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Chapter 3

Time Is Money

dpa/Corbis

Learning Objectives

A�er studying this chapter, you should be able to:

Express the �me value of money and related mathema�cs, including present and future values, principal, and
interest.
Explain the significance of compounding frequency in rela�on to future and present cash flows and effec�ve
annual percentage rates.
Iden�fy the values of common cash flow streams, including perpetui�es, annui�es, and amor�zed loans.

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Ch. 3 Introduction

The saying “�me is money” could not be more true than it is in finance. People ra�onally prefer to collect money earlier rather than later. By delaying the receipt of cash,
individuals forgo the opportunity to purchase desired goods or invest the funds to increase their wealth. The foregone interest, which could be earned if cash were received
immediately, is called the opportunity cost of delaying its receipt. Individuals require compensa�on to reimburse them for the opportunity cost of not having the funds
available for immediate investment purposes. This chapter describes how such opportunity costs are calculated. Because many business ac�vi�es require compu�ng a value
today for a series of future cash flows, the techniques presented in this chapter apply not only to finance but also to marke�ng, manufacturing, and management. Here are
examples of ques�ons that the tools introduced in this chapter can help answer:

How much should we spend on an adver�sing campaign today if it will increase sales by 5% in the future?
Is it worth buying a new computerized lathe for $120,000 if the lathe reduces material waste by 15%?
Which strategy should we employ, given their respec�ve costs and es�mated contribu�ons to future earnings?
What types of health insurance and re�rement plans are best for our employees, given the amount of money we have available?

Being able to value the cash to be received in the future—whether dividends from a share of stock, interest from a bond, or profits from a new product—is one of the
primary skills needed to run a successful business. This chapter provides you with an introduc�on to that skill.

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How do we determine the future value of $100 today?

How do we determine the future value of $100 today at a 10% interest rate?

3.1 The Time Value of Money

The �me value of money and the mathema�cs associated with it provide important tools for comparing the rela�ve values of cash flows received at different �mes. Just as a
hammer may be the most useful item in a carpenter’s toolbox, �me value of money mathema�cs is indispensable to a financial manager.

Recall from Chapter 1 that to increase shareholder wealth managers must make investments that have greater value than their costs. O�en, such investments require an
immediate cash outlay, like buying a new delivery truck. The investment (the truck) then produces cash flows for the corpora�on in the future (delivery fee income, increased
sales, lower delivery costs, etc.). To determine whether the future cash flows have greater value than the ini�al cost of the truck, managers must be able to calculate the
present value of the future stream of cash flows produced by this investment. Let’s take a closer look at the �me value of money in ac�on.

Present and Future Value

Suppose a friend owes you $100 and the payment is due today. You receive a phone call from this friend, who says she would like to delay paying you for one year. You may
reasonably demand a higher future payment, but how much more should you receive? The situa�on is illustrated in Figure 3.1 as a �meline.

Figure 3.1: Determining future value

In this diagram “now,” the present �me, is assigned t = 0, or �me zero. One year from now is assigned t = 1. The present value of the cash payment is $100 and is denoted
PV0. Its future value at t = 1 is denoted as FV1. To find the amount that you could demand for deferring receipt of the money by one year, you must solve for FV1, the future

value of $100 one year from now. The FV1 value will depend on the opportunity cost of forgoing immediate receipt of $100. You know, for instance, that if you had the

money today you could deposit the $100 in a bank account earning 5% interest annually. However, you know from Chapter 2 that value depends on risk. In your judgment,
your friend is less likely to pay you next year than is the bank. Therefore, you will increase the rate of interest to reflect the addi�onal risk that you think is inherent in the
loan to your friend.

Suppose that you decide that a 10% annual rate of interest is appropriate. The amount of the future payment, FV1, will be the original principal plus the interest that could

be earned at the 10% annual rate. Algebraically, you can solve for FV1, being careful always to convert percentages to decimals when doing arithme�c calcula�ons,

FV1 = $100 + $100 (0.10)

Factoring $100 from the right-hand side of the equa�on, we have

FV1 = $100(1 + 0.10) = $100(1.10) = $110

You may demand a $110 payment at t = 1 in lieu of an immediate $100 payment because these two amounts have equivalent value. No�ce that if you had deposited the
$100 in the bank, you would have only $105 a�er one year. The higher the interest rate, the faster the amount will grow.

Let’s say that your friend agrees to this interest rate but asks to delay payment for two years. The new scenario is illustrated in Figure 3.2.

Figure 3.2: Determining the future value of $100 at %10 interest

Now we must find FV2, the future value of the payment 2 years from today. Since we know FV1 = $110 and we know the interest rate is 10%, we can solve for FV2 by

recognizing that FV2 will equal FV1 plus the interest that could be earned on FV1 during the second year.

FV2 = FV1 + FV1(0.10) = $110 + ($110)(0.10) = $110(1 + 0.10) = $121

You may demand a $121 payment at t = 2 because its �me value is equivalent to either $110 at t = 1 or $100 at t = 0, given the 10% interest rate.

Simple and Compound Interest

We just showed that, at a 10% annual interest rate, $100 today is equivalent to $110 a year from now and $121 in 2 years. Now, we look at how compound and simple
interest affect the �me value of money. Look at the �meline shown in Figure 3.3.Processing math: 0%

How does 10% compound interest impact the value of $100 a�er two years?

Even when lending money to a friend, it’s
important to iron out the details, agree
upon the terms of repayment, and figure
out the �me value of the money.

The Agency Collec�on/Ge�y Images

Figure 3.3: The future value of $100 at 10% compound interest

This result may be generalized using the following formulas,

(3.1) FV1 = PV0(1 + r)

(3.2) FV2 = PV0(1 + r)
2

where r is the interest rate.

Equa�on (3.2) can be restated as

(3.3) FV2 = PV0(1 + 2r) + PV0(r
2)

Equa�on (3.3) is broken down in a special way. The first term on the right side of the equal sign, PV0 (1 + 2r), would yield $120 given the informa�on we have used in our

example. The second term, PV0 (), yields $1. The value $120 equals your original principal ($100) plus the amount of interest earned ($20) if your friend paid simple interest.

Simple interest means that the same dollar amount of interest is received every period. For example, if you withdraw interest earned during each year at the end of that
year, you would earn simple interest. In this case, you would receive $10 interest payments at the end of years 1 and 2, totaling $20. If, on the other hand, your friend
credited (but did not pay) interest to you every year, then you would earn interest during year 2 on the interest credited to you at the end of year 1. Earning interest on
previously earned interest is known as compounding. Thus, you would earn an extra dollar, a total of $121, over the two-year period with interest compounded annually. The
example assumed annual compounding since nearly all transac�ons are now based on compound rather than simple interest. This problem is demonstrated in the Applying
Finance: Annual Compounding feature. See Appendix A (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/appendix#appendix) for informa�on on se�ng up your
calculator, and addi�onal financial applica�on problems.

Applying Finance: Annual Compounding

Future Value With Annual Compounding: To solve the problem we just looked at with a financial calculator or Excel is straigh�orward.

To Solve Using TI Business Analyst

A�er clearing the calculator, use the following inputs:

100 [+/−] [PV]

2 [N]

0 [PMT]

10 [I/Y]

[CPT] [FV]Processing math: 0%

https://content.ashford.edu/books/AUBUS650.13.1/sections/appendix#appendix

Many of the contracts on the bills we pay involve compound
interest, such as with our credit cards. Can you think of other
examples of where compound interest is u�lized?

Rob Lewine/Ge�y Images

= $121.00

Note: These may be input in any order so long as the FV and Compute are at the end. We entered 100 as a nega�ve number. Think of it as $100 going away (you are
giving it to the bank) and the $121 is being received (the bank is giving it back to you), so the two cash flows will have opposite signs. If you enter 100 [+/–] [PV] in this
problem, then your answer will be a posi�ve 121. If you entered the PV of $100 as a posi�ve number, then the FV displayed would be signed nega�ve.

To Solve Using Excel

Use the FV func�on. The inputs for this func�on are: FV(RATE,NPER,PMT,PV,TYPE)

RATE: Interest rate per period as a %

NPER: Number of compounding periods

PMT: Any periodic payment (for the FV of a single cash flow this would be zero)

PV: Present value

TYPE: 0 if payments are made at the end of the period (most common) and 1 if payments are made at the beginning of the period

= FV(10%,2,0,-100,0)

= $121.00

Note: Financial func�ons in Excel require that cash inflows and cash ou�lows have different arithme�c signs. We signed the PV (the amount you put in the bank today)
nega�ve because it is flowing away from you and into the bank. The result ($121.00) is posi�ve because that is a cash flow to you. Commas separate the inputs, so you
cannot enter numbers with commas separa�ng thousands (e.g., $1,000). Nor can you include dollar signs ($).

Let’s extend this example to 20 years to be�er show the difference between simple and compound interest. At a 10% interest rate using simple interest our original deposit
of $100 would grow into $300 over 20 years. This growth is based on receiving $10 of interest each year.

$100 + (20 × $10) = $100 + $200 = $300

Compare this to the result from compound interest.

FV20 = PV0(1 + r)
20 = 100(1 + 0.10)20 = $672.75

The difference between simple and compound interest is $372.75 over 20 years!

Not all compounding is done on an annual basis, however. Some�mes interest is added to an account every six
months (semiannual compounding). Other contracts call for quarterly, monthly, or daily compounding. As you
will see in the next sec�on, the frequency of compounding can make a big difference when the �me value of
money is calculated.

Understanding Compound Interest

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Compound interest allows earned money to earn more money. Interest can
be compounded daily, monthly, or annually. Why does a financial manager
need to understand the difference in compounding periods?

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How does semiannual interest impact the value of $100 a�er two years?

3.2 Valuing a Single Cash Flow

As men�oned in the previous sec�on, how o�en a loan’s interest is compounded changes how we determine the �me value of money. There are different compounding
periods: mortgage or car loans use monthly compounding; corporate bonds that pay interest semiannually use semiannual compounding; some cer�ficates of deposit use
con�nuous compounding; and many credit cards use daily compounding. In this sec�on, we will show how different compounding periods affect the �me value of money
formula used.

Con�nuing our example from Sec�on 3.1, let us suppose that your friend who wishes to delay paying you agrees to a 10% annual rate of interest over the two-year period
and will allow you to compound interest semiannually. What will you be paid in two years given this agreement? Semiannual compounding means that interest will be
credited to you every six months, based on half of the annual rate. In effect you will be earning a 5% semiannual rate of interest over four six-month periods. In other words,
the periodic interest rate will be half the annual rate because you are using semiannual compounding and you will be earning interest for four �me periods (n = 1 through
4), each period being one-half year long. The new situa�on is illustrated in Figure 3.4.

Figure 3.4: Semiannual compounding

Here, FV1 is the future value of the $100 at the end of period 1 (the first six months). As before, FV1 equals the $100 beginning principal plus interest earned over the six

months at the 5% semiannual interest rate.

FV1 = $100 + $100(0.05) = $100(1 + 0.05) = $105

Therefore, at the end of period 1 (at n = 1), the principal balance you are owed will be $105. FV2 will be equal to the principal at the beginning of period 2 plus interest

earned during period 2.

FV2 = $105 + $105(0.05) = $110.25

Note that we could subs�tute [$100(1.05)] for $105 in the previous equa�on. Doing so, FV2 could be expressed as follows:

FV2 = $105(1.05) = [$100(1.05)](1.05) = $100(1.05)
2

Following this pa�ern, finding FV3 and FV4 is straigh�orward.

FV3 = $100(1.05)
3 = $115.76

FV4 = $100(1.05)
4 = $121.55

The final equa�on gives the answer we seek. The future value at the end of four six-month periods is $121.55. Changing from annual compounding to semiannual
compounding has increased the future value of your friend’s obliga�on to you by $0.55. The addi�onal interest earned from semiannual compounding, $0.55, doesn’t seem
like much but imagine a firm borrowing $100 million; then the amount earned from compounding—the interest earned on previous interest—can turn into tens of thousands
of dollars.

The Future Value of a Single Cash Flow

The pa�ern established here may be generalized into the formula for the future value of a single cash flow using compound interest.

(3.4) FVn = PV0(1 + r)
n

where

FVn = the future value at the end of n �me periods

PV0 = the present value of the cash flow

r = the periodic interest rate

which equals the annual nominal rate divided by the number of compounding periods per year,

n = the number of compounding periods un�l maturity, or

n = (number of years un�l maturity)(compounding periods per year)

It is cri�cal when using this formula to be certain that r and n agree with each other. If, for example, you are finding the future value of $100 a�er six years and the annual
rate is 18%, compounded monthly, then the appropriate r is 1.5% per month (18%/12 = 1.5%) and n is 72 months (6 years �mes 12 months per year = 72 months). Students

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o�en adjust the interest rate and then forget to adjust the number of periods (or vice versa)! The answer to this problem is

FV72 = $100(1.015)
72 = $292.12

For quarterly compounding, you would divide the annual rate by 4 and mul�ply the number of years by 4. So $100 a�er six years and an annual rate of 18% with quarterly
compounding would be found using a periodic interest rate of 4.5% (18%/4) and 24 periods (6 years �mes 4 periods per year).

FV24 = $100(1.045)
24 = $287.60

For simple interest, without compounding, the future value is simply equal to the annual interest earned, �mes the number of years, plus the original principal. The formula
for the future value of a single cash flow using simple interest is

= PV0 + (n)(PV0)(r) = PV0(1 + nr)

where

= the future value at the end of n periods using simple interest

n = the number of periods un�l maturity (Generally n simply equals the number of years because there is no adjustment for compounding periods.)

r = the periodic rate (which also usually equals the annual rate because there is no adjustment for compounding periods)

For the previous example, the future value of $100 invested for 6 years in an account paying 18% per year using simple interest is

= $100[1 + (6)(0.18)] = $208.00

The adjustment process we just discussed works for all compounding periods except one: con�nuous compounding. We won’t go into the details of the math; we will just
show the result.

