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Part

2
Important
Financial
Concepts

Chapter 4
Time Value of Money

Chapter 5
Risk and Return

Chapter 6
Interest Rates
and Bond Valuation

Chapter 7
Stock Valuation
Chapter Across the Disciplines


4
Why This Chapter Matters To You
Accounting: You need to understand
time-value-of-money calculations in order
to account for certain transactions such
as loan amortization, lease payments, and
bond interest rates.

Time Value Information systems: You need to under-
stand time-value-of-money calculations in
order to design systems that optimize the
of Money firm™s cash flows.
Management: You need to understand
time-value-of-money calculations so that
you can plan cash collections and dis-
bursements in a way that will enable the
firm to get the greatest value from its
money.
Marketing: You need to understand time
value of money because funding for new
programs and products must be justified
LEARNING GOALS financially using time-value-of-money
techniques.
Discuss the role of time value in fi-
LG1
nance, the use of computational tools, Operations: You need to understand time
and the basic patterns of cash flow. value of money because investments in
new equipment, in inventory, and in pro-
Understand the concepts of future
LG2
duction quantities will be affected by time-
and present value, their calculation
value-of-money techniques.
for single amounts, and the relation-
ship of present value to future value.
Find the future value and the present
LG3
value of an ordinary annuity, and find
the present value of a perpetuity.
Calculate both the future value and
LG4
the present value of a mixed stream of
cash flows.
Understand the effect that compound-
LG5
ing interest more frequently than
annually has on future value and on
the effective annual rate of interest.
Describe the procedures involved in
LG6
(1) determining deposits to accumu-
late a future sum, (2) loan amortiza-
tion, (3) finding interest or growth
rates, and (4) finding an unknown
number of periods.
130
131
CHAPTER 4 Time Value of Money



B ecause we view the firm as a going concern, we assess the decisions of its
financial managers, and ultimately the value of the firm itself, in light of its
cash flows. The opportunity to earn interest on the firm™s funds makes the timing
of its cash flows important, because a dollar received in the future in not the same
as a dollar received today. Thus, money has a time value, which affects every-
one”individuals, businesses, and government. In this chapter we explore the
concepts related to the time value of money.



The Role of Time Value in Finance
LG1


Financial managers and investors are always confronted with opportunities to
earn positive rates of return on their funds, whether through investment in
attractive projects or in interest-bearing securities or deposits. Therefore, the tim-
ing of cash outflows and inflows has important economic consequences, which
financial managers explicitly recognize as the time value of money. Time value is
based on the belief that a dollar today is worth more than a dollar that will be
received at some future date. We begin our study of time value in finance by con-
sidering the two views of time value”future value and present value, the compu-
tational tools used to streamline time value calculations, and the basic patterns of
cash flow.


Future Value versus Present Value
Financial values and decisions can be assessed by using either future value or pres-
ent value techniques. Although these techniques will result in the same decisions,
they view the decision differently. Future value techniques typically measure cash
flows at the end of a project™s life. Present value techniques measure cash flows at
the start of a project™s life (time zero). Future value is cash you will receive at a
given future date, and present value is just like cash in hand today.
A time line can be used to depict the cash flows associated with a given
time line
A horizontal line on which time investment. It is a horizontal line on which time zero appears at the leftmost end
zero appears at the leftmost end
and future periods are marked from left to right. A line covering five periods (in
and future periods are marked
this case, years) is given in Figure 4.1. The cash flow occurring at time zero and
from left to right; can be used to
that at the end of each year are shown above the line; the negative values repre-
depict investment cash flows.
sent cash outflows ($10,000 at time zero) and the positive values represent cash
inflows ($3,000 inflow at the end of year 1, $5,000 inflow at the end of year 2,
and so on).


FIGURE 4.1
Time Line
“$10,000 $3,000 $5,000 $4,000 $3,000 $2,000
Time line depicting an invest-
ment™s cash flows
0 1 2 3 4 5
End of Year
132 PART 2 Important Financial Concepts


FIGURE 4.2
Compounding
Compounding
and Discounting
Future
Time line showing Value
compounding to find future
value and discounting to find
present value
“$10,000 $3,000 $5,000 $4,000 $3,000 $2,000

0 1 2 3 4 5
End of Year

Present
Value

Discounting




Because money has a time value, all of the cash flows associated with an
investment, such as those in Figure 4.1, must be measured at the same point in
time. Typically, that point is either the end or the beginning of the investment™s
life. The future value technique uses compounding to find the future value of each
cash flow at the end of the investment™s life and then sums these values to find the
investment™s future value. This approach is depicted above the time line in
Figure 4.2. The figure shows that the future value of each cash flow is measured
at the end of the investment™s 5-year life. Alternatively, the present value tech-
nique uses discounting to find the present value of each cash flow at time zero
and then sums these values to find the investment™s value today. Application of
this approach is depicted below the time line in Figure 4.2.
The meaning and mechanics of compounding to find future value and of dis-
counting to find present value are covered in this chapter. Although future value
and present value result in the same decisions, financial managers”because they
make decisions at time zero”tend to rely primarily on present value techniques.



Computational Tools
Time-consuming calculations are often involved in finding future and present val-
ues. Although you should understand the concepts and mathematics underlying
these calculations, the application of time value techniques can be streamlined.
We focus on the use of financial tables, hand-held financial calculators, and com-
puters and spreadsheets as aids in computation.

Financial Tables
Financial tables include various future and present value interest factors that sim-
plify time value calculations. The values shown in these tables are easily devel-
oped from formulas, with various degrees of rounding. The tables are typically
133
CHAPTER 4 Time Value of Money


FIGURE 4.3
Interest Rate
Financial Tables
Period 1% 2% 10% 50%
20%
Layout and use
of a financial table 1
2
3


X.XXX
10


20


50




indexed by the interest rate (in columns) and the number of periods (in rows).
Figure 4.3 shows this general layout. The interest factor at a 20 percent interest
rate for 10 years would be found at the intersection of the 20% column and the
10-period row, as shown by the dark blue box. A full set of the four basic finan-
cial tables is included in Appendix A at the end of the book. These tables are
described more fully later in the chapter.


