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