# Homework Solution: Investment Science

```Homework Solution: Investment Science
Solution Problem 7.2: A Small World
Consider a world in which there are only two risky assets, E and F&szlig; and a
risk free asset J &THORN; The two risk assets are in equal supply in the market; that
is,
&lt;Q
&quot;
œ &ETH;&lt;E  &lt;F &Ntilde;.
#
The following information is known: &lt;0 œ !&THORN;&quot;!&szlig; 5E# œ !&THORN;!%&szlig; 5EF œ &THORN;!&quot;&szlig;
5F# œ !&THORN;!#&szlig; and &lt;Q œ !&THORN;&quot;)&THORN; Note:
[email protected]&ETH;\&szlig; ]  ^&Ntilde; œ [email protected]&ETH;\&szlig; ] &Ntilde;  [email protected]&ETH;\&szlig; ^&Ntilde;
[email protected]&ETH;\&szlig; \&Ntilde; œ Z +&lt;&Ograve;\&Oacute;&szlig; [email protected]&ETH;&lt;\&szlig; =] &Ntilde; œ &lt;[email protected]&ETH;\&szlig; ] &Ntilde;
EMSE 6992
Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 1
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
#
+&THORN; Find a general expression for 5Q
&szlig; &quot;E and &quot;F &THORN;
&lt;Q
&quot;
&quot; #
#
œ &ETH;&lt;E  &lt;F &Ntilde; &Ecirc; 5Q œ &ETH;5E  #5EF  5F# &Ntilde;
#
%
&quot;
&quot; #
&lt;Q œ &ETH;&lt;E  &lt;F &Ntilde; &Ecirc; 5EQ œ [email protected]&ETH;&lt;E &szlig; &lt;Q &Ntilde; œ &ETH;5E  5EF &Ntilde; &Ecirc;
#
#
&quot;
#
#
5
5
&ETH;

&Ntilde;
5
5EQ
&quot;
EF
E
#
E  5EF
&quot;E œ # œ &quot; #
œ ‚ #
#
#
#
5Q
5
5
5

#

5
5
5
&ETH;

#

&Ntilde;
EF
EF
F
E
F
E
%
&quot;
&quot;
&lt;Q œ &ETH;&lt;E  &lt;F &Ntilde; &Ecirc; 5FQ œ [email protected]&ETH;&lt;F &szlig; &lt;Q &Ntilde; œ &ETH;5EF  5F# &Ntilde; &Ecirc;
#
#
&quot;
#
5
5
&ETH;

5FQ
5EF  5F#
&quot;
EF
F&Ntilde;
#
&quot;F œ # œ &quot; #
œ ‚ #
#
#
#
5Q
5
5
5

#

5
5
5
&ETH;

#

&Ntilde;
EF
EF
F
E
F
E
%
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Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 2
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
,&THORN;
&amp;
&lt;E œ &lt;J  &quot;E &ETH;&lt;Q  &lt;0 &Ntilde; &Ecirc; &lt;E œ !&THORN;&quot;!  &ETH;!&THORN;&quot;)  !&THORN;&quot;!&Ntilde; œ #!%
%
&ETH;=// IB-/6 W:&lt;/+.=2//&gt;&Ntilde;
&lt;F œ &lt;J  &quot;F &ETH;&lt;Q
\$
 &lt;0 &Ntilde; &Ecirc; &lt;F œ !&THORN;&quot;!  &ETH;!&THORN;&quot;)  !&THORN;&quot;!&Ntilde; œ &quot;'%
%
&ETH;=// IB-/6 W:&lt;/+.=2//&gt;&Ntilde;
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Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
Page 3
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
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 &quot;\$ . 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
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
G+6-?6+&gt;398= 38 IB-/6 W:&lt;/+.=2//&gt;
&ETH;+&Ntilde; What is the expected return of the market portfolio?
&lt;Q
&quot;&amp;!
\$!!
&quot;
#
œ AE &lt;E  AF &lt;F œ
&lt;E 
&lt;F œ &lt;E  &lt;F &cedil; &quot;\$%.
%&amp;!
%&amp;!
\$
\$
&ETH;,&Ntilde; 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# &cedil; !&THORN;!)&quot;
5Q &cedil; !&THORN;!*
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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
&ETH;-&Ntilde; What is the beta of stock E?
5EQ
&quot;E œ # &szlig; 5EQ œ AE 5E#  AF 5EF
5Q
œ AE 5E#  AF 3EF 5E 5F &cedil; &quot;&THORN;#)'\$
5FQ
&quot;F œ # &szlig; 5FQ œ AF 5F#  AE 5EF
5Q
œ AF 5F#  AE 3EF 5E 5F &cedil; !&THORN;)&amp;&quot;*
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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
&ETH;.&Ntilde; What is the risk free rate in Simpleland?
