1 IV. Calculating the Efficient Frontier. 2 Calculating the Efficient Frontier 1)Short sales are allowed, riskless lending and borrowing is allowed 2)Short.
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1
IV. Calculating the Efficient Frontier
2
Calculating the Efficient Frontier
1) Short sales are allowed, riskless lending and borrowing is allowed
2) Short sales are allowed , no riskless borrowing and lending
3) Short sales disallowed, riskless lending and borrowing allowed
4) Short sales disallowed , no riskless lending and borrowing
Special CasesRiskless borrowing and lending at different rate Lintner short sales – a more realistic case
3
1
12
1 2
2
2 2j
1 1 1
1 22 1
P roduct Rule
12
( ) ( )
[
F F
]
X
d ( ) ( )( ) ( )
N
j
j
F P
N N N
j F j j k jkj j k
k j
R
X
X X
Rp
R R X X
dF X dF XF X F X
dX dX dX
4
1
1 32 2
212 j
2 2
22
p
i2
1
j 1
j i
X2 2
0
Change in variables
R Z
i F P Fi i
P P
P Fj F j i j
j P
Fi
P
i
ij
d
dX j
j
j i
N
R R R RX
d R RR R X X
dX
RX
N
5
2
i i
Lintner Equations
N
j 1
j i
For i=1...n
X , 1, X
X
i F i i
i ii
i
j ijR R Z Z
Z Z
Z
6
F12
13
23
12
R1. 14 6 .5 5
2. 8 3 .2
3. 20 15 .4
R
(6)(3)(.5) 9
13
23
21 1 1 2 1,2 3 1,3
22 1 1,2 2 2 3 2,3
23 1 1,2 2 2,3 3 3
(6)(15)(.2) 18
(3)(15)(.4) 18
F
F
F
R R Z Z Z
R R Z Z Z
R R Z Z Z
7
1 2 3
1 2 3
1 2 3
1 1
3 3
a
b
15=18Z
9 36 9 18
3=9Z 9 18
+18Z +225Z c
Solve Simultaneously
2a-b 27Z 6 Z
9
12b 189Z 9 Z
21
Z Z Z
Z Z
c
8
2
2
2
2
36
9
Plug in A
2 19 18 9
9 21
188 9 9
21
3
2
1
63
Z
Z
Z
Z
9
2
1 2 3
14 1
63 6318 181 263 63
3
6318363
14 1
18 18
3
18
Remember
14 1 3 18, Z , Z , .285
63 63 63 63
X , X ,
X
p Fi
P
R
Z
RZ
Z
10
2 2
2
2
14 1 3 264 2(14) (8) (20) 14
18 18 18 18 3
14 1 3 14 1 14 3 1 3(36) (9) (225) 2 9 2 18 2 18
18 18 18 18 18 18 18 18 18
7056 9 2025 252 1512 108 10, 96 =
324 324 324 324 324 324
p
P
R
P
2
17418
60918
2 609
324 18
264 90 174 R
18 18 18
R 174 .285
609
P F
F
P
R
R
11
If short sales are allowed:
• Portfolios of efficient portfolios are efficient
• The entire efficient frontier is a portfolio of any two portfolios on the efficient frontier
• Each security has zero weight in at most 1 efficient portfolio
12
11
1
22 2i
1
MAKE
XN N
1 i=1
N
1
SOLUTION WITH SHORT SALES
( ) =
2
P F
P
F
i i j ij
N
j i
X
Xi
i
R R
R R
X
i
1j=1
FOR
for i=1,N
N EQUATIONS FOR N UNKNOWNS
SOLUTION TO SYSTEM OF SIMULATANEOUS EQUATIONS
N
j=1
N
d = 0 i=1,N
R =
R =
i
F j ij
F j ij
X
dX
R
ZR
11
1
z
z
x
13
*
1
2 2i j
i= 1
P
i
=
SOLUTION WITH NO SHORT SALES
1
i=1
MINIMIZE X
SUBJECT TO
i 1
2
= R
1
X
xi i iji
ii
j iX
Xi
Rx
*
i
FOR i=1,....,N
SOLVE FOR ALTERNATIVE VALUES OF
QUADRATIC PRGAMMING PROBLEM
R
0
14
15
i
i
i
d θ0
dX
Khun Tucker Conditions
X 0
X 0
0
i
i
i
16
17
18
19
20
21
22
FORECAST OF FUTURE CAPITAL MARKET BEHAVIOR
FORCAST OF SECURITY TYPES AND OR
INDIVIDUAL SECURITIES USUALLY EMPLOY
AS A BASIS ONE OF THE FOLLOWING
FORECASTS:
1. INFLATION
2. TREASURY BILLS
3. TREASURY BONDS
23
INFLATION 33.3%
DIFFERENTIAL FOR BILLS .5%
BILL RATE 3.8%
DIFFERENTIAL FOR CORPS. 4.8%
CORPORATE RATE 8.6%
DIFFERENTIAL FOR STOCKS 5.1%
STOCK RATE 13.7%
DIFFERENTIAL FOR SMALL STOCKS 9.1%
SMALL STOCK RATE 22.8%
24
25
26
27
28
Short sales revisited - Broker dealer
In fact when short sell don't get proceeds to
invest in risky securities
Can't short sell an unlimited amount of
securities
Can get proceeds to invest in
i
i F
i
T-bill rate
Must put up capital but can earn T-bill rate
Limited to original capital plus lending and
borrowing
When short sell X is negative
get X ( 2 )
Limited by capital long XiR R
i
1
j=k+11
N
X
2
- short = 1,
or X 1
i Fp i i i
K
iX R RR X R
29
i F i1 1
P F i1 1
i1
i
N
N
N
X
X
X
but X 1 so R
R R ( ) ( )
( )
Same as original equation except X 1
Just scale the Z's differently
K
F i Fi i k
K
i F i i Fi i k
p F i Fi
R X R
XR R R R
R R R R
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