Scenario

Your Project 4 business report will focus on which assets to invest in or whether to disinvest. You will identify the types of assets LGI should eventually acquire and/or renew. You will also determine the investments that will help LGI improve its operations and generate the cash flow needed to improve its bottom line.

Complete the analysis calculation for the project:

Download the

Project 4 Excel Workbook

, click the Instructions tab, and read the instructions.

Calculate and evaluate the investment choices using the worksheets.

If you would like instructor feedback on this step, submit your Excel file to the Assignments folder as a milestone by the end of Week 7. This is optional. If you choose to submit the milestone, you will receive instructor feedback you may use to make corrections before submitting your final Project 4. To distinguish the milestone submission from the file you will submit in Step 5, label your file as follows: P4_milestone_lastname_Calculation_date

Prepare the Analysis Report for Project 4You have developed an in-depth understanding of LGI’s operating efficiency as it relates to costing and its impact on the bottom line. You feel confident that your investment choices will positively boost LGI’s productivity and improve the company’s operations. LGI will finally be on a path of a sustainable future. Answer the questions in the Project 4 Questions – Report Template document. Prepare your analysis report including recommendations for how the company can improve its financial situation.Complete the analysis report:Download the Project 4 Questions – Report Template 01292024

Project 4 Questions – Report Template

Instructions: Answer the five questions below based on the information you developed in the

Excel workbook. Support your reasoning from the readings in Project 4, Step 1, and the

discussion in Project 4, Step 3. Be sure to cite any sources used in proper APA (7th ed.) format.

Provide a detailed response below each question. Use 12-point font and double spacing.

Maintain the existing margins in this document. Your final Word document, including the

questions, should be at most five pages. Include a title page in addition to the five pages. Any

tables and graphs you include are excluded from the five-page limit. Name your document as

follows: P4_Final_lastname_Report_date.

You must address all five questions and fully use the information in the Excel workbook.

You are strongly encouraged to exceed the requirements by refining your analysis. Consider

other tools and techniques that were discussed in the required and recommended reading for

Project 4.

Title Page

Name

Course and section number

Faculty name

Submission date

1. Present-value calculations, rather than future-value calculations, are the key to analysis in

the field of corporate finance. Why is this the case? Explain the importance for Largo Global

Inc. (LGI) of understanding today’s value of projected future revenues and costs.

[insert your answer here]

2. Based on your calculations in Tab 2, Question 8, which offer should LGI accept for the

Bowie plant? Explain why. Be sure to include the concepts of risk and potential return in your

discussion.

[insert your answer here]

3. The proposed sale of the Bowie plant is part of a more significant effort to divest the

company of underperforming assets. A total of $1.3 billion in assets, with a book value of

$750 million, have been identified for potential sale. Assuming that all these sales could be

accomplished in 2022, identify the significant impacts on the following:

a. Balance Sheet, especially these accounts:

• Property, plant, and equipment

• Accumulated depreciation

• Net property, plant, and equipment

b. Statement of Cash Flows, especially Long-Term Investing Activities

c. Income Statement, especially Net Income

Explain the potential impacts, both positive and negative, of these changes for LGI.

[insert your answer here]

4. Based on your calculations in Tab 3, Questions 1–4, should LGI proceed with acquiring the

robotics-based sorting and distribution equipment? Explain your reasoning. How would the

acquisition fit into the efforts to turn the company around?

[insert your answer here]

5. In Tab 3, Question 5, did the change in the discount rate make proceeding with the

purchase more or less desirable? What do you conclude from this result? Discuss the role of

discount rates in LGI’s decision-making process for capital budgeting and new asset

acquisition.

[insert your answer here]

Instructions

Answer all questions in this workbook. Be sure to read the introductory text on

tabs 1 and 3 as well as these instructions.

Keep in mind that the focus of this project is corporate finance. The

information generated by the accounting system is important; but in finance,

decisions are driven by an analysis of cash flows rather than profits.

Tab 1 contains a series of exercises on the concept of the time value of money.

These exercises do not relate directly to the issues facing LGI.

Tab 2 focuses on the concept of annuities. The first few questions do not

pertain specifically to LGI; the latter questions do.

Tab 3 pertains to whether LGI should acquire new assets that may enhance the

company’s productivity and thus improve financial performance.

061523

Time value of money (TVM) exercises

There are five variables in TVM calculations: present value, number of periods, rate

of return, regular payments, and future value. If four of the variables are known,

then the fifth can be calculated using algebra, a financial calculator, or a computer

program such as Excel.

Excel functions for the five variables are as follows:

•

•

•

•

•

PV—present value

NPER—number of periods

RATE—rate of return

PMT—regular payments

FV—future value

1. Briefly explain the meaning of the term “present value” in your own words.

2. Briefly explain the meaning of the term “future value” in your own words.

3. What is the future value in five years of $1,500 invested at an interest rate of 4.95%?

4. What is the future value of a single payment with the following characteristics?

PV

$950

NPER

6 years

RATE

5,4%

5. What is the present value of $65,000 in six years, if the relevant interest rate is 8.1%?

6. What is the present value of a single payment with the following characteristics?

NPER

11 years

RATE

5,05%

FV

$10.000

7. The present value of a payment is $4000. The future value of that payment in five years will be $4800. What is

8. What is the annual rate of return of a single payment with the following characteristics?

PV

$1.000

NPER

15 years

FV

$10.000

01292024

Tab 2 – Annuities

1. How many years would be required to pay off a loan with the following characteristics?

PV

$11.500

RATE

10,6%

PMT

$1.600 (annual payments)

2. What is the annual payment required to pay off a loan with the following characteristics?

PV

$14.700

RATE

9,95%

NPER

10 years

3. Why is FV not part of the calculations for either question 1 or question 2?

4. At what annual rate of interest is a loan with the following characteristics?

NPER

PMT

PV

17 years

$100.000

$1.000.000

For questions 5-8, LGI’s cost of capital is

8,11%

5. LGI projects the following after-tax cash flows from operations from

its aging Bowie, Maryland distribution facility (which first went on line in 1953)

over the next five years. What is the PV of these cash flows?

Year

Projected after-tax cash flows

(in $ millions)

1

(40)

2

(40)

3

(40)

4

(40)

5

(40)

6. LGI extended the analysis out for an additional 7 years, and generated the

following projections. What is the PV of these cash flows?

Projected after-tax cash flows

Year

(in $ millions)

1

(40)

2

(40)

3

(40)

4

(40)

5

(40)

6

(40)

7

(40)

8

(40)

9

(40)

10

(40)

11

(40)

12

(40)

7. The CFO asked you to undertake a more detailed analysis of the plant’s costs, noting that while

it is convenient for making calculations when projections result in data that can be treated like an annuity,

this does not always represent the most accurate estimate of future results. What is the PV of these cash flows?

Year

Projected after-tax cash flows

(in $ millions)

1

(40)

2

(50)

3

(55)

4

(60)

5

(70)

8. As part of a larger plan to sell off underperforming assets, LGI is considering selling the Bowie property

and using other existing facilities more efficiently. LGI received four preliminary offers from potential buyers for th

property. What is the PV of each offer?

PV of each offer (in $ millions)

Offer A

Offer B

Offer C

Offer D

$102.17 million, paid today

$19.85 million per year, to be paid over the next 8 years

$201.88 million, to be paid in year 8

$18.09 million per year, to be paid over the next 7 years and

a $53.05 million payment in year 8

9. From a profit maximizing point of view, which offer should LGI accept?

10. Define the term annuity in your own words. How might the concept of an annuity impact the process of

capital budgeting and new asset acquisition?

01292024

Robotics-based equipment proposal

If the Bowie plant is sold, those operations will need to shift to the main Largo facility. The CEO

sorting and distribution equipment to facilitate more cost-effective operations (and be able to handle the increase

The CFO has asked you to evaluate the cash flow projections associated with the equipment purchase proposal an

the purchase should go forward. Table 2 shows projections of the cash inflows and outflows that would occur duri

using the new equipment.

Keep the following in mind:

•

•

Depreciation. The equipment will be depreciated using the straight-line method over eight years. The projec

Taxes. The CFO estimates that company operations as a whole will be profitable on an ongoing basis. As a re

on this specific project will provide a tax benefit in the year of the loss.

Table 1 – Data

Cost of the new manufacturing equipment (at year=0)

$

Corporate income tax rate – Federal

Corporate income tax rate – State of Maryland

Discount rate for the project

191,1

26,0%

8,0%

5,98%

Table 2 – After-tax Cash Flow Timeline

Year

minus

minus

equals

(all figures in $ millions)

Projected Cash

Projected Cash

Inflows from

Outflows from

Depreciation

Operations

Operations

Expense

0

1

850,0

840,0

2

900,0

810,0

3

990,0

870,0

4

1.005,0

900,0

5

1.200,0

1.100,0

6

1.300,0

1.150,0

7

1.350,0

1.300,0

8

1.320,0

1.300,0

Projected Taxable

Income

Table 3 – Example – Computing Projected After-tax Cas

For Year 4

(all figures in $ millions)

Projected Cash Inflows from Operations

1005,0

Projected Cash Outflows from Operations

(900,0)

Depreciation Expense

(23,9)

Projected Taxable Income

81,1

times

equals

Projected Taxable Income

Corporate income tax rate – Federal

Projected Federal Income Taxes

81,1

26,0%

21,1

times

equals

Projected Taxable Income

Corporate income tax rate – State

Projected State Income Taxes

81,1

8,0%

6,5

1. Complete Table 2. Compute the projected after tax cash flows for each of years 1-8.

2. Compute the total present value (PV) of the projected after tax cash flows for years 1-8.

3. Compute the net present value (NPV) of the projected after tax cash flows for years 0-8.

4. Compute the internal rate of return (IRR) of the project.

5. The CFO believes that it is possible that the next few years will bring a very low interest rate environment.

Therefore, she has asked that you repeat the NPV calculation in question 3 showing the case where the

discount rate for the project is

5,02%

01292024

acility. The CEO is proposing to acquire robotics-based

and be able to handle the increased workload) at Largo.

e equipment purchase proposal and recommend whether

nd outflows that would occur during the first eight years

ethod over eight years. The projected salvage value is $0.

itable on an ongoing basis. As a result, any accounting loss

million

Projected Federal

Income Taxes

Projected State

Income Taxes

Projected After-tax

Cash Flows

uting Projected After-tax Cash Flows

minus

minus

equals

Projected Cash Inflows from Operations

Projected Cash Outflows from Operations

Projected Federal Income Taxes

Projected State Income Taxes

1005,0

(900,0)

(21,1)

(6,5)

w interest rate environment.

ing the case where the

Projected After-tax Cash Flows

77,4

Required Reading

Parrino, R. Kidwell, D. S., & Bates, T. W. (2012). Fundamentals of

corporate finance. Wiley.

Chapter 5: The Time Value of Money

•

Section 5.1 to 5.4

Chapter 6: Discounted Cash Flows and Valuation

•

Sections 6.1 to 6.4

Recommended Reading

•

Davis, C. E. & Davis, E. (2011). Managerial Accounting.

Wiley. Chapter 9, Capital Budgeting

Chapter 5.1-5.4

5

The Time Value of Money

AFP/Getty Images, Inc.

Learning Objectives

Explain what the time value of money is and why it is so important in the field of finance.

Explain the concept of future value, including the meaning of the terms principal, simple interest,

and compound interest, and use the future value formula to make business decisions.

Explain the concept of present value, how it relates to future value, and use the present value

formula to make business decisions.

Discuss why the concept of compounding is not restricted to money, and use the future value

formula to calculate growth rates.

When you purchase an automobile from a dealer, the decision of whether to pay cash or

finance your purchase can affect the price you pay. For example, automobile manufacturers

often offer customers a choice between a cash rebate and low-cost financing. Both

alternatives affect the cost of purchasing an automobile; but one alternative can be worth

more than the other.

To see why, consider the following. In June 2010, as the end of the model year approached,

the automobile manufacturer General Motors wanted to reduce its inventory of 2010

Yukon sport utility vehicles (SUVs) before it introduced the 2011 models. In an effort to

increase sales of the 2010 Yukon SUVs, the company offered consumers a choice between

(1) receiving $3,000 off the base price of $38,020 if they paid cash and (2) receiving 0

percent financing on a five-year loan if they paid the base price. For someone who had

enough cash to buy the car outright and did not need the cash for some other use, the

decision of whether to pay cash or finance the purchase of a Yukon depended on the rate of

return they could earn by investing the cash. On the one hand, if it was possible to earn

only a 1 percent interest rate by investing in a certificate of deposit at a bank, the buyer

was better off paying cash for the Yukon. On the other hand, if it was possible to earn 5

percent, the buyer was better off taking the financing. With a 3.36 percent rate of return,

the buyer would have been largely indifferent between the two alternatives. In Chapters

5 and 6 you will learn how to calculate the rate of return at which the buyer would be

indifferent in a situation like this.

As with most business transactions, a crucial element in the analysis of the alternatives

offered by General Motors is the value of the expected cash flows. Because the cash flows

for the two alternatives take place in different time periods, they must be adjusted to

account for the time value of money before they can be compared. A car buyer wants to

select the alternative with the cash flows that have the lowest value (price). This chapter

and the next provide the knowledge and tools you need to make the correct decision. You

will learn that at the bank, in the boardroom, or in the showroom, money has a time

value—dollars today are worth more than dollars in the future—and you must account for

this when making financial decisions.

CHAPTER PREVIEW

Business firms routinely make decisions to invest in productive assets to earn income. Some assets,

such as plant and equipment, are tangible, and other assets, such as patents and trademarks, are

intangible. Regardless of the type of investment, a firm pays out money now in the hope that the

value of the future benefits (cash inflows) will exceed the cost of the asset. This process is

what value creation is all about—buying productive assets that are worth more than they cost.

The valuation models presented in this book will require you to compute the present and future

values of cash flows. This chapter and the next one provide the fundamental tools for making these

calculations. Chapter 5 explains how to value a single cash flow in different time periods,

and Chapter 6 covers valuation of multiple cash flows. These two chapters are critical for your

understanding of corporate finance.

We begin this chapter with a discussion of the time value of money. We then look at future value,

which tells us how funds will grow if they are invested at a particular interest rate. Next, we discuss

present value, which answers the question “What is the value today of cash payments received in

the future?” We conclude the chapter with a discussion of several additional topics related to time

value calculations.

5.1 THE TIME VALUE OF MONEY

•

In financial decision making, one basic problem managers face is determining the value of

(or price to pay for) cash flows expected in the future. Why is this a problem? Consider as

an example the popular Mega MillionsTM lottery game.1 In Mega Millions, the jackpot

continues to build up until some lucky person buys a winning ticket—the payouts for a

number of jackpot winning tickets have exceeded $100 million.

If you won $100 million, headlines would read “Lucky Student Wins $100 Million Jackpot!”

Does this mean that your ticket is worth $100 million on the day you win? The answer is

no. A Mega Millions jackpot is paid either as a series of 26 payments over 25 years or as a

cash lump sum. If you win “$100 million” and choose to receive the series of payments, the

26 payments will total $100 million. If you choose the lump sum option, Mega Millions will

pay you less than the stated value of $100 million. This amount was about $50 million in

June 2010. Thus, the value, or market price, of a “$100 million” winning Mega Millions

ticket is really about $50 million because of the time value of money and the timing of the

26 cash payments. An appropriate question to ask now is, “What is the time value of

money?”

Take an online lesson on the time value of money

from TeachMeFinance.com at http://teachmefinance.com/timevalueofmoney.html.

Consuming Today or Tomorrow

The time value of money is based on the idea that people prefer to consume goods today

rather than wait to consume similar goods in the future. Most people would prefer to have

a large-screen TV today than to have one a year from now, for example. Money has a time

value because a dollar in hand today is worth more than a dollar to be received in the

future. This makes sense because if you had the dollar today, you could buy something with

it—or, instead, you could invest it and earn interest. For example, if you had $100,000, you

could buy a one-year bank certificate of deposit paying 5 percent interest and earn $5,000

interest for the year. At the end of the year, you would have $105,000 ($100,000 + $5,000 =

$105,000). The $100,000 today is worth $105,000 a year from today. If the interest rate

was higher, you would have even more money at the end of the year.

time value of money

the difference in value between a dollar in hand today and a dollar promised in the future; a

dollar today is worth more than a dollar in the future

BUILDING INTUITION THE VALUE OF MONEY CHANGES WITH TIME

The term time value of money reflects the notion that people prefer to consume things today rather

than at some time in the future. For this reason, people require compensation for deferring

consumption. The effect is to make a dollar in the future worth less than a dollar today.

Based on this example, we can make several generalizations. First, the value of a dollar

invested at a positive interest rate grows over time. Thus, the further in the future you

receive a dollar, the less it is worth today. Second, the trade-off between money today and

money at some future date depends in part on the rate of interest you can earn by

investing. The higher the rate of interest, the more likely you will elect to invest your funds

and forgo current consumption. Why? At the higher interest rate, your investment will earn

more money.

In the next two sections, we look at two views of time value—future value and present

value. First, however, we describe time lines, which are pictorial aids to help solve future

and present value problems.

Time Lines as Aids to Problem Solving

Time lines are an important tool for analyzing problems that involve cash flows over time.

They provide an easy way to visualize the cash flows associated with investment decisions.

A time line is a horizontal line that starts at time zero and shows cash flows as they occur

over time. The term time zero is used to refer to the beginning of a transaction in time

value of money problems. Time zero is often the current point in time (today).

time zero

the beginning of a transaction; often the current point in time

Exhibit 5.1 shows the time line for a five-year investment opportunity and its cash flows.

Here, as in most finance problems, cash flows are assumed to occur at the end of the period.

The project involves a $10,000 initial investment (cash outflow), such as the purchase of a

new machine, that is expected to generate cash inflows over a five-year period: $5,000 at

the end of year 1, $4,000 at the end of year 2, $3,000 at the end of year 3, $2,000 at the end

of year 4, and $1,000 at the end of year 5. Because of the time value of money, it is critical

that you identify not only the size of the cash flows, but also the timing.

If it is appropriate, the time line will also show the relevant interest rate for the problem.

In Exhibit 5.1 this is shown as 5 percent. Also, note in Exhibit 5.1 that the initial cash flow of

$10,000 is represented by a negative number. It is conventional that cash outflows from the

firm, such as for the purchase of a new machine, are treated as negative values on a time

line and that cash inflows to the firm, such as revenues earned, are treated as positive

values. The 2$10,000 therefore means that there is a cash outflow of $10,000 at time zero.

As you will see, it makes no difference how you label cash inflows and outflows as long as

you are consistent. That is, if all cash outflows are given a negative value, then all cash

inflows must have a positive value. If the signs get “mixed up”—if some cash inflows are

negative and some positive—you will get the wrong answer to any problem you are trying

to solve.

EXHIBIT 5.1 Five-year Time Line for a $10,000 Investment

Time lines help us to correctly identify the size and timing of cash flows—critical tasks in

solving time value problems. This time line shows the cash flows generated over five years

by a $10,000 investment in a situation where the relevant interest rate is 5 percent.

Financial Calculator

We recommend that students purchase a financial calculator for this course. A financial

calculator will provide the computational tools to solve most problems in the book. A

financial calculator is just an ordinary calculator that has preprogrammed future value and

present value algorithms. Thus, all the variables you need to make financial calculations

exist on the calculator keys. To solve problems, all you have to do is press the proper keys.

The instructions in this book are generally meant for Texas Instruments calculators, such

as the TI BAII Plus. If you are using an HP or Sharp calculator, consult the user’s manual for

instructions.

It may sound as if the financial calculator will solve problems for you. It won’t. To get the

correct answer to textbook or real-world problems, you must first analyze the problem

correctly and then identify the cash flows (size and timing), placing them correctly on a

time line. Only then will you enter the correct inputs into the financial calculator.

A calculator can help you eliminate computation errors and save you a great deal of time.

However, it is important that you understand the calculations that the calculator is

performing. For this reason we recommend that when you first start using a financial

calculator that you solve problems by hand and then use the calculator’s financial functions

to check your answers.

To help you master your financial calculator, throughout this chapter, we provide helpful

hints on how to best use the calculator. We also recognize that some professors or students

may want to solve problems using one of the popular spreadsheet programs. In this

chapter and a number of other chapters, we provide solutions to several problems that

lend themselves to spreadsheet analysis. In solving these problems, we used Microsoft

ExcelTM. The analysis and basic commands are similar for other spreadsheet programs. We

also provide spreadsheet solutions for additional problems on the book’s Web site. Since

spreadsheet programs are very commonly used in industry, you should make sure to learn

how to use one of these programs early in your studies and become proficient with it

before you graduate.

> BEFORE YOU GO ON

1. Why is a dollar today worth more than a dollar one year from now?

2. What is a time line, and why is it important in financial analysis?

5.2 FUTURE VALUE AND COMPOUNDING

•

The future value (FV) of an investment is what the investment will be worth after earning

interest for one or more time periods. The process of converting the initial amount into

future value is called compounding. We will define this term more precisely later. First,

though, we illustrate the concepts of future value and compounding with a simple example.

future value (FV)

the value of an investment after it earns interest for one or more periods

Single-Period Investment

Suppose you place $100 in a bank savings account that pays interest at 10 percent a year.

How much money will you have in one year? Go ahead and make the calculation. Most

people can intuitively arrive at the correct answer, $110, without the aid of a formula. Your

calculation could have looked something like this:

This approach computes the amount of interest earned ($100 × 0.10) and then adds it to

the initial, or principal, amount ($100). Notice that when we solve the equation, we factor

out the $100. Recall from algebra that if you have the equation y = c + (c × x), you can factor

out the common term c and get y = c × (1 + x). By doing this in our future value calculation,

we arrived at the term (1 + 0.10). This term can be stated more generally as (1 + i),

where i is the interest rate. As you will see, this is a pivotal term in all time value of money

calculations.

Let’s use our intuitive calculation to generate a more general formula. First, we need to

define the variables used to calculate the answer. In our example $100 is the principal

amount (P0), which is the amount of money deposited (invested) at the beginning of the

transaction (time zero); the 10 percent is the simple interest rate (i); and the $110 is the

future value (FV1) of the investment after one year. We can write the formula for a singleperiod investment as follows:

Looking at the formula, we more easily see mathematically what is happening in our

intuitive calculation. P0 is the principal amount invested at time zero. If you invest for one

period at an interest rate of i, your investment, or principal, will grow by (1 + i) per dollar

invested. The term (1 + i) is the future value interest factor—often called simply the future

value factor—for a single period, such as one year. To test the equation, we plug in our

values:

Good, it works!

Two-Period Investment

We have determined that at the end of one year (one period), your $100 investment has

grown to $110. Now let’s say you decide to leave this new principal amount (FV1) of $110

in the bank for another year earning 10 percent interest. How much money would you have

at the end of the second year (FV2)? To arrive at the value for FV2, we multiply the new

principal amount by the future value factor (1 + i). That is, FV2 = FV1 × (1 + i). We then

substitute the value of FV1 (the single-period investment value) into the equation and

algebraically rearrange terms, which yields FV2 = P0 × (1 + i)2. The mathematical steps to

arrive at the equation for FV2 are shown in the following; recall that FV1 = P0 × (1 + i):

The future value of your $110 at the end of the second year (FV2) is as follows:

Another way of thinking of a two-period investment is that it is two single-period

investments back-to-back. From that perspective, based on the preceding equations, we

can represent the future value of the deposit held in the bank for two years as follows:

Turning to Exhibit 5.2, we can see what is happening to your $100 investment over the two

years we have already discussed and beyond. The future value of $121 at year 2 consists of

three parts. First is the initial principal of $100 (first row of column 2). Second is the $20

($10 + $10 = $20) of simple interest earned at 10 percent for the first and second years

(first and second rows of column 3). Third is the $1 interest earned during the second year

(second row of column 4) on the $10 of interest from the first year ($10 × 0.10 = $1.00).

This is called interest on interest. The total amount of interest earned is $21 ($10 + $11 =

$21), which is shown in column 5 and is called compound interest.

EXHIBIT 5.2 Future Value of $100 at 10 Percent

With compounding, interest earned on an investment is reinvested so that in future

periods, interest is earned on interest as well as on the principal amount. Here, interest on

interest begins accruing in year 2.

We are now in a position to formally define some important terms already mentioned in

our discussion. The principal is the amount of money on which interest is paid. In our

example, the principal amount is $100. Simple interest is the amount of interest paid on

the original principal amount. With simple interest, the interest earned each period is paid

only on the original principal. In our example, the simple interest is $10 per year or $20 for

the two years. Interest on interest is the interest earned on the reinvestment of previous

interest payments. In our example, the interest on interest is $1. Compounding is the

process by which interest earned on an investment is reinvested so that in future periods,

interest is earned on the interest previously earned as well as the principal. In other words,

with compounding, you are able to earn compound interest, which consists of both simple

interest and interest on interest. In our example, the compound interest is $21.

principal

the amount of money on which interest is paid

simple interest

interest earned on the original principal amount only

interest on interest

interest earned on interest that was earned in previous periods

compounding

the process by which interest earned on an investment is reinvested, so in future periods

interest is earned on the interest as well as the principal

compound interest

interest earned both on the original principal amount and on interest previously earned

The Future Value Equation

Let’s continue our bank example. Suppose you decide to leave your money in the bank for

three years. Looking back at equations for a single-period and two-period investment, you

can probably guess that the equation for the future value of money invested for three years

would be:

With this pattern clearly established, we can see that the general equation to find the future

value after any number of periods is as follows:

which is often written as:

where:

Let’s test our general equation. Say you leave your $100 invested in the bank savings

account at 10 percent interest for five years. How much would you have in the bank at the

end of five years? Applying Equation 5.1 yields the following:

Exhibit 5.2 shows how the interest is earned on a year-by-year basis. Notice that the total

compound interest earned over the five-year period is $61.05 (column 5) and that it is

made up of two parts: (1) $50.00 of simple interest (column 3) and (2) $11.05 of interest

on interest (column 4). Thus, the total compound interest can be expressed as follows:

The simple interest earned is $100 × 0.10 = $10.00 per year, and thus, the total simple

interest for the five-year period is $50.00 (5 years × $10.00 = $50.00). The remaining

balance of $11.05 ($61.05 − $50.00 = $11.05) comes from earning interest on interest.

CNNMoney’s Web site has a savings calculator

at http://cgi.money.cnn.com/tools/savingscalc/savingscalc.html.

A helpful equation for calculating the simple interest can be derived by using the equation

for a single-period investment and solving for the term FV1 − P0, which is equal to the

simple interest.2 The equation for the simple interest earned (SI) is:

where:

Thus, the calculation for simple interest is:3

Exhibit 5.3 shows graphically how the compound interest in Exhibit 5.2 grows. Notice that

the simple interest earned each year remains constant at $10 per year but that the amount

of interest on interest increases every year. The reason, of course, is that interest on

interest increases with the cumulative interest that has been earned. As more and more

interest is earned, the compounding of interest accelerates the growth of the interest on

interest and therefore the total interest earned.

EXHIBIT 5.3 How Compound Interest Grows on $100 at 10 Percent

The amount of simple interest earned on $100 invested at 10 percent remains constant at

$10 per year, but the amount of interest earned on interest increases each year. As more

and more interest builds, the effect of compounding accelerates the growth of the total

interest earned.

EXHIBIT 5.4 Future Value of $1 for Different Periods and Interest Rates

The higher the interest rate, the faster the value of an investment will grow, and the larger

the amount of money that will accumulate over time. Because of compounding, the growth

over time is not linear but exponential—the dollar increase in the future value is greater in

each subsequent period.

An interesting observation about Equation 5.1 is that the higher the interest rate, the faster

the investment will grow. This fact can be seen in Exhibit 5.4, which shows the growth in

the future value of $1.00 at different interest rates and for different time periods into the

future. First, notice that the growth in the future value over time is not linear, but

exponential. The dollar value of the invested funds does not increase by the same dollar

amount from year to year. It increases by a greater amount each year. In other words, the

growth of the invested funds is accelerated by the compounding of interest. Second, the

higher the interest rate, the more money accumulated for any time period. Looking at the

right-hand side of the exhibit, you can see the difference in total dollars accumulated if you

invest a dollar for 10 years: At 5 percent, you will have $1.63; at 10 percent, you will have

$2.59; at 15 percent, you will have $4.05; and at 20 percent, you will have $6.19. Finally, as

you should expect, if you invest a dollar at 0 percent for 10 years, you will only have a

dollar at the end of the period.

The Future Value Factor

To solve a future value problem, we need to know the future value factor, (1 + i)n.

Fortunately, almost any calculator suitable for college-level work has a power key

(the yx key) that we can use to make this computation. For example, to compute (1.08)10, we

enter 1.08, press the yx key and enter 10, and press the = key. The number 2.159 should

emerge. Give it a try with your calculator.

Alternatively, we can use future value tables to find the future value factor at different

interest rates and maturity periods. Exhibit 5.5 is an example of a future value table. For

example, to find the future value factor (1.08)10, we first go to the row corresponding to 10

years and then move along the row until we reach the 8 percent interest column. The entry

is 2.159, which is identical to what we found when we used a calculator. This comes as no

surprise, but we sometimes find small differences between calculator solutions and future

value tables due to rounding differences. Exhibit A.1 at the end of the book provides a more

comprehensive version of Exhibit 5.5.

Future value tables (and the corresponding present value tables) are rarely used today,

partly because they are tedious to work with. In addition, the tables show values for only a

limited number of interest rates and time periods. For example, what if the interest rate on

your $100 investment was not a nice round number such as 10 percent but was 10.236

percent? You would not find that number in the future value table. In spite of their

shortcomings, these tables were very commonly used in the days before financial

calculators and spreadsheet programs were readily available. You can still use them—for

example, to check the answers from your computations of future value factors.

EXHIBIT 5.5 Future Value Factors

To find a future value factor, simply locate the row with the appropriate number of periods

and the column with the desired interest rate. The future value factor for 10 years at 8

percent is 2.159.

Applying the Future Value Formula

Next, we will review a number of examples of future value problems to illustrate the typical

types of problems you will encounter in business and in your personal life.

The Power of Compounding

Our first example illustrates the effects of compounding. Suppose you have an opportunity

to make a $5,000 investment that pays 15 percent per year. How much money will you

have at the end of 10 years? The time line for the investment opportunity is:

where the $5,000 investment is a cash outflow and the future value you will receive in 10

years is a cash inflow.

