University of Hohenheim
Chair of Banking and Financial Services
Portfolio Management
Summer Term 2011
Exercise 1:
Basics of Portfolio Selection Theory
Prof. Dr. Hans-Peter Burghof / Katharina Nau
Slides: c/o Marion Schulz/ Robert Härtl
Basics of Portfolio Selection Theory  Exercise 1
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) = 22 = 0,08 and cov(r1,r2) = 12 = 0,02.
a)
Calculate the minimal variance portfolio for a given expected portfolio return
of μ P
a)
 25%
. What is the variance and the expected value of this portfolio?
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  (, )  1,25  0,75( 2   2 ) realize?
Basics of Portfolio Selection Theory: Exercise 1
1
Solution Question 1
Part a)
Expected portfolio value:
p  x1  1  x 2  2  x1  1  (1  x1 )  2  0,1 x1  0,3
Calculation of the portfolio weights:
p  0,25  0,1  x1  0,3
x1  0,5
x 2  0,5
Basics of Portfolio Selection Theory: Exercise 1
2
Solution Question 1
Part b)
Calculation of the portfolio variance:
N
N
   x i x jσij
2
p
i 1 j1
 2p  x12  12  x 22   22  2  x1  x 2  1, 2
 2p,x1 0,5  0,04
Standard deviation:
 p, x1 0,5   2p, x1 0,5  0,2
Basics of Portfolio Selection Theory: Exercise 1
3
Solution Question 1
Part c)
What is the expected value depending on the given variance?
 p (2p )
p  0,1 x1  0,3
Calculation of x1:
 2p  x12  12  x 22   22  2  x1  x 2  1, 2
 2p  x12  12  (1  x1 ) 2   22  2  x1  (1  x1 )  1, 2
 2p  0,08x12  0,12x1  0,08
c1)
x11, 2 
0,12  0,122  4  0,08 (0,08   p2 )
2  0,08
Basics of Portfolio Selection Theory: Exercise 1

0,12   0,0112 0,32 p2
0,16
4
Solution Question 1
Part c)
 p2
 2  x1   12  2  (1  x1 )   22  2  (1  2  x1 )   1, 2  0
x1
 22   1, 2
x1  2
 0,75
2
 1   2  2   1, 2
Thus, on the efficient frontier we receive:
x1  0,75
This means a reduction of equation c1) to:
x1 
0,12   0,0112 0,32 2p
0,16
Accordingly, the equation of the efficient frontier is:
p  0,1
0,12   0,0112 0,32 2p
0,16
Basics of Portfolio Selection Theory: Exercise 1
 0,3  0,225
 0,0112 0,32 2p
1,6
5
Solution Question 1
Part d)
Utility function:
(, )  1,25  0,75(2  2 )
Maximization:
 p
 p
 p
 p2
 1,25 
 0,75  2   p 
 0,75 
0
x1
x1
x1
x1
 p  0,1  x1  0,3
 2p  0,08x12  0,12  x1  0,08
 p
 2p
x1
 0,1
Basics of Portfolio Selection Theory: Exercise 1
x1
 0,16  x1  0,12
6
Solution Question 1
Part d)
 p
x1
 0,125 0,15 (0,1 x1  0,3)  0,75 (0,16 x1  0,12)  0
 0,135 x1  0,01  0
Utility maximizing portfolio:
x1 
2
27
 0, 074
 p  0,2926
 2p  0,0716
 p  0,2675
 p  0,2478
Basics of Portfolio Selection Theory: Exercise 1
7
Solution Question 1
Graphical solution for question 1
μP
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,05
0,1
0,15
Basics of Portfolio Selection Theory: Exercise 1
0,2
0,25
0,3
0,35
0,4
0,45
σP
8
Continuation of Question 1
Stock’s portfolio risks:
PR i 
cov(ri ,rp )
σP

σiσPρiP
 σiρiP
σP
Firstly, the cov(ri, rp) must be calculated:
i,p  E(ri  ri )  (rp  rp )  E[(ri  ri )  (( x1  r1  x 2  r2 )  ( x1  r1  x 2  r2 )]

 

rp
1,p  E[(r1  r1 )  ((x1  r1  x1  r1 )  ( x 2  r2  x 2  r2 )]
rp
 E[ x1  (r1  r1 )2 ]  E[ x 2  (r1  r1 )  (r2  r2 )]
1,p  x1  12  x 2  1,2
 2,p  x 2   22  x1  1,2
In the numerical example of part a)
1,p  0,03
 2,p  0,05
Basics of Portfolio Selection Theory: Exercise 1
9
Continuation of Question 1
Stock’s portfolio risks:
PR i 
cov(ri ,rp )
PR 1 
σP
0,03
 0,15
0,04
0,05
PR 2 
 0,25
0,04
Basics of Portfolio Selection Theory: Exercise 1
10
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
μ P  25%
. 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 (, )  1,25  0,75(   ) realize?
2
Basics of Portfolio Selection Theory: Exercise 1
2
11
Solution Question 2
Part a)
L   p2  [ x11  x2 2  (1 - x1 - x2 )rf - 0,25]
 p2  x12 12  x22 22  2 x1 x2 1, 2
 L  0,04x12  0,08x22  0,04x1 x2  [0,1x1  0,2 x 2 -0,15]
L
1.)
 0,08x1  0,04x2  0,1  0
x1
  0,8x1  0,4x2
δL
2.)
 0,16x2  0,04x1  0,2 λ  0
δx2
  0,2x1  0,8x2
Basics of Portfolio Selection Theory: Exercise 1
12
Solution Question 2
Part a)
(1)  (2)
- 0,8x1  0,4 x2  0,2 x1  0,8x2
2
x1 = x 2
3
L
3.)
 0,1x1  0,2 x2 - 0,15  0

