CHAPTER 5
INTRODUCTION TO VALUATION: THE
TIME VALUE OF MONEY
Answers to Concepts Review and Critical Thinking Questions
2. Compounding refers to the growth of a dollar amount through time via reinvestment of interest
6. It’s a reflection of the time value of money. TMCC gets to use the $24,099. If TMCC uses it wisely,
7. This will probably make the security less desirable. TMCC will only repurchase the security prior to
8. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to
10. The price would be higher because, as time passes, the price of the security will tend to rise toward
CHAPTER 5 – 2
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. The time line for the cash flows is:
0 8
$7,5
00
FV
The simple interest per year is:
So, after 8 years you will have:
With compound interest we use the future value formula:
FV = PV(1 + r)t
The difference is:
2. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
0 11
$2,3
28
FV
0 7
$7,5
13
FV
CHAPTER 5 – 3
0 14
$74,3
81
FV
0 16
$192,0
50
FV
3. To find the PV of a lump sum, we use:
PV = FV/(1 + r)t
0 13
PV $16,8
32
0 4
PV $48,3
18
0 29
PV $886,0
73
0 40
PV $550,1
64
4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
CHAPTER 5 – 4
0 5
–
$181
$317
0 17
–
$335
$1,0
80
0 13
–
$48,00
0
$185,3
82
0 30
–
$40,35
3
$531,6
18
5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV/PV)/ln(1 + r)
0 ?
–$560 $1,389
0 ?
–$810 $1,821
0 ?
–
$18,40
$289,7
15
CHAPTER 5 – 5
0
0 ?
–
$21,50
0
$430,2
58
CHAPTER 5 – 6
6. The time line is:
0 18
–
$73,00
0
$345,0
00
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
Solving for r, we get:
r = (FV/PV)1/t – 1
7. To find the length of time for money to double, triple, etc., the present value and future value are
irrelevant as long as the future value is twice the present value for doubling, three times as large for
tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the
same answer since they are the inverse of each other. We will use the FV formula, that is:
Solving for t, we get:
The length of time to double your money is:
0 ?
–$1 $2
FV = $2 = $1(1.061)t
The length of time to quadruple your money is:
0 ?
–$1 $4
Notice that the length of time to quadruple your money is twice as long as the time needed to double
your money (the slight difference in these answers is due to rounding). This is an important concept
of time value of money.
CHAPTER 5 – 7
CHAPTER 5 – 8
8. The time line is:
0 16
–
$354,9
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
9. The time line is:
0 ?
00
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV/PV)/ln(1 + r)
10. The time line is:
0 20
PV $415,000,000
To find the PV of a lump sum, we use:
PV = FV/(1 + r)t
CHAPTER 5 – 9
11. The time line is:
0 80
PV $2,000,0
00
To find the PV of a lump sum, we use:
PV = FV/(1 + r)t
12. The time line is:
0 115
$50 FV
To find the FV of a lump sum, we use:
FV = PV(1 + r)t
13. The time line is:
0 121
–$150 $1,800,0
00
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
To find the FV of the first prize in 2040, we use:
0 24
$1,800,0
00
FV
FV = PV(1 + r)t
CHAPTER 5 – 10
14. The time line is:
0 4
0
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
Notice that the interest rate is negative. This occurs when the FV is less than the PV.
Intermediate
15. The time line from minting to the first sale is:
0 192
–$15 $430,000
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
The time line from the first sale to the second sale is:
0 35
–
$430,000
$4,582,5
00
CHAPTER 5 – 11
Solving for r, we get:
r = (FV/PV)1/t – 1
The time line from minting to the second sale is:
0 227
–$15 $4,582,5
00
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t – 1
16. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
Solving for r, we get:
a. The time line is:
0 20
–$50 $100
r = (FV/PV)1/t – 1
b. The time line is:
0 10
–$50 FV
FV = PV(1 + r)t
CHAPTER 5 – 12
CHAPTER 5 – 13
c. The time line is:
0 10
–$50.50 $100
r = (FV/PV)1/t – 1
17. The time line is:
0 9
PV $245,000
To find the PV of a lump sum, we use:
PV = FV/(1 + r)t
18. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
0 45
$5,500 FV
0 35
$5,500 FV
Better start early!
19. The time line is:
0 2 8
$20,00
0
FV
CHAPTER 5 – 14
We need to find the FV of a lump sum. However, the money will only be invested for six years, so
the number of periods is six.
FV = PV(1 + r)t
CHAPTER 26 – 15