Financial Calculators
Financial calculators also can be used for time value computations. Generally,
financial calculators include numerous preprogrammed financial routines. This
chapter and those that follow show the keystrokes for calculating interest factors
and making other financial computations. For convenience, we use the impor-
tant financial keys, labeled in a fashion consistent with most major financial
calculators.
We focus primarily on the keys pictured and defined in Figure 4.4. We typi-
cally use four of the first five keys shown in the left column, along with the
compute (CPT) key. One of the four keys represents the unknown value being



FIGURE 4.4
N ” Number of periods
Calculator Keys N
I ” Interest rate per period
Important financial keys I
on the typical calculator PV ” Present value
PV
PMT ” Amount of payment (used only for annuities)
PMT
FV ” Future value
FV
CPT ” Compute key used to initiate financial calculation
CPT
once all values are input
134 PART 2 Important Financial Concepts


calculated. (Occasionally, all five of the keys are used, with one representing the
unknown value.) The keystrokes on some of the more sophisticated calculators
are menu-driven: After you select the appropriate routine, the calculator
prompts you to input each value; on these calculators, a compute key is not
needed to obtain a solution. Regardless, any calculator with the basic future and
present value functions can be used in lieu of financial tables. The keystrokes
for other financial calculators are explained in the reference guides that accom-
pany them.
Once you understand the basic underlying concepts, you probably will want
to use a calculator to streamline routine financial calculations. With a little
practice, you can increase both the speed and the accuracy of your financial
computations. Note that because of a calculator™s greater precision, slight differ-
ences are likely to exist between values calculated by using financial tables and
those found with a financial calculator. Remember that conceptual understand-
ing of the material is the objective. An ability to solve problems with the aid of
a calculator does not necessarily reflect such an understanding, so don™t just set-
tle for answers. Work with the material until you are sure you also understand
the concepts.


Computers and Spreadsheets
Like financial calculators, computers and spreadsheets have built-in routines that
simplify time value calculations. We provide in the text a number of spreadsheet
solutions that identify the cell entries for calculating time values. The value for
each variable is entered in a cell in the spreadsheet, and the calculation is pro-
grammed using an equation that links the individual cells. If values of the vari-
ables are changed, the solution automatically changes as a result of the equation
linking the cells. In the spreadsheet solutions in this book, the equation that
determines the calculation is shown at the bottom of the spreadsheet.
The ability to use spreadsheets has become a prime skill for today™s managers.
As the saying goes, “Get aboard the bandwagon, or get run over.” The spreadsheet
solutions we present in this book will help you climb up onto that bandwagon!


Basic Patterns of Cash Flow
The cash flow”both inflows and outflows”of a firm can be described by its gen-
eral pattern. It can be defined as a single amount, an annuity, or a mixed stream.

Single amount: A lump-sum amount either currently held or expected at
some future date. Examples include $1,000 today and $650 to be received at
the end of 10 years.

Annuity: A level periodic stream of cash flow. For our purposes, we™ll work
primarily with annual cash flows. Examples include either paying out or
receiving $800 at the end of each of the next 7 years.

Mixed stream: A stream of cash flow that is not an annuity; a stream of
unequal periodic cash flows that reflect no particular pattern. Examples
include the following two cash flow streams A and B.
135
CHAPTER 4 Time Value of Money



Mixed cash flow stream

End of year A B

1 $ 100 $ 50
2 800 100
3 1,200 80
4 1,200 60
5 1,400
6 300



Note that neither cash flow stream has equal, periodic cash flows and that A
is a 6-year mixed stream and B is a 4-year mixed stream.

In the next three sections of this chapter, we develop the concepts and tech-
niques for finding future and present values of single amounts, annuities, and
mixed streams, respectively. Detailed demonstrations of these cash flow patterns
are included.


Review Questions

4“1 What is the difference between future value and present value? Which
approach is generally preferred by financial managers? Why?
4“2 Define and differentiate among the three basic patterns of cash flow: (1) a
single amount, (2) an annuity, and (3) a mixed stream.




Single Amounts
LG2

The most basic future value and present value concepts and computations con-
cern single amounts, either present or future amounts. We begin by considering
the future value of present amounts. Then we will use the underlying concepts to
learn how to determine the present value of future amounts. You will see that
although future value is more intuitively appealing, present value is more useful
in financial decision making.


Future Value of a Single Amount
Imagine that at age 25 you began making annual purchases of $2,000 of an invest-
ment that earns a guaranteed 5 percent annually. At the end of 40 years, at age 65,
you would have invested a total of $80,000 (40 years $2,000 per year). Assum-
ing that all funds remain invested, how much would you have accumulated at the
end of the fortieth year? $100,000? $150,000? $200,000? No, your $80,000
would have grown to $242,000! Why? Because the time value of money allowed
your investments to generate returns that built on each other over the 40 years.
136 PART 2 Important Financial Concepts


The Concept of Future Value
compound interest
Interest that is earned on a given
We speak of compound interest to indicate that the amount of interest earned on
deposit and has become part of
a given deposit has become part of the principal at the end of a specified period.
the principal at the end of a
specified period. The term principal refers to the amount of money on which the interest is paid.
Annual compounding is the most common type.
principal
The amount of money on which The future value of a present amount is found by applying compound interest
interest is paid.
over a specified period of time. Savings institutions advertise compound interest
future value returns at a rate of x percent, or x percent interest, compounded annually, semi-
The value of a present amount at
annually, quarterly, monthly, weekly, daily, or even continuously. The concept of
a future date, found by applying
future value with annual compounding can be illustrated by a simple example.
compound interest over a
specified period of time.

