© goodluz/shutterstock.com time value of money 14e ch05.pdfwere 8.6 workers supporting each...

Will You Be Able to Retire?

Your reaction to that question is probably, “Firstthings first! I’m worried about getting a job, notabout retiring!” However, understanding the retire-ment situation can help you land a job because(1) this is an important issue today; (2) employerslike to hire people who know what’s happening inthe real world; and (3) professors often test on thetime value of money with problems related tosaving for future purposes (including retirement).

A recent study by the Employee BenefitResearch Institute suggests that many U.S. work-ers are not doing enough to prepare for retire-ment. The survey found that 60% of workers hadless than $25,000 in savings and investments (notincluding the values of their homes and definedbenefit plans). Equally concerning, 24% of thosesurveyed said they had “no confidence” that they

would be able to retire comfortably.1 Unfortu-nately, there is no easy solution. In order to reachtheir retirement goals, many current workers willneed to work longer, spend less and save more,and hopefully earn higher returns on their cur-rent savings.

Historically, many Americans have relied onSocial Security as an important source of theirretirement income. However, given currentdemographics, it is likely that this important pro-gram will need to be restructured down the roadin order to maintain its viability. Although theaverage personal savings rate in the UnitedStates had edged up in recent years, in 2013 itwas still at a fairly low level of 4.5%.2 In addition,the ratio of U.S. workers to retirees has steadilydeclined over the past half century. In 1955, there

For an interesting website thatlooks at global savings rates,refer to http://www.gfmag.com/tools/global-database/economic-data/12065-household-savings-rates.html#axzz20fpvnu2g.

© Goodluz/Shutterstock.com

CHAPTER

5Time Value of Money

1Refer to Ruth Helman et al., “The 2014 Retirement Confidence Survey: Confidence Rebounds—for Those withRetirement Plans,” Employee Benefit Research Institute, no. 397, March 2014, ebri.org/survey/rcs.2Refer to the U.S. Department of Commerce: Bureau of Economic Analysis, Personal Saving Rate: January 1,1959–January 1, 2014, http://research.stlouisfed.org/fred2/data/PSAVERT.txt.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

were 8.6 workers supporting each retiree, but by 1975, thatnumber had declined to 3.2 workers for every one retiree. From1975 through 2009, the ratio remained between 3.0 and 3.4workers for every retiree. Current projections show this ratiosignificantly declining in the years ahead—the forecast is for 2.1workers per retiree in 2035 and 2.0 workers per retiree in 2085.3

With so few people paying into the Social Security system andso many drawing funds out, Social Security is going to be inserious trouble. In fact, for the first time since its inception, in2010 (and seven years ahead of schedule), Social Security was inthe red—paying out more in benefits than it received in payroll

tax revenues. Considering these facts, many people may havetrouble maintaining a reasonable standard of living after theyretire, and many of today’s college students will have to supporttheir parents.

This is an important issue for millions of Americans, butmany don’t know how to deal with it. Most Americans havebeen ignoring what is most certainly going to be a hugepersonal and social problem. However, if you study this chap-ter carefully, you can use the tools and techniques presentedhere to avoid the trap that has caught, and is likely to catch,so many people.

Time value analysis has many applications, including planning for retirement, valuingstocks and bonds, setting up loan payment schedules, and making corporate decisionsregarding investing in new plants and equipment. In fact, of all financial concepts, timevalue of money is the single most important concept. Indeed, time value analysis is usedthroughout the book; so it is vital that you understand this chapter before continuing.

You need to understand basic time value concepts, but conceptual knowledge willdo you little good if you can’t do the required calculations. Therefore, this chapter isheavy on calculations. Most students studying finance have a financial or scientificcalculator; some also own or have access to a computer. One of these tools is necessaryto work many finance problems in a reasonable length of time. However, whenstudents begin reading this chapter, many of them don’t know how to use the timevalue functions on their calculator or computer. If you are in that situation, you will findyourself simultaneously studying concepts and trying to learn to use your calculator,and you will need more time to cover this chapter than you might expect.4

When you finish this chapter, you should be able to

• Explain how the time value of money works and discuss why it is such animportant concept in finance.

• Calculate the present value and future value of lump sums.• Identify the different types of annuities, calculate the present value and future

value of both an ordinary annuity and an annuity due, and calculate the relevantannuity payments.

Excellent retirementcalculators are availableat money.msn.com/retirement, ssa.gov, andchoosetosave.org/calculators. These calculatorsallow you to inputhypothetical retirementsavings information; theprogram then shows if currentretirement savings will besufficient to meet retirementneeds.

3Refer to the U.S. Social Security Administration, 2013 Annual Report of the Board of Trustees of the FederalOld-Age and Survivors Insurance and Federal Disability Insurance Trust Funds, Table IV.B2.4Calculator manuals tend to be long and complicated, partly because they cover a number of topics thataren’t required in the basic finance course. We provide tutorials for the most commonly used calculatorson the textbook website and you can access these by going to www.cengagebrain.com and searchingISBN 9781285867977. The tutorials are keyed to this chapter, and they show exactly how to do therequired calculations. If you don’t know how to use your calculator, go to the textbook’s website, find therelevant tutorial, and work through it as you study the chapter.

140 Part 2 Fundamental Concepts in Financial Management


• Calculate the present value and future value of an uneven cash flow stream.You will use this knowledge in later chapters that show how to value commonstocks and corporate projects.

• Explain the difference between nominal, periodic, and effective interest rates.An understanding of these concepts is necessary when comparing rates ofreturns on alternative investments.

• Discuss the basics of loan amortization and develop a loan amortizationschedule that you might use when considering an auto loan or home mortgageloan.

5-1 TIME LINES

The first step in time value analysis is to set up a time line, which will help youvisualize what’s happening in a particular problem. As an illustration, considerthe following diagram, where PV represents $100 that is on hand today, and FV isthe value that will be in the account on a future date:

0 21 35%

Periods

Cash PV = $100 FV = ?

The intervals from 0 to 1, 1 to 2, and 2 to 3 are time periods such as years ormonths. Time 0 is today, and it is the beginning of Period 1; Time 1 is one periodfrom today, and it is both the end of Period 1 and the beginning of Period 2; andso forth. Although the periods are often years, periods can also be quarters ormonths or even days. Note that each tick mark corresponds to both the end of oneperiod and the beginning of the next one. Thus, if the periods are years, the tickmark at Time 2 represents the end of Year 2 and the beginning of Year 3.

Cash flows are shown directly below the tick marks, and the relevant interestrate is shown just above the time line. Unknown cash flows, which you are tryingto find, are indicated by question marks. Here the interest rate is 5%; a single cashoutflow, $100, is invested at Time 0; and the Time 3 value is an unknown inflow.In this example, cash flows occur only at Times 0 and 3, with no flows at Times 1or 2. Note that in our example, the interest rate is constant for all three years. Thatcondition is generally true; but if it were not, we would show different interestrates for the different periods.

Time lines are essential when you are first learning time value concepts, buteven experts use them to analyze complex finance problems; and we use themthroughout the book. We begin each problem by setting up a time line to illustratethe situation, after which we provide an equation that must be solved to find theanswer. Then we explain how to use a regular calculator, a financial calculator,and a spreadsheet to find the answer.

SELF TESTDo time lines deal only with years, or can other time periods be used?

Set up a time line to illustrate the following situation: You currently have $2,000in a 3-year certificate of deposit (CD) that pays a guaranteed 4% annually.

Time LineAn important tool used intime value analysis; it is agraphical representationused to show the timingof cash flows.

Chapter 5 Time Value of Money 141


5-2 FUTURE VALUES

A dollar in hand today is worth more than a dollar to be received in the futurebecause if you had it now, you could invest it, earn interest, and own more than adollar in the future. The process of going to future value (FV) from present value(PV) is called compounding. For an illustration, refer back to our 3-year time line,and assume that you plan to deposit $100 in a bank that pays a guaranteed5% interest each year. How much would you have at the end of Year 3? We firstdefine some terms, and then we set up a time line to show how the future value iscalculated.

PV = Present value, or beginning amount. In our example, PV = $100.FVN = Future value, or ending amount, of your account after N periods.

Whereas PV is the value now, or the present value, FVN is the valueN periods into the future, after the interest earned has been added tothe account.

CFt = Cash flow. Cash flows can be positive or negative. The cash flow fora particular period is often given as a subscript, CFt, where t is theperiod. Thus, CF0 = PV = the cash flow at Time 0, whereas CF3 is thecash flow at the end of Period 3.

I = Interest rate earned per year. Sometimes a lowercase i is used. Interestearned is based on the balance at the beginning of each year, andwe assume that it is paid at the end of the year. Here I = 5% or,expressed as a decimal, 0.05. Throughout this chapter, we designatethe interest rate as I because that symbol (or I/YR, for interest rateper year) is used on most financial calculators. Note, though, that inlater chapters, we use the symbol r to denote rates because r (forrate of return) is used more often in the finance literature. Note toothat in this chapter we generally assume that interest payments areguaranteed by the U.S. government; hence, they are certain. In laterchapters, we consider risky investments, where the interest rateearned might differ from its expected level.

INT = Dollars of interest earned during the year = Beginning amount × I.In our example, INT = $100(0.05) = $5.

N = Number of periods involved in the analysis. In our example, N = 3.Sometimes the number of periods is designated with a lowercase n,so both N and n indicate the number of periods involved.

We can use four different procedures to solve time value problems.5 Thesemethods are described in the following sections.

5-2A STEP-BY-STEP APPROACH

The time line used to find the FV of $100 compounded for 3 years at 5%, alongwith some calculations, is shown. Multiply the initial amount and each succeedingamount by 1 I 1 05 :

5A fifth procedure, using tables that show “interest factors,” was used before financial calculators andcomputers became available. Now, though, calculators and spreadsheet applications such as MicrosoftExcel are programmed to calculate the specific factor needed for a given problem and then to use it tofind the FV. This is more efficient than using the tables. Moreover, calculators and spreadsheets canhandle fractional periods and fractional interest rates, such as the FV of $100 after 3.75 years when theinterest rate is 5.375%, whereas tables provide numbers only for whole periods and rates. For thesereasons, tables are not used in business today; hence, we do not discuss them in the text.

Future Value (FV)The amount to which acash flow or series of cashflows will grow over agiven period of time whencompounded at a giveninterest rate.

Present Value (PV)The value today of afuture cash flow or seriesof cash flows.

CompoundingThe arithmetic process ofdetermining the finalvalue of a cash flow orseries of cash flows whencompound interest isapplied.



Time

Amount at beginning of period $100.00

0 215% 3

$105.00 $110.25 $115.76

You start with $100 in the account—this is shown at t 0:

• You earn $100 0 05 $5 of interest during the first year, so the amount at theend of Year 1 or t 1 is $100 $5 $105.

• You begin the second year with $105, earn 0 05 $105 $5 25 on the nowlarger beginning-of-period amount, and end the year with $110 25. Interestduring Year 2 is $5 25; and it is higher than the first year’s interest, $5 00,because you earned $5 0 05 $0 25 interest on the first year’s interest. This iscalled compounding, and interest earned on interest is called compoundinterest.

• This process continues; and because the beginning balance is higher eachsuccessive year, the interest earned each year increases.

• The total interest earned, $15 76, is reflected in the final balance, $115 76.The step-by-step approach is useful because it shows exactly what is happening.However, this approach is time consuming, especially when a number of years areinvolved; so streamlined procedures have been developed.

5-2B FORMULA APPROACH

In the step-by-step approach, we multiply the amount at the beginning of eachperiod by 1 I 1 05 . If N 3, we multiply by 1 I three different times,which is the same as multiplying the beginning amount by 1 I 3. This conceptcan be extended, and the result is this key equation:

FVN PV 1 I N 5.1

We can apply Equation 5.1 to find the FV in our example:

FV3 $100 1 05 3 $115 76

Equation 5.1 can be used with any calculator that has an exponential function,making it easy to find FVs no matter how many years are involved.

