19878 28w p001- · pdf file28w-2 • web chapter 28 time value of money the thomsonnow web...

28C H A P T E R

28W-1

Time Value of Money1

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In Chapter 1, we saw that the primary objective of financial management is to maximize thevalue of the firm’s stock. We also saw that stock values depend in part on the timing of thecash flows investors expect to receive from an investment—a dollar expected soon is worthmore than a dollar expected in the distant future. Therefore, it is essential for financial man-agers to have a clear understanding of the time value of money and its impact on stockprices. These concepts are discussed in this chapter, where we show how the timing of cashflows affects asset values and rates of return.

The principles of time value analysis have many applications, ranging from setting upschedules for paying off loans to decisions about whether to acquire new equipment. In fact,of all the concepts used in finance, none is more important than the time value of money,which is also called discounted cash flow (DCF) analysis. Since this concept is used throughoutthe remainder of the book, it is vital that you understand the material in this chapter beforeyou move on to other topics.

1This Web chapter was written on the assumption that most students will have a financial calculator or personal com-puter. Calculators are relatively inexpensive, and students who cannot use them run the risk of being deemed obsoleteand uncompetitive before they even graduate. Therefore, the chapter has been written to include a discussion of financialcalculator solutions along with computer solutions using Excel.

Note also that tutorials on how to use both Excel and several Hewlett-Packard, Texas Instruments, and Sharp calcu-lators are provided in the Technology Supplement to this book, which is available to adopting instructors.

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28W-2 • Web Chapter 28 Time Value of Money

The ThomsonNOW Web site

contains an Excel file that

will guide you through the

chapter’s calculations. The

file for this chapter is IFM9

Ch28 Tool Kit.xls, and we

encourage you to open the

file and follow along as you

read the chapter.

TIME LINES

One of the most important tools in time value analysis is the time line, which isused by analysts to help visualize what is happening in a particular problem andthen to help set up the problem for solution. To illustrate the time line concept,consider the following diagram:

Time: 0 21 3 4 5

Time 0 is today; Time 1 is one period from today, or the end of Period 1; Time 2is two periods from today, or the end of Period 2; and so on. Thus, the numbersabove the tick marks represent end-of-period values. Often the periods are years,but other time intervals such as semiannual periods, quarters, months, or evendays can be used. If each period on the time line represents a year, the intervalfrom the tick mark corresponding to 0 to the tick mark corresponding to 1 wouldbe Year 1, the interval from 1 to 2 would be Year 2, and so on. Note that eachtick mark corresponds to the end of one period as well as the beginning of thenext 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 just passed.

Cash flows are placed directly below the tick marks, and interest rates areshown directly above the time line. Unknown cash flows, which you are trying tofind in the analysis, are indicated by question marks. Now consider the followingtime line:

0 21 35%

Cash flows:

Time:

–100 ?

0 215%

–100 ?

10%

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 anunknown inflow. Since the initial $100 is an outflow (an investment), it has aminus sign. Since the Period 3 amount is an inflow, it does not have a minus sign,which implies a plus sign. Note that no cash flows occur at Times 1 and 2. Notealso that we generally do not show dollar signs on time lines to reduce clutter.

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 percentduring the second period. If the interest rate is constant in all periods, we show itonly in the first period, but if it changes, we show all the relevant rates on thetime line.

Time lines are essential when you are first learning time value concepts, buteven experts use time lines to analyze complex problems. We use time linesthroughout the book, and you should get into the habit of using them when youwork problems.

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Web Chapter 28 Time Value of Money • 28W-3

Draw a three-year time line to illustrate the following situation: (1) An outflow of $10,000occurs at Time 0. (2) Inflows of $5,000 then occur at the ends of Years 1, 2, and 3. (3)The interest rate during all three years is 10 percent.

FUTURE VALUE

A dollar in hand today is worth more than a dollar to be received in the futurebecause, if you have it now, you can invest it, earn interest, and end up with morethan one dollar in the future. The process of going from today’s values, or presentvalues (PVs), to future values (FVs) is called compounding. To illustrate, suppose

Self-Test Question

In Chapter 1, we told you that managersshould strive to make their firms more valu-able, and that the value of a firm is deter-mined by the size, timing, and risk of its freecash flows (FCF). Recall that free cash flowsare the cash flows available for distribution toall of a firm’s investors (stockholders and cred-itors) and that the weighted average cost of

capital is the average rate of return requiredby all of the firm’s investors. We showed youa formula, highlighted below, for calculatingvalue. The formula takes future cash flowsand adjusts them to show how much thosefuture risky cash flows are worth today. Itrelies on time value of money concepts, whichwe explain in this chapter.

C O R P O R A T E V A L U A T I O N A N D T H E T I M E V A L U E O F M O N E Y

Free Cash Flows (FCF)

Weighted AverageCost of Capital (WACC)

Value of the Firm

SalesRevenues

OperatingCosts and

Taxes

Required NewInvestments

in Operations

FinanceDecisions

InterestRates

FirmRisk

MarketRisk

FCF1

(1 � WACC)1Value �

FCF2

(1 � WACC)2�

FCF3

(1 � WACC)3� � � � � �

FCF∞(1 � WACC)∞

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you deposit $100 in a bank that pays 5 percent interest each year. How muchwould you have at the end of one year? To begin, we define the following terms:

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 earnedis based on the balance at the beginning of each year, and we assumethat it is paid at the end of the year. Here i � 5%, or, expressed as adecimal, i � 0.05. Throughout this chapter, we designate the interestrate as i (or I) because that symbol is used on most financial calculators.Note, though, that elsewhere in the text we use the symbol r to denoteinterest rates because r 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, FVn is the value nyears into the future, after the interest earned has been added to theaccount.

n � number of periods involved in the analysis. Here n � 1.

In our example, n � 1, so FVn can be calculated as follows:

FVn � FV1 � PV � INT� PV � PV(i)� PV(1 � i)� $100(1 � 0.05) � $100(1.05) � $105

Thus, the future value (FV) at the end of one year, FV1, equals the present valuemultiplied by 1 plus the interest rate, so you will have $105 after one year.

What would you end up with if you left your $100 in the account for fiveyears? Here is a time line set up to show the amount at the end of each year:

0 21 3 4 55%

FV1 = ? FV5 = ?FV4 = ?FV3 = ?FV2 = ?–100

5.00 5.25 5.51 5.79 6.08

105.00 110.25 115.76 121.55 127.63

Initial deposit:

Interest earned:

Amount at the end ofeach period = FVn:

Note the following points: (1) You start by depositing $100 in the account—this isshown as an outflow at t � 0. (2) You earn $100(0.05) � $5 of interest duringthe 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 now larger amount,and end the second year with $110.25. Your interest during Year 2, $5.25, ishigher than the first year’s interest, $5, because you earned $5(0.05) � $0.25interest on the first year’s interest. (4) This process continues, and because thebeginning balance is higher in each succeeding year, the annual interest earnedincreases. (5) The total interest earned, $27.63, is reflected in the final balance att � 5, $127.63.

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

FV2 � FV1(1 � i)� PV(1 � i)(1 � i)� PV(1 � i)2

� $100(1.05)2 � $110.25

Continuing, the balance at the end of Year 3 is

FV3 � FV2(1 � i)� PV(1 � i)2(1 � i)� PV(1 � i)3

� $100(1.05)3 � $115.76

and

FV5 � $100(1.05)5 � $127.63

In general, the future value of an initial lump sum at the end of n years can befound by applying Equation 28-1:

FVn � PV(1 � i)n � PV(FVIFi,n) | 28-1 |

The last term in Equation 28-1 defines the future value interest factor for i and n,FVIFi,n, as (1 � i)n. This provides a shorthand way to refer to the actual formulain Equation 28-1.

Equation 28-1 and most other time value of money equations can be solved inthree ways: numerically with a regular calculator, with a financial calculator, orwith a computer spreadsheet program. Most work in financial management willbe done with a financial calculator or on a computer, but when learning basic con-cepts it is best to also work the problem numerically with a regular calculator.

NUMERICAL SOLUTION

We can use a regular calculator and either multiply (1 � i) by itself n � 1 times orelse use the exponential function to raise (1 � i) to the nth power. With most cal-culators, you would enter 1 � i � 1.05 and multiply it by itself four times, or elseenter 1.05, then press the yx (exponential) function key, and then enter 5. In eithercase, your answer would be 1.2763 (if you set your calculator to display four dec-imal places), which you would multiply by $100 to get the final answer,$127.6282, which would be rounded to $127.63.

In certain problems, it is extremely difficult to arrive at a solution using a regularcalculator. We will tell you this when we have such a problem, and in these cases wewill not show a numerical solution. Also, at times we show the numerical solutionjust below the time line, as a part of the diagram, rather than in a separate section.

FINANCIAL CALCULATOR SOLUTION

A version of Equation 28-1, along with a number of other equations, has beenprogrammed directly into financial calculators, and these calculators can be used

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2Here we assume that compounding occurs once each year. Most calculators have a setting that can be used to designatethe number of compounding periods per year. For example, the HP-10B comes preset with payments at 12 per year. Youwould 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 thegold key, and then pressing the P/YR key.

to find future values. Note that calculators have five keys that correspond to thefive most commonly used time value of money variables:

FVFVPMTPMTPVPVIINN

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 ofequal, 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 ofthe calculator, while on others they are shown on a screen after going into thetime value of money (TVM) menu. To find the future value of $100 after fiveyears at 5 percent, most financial calculators solve a version of Equation 28-2:

PV(1 � i)n � FVn � 0 | 28-2 |

The equation has four variables, FVn, PV, i, and n. If we know any three, we cansolve for the fourth. In our example, we enter N � 5, I � 5, PV � �100, andPMT � 0. Then, when we press the FV key, we get the answer, FV � 127.63(rounded to two decimal places).2

Notice that either PV or FVn in Equation 28-2 must be negative to make theequation true, assuming nonnegative interest rates. Thus, most financial calculatorsrequire that all cash flows be designated as either inflows or outflows, with out-flows being entered as negative numbers. In our illustration, you deposit, or put in,the initial amount (which is an outflow to you) and you take out, or receive, theending amount (which is an inflow to you). Therefore, you enter the PV as �100.Enter the �100 by keying in 100 and then pressing the “change sign” or �/� key.(If you entered 100, then the FV would appear as �127.63.) Also, on some calcu-lators you are required to press a “Compute” key before pressing the FV key.

Sometimes the convention of changing signs can be confusing. For example, ifyou have $100 in the bank now and want to find out how much you will have

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after five years if your account pays 5 percent interest, the calculator will give youa negative answer, in this case �127.63, because the calculator assumes you aregoing to withdraw the funds. This sign convention should cause you no problem ifyou think about what you are doing.

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

SPREADSHEET SOLUTION

Spreadsheet programs are ideally suited for solving many financial problems,including time value of money problems.3 With very little effort, the spreadsheetitself becomes a time line. Here is how the problem would look in a spreadsheet:

3In this section, and in other sections and chapters, we discuss spreadsheet solutions to various financial problems. Tohelp you with these calculations, we have placed a file, IFM9 Ch28 Tool Kit.xls, on the ThomsonNOW Web site thatdoes the various calculations using Excel. We highly recommend that you go through the models. This will give you prac-tice with Excel, which will help tremendously in later courses, in the job market, and in the workplace. Also, goingthrough the models will enhance your understanding of financial concepts.

