Download Chapter 4

Chapter 3
Mean-Variance Analysis, CAPM, APT
1
n
~
Mean : E ( X )   pi X i
i 1
~
~ 2
Variance : Var( X )  E[ X i  E ( X )]
~
 X  Var( X )
~
Semi var iance : S .V .( X )  E[( X i ) 2 ], where
~
~
X i  X i  E ( X ) if X i  E ( X )
~
0
if X i  E ( X )
2
A.Two Asset Portfolio
~
~
~
E ( RP )  aE ( X )  (1  a ) E (Y )
~
~
~
~ ~
2
2
Var ( RP )  a Var ( X )  (1  a ) Var (Y )  2a (1  a )Cov( X , Y )
~
 P  Var ( RP )
~ ~
Cov( X , Y )
 XY 
 1   XY  1
 X Y
3
A.Two Asset Portfolio
1.  XY  1
~
~
~
E ( RP )  aE ( X )  (1  a ) E (Y )
~
Var ( RP )  a 2 X2  (1  a ) 2  Y2  2a (1  a ) X  Y  XY
 a 2 X2  (1  a ) 2  Y2  2a (1  a ) X  Y
 [a X  (1  a ) Y ]2
~
 ( RP )  a X  (1  a ) Y  a ( X   Y )   Y
 P Y
a
 X Y
4
A.Two Asset Portfolio
~
~
~
E ( RP )  aE ( X )  (1  a ) E (Y )

 P  Y
~  P
~
E( X )  X
E (Y )
 X  Y
 X  Y
~
~
~
~
 X E (Y )   Y E ( X ) E ( X )  E (Y )


P
 X Y
 X  Y
5
A.Two Asset Portfolio
2.  XY
 1
~
~
~
E ( RP )  aE ( X )  (1  a ) E (Y )
~
Var ( RP )  a 2 X2  (1  a ) 2  Y2  2a (1  a ) X  Y  XY
 a 2 X2  (1  a ) 2  Y2  2a (1  a ) X  Y
 [a X  (1  a ) Y ]
2
~
 ( RP )  a X  (1  a ) Y
~
Let  ( RP )  0 a 
Y
 X  Y
6
A.Two Asset Portfolio
~
 ( RP )  a X  (1  a ) Y
~
 ( RP )  (1  a ) Y  a X
if a 
Y
 X  Y
Y
if a 
 X  Y
7
A.Two Asset Portfolio
3.  1   XY  1
~
~
~
E ( RP )  aE ( X )  (1  a ) E (Y )
~
Var( RP )  a 2 X2  (1  a ) 2  Y2  2a (1  a ) X  Y  XY
~
dVar( RP )
 2a X2  2(1  a ) Y2  2 X  Y  XY  4a X  Y  XY
da
0
 Y2   X  Y  XY
a 2
 X   Y2  2 X  Y  XY
8
B.Many Assets Portfolio
(Rf. Markowitz, Portfolio Selection,1992)
n
~
E ( RP )   wi E ( Ri )
i 1
n
n
~
Var ( RP )   wi w j i j
i 1 j 1
~
~
 ( RP )  Var ( RP )
9
B.Many Assets Portfolio
n
n
Min  P2   Qi Q j i j
i 1 j 1
n
S .T .  Qi  i   C
i 1
n
Q
i
1
i 1
n
n
n
n
i 1
i 1
L   QiQ j i j  L1 ( Qi i  C )  L2 ( Qi  1)
i 1 j 1
10
B.Many Assets Portfolio
1.
2.
Minimum Variance Opportunity Set
The locus of risk and return combination offered by portfolio of risky
assets that yields the minimum variance for a given rate of return
Efficient Set (Efficient Frontier)
The set of mean-variance choices from the investment opportunity set
where for a given variance no other investment opportunity offers a
higher return.
11
C.Capital Market Line(CML)
1.
Optimal Portfolio Choice(The efficient set) for a risk
averse investor
»
»
»
»
B:Equilibrium Point? MRS E  MRTE  Take less risk
E:Equilibrium Point?Efficient Portfolio?
D:Equilibrium Point?
C:Equilibrium Point? MRS E  MRT E
12
C.Capital Market Line(CML)
2.
Optimal Portfolio Choice for a different risk averse
investors
i
B : for i, potimal portfolio MRS E  MRT Ei
for ii , MRS Eii  MRT Eii
for iii , MRS Eiii  MRT Eiii
C : for i, MRS Ei   MRT Ei
for ii , MRS Eii  MRT Eii
for iii , MRS Eiii  MRT Eiii
D : for i, MRS Ei   MRT Ei
for ii , MRS Eii  MRT Eii
for iii , MRS Eiii  MRT Eiii
13
C.Capital Market Line(CML)
» A:Utility Maximization?
MRS E  MRT E  Take less risk
» B:Utility Maximization? No Capital Market?
With Capital Market?
MRS E  MRT E
» C: Utility Maximization?
MRS E 
MRS
i
E
E ( Rm )  R f
m
 MRS
ii
E
 MRS
iii
E