You can find a compound interest calculator

at SmartMoney.com: http://www.smartmoney.com/compoundcalc.

We can apply Equation 5.1 to find the future value of $5,000 invested for 10 years at 15

percent interest. We want to multiply the original principal amount (PV) times the

appropriate future value factor for 10 years at 15 percent, which is (1 + 0.15)10; thus:

Now let’s determine how much of the interest is from simple interest and how much is

from interest on interest. The total compound interest earned is $15,227.79 ($20,227.79 −

$5,000.00 = $15,227.79). The simple interest is the amount of interest paid on the original

principal amount: SI = P0 × i = $5,000 × 0.15 = $750 per year, which over 10 years is $750 ×

10 = $7,500. The interest on interest must be the difference between the total compound

interest earned and the simple interest: $15,227.79 − $7,500 = $7,727.79. Notice how

quickly the value of an investment increases and how the reinvestment of interest

earned—interest on interest—impacts that total compound interest when the interest

rates are high.

APPLICATION 5.1 LEARNING BY DOING

The Power of Compounding

PROBLEM: Your wealthy uncle passed away, and one of the assets he left to you was a savings

account that your great-grandfather had set up 100 years ago. The account had a single deposit of

$1,000 and paid 10 percent interest per year. How much money have you inherited, what is the

total compound interest, and how much of the interest earned came from interest on interest?

APPROACH: We first determine the value of the inheritance, which is the future value of $1,000

retained in a savings account for 100 years at a 10 percent interest rate. Our time line for the

problem is:

To calculate FV100, we begin by computing the future value factor. We then plug this number into the

future value formula (Equation 5.1) and solve for the total inheritance. Once we have computed

FV100, we calculate the total compound interest and the total simple interest and find the difference

between these two numbers, which will give us the interest earned on interest.

SOLUTION:

First, we find the future value factor:

Then we find the future value:

Your total inheritance is $13,780,612. The total compound interest earned is this amount less the

original $1,000 investment, or $13,779,612:

The total simple interest earned is calculated as follows:

The interest earned on interest is the difference between the total compound interest earned and

the simple interest:

That’s quite a difference!

As Learning by Doing Application 5.1 indicates, the relative importance of interest earned

on interest is especially great for long-term investments. For many people, retirement

savings include the longest investments they will make. As you might expect, interest

earned on interest has a great impact on how much money people ultimately have for their

retirement. For example, consider someone who inherits and invests $10,000 on her 25th

birthday and earns 8 percent per year for the next 40 years. This investment will grow to:

BUILDING INTUITION COMPOUNDING DRIVES MUCH OF THE EARNINGS ON LONG-TERM

INVESTMENTS

The earnings from compounding drive much of the return earned on a long-term investment. The

reason is that the longer the investment period, the greater the proportion of total earnings from

interest earned on interest. Interest earned on interest grows exponentially as the investment

period increases.

by the investor’s 65th birthday. In contrast, if the same individual waited until her 35th

birthday to invest the $10,000, she would have only:

when she turned 65.

Of the $116,618.65 difference in these amounts, the difference in simple interest accounts

for only $8,000 (10 years × $10,000 × 0.08 = $8,000). The remaining $108,618.65 is

attributable to the difference in interest earned on interest. This example illustrates both

the importance of compounding for investment returns and the importance on getting

started early when saving for retirement. The sooner you start saving, the better off you

will be when you retire.

Compounding More Frequently Than Once a Year

Interest can, of course, be compounded more frequently than once a year. In Equation 5.1,

the term n represents the number of periods and can describe annual, semiannual,

quarterly, monthly, or daily payments. The more frequently interest payments are

compounded, the larger the future value of $1 for a given time period. Equation 5.1 can be

rewritten to explicitly recognize different compounding periods:

Moneychimp.com provides a compound interest calculator

at http://www.moneychimp.com/calculator/compound_interest_calculator.htm.

where m is the number of times per year that interest is compounded and n is the number

of periods specified in years.

Let’s say you invest $100 in a bank account that pays a 5 percent interest rate semiannually

(2.5 percent twice a year) for two years. In other words, the annual rate quoted by the bank

is 5 percent, but the bank calculates the interest it pays you based on a six-month rate of

2.5 percent. In this example there are four six-month periods, and the amount of principal

and interest you would have at the end of the four periods would be:

It is not necessary to memorize Equation 5.2; using Equation 5.1 will do fine. All you have

to do is determine the interest paid per compounding period (i/m) and calculate the total

number of compounding periods (m × n) as the exponent for the future value factor. For

example, if the bank compounds interest quarterly, then both the interest rate and

compounding periods must be expressed in quarterly terms: (i/4) and (4 × n).

If the bank in the above example paid interest annually instead of semiannually, you would

have:

at the end of the two-year period. The difference between this amount and the $110.38

above is due to the additional interest earned on interest when the compounding period is

shorter and the interest payments are compounded more frequently.

During the late 1960s, the effects of compounding periods became an issue in banking. At

that time, the interest rates that banks and thrift institutions could pay on consumer

savings accounts were limited by regulation. However, financial institutions discovered

they could keep their rates within the legal limit and pay their customers additional

interest by increasing the compounding frequency. Prior to this, banks and thrifts had paid

interest on savings accounts quarterly. You can see the difference between quarterly and

daily compounding in Learning by Doing Application 5.2.

APPLICATION 5.2 LEARNING BY DOING

Changing the Compounding Period

PROBLEM: Your grandmother has $10,000 she wants to put into a bank savings account for five

years. The bank she is considering is within walking distance, pays 5 percent annual interest

compounded quarterly (5 percent per year/4 quarters per year = 1.25 percent per quarter), and

provides free coffee and doughnuts in the morning. Another bank in town pays 5 percent interest

compounded daily. Getting to this bank requires a bus trip, but your grandmother can ride free as a

senior citizen. More important, though, this bank does not serve coffee and doughnuts. Which bank

should your grandmother select?

APPROACH: We need to calculate the difference between the two banks’ interest payments. Bank

A, which compounds quarterly, will pay one-fourth of the annual interest per quarter, 0.05/4 =

0.0125, and there will be 20 compounding periods over the five-year investment horizon (5 years ×

4 quarters per year = 20 quarters). The time line for quarterly compounding is as follows:

Bank B, which compounds daily, has 365 compounding periods per year. Thus, the daily interest

rate is 0.000137 (0.05/365 = 0.000137), and there are 1,825 (5 years × 365 days per year = 1,825

days) compounding periods. The time line for daily compounding is:

We use Equation 5.2 to solve for the future values the investment would generate at each bank. We

then compare the two.

SOLUTION:

Bank A:

Bank B:

With daily compounding, the additional interest earned by your grandmother is $19.66:

Given that the interest gained by daily compounding is less than $20, your grandmother should

probably select her local bank and enjoy the daily coffee and doughnuts. (If she is on a diet, of

course, she should take the higher interest payment and walk to the other bank).

It is worth noting that the longer the investment period, the greater the additional interest earned

from daily compounding vs. quarterly compounding. For example, if $10,000 was invested for 40

years instead of five years, the additional interest would increase to $900.23. (You should confirm

this by doing the calculation.)

Continuous Compounding

We can continue to divide the compounding interval into smaller and smaller time periods,

such as minutes and seconds, until, at the extreme, we would compound continuously. In

this case, m in Equation 5.2 would approach infinity (∞). The formula to compute the

future value for continuous compounding (FV∞) is stated as follows:

where e is the exponential function, which has a known mathematical value of about

2.71828, n is the number of periods specified in years, and i is the annual interest rate.

Although the formula may look a little intimidating, it is really quite easy to apply. Look for

a key on your calculator labeled ex. If you don’t have the key, you still can work the problem.

Let’s go back to the example in Learning by Doing Application 5.2, in which your

grandmother wants to put $10,000 in a savings account at a bank. How much money would

she have at the end of five years if the bank paid 5 percent annual interest compounded

continuously? To find out, we enter these values into Equation 5.3:

If your calculator has an exponent key, all you have to do to calculate e0.25 is enter the

number 0.25, then hit the ex key, and the number 1.284025 should appear (depending on

your calculator, you may have to press the equal [=] key for the answer to appear). Then

multiply 1.284025 by $10,000, and you’re done! If your calculator does not have an

exponent key, then you can calculate e0.25 by inputting the value of e (2.71828) and raising it

to the 0.25 power using the yx key, as described earlier in the chapter.

Let’s look at your grandmother’s $10,000 bank balance at the end of five years with several

different compounding periods: yearly, quarterly, daily, and continuous:4

Notice that your grandmother’s total earnings get larger as the frequency of compounding

increases, as shown in column 2, but the earnings increase at a decreasing rate, as shown in

column 4. The biggest gain comes when the compounding period goes from an annual

interest payment to quarterly interest payments. The gain from daily compounding to

continuous compounding is small on a modest savings balance such as your grandmother’s.

Twenty-two cents over five years will not buy grandmother a cup of coffee, let alone a

doughnut. However, for businesses and governments with mega-dollar balances at

financial institutions, the difference in compounding periods can be substantial.

EXAMPLE 5.1 DECISION MAKING

Which Bank Offers Depositors the Best Deal?

SITUATION: You have just received a bonus of $10,000 and are looking to deposit the money in a

bank account for five years. You investigate the annual deposit rates of several banks and collect the

following information:

You understand that the more frequently interest is earned in each year, the more you will have at

the end of your five-year investment horizon. To determine which bank you should deposit your

money in, you calculate how much money you will have at the end of five years at each bank. You

apply Equation 5.2 and come up with the following results. Which bank should you choose?

DECISION: Even though you might expect Bank D’s daily compounding to result in the highest

value, the calculations reveal that Bank B provides the highest value at the end of five years. Thus,

you should deposit your money in Bank B because its higher rate offsets the more frequent

compounding at Banks C and D.

Calculator Tips for Future Value Problems

As we have mentioned, all types of future value calculations can be done easily on a

financial calculator. Here we discuss how to solve these problems, and we identify some

potential problem areas to avoid.

A financial calculator includes the following five basic keys for solving future value and

present value problems:

The keys represent the following inputs:

N is the number of periods. The periods can be years, quarters, months, days, or

some other unit of time.