&lt;E  &lt;0 œ &quot;E &ETH;&lt;Q  &lt;0 &Ntilde; &Iacute; &lt;0 œ
&lt;F  &lt;0 œ &quot;F &ETH;&lt;Q
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Chapter 7
&lt;E  &quot;E &lt;Q
&cedil; !&THORN;!'#&amp;
&quot;  &quot;E
&lt;F  &quot;F &lt;Q
 &lt;0 &Ntilde; &Iacute; &lt;0 œ
&cedil; !&THORN;!'#&amp;
&quot;  &quot;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 &lt;&ETH;G œ \$20M&Ntilde; œ !&THORN;&amp;!
 T &lt;&ETH;G œ \$&quot;'M&Ntilde; œ !&THORN;&amp;!
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.
&ETH;+&Ntilde;. What is the expected rate of return? Let &lt; be the random rate of
return, then with the independence of G and T it follows that
T
T
&quot;
&lt;T œ
 &quot; &Ecirc; &lt;T œ I&Ograve;&lt;T &Oacute; œ I&Ograve;  &quot;&Oacute; œ I&Ograve;T &Oacute;I&Ograve; &Oacute;  &quot;
G
G
G
&quot;
&quot;
*
(
œ #% ‚ &ETH;!&THORN;&amp; ‚
 !&THORN;&amp; ‚ &Ntilde;  &quot; œ #%&ETH;
 &quot;&Ntilde; œ
œ \$&amp;%
#!
&quot;'
&quot;'!
#!
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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
&ETH;,&Ntilde; It is determined that the final sales price is correlated with the
market return as follows:
#
I&Ograve;&ETH;T  T &Ntilde;&ETH;&lt;Q  &lt;Q )] œ \$20Million ‚ 5Q
.
What is the beta of the tv set production?
5T Q œ I&Ograve;&ETH;&lt;T  &lt;T &Ntilde;&ETH;&lt;Q  &lt;Q )]
T
T
œ I&Ograve;&ETH;  &Ntilde;&ETH;&lt;Q  &lt;Q )]&szlig; (using independence of with T and M)
G
G
&quot;
*
* #
#
œ I&Ograve; &Oacute;I&Ograve;&ETH;T  T &Ntilde;&ETH;&lt;Q  &lt;Q )] œ
‚ 20 ‚ 5Q œ 5Q
G
&quot;'!
)
Hence,
&quot;T œ
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Chapter 7
* #
) 5Q
#
5Q
: Investment Science - Leunberger
: The Capital Asset Pricing Model
*
œ &THORN;
)
Page 9
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
&ETH;-&Ntilde; Assume that &lt;0 œ *% and &lt;Q œ \$\$%. Is this an acceptable
project based on a CAPM criterion? What is the excess rate of return
&ETH;  or  ) above the return predicted by the CAPM?
Following CAPM we have
&lt;T œ &quot; &ETH;&lt;Q  &lt;0 &Ntilde;  &lt;0 œ
*
œ ‚ &ETH;!&THORN;\$\$  !&THORN;!*&Ntilde;  !&THORN;!*
)
* #%
*
œ

œ !&THORN;\$'
) &quot;!! &quot;!!
From &ETH;+&Ntilde; it followed &lt;T œ !&THORN;\$&amp;. Hence, according to CAPM the return
should be &quot;% higher. Thus, the project is not acceptable, but it is close.
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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
Solution Problem 7.4: Quick CAPM Derivation
Given risky assets &lt;5 œ &quot;&szlig; &aacute; 8&THORN; Derive the CAPM formula
&lt;5  &lt;0 œ &quot;5 &ETH;&lt;Q  &lt;0 &Ntilde;&szlig; 5 œ &quot;&szlig; &aacute; &szlig; 8
by using Equation &ETH;'&THORN;*&Ntilde; in Chapter 6. We have &lt;Q œ  A3 &lt;3 .
8
3œ&quot;
r
M
5
4
rf
θ
3
2
1
σr
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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
By equation &ETH;'&THORN;*&Ntilde; the weights A3 satisfy the following equation=.