SIMPLE VERSUS COMPOUND INTEREST

Interest earned on the interest earned in prior periods, aswas true in our example and is always true when we applyEquation 5.1, is called compound interest. If interest is notearned on interest, we have simple interest. The formulafor FV with simple interest is FV = PV + PV(I)(N); so in ourexample, FV would have been $100 + $100(0.05)(3) =$100 + $15 = $115 based on simple interest. Most financialcontracts are based on compound interest; but in legalproceedings, the law often specifies that simple interestmust be used. For example, Maris Distributing, a companyfounded by home-run king Roger Maris, won a lawsuitagainst Anheuser-Busch (A-B) because A-B had breached acontract and taken away Maris’s franchise to sell Budweiser

beer. The judge awarded Maris $50 million plus interest at10% from 1997 (when A-B breached the contract) until thepayment was actually made. The interest award was basedon simple interest, which as of 2005 (when a settlement wasreached between A-B and the Maris family) had raised thetotal from $50 million to $50 million + 0.10($50 million)(8 years) = $90 million. (No doubt the sheer size of thisaward and the impact of the interest, even simple interest,influenced A-B to settle.) If the law had allowed compoundinterest, the award would have totaled ($50 million)(1.10)8 =$107.18 million, or $17.18 million more. This legal proceduredates back to the days before calculators and computers.The law moves slowly!

Simple InterestOccurs when interest isnot earned on interest.

Compound InterestOccurs when interest isearned on prior periods’interest.

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5-2C FINANCIAL CALCULATORS

Financial calculators are extremely helpful in working time value problems. Theirmanuals explain calculators in detail; and on the textbook’s website, we providesummaries of the features needed to work the problems in this book for severalpopular calculators. Also see the box entitled “Hints on Using Financial Calcula-tors,” on page 146, for suggestions that will help you avoid common mistakes. Ifyou are not yet familiar with your calculator, we recommend that you workthrough the tutorial as you study this chapter.

First, note that financial calculators have five keys that correspond to the fivevariables in the basic time value equations. We show the inputs for our textexample above the respective keys and the output, the FV, below its key. Becausethere are no periodic payments, we enter 0 for PMT. We describe the keys in moredetail after this calculation.

FVPVI/YRN PMT

0–1003 5

115.76

Where:N = Number of periods. Some calculators use n rather than N.

I/YR = Interest rate per period. Some calculators use i or I rather than I YR.PV = Present value. In our example, we begin by making a deposit, which is

an outflow (the cash leaves our wallet and is deposited at one of manyfinancial institutions); so the PV should be entered with a negative sign.On most calculators, you must enter the 100, then press the − key toswitch from 100 to 100. If you enter −100 directly, 100 will besubtracted from the last number in the calculator, giving you an incor-rect answer.

PMT = Payment. This key is used when we have a series of equal, or constant,payments. Because there are no such payments in our illustrativeproblem, we enter PMT 0. We will use the PMT key when wediscuss annuities later in this chapter.

FV = Future value. In this example, the FV is positive because we enteredthe PV as a negative number. If we had entered the 100 as a positivenumber, the FV would have been negative.

As noted in our example, you enter the known values N, I YR, PV, and PMTand then press the FV key to get the answer, 115 76. Again, note that if you enter thePV as 100 without a minus sign, the FV will be shown on the calculator display as anegative number. The calculator assumes that either the PV or the FV is negative.This should not be confusing if you think about what you are doing. When PMT iszero, it doesn’t matter what sign you enter for PV as your calculator will auto-matically assign the opposite sign to FV. We will discuss this point in greater detaillater in the chapter when we cover annuities.

5-2D SPREADSHEETS6

Students generally use calculators for homework and exam problems; but inbusiness, people generally use spreadsheets for problems that involve the timevalue of money TVM . Spreadsheets show in detail what is happening, and they

6If you have never worked with spreadsheets, you may choose to skip this section. However, youmight want to read through it and refer to this chapter’s Excel model to get an idea of howspreadsheets work.



help reduce both conceptual and data-entry errors. The spreadsheet discussioncan be skipped without loss of continuity, but if you understand the basics ofExcel and have access to a computer, we recommend that you read through thissection. Even if you aren’t familiar with spreadsheets, the discussion will still giveyou an idea of how they operate.

We used Excel to create Table 5.1, which is part of the spreadsheet model thatcorresponds to this chapter. Table 5.1 summarizes the four methods of finding theFV and shows the spreadsheet formulas toward the bottom. Note that spread-sheets can be used to do calculations, but they can also be used like a wordprocessor to create exhibits like Table 5.1, which includes text, drawings, andcalculations. The letters across the top designate columns; the numbers to the leftdesignate rows; and the rows and columns jointly designate cells. Thus, C14 is thecell in which we specify the $100 investment; C15 shows the interest rate; andC16 shows the number of periods. We then created a time line on rows 17 to 19;and on row 21, we have Excel go through the step-by-step calculations, multi-plying the beginning-of-year values by 1 I to find the compounded value atthe end of each period. Cell G21 shows the final result. Then on row 23, weillustrate the formula approach, using Excel to solve Equation 5.1 and find theFV, $115 76. Next, on rows 25 to 27, we show a picture of the calculator solution.Finally, on rows 30 and 31, we use Excel’s built-in FV function to find the answersgiven in cells G30 and G31. The G30 answer is based on fixed inputs, while theG31 answer is based on cell references, which makes it easy to change inputs andsee the effects on the output.

For example, if you want to quickly see how the future value changes ifthe interest rate is 7% instead of 5%, all you need to do is change cell C15 to 7%.Looking at cell G30, you will immediately see that the future value is now$122 50.

Students can download the Excelchapter models from the studentcompanion site on the text’s website.Once downloaded onto yourcomputer, retrieve the Excel chaptermodel and follow along as you readthis chapter.

Summary of Future Value Calculations T A B L E 5 . 1

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If you are using Excel, there are a few things to keep in mind:

• When calculating time value of money problems in Excel, interest rates areentered as percentages or decimals (e.g., 05 or 5%). However, when using thetime value of money function on most financial calculators you generallyenter the interest rate as a whole number (e.g., 5).

• When calculating time value of money problems in Excel, the abbreviation forthe number of periods is nper, whereas for most financial calculators theabbreviation is simply N. Throughout the text, we will use these termsinterchangeably.

• When calculating time value of money problems in Excel, you will oftenbe prompted to enter Type. Type refers to whether the payments come atthe end of the year (in which case Type 0, or you can just omit it), or atthe beginning of the year (in which case Type 1). Most financialcalculators have a BEGIN/END mode function that you toggle on or off toindicate whether the payments come at the beginning or at the end of theperiod.

Table 5.1 demonstrates that all four methods get the same result, but they usedifferent calculating procedures. It also shows that with Excel, all inputs areshown in one place, which makes checking data entries relatively easy. Finally,it shows that Excel can be used to create exhibits, which are quite important in thereal world. In business, it’s often as important to explain what you are doing as itis to “get the right answer,” because if decision makers don’t understand youranalysis, they may reject your recommendations.

HINTS ON USING FINANCIAL CALCULATORSWhen using a financial calculator, make sure it is set up as indicated here. Refer to yourcalculator manual or to our calculator tutorial on the text’s website for information on settingup your calculator.

• One payment per period. Many calculators “come out of the box,” assuming that 12payments are made per year; that is, monthly payments. However, in this book, wegenerally deal with problems in which only one payment is made each year. Therefore,you should set your calculator at one payment per year and leave it there. See our tutorial oryour calculator manual if you need assistance.

• End mode. With most contracts, payments are made at the end of each period. However,some contracts call for payments at the beginning of each period. You can switchbetween “End Mode” and “Begin Mode,” depending on the problem you are solving.Because most of the problems in this book call for end-of-period payments, you shouldreturn your calculator to End Mode after you work a problem where payments are made atthe beginning of periods.

• Negative sign for outflows. Outflows must be entered as negative numbers. This generallymeans typing the outflow as a positive number and then pressing the +/− key to convertfrom + to − before hitting the enter key.

• Decimal places. With most calculators, you can specify from 0 to 11 decimal places. Whenworking with dollars, we generally specify two decimal places. When dealing withinterest rates, we generally specify two places after the decimal when the rate isexpressed as a percentage (e.g., 5.25%), but we specify four decimal places when therate is expressed as a decimal (e.g., 0.0525).

• Interest rates. For arithmetic operations with a nonfinancial calculator, 0.0525 must beused; but with a financial calculator and its TVM keys, you must enter 5.25, not 0.0525,because financial calculators assume that rates are stated as percentages.



5-2E GRAPHIC VIEW OF THE COMPOUNDING PROCESS

Figure 5.1 shows how a $1 investment grows over time at different interest rates.We made the curves by solving Equation 5.1 with different values for N and I. Theinterest rate is a growth rate: If a sum is deposited and earns 5% interest per year,

Q U E S T I O N :At the beginning of your freshman year, your favorite aunt and uncle deposit $10,000 into a4-year bank certificate of deposit (CD) that pays 5% annual interest. You will receive themoney in the account (including the accumulated interest) if you graduate with honors in 4years. How much will there be in the account after 4 years?

A N S W E R :Using the formula approach, we know that FVN PV 1 l N. In this case, you know thatN 4, PV $10,000, and I 0 05. It follows that the future value after 4 years will beFV4 $10,000 1 05 4 $12,155.06. Alternatively, using the calculator approach we canset the problem up as follows:

FVPVI/YRN PMT

0–1000054

12,155.06

Finally, we can use Excel’s FV function:

=FV(0.05,4,0,–10000)FV(rate, nper, pmt, [pv], [type])

Here we find that the future value equals $12,155.06.

F I G U R E 5 . 1 Growth of $1 at Various Interest Rates andTime Periods

Future Value of $1

8 9 106 74 521 30

1.00

2.00

3.00

4.00

5.00

6.00

I = 0%

I = 5%

I = 10%

I = 20%

Periods

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the funds on deposit will grow by 5% per year. Note also that time value conceptscan be applied to anything that grows—sales, population, earnings per share, orfuture salary.

5-3 PRESENT VALUES

Finding a present value is the reverse of finding a future value. Indeed, we simplysolve Equation 5.1, the formula for the future value, for the PV to produce the basicpresent value formula, Equation 5.2:

Future value FVN PV 1 I N 5.1

Present value PVFVN

1 I N 5.2

We illustrate PVs with the following example. A broker offers to sell you aTreasury bond that will pay $115 76 three years from now. Banks are currentlyoffering a guaranteed 5% interest on 3-year certificates of deposit (CDs); and if youdon’t buy the bond, you will buy a CD. The 5% rate paid on the CDs is defined asyour opportunity cost, or the rate of return you could earn on an alternativeinvestment of similar risk. Given these conditions, what’s the most you shouldpay for the bond? We answer this question using the four methods discussed inthe last section—step-by-step, formula, calculator, and spreadsheet. Table 5.2summarizes the results.

First, recall from the future value example in the last section that if youinvested $100 at 5%, it would grow to $115 76 in 3 years. You would also have$115 76 after 3 years if you bought the T-bond. Therefore, the most you shouldpay for the bond is $100—this is its “fair price.” If you could buy the bond for lessthan $100, you should buy it rather than invest in the CD. Conversely, if its pricewas more than $100, you should buy the CD. If the bond’s price was exactly $100,you should be indifferent between the T-bond and the CD.

SELF TESTExplain why this statement is true: A dollar in hand today is worth more thana dollar to be received next year.

What is compounding? What’s the difference between simple interest andcompound interest? What would the future value of $100 be after 5 years at10% compound interest? At 10% simple interest? ($161.05, $150.00)

Suppose you currently have $2,000 and plan to purchase a 3-year certificate ofdeposit (CD) that pays 4% interest compounded annually. How much will youhave when the CD matures? How would your answer change if the interest ratewere 5% or 6% or 20%? ($2,249.73, $2,315.25, $2,382.03, $3,456.00. Hint:With a calculator, enter N = 3, I/YR = 4, PV = −2000, and PMT = 0; then pressFV to get 2,249.73. Enter I/YR = 5 to override the 4%, and press FV again toget the second answer. In general, you can change one input at a time tosee how the output changes.)

A company’s sales in 2015 were $100 million. If sales grow at 8%, what will theybe 10 years later, in 2025? ($215.89 million)

How much would $1 growing at 5% per year be worth after 100 years? Whatwould the FV be if the growth rate were 10%? ($131.50, $13,780.61)

Opportunity CostThe rate of return youcould earn on analternative investment ofsimilar risk.



The $100 is defined as the present value, or PV, of $115 76 due in 3 years whenthe appropriate interest rate is 5%. In general, the present value of a cash flow due Nyears in the future is the amount which, if it were on hand today, would grow to equal thegiven future amount. Because $100 would grow to $115 76 in 3 years at a 5% interestrate, $100 is the present value of $115 76 due in 3 years at a 5% rate. Findingpresent values is called discounting; and as noted above, it is the reverse ofcompounding—if you know the PV, you can compound to find the FV, while ifyou know the FV, you can discount to find the PV.