• i is the interest rate per period, expressed as a percentage.

• PV is the present value or the original principal (P0).

• PMT is the amount of any recurring payment.

• FV is the future value.

Given any four of these inputs, the financial calculator will solve for the fifth. Note that the

interest rate key i differs with different calculator brands: Texas Instruments uses the I/Y

key, Hewlett-Packard an i, %i, or I/Y key, and Sharp the i key.

•

For future value problems, we need to use only four of the five keys: N for the number of

periods, i for the interest rate (or growth rate), PV for the present value (at time zero), and

FV for the future value in n periods. The PMT key is not used at this time, but, when doing a

problem, always enter a zero for PMT to clear the register.5

USING EXCEL TIME VALUE OF MONEY

Spreadsheet computer programs are a popular method for setting up and solving finance and

accounting problems. Throughout this book, we will show you how to structure and calculate some

problems using the Microsoft Excel spreadsheet program. Spreadsheet programs are like your

financial calculator but are especially efficient at doing repetitive calculations. For example, once

the spreadsheet program is set up, it will allow you to make computations using preprogrammed

formulas. Thus, you can simply change any of the input cells, and the preset formula will

automatically recalculate the answer based on the new input values. For this reason, we

recommend that you use formulas whenever possible.

We begin our spreadsheet applications with time value of money calculations. As with the financial

calculator approach, there are five variables used in these calculations, and knowing any four of

them will let you calculate the fifth one. Excel has already preset formulas for you to use. These are

as follows:

Solving for

Formula

Present Value

= PV (RATE, NPER, PMT, FV)

Future Value

= FV (RATE, NPER, PMT, PV)

Discount Rate

= RATE (NPER, PMT, PV, FV)

Payment

= PMT (RATE, NPER, PV, FV)

Number of Periods

= NPER (RATE, PMT, PV, FV)

To enter a formula, all you have to do is type in the equal sign, the abbreviated name of the variable

you want to compute, and an open parenthesis, and Excel will automatically prompt you to enter

the rest of the variables. Here is an example of what you would type to compute the future value:

1. =

2. FV

3. (

Here are a few important things to note when entering the formulas: (1) be consistent with signs

for cash inflows and outflows; (2) enter the rate of return as a decimal number, not a percentage;

and (3) enter the amount of an unknown payment as zero.

To see how a problem is set up and how the calculations are made using a spreadsheet, let’s return

to Learning by Doing Application 5.2. The spreadsheet for that application is on the left.

To solve a future value problem, enter the known data into your calculator. For example, if

you know that the number of periods is five, key in 5 and press the N key. Repeat the

process for the remaining known values. Once you have entered all of the values you know,

then press the key for the unknown quantity, and you have your answer. Note that with

some calculators, including the TI BAII Plus, you get the answer by first pressing the key

labeled CPT (compute).

Let’s try a problem to see how this works. Suppose we invest $5,000 at 15 percent for 10

years. How much money will we have in 10 years? To solve the problem, we enter data on

the keys as displayed in the following calculation and solve for FV. Note that the initial

investment of $5,000 is a negative number because it represents a cash outflow. Use the

+/− key to make a number negative.

EXHIBIT 5.6 Tips for Using Financial Calculators

Following these tips will help you avoid problems that sometimes arise in solving time

value of money problems with a financial calculator.

Use the Correct Compounding Period. Make sure that your calculator is set to compound one

payment per period or per year. Because financial calculators are often used to compute monthly

payments, some will default to monthly payments unless you indicate otherwise. You will need to

consult your calculator’s instruction manual because procedures for changing settings vary by

manufacturer. Most of the problems you will work in other chapters of the book will compound

annually.

Clear the Calculator Before Starting. Be sure you clear the data from the financial register before

starting to work a problem because most calculators retain information between calculations. Since

the information may be retained even when the calculator is turned off, turning the calculator off

and on will not solve this problem. Check your instruction manual for the correct procedure for

clearing the financial register of your calculator.

Negative Signs on Cash Outflows. For certain types of calculations, it is critical that you input a

negative sign for all cash outflows and a positive sign for all cash inflows. Otherwise, the calculator

cannot make the computation, and the answer screen will display some type of error message.

Putting a Negative Sign on a Number. To create a number with a negative sign, enter the number

first and then press the “change of sign key.” These keys are typically labeled “CHS” or “+/−”.

Interest Rate as a Percentage. Most financial calculators require that interest rate data be entered

in percentage form, not in decimal form. For example, enter 7.125 percent as 7.125 and not

0.07125. Unlike nonfinancial calculators, financial calculators assume that rates are stated as

percentages.

Rounding off Numbers. Never round off any numbers until all your calculations are complete. If

you round off numbers along the way, you can generate significant rounding errors.

Adjust Decimal Setting. Most calculators are set to display two decimal places. You will find it

convenient at times to display four or more decimal places when making financial calculations,

especially when working with interest rates or present value factors. Again, consult your

instruction manual.

Have Correct BEG or END mode. In finance, most problems that you solve will involve cash

payments that occur at the end of each time period, such as with the ordinary annuities discussed

in Chapter 6. Most calculators normally operate in this mode, which is usually designated as “END”

mode. However, for annuities due, which are also discussed in Chapter 6, the cash payments occur

at the beginning of each period. This setting is designated as the “BEG” mode. Most leases and rent

payments fall into this category. When you bought your financial calculator, it was set in the END

mode. Financial calculators allow you to switch between the END and BEG modes.

If you did not get the correct answer of $20,227.79, you may need to consult the instruction

manual that came with your financial calculator. However, before you do that, you may

want to look through Exhibit 5.6, which lists the most common problems with using

financial calculators. Also, note again that PMT is entered as zero to clear the register.

One advantage of using a financial calculator is that if you have values for any three of the

four variables in Equation 5.1, you can solve for the remaining variable at the press of a

button. Suppose that you have an opportunity to invest $5,000 in a bank and that the bank

will pay you $20,227.79 at the end of 10 years. What interest rate does the bank pay? The

time line for our situation is as follows:

We know the values for N (10 years), PV ($5,000), and FV ($20,227.79), so we can enter

these values into our financial calculator:

Press the interest rate (i) key, and 15.00 percent appears as the answer. Notice that the

cash outflow ($5,000) was entered as a negative value and the cash inflow ($20,227.79) as

a positive value. If both values were entered with the same sign, your financial calculator

algorithm could not compute the equation, yielding an error message. Go ahead and try it.

> BEFORE YOU GO ON

1. What is compounding, and how does it affect the future value of an investment?

2. What is the difference between simple interest and compound interest?

3. How does changing the compounding period affect the amount of interest earned on an

investment?

5.3 PRESENT VALUE AND DISCOUNTING

•

In our discussion of future value, we asked the question “If you put $100 in a bank savings

account that pays 10 percent annual interest, how much money would accumulate in one

year?” Another type of question that arises frequently in finance concerns present value.

This question asks, “What is the value today of a cash flow promised in the future?” We’ll

illustrate the present value concept with a simple example.

Single-Period Investment

Suppose that a rich uncle gives you a bank certificate of deposit (CD) that matures in one

year and pays $110. The CD pays 10 percent interest annually and cannot be redeemed

until maturity. Being a student, you need the money and would like to sell the CD. What

would be a fair price if you sold the CD today?

From our earlier discussion, we know that if we invest $100 in a bank at 10 percent for one

year, it will grow to a future value of $110 = $100 × (1 + 0.10). It seems reasonable to

conclude that if a CD has an interest rate of 10 percent and will have a value of $110 a year

from now, it is worth $100 today.

More formally, to find the present value of a future cash flow, or its value today, we

“reverse” the compounding process and divide the future value ($110) by the future value

factor (1 + 0.10). The result is $100 = $110/(1 + 0.10), which is the same answer we

derived from our intuitive calculation. If we write the calculations above as a formula, we

have a one-period model for calculating the present value of a future cash flow:

The numerical calculation for the present value (PV) from our one-period model follows:

discounting

the process by which the present value of future cash flows is obtained

discount rate

the interest rate used in the discounting process to find the present value of future cash

flows

present value (PV)

the current value of future cash flows discounted at the appropriate discount rate

We have noted that while future value calculations involve compounding an amount

forward into the future, present value calculations involve the reverse. That is, present

value calculations involve determining the current value (or present value) of a future cash

flow. The process of calculating the present value is called discounting, and the interest

rate i is known as the discount rate. Accordingly, the present value (PV) can be thought

of as the discounted value of a future amount. The present value is simply the current value

of a future cash flow that has been discounted at the appropriate discount rate.

Just as we have a future value factor, (1 + i), we also have a present value factor, which is

more commonly called the discount factor. The discount factor, which is 1/(1 + i), is the

reciprocal of the future value factor. This expression may not be obvious in the equation

above, but note that we can write that equation in two ways:

These equations amount to the same thing; the discount factor is explicit in the second one.

Multiple-Period Investment

Now suppose your uncle gives you another 10 percent CD, but this CD matures in two years

and pays $121 at maturity. Like the other CD, it cannot be redeemed until maturity. From

the previous section, we know that if we invest $100 in a bank at 10 percent for two years,

it will grow to a future value of $121 = $100 × (1 + 0.10)2. To calculate the present value, or

today’s price, we divide the future value ($121) by the future value factor (1 + 0.10)2. The

result is $100 = $121/(1 + 0.10)2.

If we write the calculations we made as an equation, the result is a two-period model for

computing the present value of a future cash flow:

Plugging the data from our example into the equation yields no surprises:

By now, you know the drill. We can extend the equation to a third year, a fourth year, and

so on:

The Present Value Equation

Given the pattern shown above, we can see that the general formula for the present value

is:6

where:

Note that Equation 5.4 can be written in slightly different ways, which we will sometimes

do in the book. The first form, introduced earlier, separates out the discount factor, 1/(1

+ i):

In the second form, DFn is the discount factor for the nth period: DFn = 1/(1 + i)n:

Future and Present Value Equations Are the Same

By now, you may have recognized that the present value equation, Equation 5.4, is just a

restatement of the future value equation, Equation 5.1. That is, to get the future value (FVn)

of funds invested for n years, we multiply the original investment by (1 + i)n. To find the

present value of a future payment (PV), we divide FVn by (1 + i)n. Stated another way, we

can start with the future value equation (Equation 5.1), FVn = PV × (1 + i)n and then solve it

for PV; the resulting equation is the present value equation (Equation 5.4), PV = FVn/(1

+ i)n.