(3.6) FVn = PV0(e
rn)

The le�er e is one of those special numbers in mathema�cs that is assigned its own name. (Another one is π, which you may remember is approximately equal to 3.14.) The
number e is approximately equal to 2.72 (more precisely 2.71828183). The exponent rn in Equa�on (3.6) doesn’t need to be adjusted. In our example of 18% for six years, rn
will be the same with or without any of those adjustments we discussed. We always do math with decimals rather than percentages, so the rn exponent for 18% would be
0.18 × 6 = 1.08 as is 1.5% (0.015 × 72) or 4.5% (0.045 × 24). Most calculators have an key, which makes compu�ng con�nuous compounding fairly easy.

Monthly compounding yielded a future value a�er six years of $292.12, or $84.12 more than simple interest in this example. Table 3.1 illustrates the future value of $100,
bearing 18% annual interest, with different compounding assump�ons. Use your calculator to replicate the solu�ons illustrated below. Be sure your n and r agree (e.g., both
are monthly, yearly, etc.), and always be sure you express percentages as decimals before doing any calcula�ons. You should prac�ce with your calculator un�l your answers
match those given. For more applica�ons, refer to the Applying Finance: Future Value feature box.

Table 3.1: The future value of $100 a�er six years at 18% annual interest, various compounding periods

Compounding assump�on n r FVn

Annual 6 0.18 $269.96

Semiannual 12 0.09 $281.27

Quarterly 24 0.045 $287.60

Monthly 72 0.015 $292.12

Weekly 312 0.00346 $293.92

Daily 2,190 0.000493 $294.39

Con�nuous ∞ $294.47

Applying Finance: Future Value

Future Value of Single Cash Flow:

If you put $400 in the bank today at 12% per year, leave it there for five years, what will be the balance at the end of the �me period?

To Solve Using TI Business Analyst

400 [PV]

5 [N]

0 [PMT]

12 [I/Y]

[CPT] [FV]

= $704.9366Processing math: 0%

How much can you borrow today at 12% compounded monthly for 36 months, if you plan to pay it back with
your $1000 bonus at the end of the loan?

Note: These may be input in any order so long as the FV and Compute are at the end. Also, the calculator register will show the answer as a nega�ve 704.9366 since you
entered 400 as a posi�ve number. Think of it as 400 is cash going one way (you are giving it to the bank) and the 704 is going the opposite direc�on (the bank is giving it
back to you), so the two cash flows will have opposite signs. If you enter 400 [+/–] [PV] in this problem, then your answer will be a posi�ve 704.9366. It does not ma�er
which way you do this.

To Solve Using Excel
Use the FV func�on. The inputs for this func�on are: FV(RATE,NPER,PMT,PV,TYPE)
RATE: Interest rate per period as a %
NPER: Number of compounding periods
PMT: Any periodic payment (for the FV of a single cash flow this would be zero)
PV: Present value
TYPE: 0 if payments are made at the end of the period (most common) and 1 if payments are made at the beginning of the period
If you put $400 in the bank today at 12% per year, leave it there for five years, what will be the balance at the end of the �me period?

=FV(12%,5,0,-400,0)

=$704.94

Note: Financial func�ons in Excel require that cash inflows and cash ou�lows have different arithme�c signs. We signed the PV (the amount you put in the bank today
nega�ve because it is flowing away from you and into the bank. The result ($704.94) is posi�ve because that is a cash flow to you. The inputs are separated by commas,
so you cannot enter numbers with commas separa�ng thousands (e.g., $1,000). Nor can you include dollar signs ($).

The Present Value of a Single Cash Flow

We have solved for the future value of a current cash flow. O�en, we must solve for the present value of a future cash flow, solving for PV rather than FV. You can think of
the present value as the amount that you have to put in the bank today to have some specific amount in the future. A higher interest rate causes a deposit to grow faster, so
the higher the interest rate the smaller the amount of money that has to be deposited today to achieve a desired future amount. Similarly, the longer the �me un�l a future
cash flow is collected, the smaller the amount deposited has to be. This is because the ini�al deposit has more �me to grow.

Suppose, for example, you are going to receive a bonus of $1,000 in three years. You could really use some cash today and are able to borrow from a bank that would
charge you an annual interest rate of 12%, compounded monthly. You decide to borrow as much as you can now such that you will s�ll be able to pay off the loan in three
years using the $1,000 bonus. In essence, you wish to solve for the present value of a $1,000 future value, knowing the interest rate (12% per year, compounded monthly)
and the term of the loan (3 years, or 36 monthly compounding periods). Figure 3.5 is a �meline illustra�ng the problem. This problem is also prac�ced in the Applying
Finance: Present Value feature.

Figure 3.5: Determining present value of $1000 in the future

Applying Finance: Present Value

Present Value of Single Cash Flow: How much money would you have to put in the bank today at 12% per year, with monthly compounding, to have $1,000 in exactly
three years?

To Solve Using TI Business Analyst

Clear TVM worksheet

2nd [CLR TVM]

2nd [Quit]

Clear CF worksheetProcessing math: 0%

2nd [CLR WORK]

2nd [Quit]

Set the compounding period to monthly

2nd [P/Y] 12 [enter]

2nd [Quit]

1000 [FV]

36 [N]

0 [PMT]
12 [I/Y]

[CPT] [PV]

= $698.92

To Solve Using Excel

Use the PV func�on with the format: PV(RATE,NPER,PMT,FV,TYPE).

The inputs for this example would be:

=PV(1%,36,0,1000,0)

= –$698.92

In this case n = 36, r = 1%, and is known, whereas PV0 is unknown. We may s�ll use Equa�on (3.4), subs�tu�ng in the known quan��es and using some algebra.

(3.4) FVn = PV0(1 + r)
n

$1,000 = PV0(1.01)
36

You could borrow $698.92 today and fully pay off the loan, given the bank’s terms, in three years using your $1,000 bonus. We can generalize the last expression into the
formula for the present value of a single cash flow with compound interest. Solving for the present value of a future cash flow is also known as discoun�ng. In fact,
compounding and discoun�ng are flip sides of the same coin. Compounding is used to express a value at a future date given a rate of interest. Discoun�ng involves
expressing a future value as an equivalent amount at an earlier date.

This formula is also called the discoun�ng formula for a single future cash flow.

(3.7)

The variables PV0, FVn, n, and r are defined exactly as they are in the future value formula because both formulas are really the same, just solved for different unknowns.

To find the present value with con�nuous compounding, we would use

(3.8)

Table 3.2 solves for the present, or discounted, value of a $1,000 cash flow to be received in 1 year at a 12% per year discount rate using different compounding periods. You
should be able to replicate these solu�ons on your calculator.

Table 3.2: The present value of $1,000 to be received in 1 year discounted at 12% annual interest, various compounding periods

Compounding assump�on N R PVN

Annual 6 0.12 $892.86

Semiannual 12 0.06 $890.00

Quarterly 24 0.03 $888.49

Monthly 72 0.01 $887.45

Weekly 312 0.00231 $887.04

Daily 2,190 0.000329 $886.94

Con�nuous ∞ $886.92

Present and future value formulas are very useful because they may be used to solve a variety of problems. Suppose you make a $500 deposit in a bank today and you want
to know how long it will take your account to double in value, assuming that the bank pays 8% interest per year, compounded annually (shown in Figure 3.6). Here, you are
solving for the number of �me periods. You may subs�tute the known quan��es PV0 = $500, FVn = $1,000, r = 0.08 into either formula and solve for n:Processing math: 0%

How many �me periods must pass to double your investment of $500 at 8% interest compounded annually?

What is the interest rate on an investment of $200 that results in a single payment of $275 in five years?

Figure 3.6: Determining number of �me periods

(3.7) PV0 = FVn(1 + r)
−n

$500 = $1,000(1.08)−n

(1.08)n = $1,000/$500

(1.08)n = 2

At this point, without using logarithms, you must use trial and error to solve for n. Suppose you try n = 10 as your first guess for n:

(1.08)10 = 2.1589

This value yields a number higher than our objec�ve of 2. Therefore, try n = 9 because a lower value of n will yield a lower answer:

(1.08)9 = 1.999

which is close enough. In nine years, the balance in your account will double.

Suppose the account earned 8% per year compounded monthly. To find the �me un�l the account’s balance doubled, you would convert the interest rate below to reflect
monthly compounding,

and then solve for the number of compounding periods.

(3.7) PV0 = FVn(1 + r)
−n

$500 = $1,000(1.00667)−n

(1.00667)n = 2

Using trial and error, you get the answer n = 105. This should be interpreted as 105 months because you are dealing with monthly compounding periods. Thus, in 8.75 years
the account will double in value when using monthly rather than annual compounding.

This example illustrates an important lesson. It takes less �me to achieve a desired amount of wealth with more frequent compounding at a given nominal interest rate. It is
no surprise that borrowers prefer less frequent compounding, while lenders prefer compounding as frequently as possible. The difference between compounding frequencies
offered at various banks makes shopping around worthwhile whether you are a borrower or a saver.

Finding Interest Rates

Another type of problem is solving for the interest rate. This �me let’s suppose that an investment cos�ng $200 will make a single payment of $275 in 5 years. What is the
interest rate such an investment will yield? Subs�tute n = 5 into the formula and solve for r.

Figure 3.7 shows a �meline for finding the interest rate that equates a $200 deposit to a future value of $275 in 5 years.

Figure 3.7: Determining interest rate

We can use Equa�on (3.7) to find r.

(3.7) PV0 = FVn(1 + r)
−n

$200 = $275(1 + r)−n

(1 + r)5 = 1.375

1 + r = (1.375)⅕

1 + r = 1.3750.20

r = (1.375)0.20 − 1

r = 0.06576Processing math: 0%

The answer, r = 0.06576, is based on an annual compound rate because we assumed n = 5 years. It is also expressed as a decimal and could be re-expressed as a percentage,
6.576% per year compounded annually. See the Applying Finance: Finding Interest Rates feature for prac�ce with this problem.

Applying Finance: Finding Interest Rates

Annual Compound Interest Rate: If a $200 deposit grows into $275 in five years, what is the annual compound interest rate?

To Solve Using TI Business Analyst

(Make sure to set P/Y to 1).

275 [FV]

5 [N]
0 [PMT]

200 [+/–] [PV]

[CPT] [I/Y]

= 6.58

To Solve Using Excel

Use the Rate func�on with the format: PV(NPER,PMT,PV,FV,TYPE,GUESS).

The inputs for this example would be:

= RATE(5,0,-200,275,,)

= 6.5763%

Effective Annual Percentage Rate

As you have seen, the frequency of compounding is important. Truth-in-lending laws now require that financial ins�tu�ons reveal the effec�ve annual percentage rate (EAR)
to customers so that the true cost of borrowing is explicitly stated. Before this legisla�on, banks could quote customers annual interest rates without revealing the
compounding period. Such a lack of disclosure can be costly to borrowers. For example, borrowing at a 12% yearly rate from bank A may be more costly than borrowing
from bank B, which charges 12.1% yearly, if bank A compounds interest daily and bank B compounds semiannually. Both 12% and 12.1% are nominal rates—they reveal the
rate “in name only” but not in terms of the true economic cost. To find the effec�ve annual rate, divide the nominal annual percentage rate (APR) by the number of
compounding periods per year and add 1; then raise this sum to an exponent equal to the number of compounding periods per year. Finally, subtract 1 from this result.

(3.9)

For our example,

Thus, if you are a borrower, you would prefer to borrow from bank B despite its higher APR. The lower EAR translates into a lower cost over the life of the loan. The
disclosure of EARs makes comparison shopping for rates much easier.

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The charity should bid no more than $6,153.07 for the hot dog stand, given the stream of expected cash
flows.

In this case, interest is deferred un�l a�er the first �me period. This is not always true of future values of
cash flow streams.

3.3 Valuing Multiple Cash Flows

Many problems in finance involve finding the �me value of mul�ple cash flows. Consider the following problem. A charity has the opportunity to purchase a used mobile hot
dog stand being sold at an auc�on. The charity would use the hot dog stand to raise money at special events held in the summer each year (at the county fair, baseball and
soccer games, etc.). The old hot dog stands will only last two years and then will be worthless. The charity es�mates that, a�er all opera�ng expenses, the stand will produce
cash flows of $1,000 in both June and July in each of the next two years and cash flows of $1,500 in each of the next two Augusts. The auc�on takes place January 1, and
the charity requires that its fundraising projects return 12% on their invested funds. How much should the charity bid for the hot dog stand? The strategy for solving this
problem is shown in Figure 3.8.

Figure 3.8: Determining the present value of mul�ple cash flows

The present value of the stream of cash flows the stand is expected to produce is found by applying Equa�on (3.7) to each of the six future cash flows. Note that 1% is used
as the periodic rate (12% per year/12 months) because cash flows are spaced in monthly intervals. The charity should bid a maximum of $6,153.07 for the hot dog stand.
Given the level of expected cash flows, paying more than this amount would result in the charity earning a lower return than its 12% objec�ve.

The hot dog stand example illustrates the general formula for finding the present value of any cash flow stream,

(3.10)

where

n = the number of compounding periods from �me 0
CFn = the cash flow to be received exactly n compounding periods from �me 0 (e.g., CF1 is the cash flow received at the end of period 1, etc.)

r = the periodic interest rate
N = the number of periods un�l the last cash flow

The future value formula for a cash flow stream is also found by finding the future value of each individual cash flow and summing. Terms in the formula are defined as in
the present value formula.

(3.11) FVN = CF1(1 + r)
N − 1 + CF2(1 + r)

N − 2 + … + CFN

You may ques�on why in Equa�on (3.11) the first term is raised to the exponent N – 1 and why the last term is not mul�plied by an interest factor. This situa�on may be
clarified by using a �meline, as shown in Figure 3.9. The last cash flow (CFn) occurs at the end of the last �me period and therefore earns no interest.

Figure 3.9: Determining the future value of cash flow streams

As the �meline shows, CF1 will earn interest for N – 1 period, but CFn earns no interest and is simply added to the other sums to find the total future value. By conven�on,

we assume that the cash flows from investments do not start immediately but are deferred un�l the end of the first period. This is not always the case, however.
Prac��oners must carefully analyze any problem to be certain exactly when cash flows will occur. A �meline is a useful aid in modeling when the cash flows from a project
will occur.

PerpetuitiesProcessing math: 0%

What is the present value of a perpetuity that pays $50 a year forever, with a 10% return on investment?

What is the present value of an annuity that pays $50 semiannually for two years at 10% annual interest?

