Chapter 5: Time Value of Money
Learning Objectives
79
Chapter 5
Time Value of Money
Learning Objectives
After reading this chapter, students should be able to:
◆ Explain how the time value of money works and discuss why it is such an important concept in finance.
◆ Calculate the present value and future value of lump sums.
◆ Identify the different types of annuities, calculate the present value and future value of both an
ordinary annuity and an annuity due, and calculate the relevant annuity payments.
◆ Calculate the present value and future value of an uneven cash flow stream. You will use this
knowledge in later chapters that show how to value common stocks and corporate projects.
◆ Explain the difference between nominal, periodic, and effective interest rates. An understanding of
these concepts is necessary when comparing rates of returns on alternative investments.
◆ Discuss the basics of loan amortization and develop a loan amortization schedule that you might use
when considering an auto loan or home mortgage loan.
Lecture Suggestions
We regard Chapter 5 as the most important chapter in the book, so we spend a good bit of time on it. We
approach time value in three ways. First, we try to get students to understand the basic concepts by use of
time lines and simple logic. Second, we explain how the basic formulas follow the logic set forth in the time
lines. Third, we show how financial calculators and spreadsheets can be used to solve various time value
problems in an efficient manner. Once we have been through the basics, we have students work problems
and become proficient with the calculations and also get an idea about the sensitivity of output, such as
present or future value, to changes in input variables, such as the interest rate or number of payments.
Some instructors prefer to take a strictly analytical approach and have students focus on the
formulas themselves. The argument is made that students treat their calculators as “black boxes,” and that
they do not understand where their answers are coming from or what they mean. We disagree. We think
that our approach shows students the logic behind the calculations as well as alternative approaches, and
because calculators are so efficient, students can actually see the significance of what they are doing better
if they use a calculator. We also think it is important to teach students how to use the type of technology
(calculators and spreadsheets) they must use when they venture out into the real world.
Our research suggests that the best calculator for the money for most students is the HP-10BII.
Finance and accounting majors might be better off with a more powerful calculator, such as the HP-17BII.
We recommend these two for people who do not already have a calculator, but we tell them that any
financial calculator that has an IRR function will do.
We also tell students that it is essential that they work lots of problems, including the end–of–
chapter problems. We emphasize that this chapter is critical, so they should invest the time now to get the
material down. We stress that they simply cannot do well with the material that follows without having this
material down cold. Bond and stock valuation, cost of capital, and capital budgeting make little sense, and
one certainly cannot work problems in these areas, without understanding time value of money first.
We base our lecture on the integrated case. The case goes systematically through the key points
in the chapter, and within a context that helps students see the real world relevance of the material in the
chapter. We ask the students to read the chapter, and also to “look over” the case before class. However,
our class consists of about 1,000 students, many of whom view the lecture on TV, so we cannot count on
them to prepare for class. For this reason, we designed our lectures to be useful to both prepared and
unprepared students.
Since we have easy access to computer projection equipment, we generally use the
PowerPoint
slides as the core of our lectures. We make these slides available to our students, and we strongly suggest
Chapter 5: Time Value of Money
Lecture Suggestions
81
provide an additional example, or the like, we use post-it notes attached at the proper spot. The
advantages of this system are (1) that we have a carefully structured lecture that is easy for us to prepare
DAYS ON CHAPTER: 4 OF 56 DAYS (50-minute periods)
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Answers and Solutions
Chapter 5: Time Value of Money
Answers to End-of-Chapter Questions
5-1 The opportunity cost is the rate of interest one could earn on an alternative investment with a risk
5-2 True. The second series is an uneven cash flow stream, but it contains an annuity of $400 for 8
5-3 True, because of compounding effects—growth on growth. The following example demonstrates
5-7 The annual percentage rate (APR) is the periodic rate times the number of periods per year. It is
also called the nominal, or stated, rate. With the “Truth in Lending” law, Congress required that
Chapter 5: Time Value of Money
Answers and Solutions
83
Solutions to End-of–Chapter Problems
5-1 0 1 2 3 4 5
| | | | | |
PV = 10,000 FV5 = ?