E ( Rm )  R f
m
 MRT E
14
D.Capital Asset Pricing Model (CAPM)
Treynor[1961], Sharpe[1963], Lintner[1965], Mosson[1966]
1. Assumptions
1) Risk-averse investors, expected utility maximization
2) Price-taker investor, Homogenous expectation, JointNormal distribution
3) Risk-free rate
4) Marketable and Perfectly divisible assets.
5) Frictionless market and No information costs
6) No market imperfections.
15
D.Capital Asset Pricing Model (CAPM)
2. Derivation of CAPM
~
~
~
E ( RP )  aE ( Ri )  (1  a) E ( Rm )
1
~
 ( RP )  [a 2 i2  (1  a) 2  m2  2a(1  a ) im ] 2
~
E ( RP )
~
~
 E ( Ri )  E ( Rm )
a
~
1

 ( RP ) 1 2 2
2 2
 [a  i  (1  a )  m  2a (1  a) im ] 2
a
2
[2a i2  2 m2  2a m2  2 im  4a im ]
~
E ( RP )
~
~
 E ( Ri )  E ( Rm )
a a 0
~
 im   m2
 ( RP )
1 2  12
2
 ( m ) (2 m  2 im ) 
a a 0 2
m
16
D.Capital Asset Pricing Model (CAPM)
In equilibrium,
~
E ( RP )
~
 ( RP )
a
~
~
~
E
(
R
E ( Ri )  E ( Rm )
m )  Rf


2
 im   m
m
m

~
~
E ( Ri )  R f  [ E ( Rm )  R f ] im2
m
a a 0
~
~
or E ( Ri )  R f  [ E ( Rm )  R f ]   i  SML
~ ~
 im cov( Ri , Rm )
Where  i  2 
~
m
Var ( Rm )
17
D.Capital Asset Pricing Model (CAPM)
18
D.Capital Asset Pricing Model (CAPM)
– Two-fund Separation Theorem
• Each investor will have a utility-maximization portfolio
that is a combination of the risk free asset and a
portfolio of risky assets that is determined by the line
drawn from the risk free rate of return tangent to the
investor’s efficient set of risky assets
19
D.Capital Asset Pricing Model (CAPM)
– Capital Market Line and Mutual Fund Theorem
• If investors have homogenous beliefs, then they all
hold the same mutual fund and
wi 
vi
n
v
i 1
i
• They all have the same linear efficient set called the
CML
~
E ( RP )  R f 
~
E ( Rm )  R f
m
 P
(
~
E ( Rm )  R f
m

~
E ( RP )  R f
P
20
)
D.Capital Asset Pricing Model (CAPM)
Model : R 'jt   0   1  j   jt
Re sults : 1. 0  0
2. j is the only factor ?
»
»
»
»
Basu [1977]:P/E
Litzenberger & Ramaswamy[1979]:Dividend
Banz[1981]: Size Effect
Keim[1983]:January Effect
21
E.Arbitrage Pricing Theory(APT)
1. Assumptions :
Ross[1976]
1) Risk-averse Investors
2) Homogeneous expectation of k-factor return generating
process
~
~
~
~ ~
~
R  E( R )  b F    b F  
i
i
i
i1 1
ik
k
i
3) Perfect Market
4) Number of assets,N > Number of factors,k
~
~

5) Idiosyncratic risk,  i is independent of all factors and j
~
~ ~
E (~ , ~ )  0 E (~ , F )  0 E ( F , F )  0
i
j
i
j
i
j
~
E (~i , Fi )  0
22
E.Arbitrage Pricing Theory(APT)
2. Model
1) Arbitrage portfolio in Equilibrium
No Wealth change
No additional return
No additional risk
A. No change in wealth
n
w
i 1
i
0
B. No additional
return
n
~
~
R p   wi Ri
i 1
n
n
n
~
~
~ n ~
  wi E ( Ri )   wi bi1 Fi     wi bik Fi   wi i
i 1
i 1
i 1
i 1
0
23
E.Arbitrage Pricing Theory(APT)
C. No additional risk (No systematic risk, No unsystematic risk)
condition w  1
i
n
n
n
wb
i 1
a.
i i
Unsystematic Risk
2
n

1
1
 2p   wi2 i2  ( ) 2  i2   i
n i 1 n
i 1
i 1 n
1
(Average residual variance)
  i2
n
1
lim  i  0
n  n
(∵ Law of large number)
n
n
24
E.Arbitrage Pricing Theory(APT)
b. Systematic Rick
n
n
n
~
~
~
~ n ~
R p   wi E ( Ri )   wi bi1 Fi     wi bik Fi   wi i
i 1
i 1
i 1
i 1
n
n
n
~
~
~
  wi E ( Ri )   wi bi1 Fi     wi bik Fi
i 1
i 1
i 1
n
~
  wi E ( Ri )
i 1
25
E.Arbitrage Pricing Theory(APT)
2) Derivation
of
APT
n
n
w
i 1
i
 0  ( wi )e  0
n
wb
i 1
i ik
i 1
0
n
~
w
E
(
R
 i i)  0
i 1
~
 E ( Ri )  0  1bi1    k bik
 R f  1bi1    k bik
~
 E ( Ri )  R f  1bi1    k bik
~
 E ( Ri )  R f  [ 1  R f ]  bi1    [ k  R f ]  bik
~ ~
cov( Ri ,  k )
where, bik 
~
Var ( k )
26
E.Arbitrage Pricing Theory(APT)
27
E.Arbitrage Pricing Theory(APT)
3) Advantages of APT
A. No assumption of normal distribution
B. No efficient market portfolio
C. Asset pricing is dependent on many factors
4) Empirical of APT: Chen, Roll and Ross (1983)
A.
B.
C.
D.
Industrial Production
Changes in default risk premium
Twists in the yield curve
Unexpected inflation
28