Some special pa�erns of cash flows are frequently encountered in finance. The nature of these pa�erns allows the general formulas to be simplified to a more concise form.
The first special case is that of perpetui�es. These are cash flow streams where equal cash flow amounts are uniformly spaced in �me (every year, every month, etc.).
Perpetuity means that these payments con�nue forever. To illustrate, suppose an investment is expected to pay $50 every year forever. Investors require a return of 10% on
this investment. What should be its current price? Recognizing that today’s price should equal the present value of the investment’s future cash flows, the problem is
illustrated using a �meline in Figure 3.10.

Figure 3.10: Determining the present value of a perpetuity

The arrow indicates that these cash flows con�nue into the future indefinitely. This poses a problem: If there are an infinite number of cash flows, how can we find all of
their present values? Let’s consider the algebraic expression of this problem.

Summing this geometric series and using some algebra yields the following formula for the present value of a perpetuity:

(3.12)

Note that there is no subscript a�ached to CF because all the cash flows are the same.

Therefore, there is no need to dis�nguish CF1 from CF2, and so on. Let’s apply the formula to the example. CF = $50, r = 0.10, and

Annuities

Of all the special pa�erns of cash flow streams, annui�es are the most common. As we shall see, millions of fixed-rate home mortgages are annui�es. Re�rement payments,
bond interest payments, automobile loan payments, and lo�ery jackpot payoffs all o�en fit the annuity pa�ern.

An annuity is a stream of equally sized cash flows, equally spaced in �me, which end a�er a fixed number of payments. Thus, annui�es are like perpetui�es, except they do
not go on forever. The present value of an annuity can be found by summing the present values of all the individual cash flows.

(3.10)

Here N is the number of cash flows being paid and CF is the uniform amount of each cash flow. Solving for CF0 using Equa�on (3.10) would be a �me-consuming problem if

n were large. However, because the right-hand side of the equa�on is yet another geometric series, it can be simplified to yield the formula for finding the present value of
an annuity.

(3.13)

To convince you that Equa�ons (3.10) and (3.13) are equivalent, let’s work an example using both approaches. Suppose you wished to know the present value of a stream of
$50 payments made semiannually over the next two years. The first payment is scheduled to begin six months from today, and the annual rate of interest is 10%. The
problem is illustrated with a �meline in Figure 3.11.

Figure 3.11: Determining the present value of an annuity

Using Equa�on (3.10), and recognizing that r = 5% = 0.05 semiannually, this problem may be solved as follows:

Alterna�vely, Equa�on (3.13) could be used to solve the same problem.

It may appear that using Equa�on (3.10) is just as �me-consuming as using Equa�on (3.13), but consider the work involved had there been 300 payments rather than 4.Processing math: 0%

Payments occur one period sooner in an annuity due, as opposed to an ordinary annuity.

What is the future value of an ordinary annuity that pays $100 a month for two years at 12% annual interest?

What is the future value of an annuity due that pays $100 a month for two years at 12% annual interest?

The problem just solved is an example of an ordinary annuity because cash flows commence at the end of the first period. Most loans require interest payments at the end
of each period. Rent, on the other hand, is usually payable in advance. Annui�es in which cash flows are made at the beginning of each period are called annui�es due.
Leases are usually structured as an annuity due; you make a payment before you get use of the asset. But loans are structured as regular annui�es because some interest
has to build up before a payment is made. Let’s change the example we just worked slightly to require that the cash flows be made at the beginning of each period.

The �meline in Figure 3.12 shows that in a four-payment annuity due, each payment occurs one period sooner than in an otherwise similar ordinary annuity. Because of this
characteris�c, each cash flow is discounted for one less period when finding the PV of an annuity due.

Figure 3.12: An ordinary annuity versus an annuity due

The formula for finding the present value of an annuity due is

(3.14)

This is simply the formula for an ordinary annuity �mes 1 + r, which adjusts for one less discoun�ng period. Thus, it is usually easier to find the PV of an ordinary annuity
and mul�ply �mes 1 + r when solving for the PV of an annuity due.

(3.15)

Now suppose you save $100 each month for two years in an account paying 12% interest annually, compounded monthly. What will be the balance in the account at the end
of two years if you make your first deposit at the end of this month? Figure 3.13 illustrates this problem with a �meline.

Figure 3.13: Determining the future value of an ordinary annuity

In this case we are trying to solve for the future value of an ordinary annuity.

FV24 = $100(1.01)
23 + 100(1.01)22 + . . . + $100

Solving our problem in this manner would take considerable �me. Fortunately, the future value of an annuity is also a geometric series, which can be simplified.

The formula for the future value of an ordinary annuity is

(3.16)

Subs�tu�ng the values for our example into Equa�on (3.16) yields the solu�on

If the first deposit were made immediately, our problem would be one of finding the future value of an annuity due. Figure 3.14 illustrates this problem using a �meline.

Figure 3.14: Determining the future value of an annuity due

Each cash flow in an annuity due earns one addi�onal period’s interest compared to the future value of an ordinary annuity. Thus, the future value of an annuity due is equal
to the future value of an ordinary annuity �mes 1 + r.

(3.17) Processing math: 0%

In an amor�zed loan—such as a typical home loan—the interest por�on of the loan payment gets smaller
over �me, so more principal is repaid with each payment, and the amount paid each month remains the
same: $804.62. A�er 30 years, the $100,000 loan will be paid off, plus the 9% annual interest.

Home mortgages are an important example of an amor�zed loan.
Can you think of any other examples of this type of loan?

Ge�y Images News/Ge�y Images

The future value of the deposits would therefore increase to $2,724.32 if they were made at the beginning of each period. No�ce the adjustment from an ordinary annuity
to an annuity due is the same whether you are solving for PV or FV [compare Equa�ons (3.15) and (3.17)]. Note that both the present value and the future value of an
annuity due are always larger than an otherwise similar ordinary annuity.

Application: Loan Amortization

Many loans, such as home mortgages, require a series of equal payments made to the lender. Each payment is
for an amount large enough to cover both the interest owed for the period as well as some principal. In the
early stages of the loan, most of each payment covers interest owed by the borrower and very li�le is used to
reduce the loan balance. Later in the loan’s life, the small principal reduc�ons have added up to a sum that has
significantly reduced the amount owed. Thus, as �me passes, less of each payment is applied toward interest
and increasing amounts are paid on the principal. This type of loan is called an amor�zed loan. The final
payment just covers both the remaining principal balance and the interest owed on that principal. An
amor�zed loan is a direct applica�on of the present value of an annuity. The original amount borrowed is the
present value of the annuity (PV0), while loan payments are the annuity’s cash flows (CFs).

If you borrow $100,000 to buy a house, what will your monthly payments be on a 30-year mortgage if the
interest rate is 9% per year? For this problem, the formula for finding the present value of an annuity is used
[Equa�on (3.13)]. The present value is the loan amount (PV0 = $100,000), there are 360 payments (N = 360),

and the monthly interest rate is 0.75% (9%/12 months). The payment amount (CF) is determined as follows:

(3.13)

A stream of 360 monthly payments of $804.62 will cover the interest owed each month and will pay off the en�re $100,000 loan as well. Figure 3.15 illustrates how the
amount of each payment applied toward principal increases over �me, with a corresponding decrease in interest expense. As shown in Figure 3.15, $750.00 of the first
payment is used to pay the interest owed the lender for the use of $100,000 during the first month at the 0.75% monthly rate. $54.62 of the first payment will be applied
toward the principal. Thus, for the second month of the loan only $99,945.38 is owed. This reduces the amount of interest owed during the second month and increases the
second month’s principal reduc�on. This pa�ern con�nues un�l the last payment when, as seen in Figure 3.15, only a $798.63 principal balance is remaining. The last
month’s interest on this balance is $5.99. Therefore, the last $804.62 payment will just pay off the loan and pay the last month’s interest, too. Note that the ending balance
(or the FV) of the loan equals zero a�er the last payment is made, so the loan is completely paid off with the last payment.

Figure 3.15: Components of an amor�zed loan over �me

Table 3.3 is an amor�za�on table showing principal and interest payments on a five-year, $10,000 loan, amor�zed using a 10% rate compounded annually. An amor�za�on
table is useful because it can be used to find the unpaid balance owed on a loan a�er some payments have been made. Using Table 3.3, a borrower would know, for
example, that $4,578.32 would be necessary to pay off the loan a�er the third annual payment is made.

Table 3.3: Loan amor�za�on table for $10,000 borrowed at 10% interest annually compounded for five years

Year Beginning principal balance Total payment Interest Principal reduc�on Ending principal balance
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Crea�ng an amor�za�on table helps you keep track of monthly balance,
principal payment, and interest payments. The video uses a TI Inspire to
create the table. What are some other tools you could use to create an
amor�za�on table?

1 $10,000 $2,637.97 $1,000 $1,637.97 $8,362.03

2 $8,362.03 $2,637.97 $836.20 $1,803.77 $6,560.26

3 $6,560.26 $2,637.97 $656.03 $1,981.94 $4,578.32

4 $4,578.32 $2,637.97 $457.83 $2,180.14 $2,398.18

5 $2,398.18 $2,637.97 $239.79 $2,398.18 $0

Note: Each year’s beginning balance equals the previous year’s ending balance. Each year’s interest equals the rate mul�plied by the total loan amount. Each year’s principal reduc�on equals the
total payment minus the amount applied toward the interest. Each year’s ending principal balance equals the beginning balance minus the principal reduc�on.

Amor�za�on Table for Month to Month Payments

Field Trip: Loan Amor�za�on

Bankrate.com provides a mortgage amor�za�on schedule calculator that allows users to build an amor�za�on table using their own data.

Visit: h�p://www.bankrate.com/calculators/mortgages/amor�za�on-calculator.aspx (h�p://www.bankrate.com/calculators/mortgages/amor�za�on-calculator.aspx)

Experiment with different mortgage amounts, terms, and interest rates to see how they affect your monthly payments. You can even experiment with adding addi�onal
payments to change the pay-off date of the loan.

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http://www.bankrate.com/calculators/mortgages/amortization-calculator.aspx

Ch. 3 Conclusion

Chapter 3 has covered much of the topic of the �me value of money. Next, the concepts and techniques introduced here will be applied to finding the value of stocks,
bonds, and other securi�es. Before that, however, it is best to prac�ce the newly acquired skills. The authors cannot overemphasize the importance of mastering �me value
mathema�cs. Therefore, as you do your homework, make sure you feel confident in your ability. If you are not, now is a good �me to ask your instructor for assistance.

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Ch. 3 Learning Resources

Key Ideas

The foregone interest, which could be earned if cash were received immediately, is called the opportunity cost of delaying its receipt.
The �me value of money and the mathema�cs associated with it provide important tools for comparing the rela�ve values of cash flows received at different �mes.
You can think of the present value as the amount that you have to put in the bank today to have some specific amount in the future.
How frequently a loan’s interest is compounded changes how we determine the �me value of money.
An amor�zed loan is a direct applica�on of the present value of an annuity.

Key Equa�ons

Cri�cal Thinking Ques�ons

1. Suppose you own some land, purchased by your father 20 years ago for $5,000. You are able to trade this land for a brand new Corve�e sports car. What economic
opportunity might you forego if you proceed with the trade? How would you es�mate the opportunity cost of proceeding with the trade?

2. The Corve�e dealership from Ques�on 1 is also willing to trade the car for an IOU you own that promises to pay you $2,000 at the end of each year for the next 10 years and
$20,000 when it matures at the end of the 10-year period. Investors are currently valuing such IOUs using a 6% discount rate. What economic opportunity might you lose if
you make the trade? How would you calculate the opportunity cost of the trade?

3. If the market for new automobiles and the real estate and bond markets are all efficient, what do you think you would discover about the opportunity costs of the trade in
Ques�ons 1 and 2?

4. Usually we compute present values using a constant interest rate. But we know that interest rates vary over �me, and it is impossible to know what the interest rate will be in
10 or 20 years. Why is using the current interest rate a good approach? Or would we be be�er off to simply ignore cash flows arriving beyond the period for which we have
reasonable interest rate es�mates? Explain your answer.

5. We discussed the EAR (effec�ve annual percentage rate). Private student loans o�en are structured so no payments are necessary while a student is in school (and for 6
months a�er). However, interest does accrue during this period. This interest is then added to the principal amount of the loan once the grace period ends. For example, you
borrow $10,000 at 6% when you start a two-year graduate program. The interest is $50 per month. You complete your degree and take advantage of some of the
postgraduate grace period and then begin making payments 25 months a�er the loan began. The new principal is now $10,000 plus the capitalized interest of $1,250 or
$11,250. Lenders don’t state an effec�ve annual rate because of the uncertainty associated with the amount of capitalized interest. Does this seem fair? Can you think of a
way that this could be expressed so student borrowers understand what they are commi�ng to when they get a private student loan with a capitalized interest feature?

Key Terms

Click on each key term to see the defini�on.

amor�zed loan
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A loan that is paid off in equal periodic payments. Automobile loans and home mortgages are o�en amor�zed loans.

annui�es due
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A finite stream of cash flows of a fixed amount, equally spaced in �me where payments are made at the beginning of each period.

annuity
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A finite stream of cash flows of a fixed amount, equally spaced in �me.

compounding
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Earning interest on previously earned interest.Processing math: 0%

https://content.ashford.edu/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter/books/AUBUS650.13.1/sections/front_matter#

discoun�ng
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Solving for the present value of a future cash flow.

discount rate
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The interest rate used to find the present value of a future payment or series of payments. For many investments, investors’ required return is the discount rate used to find
the present value.

effec�ve annual percentage rate (EAR)
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The annualized compound rate of interest.

future value
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A cash flow, or stream of cash flows, re-expressed as an equivalent amount at some future date.

interest
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The amount of money paid by a borrower to a tender for the use of the borrowed principal. The rate is expressed as a percentage of the principal owed.

nominal annual percentage rate (APR)
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The stated interest rate per year without considering the effect of compounding.

nominal rates
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The stated rate or yield that reflects expecta�ons about infla�on.

opportunity cost
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The amount of the highest valued forgone alterna�ve.

ordinary annuity
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A finite stream of cash flows of a fixed amount, equally spaced in �me, where payment are made at the end of each period.

periodic interest rate
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The rate of interest expressed per period, e.g., per month (12 periods per year); quarterly (4 periods per year); semi-annually (twice per year); weekly (52 periods per year);
bi-annually (once every two years), etc.

perpetui�es
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An infinite stream of equal cash flows, each equally spaced in �me.

present value
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A future cash flow, or stream of cash flows, re-expressed as an equivalent current amount of money.

principal
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The amount of money borrowed.

simple interest
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The number of years mul�plied by the interest rate mul�plied by the amount originally invested.