5-2 0 5 10 15 20
5-3 0 18
5-4 0 N = ?
| |
5-5 0 1 2 N – 2 N – 1 N
| | | • • • | | |
10%
7%
6.5%
12%
I/YR = ?
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Answers and Solutions
Chapter 5: Time Value of Money
5-6 Ordinary annuity:
0 1 2 3 4 5
| | | | | |
300 300 300 300 300
FVA5 = ?
5-7 0 1 2 3 4 5 6
| | | | | | |
100 100 100 200 300 500
PV = ? FV = ?
5-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV = 0.
Solve for PMT = $444.89.
7%
8%
Chapter 5: Time Value of Money
Answers and Solutions
85
5-9 a. 0 1
| | $500(1.06) = $530.00.
-500 FV = ?
b. 0 1 2
| | | $500(1.06)2 = $561.80.
-500 FV = ?
c. 0 1
| | $500(1/1.06) = $471.70.
PV = ? 500
d. 0 1 2
| | | $500(1/1.06)2 = $445.00.
PV = ? 500
5-10 a. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | | $500(1.06)10 = $895.42.
-500 FV = ?
b. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | | $500(1.12)10 = $1,552.92.
-500 FV = ?
6%
6%
6%
6%
6%
6%
12%
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Answers and Solutions
Chapter 5: Time Value of Money
d. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
PV = ? 1,552.90
e. The present value is the value today of a sum of money to be received in the future. For
5-11 a. 2009 2010 2011 2012 2013 2014
| | | | | |
-6 12 (in millions)
5-12 These problems can all be solved using a financial calculator by entering the known values shown
on the time lines and then pressing the I/YR button.
a. 0 1
| |
+700 –749
?
12%
I/YR = ?
Chapter 5: Time Value of Money
Answers and Solutions
87
c. 0 10
| |
+85,000 -201,229
5-13 a. ?
| |
-200 400
100%
5-14 a. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400
FV = ?
b. 0 1 2 3 4 5
| | | | | |
200 200 200 200 200
FV = ?
I/YR = ?
7%
10%
5%
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Answers and Solutions
Chapter 5: Time Value of Money
c. 0 1 2 3 4 5
| | | | | |
400 400 400 400 400
FV = ?
d. To solve part d using a financial calculator, repeat the procedures discussed in parts a, b, and c,
but first switch the calculator to “BEG” mode. Make sure you switch the calculator back to “END”
mode after working the problem.
1. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400 FV = ?
5%
5-15 a. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
PV = ? 400 400 400 400 400 400 400 400 400 400
0%
0%
10%
10%
Chapter 5: Time Value of Money
Answers and Solutions
89
d. 1. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400
PV = ?
2. 0 1 2 3 4 5
| | | | | |
200 200 200 200 200
PV = ?
3. 0 1 2 3 4 5
| | | | | |
400 400 400 400 400
PV = ?
5-17 0 1 2 3 4 30
| | | | | • • • |
85,000 -8,273.59 -8,273.59 -8,273.59 -8,273.59 -8,273.59
5-18 a. Cash Stream A Cash Stream B
0 1 2 3 4 5 0 1 2 3 4 5
| | | | | | | | | | | |
PV = ? 100 400 400 400 300 PV = ? 300 400 400 400 100
I/YR = ?
