07 time value of money

Time Line An important tool used in time value of money analysis; it is a graphical representation used to show the timing of cash flows... FV5 $1001.055 $127.63.In general, the future va

SOURCE: © Zigy Kaluzny/Tony Stone Images.

requirement would increase to $110,765 in 10 years and

to $180,424 in 20 years If inflation were 7 percent, her Year 20 requirement would jump to $263,139! How much wealth would Ms Jones need at retirement to maintain her standard of living, and how much would she have had to save during each working year to accumulate that wealth?

The answer depends on a number of factors, including the rate she could earn on her savings, the inflation rate, and when her savings program began.

Also, the answer would depend on how much she will receive from Social Security and from her corporate retirement plan, if she has one (She might not receive much from Social Security unless she is really down and out.) Note, too, that her plans could be upset if the inflation rate increased, if the return on her savings changed, or if she lived beyond 20 years.

Fortune and other organizations have done studies

relating to the retirement issue, using the tools and techniques described in this chapter The general conclusion is that most Americans have been putting their heads in the sand — many of us have been ignoring what is almost certainly going to be a huge personal and social problem But if you study this chapter carefully, you can avoid the trap that seems to

be catching so many people ■

our reaction to the question in the title ofthis

vignette is probably, “First things first! I’m

worried about getting a job, not retiring!”

However, an awareness ofthe retirement situation

could help you land a job because (1) this is an

important issue today, (2) employers prefer to hire

people who know the issues, and (3) professors often

test students on time value ofmoney with problems

related to saving for some future purpose, including

retirement So read on.

A recent Fortune article began with some interesting

facts: (1) The U.S savings rate is the lowest of any

industrial nation (2) The ratio of U.S workers to

retirees, which was 17 to 1 in 1950, is now down to 3.2

to 1, and it will decline to less than 2 to 1 after 2020.

(3) With so few people paying into the Social Security

System, and so many drawing funds out, Social Security

may soon be in serious trouble The article concluded

that even people making $85,000 per year will have

trouble maintaining a reasonable standard of living after

they retire, and many of today’s college students will

have to support their parents.

If Ms Jones, who earns $85,000, retires in 2001,

expects to live for another 20 years after retirement,

and needs 80 percent of her pre-retirement income, she

would require $68,000 during 2001 However, if

inflation amounts to 5 percent per year, her income

W I L L Y O U B E A B L E

T O R E T I R E ?

$ Y

289

Note also that tutorials on how to use several Hewlett-Packard, Texas Instruments, and Sharp calculators are

provided in the Technology Supplement to this book, which is available to adopting instructors We also discuss

spreadsheets briefly in the chapter, and a more complete discussion is contained in the file 07MODEL.xls on the

CD-ROM that accompanies the book The spreadsheet material is also set up so that it can be either covered or

skipped.

In Chapter 1, we saw that the primary goal of financial management is to mize the value of the firm’s stock We also saw that stock values depend in part

maxi-on the timing of the cash flows investors expect to receive from an investment —

a dollar expected soon is worth more than a dollar expected in the distant future.Therefore, it is essential for financial managers to have a clear understanding ofthe time value of money and its impact on stock prices These concepts are dis-cussed in this chapter, where we show how the timing of cash flows affects assetvalues and rates of return

The principles of time value analysis have many applications, ranging from ting up schedules for paying off loans to decisions about whether to acquire new

set-equipment In fact, of all the concepts used in finance, none is more important than

the time value of money, also called discounted cash flow (DCF) analysis Since this

concept is used throughout the remainder of the book, it is vital that you stand the material in this chapter before you move on to other topics ■

under-T I M E L I N E S

One of the most important tools in time value analysis is the time line, which

is used by analysts to help visualize what is happening in a particular problemand then to help set up the problem for solution To illustrate the time lineconcept, consider the following diagram:

Time 0 is today; Time 1 is one period from today, or the end of Period 1; Time

2 is two periods from today, or the end of Period 2; and so on Thus, the bers above the tick marks represent end-of-period values Often the periods areyears, but other time intervals such as semiannual periods, quarters, months, oreven days can be used If each period on the time line represents a year, the in-terval from the tick mark corresponding to 0 to the tick mark corresponding to

num-1 would be Year num-1, the interval from num-1 to 2 would be Year 2, and so on Notethat each tick mark corresponds to the end of one period as well as the begin-ning of the next period In other words, the tick mark at Time 1 represents the

end of Year 1, and it also represents the beginning of Year 2 because Year 1 has

will need to scroll down the page until

you see retirement calculators — there

are 17 of them, each designed to answer

a different question Each one provides

results, graphs, and explanations.

Time Line

An important tool used in time

value of money analysis; it is a

graphical representation used to

show the timing of cash flows.

fol-Here the interest rate for each of the three periods is 5 percent; a single amount

(or lump sum) cash outflow is made at Time 0; and the Time 3 value is an known inflow Since the initial $100 is an outflow (an investment), it has a

un-minus sign Since the Period 3 amount is an inflow, it does not have a un-minussign, which implies a plus sign Note that no cash flows occur at Times 1 and

2 Note also that we generally do not show dollar signs on time lines to reduceclutter

Now consider a different situation, where a $100 cash outflow is made today,and we will receive an unknown amount at the end of Time 2:

Here the interest rate is 5 percent during the first period, but it rises to 10 cent during the second period If the interest rate is constant in all periods, weshow it only in the first period, but if it changes, we show all the relevant rates

per-on the time line

Time lines are essential when you are first learning time value concepts, buteven experts use time lines to analyze complex problems We will be using timelines throughout the book, and you should get into the habit of using themwhen you work problems

?

