Business Law Question

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This is two assi

g

nment part ( send as two seperate documents). Please do not use AI and make sure its plagarism free

1

st assignment is:

ANSWER ALL QUESTIONS:

3

>1. EXPLORE THE WEBSITE OF YAHOO FINANCE AND PICK UP ANY TOPIC OF YOUR INTEREST RELATED TO FINANCE THEN WRITE ONE PAGE REPORT ON THE TOPIC.

2

. VISIT THE WEBSITE AAII.COM AND THEN EXPLORE DIFFERENT TABS. SEE WHAT INFO YOU FOUND AND THEN WRITE PAGE AND HALF.

WWW.AAII.COM

3. DISCUSS WHAT IS AMORTIZATION? WHAT ARE AMORTIZED LOANS? ALSO FIND AMORTIZATION CALCULATOR ON THE INTERNET AND REPORT YOUR FINDINGS AS TO WHAT INFORMATION IT GAVE YOU.

REFERENCES ARE MANDATORY.

%

in the next year, drop to

from

1 to Year 2, and drop to a constant 5% for Year 2 and all subsequent yea

. Hamilton has just paid a dividend of

and its stock has a required return of 11%.

$2.50

rs

30%

5%

g30%15%5%5%Year0123

(1

g) = D3

=

gL

=

P2+D2

P1=+

P1=

=

/P0

/

Dividend yield =

/P0

Cap. Gain yield=/

Cap. Gain yield=

Cap. Gain yield=

–Dividend yield

Cap. Gain yield=–

Cap. Gain yield=

1. Find the expected dividend yield.

Dividend yield =D2/P1

Dividend yield =/Dividend yield =2. Find the expected capital gains yield.

Cap. Gain yield=

/P1

Cap. Gain yield=/Cap. Gain yield=Alternatively, the capital gains yield can be calculated by simply subtracting the dividend yield from the total expected return.Cap. Gain yield=Expected return–Dividend yieldCap. Gain yield=–Cap. Gain yield=

Hamilton Landscaping’s dividend growth rate is expected to be		3	0	1			5%	Year	rs	$2.50
a. What is Hamilton’s estimated stock price today?
D0
11.0%
g0,1	Short-run g; for Year 1 only.
g1,2	15%	Short-run g; for Year 2 only.
gL	Long-run g; for Year 3 and all following years.
Dividend
PV of dividends and PV of horizon value
			=		D2		+
= Horizon value =	P2
= rs				–
=		P0
a. What is Hamilton’s estimated stock price for Year 1?
				P1
(1 + rs)
b. If you bought the stock at Year 0, what your expected dividend yield and capital gains for the upcoming year?
	1. Find the expected dividend yield.
				Dividend yield
D1
		Dividend yield =
	2. Find the expected capital gains yield.
Use the estimated price for Year 1, P1, to find the expected gain.
											Cap. Gain yield=
(P1 – P0)
	Alternatively, the capital gains yield can be calculated by simply subtracting the dividend yield from the total expected return.
	Expected return
c. What your expected dividend yield and capital gains for the second year (from Year 1 to Year 2)? Why aren’t these the same as for the first year?
Use the estimated price for Year 2, P2, to find the expected gain.
(P2 – P1)

2nd assignment is:

I need you to write a brief summary of the important concepts learned from the file below:

Also use the file below for both assignments.

Chapter 5
Introduction
Will You Be Able to Retire?
Your reaction to that question is probably, “First things first! I’m worried about getting a job,
not about retiring!” However, understanding the retirement situation can help you land a job
because
(1)
this is an important issue today,
(2)
employers like to hire people who know what’s happening in the real world and,
(3)
professors often test on the time value of money with problems related to saving for future
purposes (including retirement).
A recent study by the Employee Benefit Research Institute suggests that many U.S. workers
are not doing enough to prepare for retirement. The survey found that 40% of workers had
less than $25,000 in savings and investments (not including the values of their homes and
defined benefit plans). This includes 19% who have less than $1,000 in savings. Equally
concerning, only 67% of those surveyed said they were confident that they would be able to
retire comfortably. Unfortunately, there is no easy solution. In order to reach their
retirement goals, many current workers will need to work longer, spend less and save more,
and hopefully earn higher returns on their current savings.
For an interesting website that looks at global savings rates, refer to gfmag.com/globaldata/economic-data/916lqg-household-saving-rates.
Historically, many Americans have relied on Social Security as an important source of their
retirement income. However, given current demographics, it is likely that this important
program will need to be restructured down the road in order to maintain its viability. The
average personal savings rate in the United States has risen in recent years, in December
2019, it was 7.6%. However, that is still below the savings level during the 1960s and
1970s—reaching a peak of 17.3% in May 1975. In addition, the ratio of U.S. workers to
retirees has steadily declined over the past half century. In 1955, there were 8.6 workers
supporting each retiree, but by 1975, that number had declined to 3.2 workers for every one
retiree. From 1975 through 2016, the ratio remained between 2.8 and 3.4 workers for every
retiree. Current projections show this ratio significantly declining in the years ahead—the
forecast is for 2.3 workers per retiree in 2035 and 2.1 workers per retiree in 2095. With so
few people paying into the Social Security system and so many drawing funds out, Social
Security is going to be in serious trouble. In fact, for the first time since its inception, in 2010
(and 7 years ahead of schedule), Social Security was in the red—paying out more in benefits
than it received in payroll tax revenues. Considering these facts, many people may have
trouble maintaining a reasonable standard of living after they retire, and many of today’s
college students will have to support their parents.
This is an important issue for millions of Americans, but many don’t know how to deal with
it. Most Americans have been ignoring what is most certainly going to be a huge personal
and social problem. However, if you study this chapter carefully, you can use the tools and
techniques presented here to avoid the trap that has caught, and is likely to catch, so many
people.
Putting Things in Perspective
Time value analysis has many applications, including planning for retirement, valuing stocks
and bonds, setting up loan payment schedules, and making corporate decisions regarding
investing in new plants and equipment. In fact, of all financial concepts, time value of money
is the single most important concept. Indeed, time value analysis is used throughout the
book; so it is vital that you understand this chapter before continuing.
Excellent retirement calculators are available at msn.com/enus/money/tools/retirementplanner, ssa.gov/retire, and choosetosave.org/calculators. These
calculators allow you to input hypothetical retirement savings information; the program then
shows if current retirement savings will be sufficient to meet retirement needs.
You need to understand basic time value concepts, but conceptual knowledge will do you
little good if you can’t do the required calculations. Therefore, this chapter is heavy on
calculations. Most students studying finance have a financial or scientific calculator; some
also own or have access to a computer. One of these tools is necessary to work many finance
problems in a reasonable length of time. However, when students begin reading this
chapter, many of them don’t know how to use the time value functions on their calculator or
computer. If you are in that situation, you will find yourself simultaneously studying concepts
and trying to learn how to use your calculator, and you will need more time to cover this
chapter than you might expect.
When you finish this chapter, you should be able to do the following:
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.
5-1. Time Lines
The first step in time value analysis is to set up a time line, which will help you visualize
what’s happening in a particular problem. As an illustration, consider the following diagram,
where PV represents $100 that is on hand today, and FV is the value that will be in the
account on a future date:
Details
The intervals from 0 to 1, 1 to 2, and 2 to 3 are time periods such as years or months. Time 0
is today, and it is the beginning of Period 1; Time 1 is one period from today, and it is both
the end of Period 1 and the beginning of Period 2; and so forth. Although the periods are
often years, periods can also be quarters or months or even days. Note that each tick mark
corresponds to both the end of one period and the beginning of the next one. Thus, if the
periods are years, the tick mark at Time 2 represents the end of Year 2 and the beginning of
Year 3.
Cash flows are shown directly below the tick marks, and the relevant interest rate is shown
just above the time line. Unknown cash flows, which you are trying to find, are indicated by
question marks. Here the interest rate is 5%; a single cash outflow, $100, is invested at Time
0; and the Time 3 value is an unknown inflow. In this example, cash flows occur only at
Times 0 and 3, with no flows at Times 1 or 2. Note that in our example, the interest rate is
constant for all three years. That condition is generally true, but if it were not, we would
show different interest rates for the different periods.
Time lines are essential when you are first learning time value concepts, but even experts
use them to analyze complex finance problems—and we use them throughout the book. We
begin each problem by setting up a time line to illustrate the situation, after which we
provide an equation that must be solved to find the answer. Then we explain how to use a
regular calculator, a financial calculator, and a spreadsheet to find the answer.
SelfTest
Do time lines deal only with years, or can other time periods be used?
Set up a time line to illustrate the following situation: You currently have $2,000 in a 3-year
certificate of deposit (CD) that pays a guaranteed 4% annually.
5-2. Future Values
A dollar in hand today is worth more than a dollar to be received in the future because if you
had it now, you could invest it, earn interest, and own more than a dollar in the future. The
process of going to future value (FV) from present value (PV) is called compounding. For an
illustration, refer back to our 3-year time line, and assume that you plan to deposit $100 in a
bank that pays a guaranteed 5% interest each year. How much would you have at the end of
Year 3? We first define some terms, and then we set up a time line to show how the future
value is calculated.
PV
=
Present value, or beginning amount. In our example,
PV
=
$
100
.
FV
N
=
Future value, or ending amount, of your account after N periods. Whereas PV is the value
now, or the present value,
FV
N
is the value N periods into the future, after the interest earned has been added to the
account.
CF
t
=
Cash flow. Cash flows can be positive or negative. The cash flow for a particular period is
often given as a subscript,
CF
t
, where t is the period. Thus,
CF
0
=
PV
=
the cash flow at Time
0
, whereas
CF
3
is the cash flow at the end of Period 3.
I
=
Interest rate earned per year. Sometimes a lowercase i is used. Interest earned is based on
the balance at the beginning of each year, and we assume that it is paid at the end of the
year. Here
I
=
5
%
or, expressed as a decimal, 0.05. Throughout this chapter, we designate the interest rate as I
because that symbol (or I/YR, for interest rate per year) is used on most financial calculators.
Note, though, that in later chapters, we use the symbol r to denote rates because r (for rate
of return) is used more often in the finance literature. Note too that in this chapter we
generally assume that interest payments are guaranteed by the U.S. government; hence,
they are certain. In later chapters, we consider risky investments, where the interest rate
earned might differ from its expected level.
INT
=
Dollars of interest earned during the year
=
Beginning amount
×
I
. In our example,
INT
=
$
100
(
0.05
)
=
$
5
.
N
=
Number of periods involved in the analysis. In our example,
N
=
3
. Sometimes the number of periods is designated with a lowercase n, so both N and n
indicate the number of periods involved.
We can use four different procedures to solve time value problems. These methods are
described in the following sections.
5-2A. Step-by-Step Approach
The time line used to find the FV of $100 compounded for 3 years at 5%, along with some
calculations, is shown. Multiply the initial amount and each succeeding amount by
(
1
+
I
)
=
(
1.05
)
:
Details
You start with $100 in the account—this is shown at
t
=
0
:
You earn
$
100
(
0.05
)
=
$
5
of interest during the first year, so the amount at the end of Year 1 (or
t
=
1
) is
$
100
+
$
5
=
$
105
.
You begin the second year with $105, earn
0.05
(
$
105
)
=
$
5.25
on the now larger beginning-of-period amount, and end the year with $110.25. Interest
during Year 2 is $5.25, and it is higher than the first year’s interest, $5.00, because you
earned
$
5
(
0.05
)
=
$
0.25
interest on the first year’s interest. This is called “compounding,” and interest earned on
interest is called “compound interest.”
This process continues, and because the beginning balance is higher each successive year,
the interest earned each year increases.
The total interest earned, $15.76, is reflected in the final balance, $115.76.
The step-by-step approach is useful because it shows exactly what is happening. However,
this approach is time-consuming, especially when a number of years are involved; so
streamlined procedures have been developed.
5-2B. Formula Approach
In the step-by-step approach, we multiply the amount at the beginning of each period by
(
1
+
I
)
=
(
1.05
)
. If
N
=
3
, we multiply by
(
1
+
I
)
three different times, which is the same as multiplying the beginning amount by
(
1
+
I
)
3
. This concept can be extended, and the result is this key equation:
5.1
FV
N
=
PV
(
1
+
I
)
N
We can apply Equation 5.1 to find the FV in our example:
FV
3
=
$
100
(
1.05
)
3
=
$
115.76
Equation 5.1 can be used with any calculator that has an exponential function, making it
easy to find FVs no matter how many years are involved.
5-2C. Financial Calculators
Financial calculators are extremely helpful in working time value problems. Their manuals
explain calculators in detail; on the student companion site, we provide summaries of the
features needed to work the problems in this book for several popular calculators. Also see
the box titled, “Hints on Using Financial Calculators”, for suggestions that will help you avoid
common mistakes. If you are not yet familiar with your calculator, we recommend that you
work through the tutorial as you study this chapter.
Simple Versus Compound Interest
Interest earned on the interest earned in prior periods, as was true in our example and is
always true when we apply Equation 5.1, is called compound interest. If interest is not
earned on interest, we have simple interest. The formula for FV with simple interest is
FV
=
PV
+
PV
(
I
)
(
N
)
; so in our example, FV would have been
$
100
+
$
100
(
0.05
)
(
3
)
=
$
100
+
$
15
=
$
115
based on simple interest. Most financial contracts are based on compound interest, but in
legal proceedings, the law often specifies that simple interest must be used. For example,
Maris Distributing, a company founded by home-run king Roger Maris, won a lawsuit against
Anheuser-Busch (A-B) because A-B had breached a contract and taken away Maris’s
franchise to sell Budweiser beer. The judge awarded Maris $50 million plus interest at 10%
from 1997 (when A-B breached the contract) until the payment was actually made. The
interest award was based on simple interest, which as of 2005 (when a settlement was
reached between A-B and the Maris family) had raised the total from $50 million to
$
50
million
+
0.10
(
$
50
million
)
(
8
years
)
=
$
90
million
. (No doubt the sheer size of this award and the impact of the interest, even simple interest,
influenced A-B to settle.) If the law had allowed compound interest, the award would have
totaled
(
$
50
million
)
×
(
1.10
)
8
=
$
107.18
million
, or $17.18 million more. This legal procedure dates back to the days before calculators and
computers. The law moves slowly!
First, note that financial calculators have five keys that correspond to the five variables in the
basic time value equations. We show the inputs for our text example above the respective
keys and the output, the FV, below its key. Because there are no periodic payments, we
enter 0 for PMT. We describe the keys in more detail after this calculation.
Details
Where:
N
=
Number of periods. Some calculators use n rather than N.
I/YR
=
Interest rate per period. Some calculators use i or I rather than I/YR.
PV
=
Present value. In our example, we begin by making a deposit, which is an outflow (the cash
leaves our wallet and is deposited at one of many financial institutions); so the PV should be
entered with a negative sign. On most calculators, you must enter the 100, then press the
+/− key to switch from +100 to −100. If you enter −100 directly, 100 will be subtracted from
the last number in the calculator, giving you an incorrect answer.
PMT
=
Payment. This key is used when we have a series of equal, or constant, payments. Because
there are no such payments in our illustrative problem, we enter
PMT
=
0
. We will use the PMT key when we discuss annuities later in this chapter.
FV
=
Future value. In this example, the FV is positive because we entered the PV as a negative
number. If we had entered the 100 as a positive number, the FV would have been negative.
Resource
Students can download the Excel chapter models from the student companion site on
cengage.com. Once downloaded onto your computer, retrieve the Chapter 5 Excel model
and follow along as you read this chapter.
As noted in our example, you enter the known values (N, I/YR, PV, and PMT) and then press
the FV key to get the answer, 115.76. Again, note that if you enter the PV as 100 without a
minus sign, the FV will be shown on the calculator display as a negative number. The
calculator assumes that either the PV or the FV is negative. This should not be confusing if
you think about what you are doing. When PMT is zero, it doesn’t matter what sign you
enter for PV as your calculator will automatically assign the opposite sign to FV. We will
discuss this point in greater detail later in the chapter when we cover annuities.
5-2D. Spreadsheets
Students generally use calculators for homework and exam problems, but in business,
people generally use spreadsheets for problems that involve the time value of money (TVM).
Spreadsheets show in detail what is happening, and they help reduce both conceptual and
data-entry errors. The spreadsheet discussion can be skipped without loss of continuity, but
if you understand the basics of Excel and have access to a computer, we recommend that
you read through this section. Even if you aren’t familiar with spreadsheets, the discussion
will still give you an idea of how they operate.
We used Excel to create Table 5.1, which is part of the spreadsheet model that corresponds
to this chapter. Table 5.1 summarizes the four methods of finding the FV and shows the
spreadsheet formulas toward the bottom. Note that spreadsheets can be used to do
calculations, but they can also be used like a word processor to create exhibits like Table 5.1,
which includes text, drawings, and calculations. The letters across the top designate
columns; the numbers to the left designate rows; and the rows and columns jointly
designate cells. Thus, C14 is the cell in which we specify the 2$100 investment; C15 shows
the interest rate; and C16 shows the number of periods. We then created a time line on
rows 17 to 19, and on row 21, we have Excel go through the step-by-step calculations,
multiplying the beginning-of-year values by
(
1
+
I
)
to find the compounded value at the end of each period. Cell G21 shows the final result.
Then on row 23, we illustrate the formula approach, using Excel to solve Equation 5.1 and
find the FV, $115.76. Next, on rows 25 to 27, we show a picture of the calculator solution.
Finally, on rows 30 and 31, we use Excel’s built-in FV function to find the answers given in
cells G30 and G31. The G30 answer is based on fixed inputs, while the G31 answer is based
on cell references, which makes it easy to change inputs and see the effects on the output.
Table 5.1 Summary of Future Value Calculations
Details
For example, if you want to quickly see how the future value changes if the interest rate is
7% instead of 5%, all you need to do is change cell C15 to 7%. Looking at cell G31, you will
immediately see that the future value is now $122.50.
Hints on Using Financial Calculators
When using a financial calculator, make sure it is set up as indicated here. Refer to your
calculator manual or to our calculator tutorial on the student companion site for information
on setting up your calculator.
One payment per period. Many calculators “come out of the box,” assuming that 12
payments are made per year; that is, monthly payments. However, in this book, we generally
deal with problems in which only one payment is made each year. Therefore, you should set
your calculator at one payment per year and leave it there. See our tutorial or your
calculator manual if you need assistance.
End mode. With most contracts, payments are made at the end of each period. However,
some contracts call for payments at the beginning of each period. You can switch between
“End Mode” and “Begin Mode,” depending on the problem you are solving. Because most of
the problems in this book call for end-of-period payments, you should return your calculator
to End Mode after you work a problem where payments are made at the beginning of
periods.
Negative sign for outflows. Outflows must be entered as negative numbers. This generally
means typing the outflow as a positive number and then pressing the +/− key to convert
from + to − before hitting the enter key.
Decimal places. With most calculators, you can specify from 0 to 11 decimal places. When
working with dollars, we generally specify two decimal places. When dealing with interest
rates, we generally specify two places after the decimal when the rate is expressed as a
percentage (e.g., 5.25%), but we specify four decimal places when the rate is expressed as a
decimal (e.g., 0.0525).
Interest rates. For arithmetic operations with a nonfinancial calculator, 0.0525 must be used,
but with a financial calculator and its TVM keys, you must enter 5.25, not 0.0525, because
financial calculators assume that rates are stated as percentages.
If you are using Excel, keep a few things in mind:
When calculating time value of money problems in Excel, interest rates are entered as
percentages or decimals (e.g., 5% or .05). However, when using the time value of money
function on most financial calculators you generally enter the interest rate as a whole
number (e.g., 5).
When calculating time value of money problems in Excel, the abbreviation for the number of
periods is nper, whereas for most financial calculators the abbreviation is simply N.
Throughout the text, we will use these terms interchangeably.
When calculating time value of money problems in Excel, you will often be prompted to
enter Type. Type refers to whether the payments come at the end of the year (in which case
Type
=
0
, or you can just omit it) or at the beginning of the year (in which case
Type
=
1
). Most financial calculators have a BEGIN/END mode function that you toggle on or off to
indicate whether the payments come at the beginning or at the end of the period.
Table 5.1 demonstrates that all four methods get the same result, but they use different
calculating procedures. It also shows that with Excel, all inputs are shown in one place,
which makes checking data entries relatively easy. Finally, it shows that Excel can be used to
create exhibits, which are quite important in the real world. In business, it’s often as
important to explain what you are doing as it is to “get the right answer” because if decision
makers don’t understand your analysis, they may reject your recommendations.
5-2E. Graphic View of the Compounding Process
Figure 5.1 shows how a $1 investment grows over time at different interest rates. We made
the curves by solving Equation 5.1 with different values for N and I. The interest rate is a
growth rate: If a sum is deposited and earns 5% interest per year, the funds on deposit will
grow by 5% per year. Note also that time value concepts can be applied to anything that
grows—sales, population, earnings per share, or future salary.
Figure 5.1 Growth of $1 at Various Interest Rates and Time Periods
Details
Quick Question
Question
At the beginning of your freshman year, your favorite aunt and uncle deposit $10,000 into a
4-year bank certificate of deposit (CD) that pays 5% annual interest. You will receive the
money in the account (including the accumulated interest) if you graduate with honors in 4
years. How much will there be in the account after 4 years?
Answer
Using the formula approach, we know that
FV
N
=
PV
(
1
+
I
)
N
. In this case, you know that
N
=
4
,
PV
=
$
10
,
000
, and
I
=
0.05
. It follows that the future value after 4 years will be
FV
4
=
$
10
,
000
(
1.05
)
4
=
$
12,155.06
. Alternatively, using the calculator approach we can set the problem up as follows:
Details
Finally, we can use Excel’s FV function:
Details
Here we find that the future value equals $12,155.06.
SelfTest
Explain why this statement is true: A dollar in hand today is worth more than a dollar to be
received next year.
What is compounding? What’s the difference between simple interest and compound
interest? What would the future value of $100 be after 5 years at 10% compound interest?
At 10% simple interest? ($161.05, $150.00)
Suppose you currently have $2,000 and plan to purchase a 3-year certificate of deposit (CD)
that pays 4% interest compounded annually. How much will you have when the CD matures?
How would your answer change if the interest rate were 5% or 6% or 20%? ($2,249.73,
$2,315.25, $2,382.03, $3,456.00. Hint: With a calculator, enter
�
=
3
,
�
/
YR
=
4
,
PV
=
2000
and
PMT
=
0
; then press FV to get 2,249.73. Enter
I
/
YR
=
5
to override the 4%, and press FV again to get the second answer. In general, you can change
one input at a time to see how the output changes.)
A company’s sales in 2021 were $100 million. If sales grow at 8%, what will they be 10 years
later, in 2031? ($215.89 million)
How much would $1 growing at 5% per year be worth after 100 years? What would the FV
be if the growth rate were 10%? ($131.50, $13,780.61)
5-3. Present Values
Finding a present value is the reverse of finding a future value. Indeed, we simply solve
Equation 5.1, the formula for the future value, for the PV to produce the basic present value
formula, Equation 5.2:
5.1
Future value
=
FV
N
=
PV
(
1
+
I
)
N
5.2
Present value
=
PV
=
FV
N
(
1
+
I
)
N
We illustrate PVs with the following example. A broker offers to sell you a Treasury bond that
will pay $115.76 three years from now. Banks are currently offering a guaranteed 5% interest
on 3-year certificates of deposit (CDs), and if you don’t buy the bond, you will buy a CD. The
5% rate paid on the CDs is defined as your opportunity cost, or the rate of return you could
earn on an alternative investment of similar risk. Given these conditions, what’s the most
you should pay for the bond? We answer this question using the four methods discussed in
the last section—step-by-step, formula, calculator, and spreadsheet. Table 5.2 summarizes
the results.
Table 5.2 Summary of Present Value Calculations
Details
First, recall from the future value example in the last section that if you invested $100 at 5%,
it would grow to $115.76 in 3 years. You would also have $115.76 after 3 years if you bought
the T-bond. Therefore, the most you should pay for the bond is $100—this is its “fair price.”
If you could buy the bond for less than $100, you should buy it rather than invest in the CD.
Conversely, if its price was more than $100, you should buy the CD. If the bond’s price was
exactly $100, you should be indifferent between the T-bond and the CD.
The $100 is defined as the present value, or PV, of $115.76 due in 3 years when the
appropriate interest rate is 5%. In general, the present value of a cash flow due N years in
the future is the amount that, if it were on hand today, would grow to equal the given future
amount. Because $100 would grow to $115.76 in 3 years at a 5% interest rate, $100 is the
present value of $115.76 due in 3 years at a 5% rate. Finding present values is called
discounting, and as previously noted, it is the reverse of compounding—if you know the PV,
you can compound to find the FV, while if you know the FV, you can discount to find the PV.
The top section of Table 5.2 calculates the PV using the step-by-step approach. When we
found the future value in the previous section, we worked from left to right, multiplying the
initial amount and each subsequent amount by
(
1
+
I
)
. To find present values, we work backward, or from right to left, dividing the future value
and each subsequent amount by
(
1
+
I
)
. This procedure shows exactly what’s happening, which can be quite useful when you are
working complex problems. However, it’s inefficient, especially when you are dealing with a
large number of years.
With the formula approach, we use Equation 5.2, simply dividing the future value by
(
1
+
I
)
N
. This is more efficient than the step-by-step approach, and it gives the same result.
Equation 5.2 is built into financial calculators, and as shown in Table 5.2, we can find the PV
by entering values for N, I/YR, PMT, and FV and then pressing the PV key. Finally, Excel’s PV
function can be used:
Details
It is essentially the same as the calculator and solves Equation 5.2.
The fundamental goal of financial management is to maximize the firm’s value, and the
value of a business (or any asset, including stocks and bonds) is the present value of its
expected future cash flows. Because present value lies at the heart of the valuation process,
we will have much more to say about it in the remainder of this chapter and throughout the
book.
5-3A. Graphic View of the Discounting Process
Figure 5.2 shows that the present value of a sum to be received in the future decreases and
approaches zero as the payment date is extended further into the future and that the
present value falls faster at higher interest rates. At relatively high rates, funds due in the
future are worth very little today, and even at relatively low rates, present values of sums
due in the very distant future are quite small. For example, at a 20% discount rate, $1 million
due in 100 years would be worth only $0.0121 today. This is because $0.0121 would grow to
$1 million in 100 years when compounded at 20%.
Figure 5.2 Present Value of $1 at Various Interest Rates and Time Periods
Details
SelfTest
What is discounting, and how is it related to compounding? How is the future value equation
(Equation 5.1) related to the present value equation (Equation 5.2)?
How does the present value of a future payment change as the time to receipt is
lengthened? As the interest rate increases?
Suppose a U.S. government bond promises to pay $2,249.73 three years from now. If the
going interest rate on 3-year government bonds is 4%, how much is the bond worth today?
How much is it worth today if the bond matured in 5 years rather than 3? How much is it
worth today if the interest rate on the 5-year bond was 6% rather than 4%? ($2,000,
$1,849.11, $1,681.13)
How much would $1,000,000 due in 100 years be worth today if the discount rate was 5%? If
the discount rate was 20%? ($7,604.49, $0.0121)
5-4. Finding the Interest Rate, I
Thus far we have used Equations 5.1 and 5.2 to find future and present values. Those
equations have four variables, and if we know three of the variables, we can solve for the
fourth. Thus, if we know PV, I, and N, we can solve Equation 5.1 for FV, while if we know FV,
I, and N, we can solve Equation 5.2 to find PV. That’s what we did in the preceding two
sections.
Now suppose we know PV, FV, and N and want to find I. For example, suppose we know that
a given bond has a cost of $100 and that it will return $150 after 10 years. Thus, we know PV,
FV, and N, and we want to find the rate of return we will earn if we buy the bond. Here’s the
situation:
FV
=
PV
(
1
+
I
)
N
$
150
=
$
100
(
1
+
I
)
10
$
150
/
$
100
=
(
1
+
I
)
10
1.5
=
(
1
+
I
)
10
Unfortunately, we can’t factor I out to produce as simple a formula as we could for FV and
PV. We can solve for I, but it requires a bit more algebra. However, financial calculators and
spreadsheets can find interest rates almost instantly. Here’s the calculator setup:
Details
Enter
N
=
10
,
PV
=
-$100
,
PMT
=
0
, because there are no payments until the security matures, and
FV
=
150
. Then when you press the I/YR key, the calculator gives the answer, 4.14%. You would get
this same answer using the RATE function in Excel:
Details
Here we find that the interest rate is equal to 4.14%.
SelfTest
The U.S. Treasury offers to sell you a bond for $585.43. No payments will be made until the
bond matures 10 years from now, at which time it will be redeemed for $1,000. What
interest rate would you earn if you bought this bond for $585.43? What rate would you earn
if you could buy the bond for $550? For $600? (5.5%, 6.16%, 5.24%)
Microsoft earned $1.42 per share in 2007. Ten years later in 2017 it earned $3.08. What was
the growth rate in Microsoft’s earnings per share (EPS) over the 10-year period? If EPS in
2017 had been $2.40 rather than $3.08, what would the growth rate have been? (8.05%,
5.39%)
5-5. Finding the Number of Years, N
We sometimes need to know how long it will take to accumulate a certain sum of money,
given our beginning funds and the rate we will earn on those funds. For example, suppose
we believe that we could retire comfortably if we had $1 million. We want to find how long it
will take us to acquire $1 million, assuming we now have $500,000 invested at 4.5%. We
cannot use a simple formula—the situation is like that with interest rates. We can set up a
formula that uses logarithms, but calculators and spreadsheets find N very quickly. Here’s
the calculator setup:
Details
Enter
I
/
YR
=
4.5
,
PV
=
500000
,
PMT
=
0
, and
FV
=
1000000
. Then when you press the N key, you get the answer, 15.7473 years. If you plug
N
=
15.7473
into the FV formula, you can prove that this is indeed the correct number of years:
FV
=
PV
(
1
+
I
)
N
=
$
500
,
000
(
1.045
)
15.7473
=
$
1
,
000
,
000
You can also use Excel’s NPER function:
Details
Here we find that it will take 15.7473 years for $500,000 to double at a 4.5% interest rate.
SelfTest
How long would it take $1,000 to double if it was invested in a bank that paid 6% per year?
How long would it take if the rate was 10%? (11.9 years, 7.27 years)
Microsoft’s 2019 earnings per share were $5.06, and its growth rate during the prior 10
years was 12.06% per year. If that growth rate was maintained, how long would it take for
Microsoft’s EPS to double? (6.09 years)
5-6. Annuities
Thus far we have dealt with single payments, or “lump sums.” However, many assets provide
a series of cash inflows over time, and many obligations, such as auto, student, and
mortgage loans, require a series of payments. When the payments are equal and are made
at fixed intervals, the series is an annuity. For example, $100 paid at the end of each of the
next 3 years is a 3-year annuity. If the payments occur at the end of each year, the annuity is
an ordinary (or deferred) annuity. If the payments are made at the beginning of each year,
the annuity is an annuity due. Ordinary annuities are more common in finance; so when we
use the term annuity in this book, assume that the payments occur at the ends of the
periods unless otherwise noted.
Here are the time lines for a $100, 3-year, 5% ordinary annuity and for an annuity due. With
the annuity due, each payment is shifted to the left by one year. A $100 deposit will be made
each year, so we show the payments with minus signs:
Details
As we demonstrate in the following sections, we can find an annuity’s future and present
values, the interest rate built into annuity contracts, and the length of time it takes to reach
a financial goal using an annuity. Keep in mind that annuities must have constant payments
at fixed intervals for a specified number of periods. If these conditions don’t hold, then the
payments do not constitute an annuity.
SelfTest
What’s the difference between an ordinary annuity and an annuity due?
Why would you prefer to receive an annuity due for $10,000 per year for 10 years than an
otherwise similar ordinary annuity?
5-7. Future Value of an Ordinary Annuity
The future value of an annuity can be found using the step-by-step approach or using a
formula, a financial calculator, or a spreadsheet. As an illustration, consider the ordinary
annuity diagrammed earlier, where you deposit $100 at the end of each year for 3 years and
earn 5% per year. How much will you have at the end of the third year? The answer,
$315.25, is defined as the future value of the annuity,
FVA
N
; it is shown in Table 5.3.
Table 5.3 Summary: Future Value of an Ordinary Annuity
Details
As shown in the step-by-step section of the table, we compound each payment out to Time
3, then sum those compounded values to find the annuity’s FV,
FVA
3
=
$
315.25
. The first payment earns interest for two periods, the second payment earns interest for
one period, and the third payment earns no interest at all because it is made at the end of
the annuity’s life. This approach is straightforward, but if the annuity extends out for many
years, the approach is cumbersome and time-consuming.
As you can see from the time line diagram, with the step-by-step approach, we apply the
following equation, with
N
=
3
and
I
=
5
%
:
FVA
N
=
PMT
(
1
+
I
)
N
1
+
PMT
(
1
+
I
)
N
2
+
PMT
(
1
+
I
)
N
3
=
$
100
(
1.05
)
2
+
$
100
(
1.05
)
1
+
$
100
(
1.05
)
0
=
$
315.25
We can generalize and streamline the equation as follows:
5.3
FVA
N
=
PMT
(
1
+
I
)
N
1
+
PMT
(
1
+
I
)
N
2
+
PMT
(
1
+
I
)
N
3
+
⋯
+
PMT
(
1
+
I
)
0
=
PMT
[
(
1
+
I
)
N
1
I
]
The first line shows the equation in its long form. It can be transformed to the second form
on the last line, which can be used to solve annuity problems with a nonfinancial calculator.
This equation is also built into financial calculators and spreadsheets. With an annuity, we
have recurring payments; hence, the PMT key is used. Here’s the calculator setup for our
illustrative annuity:
Details
We enter
PV
=
0
because we start off with nothing, and we enter
PMT
=
100
because we plan to deposit this amount in the account at the end of each year. When we
press the FV key, we get the answer,
FVA
3
=
315.25
.
Because this is an ordinary annuity, with payments coming at the end of each year, we must
set the calculator appropriately. As noted earlier, calculators “come out of the box” set to
assume that payments occur at the end of each period, that is, to deal with ordinary
annuities. However, there is a key that enables us to switch between ordinary annuities and
annuities due. For ordinary annuities the designation is “End Mode” or something similar,
while for annuities due the designation is “Begin” or “Begin Mode” or “Due” or something
similar. If you make a mistake and set your calculator on Begin Mode when working with an
ordinary annuity, each payment will earn interest for one extra year. That will cause the
compounded amounts, and thus the FVA, to be too large.
The last approach in Table 5.3 shows the spreadsheet solution using Excel’s built-in function.
We can put in fixed values for N, I, PV, and PMT or set up an Input Section where we assign
values to those variables, and then input values into the function as cell references. Using
cell references makes it easy to change the inputs to see the effects of changes on the
output.
Quick Question
Question
Your grandfather urged you to begin a habit of saving money early in your life. He suggested
that you put $5 a day into an envelope. If you follow his advice, at the end of the year you
will have $1,825
(
365
×
$
5
)
. Your grandfather further suggested that you take that money at the end of the year and
invest it in an online brokerage mutual fund account that has an annual expected return of
8%.
You are 18 years old. If you start following your grandfather’s advice today, and continue
saving in this way the rest of your life, how much do you expect to have in the brokerage
account when you are 65 years old?
Answer
This problem is asking you to calculate the future value of an ordinary annuity. More
specifically, you are making 47 payments of $1,825, where the annual interest rate is 8%.
To quickly find the answer, enter the following inputs into a financial calculator:
N
=
47
;
I
/
YR
=
8
;
PV
=
0
; and
PMT
=
1825
. Then solve for the FV of the ordinary annuity by pressing the FV key,
FV
=
$
826,542.78
.
In addition, we can use Excel’s FV function:
Details
Here we find that the future value is $826,542.78.
You can see your grandfather is right—it definitely pays to start saving early!
SelfTest
For an ordinary annuity with five annual payments of $100 and a 10% interest rate, how
many years will the first payment earn interest? What will this payment’s value be at the
end? Answer this same question for the fifth payment. (4 years, $146.41, 0 years, $100)
Assume that you plan to buy a condo 5 years from now, and you estimate that you can save
$2,500 per year. You plan to deposit the money in a bank account that pays 4% interest, and
you will make the first deposit at the end of the year. How much will you have after 5 years?
How much will you have if the interest rate is increased to 6% or lowered to 3%?
($13,540.81, $14,092.73, $13,272.84)
5-8. Future Value of an Annuity Due
Because each payment occurs one period earlier with an annuity due, all of the payments
earn interest for one additional period. Therefore, the FV of an annuity due will be greater
than that of a similar ordinary annuity. If you went through the step-by-step procedure, you
would see that our illustrative annuity due has an FV of $331.01 versus $315.25 for the
ordinary annuity.
With the formula approach, we first use Equation 5.3; however, because each payment
occurs one period earlier, we multiply the Equation 5.3 result by
(
1
+
I
)
:
5.4
FVA
due
=
FVA
ordinary
(
1
+
I
)
Thus, for the annuity due,
FVA
due
=
$
315.25
(
1.05
)
=
$
331.01
, which is the same result when the period-by-period approach is used. With a calculator,
we input the variables just as we did with the ordinary annuity, but now we set the
calculator to Begin Mode to get the answer, $331.01.
SelfTest
Why does an annuity due always have a higher future value than an ordinary annuity?
If you calculated the value of an ordinary annuity, how could you find the value of the
corresponding annuity due?
Assume that you plan to buy a condo 5 years from now, and you need to save for a down
payment. You plan to save $2,500 per year (with the first deposit made immediately), and
you will deposit the funds in a bank account that pays 4% interest. How much will you have
after 5 years? How much will you have if you make the deposits at the end of each year?
($14,082.44, $13,540.81)
5-9. Present Value of an Ordinary Annuity
The present value of an annuity,
PVA
N
, can be found using the step-by-step, formula, calculator, or spreadsheet method. Look back
at Table 5.3. To find the FV of the annuity, we compounded the deposits. To find the PV, we
discount them, dividing each payment by
(
1
+
I
)
t
. The step-by-step procedure is diagrammed as follows:
Details
Equation 5.5 expresses the step-by-step procedure in a formula. The bracketed form of the
equation can be used with a scientific calculator, and it is helpful if the annuity extends out
for a number of years:
5.5
PVA
N
=
PMT
/
(
1
+
I
)
1
+
PMT
/
(
1
+
I
)
2
+
⋯
+
PMT
/
(
1
+
I
)
N
=
PMT
[
1
1
(
1
+
I
)
N
I
]
=
$
100
×
[
1
1
/
(
1.05
)
3
]
/
0.05
=
$
272.32
Calculators are programmed to solve Equation 5.5, so we merely input the variables and
press the PV key, making sure the calculator is set to End Mode. The calculator setup follows
for both an ordinary annuity and an annuity due. Note that the PV of the annuity due is
larger because each payment is discounted back one less year. Note too that you can find
the PV of the ordinary annuity and then multiply by
(
1
+
I
)
=
1.05
, calculating
$
272.32
(
1.05
)
=
$
285.94
, the PV of the annuity due.
Details
Quick Question
Question
You just won the Florida lottery. To receive your winnings, you must select ONE of the two
following choices:
You can receive $1,000,000 a year at the end of each of the next 30 years.
You can receive a one-time payment of $15,000,000 today.
Assume that the current interest rate is 6%. Which option is most valuable?
Answer
The most valuable option is the one with the largest present value. You know that the
second option has a present value of $15,000,000, so we need to determine whether the
present value of the $1,000,000, 30-year ordinary annuity exceeds $15,000,000.
Using the formula approach, we see that the present value of the annuity is
PVA
N
=
PMT
[
1
1
(
1
+
I
)
N
I
]
=
$
1
,
000
,
000
[
1
1
(
1.06
)
30
0.06
]
=
$
13,764,831.15
Alternatively, using the calculator approach, we can set up the problem as follows:
Details
Finally, we can use Excel’s PV function:
Details
Here we find that the present value is $13,764,831.15.
Because the present value of the 30-year annuity is less than $15,000,000, you should
choose to receive your winnings as a onetime up-front payment.
SelfTest
Why does an annuity due have a higher present value than a similar ordinary annuity?
If you know the present value of an ordinary annuity, how can you find the PV of the
corresponding annuity due?
What is the PVA of an ordinary annuity with 10 payments of $100 if the appropriate interest
rate is 10%? What would the PVA be if the interest rate was 4%? What if the interest rate
was 0%? How much would the PVA values be if we were dealing with annuities due?
($614.46, $811.09, $1,000.00, $675.90, $843.53, $1,000.00)
Assume that you are offered an annuity that pays $100 at the end of each year for 10 years.
You could earn 8% on your money in other investments with equal risk. What is the most
you should pay for the annuity? If the payments began immediately, how much would the
annuity be worth? ($671.01, $724.69)
5-10. Finding Annuity Payments, Periods, and Interest Rates
We can find payments, periods, and interest rates for annuities. Here five variables come
into play: N, I, PMT, FV, and PV. If we know any four, we can find the fifth.
5-10A. Finding Annuity Payments, PMT
Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose
further that we can earn a return of 6% on our savings, which are currently zero. Thus, we
know that
FV
=
10
,
000
,
PV
=
0
,
N
=
5
, and
I
/
YR
=
6
. We can enter these values in a financial calculator and press the PMT key to find how large
our deposits must be. The answer will, of course, depend on whether we make deposits at
the end of each year (ordinary annuity) or at the beginning (annuity due). Here are the
results for each type of annuity:
Ordinary Annuity:
Details
We can also use Excel’s PMT function:
Details
Because the deposits are made at the end of the year, we can leave “type” blank. Here we
find that an annual deposit of $1,773.96 is needed to reach your goal.
Annuity Due:
Details
Alternatively, Excel’s PMT function can be used to calculate the annual deposit for the
annuity due:
Details
Because the deposits are now made at the beginning of the year, enter 1 for type. Here we
find that an annual deposit of $1,673.55 is needed to reach your goal.
Thus, you must save $1,773.96 per year if you make deposits at the end of each year, but
only $1,673.55 if the deposits begin immediately. Note that the required annual deposit for
the annuity due can also be calculated as the ordinary annuity payment divided by
(
1
+
I
)
:
$
1
,
773.96
/
1.06
=
$
1
,
673.55
.
5-10B. Finding the Number of Periods, N
Suppose you decide to make end-of-year deposits, but you can save only $1,200 per year.
Again assuming that you would earn 6%, how long would it take to reach your $10,000 goal?
Here is the calculator setup:
Details
With these smaller deposits, it would take 6.96 years to reach your $10,000 goal. If you
began the deposits immediately, you would have an annuity due, and N would be a bit
smaller, 6.63 years.
You can also use Excel’s NPER function to arrive at both of these answers. If we assume endof-year payments, Excel’s NPER function looks like this:
Details
Here we find that it will take 6.96 years to reach your goal.
If we assume beginning-of-year payments, Excel’s NPER function looks like this:
Details
Here we find that it will take only 6.63 years to reach your goal.
5-10C. Finding the Interest Rate, I
Now suppose you can save only $1,200 annually (assuming end-of-year deposits), but you
still need the $10,000 in 5 years. What rate of return would enable you to achieve your goal?
Here is the calculator setup:
Details
Excel’s RATE function will arrive at the same answer:
Details
Here we find that the interest rate is 25.78%.
You must earn a whopping 25.78% to reach your goal. About the only way to earn such a
high return would be to invest in speculative stocks or head to the casinos in Las Vegas. Of
course, investing in speculative stocks and gambling aren’t like making deposits in a bank
with a guaranteed rate of return, so there’s a good chance you’d end up with nothing. You
might consider changing your plans—save more, lower your $10,000 target, or extend your
time horizon. It might be appropriate to seek a somewhat higher return, but trying to earn
25.78% in a 6% market would require taking on more risk than would be prudent.
It’s easy to find rates of return using a financial calculator or a spreadsheet. However, to find
rates of return without one of these tools, you would have to go through a trial-and-error
process, which would be very time-consuming if many years were involved.
SelfTest
Suppose you inherited $100,000 and invested it at 7% per year. What is the most you could
withdraw at the end of each of the next 10 years and have a zero balance at Year 10? How
much could you withdraw if you made withdrawals at the beginning of each year?
($14,237.75, $13,306.31)
If you had $100,000 that was invested at 7% and you wanted to withdraw $10,000 at the
end of each year, how long would your funds last? How long would they last if you earned
0%? How long would they last if you earned the 7% but limited your withdrawals to $7,000
per year? (17.8 years, 10 years, forever)
Your uncle named you beneficiary of his life insurance policy. The insurance company gives
you a choice of $100,000 today or a 12-year annuity of $12,000 at the end of each year.
What rate of return is the insurance company offering? (6.11%)
Assume that you just inherited an annuity that will pay you $10,000 per year for 10 years,
with the first payment being made today. A friend of your mother offers to give you $60,000
for the annuity. If you sell it, what rate of return would your mother’s friend earn on his
investment? If you think a “fair” return would be 6%, how much should you ask for the
annuity? (13.70%, $78,016.92)
5-11. Perpetuities
A perpetuity is simply an annuity with an extended life. Because the payments go on forever,
you can’t apply the step-by-step approach. However, it’s easy to find the PV of a perpetuity
with a formula found by solving Equation 5.5 with N set at infinity:
5.6
PV of a perpetuity
=
PMT
I
Let’s say, for example, that you buy preferred stock in a company that pays you a fixed
dividend of $2.50 each year the company is in business. If we assume that the company will
go on indefinitely, the preferred stock can be valued as a perpetuity. If the discount rate on
the preferred stock is 10%, the present value of the perpetuity, the preferred stock, is $25:
PV of a perpetuity
=
$
2.50
0.10
=
$
25
SelfTest
What’s the present value of a perpetuity that pays $1,000 per year beginning 1 year from
now, if the appropriate interest rate is 5%? What would the value be if payments on the
annuity began immediately? ($20,000, $21,000. Hint: Just add the $1,000 to be received
immediately to the value of the annuity.)
5-12. Uneven Cash Flows
The definition of an annuity includes the words constant payment—in other words,
annuities involve payments that are equal in every period. Although many financial decisions
involve constant payments, many others involve uneven, or nonconstant, cash flows. For
example, the dividends on common stocks typically increase over time, and investments in
capital equipment almost always generate uneven cash flows. Throughout the book, we
reserve the term payment (PMT) for annuities with their equal payments in each period and
use the term cash flow
(
CF
t
)
to denote uneven cash flows, where t designates the period in which the cash flow occurs.
There are two important classes of uneven cash flows:
(1)
a stream that consists of a series of annuity payments plus an additional final lump sum and