�me value of money
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The idea that, holding all else constant, people prefer to receive a given amount of money today rather than in the future.

Web ResourcesProcessing math: 0%

The importance of the �me value of money concept is discussed here: h�p://www.qfinance.com/cash-flow-management-calcula�ons/�me-value-of-money
(h�p://www.qfinance.com/cash-flow-management-calcula�ons/�me-value-of-money)

The effec�ve interest rate concept can be applied to mortgage interest rates, too. The Motley Fool website shows how to compute an effec�ve a�er-tax interest rate here:
h�p://wiki.fool.com/How_to_Calculate_an_Effec�ve_Mortgage_Rate_With_a_Tax_ Writeoff (h�p://wiki.fool.com/How_to_Calculate_an_Effec�ve_Mortgage_Rate_With_a_Tax_ Writeoff)

Mortgage agreements have a stated note rate, which determines the interest component of each payment, and an annual percentage rate (APR). The APR is almost always
higher than the note rate because it includes other costs associated with acquiring a mortgage for a home: origina�on fee, points, prepaid interest, and insurance. Here is a
descrip�on of the APR: h�p://www.americanloansearch.com/info-apr.htm (h�p://www.americanloansearch.com/info-apr.htm)

For informa�on on mortgage rates, mortgage calculators, and historic rate informa�on, visit: h�p://www.mortgagenewsdaily.com/mortgage_rates/
(h�p://www.mortgagenewsdaily.com/mortgage_rates/)

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http://www.qfinance.com/cash-flow-management-calculations/time-value-of-money

http://wiki.fool.com/How_to_Calculate_an_Effective_Mortgage_Rate_With_a_Tax_%20Writeoff

http://www.americanloansearch.com/info-apr.htm

http://www.mortgagenewsdaily.com/mortgage_rates/

Chapter 6

Capital Budgeting: Investing to Create Value

Imaginechina/

Associated Press

Learning Objectives

A�er studying this chapter, you should be able to:

Describe the significance of corporate investments in crea�ng value.
Explain how iden�fying and classifying poten�al projects plays into project selec�on.
Es�mate project cash flows.
Show how to select independent projects using NPV and IRR.
Describe how to select mutually exclusive projects that maximize value.
Iden�fy the significance and different types of op�ons and how to adjust for the op�on effect.

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Ch. 6 Introduction

Throughout this text we have stressed the importance of crea�ng value for corporate shareholders. We also indicated that the greatest opportunity for crea�ng value lay in
the inves�ng ac�vi�es of companies. In the context of the financial balance sheet, these are le�-hand side ac�vi�es. The poten�al payoffs on successful investments prompt
ingenious efforts to develop new products, build exis�ng products at lower cost, improve product quality, and devise new marke�ng strategies. For example, the advent of e-
commerce allows small companies to sell products worldwide and large companies to supplement or possibly supplant tradi�onal distribu�on channels. E-commerce has, in
turn, spawned companies that design and manage websites, provide Internet services and make encryp�on so�ware.

Some product developments create virtually new industries. Consider the spectacular growth in wireless communica�ons that was made possible by the blending of satellite
and digital technologies. In an effort to gain a compe��ve advantage, network providers have expanded their coverage areas, improved transmission quality, added services,
and cut prices. Similarly, cell phone manufacturers embrace the latest in digital technology as they vie for market share.

In this chapter, we address the fundamentals of corporate inves�ng. First, we discuss product market opportuni�es created by imperfect compe��on. Next, we develop some
guidelines for iden�fying and selec�ng investment opportuni�es. We then examine the investment decision itself, paying special a�en�on to decision criteria and discounted
cash flows. Finally, we discuss op�ons that are intrinsic to many corporate investments.

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Shareholders must pay a�en�on to the product market when
deciding on investment opportuni�es because compe�tors can

6.1 Corporate Investments and Value Creation

We will draw upon several important ideas covered thus far in the text in our discussion of corporate inves�ng:

It is cash flows, not income or earnings, that measure the success of a business or investment.
The value of cash flows depends on when they are paid or received.
The effect of �ming on the value of future cash flow is incorporated into the discount rate.
The appropriate discount rate is the investors’ required rate of return.
This required rate of return is a func�on of risk.

Investors buy bonds and stocks that represent claims against future corporate cash flows. Corporate investments must, therefore, generate at least enough cash flow to
provide all investors with their required returns. If investments generate less than the required return, the value of the company’s securi�es—and, therefore, the value of the
company—will decline. Of course, investments are made in the hope that they will produce enough cash to pay off creditors with enough le� over to increase returns to
shareholders.

Investing in Fixed Assets

Depending on the industry, much corporate investment is in long-term, or fixed, assets. These assets can be classified as tangible (machinery, real estate) or intangible
(copyrights, patents, contracts). Tradi�onal capital-intensive industries invest in factories that manufacture durable goods, such as metals, chemicals, transporta�on
equipment, and machinery. However, virtually all companies, not just manufacturers, have fixed assets. Retailers either own or lease stores. R&D firms have laboratories and
patents. Book and music publishers have copyrights and, perhaps, long-term contracts with writers and musicians. One of the best-known assets is the secret formula for
Coca-Cola. Whether tangible or intangible, fixed assets are essen�al to the long-run viability of a firm.

Identifying Asset Value: NPV and IRR

The ability to iden�fy which assets are expected to add value to the firm is central to the financial management role. In this chapter, we explore this selec�on process (called
capital budge�ng) in some detail. Essen�ally, to iden�fy value-crea�ng projects, businesses use either the net present value (NPV) or internal rate of return (IRR) criteria.

Net present value measures the dollar value added to the firm by the investment. The NPV of an investment is the present value of the future cash flows minus the ini�al
investment.

Net present value = Present value of future cash flows –

Ini�al investment

NPV directly measures the present value of the cash flows a project is expected to generate. It then compares this value to the project’s cost. If the project value is expected
to exceed the cost, the project should be pursued.

The IRR criteria compares the IRR (expected return) for a project to the required return for investors, given the project’s risk. If the expected return exceeds that requirement,
then the project should be pursued.

We will look at the equa�ons for finding NPV and IRR later in this chapter. For now, it is important to know that companies can add value to the business and increase
owners’ wealth by pursuing posi�ve NPV projects, or project’s whose IRR exceeds its required return. With this objec�ve in mind, we will begin our discussion of corporate
investments.

Product Market Opportunities

The financial model of the corpora�on is based on the premise that product markets provide valuable investment opportuni�es for firms. Firms that iden�fy and exploit
these opportuni�es create value because they do what their shareholders individually cannot do. The search for investment opportuni�es occurs within the overall mission
and strategic plan of the corpora�on.

Investment opportuni�es are o�en short-lived because successful products a�ract compe�tors. For example,
the success of Starbuck’s coffee spawned many purveyors of specialty coffees and espresso, and BlackBerry was
supplanted by the iPhone a�er a few years of market dominance.

Compe��on, or the threat of compe��on, means that firms must not only remain alert for new opportuni�es
but also try to protect their exis�ng markets. For example, major airlines on occasion have used some�mes
illegal predatory pricing to discourage low-cost “no-frills” airlines from serving their hub ci�es. Even seemingly
entrenched firms may be vulnerable to compe��on. Before Japanese autos entered the market, the United
States was the nearly exclusive turf of the big three American automakers.

Firms have a number of weapons with which to fend off compe�tors. Patents and copyrights protect, for a
�me, valuable intellectual property, such as inven�ons, publica�ons, and computer so�ware. Some�mes
protec�ng a compe��ve posi�on requires investment. For example, McDonald’s a�empted to forestall
compe��on by being the first to buy choice restaurant loca�ons. Inves�ng in a modern plant may lower
produc�on costs or increase product quality. Some industries invest heavily in promo�on. Athle�c apparel
manufacturers engage in a frenzied compe��on to sign hot sports stars to expensive long-term contracts.

Seeking out and successfully pursuing valuable investments places great demands on management. There are
many poten�al hazards. Managers may fail to recognize opportuni�es, or they may chase opportuni�es that do

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quickly overshadow a leading product as demonstrated by iPhone’s
dominance over BlackBerry.

Associated Press

not exist. For example, a market may appear to be a�rac�ve because it produces extraordinarily high profits.
Yet a closer examina�on reveals that exis�ng producers hold patents on key technologies or may control supply
sources or distribu�on channels.

When an apparent investment opportunity reveals itself, managers should ask the following ques�ons:

If this is a genuine opportunity, why is there not greater compe��on in this market?
Are compe�ng products on the horizon that may reduce market demand?
Are there costly barriers to entry?
Is the current compe��ve posture likely to remain over the long haul?
Are market forces already at work to increase compe��on?
Will the corpora�on be able to protect its investment by keeping compe�tors at bay?

Taking reasonable precau�ons should actually encourage inves�ng by making poor investments less likely. Companies that have a record of successful inves�ng may be more
aggressive in searching for new opportuni�es than those that have experienced recent or costly failures.

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The compe��ve advantage with a commodity is generally the price.
Differen�a�on strategies, such as performance, customer needs, tailoring
products, flexibility, and trust also play major roles. What would you say is
the differen�a�on strategy for a company like Amazon or Target?

High-end retailers like Neiman Marcus focus on a differen�a�on
strategy that emphasizes style, quality, and service.

Associated Press

6.2 Project Selection

Each firm must develop a compe��ve strategy for exploi�ng market opportuni�es. The most common of these
strategies are cost leadership and differen�a�on. Low-cost producers can undercut their compe�tor’s prices;
Wal-Mart, for example, uses this strategy. On the other hand, a differen�a�on strategy may take many forms. A
company may offer higher quality, a func�onally dis�nct product, or be�er service. Among clothing retailers,
there are Nordstrom and Neiman Marcus (quality and service) and L. L. Bean (func�onally dis�nct).
Differen�a�on frequently prompts large investments in adver�sing. In 1999 CNET, Inc. launched a $100 million
ad campaign to promote their technology website, even though the investment would wipe out their posi�ve
cash flows. A firm’s compe��ve strategy determines where it looks for market opportuni�es. For example, Wal-
Mart would not be likely to focus on product quality and service if that jeopardized its posi�on as a cost
leader.

A firm’s compe��ve strategy guides its strategic planning. These strategic plans are then translated into
investments. The process of transla�ng plans into investments begins with iden�fying a set of poten�al
projects. This requires the following steps:

Step 1: Iden�fy possible projects that fit into the corporate strategic plan or mission.

Step 2: Classify projects by size and purpose so that management a�en�on can be directed to those that are most important.

Step 3: Eliminate or integrate projects that are in some way dependent on other projects.

Let’s look at these more closely.

Compe��ve Strategies

Identifying Potential Projects

Ideally, a company’s search for investment opportuni�es would transcend its tradi�onal products and markets. For example, a company doing business in the United States
may consider overseas markets. A bank might consider providing computer services. A manufacturer of industrial equipment might also consider making consumer products.

Classifying Projects

Companies o�en find it useful to categorize poten�al projects by their size and the company’s experience with such projects. Large projects, with which the company has
li�le experience, require careful scru�ny. An example of such a project would be an American company inves�ng for the first �me in a less-developed country. At the other
extreme are rou�ne investments such as replacing a worn-out machine. Management resources are finite. By confining the search to projects that fit the company’s mission
and then classifying them, management can direct its a�en�on to a rela�vely few, crucial projects.

Projects that may seem risky and deserving of great management scru�ny must be judged in the context of exis�ng company opera�ons. Table 6.1 provides a representa�ve
scheme for classifying projects according to the amount of management a�en�on required.

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Foreign investment is one example of a diversifica�on project. Today, it is
not only countries like the U.S. and Great Britain that are expanding
overseas; industrial pioneer, China, is inves�ng money into the crumbling
economy of the Congo. How does China’s involvement in foreign investment
impact compe��ve markets?

Table 6.1: Project types and management oversight

Project type Descrip�on Management a�en�on required Example

Replacement
projects

Update or upgrade exis�ng capacity. Senior management typically does not make
decisions on these rela�vely rou�ne investments.

Replacing worn-out or obsolete
machinery and equipment.

Expansion
projects

Used to expand exis�ng capacity, such
as adding new machinery or
equipment to increase output.

Require only moderate management scru�ny
because capacity expansion is a response to
increased or an�cipated demand.

Retailers lease larger facili�es or
open addi�onal stores.

Diversifica�on,
or dispersion,
projects

Add new products or new regions to a
company’s opera�ons.

Demands on management may vary, depending
on how related the new products or regions are
to exis�ng ones.

Ini�al overseas expansion of a
domes�c corpora�on (high level of
management a�en�on).
Inves�ng in freight cars to lease to
private carriers (lower level of
management a�en�on).

Chinese Investment in the Congo

There are two other investment categories that don’t fit neatly into a risk classifica�on. One is investment mandated by law, such as pollu�on control equipment to comply
with environmental regula�ons and plant improvements to conform to occupa�onal safety and health codes. The other is investment in other companies. Mergers and
acquisi�ons are risky in part because they combine corporate cultures. Most mergers expand exis�ng capacity, diversify product lines, or extend opera�ons to new regions.

Eliminating Project Dependencies

When we first iden�fy poten�al investments, we may include projects that are either complementary or mutually exclusive. Pipelines to bring crude oil to the refinery and
transport refined petroleum to ports or markets must accompany a new oil refinery. It makes li�le sense to evaluate the refinery separately from the pipelines. These are
complementary projects and should be considered a single investment.

Mutually exclusive projects are subs�tutes for each other, requiring either/or decisions. There may be alterna�ve types of pipelines that can be built to serve the oil refinery,
or rail cars or barges may be used in place of pipelines. The company must select the best op�on for each task and discard the others.

Once the company’s financial analysts have combined complementary projects and chosen among mutually exclusive projects, those that remain are independent projects.
Independent projects all have equal status, meaning that the company may invest in all, none, or any combina�on of projects, knowing that each investment decision does
not affect the others. This greatly simplifies the analysis and allows management to focus on the process of crea�ng wealth. As is usual in business, even though simplifying
assump�ons aids our analysis, we must deal at some point with less simple reali�es. In truth, individual projects must be viewed in the context of the por�olio of
investments.

