basics of portfolio selection theory exercise 1

Basics of Portfolio Selection Theory Exercise 1

University of HohenheimChair of Banking and Financial Services

Portfolio ManagementSummer Term 2011

Exercise 1:

Basics of Portfolio Selection TheoryProf. Dr. Hans-Peter Burghof / Katharina Nau

Slides: c/o Marion Schulz/ Robert Härtl

Question 1

Question 1

An investor is supposed to set up a portfolio including share 1 and 2. It is E(r1) = 1 = 0,2

the expected return of share 1 and E(r2) = 2= 0,3 the expected return of share 2.

Moreover, it is var(r1) = 12 = 0,04, var(r2) = 2

2 = 0,08 and cov(r1,r2) = 12 = 0,02.

a) Calculate the minimal variance portfolio for a given expected portfolio return

of . What is the variance and the expected value of this portfolio?

a) Determine the equation of the efficient frontier that can be calculated as the

combination of both shares.

b) Which efficient portfolio should an utility-maximizing investor with a preference function

of realize?)(75,025,1),( 22

Basics of Portfolio Selection Theory: Exercise 1 2

%25μP

Solution Question 1Part a)

Expected portfolio value:

Calculation of the portfolio weights:

3,0x1,0)x1(xxx 121112211p

5,0x

5,0x

3,0x1,025,0

2

1

1p

3Basics of Portfolio Selection Theory: Exercise 1

Calculation of the portfolio variance:

Standard deviation:

Solution Question 1 Part b)

N

1i

N

1jijji

2p σxx

04,0

xx2xx2

5,0x,p

2,12122

22

21

21

2p

1

2,025,0x,p5,0x,p 11


What is the expected value depending on the given variance?

Calculation of x1:

c1)

Solution Question 1 Part c)

)( 2pp

3,0x1,0 1p

08,0x12,0x08,0

)x1(x2)x1(x

xx2xx

121

2p

2,11122

21

21

21

2p

2,12122

22

21

21

2p

16,0

32,00112,012,0

08,02

)08,0(08,0412,012,0 222

1 2,1

ppx


Solution Question 1Part c)

75,02

0)21(2)1(22

2,1

2

2

2

1

2,1

2

2

1

2,11

2

21

2

11

1

2

x

xxxxp

Thus, on the efficient frontier we receive:

This means a reduction of equation c1) to:

Accordingly, the equation of the efficient frontier is:

75,0x1

16,0

32,00112,012,0x

2p

1

6,1

32,00112,0225,03,0

16,0

32,00112,012,01,0

2p

2p

p


Utility function:

Maximization:

Solution Question 1Part d)

)(75,025,1),( 22

075,0275,025,11

2

111

xxxxpp

p

pp

1,0x

3,0x1,0

1

p

1p

12,0x16,0x

08,0x12,0x08,0

11

2p

121

2p


Utility maximizing portfolio:


001,0x135,0

0)12,0x16,0(75,0)3,0x1,0(15,0125,0x

1

111

p

2478,0

2675,0

0716,0

2926,0

074,0x

p

p

2p

p

272

1


Solution Question 1

Graphical solution for question 1

0

0,1

0,2

0,3

0,4

0,5

0,6

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45

μP

σP


Continuation of Question 1

Stock’s portfolio risks:

Firstly, the cov(ri, rp) must be calculated:

In the numerical example of part a)

iPiP

iPPi

P

pii ρσ

σ

ρσσ

σ

)r,cov(rPR

])rxrx()rxrx(()rr[(E)rr()rr(E

pp rr

iippiip,i 22112211

2112222

2122111

221122

111

22221111111

,p,

,p,

p,

xx

xx

)]rr()rr(x[E])rr(x[E

)]rxrx()rxrx(()rr[(E

050

030

2

1

,

,

p,

p,


Stock’s portfolio risks:

Continuation of Question 1

P

pii σ

)r,cov(rPR

150040

0301 ,

,

,PR

250040

0502 ,

,

,PR


Question 2

Question 2

In addition to stock 1 and 2 with E(r1)=1=0,2, E(r2)= 2=0,3, var(r1)= 12=0,04,

var(r2)= 22 =0,08 and cov(r1,r2)=12=0,02, now there is a capital market providing the

opportunity to invest and raise unlimited capital at a risk-free interest rate of rf = 0,1.

a) Calculate the minimal variance portfolio for an expected value of the portfolio return of

. What is the variance of this portfolio?

b) Calculate the variance and expected value of the tangential portfolio.

c) Find out the equation for the efficient frontier, which can be calculated by combining both

stocks and the risk-free investment.

d) How high are the portfolio-risks of stock 1 and 2 in the portfolio selected in a)? How does

they correspond to each other?

e) Which of the efficient portfolios should a utility-maximizing investor with a preference

function of realize?)(75,025,1),( 22


%25μP


]15,0-2,01,0[04,008,004,0

2

]25,0-)--1([

212122

21

2,12122

22

21

21

2

2122112

xxxxxxL

xxxx

rxxxxL

p

fp

01,004,008,0.)1 211

xxx

L

21 4,08,0 xx

0200401602 122

λ,x,x,δx

δL.)

