Σ(r) MVP E(r) Standard deviation of return C A B Expected return % 15% 10%

σ(r)

MVP

E(r)

Standard deviation of return

C

A

B

Expecte

d re

turn

%

15%

10%


E(r)

σ(r)10% 20% 30% 40% 50%

Exp

ecte

d re

turn

%

MVP

N

Consolidated

Z

Q

W

Acme

Brown

XA

Portfolio weight in stock B

1.00

1.00

-1.00

-1.00

Q

U

S

L

Y’

XB

Portfolio weight in stock A

0

X

XA


1.00

1.00

-1.00

-1.00

Q

U

S

L

Y’

XB


0

X

T

R

1.00

1.00

-1.00

-1.00

XB


0XA


Isoexpected return line

10%12%

16%

18%

20%

B

XB

MVP

-1.00 0 1.00XA

30%

28%

21%

26%

17%

C

W

X



XB

MVP

-1.00

0

1.00XA

30%

28%

21%

26%

17%

Q

WN

Z

0.5

1.00

Y

Critical Line

-0.5

Isovariance ellipse

Isoexpected return line

10%12%16%

18%20%

D

E

F

-1.00

-0.5

XA


1.0

1.0

-1.00

-1.00

Q

S

Z

XB


0 .5-.5

R

T


E(r)

σ(r)10% 20% 30% 40% 50%

Exp

ecte

d re

turn

%

MVP

N

Consolidated

Z

Q

W

Acme

Brown

XA


1.0

1.0

-1.00

-1.00

Q

S

Z

XB

0 .5-.5

R

TAC

B


E(r)

σ(r) 10% 20% 30% 40% 50%

Exp

ecte

d re

turn

%

MVP

N

CS

Z

Q

W

AT

BR

資本市場線 (CML)

一個無風險資產， N 個風險性資產 ( 投資組合M) 下的效率前緣

Pr ＝

Fx

Fr ＋

Mx

Mr ＝(1－

Mx )

Fr ＋

Mx

Mr (1)

2

p ＝ 22

MMx ,

Mx ＝

M

p (2) 將(2) 代入(1)

可得 )(PrE ＝

Fr ＋〔

M

)(

FMrrE 〕

P

CML v. s. SML(CAPM)

CML:

SML:

)(PrE ＝

Fr ＋〔

M

)(

FMrrE

〕P

E (Jr )＝

Fr ＋﹝E (

Mr )－

Fr ﹞

J

10% =γFγF

E (γ)E (γ)

E (γM )M

σ (γ)o o

Capita

l mar

ket l

ine

Exp

ecte

d r

etu

rn

15% = E (γM )

Securit

y m

arke

t lin

e

17.5%

β

Standard deviation 1 1.5Beta

A

M

E (γM) ﹣γF

FIGURE 8.8 CAPITAL MARKET LINE FIGURE 8.9 SECURITY MARKET LINE

βM = 1.00

CAPM v.s.Market Model

同 : 均與市場組合報酬有關異 :1. 截距項 2. 均衡概念 (CAPM); 無 3. 殘差項相關 (CAPM); 獨立

E (Jr )＝

Fr ＋﹝E (

Mr )－

Fr ﹞

J

tJr

,＝

JA ＋

J

tMr

,＋

tJ ,

(Fr ,

JA )

SML v.s. Characteristic lineE (

Jr )＝

Fr ＋﹝E (

Mr )－

Fr ﹞

J

=Fr (1－

J )＋

J E (

Mr )

tJr

,＝

Fr (1－

J )＋

J

tMr

,＋

tJ ,

＝JA ＋

J

tMr

,＋

tJ ,

if Mr ＝

Fr then E (

Jr )＝

Fr

all Characteristic lines 均相交於共同點(Fr ,

Fr )

C

FIGURE 8.10 CHARACTERISTIC LINES UNDER THE CAPITAL ASSET PRICING MODEL

E(γA )

E(γB )

γJ A

B

E(γM )γF

10% =γF = E(γB )