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Purchasing new pipelines for a new oil refinery is an example of a
complementary project because they are interconnected. Can you
think of any other examples of complementary projects?

Hans-Peter Merten/Digital Vision/ Ge�y Images

6.3 Estimating Project Cash Flows

Once a company’s financial analysts have iden�fied an array of independent projects, they must evaluate each as a poten�al investment. First, they must es�mate cash flows
that are associated with the project. These include the ini�al investment, opera�ng income and expenses spread over the life of the project, and project termina�on.
Opera�ng and termina�on cash flows are discounted, and their present value is then compared to the ini�al investment. If the present value of the future cash flows is
greater than the ini�al investment, the investment has a posi�ve net present value. A posi�ve net present value indicates that the investment will add value to the company.

Recall from Sec�on 6.1 that

Net present value = Present value of future cash flows – Ini�al investment

In order to calculate net present value, we must es�mate the amount and �ming of the investment’s cash flows. In this sec�on, we provide some ground rules for es�ma�ng
cash flows and then show how they are used in a discounted cash flow model.

Consider Only Incremental Cash Flows

The most difficult part of project analysis is iden�fying and quan�fying cash flows related to the project. Here, the guiding principle is to include only incremental cash flows,
defined as the change in corporate cash flows a�ributable to the project. This seems simple enough, but these cash flows can be elusive. Even the cost of some projects may
be impossible to pin down. Consider how difficult it is to es�mate the completed cost of an office building or plant that may take years to complete. Some cash flows may
escape a�en�on altogether, such as the effect of one project on another project’s cash flows. Here are a few guidelines for iden�fying incremental cash flows:

Beware of allocated costs, such as corporate overhead. Usually allocated costs do not change as a result of taking on projects. For example, a new project may use exis�ng
idle capacity on the company’s computer network. Assuming that there is no alterna�ve use for the network capacity and support staff, there is no incremental cost. So
nothing should be allocated to the new project. On the other hand, the project’s demand for network services may compel the company to add capacity. In this case, the
cost incurred is incremental and should be included in the project.
Consider the opportunity costs of currently owned resources. Take for example, a plant built on land owned by the company. The land entails no out-of-pocket costs;
however, it is not a free resource because it has alterna�ve uses. The analyst must consider, as the cost of the land, the income that could be produced from its next-best
use. Perhaps it could become a parking lot, be sold, be leased, or be used for growing tomatoes.
Ignore sunk costs. A sunk cost is money that has already been spent and cannot be recovered. However, it can be difficult (on many levels) to abandon projects on which
a great deal of money has been spent. Abandonment has its own costs, and some�mes finishing a project that may have been unwise to begin with is the only way to
recover at least some of its costs. The analyst must consider the incremental costs and revenues of comple�ng a project.
Consider incidental effects of the project. A new product may reduce sales of other company products. For example, a retailer that opens a second loca�on in the same
town will lose some customers to the new store. There may be posi�ve incidental effects as well. For instance, large airlines subsidize small feeder airlines to deliver
passengers to and from their hub airports. For the feeder airlines, these incidental effects make them viable. Iden�fying and cos�ng all incidental effects of a project is
easier said than done. Projects that depend on incidental effects may be very risky and should be taken on with cau�on.

Cash Flow Categories

It is convenient to categorize project cash flows by their �ming, that is, when they occur. The ini�al investment occurs at the beginning of the project’s life, opera�ng income
and expense are annual cash flows occurring during the project’s life, and termina�on cash flows occur when the project ends. Each category includes cash flows from
different sources. Figure 6.1 outlines these categories, and they are discussed in further detail below.

Figure 6.1: Cash flow categories over a project’s life�me

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Project cash flows can be categorized across the life of the project. Ini�al investment cash flows occur at the
beginning of the project’s life, opera�ng income and expense flows occur during the project, and termina�on
cash flows occur at the end of the project.

Wages for employees are a cash ou�low
opera�ng expense. What addi�onal
examples of opera�ng expenses can you
think of?

Jus�n Guariglia/Na�onal Geographic Stock

Ini�al Investment

Project cost (cash ou�low) may include transporta�on, insurance, setup, employee training, and prepaid maintenance. For some projects, it may also include infrastructure
costs such as roads and u�li�es. Also included may be planning and design costs such as architectural fees. Be careful to not include sunk infrastructure, planning, and design
costs. For simple projects, the ou�low occurs at the present �me (t = 0). However, large projects, such as plant construc�on, may take several years to complete.

Investment tax credits (cash inflow) reduce taxes paid in some propor�on to the project cost. From �me to �me, governments provide tax credits for certain kinds of
investments. Currently, there is no general investment tax credit ( ITC) in the United States, but there are ITCs in other countries.

Change in net working capital (cash ou�low or inflow) may be required by expansion, diversifica�on, and dispersion projects. Increased inventories and receivables may be
needed to support increased produc�on. These current assets are �ed to the investment and are therefore incremental costs. Generally, we assume that this increased
working capital is reduced to its prior level on termina�on of the project, resul�ng in a decrease, or recovery of net working capital. Some investments may actually reduce
the need for net working capital. For example, a new produc�on facility may employ just-in-�me inventory control, reducing the need for inventory stocks.

Sale of exis�ng asset (cash inflow) generally occurs only for replacement projects.

Tax effect of asset sale (cash inflow or ou�low) must be considered when the sale price of the asset is greater than its depreciated book value and the company owes tax on
the difference. If the sale price is less than the book value, the loss reduces the company’s taxable income.

Opera�ng Income and Expense

Opera�ng income and expense are annual revenues and expenses occurring during the opera�ng life of the project. Of the three
categories, the incremental cash flows associated with opera�ons are the most difficult to iden�fy.

Cash revenues (cash inflow) include sales and other incidental income. These cash flows are usually not an annuity because unit sales
and prices will not be constant from year to year.

Cash expenses (cash ou�low) include materials, labor, fuel or power, maintenance, rents, contract services, and any number of other
incremental costs. As with sales, they normally vary from period to period. Replacement projects may reduce expenses, producing
cash savings. These are all opera�ng expenses and do not include interest or other capital costs. Costs of capital are included in the
discount rate.

Deprecia�on (cash inflow) of fixed assets is noncash, tax deduc�ble expense. The tax saving is the only cash flow resul�ng from
deprecia�on. For a replacement project, only the change in deprecia�on between the new and old projects is relevant.

Project Termina�on

Income from project sale (cash inflow) results from assets that have economic value beyond the life of the project. They may be sold
intact, in parts, or as scrap. Companies o�en plan to resell assets a�er a specified period. Their resale or terminal value may add
significantly to a project’s value.

Tax effect of project sale (cash inflow or ou�low) is treated in the same way as that on the sale of an exis�ng asset.

Recovery of net working capital (cash inflow or ou�low) occurs at the termina�on of a project, when the ini�al change in net working
capital is reversed in order to return to the original net working capital posi�on.

Cash Flow Calculations

Now, we look at specific calcula�ons for two types of project cash flows.
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The Tax Effect of Asset Sales

If an asset is sold for more than its depreciated book value, the difference between the sale price and book value is a taxable capital gain. The tax that must be paid equals
the gain �mes the marginal corporate tax rate. A capital loss resul�ng from a sale price less than book value reduces the company’s tax, assuming that it has other taxable
income. The tax treatment of asset sales applies to both ini�al investment and termina�on cash flows. To illustrate, consider a project in which exis�ng equipment is to be
replaced by new equipment cos�ng $10,000 (shown in Table 6.2). The exis�ng equipment may be sold for $3,000, but it has a depreciated book value of $2,500, crea�ng a
$500 taxable gain on the sale. The tax rate is 34%. Cash flows are starred (*).

Table 6.2: Tax effect of asset sale

Data Category Value

*Project cost ($10,000)

*Sales price of exis�ng asset $3,000

Book value of exis�ng asset $2,500

Gain (Loss) $500

Tax effect of sale (gain x tax rate) ($170)

To determine the ini�al cash flow, we add the tax effect to the project cost and sale price:

Ini�al cash flow = Project cost + Sale price + Tax effect

Ini�al cash flow = (10,000) + 3,000 + (170) = ($7,170)

Some or all of the gain on an asset sale actually represents the recapture of deprecia�on. If the tax rate on capital gains and losses is the same as the tax rate on income,
the source of the gain is immaterial. However, if the capital gain tax rate is less than that on ordinary income, then deprecia�on recapture must be calculated and the
appropriate tax rate applied.

Opera�ng Cash Flows

The es�mates of annual opera�ng income and expenses must be converted to opera�ng cash flows. This is done using the net income approach to calcula�ng cash flows.

Step 1: Calculate taxable income. Earnings before tax (EBT) equal cash revenues minus opera�ng expenses and deprecia�on.

EBT = S – E – dep

Step 2: Calculate corporate income tax. Corporate tax is the product of EBT and the marginal corporate tax rate (tx).

Step 3: Calculate net income or earnings a�er tax (EAT) by subtrac�ng the tax from the EBT. The final step is to calculate opera�ng cash flow (OCF) by adding deprecia�on to
net income.

OCF = EAT + dep

Project cash flows and the calcula�on of opera�ng cash flow are summarized in Table 6.3.

Table 6.3: Classifying project cash flows

Cash flow Classifica�on

Ini�al investment

Project cost Ou�low

Investment tax credit Inflow

Change in net working capital Ou�low/inflow

Sale of asset Inflow

Tax effect of sale Ou�low/inflow

Opera�ng cash flows

Cash revenues (S) Inflow

Cash expenses (E) Ou�low

Deprecia�on (dep) Noncash expense

Tax Ou�low

Calcula�ons for opera�ng cash flows
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Earnings before tax (EBT) S – E – dep

Corporate tax EBT × tx

Earnings a�er tax EAT = EBT − tax

The Challenge of Estimating Project Cash Flows

These guidelines for es�ma�ng project cash flows do not capture the difficulty of actually gathering informa�on and producing es�mates. A single independent project may
include building a plant, installing produc�on equipment, buying trucks, and training workers. Such a project involves gathering and si�ing large quan��es of informa�on.
Incomplete or inaccurate informa�on may lead to an incorrect decision. The large amounts of capital required by many projects make the cost of incorrect decisions that
much greater.

Some cash flows, such as equipment costs and taxes, are rela�vely easy to es�mate because their costs are explicit. Future cash flows that are dependent on the success of
the project require more sophis�cated and �me-consuming es�mates. Table 6.4 divides project cash flows into two categories: those that are fairly easy to es�mate and
those that are more difficult. Keep in mind that these categories are guidelines only, not absolutes.

Table 6.4: Es�ma�ng project cash flows

Step in project life cycle Less difficult to es�mate More difficult to es�mate

Ini�al investment Project cost
Investment tax credit
Sale of exis�ng asset
Tax effect of asset sale

Change in net working capital

Opera�ng cash flows Deprecia�on Cash revenues
Expenses

Project termina�on Income from sale of project
Tax effect of project sale
Recovery of net working capital

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6.4 Selecting Independent Projects

As discussed earlier, independent projects are those that do not affect a company’s other projects. Once an independent project has been iden�fied, we must determine
whether or not it is a worthwhile endeavor. Recall that net present value plays a key role in project selec�on by determining expected cash flows.

An independent project should be taken if its NPV is posi�ve (NPV > 0). NPV is a direct measure of the project’s contribu�on to firm value. Even projects with small NPVs
should be taken, at least in principle. Any posi�ve NPV project is expected to produce a cash flow in excess of that needed to provide investors with their required rates of
return. Cash flow from a project whose NPV = 0 should provide these required returns; however, it would produce no residual cash flow to increase shareholder wealth.
Therefore, shareholders would be indifferent toward the project.

Next we will examine the equa�ons used to calculate net present value and internal rate of return for an independent project.

Calculating NPV and IRR for Independent Projects

Recall that net present value is equal to present value of future cash flows minus the ini�al investment (NPV = PV – II). Present value (PV) is the summa�on of the
discounted opera�ng cash flows (OCF) plus the discounted terminal cash flows (TCF). The formula for calcula�ng NPV is

Table 6.5 breaks down the components of Equa�on (6.1) and Equa�on (6.2).

Table 6.5: Variables in NPV and IRR equa�ons

Variable Value

II Ini�al investment

OCFt Opera�ng cash flows in year t

TCF Terminal cash flows

t Year

N Life span (in years) of the project

R(r) Project required rate of return

Remember that internal rate of return is the expected rate of return on a project. IRR is found by solving for the discount rate that equates the present value of future cash
inflows to the project cost. To calculate IRR, we use trial and error to find the discount rate that sa�sfies the condi�on PV = II, which is equivalent to NPV = 0. Solving for a
project’s IRR is the func�onal equivalent of solving for a bond’s yield to maturity.

For a single future cash flow or a mul�ple period annuity cash flow, IRR can be solved algebraically. However, project cash flows are seldom annui�es, leaving us no choice
but to find IRR through trial and error. This is a tedious process without the assistance of a calculator or computer.

The formula for IRR is

Comparing NPV and IRR Results on Independent Projects

If a project is found to have a posi�ve net present value, it will also have an internal rate of return greater than its required rate. On the other hand, if the project NPV is
found to be nega�ve (less than 0), then the IRR would also be less than the required rate of return. Symbolically, the rela�onship between NPV and IRR may be stated

If NPV > 0, then IRR > R(r).

If NPV = 0, then IRR = R(r).

If NPV < 0, then IRR < R(r).

Although IRR does not directly measure the project’s contribu�on to firm wealth, we see the equivalency of NPV and IRR decision rules (if NPV > 0, then IRR > R(r), etc.). This
equivalency leads to an important point: For independent projects, both NPV and IRR analyses will cause us to accept and reject the same projects. Therefore, it does not
ma�er which method of analysis we choose. Both use discounted cash flows and the required rate of return in the investment decision. Net present value uses the required
rate of return as a discount rate and produces a dollar value for the project; IRR uses the required return as a hurdle rate, or reference point against which to compare the
project’s internal rate of return.

Now, let’s apply what we have learned about determining project value.