See IFM9 Ch28 Tool Kit.xls.

4

3

2

1

FECB GDA

Interest rate

Cash flow

Time

Future value

0.05

–100

0 5

127.63

4

121.55

3

115.76

2

110.25

1

105.00

Cell B1 shows the interest rate, entered as a decimal number, 0.05. Row 2 showsthe periods for the time line. With Microsoft Excel, you could enter 0 in Cell B2,then the formula ��B2��1 in Cell C2, and then copy this formula into Cells D2through G2 to produce the time periods shown in Row 2. Note that if your timeline had many years, say, 50, you would simply copy the formula across morecolumns.

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. CellC4 contains the formula for Equation 28-1. The formula could be written as ��$B$3*(1��.05)^C2, but we wrote it as ��$B$3*(1��$B$1)^C2, which gives usthe flexibility to change the interest rate in Cell B1 to see how the future valuechanges with changes in interest rates. Note that the formula has a minus sign forthe PV (which is in Cell B3) to account for the minus sign of the cash flow. This for-mula 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 OKto bring up the FV dialog box. Then enter B1 or .05 for Rate, G2 or 5 for Nper, 0or leave blank for Pmt because there are no periodic payments, B3 or �100 for

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Pv, and 0 or leave blank for Type to indicate that payments occur at the end of theperiod. 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(.05,5,0, � 100,0). Rather than insert numbers, you could input cell references for Rate,Nper, Pmt, and Pv. Either way, when Excel sees the equation, it knows to use ourEquation 28-2 to fill in the specified arguments, and to deposit the result in the cellwhere the cursor was located when you began the process. If someone really knowswhat they are doing and has memorized the formula, they can skip both the timeline and the function wizard and just insert data into the formula to get the answer.But until you become an expert, we recommend that you use time lines to visualizethe problem and the function wizard to complete the formula.

Comparing the Three ProceduresThe first step in solving any time value problem is to understand the verbaldescription 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,90 percent of success is correctly setting up the time line.

After you diagram the problem on a time line, your next step is to pick anapproach to solve the problem. Which of the three approaches should you use—numerical, financial calculator, or spreadsheet? In general, you should use the easi-est approach. But which is easiest? The answer depends on the particular situation.

All business students should know Equation 28-1 by heart and should alsoknow how to use a financial calculator. So, for simple problems such as finding

Suppose you are 26 years old and justreceived your MBA. After reading the intro-duction to this chapter, you decide to startinvesting in the stock market for your retire-ment. Your goal is to have $1 million whenyou retire at age 65. Assuming you earn a10 percent annual rate on your stock invest-ments, how much must you invest at the endof each year in order to reach your goal?

The answer is $2,490.98, but thisamount depends critically on the returnearned on your investments. If returns dropto 8 percent, your required annual contribu-tions would rise to $4,185.13, while ifreturns rise to 12 percent, you would needto put away only $1,461.97 per year.

What if you are like most of us and waituntil later to worry about retirement? If you

wait until age 40, you will need to save$10,168 per year to reach your $1 milliongoal, 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 per-cent, the required amount will be $36,830per year.

While $1 million may seem like a lot ofmoney, it won’t be when you get ready toretire. If inflation averages 5 percent a yearover the next 39 years, your $1 million nestegg will be worth only $116,861 in today’sdollars. At an 8 percent rate of return, andassuming you live for 20 years after retire-ment, your annual retirement income intoday’s dollars would be only $11,903before taxes. So, after celebrating gradua-tion and your new job, start saving!

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

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the future value of a single payment, it is probably easiest and quickest to useeither the numerical approach or a financial calculator.

For problems with more than a couple of cash flows, the numerical approachis usually too time consuming, so here either the calculator or spreadsheetapproaches 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 see howchanges in an input such as the interest rate affect the future value, the spread-sheet approach is generally more efficient. If the problem has many irregular cashflows, or if you want to analyze many scenarios with different cash flows, then thespreadsheet approach is definitely the most efficient. The important thing is thatyou understand the various approaches well enough to make a rational choice,given the nature of the problem and the equipment you have available. In anyevent, you must understand the concepts behind the calculations and know howto set up time lines in order to work complex problems. This is true for stock andbond valuation, capital budgeting, lease analysis, and many other important finan-cial problems.

Graphic View of the Compounding Process: GrowthFigure 28-1 shows how $1 (or any other lump sum) grows over time at variousinterest rates. We generated the data and then made the graph with a spreadsheetmodel in the file IFM9 Ch28 Tool Kit.xls. The higher the rate of interest, the fasterthe rate of growth. The interest rate is, in fact, a growth rate: If a sum is depositedand earns 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 anything thatis growing—sales, population, earnings per share, or your future salary.

1.0

2.0

3.0

4.0

5.0Future Value of $1

Periods

i = 0%

0 642 108

i = 5%

i = 10%

i = 15%

Relationships among Future Value, Growth, Interest Rates, and TimeFigure 28-1

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Explain what is meant by the following statement: “A dollar in hand today is worth morethan a dollar to be received next year.”

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

Explain the following equation: FV1 � PV � INT.You deposit $100 in an account that pays 5 percent interest annually. Write an equation that

shows how much money you will have at the end of three years.What are the five TVM (time value of money) input keys on a financial calculator?

PRESENT VALUE

Suppose you have some extra cash, and you have a chance to buy a low-risk secu-rity 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 you regardthe security as being exactly as safe as a CD. The 5 percent rate is defined as youropportunity cost rate, or the rate of return you could earn on an alternative invest-ment of similar risk. How much should you be willing to pay for the security?

From the future value example presented in the previous section, we saw thatan initial amount of $100 invested at 5 percent per year would be worth $127.63at the end of five years. As we will see in a moment, you should be indifferentbetween $100 today and $127.63 at the end of five years. The $100 is defined asthe present value, or PV, of $127.63 due in five years when the opportunity costrate is 5 percent. If the price of the security were less than $100, you should buyit, because its price would then be less than the $100 you would have to spend ona similar-risk alternative to end up with $127.63 after five years. Conversely, if thesecurity cost more than $100, you should not buy it, because you would have toinvest only $100 in a similar-risk alternative to end up with $127.63 after fiveyears. If the price were exactly $100, then you should be indifferent—you couldeither buy the security or turn it down. Therefore, $100 is defined as the security’sfair, or equilibrium, value.

In general, the present value of a cash flow due n years in the future is theamount 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, $100is the present value of $127.63 due in five years when the opportunity cost rate is5 percent.

Finding present values is called discounting, and it is the reverse of com-pounding—if you know the PV, you can compound to find the FV, while if youknow the FV, you can discount to find the PV. When discounting, you should fol-low these steps:

Time Line:

Equation:

To develop the discounting equation, we begin with the future value equation,Equation 28-1:

0 21 3 4 5

PV = ?

5%

127.63

Self-Test Questions

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FVn � PV(1 � i)n | 28-1 |

Next, we solve it for PV in several equivalent forms:

| 28-3 |

The last form of Equation 28-3 recognizes that the present value interest factor fori and n, PVIFi,n, is shorthand for the formula in parentheses in the second versionof the equation.

1. NUMERICAL SOLUTION

Divide $127.63 by 1.05 five times, or by (1.05)5, to find PV � $100.

2. FINANCIAL CALCULATOR SOLUTION

Enter N � 5, I � 5, PMT � 0, and FV � 127.63, and then press PV to get PV � �100.

3. SPREADSHEET SOLUTION

FVFV127.63

PMTPMT

0

PVPV= –100

II

5

NN5

Output:

Inputs:

0 21 3 4 5

–100

5%

127.63105.00 121.55115.76110.25=÷ 1.05÷ 1.05 ÷ 1.05÷ 1.05÷ 1.05

= = = =

PV �FVn

(1 � i ) n� FVn a 1

1 � ib n

� FVn (PVIFi,n )

4

3

2

1

FECB GDA

Interest rate

Cash flow

Time

Present value

0.05

100

0 5

127.63

4

0

3

0

2

0

1

0

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You could enter the spreadsheet version of Equation 28-3 in Cell B4, ��127.63/(1��0.05)^5, but you could also use the built-in spreadsheet PV function. In Excel,you would put the cursor on Cell B4, then click the function wizard, indicate thatyou want a Financial function, scroll down, and double click PV. Then, in the dia-log box, enter B1 or 0.05 for Rate, G2 or 5 for Nper, 0 for Pmt (because there areno annual payments), G3 or 127.63 for Fv, and 0 (or leave blank) for Typebecause the cash flow occurs at the end of the year. Then, press OK to get theanswer, PV � �$100.00. Note that the PV function returns a negative value, thesame as the financial calculator.

Graphic View of the Discounting ProcessFigure 28-2 shows how the present value of $1 (or any other sum) to be receivedin the future diminishes as the years to receipt and the interest rate increase. Thegraph shows (1) that the present value of a sum to be received at some future datedecreases and approaches zero as the payment date is extended further into thefuture, and (2) that the rate of decrease is greater the higher the interest (discount)rate. At relatively high interest rates, funds due in the future are worth very littletoday, and even at a relatively low discount rate, the present value of a sum due inthe very distant future is quite small. For example, at a 20 percent discount rate,$1 million due in 100 years is worth approximately 1 cent today. (However,1 cent would grow to almost $1 million in 100 years at 20 percent.)

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 future change as the time is

extended and the interest rate increased?


Self-Test Questions

0

0.25

0.50

0.75

1.00

642

Present Value of $1

Periods

i = 0%

108

i = 5%

i = 10%

i = 15%

Relationships among Present Value, Interest Rates, and TimeFigure 28-2

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SOLVING FOR INTEREST RATE AND TIME

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 either theFV or the PV:

FVn � PV(1 � i)n | 28-1 |

There are four variables in the equation—PV, FV, i, and n—and if you know the val-ues of any three, you can find the value of the fourth. Thus far, we have always givenyou the interest rate (i) and the number of years (n), plus either the PV or the FV. Inmany situations, though, you will need to solve for either i or n, as we discuss below.

Solving for iSuppose you can buy a security at a price of $78.35, and it will pay you $100 afterfive years. Here you know PV, FV, and n, and you want to find i, the interest rateyou would earn if you bought the security. Such problems are solved as follows:

Time Line:

Equation:

FVn � PV(1 � i)n | 28-1 |$100 � $78.35(1 � i)5. Solve for i.


Use Equation 28-1 to solve for i:

$100 � $78.35(1 � i)5

(1 � i)5 � 1.2761 � i � (1.276)(1/5)

1 � i � 1.050i � 0.05 � 5%

Therefore, the interest rate is 5 percent.


FVFV100

PMTPMT

0

PVPV

–78.35

II= 5.0

NN5

Output:

Inputs:

$100

$78.35� (1 � i ) 5

0 21 3 4 5

–78.35

i = ?

100

19878_28W_p001-047.qxd 3/14/06 1:16 PM


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 for anyvalue of n, including fractional values.



3

2

1

FECB GDA

Time

Interest rate

Cash flow

0

5%

–78.35

5

100

4

0

3

0

2

0

1

0

Most spreadsheets have a built-in function to find the interest rate. In Excel, youwould put the cursor on Cell B3, then click the function wizard, indicate that youwant a Financial function, scroll down to RATE, and click OK. Then, in the dialogbox, enter G1 or 5 for Nper, 0 for Pmt because there are no periodic payments, B2or �78.35 for Pv, G2 or 100 for Fv, 0 for type, and leave “Guess” blank to letExcel decide where to start its iterations. Then, when you click OK, Excel solves forthe interest rate, 5.00 percent. Excel also has other procedures that could be used tofind the 5 percent, but for this problem the RATE function is easiest to apply.

Solving for nSuppose you invest $78.35 at an interest rate of 5 percent per year. How long willit take your investment to grow to $100? You know PV, FV, and i, but you do notknow n, the number of periods. Here is the situation:

Time Line:

Equation:

FVn � PV(1 � i)n | 28-1 |$100 � $78.35(1.05)n. Solve for n.


Use Equation 28-1 to solve for n:

$100 � $78.35(1 � 0.05)n

Transform to

$100/$78.35 � 1.276 � (1 � 0.05)n

0 21 n – 1 n = ?

–78.35

5%

100

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Take the natural log of both sides, and then solve for n:

n LN(1.05) � LN(1.276)n � LN(1.276)/LN(1.05)

Find the logs with a calculator, and complete the solution:

n � 0.2437/0.0488� 4.9955 � 5.0

Therefore, 5 is the number of years it takes for $78.35 to grow to $100 if theinterest rate is 5 percent.


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


FVFV100

PMTPMT

0

PVPV

–78.35

II

5

NN

= 5.0Output:

Inputs:

4

3

2

1

FECB GDA

Time

Interest rate

Cash flow

Payment

5.00

5%

0 ?

100

. . .

. . .

3

0

2

0

1

0

5 N

0

–78.35

Most spreadsheets have a built-in function to find the number of periods. InExcel, you would put the cursor on Cell B5, then click the function wizard, indi-cate that you want a Financial function, scroll down to NPER, and click OK.Then, in the dialog box, enter B3 or 5% for Rate, 0 for Pmt because there are noperiodic payments, B2 or �78.35 for Pv, G2 or 100 for Fv, and 0 for Type. Whenyou click OK, Excel solves for the number of periods, 5.

Assuming that you are given PV, FV, and the time period, n, write out an equation that canbe used to determine the interest rate, i.

Assuming that you are given PV, FV, and the interest rate, i, write out an equation that canbe used to determine the time period, n.


Self-Test Questions

19878_28W_p001-047.qxd 3/14/06 1:16 PM


FUTURE VALUE OF AN ANNUITY

An annuity is a series of equal payments made at fixed intervals for a specifiednumber of periods. For example, $100 at the end of each of the next three years isa three-year annuity. The payments are given the symbol PMT, and they can occurat either the beginning or the end of each period. If the payments occur at the endof 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 upas ordinary annuities. If payments are made at the beginning of each period, theannuity is an annuity due. Rental payments for an apartment, life insurance pre-miums, and lottery payoffs are typically set up as annuities due. Since ordinaryannuities are more common in finance, when the term “annuity” is used in thisbook, you should assume that the payments occur at the end of each period unlessotherwise noted.

Ordinary AnnuitiesAn ordinary, or deferred, annuity consists of a series of equal payments made at theend of each period. If you deposit $100 at the end of each year for three years in asavings account that pays 5 percent interest per year, how much will you have atthe end of three years? To answer this question, we must find the future value ofthe annuity, FVAn. Each payment is compounded out to the end of Period n, andthe sum of the compounded payments is the future value of the annuity, FVAn.

Time Line:

Here we show the regular time line as the top portion of the diagram, but we alsoshow how each cash flow is compounded to produce the value FVAn in the lowerportion of the diagram.

Equation:

FVAn � PMT(1 � i)n�1 � PMT(1 � i)n�2 � PMT(1 � i)n�3 � . . . � PMT(1 � i)0

| 28-4 |

� PMT a (1 � i ) n � 1

ib

� PMTan

t�1(1 � i ) n�t

0 21 35%

100 100 100105110.25

FVA3 = 315.25

� PMT(FVIFAi,n)

19878_28W_p001-047.qxd 3/14/06 1:16 PM


The first line of Equation 28-4 represents the application of Equation 28-1 to eachindividual payment of the annuity. In other words, each term is the compoundedamount of a single payment, with the superscript in each term indicating the num-ber of periods during which the payment earns interest. For example, because thefirst annuity payment was made at the end of Period 1, interest would be earnedin Periods 2 through n only, so compounding would be for n � 1 periods ratherthan n periods. Compounding for the second payment would be for Period 3through Period n, or n � 2 periods, and so on. The last payment is made at theend of the annuity’s life, so there is no time for interest to be earned.

The second line of Equation 28-4 is just a shorthand version of the first form.The Greek letter � means to replace t in the formula with 1, then replace t with 2,then with 3, until you finally replace t with n, and then add all those termstogether. The third line is found by applying the algebra of geometric progressions.This form of Equation 28-4 is especially useful when no financial calculator isavailable. Finally, the fourth line shows the payment multiplied by the futurevalue interest factor for an annuity (FVIFAi,n), which is the shorthand version ofthe formula.


The lower section of the time line shows the numerical solution, which involvesusing the first line of Equation 28-4. The future value of each cash flow is found,and those FVs are summed to find the FV of the annuity, $315.25. If a long annu-ity were being evaluated, this process would be quite tedious, and in that case youprobably would use the form of Equation 28-4 found on the third line:

| 28-4 |


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 are tryingto find the PV or the FV of the annuity. In our example, you want the FV, so pressthe FV key to get the answer, $315.25. Since there is no initial payment, we inputPV � 0.

FVFV

= 315.25

PMTPMT

–100

PVPV

0

II

5

NN3

Output:

Inputs:

� $100 a (1 � 0.05) 3 � 1

0.05b � $100 (3.1525) � $315.25

FVAn � PMT a (1 � i ) n � 1

ib

19878_28W_p001-047.qxd 3/14/06 1:16 PM



4

3

2

1

ECB DA

Interest rate

Cash flow

Time

Future value

0.05

0

315.25

32

100

1

100 100

Most spreadsheets have a built-in function for finding the future value of an annu-ity. In Excel, we could put the cursor on Cell E4, then click the function wizard,Financial, FV, and OK to get the FV dialog box. Then, we would enter .05 or B1for Rate, 3 or E2 for Nper, and �100 for Pmt. (Like the financial calculatorapproach, the payment is entered as a negative number to show that it is a cashoutflow.) We would leave Pv blank because there is no initial payment, and wewould leave Type blank to signify that payments come at the end of the periods.Then, when we clicked OK, we would get the FV of the annuity, $315.25. Notethat it isn’t necessary to show the time line, since the FV function doesn’t requireyou to input a range of cash flows. Still, the time line is useful to help visualize theproblem.

Annuities DueHad the three $100 payments in the previous example been made at the beginningof each year, the annuity would have been an annuity due. On the time line, eachpayment would be shifted to the left one year, so each payment would be com-pounded for one extra year.

Time Line:

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

0 21 35%

100 100105110.25

FVA3 (Annuity due) = 331.01

100

115.76


19878_28W_p001-047.qxd 3/14/06 1:16 PM


| 28-4a |

Equation:

FVAn(Due) � PMT(1 � i)n � PMT(1 � i)n�1 � PMT(1 � i)n�2 � . . . � PMT(1 � i)

� PMT a (1 � i ) n � 1

ib (1 � i )

� PMTan

t�1(1 � i ) n�1�t

� PMT(FVIFAi,n) (1 � i)

The only difference between Equation 28-4a for annuities due and Equation 28-4for ordinary annuities is that every term in Equation 28-4a is compounded for oneextra period, reflecting the fact that each payment for an annuity due occurs oneperiod earlier than for a corresponding ordinary annuity.


The lower section of the time line shows the numerical solution using the first line ofEquation 28-4a. The future value of each cash flow is found, and those FVs aresummed to find the FV of the annuity, $331.01. Because this process is quite tediousfor long annuities, you probably would use the third line of Equation 28-4a:

| 28-4a |


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

FVFV

= 331.01

PMTPMT

–100

PVPV

0

II

5

NN3

Output:

Inputs:

BEGIN

� $100 a (1 � 0.05) 3 � 1

0.05b (1 � 0.05) � $100 (3.1525) (1.05) � $331.01

FVAn (Due) � PMT a (1 � i ) n � 1

ib (1 � i )

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Enter N � 3, I � 5, PV � 0, PMT � �100, and then press FV to get the answer,$331.01. Since most problems specify end-of-period cash flows, you shouldalways switch your calculator back to “END” mode after you work an annuitydue problem.


For the annuity due, use the FV function just as for the ordinary annuity exceptenter 1 for Type to indicate that we now have an annuity due. Then, when youclick OK, the answer $331.01 will appear.

What is the difference between an ordinary annuity and an annuity due?How do you modify the equation for determining the value of an ordinary annuity to find

the value of an annuity due?Other things held constant, which annuity has the greater future value: an ordinary annuity

or an annuity due? Why?

PRESENT VALUE OF AN ANNUITY

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 needfor the money during the next three years, so if you accept the annuity, youwould deposit the payments in a bank account that pays 5 percent interest peryear. Similarly, the lump sum payment would be deposited into a bank account.How large must the lump sum payment today be to make it equivalent to theannuity?

Ordinary AnnuitiesIf 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 solu-tion values are shown in the left column. The PV of the annuity, PVAn, is$272.32.

0 21 35%

95.2490.70

PVA3 = 272.3286.38

100100 100

Self-Test Questions

19878_28W_p001-047.qxd 3/14/06 1:16 PM


| 28-5 |

Equation:

The general equation used to find the PV of an ordinary annuity is shown below:

� PMT £1 �

1

(1 � i ) n

i

≥

� PMTan

t�1a 1

1 � ib t

PVAn � PMT a 1

1 � ib 1

� PMT a 1

1 � ib 2

� p � PMT a 1

1 � ib n

� PMT(PVIFAi,n)

The present value interest factor for an annuity for i and n, PVIFAi,n, is a short-hand notation for the formula.


The lower section of the time line shows the numerical solution, $272.32, calcu-lated by using the first line of Equation 28-5, where the present value of each cashflow is found and then summed to find the PV of the annuity. If the annuity hasmany payments, it is easier to use the third line of Equation 28-5:

PVAn � PMT £1 �

1

(1 � i ) n

i

≥

� $100 £1 �

1

(1 � 0.05) 3

0.05

≥� $100 (2.7232) � $272.32


FVFV0

PMTPMT

–100

PVPV= 272.32

II

5

NN3

Output:

Inputs:

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Enter N � 3, I � 5, PMT � �100, and FV � 0, and then press the PV key tofind the PV, $272.32.


272.324

3

2

1

ECB DA

Interest rate

Cash flow

Time

Present value

0.05

0 32

100

1

100 100

In Excel, put the cursor on Cell B4 and then click the function wizard, Financial,PV, and OK. Then enter B1 or 0.05 for Rate, E2 or 3 for Nper, �100 for Pmt, 0or leave blank for Fv, and 0 or leave blank for Type. Then, when you click 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 ordinaryannuity, and the payments constitute the annuity stream. We will examine con-stant payment loans in more depth in a later section of this chapter.

Annuities DueHad the three $100 payments in the preceding example been made at the begin-ning of each year, the annuity would have been an annuity due. Each paymentwould be shifted to the left one year, so each payment would be discounted forone less year. Here is the time line:

Time Line:

Again, we find the PV of each cash flow and then sum these PVs to find the PV ofthe annuity due. This procedure is illustrated in the lower section of the time linediagram. Since the cash flows occur sooner, the PV of the annuity due exceeds thatof the ordinary annuity, $285.94 versus $272.32.

0 21 35%

95.2490.70

PVA3 (Annuity due) = 285.94

100100100


19878_28W_p001-047.qxd 3/14/06 1:16 PM


| 28-5a |

Equation:

� PMT £1 �

1

(1 � i ) n

i

≥(1 � i )

� PMTan

t�1a 1

1 � ib t�1

PVAn (Due) � PMT a 1

1 � ib 0

� PMT a 1

1 � ib 1

� p � PMT a 1

1 � ib n�1

� PMT(PVIFAi,n)(1 � i)


The lower section of the time line shows the numerical solution, $285.94, calcu-lated by using the first line of Equation 28-5a, where the present value of eachcash flow is found and then summed to find the PV of the annuity due. If theannuity has many payments, it is easier to use the third line of Equation 28-5a:

PVAn (Due) � PMT £1 �

1

(1 � i ) n

i

≥(1 � i )

� $100 £1 �

1

(1 � 0.05) 3

0.05

≥ (1 � 0.05)

� $100(2.7232)(1 � 0.05) � $285.94


FVFV0

PMTPMT

–100

PVPV= 285.94

II

5

NN3

Output:

Inputs:

BEGIN

| 28-5a |

19878_28W_p001-047.qxd 3/14/06 1:16 PM


4This is the equation for an ordinary annuity. Calculators and spreadsheets have a slightly different equation for an annuity due.

Self-Test Questions

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 switchyour calculator back to the “END” mode.


For an annuity due, use the PV function just as for a regular annuity except enter1 rather than 0 for Type to indicate that we now have an annuity due.

Which annuity has the greater present value: an ordinary annuity or an annuity due? Why?Explain how financial calculators can be used to find the present value of annuities.

ANNUITIES: SOLVING FOR INTEREST RATE,NUMBER OF PERIODS, OR PAYMENT

Sometimes it is useful to calculate the interest rate, payment, or number of periodsfor a given annuity. For example, suppose you can lease a computer from its man-ufacturer for $78 per month. The lease runs for 36 months, with payments due atthe end of the month. As an alternative, you can buy it for $1,988.13. In eithercase, at the end of the 36 months the computer will be worth zero. You wouldlike to know the “interest rate” the manufacturer has built into the lease; if thatrate is too high, you should buy the computer rather than lease it.

Or suppose you are thinking ahead to retirement. If you save $4,000 per year atan interest rate of 8 percent, how long will it take for you to accumulate $1 million?Or, viewing the problem another way, if you earn an interest rate of 8 percent, howmuch must you save for each of the next 20 years to accumulate the $1 million?

To solve problems such as these, we can use an equation that is built intofinancial calculators and spreadsheets:4

| 28-6 |

Note that some value must be negative. There are five variables: n, i, PV,PMT, and FV. In each of the three problems above, you know four of the vari-ables. For example, in the computer leasing problem, you know that n � 36, PV � 1,988.13 (this is positive, since you get to keep this amount if you choose tolease rather than purchase), PMT � �78 (this is negative since it is what youmust pay each month), and FV � 0. Therefore, the equation is

| 28-6a |(1,988.13) (1 � i ) 36 � (�78) a (1 � i ) 36 � 1

ib � 0 � 0

PV (1 � i ) n � PMT a (1 � i ) n � 1

ib � FV � 0

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Unless you use a financial calculator or a spreadsheet, the only way to solve for iis by trial and error. However, with a financial calculator, you simply enter the values for the four known variables (N � 36, PV � 1988.13, PMT � �78, andFV � 0), and then hit the key for the unknown fifth variable, in this case, I � 2.Since this is an interest rate of 2 percent per month or 12(2%) � 24% per year,you would probably want to buy the computer rather than lease it.

It is worth pointing out that the left side of the equation cannot equal zero ifyou put both the PV and PMT as positive numbers (assuming positive interestrates). If you do this by mistake, most financial calculators will make a rude beep-ing noise, while spreadsheets will display an error message.

In an Excel spreadsheet, you would use the same RATE function that we dis-cussed earlier. In this example, enter 36 for Nper, �78 for Pmt, 1988.13 for Pv, 0for Fv, and 0 for Type: ��RATE(36,��78,1988.13,0,0). The result is again 0.02, or2 percent.

Regarding how long you must save until you accumulate $1 million, you knowi � 8%, PV � 0 (since you don’t have any savings when you start), PMT ��4,000 (it is negative since the payment comes out of your pocket), and FV �1,000,000 (it is positive since you will get the $1 million). Substituting into Equation28-6 gives:

| 28-6a |

Using algebra, you could solve for n, but it is easier to find n with a financialcalculator. Input I � 8, PV � 0, PMT � �4000, FV � 1000000, and solve for N,which is equal to 39.56. Thus, it will take almost 40 years to accumulate $1 mil-lion if you earn 8 percent interest and save only $4,000 per year. On a spread-sheet, you could use the same NPER function that we discussed earlier. In thiscase, enter 8% for Rate, �4000 for Pmt, 0 for Pv, 1000000 for Fv, and 0 forType: ��NPER(8%,��4000,0,1000000,0). The result is again 39.56.

If you plan to save for only 20 years, how much must you save each year toaccumulate $1 million? In this case, we know that n � 20, i � 8%, PV � 0, andFV � 1000000. The equation is

| 28-6a |

You could use algebra to solve for PMT, or you could use a financial calculatorand input N � 20, I � 8, PV � 0, and FV � 1000000. The result is PMT ��21,852.21. On a spreadsheet, you would use the PMT function, inputting 8%for Rate, 20 for Nper, 0 for Pv, 1000000 for Fv, and 0 for Type: ��PMT(8%,20,0,1000000,0). The result is again �21,852.21.

Write out the equation that is built into a financial calculator.Explain why a financial calculator cannot find a solution if PV, PMT, and FV all are

positive.

(0 ) (1 � 0.08) 20 � PMT a (1 � 0.08) 20 � 1

0.08b � 1,000,000 � 0

(0) (1 � 0.08) n � (�4,000) a (1 � 0.08) n � 1

0.08b � 1,000,000 � 0

Self-Test Questions

19878_28W_p001-047.qxd 3/14/06 1:16 PM


PERPETUITIES

Most annuities call for payments to be made over some finite period of time—forexample, $100 per year for three years. However, some annuities go on indefinitely,or perpetually, and these are called perpetuities. The present value of a perpetuity isfound by applying Equation 28-7:

| 28-7 |

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 pastdebts, the bonds were called consols. Suppose each consol promised to pay $100per year in perpetuity. (Actually, interest was stated in pounds.) What would eachbond be worth if the opportunity cost rate, or discount rate, was 5 percent? Theanswer is $2,000:

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

Thus, we see that the value of a perpetuity changes dramatically when interestrates change.

What happens to the value of a perpetuity when interest rates increase?What happens when interest rates decrease?

UNEVEN CASH FLOW STREAMS

The definition of an annuity includes the words constant payment—in other words,annuities involve payments that are the same in every period. Although manyfinancial decisions do involve constant payments, other important decisions involveuneven, or nonconstant, cash flows. For example, common stocks typically pay anincreasing stream of dividends over time, and fixed asset investments such as newequipment normally do not generate constant cash flows. Consequently, it is neces-sary to extend our time value discussion to include 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, andwe will use the term cash flow (CF) to denote uneven cash flows. Financial calcu-lators are set up to follow this convention, so if you are dealing with uneven cashflows, you will need to use the “cash flow register.”

PV (Perpetuity ) �$100

0.10� $1,000 if i � 10%

PV (Perpetuity ) �$100

0.05� $2,000 if i � 5%

PV (Perpetuity ) �Payment

Interest rate�

PMT

i

Self-Test Questions

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Present Value of an Uneven Cash Flow StreamThe PV of an uneven cash flow stream is found as the sum of the PVs of the indi-vidual cash flows of the stream. For example, suppose we must find the PV of thefollowing cash flow stream, discounted at 6 percent:

0 21 4 6 7

PV = ?

6%

1,000

53

0200200200200100

| 28-8 |

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

We could find the PV of each individual cash flow using the numerical, financialcalculator, or spreadsheet methods, and then sum these values to find the presentvalue of the stream. Here is what the process would look like:

� an

t�1CFt a 1

1 � ib t

� an

t�1CFt (PVIFi,t )

PV � CF1 a 1

1 � ib 1

� CF2 a 1

1 � ib 2

� p � CFn a 1

1 � ib n

0 21 4 6 76%

1,000

53

020020020020010094.34

178.00

1,413.19

167.92158.42149.45

0.00665.06

0 21 4 6 76%

1,000

53

020020020020010094.34

1,413.19

653.790.00

665.06

693.02

All we did was apply Equation 28-8, show the individual PVs in the left column ofthe diagram, and then sum these individual PVs to find the PV of the entire stream.

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, noticethat the cash flows in periods 2 through 5 represent an annuity. We can use thatfact to solve the problem in a slightly different manner:

19878_28W_p001-047.qxd 3/14/06 1:16 PM


Cash flows during Years 2 to 5 represent an ordinary annuity, and we find its PVat Year 1 (one period before the first payment). This PV ($693.02) must then bediscounted 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 have substi-tuted in all the known values of Equation 28-8, so you only need to press theNPV key to find the present value of the stream. The calculator has been pro-grammed to find the PV of each cash flow and then to sum these values to findthe 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 the cashflow 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 presentvalue of a series of positive and negative cash flows, including the cash flow attime zero.

The second point to note is that annuities can be entered into the cash flowregister more efficiently by using the Nj key.5 In this illustration, you would enterCF0 � 0, CF1 � 100, CF2 � 200, Nj � 4 (which tells the calculator that the 200occurs 4 times), CF6 � 0, and CF7 � 1000.6 Then enter I � 6 and press the NPVkey, and 1,413.19 will appear in the display. Also, note that amounts entered intothe cash flow register remain in the register until they are cleared. Thus, if youhad previously worked a problem with eight cash flows, and then moved to aproblem with only four cash flows, the calculator would simply add the cashflows from the second problem to those of the first problem. Therefore, you mustbe sure to clear the cash flow register before starting 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:

5On some calculators, you are prompted to enter the number of times the cash flow occurs, and on still other calculators,the procedures for inputting data, as we discuss next, may be different. You should consult your calculator manual or ourTechnology Supplement to determine the appropriate steps for your specific calculator.6On some calculators, instead of entering CF6 � 0 and CF7 � 1000, you enter CF3 � 0 and CF4 � 1000, since theseare the third and fourth different cash flows.


4

3

2

1

FECB GDA

Interest rate

Cash flow

Time

Present value

0.06

1,413.19

2 7

1,000

6

0

5

200

4

200

3

100

IH

0 1

200200

To find the PV of these cash flows with Excel, put the cursor on Cell B4, click thefunction wizard, click Financial, scroll down to NPV, and click OK to get the dia-log box. Then enter B1 or 0.06 for Rate and the range of cells containing the cashflows, C3:I3, for Value 1. Be very careful when entering the range of cash flows.

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| 28-9 |

| 28-10 |

With a financial calculator, you begin by entering the time zero cash flow. WithExcel, you do not include the time zero cash flow; instead, you begin with theYear 1 cash flow. Now, when you click OK, you get the PV of the stream,$1,413.19. Note that you use the PV function if the cash flows (or payments) areconstant, but the NPV function if they are not constant. Note too that one of theadvantages of spreadsheets over financial calculators is that you can see the cashflows, which makes it easy to spot any typing errors.

Notice that the corporate valuation equation shown in the box at the begin-ning of the chapter is simply a formula to find the present value of an unevenstream of cash flows where the cash flows are free cash flows (FCFs) and theinterest rate is the weighted average cost of capital (WACC):

In Chapter 5 we showed you how to solve such an equation, even if it has an infi-nite number of cash flows.

Future Value of an Uneven Cash Flow StreamThe future value of an uneven cash flow stream (sometimes called the terminalvalue) is found by compounding each payment to the end of the stream and thensumming the future values:

The future value of our illustrative uneven cash flow stream is $2,124.92:

�an

t�1CFt (1 � i ) n�t �a

n

t�1CFt (FVIFi,n�t )

FVn � CF1 (1 � i ) n�1 � CF2 (1 � i ) n�2 � p � CFn�1 (1 � i ) � CFn

� aq

t�1

FCFt

(1 � WACC) t

V �FCF1

(1 � WACC)�

FCF2

(1 � WACC) 2� p �

FCFq

(1 � WACC)q

0 21 4 6 76%

1,000

53

02002002002001000

224.72

2,124.92

238.20252.50267.65141.85

Some financial calculators have a net future value (NFV) key which, after the cashflows and interest rate have been entered, can be used to obtain the futurevalue of an uneven cash flow stream. Even if your calculator doesn’t have the

19878_28W_p001-047.qxd 3/14/06 1:16 PM


NFV feature, you can use the cash flow stream’s net present value to find its netfuture value: NFV � NPV(1 � i)n. Thus, in our example, you could find the PV ofthe stream, then find the FV of that PV, compounded for n periods at i percent. Inthe illustrative problem, find PV � 1,413.19 using the cash flow register and I �6. Then enter N � 7, I � 6, PV � �1413.19, and PMT � 0, and then press FVto find FV � 2,124.92, which equals the NFV shown on the time line above.

Solving for i with Uneven Cash Flow StreamsIt is relatively easy to solve for i numerically when the cash flows are lump sumsor annuities. However, it is extremely difficult to solve for i if the cash flows areuneven, because then you would have to go through many tedious trial-and-errorcalculations. With a spreadsheet program or a financial calculator, though, it iseasy to find the value of i. Simply input the CF values into the cash flow registerand then press the IRR key. IRR stands for “internal rate of return,” which is thepercentage return on an investment. We discuss the IRR calculation when we dis-cuss capital budgeting methods in Chapter 12.7

Give two examples of financial decisions that would typically involve uneven cash flows.(Hint: Think about a bond or a stock that you plan to hold for five years.)

What is meant by the term “terminal value”?

GROWING ANNUITIES

Normally, an annuity is defined as a series of constant payments to be receivedover a specified number of periods. However, the term growing annuity is used todescribe a series of payments that is growing at a constant rate for a specifiednumber of periods. The most common application of growing annuities is in thearea of financial planning, where someone wants to maintain a constant real, orinflation-adjusted, income over some specified number of years. For example, sup-pose a 65-year-old person is contemplating retirement, expects to live for another20 years, has $1 million of investment funds, expects to earn 10 percent on theinvestments, expects inflation to average 5 percent per year, and wants to with-draw a constant real amount per year. What is the maximum amount that he orshe can withdraw at the end of each year?

We explain in the spreadsheet model for this chapter, IFM9 Ch28 Tool Kit.xls,that the problem can be solved in three ways. (1) Use the real rate of return for I ina financial calculator. (2) Use a relatively complicated formula.8 Or (3) use a

7To obtain an IRR solution, at least one of the cash flows must have a negative sign, indicating that it is an investment.Since none of the CFs in our example were negative, the cash flow stream has no IRR. However, had we input a cost for CF0, say, �$1,000, we could have obtained an IRR, which would be the rate of return earned on the $1,000 investment. Here IRR would be 13.96 percent.8The formula used to find the payment is shown below. Other formulas can be developed to solve for n and other terms,but they are even more complex.

PVIF of a growing annuity � PVIFGA � [1 � [(1 � g)/(1 � i)]n]/[(i � g)/(1 � g)]

� 12.72

Payment � PMT � PV/PVIFGA � $1,000,000/12.72

� $78,630.64

Self-Test Questions

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9In this chapter, the term nominal rate means the stated, or quoted, annual rate as opposed to the effective annual rate,which we explain later. In both cases, though, nominal means stated, or quoted, as opposed to some adjusted rate.

spreadsheet model, with Excel’s Goal Seek feature used to find the maximum with-drawal that will leave a zero balance in the account at the end of 20 years. Thefinancial calculator approach is the easiest to use, but the spreadsheet model pro-vides the clearest picture of what is happening. Also, the spreadsheet approach canbe adapted to find other parameters of the general model, such as the maximumnumber of years a given constant income can be provided by the initial portfolio.

To implement the calculator approach, first calculate the expected real rate ofreturn as follows, where rr is the real rate and rnom is the nominal rate of return:

Real rate � rr � [(1 � rnom)/(1 � Inflation)] � 1.0

� [1.10/1.05] � 1.0 � 4.761905%

Now, with a financial calculator, input N � 20, I � 4.761905, PV � �1000000,and FV � 0, and then press PMT to get the answer, $78,630.64. Thus, a portfolioworth $1 million will provide 20 annual payments with a current dollar value of$78,630.64 under the stated assumptions. The actual payments will be growing at5 percent per year to offset inflation. The (nominal) value of the portfolio will begrowing at first and then declining, and it will hit zero at the end of the 20th year.The IFM9 Ch28 Tool Kit.xls shows all this in both tabular and graphic form.

Differentiate between a “regular” and a growing annuity.What three methods can be used to deal with growing annuities?

SEMIANNUAL AND OTHER COMPOUNDING PERIODS

In almost all of our examples thus far, we have assumed that interest is compoundedonce a year, or annually. This is called annual compounding. Suppose, however, thatyou put $100 into a bank that states that it pays a 6 percent annual interest rate butthat interest is credited each six months. This is called semiannual compounding.How much would you have accumulated at the end of one year, two years, or someother period under semiannual compounding? Note that virtually all bonds payinterest semiannually, most stocks pay dividends quarterly, and most mortgages, stu-dent loans, and auto loans require monthly payments. Therefore, it is essential thatyou understand how to deal with nonannual compounding.

Types of Interest RatesCompounding involves three types of interest rates: nominal rates, iNom; periodicrates, iPER; and effective annual rates, EAR or EFF%.

1. Nominal, or quoted, rate, iNom.9 This is the rate that is quoted by banks, brokers,and other financial institutions. So, if you talk with a banker, broker, mortgagelender, auto finance company, or student loan officer about rates, the nominalrate is the one he or she will normally quote you. However, to be meaningful, the

Self-Test Questions

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10The only exception is in situations where (1) annuities are involved and (2) the payment periods do not correspond tothe compounding periods. If an annuity is involved and if its payment periods do not correspond to the compoundingperiods—for example, if you are making quarterly payments into a bank account to build up a specified future sum, butthe bank pays interest on a daily basis—then the calculations are more complicated. For such problems, one can proceedin two alternative ways. (1) Determine the periodic (daily) interest rate by dividing the nominal rate by 360 (or 365 if thebank uses a 365-day year), then compound each payment over the exact number of days from the payment date to theterminal point, and then sum the compounded payments to find the future value of the annuity. This is what would gen-erally be done in the real world, because with a computer, it would be a simple process. (2) Calculate the EAR, as definedon the next page, based on daily compounding, then find the corresponding nominal rate based on quarterly compound-ing (because the annuity payments are made quarterly), then find the quarterly periodic rate, and then use that rate withstandard annuity procedures. The second procedure is faster with a calculator, but hard to explain and generally not usedin practice given the ready availability of computers.

quoted nominal rate must also include the number of compounding periods peryear. For example, a bank might offer 6 percent, compounded quarterly, on CDs,or a mutual fund might offer 5 percent, compounded monthly, on its moneymarket account.

The nominal rate on loans to consumers is also called the annual percent-age rate (APR). For example, if a credit card issuer quotes an annual rate of18 percent, this is the APR.

Note that the nominal rate is never shown on a time line, and it is neverused as an input in a financial calculator, unless compounding occurs onlyonce a year. If more frequent compounding occurs, you should use the peri-odic rate as discussed below.

2. Periodic rate, iPER. This is the rate charged by a lender or paid by a borrowereach period. It can be a rate per year, per six-month period, per quarter, permonth, per day, or per any other time interval. For example, a bank mightcharge 1.5 percent per month on its credit card loans, or a finance companymight charge 3 percent per quarter on installment loans. We find the periodicrate as follows:

Periodic rate, iPER � iNom/m, | 28-11 |

which implies that

Nominal annual rate � iNom � (Periodic rate)(m) | 28-12 |

Here iNom is the nominal annual rate and m is the number of compounding periodsper year. To illustrate, consider a finance company loan at 3 percent per quarter:

Nominal annual rate � iNom � (Periodic rate)(m) � (3%)(4) � 12%

or

Periodic rate � iNom/m � 12%/4 � 3% per quarter

If there is only one payment per year, or if interest is added only once a year,then m � 1, and the periodic rate is equal to the nominal rate.

The periodic rate is the rate that is generally shown on time lines and usedin calculations.10 To illustrate use of the periodic rate, suppose you invest

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$100 in an account that pays a nominal rate of 12 percent, compoundedquarterly. How much would you have after two years?

For compounding more frequently than annually, we use the followingmodification of Equation 28-1:

| 28-13 |

Time Line and Equation:

FVn � PV(l � iPER)Number of periods


Using Equation 28-13,FV � $100(1 � 0.03)8

� $126.68


Input N � 2 � 4 � 8, I � 12/4 � 3, PV � �100, and PMT � 0, and then pressthe FV key to get FV � $126.68.

Most financial calculators have a feature that allows you to set the number ofpayments per year and then use the nominal annual interest rate. However, inour experience, students make fewer errors when using the periodic rate with theircalculators set for one payment per year, so this is what we recommend.


A spreadsheet could be developed as we did earlier in the chapter in our discus-sion of the future value of a lump sum. Rows would be set up to show the interestrate, time, cash flow, and future value of the lump sum. The interest rate used inthe spreadsheet would be the periodic interest rate (iNom/m) and the number oftime periods shown would be (m)(n).

FVFV

= 126.68

PMTPMT

0

PVPV

–100

II

3

NN8

Output:

Inputs:

0 21 4 6 83%

FV = ?

53

–100Quarters

7

FVn � PV (1 � iPER ) Number of periods � PV a 1 �iNom

mb mn

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11Most financial calculators are programmed to find the EAR or, given the EAR, to find the nominal rate. This is called“interest rate conversion,” and you simply enter the nominal rate and the number of compounding periods per year andthen press the EFF% key to find the effective annual rate.

3. Effective (or equivalent) annual rate (EAR). This is the annual rate that produces the same result as if we had compounded at a given periodic rate mtimes per year. The EAR, also called EFF% (for effective percentage), is foundas follows:

| 28-14 |

You could also use the interest conversion feature of a financial calculator.11

In the EAR equation, iNom/m is the periodic rate, and m is the number of peri-ods per year. For example, suppose you could borrow using either a credit cardthat charges 1 percent per month or a bank loan with a 12 percent quoted nominalinterest rate that is compounded quarterly. Which should you choose? To answerthis question, the cost rate of each alternative must be expressed as an EAR:

Credit card loan: EAR � (1 � 0.01)12 � 1.0 � (1.01)12 � 1.0� 1.126825 � 1.0 � 0.126825 � 12.6825%

Bank loan: EAR � (1 � 0.03)4 � 1.0 � (1.03)4 � 1.0� 1.125509 � 1.0 � 0.125509 � 12.5509%

Thus, the credit card loan is slightly more costly than the bank loan. This resultshould have been intuitive to you—both loans have the same 12 percent nominalrate, yet you would have to make monthly payments on the credit card versusquarterly payments under the bank loan.

The EAR rate is not used in calculations. However, it should be used to com-pare the effective cost or rate of return on loans or investments when paymentperiods differ, as in the credit card versus bank loan example.

The Result of Frequent CompoundingSuppose you plan to invest $100 for five years at a nominal annual rate of 10 percent. What will happen to the future value of your investment if interest iscompounded more frequently than once a year? Because interest will be earned oninterest more often, you might expect the future value to increase as the frequencyof compounding increases. Similarly, you might also expect the effective annualrate to increase with more frequent compounding. As Table 28-1 shows, youwould be correct—the future value and EAR do in fact increase as the frequencyof compounding increases. Notice that the biggest increases in FV and EAR occurwhen compounding goes from annual to semiannual, and that moving frommonthly to daily compounding has a relatively small impact.

Define the nominal (or quoted) rate, the periodic rate, and the effective annual rate.Which rate should be shown on time lines and used in calculations?

EAR (or EFF% ) � a 1 �iNom

mb

m

� 1.0

Self-Test Questions

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What changes must you make in your calculations to determine the future value of anamount that is being compounded at 8 percent semiannually versus one being com-pounded annually at 8 percent?

Why is semiannual compounding better than annual compounding from a saver’s stand-point? What about a borrower’s standpoint?

The Impact of Frequent CompoundingTable 28-1

Future Value ofFrequency of Nominal Annual Effective Annual $100 InvestedCompounding Rate Rate (EAR)a for 5 Yearsb

Annual 10% 10.000% $161.05Semiannual 10 10.250 162.89Quarterly 10 10.381 163.86Monthly 10 10.471 164.53Dailyc 10 10.516 164.86

aThe EAR is calculated using Equation 28-14.bThe future value is calculated using Equation 28-13.cThe daily calculations assume 365 days per year.

People continually face important financialdecisions that require an understanding ofthe time value of money. Should we buy orlease a car? How much and how soon do weneed to save for our children’s education?What size house can we afford? Should werefinance our home mortgage? How muchmust we save in order to retire comfortably?

The answers to these questions are oftencomplicated, and they depend on a numberof factors, such as housing and educationcosts, interest rates, inflation, expected fam-ily income, and stock market returns. Hope-fully, after completing this chapter, you willhave a better idea of how to answer suchquestions. Moreover, there are a number ofonline resources available to help with finan-cial planning.

A good place to start is http://www.smartmoney.com. Smartmoney is a personalfinance magazine produced by the publish-ers of The Wall Street Journal. If you go toSmartmoney’s Web site you will find a sec-tion entitled “Tools.” This section has anumber of financial calculators, spread-sheets, and descriptive materials that covera wide range of personal finance issues.

Another good place to look is Quicken’sWeb site, http://www.quicken.com. Hereyou will find several interesting sections,especially Planning & Tax (after selecting theInvesting tab), that deal with a variety ofpersonal finance issues. Within these sec-tions you will find background articles plusspreadsheets and calculators that you canuse to analyze your own situation.

U S I N G T H E I N T E R N E T F O R P E R S O N A LF I N A N C I A L P L A N N I N G

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FRACTIONAL TIME PERIODS

In all the examples used thus far in the chapter, we have assumed that paymentsoccur at either the beginning or the end of periods, but not at some date within aperiod. However, we often encounter situations that require compounding or dis-counting over fractional periods. For example, suppose you deposited $100 in abank that adds interest to your account daily, that is, uses daily compounding, andpays a nominal rate of 10 percent with a 360-day year. How much will be in youraccount after nine months? The answer is $107.79:12

Periodic rate � iPER � 0.10/360 � 0.00027778 per dayNumber of days � 0.75(360) � 270Ending amount � $100(1.00027778)270 � $107.79

Now suppose you borrow $100 from a bank that charges 10 percent per year“simple interest,” which means annual rather than daily compounding, but youborrow the $100 for only 270 days. How much interest would you have to payfor the use of $100 for 270 days? Here we would calculate a daily interest rate,iPER, as above, but multiply by 270 rather than use it as an exponent:

Interest owed � $100(0.00027778)(270) � $7.50 interest charged

You would owe the bank a total of $107.50 after 270 days. This is the proceduremost banks actually use to calculate interest on loans, except that they generallyrequire you to pay the interest on a monthly basis rather than after 270 days.

Finally, let’s consider a somewhat different situation. Say an Internet accessfirm had 100 customers, and its customer base is expected to grow steadily at therate of 10 percent per year. What is the estimated customer base nine monthslater? This problem would be set up exactly like the bank account with daily com-pounding, and the estimate would be 107.79 customers, rounded to 108.

The most important thing in problems like these, as in all time value prob-lems, is to be careful! Think about what is involved in a logical, systematic man-ner, draw a time line if it would help you visualize the situation, and then applythe appropriate equations.

How do you solve a time value problem with a fractional time period?

AMORTIZED LOANS

One of the most important applications of compound interest involves loans thatare paid off in installments over time. Included are automobile loans, home mort-gage loans, student loans, and most business loans other than very short-termloans and long-term bonds. If a loan is to be repaid in equal periodic amounts(monthly, quarterly, or annually), it is said to be an amortized loan.13

12Here we assumed a 360-day year, and we also assumed that the nine months all have 30 days. This convention is oftenused. However, some contracts specify that actual days be used. Computers (and many financial calculators) have a built-in calendar, and if you input the beginning and ending dates, the computer or calculator would tell you the exact numberof days, taking account of 30-day months, 31-day months, and 28- or 29-day months.13The word amortized comes from the Latin mors, meaning “death,” so an amortized loan is one that is “killed off”over time.

Self-Test Question

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Loan Amortization Schedule, 6 Percent Interest RateTable 28-2

Beginning Repayment RemainingAmount Payment Interesta of Principalb Balance

Year (1) (2) (3) (2) � (3) � (4) (1) � (4) � (5)

1 $1,000.00 $ 374.11 $ 60.00 $ 314.11 $685.892 685.89 374.11 41.15 332.96 352.933 352.93 374.11 21.18 352.93 0.00

$1,122.33 $122.33 $1,000.00

aInterest is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefore, interest in Year 1 is $1,000(0.06) �$60; in Year 2 it is $685.89(0.06) � $41.15; and in Year 3 it is $352.93(0.06) � $21.18.bRepayment of principal is equal to the payment of $374.11 minus the interest charge for each year.

Table 28-2 illustrates the amortization process. A firm borrows $1,000, andthe loan is to be repaid in three equal payments at the end of each of the nextthree years. (In this case, there is only one payment per year, so years � periodsand the stated rate � periodic rate.) The lender charges a 6 percent interest rateon the loan balance that is outstanding at the beginning of each year. The firsttask is to determine the amount the firm must repay each year, or the constantannual payment. To find this amount, recognize that the $1,000 represents thepresent value of an annuity of PMT dollars per year for three years, discounted at6 percent:

Time Line:

Equation:

The same general equation used to find the PV of an ordinary annuity is shownbelow:

PVAn � PMT a 1

1 � ib 1

� PMT a 1

1 � ib 2

� p � PMT a 1

1 � ib n

0 21 36%

PMT PMT PMT1,000

� PMT(PVIFAi,n)

� PMT £1 �

1

(1 � i ) n

i

≥ � PMTa

n

t�1a 1

1 � ib t

| 28-5 |

19878_28W_p001-047.qxd 3/14/06 1:16 PM



We know the present value, the interest rate, and the number of periods. The onlyunknown variable is the payment. Using Equation 28-5, we can solve the equationfor the payment:

$1,000 � PMT £1 �

1

(1 � 0.06) 3

0.06

≥� PMT (2.6730)

PMT � $1,000/2.6730 � $374.11


Enter N � 3, I � 6, PV � 1000, and FV � 0, and then press the PMT key to findPMT � �$374.11.


The spreadsheet is ideal for developing amortization tables. The setup is similarto Table 28-2, but you would want to include “input” cells for the interest rate,principal value, and the length of the loan. This would make the spreadsheetflexible in the sense that the loan terms could be changed and a new amortizationtable would be recalculated instantly. Then use the function wizard to find thepayment. If you had I � 6% in B1, N � 3 in B2, and PV � 1000 in B3, then thefunction ��PMT(B1, B2, B3) would return a result of �$374.11.

Therefore, the firm must pay the lender $374.11 at the end of each of the nextthree years, and the percentage cost to the borrower, which is also the rate ofreturn to the lender, will be 6 percent. Each payment consists partly of interest andpartly of repayment of principal. This breakdown is given in the amortizationschedule shown in Table 28-2. The interest component is largest in the first year,and it declines as the outstanding balance of the loan decreases. For tax purposes,a business borrower or homeowner reports the interest component shown in Col-umn 3 as a deductible cost each year, while the lender reports this same amount astaxable income.

Financial calculators are programmed to calculate amortization tables—yousimply enter the input data, and then press one key to get each entry in Table 28-2.If you have a financial calculator, it is worthwhile to read the appropriate sectionof the calculator manual and learn how to use its amortization feature. As we showin the model for this chapter, with a spreadsheet such as Excel it is easy to set upand print out a full amortization schedule.

FVFV0

PMTPMT= –374.11

PVPV

1000

II

6

NN3

Output:

Inputs:

See IFM9 Ch28 Tool Kit.xls

for details.

19878_28W_p001-047.qxd 3/14/06 1:16 PM

Self-Test Questions To construct an amortization schedule, how do you determine the amount of the periodicpayments?

How do you determine the amount of each payment that goes to interest and to principal?

SUMMARY

Most financial decisions involve situations in which someone pays money at onepoint in time and receives money at some later time. Dollars paid or received attwo different points in time are different, and this difference is recognized andaccounted for by time value of money (TVM) analysis.

• Compounding is the process of determining the future value (FV) of a cashflow or a series of cash flows. The compounded amount, or future value, isequal to the beginning amount plus the interest earned.

• Future value of a single payment: FVn � PV(1 � i)n

• Discounting is the process of finding the present value (PV) of a future cash flowor a series of cash flows; discounting is the reciprocal, or reverse, of compounding.

• Present value of a single payment:

• An annuity is defined as a series of equal periodic payments (PMT) for a spec-ified number of periods.

• Future value of an annuity:

PVAn � PMT £1 �

1

(1 � i ) n

i

≥FVAn � PMTa

n

t�1(1 � i ) n�t � PMT a (1 � i ) n � 1

ib

PV �FVn

(1 � i ) i

• Present value of an annuity:

• An annuity whose payments occur at the end of each period is called an ordi-nary annuity. The formulas above are for ordinary annuities.

• If each payment occurs at the beginning of the period rather than at the end,then we have an annuity due. The PV of each payment would be larger,because each payment would be discounted back one year less, so the PV ofthe annuity would also be larger. Similarly, the FV of the annuity due wouldalso be larger because each payment would be compounded for an extra year.The following formulas can be used to convert the PV and FV of an ordinaryannuity to an annuity due:

PVA (annuity due) � PVA of an ordinary annuity � (1 � i)FVA (annuity due) � FVA of an ordinary annuity � (1 � i)

• A perpetuity is an annuity with an infinite number of payments.

• To find the PV or FV of an uneven series, find the PV or FV of each individ-ual cash flow and then sum them.

Value of perpetuity �PMT

i


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• If you know the cash flows and the PV (or FV) of a cash flow stream, you candetermine the interest rate.

• When compounding occurs more frequently than once a year, the nominalrate must be converted to a periodic rate, and the number of years must beconverted to periods:

iPER � Stated rate/Periods per yearPeriods � Years � Periods per year

The periodic rate and number of periods would be used for calculations andshown on time lines.

• If comparing the costs of loans that require payments more than once a year,or the rates of return on investments that pay interest more frequently, thenthe comparisons should be based on equivalent (or effective) rates of returnusing this formula:

• The general equation for finding the future value for any number of com-pounding periods per year is

where

iNom � quoted interest rate.m � number of compounding periods per year.n � number of years.

• An amortized loan is one that is paid off in equal payments over a specifiedperiod. An amortization schedule shows how much of each payment consti-tutes interest, how much is used to reduce the principal, and the unpaid bal-ance at each point in time.

QUESTIONS

28-1 Define each of the following terms:a. PV; i; INT; FVn; PVAn; FVAn; PMT; m; iNomb. FVIFi,n; PVIFi,n; FVIFAi,n; PVIFAi,nc. Opportunity cost rated. Annuity; lump sum payment; cash flow; uneven cash flow streame. Ordinary (deferred) annuity; annuity duef. Perpetuity; consolg. Outflow; inflow; time line; terminal valueh. Compounding; discountingi. Annual, semiannual, quarterly, monthly, and daily compoundingj. Effective annual rate (EAR); nominal (quoted) interest rate; APR; periodic ratek. Amortization schedule; principal versus interest component of a payment;

amortized loan

FVn � PV a 1 �iNom

mb mn

Effective annual rate � EAR (or EFF% ) � a 1 �iNom

mb

m

� 1.0

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28-2 What is an opportunity cost rate? How is this rate used in discounted cash flowanalysis, and where is it shown on a time line? Is the opportunity rate a singlenumber that is used in all situations?

28-3 An annuity is defined as a series of payments of a fixed amount for a specificnumber of periods. Thus, $100 a year for 10 years is an annuity, but $100 in Year1, $200 in Year 2, and $400 in Years 3 through 10 does not constitute an annuity.However, the second series contains an annuity. Is this statement true or false?

28-4 If a firm’s earnings per share grew from $1 to $2 over a 10-year period, the totalgrowth would be 100 percent, but the annual growth rate would be less than 10percent. True or false? Explain.

28-5 Would you rather have a savings account that pays 5 percent interest compoundedsemiannually or one that pays 5 percent interest compounded daily? Explain.

PROBLEMS

28-1 Find the following values, using the equations, and then work the problems using afinancial calculator to check your answers. Disregard rounding differences. (Hint: Ifyou are using a financial calculator, you can enter the known values and then pressthe appropriate key to find the unknown variable. Then, without clearing the TVMregister, you can “override” the variable that changes by simply entering a newvalue for it and then pressing the key for the unknown variable to obtain the sec-ond answer. This procedure can be used in parts b and d, and in many other situa-tions, to see how changes in input variables affect the output variable.)a. An initial $500 compounded for 1 year at 6 percent.b. An initial $500 compounded for 2 years at 6 percent.c. The present value of $500 due in 1 year at a discount rate of 6 percent.d. The present value of $500 due in 2 years at a discount rate of 6 percent.

28-2 Use equations and a financial calculator to find the following values. See the hintfor Problem 28-1.a. An initial $500 compounded for 10 years at 6 percent.b. An initial $500 compounded for 10 years at 12 percent.c. The present value of $500 due in 10 years at a 6 percent discount rate.d. The present value of $1,552.90 due in 10 years at a 12 percent discount rate

and at a 6 percent rate. Give a verbal definition of the term present value, andillustrate it using a time line with data from this problem. As a part of youranswer, explain why present values are dependent upon interest rates.

28-3 To the closest year, how long will it take $200 to double if it is deposited andearns the following rates? [Notes: (1) See the hint for Problem 28-1. (2) This prob-lem cannot be solved exactly with some financial calculators. For example, if youenter PV � �200, PMT � 0, FV � 400, and I � 7 in an HP-12C, and then pressthe N key, you will get 11 years for part a. The correct answer is 10.2448 years,which rounds to 10, but the calculator rounds up. However, the HP-10B gives thecorrect answer.]a. 7 percent.b. 10 percent.c. 18 percent.d. 100 percent.

Present and Future Valuesfor Different Periods

Present and Future Valuesfor Different Interest

Rates

Time for a Lump Sum toDouble

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28-4 Find the future value of the following annuities. The first payment in these annu-ities is made at the end of Year 1; that is, they are ordinary annuities. (Note:See the hint to Problem 28-1. Also, note that you can leave values in the TVMregister, switch to “BEG,” press FV, and find the FV of the annuity due.)a. $400 per year for 10 years at 10 percent.b. $200 per year for 5 years at 5 percent.c. $400 per year for 5 years at 0 percent.d. Now rework parts a, b, and c assuming that payments are made at the begin-

ning of each year; that is, they are annuities due.

28-5 Find the present value of the following ordinary annuities (see note to Problem28-4):a. $400 per year for 10 years at 10 percent.b. $200 per year for 5 years at 5 percent.c. $400 per year for 5 years at 0 percent.d. Now rework parts a, b, and c assuming that payments are made at the begin-

ning of each year; that is, they are annuities due.

28-6 a. Find the present values of the following cash flow streams. The appropriateinterest rate is 8 percent. (Hint: It is fairly easy to work this problem dealingwith the individual cash flows. However, if you have a financial calculator, readthe section of the manual that describes how to enter cash flows such as theones in this problem. This will take a little time, but the investment will payhuge dividends throughout the course. Note, if you do work with the cash flowregister, then you must enter CF0 � 0.)

Future Value of anAnnuity

Present Value of anAnnuity

Uneven Cash FlowStream

Effective Rate of Interest

Future Value for VariousCompounding Periods

Present Value for VariousCompounding Periods

Year Cash Stream A Cash Stream B

1 $100 $3002 400 4003 400 4004 400 4005 300 100

b. What is the value of each cash flow stream at a 0 percent interest rate?

28-7 Find the interest rates, or rates of return, on each of the following:a. You borrow $700 and promise to pay back $749 at the end of 1 year.b. You lend $700 and receive a promise to be paid $749 at the end of 1 year.c. You borrow $85,000 and promise to pay back $201,229 at the end of 10 years.d. You borrow $9,000 and promise to make payments of $2,684.80 per year for

5 years.

28-8 Find the amount to which $500 will grow under each of the following conditions:a. 12 percent compounded annually for 5 years.b. 12 percent compounded semiannually for 5 years.c. 12 percent compounded quarterly for 5 years.d. 12 percent compounded monthly for 5 years.

28-9 Find the present value of $500 due in the future under each of the following conditions:a. 12 percent nominal rate, semiannual compounding, discounted back 5 years.b. 12 percent nominal rate, quarterly compounding, discounted back 5 years.c. 12 percent nominal rate, monthly compounding, discounted back 1 year.

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28-10 Find the future values of the following ordinary annuities:a. FV of $400 each 6 months for 5 years at a nominal rate of 12 percent,

compounded semiannually.b. FV of $200 each 3 months for 5 years at a nominal rate of 12 percent,

compounded quarterly.c. The annuities described in parts a and b have the same amount of money paid

into them during the 5-year period and both earn interest at the same nominalrate, yet the annuity in part b earns $101.75 more than the one in part a overthe 5 years. Why does this occur?

28-11 Universal Bank pays 7 percent interest, compounded annually, on time deposits.Regional Bank pays 6 percent interest, compounded quarterly.a. Based on effective interest rates, in which bank would you prefer to deposit

your money?b. Could your choice of banks be influenced by the fact that you might want to

withdraw your funds during the year as opposed to at the end of the year? Inanswering this question, assume that funds must be left on deposit during theentire compounding period in order for you to receive any interest.

28-12 a. Set up an amortization schedule for a $25,000 loan to be repaid in equal install-ments at the end of each of the next 5 years. The interest rate is 10 percent.

b. How large must each annual payment be if the loan is for $50,000? Assume thatthe interest rate remains at 10 percent and that the loan is paid off over 5 years.

c. How large must each payment be if the loan is for $50,000, the interest rate is10 percent, and the loan is paid off in equal installments at the end of each ofthe next 10 years? This loan is for the same amount as the loan in part b, butthe payments are spread out over twice as many periods. Why are these pay-ments not half as large as the payments on the loan in part b?

28-13 Hanebury Corporation’s current sales are $12 million. Sales were $6 million 5years earlier.a. To the nearest percentage point, at what rate have sales been growing?b. Suppose someone calculated the sales growth for Hanebury Corporation in

part a as follows: “Sales doubled in 5 years. This represents a growth of100 percent in 5 years, so, dividing 100 percent by 5, we find the growth rateto be 20 percent per year.” Explain what is wrong with this calculation.

28-14 Washington-Pacific invests $4 million to clear a tract of land and to set out someyoung pine trees. The trees will mature in 10 years, at which time Washington-Pacific plans to sell the forest at an expected price of $8 million. What is Washington-Pacific’s expected rate of return?

28-15 A mortgage company offers to lend you $85,000; the loan calls for payments of$8,273.59 per year for 30 years. What interest rate is the mortgage companycharging you?

28-16 To complete your last year in business school and then go through law school,you will need $10,000 per year for 4 years, starting next year (that is, you willneed to withdraw the first $10,000 one year from today). Your rich uncle offers toput you through school, and he will deposit in a bank paying 7 percent interest asum of money that is sufficient to provide the 4 payments of $10,000 each. Hisdeposit will be made today.a. How large must the deposit be?b. How much will be in the account immediately after you make the first with-

drawal? After the last withdrawal?

Future Value of anAnnuity for Various

Compounding Periods

Effective versus NominalInterest Rates

Amortization Schedule

Growth Rates

Expected Rate of Return

Effective Rate of Interest

Required Lump SumPayment

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28-17 While Mary Corens was a student at the University of Tennessee, she borrowed$12,000 in student loans at an annual interest rate of 9 percent. If Mary repays$1,500 per year, how long, to the nearest year, will it take her to repay the loan?

28-18 You need to accumulate $10,000. To do so, you plan to make deposits of $1,250per year, with the first payment being made a year from today, in a bank accountthat pays 12 percent annual interest. Your last deposit will be less than $1,250 ifless is needed to round out to $10,000. How many years will it take you to reachyour $10,000 goal, and how large will the last deposit be?

28-19 What is the present value of a perpetuity of $100 per year if the appropriate dis-count rate is 7 percent? If interest rates in general were to double and the appro-priate discount rate rose to 14 percent, what would happen to the present value ofthe perpetuity?

28-20 Assume that you inherited some money. A friend of yours is working as an unpaidintern at a local brokerage firm, and her boss is selling some securities that call forfour payments, $50 at the end of each of the next 3 years, plus a payment of$1,050 at the end of Year 4. Your friend says she can get you some of these secu-rities at a cost of $900 each. Your money is now invested in a bank that pays an8 percent nominal (quoted) interest rate but with quarterly compounding. Youregard the securities as being just as safe, and as liquid, as your bank deposit, soyour required effective annual rate of return on the securities is the same as thaton your bank deposit. You must calculate the value of the securities to decidewhether they are a good investment. What is their present value to you?

28-21 Assume that your aunt sold her house on December 31 and that she took a mort-gage in the amount of $10,000 as part of the payment. The mortgage has aquoted (or nominal) interest rate of 10 percent, but it calls for payments every 6months, beginning on June 30, and the mortgage is to be amortized over 10 years.Now, 1 year later, your aunt must inform the IRS and the person who bought thehouse of the interest that was included in the two payments made during the year.(This interest will be income to your aunt and a deduction to the buyer of thehouse.) To the closest dollar, what is the total amount of interest that was paidduring the first year?

28-22 Your company is planning to borrow $1,000,000 on a 5-year, 15%, annual pay-ment, fully amortized term loan. What fraction of the payment made at the end ofthe second year will represent repayment of principal?

28-23 a. It is now January 1. You plan to make 5 deposits of $100 each, one every 6months, with the first payment being made today. If the bank pays a nominalinterest rate of 12 percent but uses semiannual compounding, how much willbe in your account after 10 years?

b. You must make a payment of $1,432.02 ten years from today. To prepare forthis payment, you will make 5 equal deposits, beginning today and for the next4 quarters, in a bank that pays a nominal interest rate of 12 percent, quarterlycompounding. How large must each of the 5 payments be?

28-24 Anne Lockwood, manager of Oaks Mall Jewelry, wants to sell on credit, giving cus-tomers 3 months in which to pay. However, Anne will have to borrow from herbank to carry the accounts payable. The bank will charge a nominal 15 percent, butwith monthly compounding. Anne wants to quote a nominal rate to her customers(all of whom are expected to pay on time) that will exactly cover her financingcosts. What nominal annual rate should she quote to her credit customers?

Repaying a Loan

Reaching a FinancialGoal

Present Value of aPerpetuity

PV and Effective AnnualRate

Loan Amortization

Loan Amortization

Nonannual Compounding

Nominal Rate of Return

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28-25 Assume that your father is now 50 years old, that he plans to retire in 10 years,and that he expects to live for 25 years after he retires, that is, until he is 85. Hewants a fixed retirement income that has the same purchasing power at the timehe retires as $40,000 has today (he realizes that the real value of his retirementincome will decline year by year after he retires). His retirement income will beginthe day he retires, 10 years from today, and he will then get 24 additional annualpayments. Inflation is expected to be 5 percent per year from today forward; hecurrently has $100,000 saved up; and he expects to earn a return on his savings of8 percent per year, annual compounding. To the nearest dollar, how much must hesave during each of the next 10 years (with deposits being made at the end of eachyear) to meet his retirement goal?

SPREADSHEET PROBLEM

28-26 Start with the partial model in the file IFM9 Ch28 P26 Build a Model.xls fromthe ThomsonNOW Web site. Answer the following questions, using a spreadsheetmodel to do the calculations.a. Find the FV of $1,000 invested to earn 10 percent after 5 years. Answer this

question by using a math formula and also by using the Excel function wizard.b. Now create a table that shows the FV at 0 percent, 5 percent, and 20 percent

for 0, 1, 2, 3, 4, and 5 years. Then create a graph with years on the horizontalaxis and FV on the vertical axis to display your results.

c. Find the PV of $1,000 due in 5 years if the discount rate is 10 percent. Again,work the problem with a formula and also by using the function wizard.

d. A security has a cost of $1,000 and will return $2,000 after 5 years. What rateof return does the security provide?

e. Suppose California’s population is 30 million people, and its population isexpected to grow by 2 percent per year. How long would it take for the popu-lation to double?

f. Find the PV of an annuity that pays $1,000 at the end of each of the next 5years if the interest rate is 15 percent. Then find the FV of that same annuity.

g. How would the PV and FV of the annuity change if it were an annuity duerather than an ordinary annuity?

h. What would the FV and the PV for parts a and c be if the interest rate were 10percent with semiannual compounding rather than 10 percent with annualcompounding?

i. Find the PV and the FV of an investment that makes the following end-of-yearpayments. The interest rate is 8 percent.

Required AnnuityPayments

Build a Model: The TimeValue of Money

j. Suppose you bought a house and took out a mortgage for $50,000. The inter-est rate is 8 percent, and you must amortize the loan over 10 years with equalend-of-year payments. Set up an amortization schedule that shows the annualpayments and the amount of each payment that goes to pay off the principal

Year Payment

1 $1002 2003 400

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and the amount that constitutes interest expense to the borrower and interestincome to the lender.

(1) Create a graph that shows how the payments are divided between interestand principal repayment over time.

(2) Suppose the loan called for 10 years of monthly payments, with the sameoriginal amount and the same nominal interest rate. What would theamortization schedule show now?

CYBERPROBLEM

Please go to the ThomsonNOW Web site to access any Cyberproblems.

PROBLEM

Please go to the ThomsonNOW Web site to access any Thomson ONE—BusinessSchool Edition problems.

Assume that you are nearing graduation and that youhave applied for a job with a local bank. As part ofthe bank’s evaluation process, you have been askedto take an examination that covers several financialanalysis techniques. The first section of the testaddresses discounted cash flow analysis. See how youwould do by answering the following questions.a. Draw time lines for (a) a $100 lump sum cash

flow at the end of Year 2, (b) an ordinary annu-ity of $100 per year for 3 years, and (c) anuneven cash flow stream of �$50, $100, $75,and $50 at the end of Years 0 through 3.

b. (1) What is the future value of an initial $100after 3 years if it is invested in an accountpaying 10 percent annual interest?

(2) What is the present value of $100 to bereceived in 3 years if the appropriate interestrate is 10 percent?

c. We sometimes need to find how long it will take asum of money (or anything else) to grow to somespecified amount. For example, if a company’s

sales are growing at a rate of 20 percent per year,how long will it take sales to double?

d. If you want an investment to double in 3 years,what interest rate must it earn?

e. What is the difference between an ordinaryannuity and an annuity due? What type of annu-ity is shown below? How would you change itto the other type of annuity?

f. (1) What is the future value of a 3-year ordi-nary annuity of $100 if the appropriateinterest rate is 10 percent?

(2) What is the present value of the annuity?(3) What would the future and present values

be if the annuity were an annuity due?g. What is the present value of the following

uneven cash flow stream? The appropriate inter-est rate is 10 percent, compounded annually.

0 21 3

100 100 100


0 21 3 4 Years

0 –50300300100

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h. (1) Define (a) the stated, or quoted, or nominalrate (iNom) and (b) the periodic rate (iPER).

(2) Will the future value be larger or smaller ifwe compound an initial amount more oftenthan annually, for example, every 6 months,or semiannually, holding the stated interestrate constant? Why?

(3) What is the future value of $100 after 5years under 12 percent annual compound-ing? Semiannual compounding? Quarterlycompounding? Monthly compounding?Daily compounding?

(4) What is the effective annual rate (EAR)?What is the EAR for a nominal rate of12 percent, compounded semiannually?Compounded quarterly? Compoundedmonthly? Compounded daily?

i. Will the effective annual rate ever be equal tothe nominal (quoted) rate?

j. (1) Construct an amortization schedule for a$1,000, 10 percent annual rate loan with 3equal installments.

(2) What is the annual interest expense for theborrower, and the annual interest income forthe lender, during Year 2?

k. Suppose on January 1 you deposit $100 in anaccount that pays a nominal, or quoted, interestrate of 11.33463 percent, with interest added(compounded) daily. How much will youhave in your account on October 1, or after 9months?

l. (1) What is the value at the end of Year 3 of thefollowing cash flow stream if the quoted

interest rate is 10 percent, compoundedsemiannually?

(2) What is the PV of the same stream?(3) Is the stream an annuity?(4) An important rule is that you should never

show a nominal rate on a time line or use itin calculations unless what condition holds?(Hint: Think of annual compounding, wheniNom � EAR � iPER.) What would be wrongwith your answer to Questions l. (1) and l. (2) if you used the nominal rate (10%)rather than the periodic rate (iNom/2 �10%/2 � 5%)?

m. Suppose someone offered to sell you a note call-ing for the payment of $1,000 fifteen monthsfrom today. They offer to sell it to you for $850.You have $850 in a bank time deposit that paysa 6.76649 percent nominal rate with daily com-pounding, which is a 7 percent effective annualinterest rate, and you plan to leave the money inthe bank unless you buy the note. The note isnot risky—you are sure it will be paid on sched-ule. Should you buy the note? Check the deci-sion in three ways: (1) by comparing your futurevalue if you buy the note versus leaving yourmoney in the bank, (2) by comparing the PV ofthe note with your current bank account, and(3) by comparing the EAR on the note versusthat of the bank account.

0 21 3 Years

100 100 1000

For a more complete discussion of the mathematicsof finance, see

Atkins, Allen B., and Edward A. Dyl, “The Lotto Jack-pot: The Lump Sum versus the Annuity,” FinancialPractice and Education, Fall/Winter 1995, pp.107–111.

Lindley, James T., “Compounding Issues Revisited,”Financial Practice and Education, Fall 1993, pp.127–129.

Shao, Lawrence P., and Stephen P. Shao, Mathemat-ics for Management and Finance (Mason, OH:South-Western, 8th ed. 1998).

To learn more about using financial calculators,see the manual that came with your calculator orsee

White, Mark A., Financial Analysis with an ElectronicCalculator, 4th ed. (Burr Ridge, IL: McGraw-Hill/Irwin, 2004).

———, “Financial Problem Solving with an Elec-tronic Calculator: Texas Instruments’ BA II Plus,”Financial Practice and Education, Fall 1993, pp.123–126.

SELECTED ADDITIONAL REFERENCES


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