21 8,02,0 xx



2121 8,02,04,08,0-)2()1( xxxx

21 x3

2=x

015,0-2,01,0.)3 21 xxL

15,0=x2,0+x30

222

0625,05625,0375,0 21 yxx

0205625037502080562500403750 222 ,*,*,*,*,,*,p

1984,0≈039375,02pp


Tangential Portfolio

From Example 1c)

Efficient frontier:

Slope of the efficient frontier in T:


6,1

32,0+0112,0-+225,0=

2p

p

σμ

T2TT

T 64,0*32,0+0112,0-

1*

2

1*

6,1

1= σ

σδσ

δμ

23200112020

T

T

T

T

,,-*,


Slope of the capital-market-line:

Solution Question 2Part b)

T

T

T

fT

σ

-rμ

1,0

6,1

32,00112,0225,0

2

--

T

2T

6,1

32,0+0112,0-+125,0

=σ

σ



T

T

T

T r-=σ

μ

δσ

δμ

T

T

T

6,1

32,00112,0125,0

2,0

2

-

0,320,0112- 2T

6,1

32,0+0112,0-+32,0+0112,0-125,0=2,0

2T2

T2T

σσσ



2T

2 32,0+0112,0-=056,0 σ

0448,0=2Tσ

21166,0≈0448,0=Tσ

26,0=6,1

0448,0*32,0+0112,0-+225,0=Tμ


2. Approach

Structure of the tangential portfolio:

whereas the tangential portfolio only includes stock 1 and stock 2 and there is no risk-free

investment or borrowing:


232

1 xx

121 xx

12232 xx

6,04,0 21 xx

26,0T 0448,0=2Tσ


Efficient frontier:


pT

fTfp

T

T

*r-

r

,

,

04480

2602

pp *0448,0

0,16+1,0= σμ

pT

fTfp

T

p

Tp

fTp

Tp

rr

r

*

*

)(

1(

222

-

r-

)r-

f

f


Comparison with the results of part 2a)


1984,0≈

*0448,0

16,0+1,0=25,0

25,0=

p

p

p

σ

σ

μ


Portfolio risks:

From Exercise 1:


p

p,i

i =PRσ

σ

2646,0≈

1323,0≈

0525,0

02625,0

2

1

,2

,1

0

,232,11222,2

0

,132,12211,1

PR

PR

xxx

xxx

p

p

rp

rp

f

f


Maximization of

Efficient frontier:

Solution Question 2Part e)

)(75,0-25,1 22ppp

0:

16010

1

,,

r)-(

p

fTp

2f

_2 )]r)-1((-))-1([( TfTp rrE

2)]([ TTrE_

-

222Tp



16,0=p

δα

δμαασ

δα

δσ0896,0=2= 2

T

2p

0=]0,0896+0,16*)0,16+(0,1*0,75[2-16,0*25,1=Φ

ααδα

δ

00,0672-0,0384-,-, 024020

α0,1056=0,176_

6,1=3

5=α



_

p 63,0=30

11=μ

_2p 412,0=

225

28=σ 2642,0≈Φ

353,0≈pσ


Graphical solution for question 2

Solution Question 2

0

0,1

0,2

0,3

0,4

0,5

0,6

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45σP

μP


Question 3

Question 3

The expected return and the standard deviation of stock 1 and stock 2 are E(r1)=1=0,25,

1=30% and E(r2)= 2=0,15, 2 =10% respectively. The correlation is -0.2.

a) Which weights should an investor assign to stock 1 and stock 2 to set up the minimum-

variance portfolio? Also compute the expected return and the variance of the portfolio.

b) Assume that in addition to the above information a risk free investment with a yield of

10% exists on the capital market. Show that the investor can now realize the same

expected return at a lower level of risk. For this purpose, calculate the risk of the

efficient portfolio based on the expected return calculated in part a) and compare it to

the minimum-variance portfolio of part a).


Solution Question 3 Part a)


%78,8

%43,16

8571,01429,0

0032,0224,0

)1(2)²1(

2

]1-[

2

21

11

2

2,111221

21

21

2

2,12122

22

21

21

2

212

p

p

p

p

p

p

xx

xx

xxxx

xxxx

xxL

Solution Question 3 Part b)


%13,8

0066151,0

1427,0643,02143,0

]0643,0-05,015,0[012,001,009,0

2

]1643,0-)--1([

2

21

212122

21

2,12122

22

21

21

2

2122112

p

p

p

fp

yxx

xxxxxxL

xxxx

rxxxxL

basics of portfolio selection theory exercise 1

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