The top section of Table 5.2 calculates the PV using the step-by-step approach.When we found the future value in the previous section, we worked from left toright, multiplying the initial amount and each subsequent amount by 1 I . Tofind present values, we work backward, or from right to left, dividing the futurevalue and each subsequent amount by 1 I . This procedure shows exactlywhat’s happening, which can be quite useful when you are working complexproblems. However, it’s inefficient, especially when you are dealing with a largenumber of years.

With the formula approach, we use Equation 5.2, simply dividing the futurevalue by 1 I N. This is more efficient than the step-by-step approach, and itgives the same result. Equation 5.2 is built into financial calculators; and as shownin Table 5.2, we can find the PV by entering values for N I YR PMT, and FV andthen pressing the PV key. Finally, Excel’s PV function,

=PV(0.05,3,0,–115.76)PV(rate, nper, pmt, [fv], [type])

can be used. It is essentially the same as the calculator and solves Equation 5.2.The fundamental goal of financial management is to maximize the firm’s

value, and the value of a business (or any asset, including stocks and bonds) isthe present value of its expected future cash flows. Because present value lies at theheart of the valuation process, we will have much more to say about it in theremainder of this chapter and throughout the book.

Summary of Present Value Calculations T A B L E 5 . 2

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DiscountingThe process of finding thepresent value of a cashflow or a series of cashflows; discounting is thereverse of compounding.



5-3A GRAPHIC VIEW OF THE DISCOUNTING PROCESS

Figure 5.2 shows that the present value of a sum to be received in the futuredecreases and approaches zero as the payment date is extended further into thefuture and that the present value falls faster at higher interest rates. At relativelyhigh rates, funds due in the future are worth very little today; and even atrelatively low rates, present values of sums due in the very distant future arequite small. For example, at a 20% discount rate, $1 million due in 100 yearswould be worth only $0 0121 today. This is because $0 0121 would grow to$1 million in 100 years when compounded at 20%.

F I G U R E 5 . 2 Present Value of $1 at Various Interest Ratesand Time Periods

Present Value of $1

40 5020 3010

0.20

0

0.40

0.60

0.80

1.00

I = 20%

I = 10%

I = 5%

I = 0%

Periods

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SELF TESTWhat is discounting, and how is it related to compounding? How is the futurevalue Equation (5.1) related to the present value Equation (5.2)?

How does the present value of a future payment change as the time to receipt islengthened? As the interest rate increases?

Suppose a U.S. government bond promises to pay $2,249.73 three years fromnow. If the going interest rate on 3-year government bonds is 4%, how much isthe bond worth today? How would your answer change if the bond matured in5 years rather than 3? What if the interest rate on the 5-year bond was 6% ratherthan 4%? ($2,000, $1,849.11, $1,681.13)

How much would $1,000,000 due in 100 years be worth today if the discountrate was 5%? If the discount rate was 20%? ($7,604.49, $0.0121)



5-4 FINDING THE INTEREST RATE, IThus far we have used Equations 5.1 and 5.2 to find future and present values.Those equations have four variables; and if we know three of the variables, we cansolve for the fourth. Thus, if we know PV, I, and N, we can solve Equation 5.1 forFV, while if we know FV I, and N, we can solve Equation 5.2 to find PV. That’swhat we did in the preceding two sections.

Now suppose we know PV FV, and N and want to find I. For example,suppose we know that a given bond has a cost of $100 and that it will return$150 after 10 years. Thus, we know PV FV, and N, and we want to find the rate ofreturn we will earn if we buy the bond. Here’s the situation:

FV PV 1 I N

$150 $100 1 I 10

$150 $100 1 I 10

1 5 1 I 10

Unfortunately, we can’t factor I out to produce as simple a formula as wecould for FV and PV. We can solve for I, but it requires a bit more algebra.7

However, financial calculators and spreadsheets can find interest rates almostinstantly. Here’s the calculator setup:

FVPVI/YRN PMT

0 150–10010

4.14

Enter N 10, PV –100, PMT 0, because there are no payments until thesecurity matures, and FV 150. Then when you press the I YR key, the calculatorgives the answer, 4 14%. You would get this same answer using the RATEfunction in Excel:

=RATE(10,0,–100,150)RATE(nper, pmt, pv, [fv], [type], [guess])

Here we find that the interest rate is equal to 4 14%.8

SELF TESTThe U.S. Treasury offers to sell you a bond for $585.43. No payments will bemade until the bond matures 10 years from now, at which time it will beredeemed for $1,000. What interest rate would you earn if you bought thisbond for $585.43? What rate would you earn if you could buy the bond for$550? For $600? (5.5%, 6.16%, 5.24%)

Microsoft earned $0.97 per share in 2003. Ten years later in 2013 it earned $2.65.What was the growth rate in Microsoft’s earnings per share (EPS) over the10-year period? If EPS in 2013 had been $2.10 rather than $2.65, what wouldthe growth rate have been? (10.57%, 8.03%)

7Raise the left side of the equation, the 1.5, to the power 1/N = 1/10 = 0.1, getting 1.0414. That numberis 1 plus the interest rate, so the interest rate is 0.0414 = 4.14%.8The RATE function prompts you to make a guess. In many cases, you can leave this blank, but if Excelis unable to find a solution to the problem, you should enter a reasonable guess, which will help theprogram converge to the correct solution.



5-5 FINDING THE NUMBER OF YEARS, NWe sometimes need to know how long it will take to accumulate a certain sum ofmoney, given our beginning funds and the rate we will earn on those funds. Forexample, suppose we believe that we could retire comfortably if we had $1million. We want to find how long it will take us to acquire $1 million, assumingwe now have $500,000 invested at 4 5%. We cannot use a simple formula—thesituation is like that with interest rates. We can set up a formula that useslogarithms, but calculators and spreadsheets find N very quickly. Here’s thecalculator setup:

FVPVI/YRN PMT

0 1000000–5000004.5

15.7473

Enter I YR 4 5, PV –500000, PMT 0, and FV 1000000. Then when youpress the N key, you get the answer, 15 7473 years. If you plug N 15 7473 intothe FV formula, you can prove that this is indeed the correct number of years:

FV PV 1 I N $500,000 1 045 15 7473 $1,000,000

You can also use Excel’s NPER function:

=NPER(0.045,0,–500000,1000000)NPER(rate, pmt, pv, [fv], [type])

Here we find that it will take 15 7473 years for $500,000 to double at a 4 5%interest rate.

5-6 ANNUITIES

Thus far we have dealt with single payments, or “lump sums.” However, manyassets provide a series of cash inflows over time; and many obligations, such asauto, student, and mortgage loans, require a series of payments. When the pay-ments are equal and are made at fixed intervals, the series is an annuity. Forexample, $100 paid at the end of each of the next 3 years is a 3-year annuity. If thepayments occur at the end of each year, the annuity is an ordinary (or deferred)annuity. If the payments are made at the beginning of each year, the annuity is anannuity due. Ordinary annuities are more common in finance; so when we use theterm annuity in this book, assume that the payments occur at the ends of theperiods unless otherwise noted.

SELF TESTHow long would it take $1,000 to double if it was invested in a bank that paid6% per year? How long would it take if the rate was 10%? (11.9 years,7.27 years)

Microsoft’s 2013 earnings per share were $2.65, and its growth rate during theprior 10 years was 10.57% per year. If that growth rate was maintained, how longwould it take for Microsoft’s EPS to double? (6.90 years)

AnnuityA series of equal paymentsat fixed intervals for aspecified number ofperiods.

Ordinary (Deferred) AnnuityAn annuity whosepayments occur at theend of each period.

Annuity DueAn annuity whosepayments occur at thebeginning of each period.



Here are the time lines for a $100, 3-year, 5% ordinary annuity and for anannuity due. With the annuity due, each payment is shifted to the left by one year.A $100 deposit will be made each year, so we show the payments with minussigns:

Annuity Due:Periods

Payments $100 $100 $100

0 15%

2 3

Ordinary Annuity:Periods

Payments $100 $100 $100

0 15% 2 3

As we demonstrate in the following sections, we can find an annuity’s futureand present values, the interest rate built into annuity contracts, and the length oftime it takes to reach a financial goal using an annuity. Keep in mind that annuitiesmust have constant payments at fixed intervals for a specified number of periods. Ifthese conditions don’t hold, then the payments do not constitute an annuity.

5-7 FUTURE VALUE OF AN ORDINARY ANNUITY

The future value of an annuity can be found using the step-by-step approach orusing a formula, a financial calculator, or a spreadsheet. As an illustration, con-sider the ordinary annuity diagrammed earlier, where you deposit $100 at the endof each year for 3 years and earn 5% per year. How much will you have at the endof the third year? The answer, $315 25, is defined as the future value of theannuity, FVAN; it is shown in Table 5.3.

As shown in the step-by-step section of the table, we compound each paymentout to Time 3, then sum those compounded values to find the annuity’sFV FVA3 $315 25. The first payment earns interest for two periods, the secondpayment earns interest for one period, and the third payment earns no interest atall because it is made at the end of the annuity’s life. This approach is straightfor-ward; but if the annuity extends out for many years, the approach is cumbersomeand time consuming.

As you can see from the time line diagram, with the step-by-step approach,we apply the following equation, with N 3 and I 5%:

FVAN PMT 1 I N 1 PMT 1 I N 2 PMT 1 I N 3

$100 1 05 2 $100 1 05 1 $100 1 05 0

$315 25

SELF TESTWhat’s the difference between an ordinary annuity and an annuity due?

Why would you prefer to receive an annuity due for $10,000 per year for 10 yearsthan an otherwise similar ordinary annuity?

FVAN

The future value of anannuity over N periods.



We can generalize and streamline the equation as follows:

FVAN PMT 1 I N 1 PMT 1 I N 2 PMT 1 I N 3 PMT 1 I 0

PMT1 I N 1

I

5.3

The first line shows the equation in its long form. It can be transformed to thesecond form on the last line, which can be used to solve annuity problems with anonfinancial calculator.9 This equation is also built into financial calculators andspreadsheets. With an annuity, we have recurring payments; hence, the PMT keyis used. Here’s the calculator setup for our illustrative annuity:

FVPVI/YRN PMT

–100 End Mode053

315.25

We enter PV 0 because we start off with nothing, and we enter PMT −100because we plan to deposit this amount in the account at the end of each year.When we press the FV key, we get the answer, FVA3 315 25.

T A B L E 5 . 3 Summary: Future Value of an Ordinary Annuity

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9The long form of the equation is a geometric progression that can be reduced to the second form.



Because this is an ordinary annuity, with payments coming at the end of eachyear, we must set the calculator appropriately. As noted earlier, calculators“come out of the box” set to assume that payments occur at the end of eachperiod, that is, to deal with ordinary annuities. However, there is a key thatenables us to switch between ordinary annuities and annuities due. For ordinaryannuities the designation is “End Mode” or something similar, while for annu-ities due the designation is “Begin” or “Begin Mode” or “Due” or somethingsimilar. If you make a mistake and set your calculator on Begin Mode whenworking with an ordinary annuity, each payment will earn interest for oneextra year. That will cause the compounded amounts, and thus the FVA, to betoo large.

The last approach in Table 5.3 shows the spreadsheet solution using Excel’sbuilt-in function. We can put in fixed values for N, I, PV, and PMT or set up anInput Section where we assign values to those variables, and then input valuesinto the function as cell references. Using cell references makes it easy to changethe inputs to see the effects of changes on the output.

Q U E S T I O N :Your grandfather urged you to begin a habit of saving money early in your life. Hesuggested that you put $5 a day into an envelope. If you follow his advice, at the end ofthe year you will have $1,825 365 $5 . Your grandfather further suggested that you takethat money at the end of the year and invest it in an online brokerage mutual fund accountthat has an annual expected return of 8%.

You are 18 years old. If you start following your grandfather’s advice today, and continuesaving in this way the rest of your life, how much do you expect to have in the brokerageaccount when you are 65 years old?

A N S W E R :This problem is asking you to calculate the future value of an ordinary annuity. Morespecifically, you are making 47 payments of $1,825, where the annual interest rate is 8%.

To quickly find the answer, enter the following inputs into a financial calculator: N 47;I YR 8; PV 0; and PMT –1825. Then solve for the FV of the ordinary annuity bypressing the FV key, FV $826,542.78.

