Homework Solution: Investment Science
Solution Problem 7.2: A Small World
Consider a world in which there are only two risky assets, E and Fß and a
risk free asset J Þ The two risk assets are in equal supply in the market; that
is,
<Q
"
œ Ð<E  <F Ñ.
#
The following information is known: <0 œ !Þ"!ß 5E# œ !Þ!%ß 5EF œ Þ!"ß
5F# œ !Þ!#ß and <Q œ !Þ")Þ Note:
G9@Ð\ß ]  ^Ñ œ G9@Ð\ß ] Ñ  G9@Ð\ß ^Ñ
G9@Ð\ß \Ñ œ Z +<Ò\Óß G9@Ð<\ß =] Ñ œ <=G9@Ð\ß ] Ñ
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
#
+Þ Find a general expression for 5Q
ß "E and "F Þ
<Q
"
" #
#
œ Ð<E  <F Ñ Ê 5Q œ Ð5E  #5EF  5F# Ñ
#
%
"
" #
<Q œ Ð<E  <F Ñ Ê 5EQ œ G9@Ð<E ß <Q Ñ œ Ð5E  5EF Ñ Ê
#
#
"
#
#
5
5
Ð

Ñ
5
5EQ
"
EF
E
#
E  5EF
"E œ # œ " #
œ ‚ #
#
#
#
5Q
5
5
5

#

5
5
5
Ð

#

Ñ
EF
EF
F
E
F
E
%
"
"
<Q œ Ð<E  <F Ñ Ê 5FQ œ G9@Ð<F ß <Q Ñ œ Ð5EF  5F# Ñ Ê
#
#
"
#
5
5
Ð

5FQ
5EF  5F#
"
EF
FÑ
#
"F œ # œ " #
œ ‚ #
#
#
#
5Q
5
5
5

#

5
5
5
Ð

#

Ñ
EF
EF
F
E
F
E
%
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
,Þ
&
<E œ <J  "E Ð<Q  <0 Ñ Ê <E œ !Þ"!  Ð!Þ")  !Þ"!Ñ œ #!%
%
Ð=// IB-/6 W:</+.=2//>Ñ
<F œ <J  "F Ð<Q
$
 <0 Ñ Ê <F œ !Þ"!  Ð!Þ")  !Þ"!Ñ œ "'%
%
Ð=// IB-/6 W:</+.=2//>Ñ
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Solution Problem 7.6: Simple Land
In Simpleland there are only two risky stocks, E and F . Whose details are
listed below.
Number of Shares
Price
Expected
Standard deviation
outstanding
per Share
rate of return
of return
Stock A
100
$1.50
15%
15%
Stock B
150
$2.00
12%
9%
Furthermore, the correlation coefficient between the returns of stocks E
and F is 3EF œ "$ . There is also a risk free asset and Simple lands satisfies
the CAPM exactly.
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Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
G+6-?6+>398= 38 IB-/6 W:</+.=2//>
Ð+Ñ What is the expected return of the market portfolio?
<Q
"&!
$!!
"
#
œ AE <E  AF <F œ
<E 
<F œ <E  <F ¸ "$%.
%&!
%&!
$
$
Ð,Ñ What is the standard deviation of the market portfolio?
#
# #
œ A#E 5E#  #AE AF 5EF  AF
5Q
5F œ
œ A#E 5E#  #AE AF 3EF 5E 5F  A#F 5F# ¸ !Þ!)"
5Q ¸ !Þ!*
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Ð-Ñ What is the beta of stock E?
5EQ
"E œ # ß 5EQ œ AE 5E#  AF 5EF
5Q
œ AE 5E#  AF 3EF 5E 5F ¸ "Þ#)'$
5FQ
"F œ # ß 5FQ œ AF 5F#  AE 5EF
5Q
œ AF 5F#  AE 3EF 5E 5F ¸ !Þ)&"*
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Ð.Ñ What is the risk free rate in Simpleland?
<E  <0 œ "E Ð<Q  <0 Ñ Í <0 œ
<F  <0 œ "F Ð<Q
EMSE 6992
Chapter 7
<E  "E <Q
¸ !Þ!'#&
"  "E
<F  "F <Q
 <0 Ñ Í <0 œ
¸ !Þ!'#&
"  "F
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 7
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Solution Problem 7.): Wizards
Let the random variable G be the cost of a tv set production. We have
T <ÐG œ $20MÑ œ !Þ&!
 T <ÐG œ $"'MÑ œ !Þ&!
Let T be the random price from selling the tv set production. We have
T œ #%Q . Furthermore, it is assumed that T and G are independent
random variables.
Ð+Ñ. What is the expected rate of return? Let < be the random rate of
return, then with the independence of G and T it follows that
T
T
"
<T œ
 " Ê <T œ IÒ<T Ó œ IÒ  "Ó œ IÒT ÓIÒ Ó  "
G
G
G
"
"
*
(
œ #% ‚ Ð!Þ& ‚
 !Þ& ‚ Ñ  " œ #%Ð
 "Ñ œ
œ $&%
#!
"'
"'!
#!
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Chapter 7
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: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Ð,Ñ It is determined that the final sales price is correlated with the
market return as follows:
#
IÒÐT  T ÑÐ<Q  <Q )] œ $20Million ‚ 5Q
.
What is the beta of the tv set production?
5T Q œ IÒÐ<T  <T ÑÐ<Q  <Q )]
T
T
œ IÒÐ  ÑÐ<Q  <Q )]ß (using independence of with T and M)
G
G
"
*
* #
#
œ IÒ ÓIÒÐT  T ÑÐ<Q  <Q )] œ
‚ 20 ‚ 5Q œ 5Q
G
"'!
)
Hence,
"T œ
EMSE 6992
Chapter 7
* #
) 5Q
#
5Q
: Investment Science - Leunberger
: The Capital Asset Pricing Model
*
œ Þ
)
Page 9
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Ð-Ñ Assume that <0 œ *% and <Q œ $$%. Is this an acceptable
project based on a CAPM criterion? What is the excess rate of return
Ð  or  ) above the return predicted by the CAPM?
Following CAPM we have
<T œ " Ð<Q  <0 Ñ  <0 œ
*
œ ‚ Ð!Þ$$  !Þ!*Ñ  !Þ!*
)
* #%
*
œ