553 -A3 œ &lt;5  &lt;0 &szlig; 5 œ &quot;&szlig; &aacute; &szlig; 8&szlig; - constant
8
&ETH;'&THORN;*&Ntilde;
3œ&quot;
From &ETH;'&THORN;*&Ntilde; we have:
A5 553 -A3  œ A5 &ETH;&lt;5  &lt;0 &Ntilde; œ A5 &lt;5  &lt;0 œ &lt;Q  &lt;0 &ETH;&quot;&Ntilde;
8
8
8
8
5œ&quot;
3œ&quot;
5œ&quot;
5œ&quot;
Moreover, since &lt;Q œ  A3 &lt;3
8
3œ&quot;
#
A5 553 -A3  œ -553 A3 A5 œ -5Q
8
5œ&quot;
EMSE 6992
Chapter 7
8
3œ&quot;
8
8
&ETH;#&Ntilde;
5œ&quot; 3œ&quot;
: 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 &ETH;&quot;&Ntilde; and &ETH;#&Ntilde; we have:
#
-5Q
œ &lt;Q
From &lt;Q œ  A3 &lt;3 , we have
&lt;Q  &lt;0
 &lt;0 &Iacute; - œ
#
5Q
&ETH;\$&Ntilde;
8
3œ&quot;
[email protected]&ETH;&lt;Q &szlig; &lt;5 &Ntilde; œ A3 [email protected]&ETH;&lt;3 &szlig; &lt;5 &Ntilde; œ 535 A3
8
3œ&quot;
8
&ETH;%&Ntilde;
3œ&quot;
Recalling &ETH;'&THORN;*&Ntilde;  553 -A3 œ &lt;5  &lt;0 &szlig; 5 œ &quot;&szlig; &aacute; &szlig; 8 we have with &ETH;%&Ntilde;
8
3œ&quot;
- ‚ 553 A3 œ &lt;5  &lt;0 &Iacute; - ‚ [email protected]&ETH;&lt;Q &szlig; &lt;5 &Ntilde; œ &lt;5  &lt;0 &szlig; 5 œ &quot;&szlig; &aacute; &szlig; 8
8
3œ&quot;
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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
Now substitution of &ETH;\$&Ntilde; yields:
&lt;Q  &lt;0
‚ [email protected]&ETH;&lt;Q &szlig; &lt;5 &Ntilde; œ &lt;5  &lt;0 &szlig; 5 œ &quot;&szlig; &aacute; &szlig; 8 &Iacute;
#
5Q
&lt;5  &lt;0 œ
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Chapter 7
[email protected]&ETH;&lt;Q &szlig; &lt;5 &Ntilde;
‚ &ETH;&lt;Q  &lt;0 &Ntilde;&szlig; 5 œ &quot;&szlig; &aacute; &szlig; 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 &THORN; Then if the market consists
of 8 assets, 3 œ &quot;&szlig; &aacute; &szlig; 8. The total amount of assets in the market equals:
&gt; œ B3 &szlig;
8
&ETH;&quot;&Ntilde;
3œ&quot;
and we obtain for the capitalization weights A3 œ B3 &Icirc;&gt; &szlig; 3 œ &quot;&szlig; &aacute; 8&THORN;
Denoting &lt;3 the random return of asset 3, we have for the market return.
&lt;Q œ A3 &lt;3 &THORN;
8
&ETH;#&Ntilde;
3œ&quot;
Under the assumption that the returns &lt;3 &szlig; 3 œ &quot;&szlig; &aacute; &szlig; 8 are mutually
uncorrelated find an expression for &quot;3 œ [email protected]&ETH;&lt;3 &szlig; &lt;7 &Ntilde;&Icirc;Z +&lt;&ETH;&lt;7 &Ntilde; in terms
of B3 &szlig; &gt; and the 53 ' s.
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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
We have from &ETH;#&Ntilde; that
Z +&lt;&ETH;&lt;7 &Ntilde; œ A3 A4 534
8
8
&ETH;\$&Ntilde;
3œ&quot; 4œ&quot;
However, when 3 &Aacute; 4 it follows from uncorrelatedness that 534 œ !. Thus
it follows from &ETH;\$&Ntilde; that:
Z +&lt;&ETH;&lt;7 &Ntilde; œ A#4 54# .
8
&ETH;%&Ntilde;
4œ&quot;
Analagously, we have from &ETH;#&Ntilde; with uncorrelatedness of returns
[email protected]&ETH;&lt;3 &szlig; &lt;7 &Ntilde; œ A4 534 œ A3 53#
8
&ETH;&amp;&Ntilde;
4œ&quot;
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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
Thus:
[email protected]&ETH;&lt;3 &szlig; &lt;7 &Ntilde;
A3 53#
&ETH;B3 &Icirc;&gt;&Ntilde;53#
B3 53#
&quot;3 œ
œ 8
œ 8
œ&gt;‚ 8
&THORN; &ETH;'&Ntilde;
Z +&lt;&ETH;&lt;7 &Ntilde;
 A#4 54#
 &ETH;B4 &Icirc;&gt;&Ntilde;# 54#
 B4# 54#
4œ&quot;
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Chapter 7
: Investment Science - Leunberger
: The Capital Asset Pricing Model
4œ&quot;
4œ&quot;
Page 17
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
```