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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 firmâ€™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 firmâ€™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-

oneâ€”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 valueâ€”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 projectâ€™s life. Present value techniques measure cash flows at

the start of a projectâ€™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

â€“$10,000 $3,000 $5,000 $4,000 $3,000 $2,000

Time line depicting an invest-

mentâ€™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

â€“$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 investmentâ€™s

life. The future value technique uses compounding to find the future value of each

cash flow at the end of the investmentâ€™s life and then sums these values to find the

investmentâ€™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 investmentâ€™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 investmentâ€™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 managersâ€”because they

make decisions at time zeroâ€”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 â€” Number of periods

Calculator Keys N

I â€” Interest rate per period

Important financial keys I

on the typical calculator PV â€” Present value

PV

PMT â€” Amount of payment (used only for annuities)

PMT

FV â€” Future value

FV

CPT â€” 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 calculatorâ€™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 donâ€™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 todayâ€™s managers.

As the saying goes, â€śGet aboard the bandwagon, or get run over.â€ť The spreadsheet

solutions we present in this book will help you climb up onto that bandwagon!

Basic Patterns of Cash Flow

The cash flowâ€”both inflows and outflowsâ€”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, weâ€™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

4â€“1 What is the difference between future value and present value? Which

approach is generally preferred by financial managers? Why?

4â€“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 accountâ€”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

Aâ€“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 â€ścompound interest tableâ€ť or a â€śtable of the future value of one dollar.â€ť

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 oneâ€™s money at the given rate. For example, to double oneâ€™s

money at a 10% annual rate of interest will take about 7.2 years (72 10 7.2). Looking at Table Aâ€“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 Aâ€“1, is 1.338. Using Equation 4.6, $800 1.338 $1,070.40. Therefore,

the future value of Janeâ€™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 â€śiâ€ť 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 â€śCal-

Input Function

800 PV

culator Useâ€ť 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 differenceâ€”in this case, $0.18â€”

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 paymentsâ€”12 payments per year. Because we work primarily with annual paymentsâ€”one payment per

yearâ€”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 flowsâ€” present value (PV) and future value (FV)â€”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 amountâ€”the amount of money that

future amountâ€”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: â€śIf 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?â€ť

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 (â€śpresent valueâ€ť) 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). Letâ€™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. Pamâ€™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 Aâ€“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 Aâ€“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 calculatorâ€™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 Aâ€“2,

into 1.0, and compare the resulting value to the future value interest factor given

in Table Aâ€“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 Aâ€“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 Aâ€“2) for 10 percent and 5 periods.

Review Questions

4â€“3 How is the compounding process related to the payment of interest on

savings? What is the general equation for future value?

4â€“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

4â€“5 What is meant by â€śthe present value of a future amountâ€ť? What is the

general equation for present value?

4â€“6 What effect does increasing the required return have on the present value

of a future amount? Why?

4â€“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 annuityâ€™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 bookâ€™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 Aâ€“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 Aâ€“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. (Weâ€™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 Aâ€“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 Aâ€“4. The factors in the table are derived by summing the present value

interest factors (in Table Aâ€“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 Aâ€“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 calculatorâ€™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 lifeâ€”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 Aâ€“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 Clarkâ€™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

4â€“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?

4â€“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 Aâ€“2 and Aâ€“4, respectively?

4â€“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 Aâ€“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 Aâ€“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 Industriesâ€™ 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 Aâ€“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 Companyâ€™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 Aâ€“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

4â€“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

ńňđ. 1 |