2
x 2 + 0,2x 2 = 0,15
30
x1  0,375
x2  0,5625
y  0,0625
p2  0,3752 * 0,04  0,56252 * 0,08  2 * 0,375* 0,5625* 0,02
 p2  0,039375  p ≈ 0,1984
Basics of Portfolio Selection Theory: Exercise 1
13
Solution Question 2
Part a)
Tangential Portfolio
From Example 1c)
Efficient frontier:
μp = 0,225+
- 0,0112+ 0,32σ p2
1,6
Slope of the efficient frontier in T:
δμT
1 1
1
=
* *
* 0,64σ T
2
δσT 1,6 2
- 0,0112+ 0,32σ T
 T
 0,2 *
T
T
- 0,0112 0,32T2
Basics of Portfolio Selection Theory: Exercise 1
14
Solution Question 2
Part b)
Slope of the capital-market-line:
μT -rf
σT

- 0,0112 0,32 T2
0,225
- 0,1
1,6
T
- 0,0112+ 0,32σ T2
0,125+
1,6
=
σT
Basics of Portfolio Selection Theory: Exercise 1
15
Solution Question 2
Part b)
δμT μT - r
=
δσT
σT
0,2 T
- 0,0112  0,32
2
T

- 0,0112 0,32 T2
0,125
1,6
T
2
0
,
0112
+
0
,
32
σ
T
0,2σ T2 = 0,125 - 0,0112+ 0,32σ T2 +
1,6
Basics of Portfolio Selection Theory: Exercise 1
16
Solution Question 2
Part b)
0,0562 = -0,0112+ 0,32σT2
σ = 0,0448
2
T
σT = 0,0448 ≈ 0,21166
- 0,0112+ 0,32* 0,0448
μT = 0,225+
= 0,26
1,6
Basics of Portfolio Selection Theory: Exercise 1
17
Solution Question 2
Part b)
2. Approach
Structure of the tangential portfolio:
x1  23 x2
whereas the tangential portfolio only includes stock 1 and stock 2 and there is no risk-free
investment or borrowing:
x1  x2  1
2
3
x2  x2  1
x1  0,4
T  0,26
x2  0,6
σ T2 = 0,0448
Basics of Portfolio Selection Theory: Exercise 1
18
Solution Question 2
Part c)
Efficient frontier:
 T  0,26
 p  T  (1 -  )rf
 T2  0,0448
 p   ( T - rf )  rf
 T - rf
  p  rf 
* p
T
0,16
μp = 0,1 +
* σp
0,0448
Basics of Portfolio Selection Theory: Exercise 1
 p2   2 *  T2
p

T
T - r f
  p  r f 
* p
T
19
Solution Question 2
Part c)
Comparison with the results of part 2a)
μp = 0,25
0,16
0,25 = 0,1 +
* σp
0,0448
σ p ≈ 0,1984
Basics of Portfolio Selection Theory: Exercise 1
20
Solution Question 2
Part d)
Portfolio risks:
P Ri =
σ i,p
σp
From Exercise 1:
 1, p  x1 12  x2 1, 2  x3 1,r
f
0
 2, p  x2 22  x1 1, 2  x3 2,r
f
0
 1, p  0,02625
 2, p  0,0525
PR1 ≈ 0,1323
PR2 ≈ 0,2646
Basics of Portfolio Selection Theory: Exercise 1
21
Solution Question 2
Part e)
Maximization of
  1,25 p - 0,75( p2   p2 )


: 0

Efficient frontier:
p  T  (1 -  )rf
p  0,1  0,16
_
  E[(  rT  (1 -  )  rf ) - (   T  (1 -  )  rf )]2
2
p
_
 E[ (rT -  T )]2
p2  2T2
Basics of Portfolio Selection Theory: Exercise 1
22
Solution Question 2
Part e)
δ μp
δα
δσp2
= 0,16
δα
= 2ασT2 = 0,0896α
δΦ
= 1,25 * 0,16 - 0,75[2 * (0,1 + 0,16α ) * 0,16 + 0,0896 α ] = 0
δα
 0,2 - 0,024 - 0,0384 - 0,0672  0
0,176= 0,1056α
_
5
α = = 1, 6
3
Basics of Portfolio Selection Theory: Exercise 1
23
Solution Question 2
Part e)
_
11
μp =
= 0,3 6
30
_
28
σ =
= 0,12 4
225
2
p
Φ ≈ 0,2642
σ p ≈ 0,353
Basics of Portfolio Selection Theory: Exercise 1
24
Solution Question 2
Graphical solution for question 2
μP
0,6
0,5
0,4
0,3
0,2
0,1
0
σP
0
0,05
0,1
0,15
Basics of Portfolio Selection Theory: Exercise 1
0,2
0,25
0,3
0,35
0,4
0,45
25
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 minimumvariance 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).
Basics of Portfolio Selection Theory: Exercise 1
26
Solution Question 3
Part a)
L   p2  [ x1  x2 - 1]
 p2  x12 12  x22 22  2 x1 x2 1, 2
 p2  x12 12  (1  x1 )² 22  2 x1 (1  x1 ) 1, 2
 p2
x1
 0,224x1  0,032  0
x1  0,1429
x2  0,8571
 p  16,43%
 p2  8,78%
Basics of Portfolio Selection Theory: Exercise 1
27
Solution Question 3
Part b)
L   p2  [ x11  x2  2  (1 - x1 - x2 )rf - 0,1643]
 p2  x12 12  x22 22  2 x1 x2 1, 2
 L  0,09x12  0,01x22  0,012x1 x2  [0,15x1  0,05x 2 -0,0643]
x1  0,2143
x2  0,643
y  0,1427
 p2  0,0066151
 p  8,13%
Basics of Portfolio Selection Theory: Exercise 1
28