Application: The Pogo Harness Project

Nine years ago, engineer Paula Bauer founded Pacific Offshore Ltd. (POL) as a supplier of high-quality hardware and gear for sailboats. Five years ago, a successful ini�al
public offering of stock provided the capital POL needed to meet demand and expand its product line. Paula is most directly involved in product development and is always
looking for ideas that can be turned into new products for the sailor. She o�en sails with her dog Pogo. A�er having to fish Pogo out of the water on several occasions, Paula

Processing math: 0%

While the Pogo harness could be considered a product sailors will
use, it is essen�ally an independent project as POL’s investment in
it does not affect any other company product.

ZUMA Press/Corbis

recognized the need for a harness that would keep Pogo on board yet give the dog some freedom to move
about the boat. Paula and her vice president, Sonny Wheeler, designed and tested a harness that met with
Pogo’s approval.

Paula and Sonny calculate that they can produce the harness and associated hardware with an investment of
$57,000 in tools and equipment. Alterna�vely, they could invest about $225,000 in automated, high-speed
machine tools that would greatly reduce unit produc�on costs at higher produc�on volume. Because the Pogo
harness is an untested product, they opt for the lower investment. It will cost an addi�onal $11,000 to slightly
alter their manufacturing facility to handle this new product. Increased sales from the Pogo harness project
should increase average receivables and inventory from $23,000 to $25,000. Distribu�on will be handled
through normal catalog and chandlery sales, but they also plan to market through pet supply stores, and the
Pogo harness will be featured on POL’s new website. Paula and Sonny agree that they will push ahead with the
website even if they decide to not produce the Pogo harness. The reconfigura�on of the manufacturing facility
will allow POL to sell some older equipment for an es�mated $7,800.

Paula es�mates that the Pogo harness will produce cash flows for five years. A�er five years, if demand
warrants, she will invest in automated equipment to cut produc�on costs. If there is insufficient demand or if
lower-cost compe�tors have flooded the market, she will cease produc�on. Either way, POL will no longer need

the exis�ng tools. Paula’s produc�on manager es�mates that the rather specialized tools will bring no more than $12,000 when they are sold in five years. Tools, equipment,
and reconfigura�on costs are depreciated on a seven-year accelerated cost-recovery schedule. Pacific Offshore’s effec�ve tax rate on income is 34%, and its tax rate on capital
gains is 28%. The projected cash flows for the Pogo harness project are shown in Table 6.6 and Table 6.7.

Table 6.6 presents data on the sale of the exis�ng equipment and the sale in Year 5 of the tools of the Pogo harness project. All project cash flows, including opera�ng cash
flows, are shown in Table 6.7.

Table 6.6: Pogo harness project: Calcula�ng the tax effect of asset sales

Data Category Cash Value

Sale of exis�ng equipment (Year 0)

Book value (book) $0

Sale price (sale)* 7,800

Capital gain or loss (gain) 7,800

Tax effect of sale (gain) × (tax)* 2,184

Sale of Pogo harness project tools (Year 5)

Original purchase price $57,000

Book value (book)** 12,717

Sale price (sale)* 12,000

Capital gain or loss (gain) (717)

Tax effect of sale (gain) × (tax)* (201)

* Designates cash flow
** At the end of Year 5, accumulated deprecia�on totals 77.69%
Book value = $57,000 x (1 – 0.07769) = $12,717
Capital gain tax rate = 28%

Table 6.7: Pogo harness project: Cash flows

Cash Flow Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

Ini�al investment

Tools and equipment ($57,000)

Plant reconfigura�on ($11,000)

Added net working capital ($2,000)

Sale of asset $7,800

Tax effect of sale $2,184

Total ($64,384)

Opera�ng cash flows

Cash revenues (S) $42,500 $49,300 $55,216 $60,185 $65,602Processing math: 0%

Expenses (E) $29,750 $29,580 $33,130 $30,093 $32,801

Deprecia�on (Dep) $9,717 $16,653 $11,893 $8,493 $6,072

Earnings before tax
(EBT = S – E – dep)

$3,033 $3,067 $10,193 $21,600 $26,729

Tax (tax=0.34)
(EBT x tax)

$1,031 $1,043 $3,466 $7,344 $9,088

Earnings a�er tax
(EAT = EBT – tax)

$2,002 $2,024 $6,728 $14,256 $17,641

Add back deprecia�on
(EAT + Dep)

$11,719 $18,677 $18,621 $22,749 $23,713

Project termina�on cash flow

Income from sale $12,000

Tax effect of sale $201

Recovering net working capital $2,000

Total $14,201

Annual deprecia�on rate 14.29% 24.49% 17.49% 12.49% 8.93%

Calcula�ng the NPV and IRR for the Pogo Harness Project

The Pogo harness project’s cash flows are summarized in Table 6.8. Pacific Offshore’s required rate of return on investments is 12.5%. For our analysis, this required return
(R(r)) is given. In Chapter 7 we will show how to es�mate the required rate of return for a project.

Table 6.8: Pogo harness project cash flow summary

Cash Flow Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

Ini�al investment ($64,384)

Opera�ng cash flows $11,719 $18,677 $18,621 $22,749 $23,713

Ending cash flows $14,201

Cash flows ($64,384) $11,719 $18,677 $18,621 $22,749 $37,914

Present value ($64,384) $10,417 $14,757 $13,078 $14,202 $21,040

NPV $9,110

IRR 17.2%

Required rate of return 12.5%

Using the data given, we can expand Equa�on (6.1) to determine the project’s NPV:

The NPV indicates that the harness project will add $9,110 in value to the company if our cash flow es�mates are correct and if 12.5% is the appropriate required rate of
return. Note that the cash flow in Year 5 is the sum of the opera�ng cash flow and termina�on cash flow ($23,713 + $14,201).

Now, let’s solve for the IRR for this project.

Solving for the discount rate, the IRR for the Pogo harness project is 17.2%. This is the rate that equates the present value of the cash flows in Years 1 through 5 to $64,384.

Comparing NPV and IRR Results for the Pogo Harness Project

As we discovered in our calcula�ons above, the IRR for the Pogo harness project is greater than its required rate of return (17.2% > 12.5%), making it a worthwhile project
for the company to pursue. If Pacific Offshore Ltd. chooses to invest in the project, it will add $9,110 to the value of the company. Stated another way, if it does not take on
the project, it will have missed an opportunity to increase firm value by that amount.

The rela�onship between NPV and the discount rate is worth exploring further. Figure 6.2 plots the NPV of the Pogo harness project at various discount rates. One of the
discount rates is the IRR, which is 17.2%. The IRR of 17.2% is the point at which NPV = 0. (This is where the line crosses the x-axis.) Note that at discount rates less than
17.2%, NPV is posi�ve, while rates above 17.2% yield nega�ve NPVs.

Figure 6.2: Pogo harness project NPV at various discount rates
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The NPV of a project declines if the project has a higher required return.

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A city that wants an upgraded stadium must decide whether to
renovate or build anew. This is an example of a mutually exclusive
project.

Ge�y Images News/Ge�y Images

6.5 Selecting Mutually Exclusive Projects

With the Pogo harness project, we saw that both NPV and IRR will lead to the correct accept or reject decision. The harmony between NPV and IRR exists because the Pogo
harness project is independent. When projects are not independent, the harmony between NPV and IRR breaks down. There are primarily two types of project dependency:
mutually exclusive projects and limited capital budget (capital ra�oning). In each case, acceptable projects (i.e., those with NPV > 0 and IRR > R(r)) must compete against one
another. This implies that some acceptable projects will not be taken. In this sec�on we deal with mutually exclusive projects.

Mutually exclusive projects compete with others, all of which are acceptable using NPV and IRR decision rules.
The analyst must choose the best of these projects and discard the rest. In most cases, both NPV and IRR will
iden�fy the same best project; that is, the project with the highest NPV will also have the highest IRR.
However, the analyst watches for condi�ons under which NPV and IRR disagree:

The �ming of cash flows differs substan�ally between projects. For example, most cash flows for one
project occur early in its life, while those for another project occur late in its life.
Projects are of substan�ally different size, meaning that one requires a much larger investment than the
other.

Timing of Project Cash Flows

Let’s look at four mutually exclusive projects to see how �ming of cash flows impacts the accord between IRR
and NPV.

Table 6.9 shows cash flows, IRR, and NPV for mutually exclusive projects A–D. The required rate of return, R(r),
equals 10%. The �ming of cash flows differ over the six-year period.

Table 6.9: Mutually exclusive projects with different �ming of cash flows

Project Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 NPV at 10% IRR

A ($9,000) $2,400 $2,400 $2,400 $2,400 $2,400 $2,400 $1,453 15.3%

B ($9,000) $500 $500 $2,000 $2,000 $6,000 $6,000 $1,849 14.6%

C ($9,000) $1,700 $1,700 $1,700 $1,700 $1,700 $1,700 ($1,596) 3.4%

D ($9,000) $5,000 $5,000 $1,000 $1,000 $300 $300 $1,468 19.4%

Both NPV and IRR indicate that project C is not acceptable; therefore, we can concentrate on the remaining three projects. Project A is an annuity, project B’s cash flows
occur mostly in the later years, and project D’s cash flows occur mostly in the early years. No�ce that the project with the late cash flows (B) has the highest NPV, and the
project with the early cash flows (D) has the highest IRR. Project A, the annuity, has neither the highest NPV nor the highest IRR. Projects A, C, and D all have IRR greater
than the R(r) of 10%; this means that they could all be considered acceptable projects, because their IRR > R(r).

Why does �ming of cash flows lead to conflicts between NPV and IRR? Because, for acceptable projects, NPV discounts cash flows at a lower rate, R(r), than does IRR. This
affects project selec�on because discount rates are also compounding rates of return.

In general, IRR favors projects whose cash flows occur mostly in the early years. NPV, which is less affected by compounding because of its lower discount rate, does not
favor projects with early cash flows. NPV favors project B with its greater dollar cash flows, even though they occur in the later years. Although this may seem like a technical
triviality, it is not if it causes disagreement between NPV and IRR.

Differences in Size of the Initial Investment

Now, let us look at four mutually exclusive projects to see how a different size in ini�al investment disrupts the harmony between IRR and NPV.

Table 6.10 shows the ini�al investment, cash flows, NPV, and IRR for mutually exclusive projects E–H. As in the previous example, the required rate of return, R(r), equals
10%.

Table 6.10: Mutually exclusive projects with different ini�al investments

Project Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 NPV at 10% IRR

E ($5,000) $1,800 $1,800 $1,800 $1,800 $1,800 $1,800 $2,839 27.7%

F ($10,000) $3,300 $3,300 $3,300 $3,300 $3,300 $3,300 $4,372 23.9%

G ($15,000) $4,500 $4,500 $4,500 $4,500 $4,500 $4,500 $4,599 19.9%

H ($20,000) $5,700 $5,700 $5,700 $5,700 $5,700 $5,700 $4,825 17.9%

We see that the largest project (H) has the largest NPV. We expect larger projects to produce greater value. Conversely, the smallest project (E) has the highest IRR. In
general, lower-cost investments tend to have higher IRR. If two investments have equal cash flows, the investment that costs the least must have the highest rate of return.

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Resolving the Conflict: Choosing NPV Over IRR

Most of the �me, an analyst will not be faced with having to resolve a conflict between NPV and IRR. Conflicts are irrelevant if projects are independent. For mutually
exclusive projects, conflicts are likely only when there are substan�al differences in �ming of cash flows or differences in project size. When conflicts arise, there are three
reasons for choosing NPV.

Reason 1: The Rate of Return. For acceptable projects, NPV assumes that project cash flows are compounded at the required rate of return, whereas IRR assumes that cash
flows are compounded at a higher rate [IRR > R(r)]. Consider project D in Table 6.8 with its IRR of 19.4%. This project generates a $5,000 cash flow in Year 1 of a six-year life.
The IRR method implicitly assumes that this $5,000 is invested at 19.4% for the remaining five years. The NPV method, on the other hand, assumes that the $5,000 is
invested at 10% over the same period. Unless the company has other investment opportuni�es that yield close to 19.4%, it is more prudent to assume that the project cash
flows will earn about 10%, which represents the company’s opportunity cost.

Reason 2: Value Crea�on. NPV is a direct measure of value. For example, project H in Table 6.9 is expected to add $4,825 in value to the company, whereas project E, which
has the highest IRR, will add only $2,839 in value.

Reason 3: Mul�ple IRRs. The cash flows for all investments follow the same basic pa�ern: An ini�al investment is followed by cash inflows, usually over a number of years.
Some�mes the ini�al investment may last for a number of years before the cash inflows begin. For most projects, once the cash inflows begin, they con�nue un�l the project
is terminated. However, there are cases where a project may require a midlife investment. A planned upgrade of a plant or equipment would be an example. The cash flow
pa�ern of such an investment is shown in Table 6.11.

Table 6.11: Sample cash flow for project with midlife investment

Year Cash flow (+ or –)

1 –

2 +

3 +

4 +

5 –

6 +

7 +

With this or any similar cash flow pa�ern involving more than one sign change, there is more than one discount rate that renders NPV = 0. Therefore, the project has more
than one IRR. For such projects, the calculated IRR is unreliable, and IRR should be abandoned in favor of NPV.

Payback as an Alternative to NPV and IRR

Before electronics gave us a hand in making complex calcula�ons, payback was the most common means of evalua�ng a project. Payback period is simply a measure of how
many years it takes a project to recoup its ini�al investment. Table 6.12 compares the payback period for projects A–D from Table 6.9 with their NPV and IRR.

Table 6.12: Comparing payback to NPV and IRR for mutually exclusive projects

Project Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Payback Years NPV at 10% IRR

A ($9,000) $2,400 $2,400 $2,400 $2,400 $2,400 $2,400 3.75 $1,453 15.3%

B ($9,000) $500 $500 $2,000 $2,000 $6,000 $6,000 4.67 $1,849 14.6%

C ($9,000) $1,700 $1,700 $1,700 $1,700 $1,700 $1,700 5.29 ($1,596) 3.4%

D ($9,000) $5,000 $5,000 $1,000 $1,000 $300 $300 1.8 $1,468 19.4%

The calcula�on of payback is quite simple. Project A in has an ini�al investment of $9,000, and its inflows are outlined in Table 6.13.