-10%

5%

γM

Market return

o

FIGURE 8.11 POSITIONS OF INDIVIDUAL STOCKS IN E(γ), σ (γ) SPACE

10% =γF Exp

ecte

d r

etu

rn 15% = E(γM )

Standard deviation

17.5%

E(γ)

σ (γ)

AA’

M

CML

15% 22.5% 30%

β = 1.50

β = 1.00

β = .50

β = .00

SML(CAPM) in (E (r), σ)

E (Jr )＝

Fr ＋﹝E (

Mr )－

Fr ﹞

J

J ＝

JM / 2

M ；

JM ＝

JM /

MJ

E (Jr )＝

Fr ＋〔

JM

FMrrE

M

)( 〕

P

FIGURE 8.12 POSITIONS OF STOCKS ACCORDING TO THEIR CORRELATION COEFFICIENTS

γF

Exp

ecte

d r

etu

rn

Standard deviation

E(γ)

σ (γ)

M

ρ = -1.00

ρ = 1.00

ρ = .50

ρ = .00

Capital market lin

e

ρ = -.50

o

資本資產定價模式 (CAPM)

Where is the price?E (

Jr )＝

Fr ＋﹝E (

Mr )－

Fr ﹞

J

0P =

)(1

)()(11

rE

DEPE

（由E ( r ) =0

011)()(

P

PDEPE ）

Pricing:

1. 預測Fr ,E (

Mr ),

J 依 CAPM得 E(r)

2. 預測 E(P1),E(D

1)再與前所得之 E(r)共同設定 P

0

FIGURE 8.13 MARKET PRESSURE TO FORCE A STOCK TO ITS EQUILIBRIUM PRICE

γF = 10%

Standard deviation

E(γ)

σ (γ)

M

Capital m

arket line

o

15%

E(γ)

BM’

.10

5%

C’

C

C’’

A

CAPM之應用投資風險評估－系統風險 ( 貝他係數 ) 投資績效評估－風險調整績效評估

Jensen Index Treynor Index Sharpe Index

資本預算評估－估計權益資金成本

風險控管個別風險；非系統風險－分散多種證券

(cross-sectional diversification) 例如，共同基金； ETF 。市場風險；非系統風險－分散不同時間點 (time-series diversification) 。

例如，定時定額投資；期貨避險。 What risk take , what return make

投資績效評估－風險調整績效評估 Jensen Index (Based on CAPM) Treynor Index Sharpe Index

Jensen Index Based on CAPM(SML)

母體

樣本

PJ = E(r

p)－﹛r

F＋﹝E(r

M)－r

F﹞

p ﹜

J =pr －﹝

Fr ＋(

Mr －

Fr )

p

﹞

FIGURE 11.4 ESTIMATED SML (BULL MARKET)

o

γP

βPPortfolio beta

Ave

rag

e p

ort

foli

o r

etu

rnB

A

Treynor Index Based on

CAPM(SML)

母體

樣本

T P =p

)(

FPrrE

P

FP

P

rrT

FIGURE 11.6 LEVERING ALPHA TO DOMINATE OMEGA

γF Exp

ecte

d r

etu

rn

E(γ)

β

M

o

A’

A’’

O’

1.501.00

Beta

Sharpe Index Based on

CAPM(CML)

母體

樣本

S P =p

)(

FPrrE

P

FP

P

rrS

FIGURE 11.7 PERFORMANCE OF ALPHA AND OMEGA ACCORDING TO THE SHARPE INDEX

γF

Exp

ecte

d r

etu

rn

Standard deviation

E(γ)

σ (γ)

M

o

A’

A

O

O’

資本預算－估計權益資金成本估計 β ，依 CAPM 決定權益資金成本 E(r) 預測投資方案現金流量，估計淨現值 (NPV)

NPV＝

N

t

in

rE

CF1 )(1

－

N

t

out

rE

CF1 )(1

獨立方案：NPV 0 ；互斥方案：選 NPV最大者

Σ(r) MVP E(r) Standard deviation of return C A B Expected return % 15% 10%

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