If Fred Moreno places $100 in a savings account paying 8% interest com-
EXAMPLE
pounded annually, at the end of 1 year he will have $108 in the account”the ini-
tial principal of $100 plus 8% ($8) in interest. The future value at the end of the
first year is calculated by using Equation 4.1:
Future value at end of year 1 $100 (1 0.08) $108 (4.1)
If Fred were to leave this money in the account for another year, he would be
paid interest at the rate of 8% on the new principal of $108. At the end of this
second year there would be $116.64 in the account. This amount would represent
the principal at the beginning of year 2 ($108) plus 8% of the $108 ($8.64) in
interest. The future value at the end of the second year is calculated by using
Equation 4.2:
Future value at end of year 2 $108 (1 0.08) (4.2)
$116.64
Substituting the expression between the equals signs in Equation 4.1 for the
$108 figure in Equation 4.2 gives us Equation 4.3:

Future value at end of year 2 $100 (1 0.08) (1 0.08) (4.3)
0.08)2
$100 (1
$116.64

The equations in the preceding example lead to a more general formula for
calculating future value.

The Equation for Future Value
The basic relationship in Equation 4.3 can be generalized to find the future value
after any number of periods. We use the following notation for the various inputs:
FVn future value at the end of period n
PV initial principal, or present value
i annual rate of interest paid. (Note: On financial calculators, I is typi-
cally used to represent this rate.)
n number of periods (typically years) that the money is left on deposit
The general equation for the future value at the end of period n is
i)n
FVn PV (1 (4.4)
137
CHAPTER 4 Time Value of Money


A simple example will illustrate how to apply Equation 4.4.

Jane Farber places $800 in a savings account paying 6% interest compounded
EXAMPLE
annually. She wants to know how much money will be in the account at the end
of 5 years. Substituting PV $800, i 0.06, and n 5 into Equation 4.4 gives
the amount at the end of year 5.
0.06)5
FV5 $800 (1 $800 (1.338) $1,070.40
This analysis can be depicted on a time line as follows:

FV5 = $1,070.40
Time line for future
value of a single
amount ($800 initial
principal, earning 6%,
PV = $800
at the end of 5 years)
0 1 2 3 4 5
End of Year


Using Computational Tools to Find Future Value
Solving the equation in the preceding example involves raising 1.06 to the fifth
power. Using a future value interest table or a financial calculator or a computer
and spreadsheet greatly simplifies the calculation. A table that provides values for
(1 i)n in Equation 4.4 is included near the back of the book in Appendix Table
A“1.1 The value in each cell of the table is called the future value interest factor.
future value interest factor
The multiplier used to calculate, This factor is the multiplier used to calculate, at a specified interest rate, the
at a specified interest rate, the
future value of a present amount as of a given time. The future value interest fac-
future value of a present amount
tor for an initial principal of $1 compounded at i percent for n periods is referred
as of a given time.
to as FVIFi,n.
i)n
Future value interest factor FVIFi,n (1 (4.5)
By finding the intersection of the annual interest rate, i, and the appropriate
periods, n, you will find the future value interest factor that is relevant to a par-
ticular problem.2 Using FVIFi,n as the appropriate factor, we can rewrite the gen-
eral equation for future value (Equation 4.4) as follows:
FVn PV (FVIFi,n) (4.6)
This expression indicates that to find the future value at the end of period n of an
initial deposit, we have merely to multiply the initial deposit, PV, by the appro-
priate future value interest factor.3


1. This table is commonly referred to as a “compound interest table” or a “table of the future value of one dollar.”
As long as you understand the source of the table values, the various names attached to it should not create confu-
sion, because you can always make a trial calculation of a value for one factor as a check.
2. Although we commonly deal with years rather than periods, financial tables are frequently presented in terms of
periods to provide maximum flexibility.
3. Occasionally, you may want to estimate roughly how long a given sum must earn at a given annual rate to double
the amount. The Rule of 72 is used to make this estimate; dividing the annual rate of interest into 72 results in the
approximate number of periods it will take to double one™s money at the given rate. For example, to double one™s
money at a 10% annual rate of interest will take about 7.2 years (72 10 7.2). Looking at Table A“1, we can see
that the future value interest factor for 10% and 7 years is slightly below 2 (1.949); this approximation therefore
appears to be reasonably accurate.
138 PART 2 Important Financial Concepts


In the preceding example, Jane Farber placed $800 in her savings account at 6%
EXAMPLE
interest compounded annually and wishes to find out how much will be in the
account at the end of 5 years.
Table Use The future value interest factor for an initial principal of $1 on
deposit for 5 years at 6% interest compounded annually, FVIF6%, 5yrs, found in
Table A“1, is 1.338. Using Equation 4.6, $800 1.338 $1,070.40. Therefore,
the future value of Jane™s deposit at the end of year 5 will be $1,070.40.
Calculator Use4 The financial calculator can be used to calculate the future
value directly.5 First punch in $800 and depress PV; next punch in 5 and depress
N; then punch in 6 and depress I (which is equivalent to “i” in our notation)6;
finally, to calculate the future value, depress CPT and then FV. The future value
of $1,070.58 should appear on the calculator display as shown at the left. On
many calculators, this value will be preceded by a minus sign ( 1,070.58). If a
minus sign appears on your calculator, ignore it here as well as in all other “Cal-
Input Function
800 PV
culator Use” illustrations in this text.7
5 N Because the calculator is more accurate than the future value factors, which
have been rounded to the nearest 0.001, a slight difference”in this case, $0.18”
6 I

will frequently exist between the values found by these alternative methods.
CPT
Clearly, the improved accuracy and ease of calculation tend to favor the use of
FV
the calculator. (Note: In future examples of calculator use, we will use only a dis-
Solution
play similar to that shown at the left. If you need a reminder of the procedures
1070.58
involved, go back and review the preceding paragraph.)
Spreadsheet Use The future value of the single amount also can be calculated as
shown on the following Excel spreadsheet.