In addition, we can use Excel’s FV function:

=FV(0.08,47,–1825,0)FV(rate, nper, pmt, [pv], [type])

Here we find that the future value is $826,542.78.

You can see your grandfather is right—it definitely pays to start saving early!



5-8 FUTURE VALUE OF AN ANNUITY DUE

Because each payment occurs one period earlier with an annuity due, all of thepayments earn interest for one additional period. Therefore, the FV of an annuitydue will be greater than that of a similar ordinary annuity. If you went throughthe step-by-step procedure, you would see that our illustrative annuity due has anFV of $331 01 versus $315 25 for the ordinary annuity.

With the formula approach, we first use Equation 5.3; but because eachpayment occurs one period earlier, we multiply the Equation 5.3 result by 1 I :

FVAdue FVAordinary 1 I 5.4

Thus, for the annuity due, FVAdue $315 25 1 05 $331 01, which is the sameresult when the period-by-period approach is used. With a calculator, we inputthe variables just as we did with the ordinary annuity; but now we set thecalculator to Begin Mode to get the answer, $331 01.

SELF TESTWhy does an annuity due always have a higher future value than an ordinaryannuity?

If you calculated the value of an ordinary annuity, how could you find the valueof the corresponding annuity due?

Assume that you plan to buy a condo 5 years from now, and you need to savefor a down payment. You plan to save $2,500 per year (with the first depositmade immediately), and you will deposit the funds in a bank account that pays4% interest. How much will you have after 5 years? How much will you have ifyou make the deposits at the end of each year? ($14,082.44, $13,540.81)

SELF TESTFor an ordinary annuity with five annual payments of $100 and a 10% interestrate, how many years will the first payment earn interest? What will this pay-ment’s value be at the end? Answer this same question for the fifth payment.(4 years, $146.41, 0 years, $100)

Assume that you plan to buy a condo 5 years from now, and you estimatethat you can save $2,500 per year. You plan to deposit the money in a bankaccount that pays 4% interest, and you will make the first deposit at the end ofthe year. How much will you have after 5 years? How will your answer change ifthe interest rate is increased to 6% or lowered to 3%? ($13,540.81, $14,092.73,$13,272.84)



5-9 PRESENT VALUE OF AN ORDINARY ANNUITY

The present value of an annuity, PVAN, can be found using the step-by-step,formula, calculator, or spreadsheet method. Look back at Table 5.3. To find the FVof the annuity, we compounded the deposits. To find the PV, we discount them,dividing each payment by 1 I t. The step-by-step procedure is diagrammed asfollows:

210 5%Periods

Payments

3

$100 $100 $100

$ 95.2490.7086.38

$272.32 = Present value of the annuity (PVAN)

Equation 5.5 expresses the step-by-step procedure in a formula. The bracketedform of the equation can be used with a scientific calculator, and it is helpful if theannuity extends out for a number of years:

PVAN PMT 1 I 1 PMT 1 I 2 PMT 1 I N

PMT

11

1 I N

I

$100 1 1 1 05 3 0 05 $272 32

5.5

Calculators are programmed to solve Equation 5.5, so we merely input thevariables and press the PV key, making sure the calculator is set to End Mode. Thecalculator setup follows for both an ordinary annuity and an annuity due. Notethat the PV of the annuity due is larger because each payment is discounted backone less year. Note too that you can find the PV of the ordinary annuity and thenmultiply by 1 I 1 05, calculating $272 32 1 05 $285 94, the PV of theannuity due.

FVPVI/YRN PMT

–100 0 End Mode(OrdinaryAnnuity)

53

272.32

FVPVI/YRN PMT

–100 0 Begin Mode(AnnuityDue)

53

285.94

PVAN

The present value of anannuity of N periods.



Q U E S T I O N :You just won the Florida lottery. To receive your winnings, you must select one of the twofollowing choices:1. You can receive $1,000,000 a year at the end of each of the next 30 years; OR2. You can receive a one-time payment of $15,000,000 today.

Assume that the current interest rate is 6%. Which option is most valuable?

A N S W E R :The most valuable option is the one with the largest present value. You know that thesecond option has a present value of $15,000,000, so we need to determine whether thepresent value of the $1,000,000 30-year ordinary annuity exceeds $15,000,000.

Using the formula approach, we see that the present value of the annuity is:

PVAN PMT1−

11 I N

I

$1,000,0001−

11 06 30

0 06

$13,764,831.15

Alternatively, using the calculator approach we can set up the problem as follows:

FVPVI/YRN PMT

06 –100000030

13,764,831.15

Finally, we can use Excel’s PV function:

=PV(0.06,30,–1000000,0)PV(rate, nper, pmt, [fv], [type])

Here we find that the present value is $13,764,831.15.

Because the present value of the 30-year annuity is less than $15,000,000, you shouldchoose to receive your winnings as a one-time upfront payment.



5-10 FINDING ANNUITY PAYMENTS, PERIODS,AND INTEREST RATES

We can find payments, periods, and interest rates for annuities. Here five vari-ables come into play: N, I, PMT, FV, and PV. If we know any four, we can findthe fifth.

5-10A FINDING ANNUITY PAYMENTS, PMT

Suppose we need to accumulate $10,000 and have it available 5 years from now.Suppose further that we can earn a return of 6% on our savings, which arecurrently zero. Thus, we know that FV 10,000, PV 0, N 5, and I YR 6.We can enter these values in a financial calculator and press the PMT key to findhow large our deposits must be. The answer will, of course, depend on whetherwe make deposits at the end of each year (ordinary annuity) or at the beginning(annuity due). Here are the results for each type of annuity:

Ordinary Annuity:

FVPVI/YRN PMT

100000 End Mode(OrdinaryAnnuity)

65

–1,773.96

We can also use Excel’s PMT function:

=PMT(0.06,5,0,10000PMT(rate, nper, pv, [fv], [type])

Because the deposits are made at the end of the year, we can leave “type” blank.Here we find that an annual deposit of $1 773 96 is needed to reach your goal.

SELF TESTWhy does an annuity due have a higher present value than a similar ordinaryannuity?

If you know the present value of an ordinary annuity, how can you find the PV ofthe corresponding annuity due?

What is the PVA of an ordinary annuity with 10 payments of $100 if theappropriate interest rate is 10%? What would the PVA be if the interest ratewas 4%? What if the interest rate was 0%? How would the PVA values differ if wewere dealing with annuities due? ($614.46, $811.09, $1,000.00, $675.90,$843.53, $1,000.00)

Assume that you are offered an annuity that pays $100 at the end of each yearfor 10 years. You could earn 8% on your money in other investments withequal risk. What is the most you should pay for the annuity? If the paymentsbegan immediately, how much would the annuity be worth? ($671.01,$724.69)



Annuity Due:

FVPVI/YRN PMT

100000 Begin Mode(AnnuityDue)

65

–1,673.55

Alternatively, Excel’s PMT function can be used to calculate the annual deposit forthe annuity due:

=PMT(0.06,5,0,10000,1)PMT(rate, nper, pv, [fv], [type])

Because the deposits are now made at the beginning of the year, enter 1 for type.Here we find that an annual deposit of $1,673 55 is needed to reach your goal.

Thus, you must save $1,773 96 per year if you make deposits at the end of eachyear, but only $1,673 55 if the deposits begin immediately. Note that the requiredannual deposit for the annuity due can also be calculated as the ordinary annuitypayment divided by 1 I $1,773 96 1 06 $1,673 55.

5-10B FINDING THE NUMBER OF PERIODS, NSuppose you decide to make end-of-year deposits, but you can save only $1,200per year. Again assuming that you would earn 6%, how long would it take toreach your $10,000 goal? Here is the calculator setup:

FVPVI/YRN PMT

10000–12000 od6

6.96

End Mode

With these smaller deposits, it would take 6 96 years to reach your $10,000 goal. Ifyou began the deposits immediately, you would have an annuity due, and Nwould be a bit smaller, 6 63 years.

You can also use Excel’s NPER function to arrive at both of these answers. Ifwe assume end-of-year payments, Excel’s NPER function looks like this:

=NPER(0.06,–1200,0,10000)NPER(rate, pmt, pv, [fv], [type])

Here we find that it will take 6 96 years to reach your goal.If we assume beginning-of-year payments, Excel’s NPER function looks like this:

=NPER(0.06,–1200,0,10000,1)NPER(rate, pmt, pv, [fv], [type])

Here we find that it will take only 6 63 years to reach your goal.

5-10C FINDING THE INTEREST RATE, INow suppose you can save only $1,200 annually, but you still need the $10,000 in5 years. What rate of return would enable you to achieve your goal? Here is thecalculator setup:

FVPVI/YRN PMT

10000–12000 End Mode5

25.78



Excel’s RATE function will arrive at the same answer:

=RATE(5,–1200,0,10000)RATE(nper, pmt, pv, [fv], [type], [guess])

Here we find that the interest rate is 25 78%.You must earn a whopping 25 78% to reach your goal. About the only way

to earn such a high return would be to invest in speculative stocks or head to thecasinos in Las Vegas. Of course, investing in speculative stocks and gamblingaren’t like making deposits in a bank with a guaranteed rate of return, so there’sa good chance you’d end up with nothing. You might consider changing yourplans—save more, lower your $10,000 target, or extend your time horizon. Itmight be appropriate to seek a somewhat higher return, but trying to earn25 78% in a 6% market would require taking on more risk than would beprudent.

It’s easy to find rates of return using a financial calculator or a spreadsheet.However, to find rates of return without one of these tools, you would have to gothrough a trial-and-error process, which would be very time consuming if manyyears were involved.

5-11 PERPETUITIES

A perpetuity is simply an annuity with an extended life. Because the payments goon forever, you can’t apply the step-by-step approach. However, it’s easy to findthe PV of a perpetuity with a formula found by solving Equation 5.5 with N set atinfinity:10

SELF TESTSuppose you inherited $100,000 and invested it at 7% per year. What is the mostyou could withdraw at the end of each of the next 10 years and have a zerobalance at Year 10? How would your answer change if you made withdrawals atthe beginning of each year? ($14,237.75, $13,306.31)

If you had $100,000 that was invested at 7% and you wanted to withdraw$10,000 at the end of each year, how long would your funds last? How longwould they last if you earned 0%? How long would they last if you earned the7% but limited your withdrawals to $7,000 per year? (17.8 years, 10 years;forever)

Your uncle named you beneficiary of his life insurance policy. The insurancecompany gives you a choice of $100,000 today or a 12-year annuity of $12,000at the end of each year. What rate of return is the insurance company offering?(6.11%)

Assume that you just inherited an annuity that will pay you $10,000 per year for10 years, with the first payment being made today. A friend of your motheroffers to give you $60,000 for the annuity. If you sell it, what rate of return wouldyour mother’s friend earn on his investment? If you think a “fair” return would be6%, how much should you ask for the annuity? (13.70%, $78,016.92)

10Equation 5.6 was found by letting N in Equation 5.5 approach infinity.

PerpetuityA stream of equalpayments at fixedintervals expected tocontinue forever.



PV of a perpetuityPMT

I 5.6

Let’s say, for example, that you buy preferred stock in a company that pays you afixed dividend of $2 50 each year the company is in business. If we assume thatthe company will go on indefinitely, the preferred stock can be valued as aperpetuity. If the discount rate on the preferred stock is 10%, the present valueof the perpetuity, the preferred stock, is $25:

PV of a perpetuity$2 500 10

$25

5-12 UNEVEN CASH FLOWS

The definition of an annuity includes the words constant payment—in other words,annuities involve payments that are equal in every period. Although many finan-cial decisions involve constant payments, many others involve uneven, or non-constant, cash flows. For example, the dividends on common stocks typicallyincrease over time, and investments in capital equipment almost always generateuneven cash flows. Throughout the book, we reserve the term payment (PMT) forannuities with their equal payments in each period and use the term cash flow(CFt) to denote uneven cash flows, where t designates the period in which the cashflow occurs.

There are two important classes of uneven cash flows: (1) a stream thatconsists of a series of annuity payments plus an additional final lump sum and(2) all other uneven streams. Bonds represent the best example of the first type,while stocks and capital investments illustrate the second type. Here are numericalexamples of the two types of flows:

4210 I = 12%Periods1. Annuity plus additional final payment:

2. Irregular cash flows:

Cash flows

3 5

$0 $100 $100$100$100 $ 1001,000

$1,100

4210 I = 12%Periods

Cash flows

3 5

$0 $100 $300 $500$300$300

SELF TESTWhat’s the present value of a perpetuity that pays $1,000 per year beginning 1year from now, if the appropriate interest rate is 5%? What would the value be ifpayments on the annuity began immediately? ($20,000, $21,000. Hint: Just addthe $1,000 to be received immediately to the value of the annuity.)

Cash Flow (CFt)This term designates acash flow that’s not partof an annuity.

Uneven (Nonconstant) CashFlowsA series of cash flowswhere the amount variesfrom one period to thenext.

Payment (PMT)This term designates equalcash flows coming atregular intervals.



We can find the PV of either stream by using Equation 5.7 and following the step-by-step procedure, where we discount each cash flow and then sum them to findthe PV of the stream:

PVCF1

1 I 1

CF2

1 I 2

CFN

1 I N

N

t 1

CFt

1 I t 5.7

If we did this, we would find the PV of Stream 1 to be $927 90 and the PV ofStream 2 to be $1 016 35.

The step-by-step procedure is straightforward; but if we have a large numberof cash flows, it is time consuming. However, financial calculators speed up theprocess considerably. First, consider Stream 1; notice that we have a 5-year, 12%ordinary annuity plus a final payment of $1 000. We could find the PV of theannuity, and then find the PV of the final payment and sum them to obtain the PVof the stream. Financial calculators do this in one simple step—use the five TVMkeys; enter the data as shown below; and press the PV key to obtain the answer,$927 90.

FVPVI/YRN PMT

10001005 12

–927.90

The solution procedure is different for the second uneven stream. Here wemust use the step-by-step approach, as shown in Figure 5.3. Even calculators andspreadsheets solve the problem using the step-by-step procedure, but they do itquickly and efficiently. First, you enter all of the cash flows and the interest rate;then the calculator or computer discounts each cash flow to find its present valueand sums these PVs to produce the PV of the stream. You must enter each cashflow in the calculator’s “cash flow register,” enter the interest rate, and then pressthe NPV key to find the PV of the stream. NPV stands for “net present value.” Wecover the calculator mechanics in the calculator tutorial, and we discuss theprocess in more detail in Chapters 9 and 11, where we use the NPV calculationto analyze stocks and proposed capital budgeting projects. If you don’t know howto do the calculation with your calculator, it would be worthwhile to review thetutorial or your calculator manual, learn the steps, and make sure you can do thiscalculation. Because you will have to learn to do it eventually, now is a good timeto begin.

F I G U R E 5 . 3 PV of an Uneven Cash Flow Stream

$ 89.29239.16213.53190.66283.71

$1,016.35 = PV of cash flow stream = Value of the asset

4210 I = 12%Periods

Cash flowsPV of CFs

3 5

$0 $100 $300$300$300 $500

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5-13 FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM

We find the future value of uneven cash flow streams by compounding ratherthan discounting. Consider Cash Flow Stream 2 in the preceding section. Wediscounted those cash flows to find the PV, but we would compound them tofind the FV. Figure 5.4 illustrates the procedure for finding the FV of the stream,using the step-by-step approach.

The values of all financial assets—stocks, bonds, and business capital invest-ments—are found as the present values of their expected future cash flows. There-fore, we need to calculate present values very often, far more often than futurevalues. As a result, all financial calculators provide automated functions forfinding PVs, but they generally do not provide automated FV functions. On therelatively few occasions when we need to find the FV of an uneven cash flowstream, we generally use the step-by-step procedure shown in Figure 5.4. Thatapproach works for all cash flow streams, even those for which some cash flowsare zero or negative.11

SELF TESTHow could you use Equation 5.2 to find the PV of an uneven stream of cashflows?

What’s the present value of a 5-year ordinary annuity of $100 plus an additional$500 at the end of Year 5 if the interest rate is 6%? What is the PV if the $100payments occur in Years 1 through 10 and the $500 comes at the end of Year10? ($794.87, $1,015.21)

What’s the present value of the following uneven cash flow stream: $0 at Time 0,$100 in Year 1 (or at Time 1), $200 in Year 2, $0 in Year 3, and $400 in Year 4 ifthe interest rate is 8%? ($558.07)

Would a typical common stock provide cash flows more like an annuity or morelike an uneven cash flow stream? Explain.

F I G U R E 5 . 4 FV of an Uneven Cash Flow Stream

$ 500.00336.00376.32421.48157.35

0.00$1,791.15

4210 I = 12%Periods

Cash flows

3 5

$0 $100 $300$300$300 $500

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11The HP 10bII+ calculator provides a shortcut to finding the FV of a cash flow stream. Enter the cashflows into the cash flow register; input the interest rate; and calculate the net present value of thestream. Once the NPV of the stream is calculated, simply press SWAP, and the calculator will displaythe FV of the cash flow stream.



5-14 SOLVING FOR I WITH UNEVEN CASH FLOWS12

Before financial calculators and spreadsheets existed, it was extremely difficult tofind I when the cash flows were uneven. With spreadsheets and financial calcu-lators, however, it’s relatively easy to find I. If you have an annuity plus a finallump sum, you can input values for N, PV, PMT, and FV into the calculator’s TVMregisters and then press the I YR key. Here is the setup for Stream 1 from Section5-12, assuming we must pay $927 90 to buy the asset. The rate of return on the$927 90 investment is 12%.

FVPVI/YRN PMT

1000100–927.905

12.00

Finding the interest rate for an uneven cash flow stream such as Stream 2 is abit more complicated. First, note that there is no simple procedure—finding therate requires a trial-and-error process, which means that a financial calculator or aspreadsheet is needed. With a calculator, we enter each CF into the cash flowregister and then press the IRR key to get the answer. IRR stands for “internal rateof return,” and it is the rate of return the investment provides. The investment isthe cash flow at Time 0, and it must be entered as a negative value. As anillustration, consider the cash flows given here, where CF0 −$1,000 is the costof the asset.

4210Periods

Cash flows

3 5

$1,000

IRR I 12.55%

$100 $300$300$300 $500

When we enter those cash flows into the calculator’s cash flow register andpress the IRR key, we get the rate of return on the $1,000 investment, 12 55%. Youget the same answer using Excel’s IRR function. This process is covered in thecalculator tutorial; it is also discussed in Chapter 11, where we study capitalbudgeting.

SELF TESTWhy are we more likely to need to calculate the PV of cash flow streams than theFV of streams?

What is the future value of this cash flow stream: $100 at the end of 1 year, $150due after 2 years, and $300 due after 3 years, if the appropriate interest rate is15%? ($604.75)

12This section is relatively technical. It can be deferred at this point, but the calculations will be requiredin Chapter 11.



5-15 SEMIANNUAL AND OTHER COMPOUNDING PERIODS

In all of our examples thus far, we assumed that interest was compounded once ayear, or annually. This is called annual compounding. Suppose, however, thatyou deposit $100 in a bank that pays a 5% annual interest rate but credits interesteach 6 months. So in the second 6-month period, you earn interest on your original$100 plus interest on the interest earned during the first 6 months. This is calledsemiannual compounding. Note that banks generally pay interest more than oncea year; virtually all bonds pay interest semiannually; and most mortgages, studentloans, and auto loans require monthly payments. Therefore, it is important tounderstand how to deal with nonannual compounding.

For an illustration of semiannual compounding, assume that we deposit $100in an account that pays 5% and leave it there for 10 years. First, consider againwhat the future value would be under annual compounding:

FVN PV 1 I N $100 1 05 10 $162 89

We would, of course, get the same answer using a financial calculator or aspreadsheet.

How would things change in this example if interest was paid semiannuallyrather than annually? First, whenever payments occur more than once a year, youmust make two conversions: (1) Convert the stated interest rate into a “periodicrate” and (2) convert the number of years into “number of periods.” The conver-sions are done as follows, where I is the stated annual rate, M is the number ofcompounding periods per year, and N is the number of years:

Periodic rate IPERStated annual rate

Number of payments per yearI M 5.8

With a stated annual rate of 5%, compounded semiannually, the periodic rateis 2 5%:

Periodic rate 5% 2 2 5%

The number of compounding periods is found with Equation 5.9:

Number of periods Number of years Periods per year NM 5.9

With 10 years and semiannual compounding, there are 20 periods:

Number of periods 10 2 20 periods

SELF TESTAn investment costs $465 and is expected to produce cash flows of $100 at theend of each of the next 4 years, then an extra lump sum payment of $200 atthe end of the fourth year. What is the expected rate of return on thisinvestment? (9.05%)

An investment costs $465 and is expected to produce cash flows of $100 at theend of Year 1, $200 at the end of Year 2, and $300 at the end of Year 3. What isthe expected rate of return on this investment? (11.71%)

Annual CompoundingThe arithmetic process ofdetermining the finalvalue of a cash flow orseries of cash flows wheninterest is added once ayear.

Semiannual CompoundingThe arithmetic process ofdetermining the finalvalue of a cash flow orseries of cash flows wheninterest is added twice ayear.



Under semiannual compounding, our $100 investment will earn 2 5% every 6months for 20 semiannual periods, not 5% per year for 10 years. The periodic rateand number of periods, not the annual rate and number of years, must be shownon time lines and entered into the calculator or spreadsheet whenever you areworking with nonannual compounding.13

With this background, we can find the value of $100 after 10 years if it is heldin an account that pays a stated annual rate of 5 0%, but with semiannualcompounding. Here’s the time line and the future value:

19210Periods

Cash flows

I = 2.5% 20

$100 PV (1 + I)N = $100(1.025)20 = FV20 = $163.86

With a financial calculator, we get the same result using the periodic rate andnumber of periods:

FVPVI/YRN PMT

0–100

163.86

2.520

The future value under semiannual compounding, $163 86, exceeds the FV underannual compounding, $162 89, because interest starts accruing sooner; thus, youearn more interest on interest.

How would things change in our example if interest was compounded quar-terly or monthly or daily? With quarterly compounding, there would beNM 10 4 40 periods and the periodic rate would be I M 5% 4 1 25%per quarter. Using those values, we would find FV $164 36. If we used monthlycompounding, we would have 10 12 120 periods, the monthly rate wouldbe 5% 12 0 416667%, and the FV would rise to $164 70. If we went to dailycompounding, we would have 10 365 3,650 periods, the daily rate wouldbe 5% 365 0 0136986% per day, and the FV would be $164 87 (based on a365-day year).

The same logic applies when we find present values under semiannual com-pounding. Again, we use Equation 5.8 to convert the stated annual rate to theperiodic (semiannual) rate and Equation 5.9 to find the number of semiannualperiods. We then use the periodic rate and number of periods in the calculations.For example, we can find the PV of $100 due after 10 years when the stated annualrate is 5%, with semiannual compounding:

Periodic rate 5% 2 2 5% per period

Number of periods 10 2 20 periods

PV of $100 $100 1 025 20 $61 03

13With some financial calculators, you can enter the annual (nominal) rate and the number ofcompounding periods per year rather than make the conversions we recommend. We prefer theconversions because they must be used on time lines and because it is easy to forget to reset yourcalculator after you change its settings, which may lead to an error on your next calculations.



We would get this same result with a financial calculator:

FVPVI/YRN PMT

0 –1002.520

61.03

If we increased the number of compounding periods from 2 (semiannual) to12 (monthly), the PV would decline to $60 72; and if we went to daily compounding,the PV would fall to $60 66.

5-16 COMPARING INTEREST RATES

Different compounding periods are used for different types of investments. Forexample, bank accounts generally pay interest daily; most bonds pay interestsemiannually; stocks pay dividends quarterly; and mortgages, auto loans, andother instruments require monthly payments.14 If we are to compare investmentsor loans with different compounding periods properly, we need to put them on acommon basis. Here are some terms you need to understand:

• The nominal interest rate (INOM), also called the annual percentage rate (APR)(or quoted or stated rate), is the rate that credit card companies, student loanofficers, auto dealers, and other lenders tell you they are charging on loans.Note that if two banks offer loans with a stated rate of 8%, but one requiresmonthly payments and the other quarterly payments, they are not chargingthe same “true” rate. The one that requires monthly payments is chargingmore than the one with quarterly payments because it will receive yourmoney sooner. So to compare loans across lenders, or interest rates earnedon different securities, you should calculate effective annual rates asdescribed here.15

• The effective annual rate, abbreviated EFF%, is also called the equivalentannual rate (EAR). This is the rate that would produce the same future value

SELF TESTWould you rather invest in an account that pays 7% with annual compoundingor 7% with monthly compounding? Would you rather borrow at 7% and makeannual or monthly payments? Why?

What’s the future value of $100 after 3 years if the appropriate interest rate is 8%compounded annually? Compounded monthly? ($125.97, $127.02)

What’s the present value of $100 due in 3 years if the appropriate interest rate is8% compounded annually? Compounded monthly? ($79.38, $78.73)

Annual Percentage Rate(APR)The periodic rate times thenumber of periods peryear.

Nominal (Quoted or Stated)Interest Rate, INOM

The contracted interestrate.

Effective (Equivalent)Annual Rate (EFF% or EAR)The annual rate of interestactually being earned, asopposed to the quotedrate.

14Some banks even pay interest compounded continuously. Continuous compounding is discussed inWeb Appendix 5A.15Note, though, that if you are comparing two bonds that both pay interest semiannually, it’s okayto compare their nominal rates. Similarly, you can compare the nominal rates on two money fundsthat pay interest daily. But don’t compare the nominal rate on a semiannual bond with the nominalrate on a money fund that compounds daily because that will make the money fund look worse than itreally is.



under annual compounding as would more frequent compounding at a givennominal rate.

• If a loan or an investment uses annual compounding, its nominal rate is alsoits effective rate. However, if compounding occurs more than once a year, theEFF% is higher than INOM.

• To illustrate, a nominal rate of 10% with semiannual compounding isequivalent to a rate of 10 25% with annual compounding because both rateswill cause $100 to grow to the same amount after 1 year. The top line in thefollowing diagram shows that $100 will grow to $110 25 at a nominal rate of10 25%. The lower line shows the situation if the nominal rate is 10% butsemiannual compounding is used.

0 Nom = EFF% = 10.25% 1

$100.00 $110.25

0 Nom = 10.00% semi; EFF% = 10.25% 1 2

$100.00 $105 $110.25

Given the nominal rate and the number of compounding periods per year, wecan find the effective annual rate with this equation:

Effective annual rate EFF% 1INOM

M

M

1 0 5.10

Here INOM is the nominal rate expressed as a decimal and M is the number ofcompounding periods per year. In our example, the nominal rate is 10%; butwith semiannual compounding, INOM 10% 0 10 and M 2. This results inEFF% 10 25%:

Effective annual rate EFF% 10 10

2

2

1 0 1025 10 25%

We can also use the EFFECT function in Excel to solve for the effective rate:

=EFFECT (0.1,2)EFFECT(nominal_rate, npery)

Here we find that the effective rate is 10 25%. NPERY refers to the number ofpayments per year. Likewise, if you know the effective rate and want to solve forthe nominal rate, you can use the NOMINAL function in Excel.16 Thus, if oneinvestment promises to pay 10% with semiannual compounding, and an equallyrisky investment promises 10 25% with annual compounding, we would be indif-ferent between the two.

16Most financial calculators are programmed to find the EFF% or, given the EFF%, to find the nominalrate. This is called interest rate conversion. You enter the nominal rate and the number of compoundingperiods per year and then press the EFF% key to find the effective annual rate. However, we generallyuse Equation 5.10 because it’s as easy to use as the interest conversion feature, and the equationreminds us of what we are really doing. If you use the interest rate conversion feature on yourcalculator, don’t forget to reset your calculator settings. Interest rate conversion is discussed in thecalculator tutorials.



Q U E S T I O N :You just received your first credit card and decided to purchase a new Apple iPad. Youcharged the iPad’s $500 purchase price on your new credit card. Assume that the nominalinterest rate on the credit card is 18% and that interest is compounded monthly.

The minimum payment on the credit card is only $10 a month. If you pay the minimum andmake no other charges, how long will it take you to fully pay off the credit card?

A N S W E R :Here we are given that the nominal interest rate is 18%. It follows that the monthly periodicrate is 1 5% 18% 12 . Using a financial calculator, we can solve for the number of monthsthat it takes to pay off the credit card.

FVPVI/YRN PMT

1.5 500 –10 0

93.11

We can also use Excel’s NPER function:

=NPER(0.015,–10,500,0)NPER(rate, pmt, pv, [fv], [type])

Here we find that it will take 93 11 months to pay off the credit card.

Note that it would take you almost 8 years to pay off your iPad purchase. Now, you seewhy you can quickly get into financial trouble if you don’t manage your credit cardswisely!

SELF TESTDefine the terms annual percentage rate (APR), effective annual rate (EFF%), andnominal interest rate (INOM).

A bank pays 5% with daily compounding on its savings accounts. Should itadvertise the nominal or effective rate if it is seeking to attract newdeposits?

By law, credit card issuers must print their annual percentage rate on theirmonthly statements. A common APR is 18% with interest paid monthly. Whatis the EFF% on such a loan? [EFF% = (1 + 0.18/12)12 − 1 = 0.1956 = 19.56%]

Some years ago banks didn’t have to reveal the rates they charged on creditcards. Then Congress passed the Truth in Lending Act that required banks topublish their APRs. Is the APR really the most truthful rate, or would the EFF% bemore truthful? Explain.



5-17 FRACTIONAL TIME PERIODS

Thus far we have assumed that payments occur at the beginning or the end ofperiods but not within periods. However, we often encounter situations thatrequire compounding or discounting over fractional periods. For example, sup-pose you deposited $100 in a bank that pays a nominal rate of 10% but addsinterest daily, based on a 365-day year. How much would you have after 9months? The answer is $107 79, found as follows:17

Periodic rate IPER 0 10 365 0 000273973 per day

Number of days 9 12 365 0 75 365 273 75, rounded to 274

Ending amount $100 1 000273973 274 $107 79

Now suppose you borrow $100 from a bank whose nominal rate is 10% peryear simple interest, which means that interest is not earned on interest. If the loanis outstanding for 274 days, how much interest would you have to pay? Here wewould calculate a daily interest rate, IPER, as just shown, but multiply it by 274rather than use the 274 as an exponent:

Interest owed $100 0 000273973 274 $7 51

You would owe the bank a total of $107 51 after 274 days. This is the procedurethat most banks use to calculate interest on loans, except that they require bor-rowers to pay the interest on a monthly basis rather than after 274 days.

SELF TESTSuppose a company borrowed $1 million at a rate of 9% simple interest,with interest paid at the end of each month. The bank uses a 360-dayyear. How much interest would the firm have to pay in a 30-day month? Whatwould the interest be if the bank used a 365-day year? [(0.09/360)(30)($1,000,000) = $7,500 interest for the month. For the 365-day year, (0.09/365)(30)($1,000,000) = $7,397.26 of interest. The use of a 360-day yearraises the interest cost by $102.74, which is why banks like to use it onloans.]

Suppose you deposited $1,000 in a credit union account that pays 7% with dailycompounding and a 365-day year. What is the EFF%, and how much could youwithdraw after seven months, assuming this is seven-twelfths of a year? [EFF% =(1 + 0.07/365)365 − 1 = 0.07250098 = 7.250098%. Thus, your account wouldgrow from $1,000 to $1,000(1.07250098)0.583333 = $1,041.67, and you couldwithdraw that amount.]

17Bank loan contracts specifically state whether they are based on a 360- or a 365-day year. If a 360-dayyear is used, the daily rate is higher, which means that the effective rate is also higher. Here weassumed a 365-day year. Also note that in real-world calculations, banks’ computers have built-incalendars, so they can calculate the exact number of days, taking account of 30-day, 31-day, and 28- or29-day months.



5-18 AMORTIZED LOANS18

An important application of compound interest involves loans that are paid off ininstallments over time. Included are automobile loans, home mortgage loans,student loans, and many business loans. A loan that is to be repaid in equalamounts on a monthly, quarterly, or annual basis is called an amortized loan.19

Table 5.4 illustrates the amortization process. A homeowner borrows $100,000on a mortgage loan, and the loan is to be repaid in five equal payments at the endof each of the next 5 years.20 The lender charges 6% on the balance at thebeginning of each year. Our first task is to determine the payment the homeownermust make each year. Here’s a picture of the situation:

0 I = 6% 3 4

PMTPMTPMTPMTPMT$100,000

521

The payments must be such that the sum of their PVs equals $100,000:

$100,000PMT

1 06 1

PMT

1 06 2

PMT

1 06 3

PMT

1 06 4

PMT

1 06 5

5

t 1

PMT

1 06 t

We could insert values into a calculator as follows to get the required payments,$23 739 64:21

FVPVI/YRN PMT

100000 065

–23,739.64

Therefore, the borrower must pay the lender $23,739 64 per year for the next5 years.

Each payment will consist of two parts—interest and repayment of principal.This breakdown is shown on an amortization schedule, such as the one in Table5.4. The interest component is relatively high in the first year, but it declines as theloan balance decreases. For tax purposes, the borrower would deduct the interestcomponent, and the lender would report the same amount as taxable income.

SELF TESTSuppose you borrowed $30,000 on a student loan at a rate of 8% and must repayit in three equal installments at the end of each of the next 3 years. How largewould your payments be; how much of the first payment would represent interest;how much would be principal; and what would your ending balance be after thefirst year? (PMT = $11,641.01; Interest = $2,400; Principal = $9,241.01; Balanceat end of Year 1 = $20,758.99)

Amortization ScheduleA table showing preciselyhow a loan will be repaid.It gives the requiredpayment on eachpayment date and abreakdown of thepayment, showing howmuch is interest and howmuch is repayment ofprincipal.

18Amortized loans are important, but this section can be omitted without loss of continuity.19The word amortized comes from the Latin mors, meaning “death”; so an amortized loan is one that is“killed off” over time.20Most mortgage loans call for monthly payments over 10 to 30 years, but we use a shorter period toreduce the calculations.21You could also factor out the PMT term; find the value of the remaining summation term (4.212364);and divide it into the $100,000 to find the payment, $23,739.64.

Amortized LoanA loan that is repaid inequal payments over itslife.



In this chapter, we worked with single payments, ordinary annuities, annuitiesdue, perpetuities, and uneven cash flow streams. One fundamental equation,Equation 5.1, is used to calculate the future value of a given amount. The equationcan be transformed to Equation 5.2 and then used to find the present value of agiven future amount. We used time lines to show when cash flows occur; and wesaw that time value problems can be solved in a step-by-step manner when wework with individual cash flows, with formulas that streamline the approach, withfinancial calculators, and with spreadsheets.

As we noted at the outset, TVM is the single most important concept in finance,and the procedures developed in Chapter 5 are used throughout this book. Timevalue analysis is used to find the values of stocks, bonds, and capital budgetingprojects. It is also used to analyze personal finance problems, such as the retire-ment issue set forth in the opening vignette. You will become more familiar withtime value analysis as you go through the book, but we strongly recommend thatyou get a good handle on Chapter 5 before you continue.

Loan Amortization Schedule, $100,000 at 6% for 5 Years T A B L E 5 . 4

Amount borrowed: $100,000Years: 5Rate: 6%PMT: −$23,739.64

BeginningAmount Payment Interesta

Repayment ofPrincipalb

EndingBalance

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

1 $100,000.00 $23,739.64 $6,000.00 $17,739.64 $82,260 362 82,260 36 23,739.64 4,935.62 18,804.02 63,456 343 63,456 34 23,739.64 3,807 38 19,932.26 43,524.084 43,524.08 23,739.64 2,611.44 21,128.20 22,395.895 22,395.89 23,739.64 1,343.75 22,395.89 0.00

Notes:aInterest in each period is calculated by multiplying the loan balance at the beginning of theyear by the interest rate. Therefore, interest in Year 1 is $100,000.00(0.06) = $6,000; in Year 2, it is$4,935.62; and so forth.bRepayment of principal is equal to the payment of $23,739.64 minus the interest charge forthe year.

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(Solutions Appear in Appendix A)

ST-1 KEY TERMS Define each of the following terms:

a. Time lineb. FVN; PV; I; INT; N; FVAN; PMT; PVAN

c. Compounding; discountingd. Simple interest; compound intereste. Opportunity costf. Annuity; ordinary (deferred) annuity; annuity due; perpetuityg. Uneven (nonconstant) cash flow; payment PMT); cash flow (CFt

h. Annual compounding; semiannual compoundingi. Nominal (quoted) interest rate; annual percentage rate APR ; effective (equivalent)

annual rate EAR or EFFj. Amortized loan; amortization schedule

ST-2 FUTURE VALUE It is now January 1, 2015. Today you will deposit $1,000 into a savingsaccount that pays 8%.

a. If the bank compounds interest annually, how much will you have in your account onJanuary 1, 2018?

b. What will your January 1, 2018, balance be if the bank uses quarterly compounding?c. Suppose you deposit $1,000 in three payments of $333 333 each on January 1 of 2016,

2017, and 2018. How much will you have in your account on January 1, 2018, based on8% annual compounding?

d. How much will be in your account if the three payments begin on January 1, 2015?e. Suppose you deposit three equal payments into your account on January 1 of 2016,

2017, and 2018. Assuming an 8% interest rate, how large must your payments be tohave the same ending balance as in part a?

ST-3 TIME VALUE OF MONEY It is now January 1, 2015; and you will need $1,000 on January 1,2019, in 4 years. Your bank compounds interest at an 8% annual rate.

a. How much must you deposit today to have a balance of $1,000 on January 1, 2019?b. If you want to make four equal payments on each January 1 from 2016 through 2019 to

accumulate the $1,000, how large must each payment be? (Note that the paymentsbegin a year from today.)

c. If your father offers to make the payments calculated in part b ($221 92) or to give you$750 on January 1, 2016 (a year from today), which would you choose? Explain.

d. If you have only $750 on January 1, 2016, what interest rate, compounded annually for3 years, must you earn to have $1,000 on January 1, 2019?

e. Suppose you can deposit only $200 each January 1 from 2016 through 2019 (4 years).What interest rate, with annual compounding, must you earn to end up with $1,000 onJanuary 1, 2019?

f. Your father offers to give you $400 on January 1, 2016. You will then make sixadditional equal payments each 6 months from July 2016 through January 2019. If yourbank pays 8% compounded semiannually, how large must each payment be for you toend up with $1,000 on January 1, 2019?

g. What is the EAR, or EFF%, earned on the bank account in part f? What is the APRearned on the account?

ST-4 EFFECTIVE ANNUAL RATES Bank A offers loans at an 8% nominal rate (its APR)but requires that interest be paid quarterly; that is, it uses quarterly compounding.Bank B wants to charge the same effective rate on its loans, but it wants to collectinterest on a monthly basis, that is, to use monthly compounding. What nominal ratemust Bank B set?



QUESTIONS

5-1 What is an opportunity cost? How is this concept used in TVM analysis, and where is itshown on a time line? Is a single number used in all situations? Explain.

5-2 Explain whether the following statement is true or false: $100 a year for 10 years is anannuity; but $100 in Year 1, $200 in Year 2, and $400 in Years 3 through 10 does notconstitute an annuity. However, the second series contains an annuity.

5-3 If a firm’s earnings per share grew from $1 to $2 over a 10-year period, the total growthwould be 100%, but the annual growth rate would be less than 10%. True or false? Explain.(Hint: If you aren’t sure, plug in some numbers and check it out.)

5-4 Would you rather have a savings account that pays 5% interest compounded semiannuallyor one that pays 5% interest compounded daily? Explain.

5-5 To find the present value of an uneven series of cash flows, you must find the PVs of theindividual cash flows and then sum them. Annuity procedures can never be of use, evenwhen some of the cash flows constitute an annuity, because the entire series is not anannuity. True or false? Explain.

5-6 The present value of a perpetuity is equal to the payment on the annuity, PMT, divided bythe interest rate, I PV PMT I. What is the future value of a perpetuity of PMT dollars peryear? (Hint: The answer is infinity, but explain why.)

5-7 Banks and other lenders are required to disclose a rate called the APR. What is this rate?Why did Congress require that it be disclosed? Is it the same as the effective annual rate? Ifyou were comparing the costs of loans from different lenders, could you use their APRs todetermine the loan with the lowest effective interest rate? Explain.

5-8 What is a loan amortization schedule, and what are some ways these schedules are used?

PROBLEMS

5-1 FUTURE VALUE If you deposit $10,000 in a bank account that pays 10% interest annually,how much will be in your account after 5 years?

5-2 PRESENT VALUE What is the present value of a security that will pay $5,000 in 20 years ifsecurities of equal risk pay 7% annually?

5-3 FINDING THE REQUIRED INTEREST RATE Your parents will retire in 18 years. Theycurrently have $250,000 saved, and they think they will need $1,000,000 at retirement.What annual interest rate must they earn to reach their goal, assuming they don’t save anyadditional funds?

5-4 TIME FOR A LUMP SUM TO DOUBLE If you deposit money today in an account that pays6 5% annual interest, how long will it take to double your money?

5-5 TIME TO REACH A FINANCIAL GOAL You have $42,180.53 in a brokerage account, and youplan to deposit an additional $5,000 at the end of every future year until your account totals$250,000. You expect to earn 12% annually on the account. How many years will it take toreach your goal?

5-6 FUTURE VALUE: ANNUITY VERSUS ANNUITY DUE What’s the future value of a 7%, 5-yearordinary annuity that pays $300 each year? If this was an annuity due, what would itsfuture value be?

5-7 PRESENT AND FUTURE VALUES OF A CASH FLOW STREAM An investment will pay $100at the end of each of the next 3 years, $200 at the end of Year 4, $300 at the end of Year 5,and $500 at the end of Year 6. If other investments of equal risk earn 8% annually, what isits present value? Its future value?

5-8 LOAN AMORTIZATION AND EAR You want to buy a car, and a local bank will lend you$20,000. The loan will be fully amortized over 5 years (60 months), and the nominal interestrate will be 12% with interest paid monthly. What will be the monthly loan payment? Whatwill be the loan’s EAR?

EasyProblems1–8



5-9 PRESENT AND FUTURE VALUES FOR DIFFERENT PERIODS Find the following values usingthe equations and then a financial calculator. Compounding/discounting occurs annually.

a. An initial $500 compounded for 1 year at 6%b. An initial $500 compounded for 2 years at 6%c. The present value of $500 due in 1 year at a discount rate of 6%d. The present value of $500 due in 2 years at a discount rate of 6%

5-10 PRESENT AND FUTURE VALUES FOR DIFFERENT INTEREST RATES Find the followingvalues. Compounding/discounting occurs annually.

a. An initial $500 compounded for 10 years at 6%b. An initial $500 compounded for 10 years at 12%c. The present value of $500 due in 10 years at 6%d. The present value of $1,552.90 due in 10 years at 12% and at 6%e. Define present value and illustrate it using a time line with data from part d. How are

present values affected by interest rates?

5-11 GROWTH RATES Shalit Corporation’s 2014 sales were $12 million. Its 2009 sales were$6 million.

a. At what rate have sales been growing?b. Suppose someone made this statement: “Sales doubled in 5 years. This represents a

growth of 100% in 5 years; so dividing 100% by 5, we find the growth rate to be 20%per year.” Is the statement correct?

5-12 EFFECTIVE RATE OF INTEREST Find the interest rates earned 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 the borrower promises to pay you $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 at the end of each year

for 5 years.

5-13 TIME FOR A LUMP SUM TO DOUBLE How long will it take $200 to double if it earns thefollowing rates? Compounding occurs once a year.

a. 7%b. 10%c. 18%d. 100%

5-14 FUTURE VALUE OF AN ANNUITY Find the future values of these ordinary annuities. Com-pounding occurs once a year.

a. $400 per year for 10 years at 10%b. $200 per year for 5 years at 5%c. $400 per year for 5 years at 0%d. Rework parts a, b, and c assuming they are annuities due.

5-15 PRESENT VALUE OF AN ANNUITY Find the present values of these ordinary annuities.Discounting occurs once a year.

a. $400 per year for 10 years at 10%b. $200 per year for 5 years at 5%c. $400 per year for 5 years at 0%d. Rework parts a, b, and c assuming they are annuities due.

5-16 PRESENT VALUE OF A PERPETUITY What is the present value of a $100 perpetuity if theinterest rate is 7%? If interest rates doubled to 14%, what would its present value be?

5-17 EFFECTIVE INTEREST RATE You borrow $85,000; the annual loan payments are $8,273 59for 30 years. What interest rate are you being charged?

5-18 UNEVEN CASH FLOW STREAM

a. Find the present values of the following cash flow streams at an 8% discount rate.

IntermediateProblems9–26



10 2 3

$0$0

$100$300

$400$400

$400$400

$400$400

$300$100

Stream AStream B

4 5

b. What are the PVs of the streams at a 0% discount rate?

5-19 FUTURE VALUE OF AN ANNUITY Your client is 40 years old. She wants to begin saving forretirement, with the first payment to come one year from now. She can save $5,000 peryear, and you advise her to invest it in the stock market, which you expect to provide anaverage return of 9% in the future.

a. If she follows your advice, how much money will she have at 65?b. How much will she have at 70?c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70. If

her investments continue to earn the same rate, how much will she be able to withdrawat the end of each year after retirement at each retirement age?

5-20 PV OF A CASH FLOW STREAM A rookie quarterback is negotiating his first NFL contract.His opportunity cost is 10%. He has been offered three possible 4-year contracts. Paymentsare guaranteed, and they would be made at the end of each year. Terms of each contract areas follows:

1 2 3

$3,000,000$5,000,000$1,000,000

$3,000,000$4,000,000$1,000,000

$3,000,000$3,000,000$1,000,000

$3,000,000$2,000,000$7,000,000

Contract 1Contract 2Contract 3

4

As his adviser, which contract would you recommend that he accept?

5-21 EVALUATING LUMP SUMS AND ANNUITIES Crissie just won the lottery, and she mustchoose among three award options. She can elect to receive a lump sum today of $61million, to receive 10 end-of-year payments of $9 5 million, or to receive 30 end-of-yearpayments of $5 5 million.

a. If she thinks she can earn 7% annually, which should she choose?b. If she expects to earn 8% annually, which is the best choice?c. If she expects to earn 9% annually, which option would you recommend?d. Explain how interest rates influence her choice.

5-22 LOAN AMORTIZATION Jan sold her house on December 31 and took a $10,000 mortgageas part of the payment. The 10-year mortgage has a 10% nominal interest rate, but it callsfor semiannual payments beginning next June 30. Next year Jan must report on Schedule Bof her IRS Form 1040 the amount of interest that was included in the two payments shereceived during the year.

a. What is the dollar amount of each payment Jan receives?b. How much interest was included in the first payment? How much repayment of

principal was included? How do these values change for the second payment?c. How much interest must Jan report on Schedule B for the first year? Will her interest

income be the same next year?d. If the payments are constant, why does the amount of interest income change over

time?

5-23 FUTURE VALUE FOR VARIOUS COMPOUNDING PERIODS Find the amount to which $500will grow under each of these conditions:

a. 12% compounded annually for 5 yearsb. 12% compounded semiannually for 5 yearsc. 12% compounded quarterly for 5 yearsd. 12% compounded monthly for 5 yearse. 12% compounded daily for 5 yearsf. Why does the observed pattern of FVs occur?



5-24 PRESENT VALUE FOR VARIOUS DISCOUNTING PERIODS Find the present value of $500due in the future under each of these conditions:

a. 12% nominal rate, semiannual compounding, discounted back 5 yearsb. 12% nominal rate, quarterly compounding, discounted back 5 yearsc. 12% nominal rate, monthly compounding, discounted back 1 yeard. Why do the differences in the PVs occur?

5-25 FUTURE VALUE OF AN ANNUITY Find the future values of the following ordinary annuities:

a. FV of $400 paid each 6 months for 5 years at a nominal rate of 12% compoundedsemiannually

b. FV of $200 paid each 3 months for 5 years at a nominal rate of 12% compounded quarterlyc. These annuities receive the same amount of cash during the 5-year period and earn interest

at the same nominal rate, yet the annuity in part b ends up larger than the one in part a.Why does this occur?

5-26 PV AND LOAN ELIGIBILITY You have saved $4,000 for a down payment on a new car. Thelargest monthly payment you can afford is $350. The loan will have a 12% APR based onend-of-month payments. What is the most expensive car you can afford if you finance it for48 months? For 60 months?

5-27 EFFECTIVE VERSUS NOMINAL INTEREST RATES Bank A pays 4% interest compoundedannually on deposits, while Bank B pays 3 5% compounded daily.

a. Based on the EAR (or EFF ), which bank should you use?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? Assume that yourfunds must be left on deposit during an entire compounding period in order to receiveany interest.

5-28 NOMINAL INTEREST RATE AND EXTENDING CREDIT As a jewelry store manager, youwant to offer credit, with interest on outstanding balances paid monthly. To carry receiv-ables, you must borrow funds from your bank at a nominal 6%, monthly compounding. Tooffset your overhead, you want to charge your customers an EAR (or EFF ) that is 2%more than the bank is charging you. What APR rate should you charge your customers?