F U T U R E VA L U E

Outflow

A cash deposit, cost, or amount

paid It has a minus sign.

out-F U T U R E VA L U E

A dollar in hand today is worth more than a dollar to be received in the futurebecause, if you had it now, you could invest it, earn interest, and end up withmore than one dollar in the future The process of going from today’s values,

or present values (PVs), to future values (FVs) is called compounding To

il-lustrate, suppose you deposit $100 in a bank that pays 5 percent interest eachyear How much would you have at the end of one year? To begin, we definethe following terms:

Compounding

The arithmetic process of

determining the final value of a

cash flow or series of cash flows

when compound interest is

applied.

PV  present value, or beginning amount, in your account Here PV  $100.

i  interest rate the bank pays on the account per year The interest earned

is based on the balance at the beginning of each year, and we assume that

it is paid at the end of the year Here i 5%, or, expressed as a decimal,

i 0.05 Throughout this chapter, we designate the interest rate as i (orI) because that symbol is used on most financial calculators Note,though, that in later chapters we use the symbol k to denote interestrates because k is used more often in the financial literature

INT  dollars of interest you earn during the year  Beginning amount  i

Here INT  $100(0.05)  $5

FVn  future value, or ending amount, of your account at the end of n years

Whereas PV is the value now, or the present value, FVnis the value n

years into the future, after the interest earned has been added to the

ac-count

n  number of periods involved in the analysis Here n  1

In our example, n  1, so FVncan be calculated as follows:

Note the following points: (1) You start by depositing $100 in the account —this is shown as an outflow at t  0 (2) You earn $100(0.05)  $5 of interestduring the first year, so the amount at the end of Year 1 (or t  1) is $100 

$5  $105 (3) You start the second year with $105, earn $5.25 on the nowlarger amount, and end the second year with $110.25 Your interest duringYear 2, $5.25, is higher than the first year’s interest, $5, because you earned

$5(0.05)  $0.25 interest on the first year’s interest (4) This process continues,and because the beginning balance is higher in each succeeding year, the annualinterest earned increases (5) The total interest earned, $27.63, is reflected inthe final balance at t  5, $127.63

Note that the value at the end of Year 2, $110.25, is equal to

The amount to which a cash flow

or series of cash flows will grow

over a given period of time when

compounded at a given interest

100 FV1  ? FV2  ? FV3  ? FV4  ? FV5  ?

1

5.00 Interest earned:

FV5 $100(1.05)5 $127.63.

In general, the future value of an initial lump sum at the end of n years can

be found by applying Equation 7-1:

Equation 7-1 and most other time value of money equations can be solved infour ways: numerically with a regular calculator, with interest tables, with a fi-nancial calculator, or with a computer spreadsheet program Most advancedwork in financial management will be done with a financial calculator or on acomputer, but when learning basic concepts it is best to work through all themethods

In certain problems, it is extremely difficult to arrive at a solution using aregular calculator We will tell you this when we have such a problem, and inthese cases we will not show a numerical solution Also, at times we show thenumerical solution just below the time line, as a part of the diagram, ratherthan in a separate section

Interest Tables (Tabular Solution)

The Future Value Interest Factor for i and n (FVIF i,n ) is defined as (1  i)n,and these factors can be found by using a regular calculator as discussed aboveand then put into tables Table 7-1 is illustrative, while Table A-3 in Appendix

A at the back of the book contains FVIFi,nvalues for a wide range of i and nvalues

Since (1  i)n FVIFi,n, Equation 7-1 can be rewritten as follows:

To illustrate, the FVIF for our five-year, 5 percent interest problem can befound in Table 7-1 by looking down the first column to Period 5, and thenlooking across that row to the 5 percent column, where we see thatFVIF5%,5 1.2763 Then, the value of $100 after five years is found as follows:

FVn PV(FVIFi,n)

 $100(1.2763)  $127.63

F U T U R E VA L U E

Future Value Interest Factor

for i and n (FVIFi,n)

The future value of $1 left on

deposit for n periods at a rate of i

percent per period.

Before financial calculators became readily available (in the 1980s), such tableswere used extensively, but they are rarely used today in the real world.

Financial Calculator Solution

Equation 7-1 and a number of other equations have been programmed directlyinto financial calculators, and these calculators can be used to find future val-ues Note that calculators have five keys that correspond to the five most com-monly used time value of money variables:

Here

N  the number of periods Some calculators use n rather than N

I  interest rate per period Some calculators use i or I/YR rather than I

PV  present value

PMT  payment This key is used only if the cash flows involve a series of

equal, or constant, payments (an annuity) If there are no periodicpayments in a particular problem, then PMT  0

FV  future value

On some financial calculators, these keys are actually buttons on the face of thecalculator, while on others they are shown on a screen after going into the timevalue of money (TVM) menu

In this chapter, we deal with equations involving only four of the variables atany one time — three of the variables are known, and the calculator then solvesfor the fourth (unknown) variable In the next chapter, when we deal withbonds, we will use all five variables in the bond valuation equation.2

Future Value Interest Factors: FVIF i,n ⴝ (1 ⴙ i)

2 The equation programmed into the calculators actually has five variables, one for each key In this chapter, the value of one of the variables is always zero It is a good idea to get into the habit of inputting

