Homework Solution: Investment Science
Solution Problem 5.2: The Road
Project # or 4 are carried out Ê B#  B% œ "
Project ' or ( are carried out Ê B'  B( œ "
Add constraint:
ÐB#  B% ÑÖ"  ÐB'  B( Ñ× œ !
Observe:
B#  B % œ " Ê B '  B ( œ "
Hence, if projects # or % are carried out, either project ' or ( must also be
carried out.
EMSE 6992
Chapter 5
: Investment Science - Leunberger
: The Term Structure of Interest Rates
Page 1
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Cost
($1,000)
NPV
($1,000)
Optimal
x-values
Cost
NPV
Goals
Road between Augen and Burger
1 Concrete, 2 lanes
2,000
4,000
0
0
0
2 Concrete, 4 lanes
3,000
5,000
0
0
0
3 Asphalt, 2 lanes
1,500
3,000
0
0
0
4 Asphalt, 4 lanes
2,200
4,300
1
2,200
4,300
500
1,000
0
0
0
6 Add Lane
1,500
1,500
1
1,500
1,500
7 New Structure
2,500
2,000
0
0
0
8 Traffic lights
100
300
0
0
0
9 Turn lanes
600
1,000
0
0
0
10 Underpass
1,000
At most one project can be selected for each major objective
2,000
1
1,000
4,700
2,000
7,800
1
Bridge at Cay Road
5 Repair Exisiting
1
Traffic Control in Downsburg
EMSE 6992
Chapter 5
: Investment Science - Leunberger
: The Term Structure of Interest Rates
Totals
1
Page 2
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Solution Problem 5.11: Growing Annuity
W œ
∞
5œ"
œ
œ
œ
"
"<

Ð"1Ñ5"
Ð"<Ñ5
"
"<

"1
"<
"1
"<
  Ð"1Ñ
∞
Ð"<Ñ5
 Ð"1Ñ5"
∞
5œ#
5#
Ð"<Ñ
 Ð"1Ñ
∞
5"

"1
"<
W Í WÐ" 
"
"<
5"
5œ#
"
"<
<1
Í W "<
œ
EMSE 6992
Chapter 5
œ
"
"<
5œ"
Ð"<Ñ5
"1
"< Ñ
ÍW œ
: Investment Science - Leunberger
: The Term Structure of Interest Rates
œ
"
"<
"
<1
Page 3
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Solution Problem 5.12: Two-Stage Growth
It is common practice in security analysis to modify the basic dividend
growth model by allowing more than one stage of growth, with the growth
factor being different in different stages. As an example, consider company
^ , which currently distributed dividend of $"!Q annually. The dividends
are expected to grow at a rate of "!% for the next & years and at a rate of
&% thereafter.
+Þ Using a dividend discount approach with an interest rate of "&%
what is the approximate value of the company?
Approximate Value: $"#(Þ*"Q (See spreadsheet)
,Þ Find a general formula for the value of a company satisfying a
two-stage growth model. Assume a growth rate of K for 5 years,
followed by a growth rate of 1 thereafter, and an initial dividend
of H! .
EMSE 6992
Chapter 5
: Investment Science - Leunberger
: The Term Structure of Interest Rates
Page 4
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
Ð"  KÑ3
H! Ð"  KÑ5 ∞ Ð"  1Ñ3

R T Z œ H! 

3
5
3
Ð"

<Ñ
Ð"

<Ñ
Ð"

<Ñ
3œ"
3œ"
5
Ð"Ñ
For the first term in Ð"Ñ we have À
5
5
Ð"  KÑ3
"K 3
"K
3



œ
œ
B
ß
where
B
œ


Ð"  <Ñ3
"<
"<
3œ"
3œ"
3œ"
5
Utilizing  B3 œ
8
3œ"
BB8"
"B
Ð"  KÑ3

œ
3
Ð"  <Ñ
3œ"
5
EMSE 6992
Chapter 5
one obtains
"K
"<

5 +1
"K
 "< 
"
"K
"<
"
œ Ð"  KÑ ‚
: Investment Science - Leunberger
: The Term Structure of Interest Rates
 "K
"< 
<K
5
Ð#Ñ
Page 5
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
For the second term in Ð#Ñ we have
∞
Ð"  1Ñ3
"1 3

œ 

3
Ð"

<Ñ
"

<
3œ"
3œ"
∞
Utilizing  B3 œ
∞
3œ"
B
"B
one obtains
Ð"  1Ñ3
"1
"
"
"<

œ
œ
‚
œ Ð"  1Ñ ‚
Ð$Ñ
"1
"1
3
Ð"  <Ñ
"  < "  "<
<1
"  "<
3œ"
∞
"1
Finally, substitution of Ð#Ñ and Ð$Ñ in Ð"Ñ yields:
EMSE 6992
Chapter 5
: Investment Science - Leunberger
: The Term Structure of Interest Rates
Page 6
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr
Homework Solution: Investment Science
"
R T Z œ H! Ð"  KÑ ‚
5
"K
 "< 
<K
H! Ð"  KÑ5
"

Ð"  1Ñ ‚
Ð"Ñ
Ð"  <Ñ5
<1
Substitution of < œ !Þ"&ß K œ !Þ"ß 1 œ !Þ!&, H! œ "! and 5 œ & yields:
R T Z œ $"#(Þ*#Q
EMSE 6992
Chapter 5
: Investment Science - Leunberger
: The Term Structure of Interest Rates
Page 7
Notes by J. Rene van Dorp
www.seas.gwu.edu/~dorpjr