(2)
all other uneven streams.
Bonds represent the best example of the first type, while stocks and capital investments
illustrate the second type. Here are numerical examples of the two types of flows:
Details
We can find the PV of either stream by using Equation 5.7 and following the step-by-step
procedure, where we discount each cash flow and then sum them to find the PV of the
stream:
5.7
PV
=
CF
1
(
1
+
I
)
1
+
CF
2
(
1
+
I
)
2
+
⋯
+
CF
N
(
1
+
I
)
N
=
∑
t
=
1
N
CF
t
(
1
+
I
)
t
If we did this, we would find the PV of Stream 1 to be $927.90 and the PV of Stream 2 to be
$1,016.35.
The step-by-step procedure is straightforward; however, if we have a large number of cash
flows, it is time-consuming. However, financial calculators speed up the process
considerably. First, consider Stream 1; notice that we have a 5-year, 12% ordinary annuity
plus a final payment of $1,000. We could find the PV of the annuity, and then find the PV of
the final payment and sum them to obtain the PV of the stream. Financial calculators do this
in one simple step—use the five TVM keys; enter the data as shown here and press the PV
key to obtain the answer, $927.90.
Details
The solution procedure is different for the second uneven stream. Here we must use the
step-by-step approach, as shown in Figure 5.3. Even calculators and spreadsheets solve the
problem using the step-by-step procedure, but they do it quickly and efficiently. First, you
enter all of the cash flows and the interest rate; then the calculator or computer discounts
each cash flow to find its present value and sums these PVs to produce the PV of the stream.
You must enter each cash flow in the calculator’s “cash flow register,” enter the interest rate,
and then press the NPV key to find the PV of the stream. NPV stands for “net present value.”
We cover the calculator mechanics in the calculator tutorial, and we discuss the process in
more detail in Chapters 9 and 11, where we use the NPV calculation to analyze stocks and
proposed capital budgeting projects. If you don’t know how to do the calculation with your
calculator, it would be worthwhile to review the tutorial or your calculator manual, learn the
steps, and make sure you can do this calculation. Because you will have to learn to do it
eventually, now is a good time to begin.
Figure 5.3 PV of an Uneven Cash Flow Stream
Details
SelfTest
How could you use Equation 5.2 to find the PV of an uneven stream of cash flows?
What’s the present value of a 5-year ordinary annuity of $100 plus an additional $500 at the
end of Year 5 if the interest rate is 6%? What is the PV if the $100 payments occur in Years 1
through 10 and the $500 comes at the end of Year 10? ($794.87, $1,015.21)
What’s the present value of the following uneven cash flow stream: $0 at Time 0, $100 in
Year 1 (or at Time 1), $200 in Year 2, $0 in Year 3, and $400 in Year 4 if the interest rate is
8%? ($558.07)
Would a typical common stock provide cash flows more like an annuity or more like an
uneven cash flow stream? Explain.
5-13. Future Value of an Uneven Cash Flow Stream
We find the future value of uneven cash flow streams by compounding rather than
discounting. Consider Cash Flow Stream 2 in the preceding section. We discounted those
cash flows to find the PV, but we would compound them to find the FV. Figure 5.4 illustrates
the procedure for finding the FV of the stream, using the step-by-step approach.
Figure 5.4 FV of an Uneven Cash Flow Stream
Details
The values of all financial assets—stocks, bonds, and business capital investments—are
found as the present values of their expected future cash flows. Therefore, we need to
calculate present values very often, far more often than future values. As a result, all
financial calculators provide automated functions for finding PVs, but they generally do not
provide automated FV functions. On the relatively few occasions when we need to find the
FV of an uneven cash flow stream, we generally use the step-by-step procedure shown in
Figure 5.4. That approach works for all cash flow streams, even those for which some cash
flows are zero or negative.
SelfTest
Why are we more likely to need to calculate the PV of cash flow streams than the FV of
streams?
What is the future value of this cash flow stream: $100 at the end of 1 year, $150 due after 2
years, and $300 due after 3 years, if the appropriate interest rate is 15%? ($604.75)
5-14. Solving for I with Uneven Cash Flows
Before financial calculators and spreadsheets existed, it was extremely difficult to find I
when the cash flows were uneven. With spreadsheets and financial calculators, however, it’s
relatively easy to find I. If you have an annuity plus a final lump sum, you can input values for
N, PV, PMT, and FV into the calculator’s TVM registers and then press the I/YR key. Here is
the setup for Stream 1 from Section 5-12, assuming we must pay $927.90 to buy the asset.
The rate of return on the $927.90 investment is 12%.
Details
Finding the interest rate for an uneven cash flow stream such as Stream 2 is a bit more
complicated. First, note that there is no simple procedure—finding the rate requires a trialand-error process, which means that a financial calculator or a spreadsheet is needed. With
a calculator, we enter each CF into the cash flow register and then press the IRR key to get
the answer. IRR stands for “internal rate of return,” and it is the rate of return the investment
provides. The investment is the cash flow at Time 0, and it must be entered as a negative
value. As an illustration, consider the cash flows given here, where
CF
0
=
$
1
,
000
is the cost of the asset.
Details
When we enter those cash flows into the calculator’s cash flow register and press the IRR
key, we get the rate of return on the $1,000 investment, 12.55%. You get the same answer
using Excel’s IRR function. This process is covered in the calculator tutorial; it is also
discussed in Chapter 11, where we study capital budgeting.
SelfTest
An investment costs $465 and is expected to produce cash flows of $100 at the end of each
of the next 4 years, then an extra lump sum payment of $200 at the end of the fourth year.
What is the expected rate of return on this investment? (9.05%)
An investment costs $465 and is expected to produce cash flows of $100 at the end of Year
1, $200 at the end of Year 2, and $300 at the end of Year 3. What is the expected rate of
return on this investment? (11.71%)
5-15. Semiannual and Other Compounding Periods
In all of our examples thus far, we assumed that interest was compounded once a year, or
annually. This is called annual compounding. Suppose, however, that you deposit $100 in a
bank that pays a 5% annual interest rate but credits interest each 6 months. So in the second
6-month period, you earn interest on your original $100 plus interest on the interest earned
during the first 6 months. This is called semiannual compounding. Note that banks generally
pay interest more than once a year; virtually all bonds pay interest semiannually; and most
mortgages, student loans, and auto loans require monthly payments. Therefore, it is
important to understand how to deal with nonannual compounding.
For an illustration of semiannual compounding, assume that we deposit $100 in an account
that pays 5% and leave it there for 10 years. First, consider again what the future value
would be under annual compounding:
FV
N
=
PV
(
1
+
I
)
N
=
$
100
(
1.05
)
10
=
$
162.89
We would, of course, get the same answer using a financial calculator or a spreadsheet.
How would things change in this example if interest was paid semiannually rather than
annually? First, whenever payments occur more than once a year, you must make two
conversions:
(1)
Convert the stated interest rate into a “periodic rate.”
(2)
Convert the number of years into “number of periods.”
The conversions are done as follows, where I is the stated annual rate, M is the number of
compounding periods per year, and N is the number of years:
5.8
Periodic rate
(
I
PER
)
=
Stated annual rate
Number of payments per year
=
1
/
M
With a stated annual rate of 5%, compounded semiannually, the periodic rate is 2.5%:
Periodic rate
=
5
%
/
2
=
2.5
%
The number of compounding periods is found with Equation 5.9:
5.9
Number of periods
=
(
Number of years
)
(
Periods per year
)
=
NM
With 10 years and semiannual compounding, there are 20 periods:
Number of periods
=
10
(
2
)
=
20
periods
Under semiannual compounding, our $100 investment will earn 2.5% every 6 months for 20
semiannual periods, not 5% per year for 10 years. The periodic rate and number of periods,
not the annual rate and number of years, must be shown on time lines and entered into the
calculator or spreadsheet whenever you are working with nonannual compounding.
With this background, we can find the value of $100 after 10 years if it is held in an account
that pays a stated annual rate of 5.0%, but with semiannual compounding. Here’s the time
line and the future value:
Details
With a financial calculator, we get the same result using the periodic rate and number of
periods:
Details
The future value under semiannual compounding, $163.86, exceeds the FV under annual
compounding, $162.89, because interest starts accruing sooner; thus, you earn more
interest on interest.
How would things change in our example if interest was compounded quarterly or monthly
or daily? With quarterly compounding, there would be
NM
=
10
(
4
)
=
40
periods and the periodic rate would be
I
/
M
=
5
%
/
4
=
1.25
%
per quarter. Using those values, we would find
FV
=
$
164.36
. If we used monthly compounding, we would have
10
(
12
)
=
120
periods, the monthly rate would be
5
%
/
12
=
0.416667
%
, and the FV would rise to $164.70. If we went to daily compounding, we would have
10
(
365
)
=
3
,
650
periods, the daily rate would be
5
%
/
365
=
0.0136986
%
per day, and the FV would be $164.87 (based on a 365-day year).
The same logic applies when we find present values under semiannual compounding. Again,
we use Equation 5.8 to convert the stated annual rate to the periodic (semiannual) rate and