Many financial calculators require that all cash flows be designated as eitherinflows or outflows, with outflows being entered as negative numbers In our il-lustration, you deposit, or put in, the initial amount (which is an outflow toyou) and you take out, or receive, the ending amount (which is an inflow toyou) If your calculator requires that you follow this sign convention, the PVwould be entered as
100 Enter the
100 by keying in 100 and then pressingthe “change sign” or /
key (If you entered 100, then the FV would appear

as
127.63.) Also, on some calculators you are required to press a “Compute”key before pressing the FV key

Sometimes the convention of changing signs can be confusing For example,

if you have $100 in the bank now and want to find out how much you will haveafter five years if your account pays 5 percent interest, the calculator will giveyou a negative answer, in this case
127.63, because the calculator assumes youare going to withdraw the funds This sign convention should cause you noproblem if you think about what you are doing

We should also note that financial calculators permit you to specify the ber of decimal places that are displayed Twelve significant digits are actuallyused in the calculations, but we generally use two places for answers when work-ing with dollars or percentages and four places when working with decimals Thenature of the problem dictates how many decimal places should be displayed

num-Spreadsheet Solution

As noted back in Chapter 2, spreadsheet programs are ideally suited for solvingmany financial problems, including time value of money problems.4With verylittle effort, the spreadsheet itself becomes a time line Here is how the prob-lem would look in a spreadsheet:

F U T U R E VA L U E

a zero for the unused variable (whose value is automatically set equal to zero when you clear the lator’s memory); if you forget to clear your calculator, inputting a zero will help you avoid trouble.

calcu-3 Here we assume that compounding occurs once each year Most calculators have a setting that can

be used to designate the number of compounding periods per year For example, the HP-10B comes preset with payments at 12 per year You would need to change it to 1 per year to get FV  127.63 With the HP-10B, you would do this by typing 1, pressing the gold key, and then pressing the P/YR key.

4 In this section, and in other sections and chapters, we discuss spreadsheet solutions to various nancial problems If a reader is not familiar with spreadsheets and has no interest in them, then

fi-these sections can be omitted For those who are interested, 07MODEL.xls is the Excel file on the CD-ROM for this chapter that does the various calculations in the chapter If you have the time,

we highly recommend that you go through the models This will give you practice with Excel, which

will help tremendously in later courses, in the job market, and in the workplace Also, going through the models will enhance your understanding of financial concepts.

(Footnote 2 continued)

Cell B1 shows the interest rate, entered as a decimal number, 0.05 Row 2

shows the periods for the time line With Microsoft Excel, you could enter 0 in

Cell B2, then the formula ⴝB2ⴙ1 in Cell C2, and then copy this formula into

Cells D2 through G2 to produce the time periods shown on Row 2 Note that

if your time line had many years, say, 50, you would simply copy the formulaacross more columns Other procedures could also be used to enter the periods.Row 3 shows the cash flows In this case, there is only one cash flow, shown inCell B3 Row 4 shows the future value of this cash flow at the end of each year.Cell C4 contains the formula for Equation 7-1 The formula could be written as

ⴝⴚ$B$3*(11.05)^C2, but we wrote it as ⴝⴚ$B$3*(1ⴙ$B$1)^C2, which

gives us the flexibility to change the interest rate in Cell B1 to see how the futurevalue changes with changes in interest rates Note that the formula has a minussign for the PV (which is in Cell B3) to account for the minus sign of the cashflow This formula was then copied into Cells D4 through G4 As Cell G4 shows,the value of $100 compounded for five years at 5 percent per year is $127.63.You could also find the FV by putting the cursor on Cell G4, then clicking thefunction wizard, then Financial, then scrolling down to FV, and then clicking

OK to bring up the FV dialog box Then enter B1 or 0.05 for Rate, G2 or 5 forNper, 0 or leave blank for Pmt because there are no periodic payments, B3 or

100 for PV, and 0 or leave blank for Type to indicate that payments occur at theend of the period Then, when you click OK, you get the future value, $127.63.Note that the dialog box prompts you to fill in the arguments in an equation

The equation itself, in Excel format, is FV(Rate,Nper,Pmt,PV,Type) FV(0.05,5,0,
100,0) Rather than insert numbers, you could input cell refer-ences for Rate, Nper, Pmt, and PV Either way, when Excel sees the equation,

it knows to use our Equation 7-1 to fill in the specified arguments, and to posit the result in the cell where the cursor was located when you began the

de-process If someone really knows what they are doing and has memorized the

formula, they can skip both the time line and the function wizard and just sert data into the formula to get the answer But until you become an expert,

in-we recommend that you use time lines to visualize the problem and the tion wizard to complete the formula

func-CO M PA R I N G T H E FO U R PR O C E D U R E S

The first step in solving any time value problem is to understand the verbal scription of the problem well enough to diagram it on a time line Woody Allensaid that 90 percent of success is just showing up With time value problems, 90percent of success is correctly setting up the time line

de-After you diagram the problem on a time line, your next step is to pick an

— numerical, tabular, financial calculator, or spreadsheet? In general, youshould use the easiest approach But which is easiest? The answer depends onthe particular situation

First, we would never recommend the tabular approach — it went out whencalculators were invented some 20 years ago Second, all business studentsshould know Equation 7-1 by heart and should also know how to use a finan-cial calculator So, for simple problems such as finding the future value of a sin-gle payment, it is probably easiest and quickest to use either the numerical ap-proach or a financial calculator