œ !Þ$'
) "!! "!!
From Ð+Ñ it followed <T œ !Þ$&. Hence, according to CAPM the return
should be "% higher. Thus, the project is not acceptable, but it is close.
EMSE 6992
Chapter 7
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: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Solution Problem 7.4: Quick CAPM Derivation
Given risky assets <5 œ "ß á 8Þ Derive the CAPM formula
<5  <0 œ "5 Ð<Q  <0 Ñß 5 œ "ß á ß 8
by using Equation Ð'Þ*Ñ in Chapter 6. We have <Q œ  A3 <3 .
8
3œ"
r
M
5
4
rf
θ
3
2
1
σr
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: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
By equation Ð'Þ*Ñ the weights A3 satisfy the following equation=.
553 -A3 œ <5  <0 ß 5 œ "ß á ß 8ß - constant
8
Ð'Þ*Ñ
3œ"
From Ð'Þ*Ñ we have:
A5 553 -A3  œ A5 Ð<5  <0 Ñ œ A5 <5  <0 œ <Q  <0 Ð"Ñ
8
8
8
8
5œ"
3œ"
5œ"
5œ"
Moreover, since <Q œ  A3 <3
8
3œ"
#
A5 553 -A3  œ -553 A3 A5 œ -5Q
8
5œ"
EMSE 6992
Chapter 7
8
3œ"
8
8
Ð#Ñ
5œ" 3œ"
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 12
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Combining Ð"Ñ and Ð#Ñ we have:
#
-5Q
œ <Q
From <Q œ  A3 <3 , we have
<Q  <0
 <0 Í - œ
#
5Q
Ð$Ñ
8
3œ"
G9@Ð<Q ß <5 Ñ œ A3 G9@Ð<3 ß <5 Ñ œ 535 A3
8
3œ"
8
Ð%Ñ
3œ"
Recalling Ð'Þ*Ñ  553 -A3 œ <5  <0 ß 5 œ "ß á ß 8 we have with Ð%Ñ
8
3œ"
- ‚ 553 A3 œ <5  <0 Í - ‚ G9@Ð<Q ß <5 Ñ œ <5  <0 ß 5 œ "ß á ß 8
8
3œ"
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Now substitution of Ð$Ñ yields:
<Q  <0
‚ G9@Ð<Q ß <5 Ñ œ <5  <0 ß 5 œ "ß á ß 8 Í
#
5Q
<5  <0 œ
EMSE 6992
Chapter 7
G9@Ð<Q ß <5 Ñ
‚ Ð<Q  <0 Ñß 5 œ "ß á ß 8
#
5Q
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 14
Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Solution Problem 7.5: Uncorrelated Assets
The total amount of asset 3 in the market is B3 Þ Then if the market consists
of 8 assets, 3 œ "ß á ß 8. The total amount of assets in the market equals:
> œ B3 ß
8
Ð"Ñ
3œ"
and we obtain for the capitalization weights A3 œ B3 Î> ß 3 œ "ß á 8Þ
Denoting <3 the random return of asset 3, we have for the market return.
<Q œ A3 <3 Þ
8
Ð#Ñ
3œ"
Under the assumption that the returns <3 ß 3 œ "ß á ß 8 are mutually
uncorrelated find an expression for "3 œ G9@Ð<3 ß <7 ÑÎZ +<Ð<7 Ñ in terms
of B3 ß > and the 53 ' s.
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
We have from Ð#Ñ that
Z +<Ð<7 Ñ œ A3 A4 534
8
8
Ð$Ñ
3œ" 4œ"
However, when 3 Á 4 it follows from uncorrelatedness that 534 œ !. Thus
it follows from Ð$Ñ that:
Z +<Ð<7 Ñ œ A#4 54# .
8
Ð%Ñ
4œ"
Analagously, we have from Ð#Ñ with uncorrelatedness of returns
G9@Ð<3 ß <7 Ñ œ A4 534 œ A3 53#
8
Ð&Ñ
4œ"
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
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Notes by J. Rene van Dorp
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Homework Solution: Investment Science
Thus:
G9@Ð<3 ß <7 Ñ
A3 53#
ÐB3 Î>Ñ53#
B3 53#
"3 œ
œ 8
œ 8
œ>‚ 8
Þ Ð'Ñ
Z +<Ð<7 Ñ
 A#4 54#
 ÐB4 Î>Ñ# 54#
 B4# 54#
4œ"
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
4œ"
4œ"
Page 17
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr