chapter 6 relation between discount factors,betas,and mean-variance frontiers

Asset Pricing Zheng Zhenlong

CHAPTER 6Relation between Discount Factors,Betas,and Mean-Variance Frontiers

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Main contents

• we will draw the connection between discount factors,mean-variance frontiers, and beta representations,then we will show how they transform between each other,because these three representations are equivalent.


Transformation between the three representations

( )p E mx( ) .p E mx eRorRRxm ,,,

)(mxEp R

)|( Xmprojx )( 2 xExR eRRR ,

)(mxEp mvbRam )(mxEp


Transformation between the three representations(2)

• . If we have an expected return-beta model with factors f , then linear in the factors satisfies .

• If a return is on the mean-variance fron-tier,then there is an expected return-beta model with that return as reference variable.

)(mxEp fbm )(mxEp


Transformation between the three representations(2)


6.1 From Discount Factors to Beta Representations

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Asset Pricing Zheng ZhenlongBeta representation using m

Multiply and divide by var(m),define ,we get:

),cov()()()(1 iii RmREmEmRE

)(

),cov(

)(

1)(

mE

Rm

mERE

ii

)(1 mE

mmi

ii

mE

m

m

RmRE ,)

)(

)var()(

)var(

),cov(()(

Asset Pricing Zheng ZhenlongTheorem

)( 2 xExR

xxi

iRE ,

)( )()(

,RERE

Ri

i

),cov()()()()(1 iiii RxRExERxEmRE

)(

)var(

)var(

),cov(

)(

1

)(

),cov(

)(

1)(

xE

x

x

Rx

xExE

Rx

xERE

iii

Asset Pricing Zheng ZhenlongProof

xxi

iRE ,

)(

)( 2 xExR

*2 *2( ) cov( , ) ( ) cov( , ) var( )( )

( ) ( ) ( ) var( ) ( )

i ii E R R R E R R R R

E RE R E R E R R E R

)(

)var()(

RE

RRE

)()(

,RERE

RR

ii


P=0(超额收益率）

Rf

P=1(收益率）

状态 1回报

状态 2回报

R*1

Re*

x*

pc

Asset Pricing Zheng ZhenlongSpecial case


6.2 From Mean-Variance Frontier to a Discount Factor and beta Representation

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)( nRRbabRam e

)()()(1 2 RbERaEmRE

)()()()()(0 2 eeee REbaREbRaEmRE

2 2

1,

( ) ( ) ( ) ( )a b

E R E R E R E R

Asset Pricing Zheng ZhenlongProof(2)

)()(

)(2

RERE

nRRm

e

ieiii nRRyx


))()(

))((()(

2

RERE

nRRynRREmxE

ieiiei

2

2

2

( ) ( ) ( )( )

( ) ( )

( )

( ) ( )

i i ii

ii

y E R y E R E nnE mx

E R E R

E nny

E R E R

n

Asset Pricing Zheng ZhenlongNote

• If the denominator is zero, i.e., if ,this construction cannot work.

• If there is a risk-free rate, we are ruling out the case

• If there is no risk-free rate, we must rule out the

case (the “constant- mimicking portfolio return”).• 证毕。

2( ( ) 1 ( )E R E R E x ）

fefmv RRRRR

*2 *( ( ) / ( ))mv eR R E R E R R


6.3Factor Models and Discount Factors

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)(1 fEfbm


1 ( ) ,0 ( )em b f E f E mR

ieRE )(


• From (6.7),

• Here we get (6.8)

• where

),cov()()(0 eee RfbREmRE

),cov()var()var(

),cov()(1 e

ee

Rfffb

RfbRE

bf )(rva


)(1, imREfbam

iiRE )(


a

bfRE

amE

Rm

mERE

)(1

)(

),cov(

)(

1)(

)()( 1 ii fREffE

a

bffE

a

a

bffEffEfRE

aa

bfRE

aRE

)(1

)()()(1)(1)(

1

1 1 1, cov( ) ( )