For problems with more than a couple of cash flows, the numerical approach

is usually too time consuming, so here either the calculator or spreadsheet proaches would generally be used Calculators are portable and quick to set up,but if many calculations of the same type must be done, or if you want to seehow changes in an input such as the interest rate affect the future value, thespreadsheet approach is generally more efficient If the problem has many ir-regular cash flows, or if you want to analyze many scenarios with different cashflows, then the spreadsheet approach is definitely the most efficient The im-portant point is that you understand the various approaches well enough tomake a rational choice, given the nature of the problem and the equipment youhave available In any event, you must understand the concepts behind the cal-culations and know how to set up time lines in order to work complex prob-lems This is true for stock and bond valuation, capital budgeting, lease analy-sis, and many other important types of problems

ap-PR O B L E M FO R M AT

To help you understand the various types of time value problems, we generallyuse a standard format First, we state the problem in words Next, we diagramthe problem on a time line Then, beneath the time line, we show the equationthat must be solved Finally, we present four alternative procedures for solvingthe equation to obtain the answer: (1) use a regular calculator to obtain a nu-merical solution, (2) use the tables, (3) use a financial calculator, or (4) use aspreadsheet program For some of the very easy problems, we will not show aspreadsheet solution, and for some difficult problems, we will not show numer-ical or tabular solutions because they are simply too inefficient

To illustrate the format, consider again our five-year, 5 percent example:

Using a regular calculator, raise 1.05 to the 5th power and multiply by $100 toget FV5 $127.63.

4 Spreadsheet Solution

5 We input PMT  0, but if you cleared the calculator before you started, the PMT register would already have been set to 0.

Cell G4 contains the formula for Equation 7-1: ⴝⴚ$B$3*(1ⴙ$B$1)^G2 or

ⴝⴚ$B$3*(1ⴙ.05)^G2 You could also use Excel’s FV function to find the

$127.63, following the procedures described in the previous section

A-3, but we generated the data and then made the graph with a spreadsheetmodel See 07MODEL.xls The higher the rate of interest, the faster the rate ofgrowth The interest rate is, in fact, a growth rate: If a sum is deposited andearns 5 percent interest, then the funds on deposit will grow at a rate of 5 per-cent per period Note also that time value concepts can be applied to anythingthat is growing — sales, population, earnings per share, your future salary, orwhatever

1.0 2.0 3.0 4.0 5.0

Explain what is meant by the following statement: “A dollar in hand today

is worth more than a dollar to be received next year.”

What is compounding? Explain why earning “interest on interest” is called

“compound interest.”

Explain the following equation: FV1 PV  INT

Set up a time line that shows the following situation: (1) Your initial deposit

is $100 (2) The account pays 5 percent interest annually (3) You want toknow how much money you will have at the end of three years

Write out an equation that could be used to solve the preceding problem.What are the five TVM (time value of money) input keys on a financial cal-culator? List them (horizontally) in the proper order

T H E P OW E R O F C O M P O U N D I N T E R E S T

P R E S E N T VA L U E

Suppose you have some extra cash, and you have a chance to buy a low-risk curity that will pay $127.63 at the end of five years Your local bank is currentlyoffering 5 percent interest on five-year certificates of deposit (CDs), and youregard the security as being exactly as safe as a CD The 5 percent rate is de-

se-fined as your opportunity cost rate, or the rate of return you could earn on an

alternative investment of similar risk How much should you be willing to payfor the security?

From the future value example presented in the previous section, we sawthat an initial amount of $100 invested at 5 percent per year would be worth

$127.63 at the end of five years As we will see in a moment, you should beindifferent between $100 today and $127.63 at the end of five years The

$100 is defined as the present value, or PV, of $127.63 due in five years

when the opportunity cost rate is 5 percent If the price of the security wereless than $100, you should buy it, because its price would then be less thanthe $100 you would have to spend on a similar-risk alternative to end up with

$127.63 after five years Conversely, if the security cost more than $100, youshould not buy it, because you would have to invest only $100 in a similar-risk alternative to end up with $127.63 after five years If the price were ex-actly $100, then you should be indifferent — you could either buy the secu-

rity or turn it down Therefore, $100 is defined as the security’s fair, or

equilibrium, value.

In general, the present value of a cash flow due n years in the future is the amount

which, if it were on hand today, would grow to equal the future amount Since $100

would grow to $127.63 in five years at a 5 percent interest rate, $100 is thepresent value of $127.63 due in five years when the opportunity cost rate is 5percent

Opportunity Cost Rate

The rate of return on the best

available alternative investment of

equal risk.

Present Value (PV)

The value today of a future cash

flow or series of cash flows.

Fair (Equilibrium) Value

The price at which investors are

indifferent between buying or

selling a security.

You are 21 years old and have just graduated from college After

reading the introduction to this chapter, you decide to start

investing in the stock market for your retirement Your goal is to

have $1 million when you retire at age 65 Assuming you earn a

10 percent annual rate on your stock investments, how much must

you invest at the end ofeach year in order to reach your goal?

The answer is $1,532.24, but this amount depends critically

on the return earned on your investments If returns drop to 8

percent, your required annual contributions would rise to

$2,801.52, while if returns rise to 12 percent, you would only

need to put away $825.21 per year.

What if you are like most of us and wait until later to worry

about retirement? If you wait until age 40, you will need to

save $10,168 per year to reach your $1 million goal, assuming you earn 10 percent, and $13,679 per year if you earn only 8 percent If you wait until age 50 and then earn 8 percent, the required amount will be $36,830 per year.

While $1 million may seem like a lot of money, it won’t be when you get ready to retire If inflation averages 5 percent a year over the next 44 years, your $1 million nest egg will be worth only $116,861 in today’s dollars At an 8 percent rate of return, and assuming you live for 20 years after retirement, your annual retirement income in today’s dollars would be only

$11,903 before taxes So, after celebrating graduation and your new job, start saving!

Finding present values is called discounting, and it is simply the reverse of

compounding — if you know the PV, you can compound to find the FV, while

if you know the FV, you can discount to find the PV When discounting, youwould follow these steps:

The term in parentheses in Equation 7-2 is called the Present Value Interest

Factor for i and n, or PVIF i,n , and Table A-1 in Appendix A contains present

value interest factors for selected values of i and n The value of PVIFi,n for

i  5% and n  5 is 0.7835, so the present value of $127.63 to be received afterfive years when the appropriate interest rate is 5 percent is $100:

PV  $127.63(PVIF5%,5)  $127.63(0.7835)  $100

3 Financial Calculator Solution

Enter N 5, I  5, PMT  0, and FV  127.63, and then press PV to get

PV
100 This is the easy way!

Present Value Interest Factor

for i and n (PVIFi,n)

The present value of $1 due n

periods in the future discounted at

i percent per period.

Discounting

The process of finding the present

value of a cash flow or a series of

cash flows; discounting is the

reverse of compounding.

4 Spreadsheet Solution

You could enter the spreadsheet version of Equation 7-2 in Cell B4,

ⴝ127.63/(1ⴙ0.05)^5, but you could also use the built-in spreadsheet PV

func-tion In Excel, you would put the cursor on Cell B4, then click the function

wiz-ard, indicate that you want a Financial function, scroll down, and double click

PV Then, in the dialog box, enter B1 or 0.05 for Rate, G2 or 5 for Nper, 0 forPmt (because there are no annual payments), G3 or 127.63 for FV, and 0 (orleave blank) for Type because the cash flow occurs at the end of the year Then,press OK to get the answer, PV  $100.00

GR A P H I C VI E W O F T H E DI S C O U N T I N G PR O C E S S

Figure 7-2 shows how the present value of $1 (or any other sum) to be received

in the future diminishes as the years to receipt and the interest rate increase.Again, the data used to plot the curves could be obtained with a calculator,but we used a spreadsheet to calculate the data and make the graph See07MODEL.xls The graph shows (1) that the present value of a sum to be re-ceived at some future date decreases and approaches zero as the payment date

is extended further into the future, and (2) that the rate of decrease is greaterthe higher the interest (discount) rate At relatively high interest rates, fundsdue in the future are worth very little today, and even at a relatively low dis-count rate, the present value of a sum due in the very distant future is quitesmall For example, at a 20 percent discount rate, $1 million due in 100 years isworth approximately 1 cent today (However, 1 cent would grow to almost $1million in 100 years at 20 percent.)

S E L F - T E S T Q U E S T I O N S

What is meant by the term “opportunity cost rate”?

What is discounting? How is it related to compounding?

How does the present value of an amount to be received in the futurechange as the time is extended and as the interest rate increases?

S O L V I N G F O R I N T E R E S T R A T E A N D T I M E

At this point, you should realize that compounding and discounting are related,and that we have been dealing with one equation that can be solved for eitherthe FV or the PV

SO LV I N G F O R i

Suppose you can buy a security at a price of $78.35, and it will pay you $100after five years Here you know PV, FV, and n, and you want to find i, the in-terest rate you would earn if you bought the security Problems such as this aresolved as follows:

2 0

0.25 0.50 0.75 1.00

5 percent The trial-and-error procedure is extremely tedious and inefficientfor most time value problems, so no one in the real world uses it.

2 Tabular Solution

FVn PV(1  i)n PV(FVIFi,n)

$100  $78.35(FVIFi,5)FVIFi,5 $100/$78.35  1.2763

Find the value of the FVIF as shown above, and then look across the Period 5row in Table A-3 until you find FVIF  1.2763 This value is in the 5% col-umn, so the interest rate at which $78.35 grows to $100 over five years is 5 per-cent This procedure can be used only if the interest rate is in the table; there-fore, it will not work for fractional interest rates or where n is not a wholenumber Approximation procedures can be used, but they are laborious and in-exact

3 Financial Calculator Solution

Enter N  5, PV 
78.35, PMT  0, and FV  100, and then press I to get

I  5% This procedure is easy, and it can be used for any interest rate or forany value of n, including fractional values

Most spreadsheets have a built-in function to find the interest rate In Excel,

you would put the cursor on Cell B3, then click the function wizard, indicatethat you want a Financial function, scroll down to Rate, and click OK Then,

in the dialog box, enter G1 or 5 Nper, 0 for Pmt because there are no periodicpayments, B2 or
78.35 for PV, G2 or 100 for FV, 0 for type, and leave

“Guess” blank to let Excel decide where to start its iterations Then, when you click OK, Excel solves for the interest rate, 5.00 percent Excel also has other

procedures that could be used to find the 5 percent, but for this problem theRate function is easiest to apply

SO LV I N G F O R n

Suppose you know that a security will provide a return of 5 percent per year,that it will cost $78.35, and that you will receive $100 at maturity, but you donot know when the security matures Thus, you know PV, FV, and i, but you

do not know n, the number of periods Here is the situation:

Now look down the 5% column in Table A-3 until you find FVIF  1.2763.This value is in Row 5, which indicates that it takes five years for $78.35 togrow to $100 at a 5 percent interest rate

3 Financial Calculator Solution

An annuity is a series of equal payments made at fixed intervals for a specified

number of periods For example, $100 at the end of each of the next three years

is a three-year annuity The payments are given the symbol PMT, and they canoccur at either the beginning or the end of each period If the payments occur

at the end of each period, as they typically do, the annuity is called an ordinary,

or deferred, annuity Payments on mortgages, car loans, and student loans are

typically set up as ordinary annuities If payments are made at the beginning of

each period, the annuity is an annuity due Rental payments for an apartment,

life insurance premiums, and lottery payoffs are typically set up as annuitiesdue Since ordinary annuities are more common in finance, when the term “an-nuity” is used in this book, you should assume that the payments occur at theend of each period unless otherwise noted

OR D I N A R Y AN N U I T I E S

An ordinary, or deferred, annuity consists of a series of equal payments made at

the end of each period If you deposit $100 at the end of each year for three

years in a savings account that pays 5 percent interest per year, how much willyou have at the end of three years? To answer this question, we must find the

future value of the annuity, FVA n Each payment is compounded out to the end

of Period n, and the sum of the compounded payments is the future value ofthe annuity, FVAn

Time Line:

105 110.25

Annuity

A series of payments of an equal

amount at fixed intervals for a

specified number of periods.

Ordinary (Deferred) Annuity

An annuity whose payments occur

at the end of each period.

Annuity Due

An annuity whose payments occur

at the beginning of each period.

FVA n

The future value of an annuity

over n periods.

Here we show the regular time line as the top portion of the diagram, but wealso show how each cash flow is compounded to produce the value FVAnin thelower portion of the diagram

be for n
1 periods rather than n periods Compounding for the second nuity payment would be for Period 3 through Period n, or n
2 periods, and

an-so on The last annuity payment is made at the end of the annuity’s life, an-so there

is no time for interest to be earned The second form of Equation 7-3 is just ashorthand version of the first Finally, the third line shows the payment multi-

plied by the Future Value Interest Factor for an Annuity (FVIFA i,n ), which

is the tabular approach

 PMT an

t 1

(1 i)n
t

F U T U R E VA L U E O F A N A N N U I T Y

Future Value Interest Factor

for an Annuity (FVIFAi,n)

The future value interest factor

for an annuity of n periods

compounded at i percent.

6 Another form for Equation 7-3a is as follows:

This form is found by applying the algebra of geometric progressions This equation is useful in situations when the required values of i and n are not in the tables and no financial calculator or computer is available.

FVIFA i,n (1 i)

n
1

three-year, $100 annuity problem, first refer to Table A-4 and look down the5% column to the third period; the FVIFA is 3.1525 Thus, the future value ofthe $100 annuity is $315.25:

FVAn PMT(FVIFAi,n)FVA3 $100(FVIFA5%,3)  $100(3.1525)  $315.25

3 Financial Calculator Solution

Note that in annuity problems, the PMT key is used in conjunction with the Nand I keys, plus either the PV or the FV key, depending on whether you aretrying to find the PV or the FV of the annuity In our example, you want the

FV, so press the FV key to get the answer, $315.25 Since there is no initial ment, we input PV  0

pay-4 Spreadsheet Solution

Most spreadsheets have a built-in function to find the future value of an

annu-ity In Excel, we could put the cursor on Cell E4, then click function wizard,

Financial, FV, and OK to get the FV dialog box Then, we would enter 0.05

or B1 for Rate, 3 or E2 for Nper, and
100 for Pmt (Like the financial lator approach, the payment is entered as a negative number to show that it is

calcu-a ccalcu-ash outflow.) We would lecalcu-ave PV blcalcu-ank beccalcu-ause there is no initicalcu-al pcalcu-ayment,and we would leave Type blank to signify that payments come at the end ofthe periods Then, when we clicked OK, we would get the FV of the annuity,

$315.25 Note that it isn’t necessary to show the time line, since the FV tion doesn’t require you to input a range of cash flows Still, the time line isuseful to help visualize the problem

func-AN N U I T I E S DU E

Had the three $100 payments in the previous example been made at the

begin-ning of each year, the annuity would have been an annuity due On the time line,

each payment would be shifted to the left one year; therefore, each paymentwould be compounded for one extra year

1 Time Line and Numerical Solution

Again, the time line is shown at the top of the diagram, and the values as culated with a regular calculator are shown under Year 3 The future value ofeach cash flow is found, and those FVs are summed to find the FV of the an-nuity due The payments occur earlier, so more interest is earned Therefore,the future value of the annuity due is larger — $331.01 versus $315.25 for theordinary annuity

cal-2 Tabular Solution

In an annuity due, each payment is compounded for one additional period, sothe future value of the entire annuity is equal to the future value of an ordinaryannuity compounded for one additional period Here is the tabular solution:

FVAn(Annuity due)  PMT(FVIFAi,n)(1  i) (7-3b)

 $100(3.1525)(1.05)  $331.01

3 Financial Calculator Solution

Most financial calculators have a switch, or key, marked “DUE” or “BEG” thatpermits you to switch from end-of-period payments (ordinary annuity) tobeginning-of-period payments (annuity due) When the beginning mode is ac-tivated, the display will normally show the word “BEGIN.” Thus, to deal withannuities due, switch your calculator to “BEGIN” and proceed as before:

BEGIN

Enter N  3, I  5, PV  0, PMT 
100, and then press FV to get the

an-swer, $331.01 Since most problems specify end-of-period cash flows, you should always

switch your calculator back to “END” mode after you work an annuity due problem.

4 Spreadsheet Solution

For the annuity due, proceed just as for the ordinary annuity except enter 1 forType to indicate that we now have an annuity due Then, when you click OK,the answer $331.01 will appear

105 110.25 115.76

331.01

FVA3 (Annuity due) 

F U T U R E VA L U E O F A N A N N U I T Y

P R E S E N T VA L U E O F A N A N N U I T Y

Suppose you were offered the following alternatives: (1) a three-year annuitywith payments of $100 or (2) a lump sum payment today You have no need forthe money during the next three years, so if you accept the annuity, you woulddeposit the payments in a bank account that pays 5 percent interest per year.Similarly, the lump sum payment would be deposited into a bank account Howlarge must the lump sum payment today be to make it equivalent to the annuity?

OR D I N A R Y AN N U I T I E S

If the payments come at the end of each year, then the annuity is an ordinaryannuity, and it would be set up as follows:

Time Line:

The regular time line is shown at the top of the diagram, and the numerical

so-lution values are shown in the left column The PV of the annuity, PVA n, is

Other things held constant, which annuity has the greater future value: an

ordinary annuity or an annuity due? Why?

Explain how financial calculators can be used to solve future value of ity problems

1  ib

t



1
1(1  i) n

i 1i
i(1 i)1 n

(footnote continues)

(7-4)

1 Numerical Solution

The present value of each cash flow is found and then summed to find the PV

of the annuity This procedure is shown in the lower section of the time line agram, where we see that the PV of the annuity is $272.32

di-2 Tabular Solution

The summation term in Equation 7-4 is called the Present Value Interest

Factor for an Annuity (PVIFA i,n ), and values for the term at different values

of i and n are shown in Table A-2 at the back of the book Here is the tion:

To find the answer to the three-year, $100 annuity problem, simply refer toTable A-2 and look down the 5% column to the third period The PVIFA is2.7232, so the present value of the $100 annuity is $272.32:

PVAn PMT(PVIFAi,n)PVA3 $100(PVIFA5%,3)  $100(2.7232)  $272.32

3 Financial Calculator Solution

P R E S E N T VA L U E O F A N A N N U I T Y

(Footnote 7 continued)

This form of the equation is useful for dealing with annuities when the values for i and n are not

in the tables and no financial calculator or computer is available.

Present Value Interest Factor

for an Annuity (PVIFAi,n)

The present value interest factor

for an annuity of n periods

In Excel, put the cursor on Cell B4 and then click the function wizard,

Finan-cial, PV, and OK Then enter B1 or 0.05 for Rate, E2 or 3 for Nper,
100 forPmt, 0 or leave blank for FV, and 0 or leave blank for Type Then, when youclick OK, you get the answer, $272.32

One especially important application of the annuity concept relates to loanswith constant payments, such as mortgages and auto loans With such loans,

called amortized loans, the amount borrowed is the present value of an ordinary

annuity, and the payments constitute the annuity stream We will examine stant payment loans in more depth in a later section of this chapter

con-AN N U I T I E S DU E

Had the three $100 payments in the preceding example been made at the

be-ginning of each year, the annuity would have been an annuity due Each

pay-ment would be shifted to the left one year, so each paypay-ment would be counted for one less year Here is the time line setup:

dis-1 Time Line and Numerical Solution

Again, we find the PV of each cash flow and then sum these PVs to find the PV

of the annuity due This procedure is illustrated in the lower section of the timeline diagram Since the cash flows occur sooner, the PV of the annuity due ex-ceeds that of the ordinary annuity, $285.94 versus $272.32

2 Tabular Solution

In an annuity due, each payment is discounted for one less period Since itspayments come in faster, an annuity due is more valuable than an ordinary an-nuity This higher value is found by multiplying the PV of an ordinary annuity

285.94

PVA3 (Annuity due) 

Switch to the beginning-of-period mode, and then enter N 3, I  5, PMT 

100, and FV  0, and then press PV to get the answer, $285.94 Again, since

most problems deal with end-of-period cash flows, don’t forget to switch your calculator back to the “END” mode.

definitely, or perpetually, and these are called perpetuities The present value

of a perpetuity is found by applying Equation 7-5.8

(7-5)

Perpetuities can be illustrated by some British securities issued after theNapoleonic Wars In 1815, the British government sold a huge bond issue andused the proceeds to pay off many smaller issues that had been floated in prioryears to pay for the wars Since the purpose of the bonds was to consolidate

past debts, the bonds were called consols Suppose each consol promised to

pay $100 per year in perpetuity (Actually, interest was stated in pounds.) Whatwould each bond be worth if the opportunity cost rate, or discount rate, was 5percent? The answer is $2,000:

PV (Perpetuity) $1000.05  $2,000 if i  5%

PV(Perpetuity) Interest ratePayment  PMTi

8 The derivation of Equation 7-5 is given in the Web/CD Extension to Chapter 5 of Eugene F.

Brigham and Phillip R Daves, Intermediate Financial Management, 7th ed (Fort Worth, TX:

Har-court College Publishers, 2002).

Perpetuity

A stream of equal payments

expected to continue forever.

Consol

A perpetual bond issued by the

British government to consolidate

past debts; in general, any

perpetual bond.