Exhibit 5.7 illustrates the relation between the future value and present value calculations

for $100 invested at 10 percent interest. You can see from the exhibit that present value

and future value are just two sides of the same coin. The formula used to calculate the

present value is really the same as the formula for future value, just rearranged.

Applying the Present Value Formula

Let’s work through some examples to see how the present value equation is used. Suppose

you are interested in buying a new BMW Sports Coupe a year from now. You estimate that

the car will cost $40,000. If your local bank pays 5 percent interest on savings deposits,

how much money will you need to save in order to buy the car as planned? The time line

for the car purchase problem is as follows:

EXHIBIT 5.7 Comparing Future Value and Present Value Calculations

The future value and present value formulas are one and the same; the present value

factor, 1/(1 + i)n, is just the reciprocal of the future value factor, (1 + i)n.

The problem is a direct application of Equation 5.4. What we want to know is how much

money you have to put in the bank today to have $40,000 a year from now to buy your

BMW. To find out, we compute the present value of $40,000 using a 5 percent discount

rate:

If you put $38,095.24 in a bank savings account at 5 percent today, you will have the

$40,000 to buy the car in one year.

Since that’s a lot of money to come up with, your mother suggests that you leave the money

in the bank for two years instead of one year. If you follow her advice, how much money do

you need to invest? The time line is as follows:

SmartMoney’s personal finance Web site provides a lot of useful information for day-today

finance dealings at http://www.smartmoney.com/pf/?nav=dropTab.

For a two-year waiting period, assuming the car price will stay the same, the calculation is:

Given the time value of money, the result is exactly what we would expect. The present

value of $40,000 two years out is lower than the present value of $40,000 one year out—

$36,281.18 compared with $38,095.24. Thus, if you are willing to leave your money in the

bank for two years instead of one, you can make a smaller initial investment to reach your

goal.

Now suppose your rich neighbor says that if you invest your money with him for one year,

he will pay you 15 percent interest. The time line is:

The calculation for the initial investment at this new rate is as follows:

Thus, when the interest rate, or discount rate, is 15 percent, the present value of $40,000 to

be received in a year’s time is $34,782.61, compared with $38,095.24 at a rate of 5 percent

and a time of one year. Holding maturity constant, an increase in the discount rate

decreases the present value of the future cash flow. This makes sense because when

interest rates are higher, it is more valuable to have dollars in hand today to invest; thus,

dollars in the future are worth less.

APPLICATION 5.3 LEARNING BY DOING

European Graduation Fling

PROBLEM: Suppose you plan to take a “graduation vacation” to Europe when you finish college in

two years. If your savings account at the bank pays 6 percent, how much money do you need to set

aside today to have $8,000 when you leave for Europe?

APPROACH: The money you need today is the present value of the amount you will need for your

trip in two years. Thus, the value of FV2 is $8,000. The interest rate is 6 percent. Using these values

and the present value equation, we can calculate how much money you need to put in the bank at 6

percent to generate $8,000. The time line is:

SOLUTION:

Thus, if you invest $7,119.97 in your savings account today, at the end of two years you will have

exactly $8,000.

The Relations among Time, the Discount Rate, and Present Value

From our discussion so far, we can see that (1) the farther in the future a dollar will be

received, the less it is worth today, and (2) the higher the discount rate, the lower the

present value of a dollar to be received in the future. Let’s look a bit more closely at these

relations.

Recall from Exhibit 5.4 that the future value of a dollar increases with time because of

compounding. In contrast, the present value of a dollar becomes smaller the farther into

the future that dollar is to be received. The reason is that the present value factor 1/(1

+ i)n is the reciprocal of the future value factor (1 + i)n. Thus, the present value of $1 must

become smaller the farther into the future that dollar is to be received. You can see this

relation in EXHIBIT 5.8, which shows the present value of $1 for various interest rates and

time periods. For example, at a 10 percent interest rate, the present value of $1 one year in

the future is 90.9 cents ($1/1.10); at two years in the future, 82.6 cents [$1/(1.10)2]; at five

years in the future, 62.1 cents [$1/(1.10)5]; and at 30 years in the future, 5.7 cents

[$1/(1.10)30]. The relation is consistent with our view of the time value of money. That is,

the longer you have to wait for money, the less it is worth today. Exhibit A.2, at the end of

the book, provides present value factors for a wider range of years and interest rates.

EXHIBIT 5.8 Present Value Factors

To locate a present value factor, find the row for the number of periods and the column for

the proper discount rate. Notice that whereas future value factors grow larger over time

and with increasing interest rates, present value factors become smaller. This pattern

reflects the fact that the present value factor is the reciprocal of the future value factor.

Exhibit 5.9 shows the present values of $1 for different time periods and discount rates. For

example, the present value of $1 discounted at 5 percent for 10 years is 61 cents, at 10

percent it is 39 cents, and at 20 percent, 16 cents. Thus, the higher the discount rate, the

lower the present value of $1 for a given time period. Exhibit 5.9 also shows that, just as

with future value, the relation between the present value of $1 and time is not linear but

exponential. Finally, it is interesting to note that if interest rates are zero, the present value

of $1 is $1; that is, there is no time value of money. In this situation, $1,000 today has the

same value as $1,000 a year from now or, for that matter, 10 years from now.

Calculator Tips for Present Value Problems

Calculating the discount factor (present value factor) on a calculator is similar to

calculating the future value factor but requires an additional keystroke on most advancedlevel calculators. The discount factor, 1/(1 + i)n, is the reciprocal of the future value factor,

(1 + i)n. The additional keystroke involves the use of the reciprocal key (1/x) to find the

discount factor. For example, to compute 1/(1.08)10, first enter 1.08, press the yx key and

enter 10, then press the equal (=) key. The number on the screen should be 2.159. This is

the future value factor. It is a calculation you have made before. Now press the 1/x key,

then the equal key, and you have the present value factor, 0.463!

EXHIBIT 5.9 Present Value of $1 for Different Time Periods and Discount Rates

The higher the discount rate, the lower the present value of $1 for a given time period. Just

as with future value, the relation between the present value and time is not linear, but

exponential.

Calculating present value (PV) on a financial calculator is the same as calculating the future

value (FVn) except that you solve for PV rather than FVn. For example, what is the present

value of $1,000 received 10 years from now at a 9 percent discount rate? To find the

answer on your financial calculator, enter the following keystrokes:

then solve for the present value (PV), which is −$422.41. Notice that the answer has a

negative sign. As we discussed previously, the $1,000 represents an inflow, and the $442.41

represents an outflow.

EXAMPLE 5.2 DECISION MAKING

Picking the Best Lottery Payoff Option

SITUATION: Congratulations! You have won the $1 million lottery grand prize. You have been

presented with several payout alternatives, and you have to decide which one to accept. The

alternatives are as follows:

$1 million today

$1.2 million lump sum in two years

$1.5 million lump sum in five years

$2 million lump sum in eight years

You are intrigued by the choice of collecting the prize money today or receiving double the amount

of money in the future. Which payout option should you choose?

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•

•

Your cousin, a stockbroker, advises you that over the long term you should be able to earn 10

percent on an investment portfolio. Based on that rate of return, you make the following

calculations:

DECISION: As appealing as the higher amounts may sound, waiting for the big payout is not

worthwhile in this case. Applying the present value formula has enabled you to convert future

dollars into present, or current, dollars. Now the decision is simple—you can directly compare the

present values. Given the above choices, you should take the $1 million today.

Future Value versus Present Value

We can analyze financial decisions using either future value or present value techniques.

Although the two techniques approach the decision differently, both yield the same result.

Both techniques focus on the valuation of cash flows received over time. In corporate

finance, future value problems typically measure the value of cash flows at the end of a

project, whereas present value problems measure the value of cash flows at the start of a

project (time zero).

Exhibit 5.10 compares the $10,000 investment decision shown in Exhibit 5.1 in terms of

future value and present value. When managers are making a decision about whether to

accept a project, they must look at all of the cash flows associated with that project with

reference to the same point in time. As Exhibit 5.10 shows, for most business decisions, that

point is either the start (time zero) or the end of the project (in this example, year 5).

In Chapter 6 we will discuss calculation of the future value or the present value of a series

of cash flows like that illustrated in Exhibit 5.10.

EXHIBIT 5.10 Future Value and Present Value Compared

Compounding converts a present value into its future value, taking into account the time

value of money. Discounting is just the reverse—it converts future cash flows into their

present value.

> BEFORE YOU GO ON

1. What is the present value and when is it used?

2. What is the discount rate? How does the discount rate differ from the interest rate in the

future value equation?

3. What is the relation between the present value factor and the future value factor?

4. Explain why you would expect the discount factor to become smaller the longer the time

to payment.

5.4 ADDITIONAL CONCEPTS AND APPLICATIONS

•

In this final section, we discuss several additional issues concerning present and future

value, including how to find an unknown discount rate, the time required for an investment

to grow by a certain amount, a rule of thumb for estimating the length of time it will take to

“double your money,” and how to find the growth rates of various kinds of investments.

Finding the Interest Rate

In finance, some situations require you to determine the interest rate (or discount rate) for

a given future cash flow. These situations typically arise when you want to determine the

return on an investment. For example, an interesting Wall Street innovation is the zero

coupon bond. These bonds are essentially loans that pay no periodic interest. The issuer

(the firm that borrows the money) makes a single payment when the bond matures (the

loan is due) that includes repayment of the amount borrowed plus all of the interest.

Needless to say, the issuer must prepare in advance to have the cash to pay off

bondholders.

Suppose a firm is planning to issue $10 million worth of zero coupon bonds with 20 years

to maturity. The bonds are issued in denominations of $1,000 and are sold for $90 each. In

other words, you buy the bond today for $90, and 20 years from now, the firm pays you

$1,000. If you bought one of these bonds, what would be your return on investment?