4. Many calculators allow the user to set the number of payments per year. Most of these calculators are preset for
monthly payments”12 payments per year. Because we work primarily with annual payments”one payment per
year”it is important to be sure that your calculator is set for one payment per year. And although most calculators
are preset to recognize that all payments occur at the end of the period, it is important to make sure that your calcu-
lator is correctly set on the END mode. Consult the reference guide that accompanies your calculator for instruc-
tions for setting these values.
5. To avoid including previous data in current calculations, always clear all registers of your calculator before
inputting values and making each computation.
6. The known values can be punched into the calculator in any order; the order specified in this as well as other
demonstrations of calculator use included in this text merely reflects convenience and personal preference.
7. The calculator differentiates inflows from outflows by preceding the outflows with a negative sign. For example,
in the problem just demonstrated, the $800 present value (PV), because it was keyed as a positive number (800), is
considered an inflow or deposit. Therefore, the calculated future value (FV) of 1,070.58 is preceded by a minus
sign to show that it is the resulting outflow or withdrawal. Had the $800 present value been keyed in as a negative
number ( 800), the future value of $1,070.58 would have been displayed as a positive number (1,070.58). Simply
stated, the cash flows” present value (PV) and future value (FV)”will have opposite signs.
139
CHAPTER 4 Time Value of Money


A Graphical View of Future Value
Remember that we measure future value at the end of the given period. Figure 4.5
illustrates the relationship among various interest rates, the number of periods
interest is earned, and the future value of one dollar. The figure shows that (1) the
higher the interest rate, the higher the future value, and (2) the longer the period
of time, the higher the future value. Note that for an interest rate of 0 percent, the
future value always equals the present value ($1.00). But for any interest rate
greater than zero, the future value is greater than the present value of $1.00.


Present Value of a Single Amount
It is often useful to determine the value today of a future amount of money. For
example, how much would I have to deposit today into an account paying 7 per-
cent annual interest in order to accumulate $3,000 at the end of 5 years? Present
present value
The current dollar value of a value is the current dollar value of a future amount”the amount of money that
future amount”the amount of would have to be invested today at a given interest rate over a specified period to
money that would have to be
equal the future amount. Present value depends largely on the investment oppor-
invested today at a given interest
tunities and the point in time at which the amount is to be received. This section
rate over a specified period to
explores the present value of a single amount.
equal the future amount.


The Concept of Present Value
The process of finding present values is often referred to as discounting cash
discounting cash flows
The process of finding present flows. It is concerned with answering the following question: “If I can earn i
values; the inverse of compound-
percent on my money, what is the most I would be willing to pay now for an
ing interest.
opportunity to receive FVn dollars n periods from today?”
This process is actually the inverse of compounding interest. Instead of find-
ing the future value of present dollars invested at a given rate, discounting deter-
mines the present value of a future amount, assuming an opportunity to earn a
certain return on the money. This annual rate of return is variously referred to



FIGURE 4.5
Future Value of One Dollar ($)




20%
Future Value 30.00
Relationship
25.00
Interest rates, time periods,
15%
and future value of one
20.00
dollar
15.00
10%
10.00
5%
5.00

0%
1.00
0 2 4 6 8 10 12 14 16 18 20 22 24
Periods
140 PART 2 Important Financial Concepts


as the discount rate, required return, cost of capital, and opportunity cost. These
terms will be used interchangeably in this text.

Paul Shorter has an opportunity to receive $300 one year from now. If he can
EXAMPLE
earn 6% on his investments in the normal course of events, what is the most he
should pay now for this opportunity? To answer this question, Paul must deter-
mine how many dollars he would have to invest at 6% today to have $300 one
year from now. Letting PV equal this unknown amount and using the same nota-
tion as in the future value discussion, we have
PV (1 0.06) $300 (4.7)
Solving Equation 4.7 for PV gives us Equation 4.8:
$300
PV (4.8)
(1 0.06)
$283.02
The value today (“present value”) of $300 received one year from today,
given an opportunity cost of 6%, is $283.02. That is, investing $283.02 today at
the 6% opportunity cost would result in $300 at the end of one year.


The Equation for Present Value
The present value of a future amount can be found mathematically by solving
Equation 4.4 for PV. In other words, the present value, PV, of some future
amount, FVn, to be received n periods from now, assuming an opportunity cost of
i, is calculated as follows:
FVn 1
PV FVn (4.9)
(1 i)n i)n
(1
Note the similarity between this general equation for present value and the
equation in the preceding example (Equation 4.8). Let™s use this equation in an
example.

Pam Valenti wishes to find the present value of $1,700 that will be received 8
EXAMPLE
years from now. Pam™s opportunity cost is 8%. Substituting FV8 $1,700, n 8,
and i 0.08 into Equation 4.9 yields Equation 4.10:
$1,700 $1,700
PV $918.42 (4.10)
(1 0.08)8 1.851
The following time line shows this analysis.

End of Year
Time line for present
0 1 2 3 4 5 6 7 8
value of a single
amount ($1,700 future
FV8 = $1,700
amount, discounted at
8%, from the end of
8 years)
PV = $918.42
141
CHAPTER 4 Time Value of Money


Using Computational Tools to Find Present Value
The present value calculation can be simplified by using a present value interest
present value interest factor
The multiplier used to calculate, factor. This factor is the multiplier used to calculate, at a specified discount rate,
at a specified discount rate, the the present value of an amount to be received in a future period. The present
present value of an amount to be
value interest factor for the present value of $1 discounted at i percent for n peri-
received in a future period.
ods is referred to as PVIFi,n.
1
Present value interest factor PVIFi,n (4.11)
i)n
(1
Appendix Table A“2 presents present value interest factors for $1. By letting
PVIFi,n represent the appropriate factor, we can rewrite the general equation for
present value (Equation 4.9) as follows:
PV FVn (PVIFi,n) (4.12)
This expression indicates that to find the present value of an amount to be re-
ceived in a future period, n, we have merely to multiply the future amount, FVn ,
by the appropriate present value interest factor.