Table 6.13: Project A cash inflows

Year Cash flow Cumula�ve cash inflow

1 $2400 $2400

2 $2400 $4800

3 $2400 $7200

4 $2400 $9600

5 $2400 $12000

6 $2400 $14400Processing math: 0%

The benefit of simple calcula�on is no
longer a viable reason to use payback
rather than NPV and IRR as advanced
so�ware and calculators now do the work
for you.

PR Newswire/Associated Press

From the cumula�ve cash flow column, we see that the $9,000 will be returned some�me during the fourth year. Through
interpola�on, we determine that payback occurs three-quarters of the way through the fourth year, making the payback period
3.75 years. Payback periods for projects B–D are similarly calculated.

Payback’s single virtue is that it is easily calculated. However, payback has some serious short-comings. First, we have not
discounted future cash flows, so the �me value of money is ignored. This flaw could be overcome by calcula�ng a payback period
using discounted annual cash flows. Discoun�ng complicates the calcula�on, elimina�ng payback’s most basic virtue. Payback also
disregards cash flows occurring a�er the payback period. Project D’s payback is 1.8 years, yet the project produces cash flows for 4
more years. If the last 4 years of Project D’s cash flows were to disappear, its payback would remain at 1.8 years although the
project’s value would have diminished.

No�ce in Table 6.12 that projects with the lowest payback period are those with the highest IRR. This is because both payback and
IRR favor projects with large early cash flows. If we are seeking projects with the greatest NPV, payback leads us down the wrong
path. No�ce also that we cannot automa�cally exclude project C, which has a payback of 5.29 years. Project C is unacceptable
because of its nega�ve NPV and IRR, which is less than the 10% required return. However, we do not know whether 5.29 years is
an unacceptably long payback period. This absence of a defini�ve criterion for accep�ng or rejec�ng a project is yet another failing
of payback. Ease of calcula�on is not a compelling enough reason to use payback in the age of electronics.

Processing math: 0%

Op�ons are more familiar than you would think, as coupons are a
type of op�on. Can you think of any other examples of everyday
op�ons?

PR Newswire/Associated Press

6.6 Options in Capital Projects

To most people, op�ons are synonymous with stock op�ons, which are contracts that investors buy and sell in the op�ons market. Stock op�ons have become big news in
recent years because they are o�en a major part of the compensa�on of execu�ves and employees. In this sec�on, we describe op�ons, explain why they are valuable, and
iden�fy op�ons that are associated with capital projects.

Characteristics of Options

All op�ons have certain features in common. First, they give the holder the right to buy or sell valuable assets at some future �me. Some, like call op�ons, specify the price
and �me period in which the asset can be purchased. Other op�ons give the holder the right to perform a certain ac�on. For example, a company may contract with a labor
union for an op�on to lay off factory workers temporarily during cyclical reduc�ons in demand for its product. The company holds a put op�on, which is the right to sell or
rid itself of unneeded labor (an asset). In return, the union may extract concessions from the company in the form of job security or compensa�on. A second feature of
op�ons is that they do not bind the holder to buy or sell. The holder will only exercise the op�on when it is in his or her interest (i.e., it is profitable to do so). A third
feature of op�ons is that the seller must honor the terms of the op�on when the holder chooses to exercise it. Many op�ons, however, carry specific expira�on dates. Once
the date has passed, the op�on has no more value than an expired lo�ery �cket, and the seller is no longer required to honor the terms.

There are two prices associated with op�ons. The first is the price paid by the buyer. This could be in the form of cash, concessions, or some other form. The second is the
price paid when the holder exercises an op�on to buy an asset. It may also be the price received when the holder of a put op�on exercises an op�on to sell. The price paid
on a call op�on or received on a put op�on is called the exercise price.

An Option Example

You may be more familiar with op�ons than you realize. Consider a pizza coupon. Suppose The Pizza Company offers a coupon that gives you the right to buy their famous
Kitchen Sink pizza that normally sells for $13.99 for $9.99, however, you must use it by November 15. This coupon is an example of a call op�on contract. The valuable asset
is the pizza, the exercise price is $9.99, and the expira�on date is November 15.

Another familiar example involves buying a house. A buyer makes an offer on a house for $200,000. The buyer accompanies the offer with an earnest money check for
$2,000. This earnest money is the price of a call op�on on the house, effec�vely taking the house off the market and giving the buyer a few days to reconsider and arrange
financing. The buyer may then exercise her op�on to buy the house for the exercise price of $200,000. If she chooses not to buy, she loses her earnest money.

The cost of the op�on on the home is explicit, $2,000. The pizza coupon is nominally free, but there is a cost (at least The Pizza Company hopes there is a cost). If you were
going to buy a Kitchen Sink pizza without the coupon, then The Pizza Company has lost the difference between the regular price and the coupon price. On the other hand,
you may be tempted to visit The Pizza Company instead of your regular pizza parlor and try a Kitchen Sink pizza. If you like it, you could become a regular customer of The
Pizza Company. If you don’t, you paid only $9.99 for the pizza.

We exercise a call op�on when the value or price of the asset exceeds the exercise price. Obviously, the pizza
coupon would be worthless if you could buy the pizza for $9.99 or less without the coupon. Conversely, we
exercise a put op�on (to sell) if the exercise price exceeds the value of the asset. O�en a put op�on can be
thought of as the opposite of a call op�on. For example, if The Pizza Company thought that it could sell all the
pizzas it could make at $13.99, it would not offer the coupon. It offers the coupon because selling pizza for
$9.99 is be�er than selling it for a lower price, giving away free pizza, or losing inventory through spoilage.

It is important to note that, because op�ons do not have to be exercised, they can never have a nega�ve
value. Op�ons are exercised only when they have value to the holder; otherwise, they simply expire. This is not
to say that an investor cannot lose money by buying an op�on. If you pay $100 for an op�on that ul�mately
proves to be worthless, you will not exercise the op�on, and you will have lost $100. If the op�on has a value
of $50, you will exercise it, losing only $50. This is an important point. If the op�on has any value at all, it
should be exercised. Doing so will at least cut your loss. Next, we examine the five factors that make op�ons
valuable.

Factors That Affect the Value of an Option

Let’s begin by examining two of the factors that make op�ons valuable. The first is the value of the op�oned
asset. The second is the exercise price. Op�ons gain value as the difference between the asset value and
exercise price widens. For a call op�on, this will occur if the asset value rises or the exercise price falls.
Returning to The Pizza Company example, if the normal price of the pizza rises to $15.99 from $13.99, the
coupon rises in value by $2.00. At $13.99, the $9.99 coupon saves you $4.00. At $15.99, it saves you $6.00. If
The Pizza Company offers another coupon for a Kitchen Sink pizza at $8.99, this coupon would be worth $1.00
more than the $9.99 coupon because the savings on a $13.99 pizza would rise from $4.00 to $5.00. If there
were a market for pizza coupons, their minimum value would equal the amount of the savings, which is the
market price minus the exercise price. It is possible that the value could be greater, depending on whether
pizza prices were expected to rise.

Summarizing the first two factors that affect op�on value:

1. All else being equal, as an op�on’s exercise price goes down, the value of a call op�on increases.Processing math: 0%

2. All else being equal, the higher the price of the underlying asset is, the higher the value of a call op�on will be.

Note as well that the coupon might have value even if the current promo�onal price of a pizza were less than the coupon price. The coupon has value, for example, if it does
not expire for a month, during which �me there is a reasonable chance that the price of pizza will rise above the coupon price. Now, suppose that the coupon will expire
tomorrow. It is unlikely that the promo�on will end by tomorrow. In that case, the coupon is essen�ally valueless. Time to expira�on, therefore, is another factor affec�ng
op�on value:

3. All else being equal, as the �me to expira�on increases, the value of a call op�on increases.

Consider a situa�on in which pizza prices are highly variable because of a shortage of mozzarella cheese. When The Pizza Company has to pay a premium for cheese, the
price of the Kitchen Sink pizza may rise to as much as $19.99. The possibility of prices greater than $13.99 makes the coupon even more valuable. People will conserve their
coupons when the price is $13.99 to use when the price is $19.99. At prices above the exercise price, the savings—and hence the value of the coupon—increases dollar for
dollar. At $19.99, the coupon is worth $10.00 ($19.99 – $9.99). When the price is $13.99, the coupon is nominally worth $4.00; however, with the price of pizza almost
certain to rise above $13.99, people are willing to pay more than $4.00 for a coupon. Price vola�lity of the asset, therefore, is another factor affec�ng op�on value.

4. All else being equal, the more vola�le the price of the underlying asset is, the more valuable the op�on will be.

The final factor that affects op�on value relates to the �me value of money lessons we learned earlier in this text. In effect, op�ons allow us to defer payment, and because
money has �me value, this deferred payment adds value to the op�on. The �me value of the deferred payment increases with the interest rate.

5. All else being equal, as interest rates rise, the value of op�on contracts will also increase.

Table 6.14 summarizes the five factors that affect the value of op�ons.

Table 6.14: Factors affec�ng the value of call op�ons

Factor 1 As an op�on’s exercise price goes down, the value of a call op�on increases.

Factor 2 The higher the price or the underlying asset, the higher the value of a call op�on.

Factor 3 As the �me to expira�on increases, the value of a call op�on increases.

Factor 4 Higher vola�lity of the price of the underlying asset increases the value of the op�on.

Factor 5 As interest rates rise, the value of op�on contracts also increases.

Real Options

Op�ons associated with capital projects are known as real op�ons. These are called real op�ons, to dis�nguish them from op�ons contracts on stocks and other securi�es.
Real op�ons are important a�ributes of many projects, although they are o�en difficult to iden�fy and value. Here, we look more closely at different types of real op�ons,
and why the valua�on of real op�ons is o�en problema�c.

Many companies have learned that taking into account the op�ons in projects significantly affects their inves�ng decisions. This op�ons approach has become a par�al
subs�tute for long-range planning, which relies on forecasts of future events. By contrast, op�ons thrive on uncertainty. Rather than relying on forecasts to select projects,
companies may seed a number of projects, recognizing that many will never be developed. They expect that those that are developed will be very profitable. By seeding a
number of projects, a company is giving itself the op�on to develop the projects that are most promising as �me goes by and markets develop.

Op�ons as a Strategic Investment. Op�ons are ideal hedges against an uncertain future, such as unforeseen changes in product demand. For example, a car company could
opt to pay an extra $3 million in design and manufacturing costs for a plant that can quickly change produc�on from one model to another. This would allow the
manufacturer to change over produc�on in just a few days, as opposed to the industry standard of several weeks. This changeover op�on represents a hedge against
unforeseen changes in demand for certain car models. It allows the company to reduce the risk associated with an unclear future, but the op�on does come with a price (in
this case, the price is $3 million).

Op�ons to Close Down and Start Up. The op�on to close a project down and then start it back up at a later �me can be valuable. This op�on can be par�cularly profitable
when dealing with products whose market prices are subject to great vola�lity, such as electricity.

The Op�on to Abandon a Project. In NPV analysis, we es�mate the life of the project. However, as men�oned earlier in this chapter, there are �mes when the project should
be abandoned or terminated prior to the end of its expected life. To determine whether the project should be abandoned in any par�cular year, we compare the salvage
value in that year to the discounted value of the project’s remaining cash flows. If the salvage value exceeds the discounted value of the remaining cash flows, then the
project should be immediately terminated. In effect, we are determining whether the project is worth more dead than alive.

To understand the abandonment op�on, consider the prin�ng press project in Table 6.15. The press ini�ally costs $60,000 and will be used to print a weekly newspaper.
Revenues are generated from adver�sing and subscrip�ons. The expected life of the project is six years. The project’s a�er-tax cash flows follow a typical life cycle pa�ern.
They rise as sales build and then fla�en and decline as compe�ng newspapers enter the market or as demand for weekly newspapers wanes. Because the prin�ng press has
many alterna�ve uses, its salvage value ini�ally declines slowly, reflec�ng wear and tear rather than obsolescence. At a discount rate of 12%, the six-year project has a
posi�ve NPV of $14,410, so the project is accepted.

Table 6.15: Cash flows for prin�ng press project

Data Category Year 1 Year 2 Year 3 Year 4 Year 5 Year 6

Opera�ng cash flows $12,000 $16,000 $18,000 $18,000 $14,000 $10,000Processing math: 0%

Salvage value $56,000 $52,000 $48,000 $44,000 $34,000 $27,000

To determine the op�mal termina�on date, we compute the NPV, including salvage value, for each year. For example, if we abandon a�er the first year, we receive the
opera�ng cash flow of $12,000 for Year 1 plus the salvage value of $56,000. Discoun�ng at 12% results in an NPV (abandon a�er year 1) of $714.29

Using this approach, the NPV of abandonment at Years 2 through 5, are

This analysis shows that the highest NPV occurs when the project ends a�er Year 4. Beyond Year 4, the present value of the addi�onal opera�ng cash flows do not
compensate for the loss in the asset’s salvage value. In other words, the cost of con�nuing exceeds the benefits of con�nuing to receive opera�ng cash flows. Being able to
abandon the project early, in Year 4, raises project value by $1,274, which is the value of the project if it is sold in Year 4 minus its value if it is sold in Year 6 (15,684 –
14,410). If the company had to buy the op�on to terminate the project before Year 6, it could pay up to this amount and shareholders would s�ll benefit. Thus, the
abandonment op�on is worth $1,274.

The value of the abandonment op�on increases with the salvage value of the asset. A primary determinant of salvage value is the number of alterna�ve uses for the asset. A
generic asset that can be used in many applica�ons is generally more valuable than a highly specialized asset with few uses. Specialized assets—machines and equipment—
tend to be produc�vely more efficient. Quite o�en a company must consider the tradeoff between a specialized asset that increases opera�ng cash flows by lowering
produc�on costs and a generic asset that increases salvage value.

Adjusting NPV for the Option Effect

Compu�ng a precise value for real op�ons is difficult because the amount and �ming of payoffs is uncertain and, in some cases, not measurable. In some situa�ons there is
no need to make the computa�on. For example, suppose that a project has a posi�ve NPV before considera�on of an abandonment op�on. In this case, the op�on merely
adds value to an already acceptable project. Unless you are faced with pursuing capital ra�oning or choosing between mutually exclusive investments, where the op�on may
change a project’s rela�ve standing among compe�ng projects, there is no reason to calculate the op�on value. Remember that for an independent project the only
requirement for acceptance is that it has a posi�ve NPV.