To find the return, we need to solve Equation 5.1, the future value equation, for i, the

interest, or discount, rate. The $90 you pay today is the PV (present value), the $1,000 you

get in 20 years is the FV (future value), and 20 years is n (the compounding period). The

resulting calculation is as follows:

The rate of return on your investment, compounded annually, is 12.79 percent. Using a

financial calculator, we arrive at the following solution:

APPLICATION 5.4 LEARNING BY DOING

Interest Rate on a Loan

PROBLEM: Greg and Joan Hubbard are getting ready to buy their first house. To help make the

down payment, Greg’s aunt offers to loan them $15,000, which can be repaid in 10 years. If Greg

and Joan borrow the money, they will have to repay Greg’s aunt the amount of $23,750. What rate

of interest would Greg and Joan be paying on the 10-year loan?

APPROACH: In this case, the present value is the value of the loan ($15,000), and the future value is

the amount due at the end of 10 years ($23,750). To solve for the rate of interest on the loan, we

can use the future value equation, Equation 5.1. Alternatively, we can use a financial calculator to

compute the interest rate. The time line for the loan, where the $15,000 is a cash inflow to Greg and

Joan and the $23,750 is a cash outflow, is as follows:

SOLUTION:

Using Equation 5.1:

Financial calculator steps:

Finding How Many Periods It Takes an Investment to Grow a Certain

Amount

Up to this point we have used variations of Equation 5.1:

to calculate the future value of an investment (FVn), the present value of an investment

(PV), and the interest rate necessary for an initial investment (the present value) to grow to

a specific value (the future value) over a certain number of periods (i). Note that Equation

5.1 has a total of four variables. You may have noticed that in all of the previous

calculations, we took advantage of the mathematical principal that if we know the values of

three of these variables we can calculate the value of the fourth.

The same principal allows us to calculate the number of periods (n) that it takes an

investment to grow a certain amount. This is a more complicated calculation than the

calculations of the values of the other three variables, but it is an important one for you to

be familiar with.

Suppose that you would like to purchase a new cross-country motorcycle to ride on dirt

trails near campus. The motorcycle dealer will finance the bike that you are interested in if

you make a down payment of $1,175. Right now you only have $1,000. If you can earn 5

percent by investing your money, how long will it take for your $1,000 to grow to $1,175?

To find this length of time, we must solve Equation 5.1, the future value equation, for n.

It will take 3.31 years for your investment to grow to $1,175. If you don’t want to wait this

long to get your motorcycle, you cannot rely on your investment earnings alone. You will

have to put aside some additional money.

Note that because n is an exponent in the future value formula, we have to take the natural

logarithm, ln (x), of both sides of the equation in the fourth line of the above series of

calculations to calculate the value of n directly. Your financial calculator should have a key

that allows you to calculate natural logarithms. Just enter the value in the parentheses and

then hit the LN key.

Using a financial calculator, we obtain the same solution.

The Rule of 72

People are fascinated by the possibility of doubling their money. Infomercials on television

tout speculative land investments, claiming that “some investors have doubled their money

in four years.” Before there were financial calculators, people used rules of thumb to

approximate difficult present value calculations. One such rule is the Rule of 72, which was

used to determine the amount of time it takes to double the value of an investment.

The Rule of 72 says that the time to double your money (TDM) approximately equals 72/i,

where i is the rate of return expressed as a percentage. Thus,

Rule of 72

a rule proposing that the time required to double money invested (TDM) approximately

equals 72/i, where i is the rate of return expressed as a percentage

Applying the Rule of 72 to our land investment example suggests that if you double your

money in four years, your annual rate of return will be 18 percent (i = 72/4 = 18).

Let’s check the rule’s accuracy by applying the future value formula to the land example.

We are assuming that you will double our money in four years, so n = 4. We did not specify

a present value or future value amount; however, doubling our money means that we will

get back $2 (FV) for every $1 invested (PV). Using Equation 5.1 and solving for the interest

rate (i), we find that i = 0.1892, or 18.92 percent.7

That’s not bad for a simple rule of thumb: 18.92 percent versus 18 percent. Within limits,

the Rule of 72 provides a quick “back of the envelope” method for determining the amount

of time it will take to double an investment for a particular rate of return. The Rule of 72 is

a linear approximation of a nonlinear function, and as such, the rule is fairly accurate for

interest rates between 5 and 20 percent. Outside these limits, the rule is not very accurate.

Compound Growth Rates

The concept of compounding is not restricted to money. Any number that changes over

time, such as the population of a city, changes at some compound growth rate. Compound

growth occurs when the initial value of a number increases or decreases each period by the

factor (1 + growth rate). As we go through the course, we will discuss many different types

of interest rates, such as the discount rate on capital budgeting projects, the yield on a

bond, and the internal rate of return on an investment. All of these “interest rates” can be

thought of as growth rates (g) that relate future values to present values.

When we refer to the compounding effect, we are really talking about what happens when

the value of a number increases or decreases by (1 + growth rate)n. That is, the future value

of a number after n periods will equal the initial value times (1 + growth rate)n. Does this

sound familiar? If we want, we can rewrite Equation 5.1 in a more general form as a

compound growth rate formula, substituting g, the growth rate, for i, the interest rate:

where:

Suppose, for example, that because of an advertising campaign, a firm’s sales increased

from $20 million in 2009 to more than $35 million in 2012. What has been the average

annual growth rate in sales? Here, the future value is $35 million, the present value is $20

million, and n is 3 since we are interested in the annual growth rate over three years. The

time line is:

Applying Equation 5.6 and solving for the growth factor (g) yields:

Thus, sales grew nearly 21 percent per year. More precisely, we could say that sales grew at

a compound annual growth rate (CAGR) of nearly 21 percent. If we use our financial

calculator, we find the same answer:

compound annual growth rate (CAGR)

the average annual growth rate over a specified period of time

Note that we enter $20 million as a negative number even though it is not a cash outflow.

This is because one value must be negative when using a financial calculator. It makes no

difference which number is negative and which is positive.

APPLICATION 5.5 LEARNING BY DOING

The Growth Rate of the World’s Population

PROBLEM: Hannah, an industrial relations major, is writing a term paper and needs an estimate of

how fast the world population is growing. In her almanac, she finds that the world’s population was

an estimated 6.9 billion people in 2010. The United Nations estimates that the population will reach

9 billion people in 2054. Calculate the annual population growth rate implied by these numbers. At

that growth rate, what will be the world’s population in 2015?

APPROACH: We first find the annual rate of growth through 2054 by applying Equation 5.6 for the

44-year period 2054–2010. For the purpose of this calculation, we can use the estimated population

of 6.9 billion people in 2010 as the present value, the estimated future population of 9 million

people as the future value, and 44 years as the number of compounding periods (n). We want to

solve for g, which is the annual compound growth rate over the 44-year period. We can then plug

the 44-year population growth rate in Equation 5.6 and solve for the world’s population in 2015

(FV5). Alternatively, we can get the answer by using a financial calculator.

SOLUTION:

Using Equation 5.6, we find the growth rate as follows:

The world’s population in 2015 is therefore estimated to be:

Using the financial calculator approach:

APPLICATION 5.6 LEARNING BY DOING

Calculating Projected Earnings

PROBLEM: IBM’s net income in 2010 was $14.83 billion. Wall Street analysts expect IBM’s earnings

to increase by 6 percent per year over the next three years. Using your financial calculator,

determine what IBM’s earnings should be in three years.

APPROACH: This problem involves the growth rate (g) of IBM’s earnings. We already know the

value of g, which is 6 percent, and we need to find the future value. Since the general compound

growth rate formula, Equation 5.6, is the same as Equation 5.1, the future value formula, we can use

the same calculator procedure we used earlier to find the future value. We enter the data on the

calculator keys as shown below, using the growth rate value for the interest rate. Then we solve for

the future value:

SOLUTION:

Concluding Comments

This chapter has introduced the basic principles of present value and future value. The

table at the end of the chapter summarizes the key equations developed in the chapter. The

basic equations for future value (Equation 5.1) and present value (Equation 5.4) are two of

the most fundamental relations in finance and will be applied throughout the rest of the

textbook.

> BEFORE YOU GO ON

1. What is the difference between the interest rate (i) and the growth rate (g) in the future

value equation?

SUMMARY OF Learning Objectives

Explain what the time value of money is and why it is so important in the field of

finance.

The idea that money has a time value is one of the most fundamental concepts in the field

of finance. The concept is based on the idea that most people prefer to consume goods

today rather than wait to have similar goods in the future. Since money buys goods, they

would rather have money today than in the future. Thus, a dollar today is worth more than

a dollar received in the future. Another way of viewing the time value of money is that your

money is worth more today than at some point in the future because, if you had the money

now, you could invest it and earn interest. Thus, the time value of money is the opportunity

cost of forgoing consumption today.

Applications of the time value of money focus on the trade-off between current dollars and

dollars received at some future date. This is an important element in financial decisions

because most investment decisions require the comparison of cash invested today with the

value of expected future cash inflows. Time value of money calculations facilitate such

comparisons by accounting for both the magnitude and timing of cash flows. Investment

opportunities are undertaken only when the value of future cash inflows exceeds the cost

of the investment (the initial cash outflow).

Explain the concept of future value, including the meaning of the terms principal,

simple interest, and compound interest, and use the future value formula to make

business decisions.

The future value is the sum to which an investment will grow after earning interest. The

principal is the amount of the investment. Simple interest is the interest paid on the

original investment; the amount of simple interest remains constant from period to period.

Compound interest includes not only simple interest, but also interest earned on the

reinvestment of previously earned interest, the so-called interest earned on interest. For

future value calculations, the higher the interest rate, the faster the investment will grow.

The application of the future value formula in business decisions is presented in Section

5.2.

Explain the concept of present value, how it relates to future value, and use the

present value formula to make business decisions.

The present value is the value today of a future cash flow. Computing the present value

involves discounting future cash flows back to the present at an appropriate discount rate.

The process of discounting cash flows adjusts the cash flows for the time value of money.

Computationally, the present value factor is the reciprocal of the future value factor, or

1/(1 + i). The calculation and application of the present value formula in business decisions

is presented in Section 5.3.

Discuss why the concept of compounding is not restricted to money, and use the

future value formula to calculate growth rates.

Any number of changes that are observed over time in the physical and social sciences

follow a compound growth rate pattern. The future value formula can be used in

calculating these growth rates.

SUMMARY OF Key Equations

Self-Study Problems

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5.1 Amit Patel is planning to invest $10,000 in a bank certificate of deposit (CD)

for five years. The CD will pay interest of 9 percent. What is the future value of

Amit’s investment?