As noted, Pam Valenti wishes to find the present value of $1,700 to be received 8
EXAMPLE
years from now, assuming an 8% opportunity cost.
Input Function
Table Use The present value interest factor for 8% and 8 years, PVIF8%, 8 yrs,
1700 FV
found in Table A“2, is 0.540. Using Equation 4.12, $1,700 0.540 $918. The
8 N
present value of the $1,700 Pam expects to receive in 8 years is $918.
8 I

Calculator Use Using the calculator™s financial functions and the inputs shown
CPT
at the left, you should find the present value to be $918.46. The value obtained
PV
with the calculator is more accurate than the values found using the equation or
Solution
the table, although for the purposes of this text, these differences are
918.46
insignificant.

Spreadsheet Use The present value of the single future amount also can be cal-
culated as shown on the following Excel spreadsheet.




A Graphical View of Present Value
Remember that present value calculations assume that the future values are mea-
sured at the end of the given period. The relationships among the factors in a
present value calculation are illustrated in Figure 4.6. The figure clearly shows
that, everything else being equal, (1) the higher the discount rate, the lower the
142 PART 2 Important Financial Concepts


FIGURE 4.6




Present Value of One Dollar ($)
1.00 0%
Present Value
Relationship
Discount rates, time periods, 0.75
and present value of one
dollar
0.50

5%
0.25
10%
15%
20%
0 2 4 6 8 10 12 14 16 18 20 22 24
Periods




present value, and (2) the longer the period of time, the lower the present value.
Also note that given a discount rate of 0 percent, the present value always equals
the future value ($1.00). But for any discount rate greater than zero, the present
value is less than the future value of $1.00.


Comparing Present Value and Future Value
We will close this section with some important observations about present val-
ues. One is that the expression for the present value interest factor for i percent
and n periods, 1/(1 i)n, is the inverse of the future value interest factor for i
percent and n periods, (1 i)n. You can confirm this very simply: Divide a pres-
ent value interest factor for i percent and n periods, PVIFi,n, given in Table A“2,
into 1.0, and compare the resulting value to the future value interest factor given
in Table A“1 for i percent and n periods, FVIFi,n,. The two values should be
equivalent.
Second, because of the relationship between present value interest factors
and future value interest factors, we can find the present value interest factors
given a table of future value interest factors, and vice versa. For example, the
future value interest factor (from Table A“1) for 10 percent and 5 periods is
1.611. Dividing this value into 1.0 yields 0.621, which is the present value inter-
est factor (given in Table A“2) for 10 percent and 5 periods.


Review Questions

4“3 How is the compounding process related to the payment of interest on
savings? What is the general equation for future value?
4“4 What effect would a decrease in the interest rate have on the future value
of a deposit? What effect would an increase in the holding period have on
future value?
143
CHAPTER 4 Time Value of Money


4“5 What is meant by “the present value of a future amount”? What is the
general equation for present value?
4“6 What effect does increasing the required return have on the present value
of a future amount? Why?
4“7 How are present value and future value calculations related?



Annuities
LG3


How much will you have at the end of 5 years if your employer withholds and
invests $1,000 of your year-end bonus at the end of each of the next 5 years, guar-
anteeing you a 9 percent annual rate of return? How much would you pay today,
given that you can earn 7 percent on low-risk investments, to receive a guaranteed
annuity $3,000 at the end of each of the next 20 years? To answer these questions, you
A stream of equal periodic cash
need to understand the application of the time value of money to annuities.
flows, over a specified time
An annuity is a stream of equal periodic cash flows, over a specified time
period. These cash flows can be
period. These cash flows are usually annual but can occur at other intervals, such
inflows of returns earned on
as monthly (rent, car payments). The cash flows in an annuity can be inflows (the
investments or outflows of funds
invested to earn future returns. $3,000 received at the end of each of the next 20 years) or outflows (the $1,000
invested at the end of each of the next 5 years).
ordinary annuity
An annuity for which the cash
flow occurs at the end of each
period.
Types of Annuities
annuity due
There are two basic types of annuities. For an ordinary annuity, the cash flow
An annuity for which the cash
occurs at the end of each period. For an annuity due, the cash flow occurs at the
flow occurs at the beginning of
beginning of each period.
each period.

Fran Abrams is choosing which of two annuities to receive. Both are 5-year,
EXAMPLE
$1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity
due. To better understand the difference between these annuities, she has listed
their cash flows in Table 4.1. Note that the amount of each annuity totals
$5,000. The two annuities differ in the timing of their cash flows: The cash flows
are received sooner with the annuity due than with the ordinary annuity.

Although the cash flows of both annuities in Table 4.1 total $5,000, the
annuity due would have a higher future value than the ordinary annuity, because
each of its five annual cash flows can earn interest for one year more than each
of the ordinary annuity™s cash flows. Similarly, the present value of the annuity
due would be greater than that of the ordinary annuity, because each annuity due
cash flow is discounted back one less year than for the ordinary annuity. In gen-
eral, both the future value and the present value of an annuity due are always
greater than the future value and the present value, respectively, of an otherwise
identical ordinary annuity.
Because ordinary annuities are more frequently used in finance, unless other-
wise specified, the term annuity is used throughout this book to refer to ordinary
annuities. In addition, discussions of annuities in this book concentrate on ordi-
WW nary annuities. For discussion and computations of annuities due, see the book™s
W
Web site at www.aw.com/gitman.
144 PART 2 Important Financial Concepts


TABLE 4.1 Comparison of Ordinary Annuity
and Annuity Due Cash Flows
($1,000, 5 Years)

Annual cash flows

End of yeara Annuity A (ordinary) Annuity B (annuity due)

0 $ 0 $1,000
1 1,000 1,000
2 1,000 1,000
3 1,000 1,000
4 1,000 1,000
5 1,000 0
Totals $5,000 $5,000
aThe ends of years 0, 1, 2, 3, 4, and 5 are equivalent to the beginnings of years
1, 2, 3, 4, 5, and 6, respectively.




