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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. - PowerPoint PPT Presentation
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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
4Basics of Portfolio Selection Theory: Exercise 1
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
5Basics of Portfolio Selection Theory: Exercise 1
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
6Basics of Portfolio Selection Theory: Exercise 1
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
7Basics of Portfolio Selection Theory: Exercise 1
Utility maximizing portfolio:
Solution Question 1Part d)
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
8Basics of Portfolio Selection Theory: Exercise 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
9Basics of Portfolio Selection Theory: Exercise 1
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,
10Basics of Portfolio Selection Theory: Exercise 1
Stock’s portfolio risks:
Continuation of Question 1
P
pii σ
)r,cov(rPR
150040
0301 ,
,
,PR
250040
0502 ,
,
,PR
11Basics of Portfolio Selection Theory: Exercise 1
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
12Basics of Portfolio Selection Theory: Exercise 1
%25μP
Solution Question 2Part a)
]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
13Basics of Portfolio Selection Theory: Exercise 1
Solution Question 2Part a)
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
14Basics of Portfolio Selection Theory: Exercise 1
Tangential Portfolio
From Example 1c)
Efficient frontier:
Slope of the efficient frontier in T:
Solution Question 2Part a)
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
,,-*,
15Basics of Portfolio Selection Theory: Exercise 1
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
16Basics of Portfolio Selection Theory: Exercise 1
Solution Question 2Part b)
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
17Basics of Portfolio Selection Theory: Exercise 1
Solution Question 2Part b)
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
18Basics of Portfolio Selection Theory: Exercise 1
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:
Solution Question 2Part b)
232
1 xx
121 xx
12232 xx
6,04,0 21 xx
26,0T 0448,0=2Tσ
19Basics of Portfolio Selection Theory: Exercise 1
Efficient frontier:
Solution Question 2Part c)
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
20Basics of Portfolio Selection Theory: Exercise 1
Comparison with the results of part 2a)
Solution Question 2Part c)
1984,0≈
*0448,0
16,0+1,0=25,0
25,0=
p
p
p
σ
σ
μ
21Basics of Portfolio Selection Theory: Exercise 1
Portfolio risks:
From Exercise 1:
Solution Question 2Part d)
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
22Basics of Portfolio Selection Theory: Exercise 1
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
23Basics of Portfolio Selection Theory: Exercise 1
Solution Question 2Part e)
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
24Basics of Portfolio Selection Theory: Exercise 1
Solution Question 2Part e)
_
p 63,0=30
11=μ
_2p 412,0=
225
28=σ 2642,0≈Φ
353,0≈pσ
25Basics of Portfolio Selection Theory: Exercise 1
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
26Basics of Portfolio Selection Theory: Exercise 1
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).
27Basics of Portfolio Selection Theory: Exercise 1
Solution Question 3 Part a)
28Basics of Portfolio Selection Theory: Exercise 1
%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)
29Basics of Portfolio Selection Theory: Exercise 1
%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