5-29 BUILDING CREDIT COST INTO PRICES Your firm sells for cash only, but it is thinking ofoffering credit, allowing customers 90 days to pay. Customers understand the time value ofmoney, so they would all wait and pay on the 90th day. To carry these receivables, youwould have to borrow funds from your bank at a nominal 12%, daily compounding basedon a 360-day year. You want to increase your base prices by exactly enough to offset yourbank interest cost. To the closest whole percentage point, by how much should you raiseyour product prices?

5-30 REACHING A FINANCIAL GOAL Erika and Kitty, who are twins, just received $30,000 each fortheir 25th birthday. They both have aspirations to become millionaires. Each plans to make a$5,000 annual contribution to her “early retirement fund” on her birthday, beginning a year fromtoday. Erika opened an account with the Safety First Bond Fund, a mutual fund that invests inhigh-quality bonds whose investors have earned 6% per year in the past. Kitty invested in theNew Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whoseinvestors have earned an average of 20% per year in the fund’s relatively short history.

a. If the two women’s funds earn the same returns in the future as in the past, how oldwill each be when she becomes a millionaire?

b. How large would Erika’s annual contributions have to be for her to become amillionaire at the same age as Kitty, assuming their expected returns are realized?

c. Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?

5-31 REQUIRED LUMP SUM PAYMENT Starting next year, you will need $10,000 annually for4 years to complete your education. (One year from today you will withdraw the first$10,000.) Your uncle deposits an amount today in a bank paying 5% annual interest, whichwill provide the needed $10,000 payments.

a. How large must the deposit be?b. How much will be in the account immediately after you make the first withdrawal?

5-32 REACHING A FINANCIAL GOAL Six years from today you need $10,000. You plan to deposit$1,500 annually, with the first payment to be made a year from today, in an account that pays

ChallengingProblems27–40



an 8% effective annual rate. Your last deposit, which will occur at the end of Year 6, will be forless than $1,500 if less is needed to reach $10,000. How large will your last payment be?

5-33 FV OF UNEVEN CASH FLOW You want to buy a house within 3 years, and you are currentlysaving for the down payment. You plan to save $5,000 at the end of the first year, and youanticipate that your annual savings will increase by 10% annually thereafter. Your expectedannual return is 7%. How much will you have for a down payment at the end of Year 3?

5-34 AMORTIZATION SCHEDULE

a. Set up an amortization schedule for a $25,000 loan to be repaid in equal installments atthe end of each of the next 3 years. The interest rate is 10% compounded annually.

b. What percentage of the payment represents interest and what percentage representsprincipal for each of the 3 years? Why do these percentages change over time?

5-35 AMORTIZATION SCHEDULE WITH A BALLOON PAYMENT You want to buy a house thatcosts $100,000. You have $10,000 for a down payment, but your credit is such that mort-gage companies will not lend you the required $90,000. However, the realtor persuades theseller to take a $90,000 mortgage (called a seller take-back mortgage) at a rate of 7%,provided the loan is paid off in full in 3 years. You expect to inherit $100,000 in 3 years;but right now all you have is $10,000, and you can afford to make payments of no morethan $7,500 per year given your salary. (The loan would call for monthly payments, butassume end-of-year annual payments to simplify things.)

a. If the loan was amortized over 3 years, how large would each annual payment be?Could you afford those payments?

b. If the loan was amortized over 30 years, what would each payment be? Could youafford those payments?

c. To satisfy the seller, the 30-year mortgage loan would be written as a balloon note,which means that at the end of the third year, you would have to make the regularpayment plus the remaining balance on the loan. What would the loan balance be atthe end of Year 3, and what would the balloon payment be?

5-36 NONANNUAL COMPOUNDING

a. You plan to make five deposits of $1,000 each, one every 6 months, with the first paymentbeing made in 6 months. You will then make no more deposits. If the bank pays 4% nominalinterest, compounded semiannually, how much will be in your account after 3 years?

b. One year from today you must make a payment of $10,000. To prepare for this payment,you plan to make two equal quarterly deposits (at the end of Quarters 1 and 2) in a bankthat pays 4% nominal interest compounded quarterly. How large must each of the twopayments be?

5-37 PAYING OFF CREDIT CARDS Simon recently received a credit card with an 18% nominalinterest rate. With the card, he purchased an Apple iPhone 5 for $372 71. The minimumpayment on the card is only $10 per month.

a. If Simon makes the minimum monthly payment and makes no other charges, howmany months will it be before he pays off the card? Round to the nearest month.

b. If Simon makes monthly payments of $35, how many months will it be before he paysoff the debt? Round to the nearest month.

c. How much more in total payments will Simon make under the $10-a-month plan thanunder the $35-a-month plan? Make sure you use three decimal places for N.

5-38 PV AND A LAWSUIT SETTLEMENT It is now December 31, 2014 t 0 , and a jury just foundin favor of a woman who sued the city for injuries sustained in a January 2013 accident. Sherequested recovery of lost wages plus $100,000 for pain and suffering plus $20,000 for legalexpenses. Her doctor testified that she has been unable to work since the accident and that shewill not be able to work in the future. She is now 62, and the jury decided that she would haveworked for another three years. She was scheduled to have earned $34,000 in 2013. (Tosimplify this problem, assume that the entire annual salary amount would have been receivedon December 31, 2013.) Her employer testified that she probably would have received raises of3% per year. The actual payment for the jury award will be made on December 31, 2015. Thejudge stipulated that all dollar amounts are to be adjusted to a present value basis onDecember 31, 2015, using a 7% annual interest rate and using compound, not simple, interest.Furthermore, he stipulated that the pain and suffering and legal expenses should be based on aDecember 31, 2014, date. How large a check must the city write on December 31, 2015?

5-39 REQUIRED ANNUITY PAYMENTS Your father is 50 years old and will retire in 10 years. Heexpects to live for 25 years after he retires, until he is 85. He wants a fixed retirement incomethat has the same purchasing power at the time he retires as $40,000 has today. (The real



value of his retirement income will decline annually after he retires.) His retirement incomewill begin the day he retires, 10 years from today, at which time he will receive 24 additionalannual payments. Annual inflation is expected to be 5%. He currently has $100,000 saved,and he expects to earn 8% annually on his savings. How much must he save during each ofthe next 10 years (end-of-year deposits) to meet his retirement goal?

5-40 REQUIRED ANNUITY PAYMENTS A father is now planning a savings program to put hisdaughter through college. She is 13, plans to enroll at the university in 5 years, and shouldgraduate 4 years later. Currently, the annual cost (for everything—food, clothing, tuition, books,transportation, and so forth) is $15,000, but these costs are expected to increase by 5% annually.The college requires total payment at the start of the year. She now has $7,500 in a collegesavings account that pays 6% annually. Her father will make six equal annual deposits into heraccount; the first deposit today and the sixth on the day she starts college. How large must eachof the six payments be? [Hint: Calculate the cost (inflated at 5%) for each year of college and findthe total present value of those costs, discounted at 6%, as of the day she enters college. Thenfind the compounded value of her initial $7,500 on that same day. The difference between thePV of costs and the amount that would be in the savings account must be made up by thefather’s deposits, so find the six equal payments that will compound to the required amount.]

COMPREHENSIVE/SPREADSHEET PROBLEM

5-41 TIME VALUE OF MONEY Answer the following questions:

a. Assuming a rate of 10% annually, find the FV of $1,000 after 5 years.b. What is the investment’s FV at rates of 0%, 5%, and 20% after 0, 1, 2, 3, 4, and 5 years?c. Find the PV of $1,000 due in 5 years if the discount rate is 10%.d. What is the rate of return on a security that costs $1,000 and returns $2,000 after 5

years?e. Suppose California’s population is 36 5 million people and its population is expected to

grow by 2% annually. How long will it take for the population to double?f. Find the PV of an ordinary annuity that pays $1,000 each of the next 5 years if the

interest rate is 15%. What is the annuity’s FV?g. How will the PV and FV of the annuity in part f change if it is an annuity due?h. What will the FV and the PV be for $1,000 due in 5 years if the interest rate is 10%,

semiannual compounding?i. What will the annual payments be for an ordinary annuity for 10 years with a PV of

$1,000 if the interest rate is 8%? What will the payments be if this is an annuity due?j. Find the PV and the FV of an investment that pays 8% annually and makes the

following end-of-year payments:

10 2 3

$400$200$100

k. Five banks offer nominal rates of 6% on deposits, but A pays interest annually; B payssemiannually; C pays quarterly; D pays monthly; and E pays daily.1. What effective annual rate does each bank pay? If you deposit $5,000 in each bank

today, how much will you have in each bank at the end of 1 year? 2 years?2. If all of the banks are insured by the government (the FDIC) and thus are equally

risky, will they be equally able to attract funds? If not (and the TVM is the onlyconsideration), what nominal rate will cause all of the banks to provide the sameeffective annual rate as Bank A?

3. Suppose you don’t have the $5,000 but need it at the end of 1 year. You plan to makea series of deposits—annually for A, semiannually for B, quarterly for C, monthlyfor D, and daily for E—with payments beginning today. How large must thepayments be to each bank?

4. Even if the five banks provided the same effective annual rate, would a rationalinvestor be indifferent between the banks? Explain.

l. Suppose you borrow $15,000. The loan’s annual interest rate is 8%, and it requires fourequal end-of-year payments. Set up an amortization schedule that shows the annualpayments, interest payments, principal repayments, and beginning and ending loanbalances.



FIRST NATIONAL BANK

5-42 TIME VALUE OF MONEY ANALYSIS You have applied for a job with a local bank. As part of its evaluation process,you must take an examination on time value of money analysis covering the following questions:

a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2; (2) an ordinary annuity of $100 per year for3 years; and (3) an uneven cash flow stream of −$50, $100, $75, and $50 at the end of Years 0 through 3.

b. 1. What’s the future value of $100 after 3 years if it earns 4%, annual compounding?2. What’s the present value of $100 to be received in 3 years if the interest rate is 4%, annual compounding?

c. What annual interest rate would cause $100 to grow to $119 10 in 3 years?d. If a company’s sales are growing at a rate of 10% annually, how long will it take sales to double?e. What’s the difference between an ordinary annuity and an annuity due? What type of annuity is shown here?

How would you change it to the other type of annuity?

0 1 2 3

0 $100$100$100

f. 1. What is the future value of a 3-year, $100 ordinary annuity if the annual interest rate is 4%?2. What is its present value?3. What would the future and present values be if it was an annuity due?

g. A 5-year $100 ordinary annuity has an annual interest rate of 4%.1. What is its present value?2. What would the present value be if it was a 10-year annuity?3. What would the present value be if it was a 25-year annuity?4. What would the present value be if this was a perpetuity?

h. A 20-year-old student wants to save $5 a day for her retirement. Every day she places $5 in a drawer. At the endof each year, she invests the accumulated savings ($1,825) in a brokerage account with an expected annual returnof 8%.1. If she keeps saving in this manner, how much will she have accumulated at age 65?2. If a 40-year-old investor began saving in this manner, how much would he have at age 65?3. How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the

20-year-old investor?

i. What is the present value of the following uneven cash flow stream? The annual interest rate is 4%.

0 14% 2 3

0 $50$300$300$100

4 Years

j. 1. Will the future value be larger or smaller if we compound an initial amount more often than annually(e.g., semiannually, holding the stated (nominal) rate constant)? Why?

2. Define (a) the stated (or quoted or nominal) rate, (b) the periodic rate, and (c) the effective annual rate (EARor EFF ).

3. What is the EAR corresponding to a nominal rate of 4% compounded semiannually? Compounded quarterly?Compounded daily?

4. What is the future value of $100 after 3 years under 4% semiannual compounding? Quarterly compounding?



k. When will the EAR equal the nominal (quoted) rate?

l. 1. What is the value at the end of Year 3 of the following cash flow stream if interest is 4% compoundedsemiannually? (Hint: You can use the EAR and treat the cash flows as an ordinary annuity or use theperiodic rate and compound the cash flows individually.)

0 2 4 6 Periods

$100$100$1000

2. What is the PV?3. What would be wrong with your answer to parts l 1 and l 2 if you used the nominal rate, 4%, rather

than the EAR or the periodic rate, INOM 2 4% 2 2%, to solve the problems?

m. 1. Construct an amortization schedule for a $1,000, 4% annual interest loan with three equal installments.2. What is the annual interest expense for the borrower and the annual interest income for the lender

during Year 2?



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