Finding the Future Value of an Ordinary Annuity
The calculations required to find the future value of an ordinary annuity are illus-
trated in the following example.

Fran Abrams wishes to determine how much money she will have at the end of 5
EXAMPLE
years if he chooses annuity A, the ordinary annuity. It represents deposits of
$1,000 annually, at the end of each of the next 5 years, into a savings account
paying 7% annual interest. This situation is depicted on the following time line:


$1,311
Time line for future
1,225
value of an ordinary
1,145
annuity ($1,000 end-of-
1,070
year deposit, earning
1,000
7%, at the end of 5
$5,751 Future Value
years)

$1,000 $1,000 $1,000 $1,000 $1,000

0 1 2 3 4 5
End of Year



As the figure shows, at the end of year 5, Fran will have $5,751 in her account.
Note that because the deposits are made at the end of the year, the first deposit
will earn interest for 4 years, the second for 3 years, and so on.
145
CHAPTER 4 Time Value of Money


Using Computational Tools to Find
the Future Value of an Ordinary Annuity
Annuity calculations can be simplified by using an interest table or a financial cal-
culator or a computer and spreadsheet. A table for the future value of a $1 ordi-
nary annuity is given in Appendix Table A“3. The factors in the table are derived
by summing the future value interest factors for the appropriate number of years.
For example, the factor for the annuity in the preceding example is the sum of the
factors for the five years (years 4 through 0): 1.311 1.225 1.145 1.070
1.000 5.751. Because the deposits occur at the end of each year, they will earn
interest from the end of the year in which each occurs to the end of year 5. There-
fore, the first deposit earns interest for 4 years (end of year 1 through end of year
5), and the last deposit earns interest for zero years. The future value interest fac-
tor for zero years at any interest rate, FVIFi,0, is 1.000, as we have noted. The for-
mula for the future value interest factor for an ordinary annuity when interest is
future value interest factor
for an ordinary annuity compounded annually at i percent for n periods, FVIFAi,n, is8
The multiplier used to calculate
n
the future value of an ordinary
i)t 1
FVIFAi,n (1 (4.13)
annuity at a specified interest
t1
rate over a given period of time.
This factor is the multiplier used to calculate the future value of an ordinary
annuity at a specified interest rate over a given period of time.
Using FVAn for the future value of an n-year annuity, PMT for the amount
to be deposited annually at the end of each year, and FVIFAi,n for the appropri-
ate future value interest factor for a one-dollar ordinary annuity compounded at
i percent for n years, we can express the relationship among these variables alter-
natively as
FVAn PMT (FVIFAi,n) (4.14)
The following example illustrates this calculation using a table, a calculator, and
a spreadsheet.

As noted earlier, Fran Abrams wishes to find the future value (FVAn) at the end
EXAMPLE
of 5 years (n) of an annual end-of-year deposit of $1,000 (PMT) into an account
paying 7% annual interest (i) during the next 5 years.
Input Function
1000 PMT
Table Use The future value interest factor for an ordinary 5-year annuity at 7%
5 N
(FVIFA7%,5yrs), found in Table A“3, is 5.751. Using Equation 4.14, the $1,000
7 I
deposit 5.751 results in a future value for the annuity of $5,751.
CPT
FV
Calculator Use Using the calculator inputs shown at the left, you will find the
future value of the ordinary annuity to be $5,750.74, a slightly more precise
Solution
5750.74 answer than that found using the table.


8. A mathematical expression that can be applied to calculate the future value interest factor for an ordinary annuity
more efficiently is
1
i)n
FVIFAi,n [(1 1] (4.13a)
i
The use of this expression is especially attractive in the absence of the appropriate financial tables and of any finan-
cial calculator or personal computer and spreadsheet.
146 PART 2 Important Financial Concepts


Spreadsheet Use The future value of the ordinary annuity also can be calculated
as shown on the following Excel spreadsheet.




Finding the Present Value of an Ordinary Annuity
Quite often in finance, there is a need to find the present value of a stream of cash
flows to be received in future periods. An annuity is, of course, a stream of equal
periodic cash flows. (We™ll explore the case of mixed streams of cash flows in a
later section.) The method for finding the present value of an ordinary annuity is
similar to the method just discussed. There are long and short methods for mak-
ing this calculation.


Braden Company, a small producer of plastic toys, wants to determine the most it
EXAMPLE
should pay to purchase a particular ordinary annuity. The annuity consists of
cash flows of $700 at the end of each year for 5 years. The firm requires the
annuity to provide a minimum return of 8%. This situation is depicted on the fol-
lowing time line:


End of Year
Time line for present
0 1 2 3 4 5
value of an ordinary
annuity ($700 end-
$700 $700 $700 $700 $700
of-year cash flows,
discounted at 8%, $ 648.20
over 5 years) 599.90
555.80
514.50
476.70
Present Value $2,795.10




Table 4.2 shows the long method for finding the present value of the annuity.
This method involves finding the present value of each payment and summing
them. This procedure yields a present value of $2,795.10.
147
CHAPTER 4 Time Value of Money


TABLE 4.2 The Long Method for Finding
the Present Value of an
Ordinary Annuity

Present value
PVIF8%,na
Cash flow [(1) (2)]
Year (n) (1) (2) (3)

1 $700 0.926 $ 648.20
2 700 0.857 599.90
3 700 0.794 555.80
4 700 0.735 514.50
5 700 0.681 476.70
Present value of annuity $2,795.10

aPresent value interest factors at 8% are from Table A“2.




