金融學概論徐信忠光華-作業(yè)frommarkwitzto

Mean-Variance

TheoryThe

quadratic

program(Markowitz

1952):wT1

1Where,

e

is

the

expected

return

vector,

andV

the

variance

covarance

matrix,w

the

weight

vector,E[~r ]

the expected

return

of

the

portfolio.pmin

1

wTVw{w}

2ps.t.

wT

e

E[r~

]12FOC

:Vw

e

1

0E[rp

]

w e

0T1

wT

1

0min

L

{w,

,

}T(1

w

1)Tw e)

pTw

Vw

(E[r

]

The

Lagrangian

Approach~T

1pmultiplyin

g

by

1T

,

gives

:1

(1T

V

1e)

(1T

V

11)E[r]

(eTV

1e)

(e

V

1)w

(V

1e)

(V

11)multiplyin

g

by

eT

,

gives

:The

optimumA

eTV

11

1T

V

1eB

eTV

1eC

1T

V

11D

BC

A2where

:DpDB

AE

[

~r

]pCE[

~r

]

A

w*

g

hE[r~

]Dh

1

[C(V

1e)

A(V

11)]Dg

1

[B(V

11)

A(V

1e)]wherepA~p

q

p

qC

(E[~r

DA

1cov(r~

,

r~

)

wTVw

CovarianceC

CDCp2pTppp

p2p]

)~

(E[r~

~

cov(r,

r

)

w

Vw2A

1VarianceMinimum

Variance

PortfolioMinimum

variance

portfolio

(mvp):And

the

expected

return

is,The

covariance

of

mvp

and

any

portfoliois1/C}{

,C

CA

1CpE[r~

]

ACV

1I]

g

hA/

C

p~The

portfoliois,

w*

g

hE[rSome

ShortcutsTangent

portfolio:weights

vector

:

wM

weights

vector

:

wmvp

Minimum

variance

portfolio

(mvp):V

1IC

CA

1

}Mean

Variance

pair

:{,CV

1

(r

rf

I

)fMean

Variance

pair

:{r

wM

,

wMVwM

}A

r

CTreynor

(1961),

Sharpe

(1961,

1964),

Litner(1965),

Mossin(1966),Black(1972)風險資產(chǎn)有效前沿

PPr市場組合MCMLrPrMrf

M

PProof

1:

Tangent

Line

Geometrics一個投資組合，其中a%投資于風險資產(chǎn)i

，(1-a%)投資于市場組合，則該組合的均值和標準差為(1)(2)]1/

2

2a(1

a)rp

ari

(1

a)rM2

2M

(1

a)

2

2iiMp

[a

(4)]1/

2(3)i

MiMiM2[a2

2

(1

a)2

2

2a(1

a)

4a2a

2

2

2

2a

2

2i

M

M

iMpMirpaa

r

r利用方程(3)、(4)，當a=0時，可以得到a

0Mia

0a

p

r

rarp

2

iM M

Ma的變動對均值和標準差的影響為：在市場達到均衡時，點M處的風險－收益曲線的斜率為：M

Mri

rM

p

p

/

a

(

iM

2

)

/rp

rp

/

aa0

rM

rf

MMM(

iM

2

)

/ri

rM在點(組合)M處，CML的斜率，(rM

rf

)

/

M，必須等于曲線IMI’的斜率：CAPM與市場線(SML)期望收益率和風險之間的均衡關(guān)系為：市場線SML：(rM

rf

)M

2

iMi

fr

rNi1i

iM

2M

M

MiiM

iM

i

2

i

f i

M

fr

r

r

r

Proof

2:

Pure

AlgebraThe

tangent

portfolio

satisfyingWhere,

ei

is

a

vector

with

the

ith

element

being1

and

others

0.V

1(r

rf

I

)weights

vector

:

wM

fMean

Variance

pair

:{r

wM

,

wMVwM

}Covariance

between

the

tangent

portfolio

andsecurity

i

is:A

r

Ci

ff

ffiMA

r

CA

r

CV

1(r

r

I

) (r

r

)i

M

i

M

i

cov(r

,r

)

e

Vw

e

V

,Proof

2

cont’dWe

know

the

variance

of

the

tangent

portfolio

is,And(r

r

I

)V

1(r

r

I

)f

f(

A

rf

C)22MVM

M

w

Vw

f

fA

r

C A

r

C(r

rf

I

)V

1

V

1

(r

rf

I

)fA

r

C

(ri

rf

)

/f(

A

r

C)2i

f

f

ffA

r

C iM

2M/(r

r

I

)V

1(r

r

I

)f

f(r

r

) (r

r

I

)V

1(r

r

I

)Proof

2

cont’dWith

the

returnThat’sThe

last

step

is

to

verify(r

r

I

)V

1(r

r

I

)f

ffMf

r

rA

r

C1I

)or

r

r

r

r

i

f

i

M

f

(ri

rf

)

/(rMf

r

),iMi

2MProof

3:

Geometics

and

AlgebraCombinedMean and

variance

of

the

returns:The

optimal

portfolio

consisting

of

security

i

andmarket

portfolio

(x,

1-x)

should

be

(0,

1),

whichmeans

x=0.2

iM M

iM

2i

rM

ri

r

,

V

V

1(r

rf

I

)A

rf

Cweights

vector:

wM

Proof

3:

cont’dThat’s

wTherow

means

CAPM22xffiM

ii

M

ff

M

i

fiM

M

/(

A

r

C)fifr

rif

/(

A

r

C)M f

1

0

(r

r

)

(r

r

)

(r

r

)