21 to obtain is that each of the financialclaims issued by the firm areclaims issued by the firm are redundantredundantsecuritiessecurities..

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An alternative approach to theAn alternative approach to thedevelopment of nontrivial spanningdevelopment of nontrivial spanningtheorems is to derive a class of utilitytheorems is to derive a class of utilityfunctions for investors .functions for investors .

Such that even with arbitrary jointSuch that even with arbitrary jointprobability distributions for the availableprobability distributions for the availablesecurities,investors within the class cansecurities,investors within the class can

generate their optimal portfolios from thegenerate their optimal portfolios from thespanning portfolios.spanning portfolios.

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Let denote the set of optimal portfoliosLet denote the set of optimal portfolios

selected from by investors with strictlyselected from by investors with strictlyconcave von Neumannconcave von Neumann--Morgenstern utilityMorgenstern utilityfunctions.functions.

Theorem 2.22 There exists a portfolio withTheorem 2.22 There exists a portfolio withreturn X such that span if andreturn X such that span if andonly if , where is theonly if , where is theabsolute riskabsolute risk--aversion function for investoraversion function for investor

in .in .

u]

f

]

u]

u]( , )X R

( ) 1 ( ) 0i i

A W a bW! "i

A

i

B th b i th t t t fB th b i th t t t f

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Because the b in the statement ofBecause the b in the statement ofTheorem 2.22 does not have a subscript ,Theorem 2.22 does not have a subscript ,

therefore all investors in must havetherefore all investors in must havevirtually the same utility function.virtually the same utility function.

Cass and Stiglitz (1970) conclude: it isCass and Stiglitz (1970) conclude: it is

requirement that there be any mutualrequirement that there be any mutualfunds, and not the limitation on thefunds, and not the limitation on thenumber of mutual funds.number of mutual funds.

This is a negative report on the approachThis is a negative report on the approachto developing spanning theorems.to developing spanning theorems.

iu]

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The EndThe End

thanksthanks

ctf-chapter 2 (1)

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