Using Computational Tools to Find
the Present Value of an Ordinary Annuity
Annuity calculations can be simplified by using an interest table for the present
value of an annuity, a financial calculator, or a computer and spreadsheet.
The values for the present value of a $1 ordinary annuity are given in Appendix
Table A“4. The factors in the table are derived by summing the present value
interest factors (in Table A“2) for the appropriate number of years at the
given discount rate. The formula for the present value interest factor for an ordi-
present value interest factor
for an ordinary annuity nary annuity with cash flows that are discounted at i percent for n periods,
The multiplier used to calculate PVIFAi,n, is9
the present value of an ordinary
annuity at a specified discount n
1
PVIFAi,n (4.15)
rate over a given period of time.
i)t
(1
t1


This factor is the multiplier used to calculate the present value of an ordinary
annuity at a specified discount rate over a given period of time.
By letting PVAn equal the present value of an n-year ordinary annuity, letting
PMT equal the amount to be received annually at the end of each year, and let-
ting PVIFAi,n represent the appropriate present value interest factor for a one-
dollar ordinary annuity discounted at i percent for n years, we can express the
relationship among these variables as

PVAn PMT (PVIFAi,n) (4.16)


9. A mathematical expression that can be applied to calculate the present value interest factor for an ordinary annu-
ity more efficiently is
1 1
PVIFAi,n 1 (4.15a)
i)n
i (1
The use of this expression is especially attractive in the absence of the appropriate financial tables and of any finan-
cial calculator or personal computer and spreadsheet.
148 PART 2 Important Financial Concepts


The following example illustrates this calculation using a table, a calculator, and
a spreadsheet.


Braden Company, as we have noted, wants to find the present value of a 5-year
EXAMPLE
ordinary annuity of $700, assuming an 8% opportunity cost.

Table Use The present value interest factor for an ordinary annuity at 8% for
Input Function
5 years (PVIFA8%,5yrs), found in Table A“4, is 3.993. If we use Equation 4.16,
700 PMT
$700 annuity 3.993 results in a present value of $2,795.10.
5 N
8 I
Calculator Use Using the calculator™s inputs shown at the left, you will find the
CPT
present value of the ordinary annuity to be $2,794.90. The value obtained with
PV
the calculator is more accurate than those found using the equation or the table.
Solution
Spreadsheet Use The present value of the ordinary annuity also can be calcu-
2794.90
lated as shown on the following Excel spreadsheet.




Finding the Present Value of a Perpetuity
A perpetuity is an annuity with an infinite life”in other words, an annuity that
perpetuity
An annuity with an infinite life, never stops providing its holder with a cash flow at the end of each year (for
providing continual annual cash example, the right to receive $500 at the end of each year forever).
flow.
It is sometimes necessary to find the present value of a perpetuity. The present
value interest factor for a perpetuity discounted at the rate i is
1
PVIFAi, (4.17)
i
As the equation shows, the appropriate factor, PVIFAi, , is found simply by
dividing the discount rate, i (stated as a decimal), into 1. The validity of this
method can be seen by looking at the factors in Table A“4 for 8, 10, 20, and
40 percent: As the number of periods (typically years) approaches 50, these fac-
tors approach the values calculated using Equation 4.17: 1 0.08 12.50;
1 0.10 10.00; 1 0.20 5.00; and 1 0.40 2.50.

Ross Clark wishes to endow a chair in finance at his alma mater. The university
EXAMPLE
indicated that it requires $200,000 per year to support the chair, and the endow-
ment would earn 10% per year. To determine the amount Ross must give the
university to fund the chair, we must determine the present value of a $200,000
perpetuity discounted at 10%. The appropriate present value interest factor can
149
CHAPTER 4 Time Value of Money


be found by dividing 1 by 0.10, as noted in Equation 4.17. Substituting the
resulting factor, 10, and the amount of the perpetuity, PMT $200,000, into
Equation 4.16 results in a present value of $2,000,000 for the perpetuity. In other
words, to generate $200,000 every year for an indefinite period requires
$2,000,000 today if Ross Clark™s alma mater can earn 10% on its investments. If
the university earns 10% interest annually on the $2,000,000, it can withdraw
$200,000 a year indefinitely without touching the initial $2,000,000, which
would never be drawn upon.


Review Questions

4“8 What is the difference between an ordinary annuity and an annuity due?
Which always has greater future value and present value for identical
annuities and interest rates? Why?
4“9 What are the most efficient ways to calculate the present value of an ordi-
nary annuity? What is the relationship between the PVIF and PVIFA
interest factors given in Tables A“2 and A“4, respectively?
4“10 What is a perpetuity? How can the present value interest factor for such a
stream of cash flows be determined?




Mixed Streams
LG4


Two basic types of cash flow streams are possible: the annuity and the mixed
stream. Whereas an annuity is a pattern of equal periodic cash flows, a mixed
mixed stream
A stream of unequal periodic stream is a stream of unequal periodic cash flows that reflect no particular pat-
cash flows that reflect no partic- tern. Financial managers frequently need to evaluate opportunities that are
ular pattern.
expected to provide mixed streams of cash flows. Here we consider both the
future value and the present value of mixed streams.


Future Value of a Mixed Stream
Determining the future value of a mixed stream of cash flows is straightforward.
We determine the future value of each cash flow at the specified future date and
then add all the individual future values to find the total future value.

Shrell Industries, a cabinet manufacturer, expects to receive the following mixed
EXAMPLE
stream of cash flows over the next 5 years from one of its small customers.