10%
5%
0%
8%
8%
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Answers and Solutions
Chapter 5: Time Value of Money
5-19 a. Begin with a time line:
40 41 64 65
| | • • • | |
5,000 5,000 5,000
FV = ?
b. 40 41 69 70
| | • • • | |
5,000 5,000 5,000
FV = ?
c. 1. 65 66 67 84 85
| | | • • • | |
423,504.48 PMT PMT PMT PMT
2. 70 71 72 84 85
| | | • • • | |
681,537.69 PMT PMT PMT PMT
5-20 Contract 1: PV =
432 )10.1(
000,000,3$
)10.1(
000,000,3$
)10.1(
000,000,3$
10.1
000,000,3$ +++
9%
9%
9%
9%
Chapter 5: Time Value of Money
Answers and Solutions
91
5-21 a. If Crissie expects a 7% annual return on her investments:
1 payment 10 payments 30 payments
N = 10 N = 30
b. If Crissie expects an 8% annual return on her investments:
1 payment 10 payments 30 payments
c. If Crissie expects a 9% annual return on her investments:
1 payment 10 payments 30 payments
d. The higher the interest rate, the more useful it is to get money rapidly, because it can be
5-22 a. This can be done with a calculator by specifying an interest rate of 5% per period for 20
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Answers and Solutions
Chapter 5: Time Value of Money
b. Set up an amortization table:
Beginning Payment of Ending
Period Balance Payment Interest Principal Balance
5-23 a. 0 1 2 3 4 5
| | | | | |
-500 FV = ?
b. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
-500 FV = ?
c. 0 4 8 12 16 20
| | | | | |
-500 FV = ?
1%
12%
6%
3%
Chapter 5: Time Value of Money
Answers and Solutions
93
e. 0 365 1,825
| | • • • |
-500 FV = ?
5-24 a. 0 2 4 6 8 10
| | | | | |
PV = ? 500
b. 0 4 8 12 16 20
| | | | | |
PV = ? 500
c. 0 1 2 12
| | | • • • |
PV = ? 500
6%
3%
1%
0.0329%
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Answers and Solutions
Chapter 5: Time Value of Money
5-25 a. 0 1 2 3 9 10
| | | | • • • | |
-400 -400 -400 -400 -400
FV = ?
5-26 Using the information given in the problem, you can solve for the maximum car price attainable.
5-27 a. Bank A: INOM = Effective annual rate = 4%.
b. If funds must be left on deposit until the end of the compounding period (1 year for Bank A
6%
Chapter 5: Time Value of Money
Answers and Solutions
95
5-28 Here you want to have an effective annual rate on the credit extended that is 2% more than what
5-30 a. Using the information given in the problem, you can solve for the length of time required to
reach $1 million.
b. Using the 16.043713 year target, you can solve for the required payment:
c. Erika is investing in a relatively safe fund, so there is a good chance that she will achieve her
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Answers and Solutions
Chapter 5: Time Value of Money
5-31 a. 0 1 2 3 4
| | | | |
PV = ? -10,000 –10,000 -10,000 -10,000
5-32 0 1 2 3 4 5 6
| | | | | | |
1,500 1,500 1,500 1,500 1,500 ?
5-33 Begin with a time line:
0 1 2 3
| | | |
5,000 5,500 6,050
FV = ?
Use a financial calculator to calculate the present value of the cash flows and then determine the
future value of this present value amount:
5%
8%
7%
Chapter 5: Time Value of Money
Answers and Solutions
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5-34 a. With a financial calculator, enter N = 3, I/YR = 10, PV = -25000, and FV = 0, and solve for
b. % Interest % Principal
Year 1: $2,500/$10,052.87 = 24.87% $7,552.87/$10,052.87 = 75.13%
5-35 a. Using a financial calculator, enter N = 3, I/YR = 7, PV = -90000, and FV = 0, then solve for
c. 30-year amortization with balloon payment at end of Year 3:
Beginning Principal Ending
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Answers and Solutions
Chapter 5: Time Value of Money
5-36 a. Begin with a time line:
0 1 2 3 4 5 6 6-mos.
0 1 2 3 Years
| | | | | | |
1,000 1,000 1,000 1,000 1,000 FVA = ?
b. Here’s the time line:
0 1 2 3 4 Qtrs
| | | | |
PMT = ? PMT = ? FV = 10,000
5-37 a. Using the information given in the problem, you can solve for the length of time required to
pay off the card.
2%
1%