Equation 5.9 to find the number of semiannual periods. We then use the periodic rate and
number of periods in the calculations. For example, we can find the PV of $100 due after 10
years when the stated annual rate is 5%, with semiannual compounding:
Periodic rate
=
5
%
/
2
=
2.5
%
per period
Number of periods
=
10
(
2
)
=
20
periods
PV of
$
100
=
$
100
/
(
1.025
)
20
=
$
61.03
We would get this same result with a financial calculator:
Details
If we increased the number of compounding periods from 2 (semiannual) to 12 (monthly),
the PV would decline to $60.72; if we went to daily compounding, the PV would fall to
$60.66.
SelfTest
Would you rather invest in an account that pays 7% with annual compounding or 7% with
monthly compounding? Would you rather borrow at 7% and make annual or monthly
payments? Why?
What’s the future value of $100 after 3 years if the appropriate interest rate is 8%
compounded annually? Compounded monthly? ($125.97, $127.02)
What’s the present value of $100 due in 3 years if the appropriate interest rate is 8%
compounded annually? Compounded monthly? ($79.38, $78.73)
5-16. Comparing Interest Rates
Different compounding periods are used for different types of investments. For example,
bank accounts generally pay interest daily; most bonds pay interest semiannually; stocks pay
dividends quarterly; and mortgages, auto loans, and other instruments require monthly
payments. If we are to compare investments or loans with different compounding periods
properly, we need to put them on a common basis. Here are some terms you need to
understand:
The nominal interest rate
(
I
NOM
)
, also called the annual percentage rate (APR), is the quoted, or stated, rate that credit card
companies, student loan officers, auto dealers, and other lenders tell you they are charging
on loans. Note that if two banks offer loans with a stated rate of 8%, but one requires
monthly payments and the other quarterly payments, they are not charging the same “true”
rate. The one that requires monthly payments is charging more than the one with quarterly
payments because it will receive your money sooner. So to compare loans across lenders, or
interest rates earned on different securities, you should calculate effective annual rates as
described here.
The effective annual rate, abbreviated EFF%, is also called the equivalent annual rate (EAR).
This is the rate that would produce the same future value under annual compounding as
would more frequent compounding at a given nominal rate.
If a loan or an investment uses annual compounding, its nominal rate is also its effective
rate. However, if compounding occurs more than once a year, the EFF% is higher than
I
NOM
.
To illustrate, a nominal rate of 10% with semiannual compounding is equivalent to a rate of
10.25% with annual compounding because both rates will cause $100 to grow to the same
amount after 1 year. The top line in the following diagram shows that $100 will grow to
$110.25 at a nominal rate of 10.25%. The lower line shows the situation if the nominal rate
is 10% but semiannual compounding is used.
Details
Given the nominal rate and the number of compounding periods per year, we can find the
effective annual rate with this equation:
5.10
Effective annual rate
(
EFF
%
)
=
[
1
+
I
NOM
M
]
M
1.0
Here
I
NOM
is the nominal rate expressed as a decimal, and M is the number of compounding periods
per year. In our example, the nominal rate is 10%. But with semiannual compounding,
I
NOM
=
10
%
=
0.10
and
M
=
2
. This results in
EFF
%
=
10.25
%
:
Effective annual rate
(
EFF
%
)
=
[
1
+
0.10
2
]
2
1
=
0.1025
=
10.25
%
We can also use the EFFECT function in Excel to solve for the effective rate:
Here we find that the effective rate is 10.25%. NPERY refers to the number of payments per
year. Likewise, if you know the effective rate and want to solve for the nominal rate, you can
use the NOMINAL function in Excel. Thus, if one investment promises to pay 10% with
semiannual compounding, and an equally risky investment promises 10.25% with annual
compounding, we would be indifferent between the two.
Quick Question
Question
You just received your first credit card and decided to purchase a new Apple iPad. You
charged the iPad’s $500 purchase price on your new credit card. Assume that the nominal
interest rate on the credit card is 18% and that interest is compounded monthly.
The minimum payment on the credit card is only $10 a month. If you pay the minimum and
make no other charges, how long will it take you to fully pay off the credit card?
Answer
Here we are given that the nominal interest rate is 18%. It follows that the monthly periodic
rate is 1.5% (18%/12). Using a financial calculator, we can solve for the number of months
that it takes to pay off the credit card.
Details
We can also use Excel’s NPER function:
Details
Here we find that it will take 93.11 months to pay off the credit card.
Note that it would take you almost 8 years to pay off your iPad purchase. Now, you see why
you can quickly get into financial trouble if you don’t manage your credit cards wisely!
SelfTest
Define the terms annual percentage rate (APR), effective annual rate (EFF%), and nominal
interest rate
(
�
NOM
)
.
A bank pays 5% with daily compounding on its savings accounts. Should it advertise the
nominal or effective rate if it is seeking to attract new deposits?
By law, credit card issuers must print their annual percentage rate on their monthly
statements. A common APR is 18% with interest paid monthly. What is the EFF% on such a
loan?
(
EFF
%
=
[
1
+
0.18
/
12
]
12
1
=
0.1956
=
19.56
%
)
Fifty years ago, banks didn’t have to reveal the rates they charged on credit cards. Then
Congress passed the Truth in Lending Act that required banks to publish their APRs. Is the
APR really the most truthful rate, or would the EFF% be more truthful? Explain.
5-17. Fractional Time Periods
Thus far we have assumed that payments occur at the beginning or the end of periods but
not within periods. However, we often encounter situations that require compounding or
discounting over fractional periods. For example, suppose you deposited $100 in a bank that
pays a nominal rate of 10% but adds interest daily, based on a 365-day year. How much
would you have after 9 months? The answer is $107.79, found as follows:
Periodic rate
=
I
PER
=
0.10
/
365
=
0.000273973
per day
Number of days
=
(
9
/
12
)
(
365
)
=
0.75
(
365
)
=
273.75
,
rounded to
274
Ending amount
=
$
100
(
1.000273973
)
274
=
$
107.79
Now suppose you borrow $100 from a bank whose nominal rate is 10% per year simple
interest, which means that interest is not earned on interest. If the loan is outstanding for
274 days, how much interest would you have to pay? Here we would calculate a daily
interest rate,
I
PER
, as just shown, but multiply it by 274 rather than use the 274 as an exponent:
Interest owed
=
$
100
(
0.000273973
)
(
274
)
=
$
7.51
You would owe the bank a total of $107.51 after 274 days. This is the procedure that most
banks use to calculate interest on loans, except that they require borrowers to pay the
interest on a monthly basis rather than after 274 days.
SelfTest
Suppose a company borrowed $1 million at a rate of 9% simple interest, with interest paid at
the end of each month. The bank uses a 360-day year. How much interest would the firm
have to pay in a 30-day month? What would the interest be if the bank used a 365-day year?
(
[
0.09
/
360
]
[
30
]
[
$
1,000,000
]
=
$
7,500
interest for the month. For the 365-day year,
[
0.09
/
365
]
[
30
]
[
$
1,000,000
]
=
$
7,397.26
of interest. The use of a 360-day year raises the interest cost by $102.74, which is why banks
like to use it on loans.)
Suppose you deposited $1,000 in a credit union account that pays 7% with daily
compounding and a 365-day year. What is the EFF%, and how much could you withdraw
after 7 months, assuming this is seven-twelfths of a year?
(
EFF
%
=
[
1
+
0.07
/
365
]
365
1
=
0.07250098
=
7.250098
%
. Thus, your account would grow from $1,000 to
$
1
,
000
[
1.07250098
]
0.583333
=
$
1
,
041.67
, and you could withdraw that amount.)
5-18. Amortized Loans
An important application of compound interest involves loans that are paid off in
installments over time. Included are automobile loans, home mortgage loans, student loans,
and many business loans. A loan that is to be repaid in equal amounts on a monthly,
quarterly, or annual basis is called an Amortized Loan.
Table 5.4 illustrates the amortization process. A homeowner borrows $100,000 on a
mortgage loan, and the loan is to be repaid in five equal payments at the end of each of the
next 5 years. The lender charges 6% on the balance at the beginning of each year. Our first
task is to determine the payment the homeowner must make each year. Here’s a picture of
the situation:
Details
Table 5.4 Loan Amortization Schedule, $100,000 at 6% for 5 Years
The payments must be such that the sum of their PVs equals $100,000:
$
100
,
000
=
PMT
(
1.06
)
1
+
PMT
(
1.06
)
2
+
PMT
(
1.06
)
3
+
PMT
(
1.06
)
5
=
∑
t
=
1
5
PMT
(
1.06
)
t
We could insert values into a calculator as shown on the next page to get the required
payments, $23,739.64:
Details
Therefore, the borrower must pay the lender $23,739.64 per year for the next 5 years.
Each payment will consist of two parts—interest and repayment of principal. This
breakdown is shown on an amortization schedule, such as the one in Table 5.4. The interest
component is relatively high in the first year, but it declines as the loan balance decreases.
For tax purposes, the borrower would deduct the interest component, and the lender would
report the same amount as taxable income.
SelfTest
Suppose you borrowed $30,000 on a student loan at a rate of 8% and must repay it in three
equal installments at the end of each of the next 3 years. How large would your payments
be; how much of the first payment would represent interest, how much would be principal;
and what would your ending balance be after the first year? (
PMT
=
$
11
,
641.01
;
Interest
=
$
2
,
400
;
Principal
=
$
9
,
241.01
; Balance at end of
Year
1
=
$
20
,
758.99
)