Suppose the interest rate rose to 10 percent; what would happen to the consol’svalue? The value would drop to $1,000:

We see that the value of a perpetuity changes dramatically when interestrates change Perpetuities are discussed further in Chapter 9

The definition of an annuity includes the words constant payment — in other

words, annuities involve payments that are equal in every period Althoughmany financial decisions do involve constant payments, other important deci-sions involve uneven, or nonconstant, cash flows; for example, common stockstypically pay an increasing stream of dividends over time, and fixed asset in-vestments such as new equipment normally do not generate constant cashflows Consequently, it is necessary to extend our time value discussion to in-

clude uneven cash flow streams.

Throughout the book, we will follow convention and reserve the term

pay-ment (PMT) for annuity situations where the cash flows are equal amounts,

and we will use the term cash flow (CF) to denote uneven cash flows

Finan-cial calculators are set up to follow this convention, so if you are dealing withuneven cash flows, you will need to use the “cash flow register.”

PR E S E N T VA L U E O F A N UN E V E N CA S H FL O W ST R E A M

The PV of an uneven cash flow stream is found as the sum of the PVs of theindividual cash flows of the stream For example, suppose we must find the PV

of the following cash flow stream, discounted at 6 percent:

The PV will be found by applying this general present value equation:

3 200

4 200

5 200

6 0

7 1,000

PV  ?

Uneven Cash Flow Stream

A series of cash flows in which the

amount varies from one period to

the next.

Payment (PMT)

This term designates equal cash

flows coming at regular intervals.

Cash Flow (CF)

This term designates uneven cash

flows.

We could find the PV of each individual cash flow using the numerical, tabular,financial calculator, or spreadsheet methods, and then sum these values to findthe present value of the stream Here is what the process would look like:

All we did was to apply Equation 7-6, show the individual PVs in the left umn of the diagram, and then sum these individual PVs to find the PV of theentire stream

col-The present value of a cash flow stream can always be found by summing thepresent values of the individual cash flows as shown above However, cash flowregularities within the stream may allow the use of shortcuts For example, no-tice that the cash flows in periods 2 through 5 represent an annuity We can usethat fact to solve the problem in a slightly different manner:

Cash flows during Years 2 to 5 represent an ordinary annuity, and we find its

PV at Year 1 (one period before the first payment) This PV ($693.02) mustthen be discounted back one more period to get its Year 0 value, $653.79.Problems involving uneven cash flows can be solved in one step with mostfinancial calculators First, you input the individual cash flows, in chronologicalorder, into the cash flow register Cash flows are usually designated CF0, CF1,

CF2, CF3, and so on Next, you enter the interest rate, I At this point, you havesubstituted in all the known values of Equation 7-6, so you only need to pressthe NPV key to find the present value of the stream The calculator has beenprogrammed to find the PV of each cash flow and then to sum these values tofind the PV of the entire stream To input the cash flows for this problem, enter

0 (because CF0  0), 100, 200, 200, 200, 200, 0, 1000 in that order into thecash flow register, enter I  6, and then press NPV to obtain the answer,

$1,413.19

Two points should be noted First, when dealing with the cash flow register,the calculator uses the term “NPV” rather than “PV.” The N stands for “net,” soNPV is the abbreviation for “Net Present Value,” which is simply the net present

100

2 200

3 200

4 200

5 200

6 0

7 1,000 94.34

3 200

4 200

5 200

6 0

7 1,000 94.34

value of a series of positive and negative cash flows Our example has no negativecash flows, but if it did, we would simply input them with negative signs.9The second point to note is that annuities can be entered into the cash flowregister more efficiently by using the Nj key (On some calculators, you areprompted to enter the number of times the cash flow occurs, and on still othercalculators, the procedures for inputting data, as we discuss next, may be dif-

ferent You should consult your calculator manual or our Technology Supplement

to determine the appropriate steps for your specific calculator.) In this tion, you would enter CF0 0, CF1 100, CF2 200, Nj 4 (which tells thecalculator that the 200 occurs 4 times), CF6 0, and CF7 1000 Then enter

illustra-I  6 and press the NPV key, and 1,413.19 will appear in the display Also, notethat amounts entered into the cash flow register remain in the register untilthey are cleared Thus, if you had previously worked a problem with eight cashflows, and then moved to a problem with only four cash flows, the calculatorwould simply add the cash flows from the second problem to those of the firstproblem Therefore, you must be sure to clear the cash flow register beforestarting a new problem

Spreadsheets are especially useful for solving problems with uneven cashflows Just as with a financial calculator, you must enter the cash flows in thespreadsheet:

9 To input a negative number, type in the positive number, then press the /
key to change the sign to negative If you begin by typing the minus sign, you make the mistake of subtracting the negative number from the last number that was entered in the calculator.

To find the PV of these cash flows with Excel, put the cursor on Cell B4,

click the function wizard, click Financial, scroll down to NPV, and click OK

to get the dialog box Then enter B1 or 0.06 for Rate and the range of cellscontaining the cash flows, C3:I3, for Value 1 N stands for Net, so the NPV isthe net present value of a stream of cash flows, some of which may be negative.Now, when you click OK, you get the PV of the stream, $1,413.19 Note thatyou use the PV function if the cash flows (or payments) are constant, but theNPV function if they are not constant Note too that one of the advantages ofspreadsheets over financial calculators is that you can see the cash flows, whichmakes it easy to spot any typing errors