End of year Cash flow

1 $11,500
2 14,000
3 12,900
4 16,000
5 18,000
150 PART 2 Important Financial Concepts


If Shrell expects to earn 8% on its investments, how much will it accumulate by
the end of year 5 if it immediately invests these cash flows when they are
received? This situation is depicted on the following time time:


$15,640.00
Time line for future
17,640.00
value of a mixed
15,041.40
stream (end-of-year
17,280.00
cash flows, com-
18,000.00
pounded at 8% to
$83,601.40 Future Value
the end of year 5)

$11,500 $14,000 $12,900 $16,000 $18,000

0 1 2 3 4 5
End of Year


Table Use To solve this problem, we determine the future value of each cash
flow compounded at 8% for the appropriate number of years. Note that the first
cash flow of $11,500, received at the end of year 1, will earn interest for 4 years
(end of year 1 through end of year 5); the second cash flow of $14,000, received at
the end of year 2, will earn interest for 3 years (end of year 2 through end of year
5); and so on. The sum of the individual end-of-year-5 future values is the future
value of the mixed cash flow stream. The future value interest factors required are
those shown in Table A“1. Table 4.3 presents the calculations needed to find the
future value of the cash flow stream, which turns out to be $83,601.40.

Calculator Use You can use your calculator to find the future value of each
individual cash flow, as demonstrated earlier (page 138), and then sum the future
values, to get the future value of the stream. Unfortunately, unless you can pro-
gram your calculator, most calculators lack a function that would allow you to



TABLE 4.3 Future Value of a Mixed Stream
of Cash Flows

Number of years Future value
a
Cash flow earning interest (n) FVIF8%,n [(1) (3)]
Year (1) (2) (2) (4)

1 $11,500 5 1 4 1.360 $15,640.00
2 14,000 5 2 3 1.260 17,640.00
3 12,900 5 3 2 1.166 15,041.40
4 16,000 5 4 1 1.080 17,280.00
1.000b
5 18,000 5 5 0 18,000.00
Future value of mixed stream $83,601.40

aFuture value interest factors at 8% are from Table A“1.
bThe future value of the end-of-year-5 deposit at the end of year 5 is its present value because it earns
interest for zero years and (1 0.08)0 1.000.
151
CHAPTER 4 Time Value of Money


input all of the cash flows, specify the interest rate, and directly calculate the
future value of the entire cash flow stream. Had you used your calculator to find
the individual cash flow future values and then summed them, the future value of
Shrell Industries™ cash flow stream at the end of year 5 would have been
$83,608.15, a more precise value than the one obtained by using a financial table.

Spreadsheet Use The future value of the mixed stream also can be calculated as
shown on the following Excel spreadsheet.




If Shrell Industries invests at 8% interest the cash flows received from its cus-
tomer over the next 5 years, the company will accumulate about $83,600 by the
end of year 5.


Present Value of a Mixed Stream
Finding the present value of a mixed stream of cash flows is similar to finding the
future value of a mixed stream. We determine the present value of each future
amount and then add all the individual present values together to find the total
present value.

Frey Company, a shoe manufacturer, has been offered an opportunity to receive
EXAMPLE
the following mixed stream of cash flows over the next 5 years:


End of year Cash flow

1 $400
2 800
3 500
4 400
5 300
152 PART 2 Important Financial Concepts


If the firm must earn at least 9% on its investments, what is the most it should
pay for this opportunity? This situation is depicted on the following time line:

End of Year
Time line for present
0 1 2 3 4 5
value of a mixed
stream (end-of-year
$400 $800 $500 $400 $300
cash flows, discounted
at 9% over the corre-
$ 366.80
sponding number of
673.60
years)
386.00
283.20
195.00
Present Value $1,904.60



Table Use To solve this problem, determine the present value of each cash flow
discounted at 9% for the appropriate number of years. The sum of these individ-
ual values is the present value of the total stream. The present value interest fac-
tors required are those shown in Table A“2. Table 4.4 presents the calculations
needed to find the present value of the cash flow stream, which turns out to be
$1,904.60.

Calculator Use You can use a calculator to find the present value of each indi-
vidual cash flow, as demonstrated earlier (page 141), and then sum the present
values, to get the present value of the stream. However, most financial calcula-
tors have a function that allows you to punch in all cash flows, specify the dis-
count rate, and then directly calculate the present value of the entire cash flow
stream. Because calculators provide solutions more precise than those based on
rounded table factors, the present value of Frey Company™s cash flow stream
found using a calculator is $1,904.76, which is close to the $1,904.60 value cal-
culated before.


TABLE 4.4 Present Value of a Mixed
Stream of Cash Flows

Present value
PVIF9%,na
Cash flow [(1) (2)]
Year (n) (1) (2) (3)

1 $400 0.917 $ 366.80
2 800 0.842 673.60
3 500 0.772 386.00
4 400 0.708 283.20
5 300 0.650 195.00
Present value of mixed stream $1,904.60

aPresent value interest factors at 9% are from Table A“2.
153
CHAPTER 4 Time Value of Money


Spreadsheet Use The present value of the mixed stream of future cash flows
also can be calculated as shown on the following Excel spreadsheet.




Paying about $1,905 would provide exactly a 9% return. Frey should pay no
more than that amount for the opportunity to receive these cash flows.


Review Question

4“11 How is the future value of a mixed stream of cash flows calculated? How
is the present value of a mixed stream of cash flows calculated?



Compounding Interest
LG5

More Frequently Than Annually
Interest is often compounded more frequently than once a year. Savings institu-
tions compound interest semiannually, quarterly, monthly, weekly, daily, or even
continuously. This section discusses various issues and techniques related to these
more frequent compounding intervals.

Semiannual Compounding
Semiannual compounding of interest involves two compounding periods within
semiannual compounding
Compounding of interest over the year. Instead of the stated interest rate being paid once a year, one-half of the
two periods within the year.
stated interest rate is paid twice a year.

Fred Moreno has decided to invest $100 in a savings account paying 8% interest
EXAMPLE

. 1
( 3)



>>