( )ff b E mf

E m a a

iiRE )(


( )( ) ( )

E fp f E f p f

Asset Pricing Zheng ZhenlongFactor-mimicking

porfolios

)())|(()()( xfbExXfprojbEfxbEmxEp


')( iRE


)|(

)|(

Xfprojp

Xfprojf

i

ii

)()(,,

fEREfifi

ei


6.4 Discount Factors and Beta Models to Mean-Variance Frontier

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6.5 Three Risk-free Rate Analogues

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)()()( RERERRE

)(

1

)(

)(

)(

)()(

2

xERE

RE

RE

RRERE


R

R)(RE

)(R

)( RE

=E(R*2)/E(R*)

其长度为 *2( )E R

利用相似三角形


)()()( 22 RERERE)()( 2 RERE


)()()(

)()(

2

eRERERE

RERE

)()(

)var(

)()(

)()( 22

ee RERE

R

RERE

RERE

e

eR

RERE

RRR

)()(

)var(


Minimum-Variance Return• The risk-free rate obviously is the minimum -

variance return when it exists. When there is no risk-free rate, the minimum-variance return is

(6.15)• Taking expectations,

e

eR

RE

RERR

)(1

)(.var.min

)(1

)()(

)(1

)()()( .var.min

ee

e RE

RERE

RE

RERERE


eRRERR )( .var.min.var.min

)()()(2)()()(

)()()var(min

222222

22

eee

eee

RERERERERERE

RRERRERR

),()()(1)(0 eee RERERERE)(1

)(

eRE

RE

Asset Pricing Zheng ZhenlongConstant-Mimicking Portfolio

Return

)|1(

)|1(ˆXprojp

XprojR


ee R

RE

RERRRR

)(

)(ˆ2

R

RE

REXprojRe

)(

)()|1(

2

)(

)()|1(

2

RE

REXprojp

R

RE

RERXproj e

)(

)()|1(

2

Asset Pricing Zheng ZhenlongRisk-Free Rate

• Here we will show that if there exists a risk-free rate,then all the zero-beta return, minimum-variance return,and constant-mimicking portfolio return reduce to the risk-free rate.

• These other rates are:• Constant-mimicking:

eR

RE

RERR

)(

)(ˆ2


• Minimum-variance:

• Zero-beta:• And the risk-free rate:

(6.19)• To establish that there are all the same when

there is a risk-free rate, we need to show that:

e

eR

RE

RERR

)(1

)(.var.min

e

eR

RERE

RRR

)()(

)var(

eff RRRR

)()(

)var(

)(1

)(

)(

)( 2

ee

f

RERE

R

RE

RE

RE

RER


)()( eff RERRER

)(

)(

)(

1 2

RE

RE

xER f

)()()(

)( 2

ef RERRERE

RE

)()(

)var(

)()(

)()( 22

ee

f

RERE

R

RERE

RERER


6.6 Mean-Variance Special Cases with No Risk-Free Rate

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• There exist special cases for the equivalence theorems,that is,when the expected discount factor,price of a unit payoff,or risk-free rate is zero or infinity.

• If risk-free rate is traded or the market is complete,then it won’t be a problem; however,in an incomplete market in which no risk free rate is traded,we must pay attention to it and make it sure that

)(0 ME


The special case for a mean-variance frontier to a discount factor

eRRERER )()( 2


The special case for mean-variance frontier to a beta model• We can use any return on the mean-variance

frontier as the reference return for a single-beta representation,except the minimum-variance return.


Theorem:

mvmvmv R

mv

RiR

i RERE )()(,

)()()( eii RERERE

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)()()var()()()var(

)()()()var()var(

)(),(cov),cov(

eeie

eiei

eiemvi

RERERRERER

RERERR

RRRRRR

)(1

)(

)()(

)()(

)var(

)()(22

e

ee

e

e

e

RE

RE

RERE

RERE

R

RERE

chapter 6 relation between discount factors,betas,and mean-variance frontiers

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