5.2 Megan Gaumer expects to need $50,000 as a down payment on a house in six

years. How much does she need to invest today in an account paying 7.25

percent?

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5.3 Kelly Martin has $10,000 that she can deposit into a savings account for five

years. Bank A pays compounds interest annually, Bank B twice a year, and Bank

C quarterly. Each bank has a stated interest rate of 4 percent. What amount

would Kelly have at the end of the fifth year if she left all the interest paid on the

deposit in each bank?

5.4 You have an opportunity to invest $2,500 today and receive $3,000 in three

years. What will be the return on your investment?

5.5 Emily Smith deposits $1,200 in her bank today. If the bank pays 4 percent

simple interest, how much money will she have at the end of five years? What if

the bank pays compound interest? How much of the earnings will be interest on

interest?

Solutions to Self-Study Problems

•

5.1 Present value of Amit’s investment = PV = $10,000

Interest rate = i = 9%

Number of years = n = 5

•

5.2 Amount Megan will need in six years = FV6 = $50,000

Number of years = n = 6

Interest rate = i = 7.25%

Amount needed to be invested now = PV =?

•

5.3 Present value of Kelly’s deposit = PV = $10,000

Number of years = n = 5

Interest rate = i = 4%

Compound period (m):

Amount at the end of five years = FV5 =?

•

5.4 Your investment today = PV = $2,500

Amount to be received = FV3 = $3,000

Time of investment = n = 3

Return on the investment = i =?

•

5.5 Emily’s deposit today = PV = $1,200

Interest rate = i = 4%

Number of years = n = 5

Amount to be received = FV5 =?

a. Future value with simple interest

Simple interest per year = $1,200 × 0.04 = $48

Simple interest for 5 years = $48 × 5 = $240

FV5 = $1,200 + $240 = $1,440

b. Future value with compound interest

Simple interest = ($1,440 − $1,200) = $240

Interest on interest = $1,459.98 − $1,200 − $240 = $19.98

Critical Thinking Questions

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5.1 Explain the phrase “a dollar today is worth more than a dollar tomorrow.”

5.2 Explain the importance of a time line.

5.3 What are the two factors to be considered in time value of money?

5.4 Differentiate future value from present value.

5.5 Differentiate between compounding and discounting.

5.6 Explain how compound interest differs from simple interest.

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5.7 If you were given a choice of investing in an account that paid quarterly

interest and one that paid monthly interest, which one should you choose if they

both offer the same stated interest rate and why?

5.8 Compound growth rates are exponential over time. Explain.

5.9 What is the Rule of 72?

5.10 You are planning to take a spring break trip to Cancun your senior year.

The trip is exactly two years away, but you want to be prepared and have

enough money when the time comes. Explain how you would determine the

amount of money you will have to save in order to pay for the trip.

Chapter 6.1-6.4

6

Discounted Cash Flows and Valuation

Leon Neal/AFP/Getty Images/NewsCom

Learning Objectives

Explain why cash flows occurring at different times must be adjusted to reflect their value as of a

common date before they can be compared, and compute the present value and future value for

multiple cash flows.

Describe how to calculate the present value and the future value of an ordinary annuity and how

an ordinary annuity differs from an annuity due.

Explain what a perpetuity is and where we see them in business, and calculate the value of a

perpetuity.

Discuss growing annuities and perpetuities, as well as their application in business, and

calculate their values.

Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest

rates, and calculate the EAR.

On January 18, 2010, the Board of Directors at Cadbury PLC, the second-largest

confectionary company in the world, recommended to its stockholders that they accept a

takeover offer from Kraft Foods. The announcement ended a takeover contest that had

begun four months earlier and that had taken on many of the characteristics of the hostile

takeover contests from the 1980s. By April 2010, Cadbury PLC was no longer an

independent company.

Cadbury, founded in 1824 in Birmingham, England, was widely viewed by the British public

as a national treasure. The offer from Kraft, an American company, met with widespread

opposition from the British public, labor unions, and politicians, as well as the Cad-bury

board. It also fueled speculation that Hershey Foods, Nestlé, or both would make a

competing friendly offer. In fact, Hershey hired an investment banker and held private talks

with Cadbury about a possible deal.

In the end, however, Kraft prevailed by offering a price that neither Hershey nor Nestlé was

willing to match. Over the four-month period, Kraft raised its offer from $16.2 billion to

$18.9 billion. The final offer represented a 49.6 percent premium over the price at which

Cadbury’s stock had been trading before the contest began and attracted so much support

from key stockholders that the Cadbury board had no choice but to back down from its

opposition to the deal. The combination of Kraft and Cadbury brought together well-known

Kraft brands such as Oreo cookies, Toblerone chocolates, and Ritz crackers with Cadbury

brands such as Trident gum and Dairy Milk chocolates.

In the excitement of such a takeover contest, it is important not to lose sight of the central

question: What is the firm really worth? A company invests in an asset—a business or

a capital project—because it expects the asset to be worth more than it costs. That’s how

value is created. The value of a business is the sum of its discounted future cash flows.

Thus, the task for Kraft was to estimate the value of the cash flows that Cadbury would

generate under its ownership. Whether the $18.9 billion price tag is justified remains to be

seen. This chapter, which discusses the discounting of future cash flows, provides tools that

help answer the question of what Cadbury is worth to Kraft.

CHAPTER PREVIEW

In Chapter 5 we introduced the concept of the time value of money: Dollars today are more valuable

than dollars to be received in the future. Starting with that concept, we developed the basics of

simple interest, compound interest, and future value calculations. We then went on to discuss

present value and discounted cash flow analysis. This was all done in the context of a single cash

flow.

In this chapter, we consider the value of multiple cash flows. Most business decisions, after all,

involve cash flows over time. For example, if Hatteras Hammocks®, a North Carolina-based firm that

manufactures hammocks, swings, and rockers, wants to consider building a new factory, the

decision will require an analysis of the project’s expected cash flows over a number of periods.

Initially, there will be large cash outlays to build and get the new factory operational. Thereafter,

the project should produce cash inflows for many years. Because the cash flows occur over time, the

analysis must consider the time value of money, discounting each of the cash flows by using the

present value formula we discussed in Chapter 5.

We begin the chapter by describing calculations of future and present values for multiple cash

flows. We then examine some situations in which future cash flows are level over time: These

involve annuities, in which the cash flow stream goes on for a finite period, and perpetuities, in

which the stream goes on forever. Next, we examine annuities and perpetuities in which the cash

flows grow at a constant rate over time. These cash flows resemble common cash flow patterns

encountered in business. Finally, we describe the effective annual interest rate and compare it with

the annual percentage rate (APR), which is a rate that is used to describe the interest rate in

consumer loans.

6.1 MULTIPLE CASH FLOWS

•

We begin our discussion of the value of multiple cash flows by calculating the future value

and then the present value of multiple cash flows. These calculations, as you will see, are

nothing more than applications of the techniques you learned in Chapter 5.

Future Value of Multiple Cash Flows

In Chapter 5, we worked through several examples that involved the

future value of a lump sum of money invested in a savings account that paid 10 percent

interest per year. But suppose you are investing more than one lump sum. Let’s say you put

$1,000 in your bank savings account today and another $1,000 a year from now. If the bank

continues to pay 10 percent interest per year, how much money will you have at the end of

two years?

To solve this future value problem, we can use Equation 5.1: FVn PV (1 i)n. First, however,

we construct a time line so that we can see the magnitude and timing of the cash flows.

As Exhibit 6.1 shows, there are two cash flows into the savings plan. The first cash flow is

invested for two years and compounds to a value that is computed as follows:

EXHIBIT 6.1 Future Value of Two Cash Flows

This exhibit shows a time line for two cash flows invested in a savings account that pays 10

percent interest annually. The total amount in the savings account after two years is

$2,310, which is the sum of the future values of the two cash flows.

The second cash flow earns simple interest for a single period only and grows to:

As Exhibit 6.1 shows, the total amount of money in the savings account after two years is

the sum of these two amounts, which is $2,310 ($1,100 $1,210 $2,310).

Now suppose that you expand your investment horizon to three years and invest $1,000

today, $1,000 a year from now, and $1,000 at the end of two years. How much money will

you have at the end of three years? First, we draw a time line to be sure that we have

correctly identified the time period for each cash flow. This is shown in Exhibit 6.2. Then

we compute the future value of each of the individual cash flows using Equation 5.1. Finally,

we add up the future values. The total future value is $3,641. The calculations are as

follows:

To summarize, solving future value problems with multiple cash flows involves a simple

process. First, draw a time line to make sure that each cash flow is placed in the correct

time period. Second, calculate the future value of each cash flow for its time period. Third,

add up the future values.

Let’s use this process to solve a practical problem. Suppose you want to buy a

condominium in three years and estimate that you will need $20,000 for a down payment.

If the interest rate you can earn at the bank is 8 percent and you can save $3,000 now,

$4,000 at the end of the first year, and $5,000 at the end of the second year, how much

money will you have to come up with at the end of the third year to have a $20,000 down

payment?

EXHIBIT 6.2 Future Value of Three Cash Flows

The exhibit shows a time line for an investment program with a three-year horizon. The

value of the investment at the end of three years is $3,641, the sum of the future values of

the three separate cash flows.

The time line for the future value calculation in this problem looks like this:

To solve the problem, we need to calculate the future value for each of the cash flows, add

up these values, and find the difference between this amount and the $20,000 needed for

the down payment. Using Equation 5.1, we find that the future values of the cash flows at

the end of the third year are:

At the end of the third year, you will have $13,844.74, so you will need an additional

$6,155.26 ($20,000 − $13,844.74 = $6,155.26) at that time to make the down payment.

APPLICATION 6.1 LEARNING BY DOING

Government Contract to Rebuild a Bridge

PROBLEM: The firm you work for is considering bidding on a government contract to rebuild an

old bridge that has reached the end of its useful life. The two-year contract will pay the firm

$11,000 at the end of the second year. The firm’s estimator believes that the project will require an

initial expenditure of $7,000 for equipment. The expenses for years 1 and 2 are estimated at $1,500

per year. Because the cash inflow of $11,000 at the end of the contract exceeds the …

The price is based on these factors:

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