When a project has a slightly nega�ve NPV before considera�on of poten�ally value-enhancing op�ons, we must es�mate the op�on’s value, using the five factors that affect
op�on value as a guide. A sufficiently valuable op�on could cause us to accept a project that may otherwise be rejected. Any analysis must begin by determining whether or
not the op�on might ever be used. For example, if we know we will con�nue with a project through its en�re economic life, then the abandonment op�on has no value and
no further considera�on is needed. Of course, not all op�ons are imbedded in projects. Some must be purchased. Other op�ons, even though they may be a�ached to
projects, may add cost to the project. In these cases, the decision must be made on whether to buy the op�on, and this requires us to es�mate the op�on’s value.

Ignoring op�ons that are a�ached to investment projects means ignoring some of the projects’ poten�al value, implying that some profitable projects will be rejected.
Valuing real op�ons remains elusive, but the five factors that affect op�on value can serve as a framework for making es�mates. Considering these op�ons, even in this
rough way, helps managers iden�fy profitable investments.

A final word of cau�on: We must be careful when we modify rigorous analyses with educated guesses. It is possible for a manager to use such hasty analysis to make any
project look profitable. If a project is accepted because of the value of its a�ached op�ons, then those op�ons and the source of their value must be carefully considered.
The presence of op�ons, real or imagined, should not be used as a pretext for taking on ill-advised projects.

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Ch. 6 Conclusion

The decision to invest in long-term assets is crucial to the long-run success of a corpora�on. This investment represents the implementa�on of the corporate mission and
goals. If the company does not have a clear sense of where it is going, it may invest its resources in inappropriate product markets. In this chapter, we outlined a process for
transla�ng a corporate strategic plan into iden�fica�on of specific projects. We also suggested a method of classifying projects according to the amount of management
a�en�on required.

We then showed how to dis�ll dependent projects down to an array of independent projects, which can be evaluated using either NPV or IRR. We showed that NPV and IRR
methods of analysis are en�rely consistent with each other for independent projects but may give conflic�ng accept/reject signals when used to choose from among mutually
exclusive projects. If such conflicts arise, we should opt to select projects on the basis of NPV rather than IRR. In the final analysis, NPV gives us a direct measure of the
value added to the company by an investment project.

Finally, we showed that many investment projects also contain call op�ons on future investment opportuni�es and put op�ons on projects that may be terminated. Although
these op�ons may be difficult to value explicitly, they may nonetheless be useful enough to influence the investment decision.

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Ch. 6 Learning Resources

Key Ideas

Fixed assets can be classified as tangible (machinery, real estate) or intangible (copyrights, patents, contracts).
Net present value measures the value added to the firm by an investment. The NPV of an investment is the present value of the future cash flows minus the ini�al
investment.
The IRR criteria compares the expected return for a project to the required return for investors, given the project’s risk. If the expected return exceeds that requirement,
then the project should be pursued.
Companies may choose to undertake three types of projects: replacement, expansion, and diversifica�on.
Mutually exclusive projects are subs�tutes for each other, requiring either/or decisions.
Independent projects all have equal status, meaning that the company may invest in any knowing that each investment decision does not affect the others.
When es�ma�ng cash flows, consider only incremental cash flows by remembering to beware of allocated costs, consider the opportunity costs of currently owned
resources, ignore sunk costs, and consider incidental effects of the project.

Key Equa�ons

Cri�cal Thinking Ques�ons

1. Individuals and families, like corpora�ons, have long-term investments. What are two investments that most families have?
2. Some companies, such as Motorola, spend millions of dollars each year on employee training. The cost of this training is treated as an accoun�ng expense, but it may really

be an investment. Why might training be an investment?
3. Apple is now the highest valued company in the world at over $600 billion. Apple earned this posi�on by producing products with very high profit margins. Think about

Apple’s products and explain what Apple does to maintain such high profit margins (much higher than compe�tors). High profit margins almost always imply products with
large posi�ve NPVs. What is the source of Apple’s huge posi�ve NPVs?

4. Over the past few years, we have seen film cameras and video rental stores disappear, many book stores close, and much discussion about whether print newspapers will
survive this decade. What does this imply about project proposals that assume 10 or 15 years of high cash flows?

5. In July 2011, Nortel Networks, a now closed Canadian telecommunica�ons company, auc�oned off its patents. The auc�on brought in $4.5 billion from bidders that included
Apple, Microso�, and four other companies. Google was among the companies that were outbid. Why would these companies spend so much on patents? See Web
Resources at the end of Chapter 6 for more details about this auc�on.

Key Terms

Click on each key term to see the defini�on.

abandonment
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The op�on to terminate or sell a project before the end of its func�onal life.

a�er-tax cash flows
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The amount of cash flow remaining a�er taxes have been deducted.

allocated costs
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Costs, such as overhead costs, that do not necessarily change as a result of taking on a project.

call op�on
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The right, but not the obliga�on, to buy an asset at a specified price within a specified �me period.

SLIDE 1 OF 3

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capital budge�ng
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The process in which a business determines whether projects are worth pursuing.

complementary projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investment projects that are related such that all or none must be taken.

dispersion projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investments that add new geographic regions, including other countries, to a company’s opera�ons.

diversifica�on projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investments that add new products or product lines to a company’s opera�ons.

exercise a call op�on
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The purchase of an asset under the terms of an op�on contract.

exercise price
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Price at which an asset may be bought or sold by the owner of an op�on.

expansion projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investments that expand exis�ng capacity, such as adding new machinery or equipment to increase output.

expira�on date
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The date that an op�on to buy or sell an asset lapses.

fixed assets
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Long-term investments. They may be tangible, such as machinery and equipment, or intangible, such as patents and employee training.

hurdle rate
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A required rate of return, or reference point, against which to compare a project’s internal rate of return.

incidental effects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Indirect effects of an investment. Costs or revenues not normally associated with the investment.

incremental cash flows
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The change in corporate cash flows a�ributable to a project.

independent projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The decision to invest in any project has no impact on the decision to invest in any other project.

internal rate of return (IRR)
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The discount rate that equates the present value of an investment’s future cash flows with the investment’s cost.

mutually exclusive projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investment projects that are related such that only one can be taken.

net present value (NPV)
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The present value of future cash flows minus the ini�al investment. NPV is the present value of all cash flows connected to an investment.
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opportunity costs
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The amount of the highest valued forgone alterna�ve.

payback period
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A measure of how many years it takes to recoup the ini�al investment in a project.

put op�on
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The right, but not the obliga�on, to sell an asset at a specified price within a specified �me period.

real op�ons
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Op�ons associated with capital projects.

recovery of net working capital
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The reduc�on in net working capital associated with the termina�on of an investment.

replacement projects
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investments that update or upgrade exis�ng capacity; such as replacing worn out or obsolete machinery and equipment.

sunk costs
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A cost that has already been incurred. The cost is irretrievable. Sunk costs are not relevant in decision making.

Web Resources

A Business Week ar�cle from July 19, 2011 discusses the importance of patents to leading technology companies:
h�p://www.bloomberg.com/news/2011-07-20/patents-are-veryvaluable-tech-giants-discover-nathan-myhrvold.html (h�p://www.bloomberg.com/news/2011-07-20/patents-are-very-
valuable-tech-giants-discover-nathan-myhrvold.html)

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http://www.bloomberg.com/news/2011-07-20/patents-are-very-valuable-tech-giants-discover-nathan-myhrvold.html

Chapter 4

Time Value Applications: Security Valuation and
Expected Returns

Alan Schein Photography/Corbis

Learning Objectives

A�er studying this chapter, you should be able to:

Solve for the value of zero-coupon bonds using �me value of money mathema�cs.
Determine the value of preferred stock using �me value of money mathema�cs.
Solve for the value of constant-growth common stock using �me value of money mathema�cs.
Determine the value of nonconstant-growth common stock using �me value of money mathema�cs.
Explain how factors such as coupon rate, interest rates, and maturity affect bond values over �me.
Solve for the expected rates of return for securi�es, given their market prices and cash flow characteris�cs.

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Ch. 4 Introductio

In Chapter 3, we began the study of money’s �me value. In this chapter, we apply those basics to the valua�on of securi�es (stocks and bonds) and to solving for expected
returns from inves�ng. Along the way, we present some of the terminology and features of corporate securi�es.

The ability to solve for the value of a share of common stock is a fundamental skill for a corporate manager to have. Recall from Chapter 1 that it is management’s job to
maximize shareholders’ wealth, a task impossible to carry out without knowledge of what factors influence share prices and therefore determine the wealth of shareholders.
Common and preferred stock valua�on as well as bond valua�on are also important topics for anyone who may wish to personally invest in such securi�es. The first part of
this chapter introduces security valua�on.

Solving for expected returns is the topic that concludes Chapter 4. When price is known, it may be helpful for the manager (or the investor) to es�mate the return or yield
that can reasonably be expected from a project or investment. Such an expected return can be compared to returns offered by compe�ng projects or investments. An
investor, for example, would not want to invest in a corporate bond whose expected yield was below that of a less risky government bond.

Before beginning, let’s quickly review value. Recall from earlier chapters that value is dependent on cash flows to investors, the �ming of those cash flows, and their riskiness.
The cash flow that a security holder receives is the principal benefit of ownership. Without that benefit, the security would be nearly worthless. Cash flows from the firm to
shareholders come in the form of dividends, and for bondholders the cash received comes in the form of coupon interest payment. As we will demonstrate, shareholders also
receive cash flows from other investors when they sell their stock at a (hopefully!) higher price. This price apprecia�on is due to the expecta�on of higher future dividends,
making the claim on future cash flows more valuable.

Now we apply the �me value of money techniques introduced in Chapter 3 to the valua�on of commonly encountered securi�es.

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What is the present value of a zero-coupon bond that pays $1,000 in 20 years with a required return of 8%
annually?

One of the most common types of zero-coupon bonds is the U.S.
savings bond. Can you think of any other examples of zero-coupon
bonds?

Associated Press

4.1 Zero-Coupon Bonds: A Single Cash Flow

Corpora�ons and the government some�mes issue bonds known as zero-coupon bonds. These bonds differ from typical bonds in that they make no payments to the
bondholders un�l maturity. Let’s consider a bond that matures in 20 years, pays no coupon interest, and has a par value, or maturity value, of $1,000. That is, the investor
will receive $1,000 on the bond’s maturity date but no other cash payments during the life of the bond. If investors require an 8% annual return from this security, based on
annual compounding, what should be the selling price of the bond? The problem is illustrated with a �meline in Figure 4.1, and prac�ced in the Applying Finance: Price of a
Zero-Coupon Bond feature.

Figure 4.1: Determining the present value of a zero-coupon bond

Use Equa�on (3.7) from Chapter 3 to find the current value:

(3.7) PV0 = FVn(1 + r)
-n or FVn /(1 + r)

PV0 = $1,000(1.08)
-20 = $1,000(0.21455) + $214.55

Applying Finance: Price of a Zero-Coupon Bond

Present Value of a Zero-Coupon Bond: How much would an investor pay today for a zero-coupon bond that pays $1,000 in 20 years and earns 8% per year?

To Solve Using TI Business Analyst Calculator

1000 [FV]

8 [I/Y]

20 [N]

0 [PMT]

[CPT] [PV]

= $214.55

Note: Similar to Excel, the PV is displayed as a nega�ve. Also, you may enter the keystrokes in any order you wish so long as you enter CPT PV at the end.

To Solve Using Excel

Use the PV func�on. The inputs for this func�on are: = PV(Rate%,NPER,PMT,FV,Type)

= PV(8%,20,0,1000,0) = (214.55)

The answer displayed is nega�ve (in parentheses or red or signed nega�ve) because that is how much the investor will pay today (an ou�low or nega�ve cash flow) to
receive $1,000 in 20 years. Remember that Excel requires an ou�low and an inflow (i.e., a cash flow signed posi�ve and a cash flow signed nega�ve). When we entered
the posi�ve FV of $1,000 that meant that the PV had to be nega�ve. Cau�on: Remember that numbers cannot be entered with commas separa�ng thousands of dollars
because commas separate inputs in Excel func�ons.

The secondary market for zero-coupon bonds is very ac�ve. Suppose one is selling for $425, maturing in 14
years, at which �me it will pay $1,000 to its holder. In this case, investors would be interested in the yield-to-
maturity (YTM), or the return that the bond offers given its current market price and other characteris�cs. This
problem is illustrated in Figure 4.2, and prac�ced in the Applying Finance: Yield-to-Maturity of a Zero-Coupon
Bond feature.

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What is the YTM of a zero-coupon bond selling for $425 today, if it will pay $1,000 in 14 years?

Figure 4.2: Determining the YTM of a zero-coupon bond

To solve for r, Equa�on (3.4) from Chapter 3 could be used.

This bond is expected to yield 6.303% if held to maturity.

Applying Finance: Yield-to-Maturity of a Zero-Coupon Bond

Yield-to-Maturity of a Zero-Coupon Bond: What annual rate of return will an investor earn if she pays $425 today for a zero coupon that pays $1,000 when it matures in
14 years?

To Solve Using TI Business Analyst

14 [N], 0 [PMT], 425 [+/–] [PV], 1000 [FV], [CPT] [I/I]

Note that, like the Excel keystrokes, either the price (PV) or the par value (FV) has to be nega�ve in order to “tell” the calculator that one cash flow is going to the firm
and one is going to the investor.

To Solve Using Excel

Use the Rate func�on with the format: RATE(NPER, PMT, PV, FV, TYPE, GUESS)

=RATE(14,0, –425,1000,0,10%) = 6.303%

Note: One of the cash flows is nega�ve (the $425) and the other is posi�ve ($1000). There is no comma or dollar sign in the $1000. The TYPE is zero because we assume
interest accrues at the end of the period. GUESS can be le� out or enter something that seems reasonable. The display of the answer can be adjusted to show more or
fewer decimal places by forma�ng the cells.

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Public u�lity companies like Pacific Gas &
Electric are among the main issuers of
preferred stocks. What do you think are
the benefits of preferred stocks?

Ge�y Images News/Ge�y Images

4.2 Preferred Stock: A Perpetuity

The most common type of perpetuity is preferred stock. Preferred stock generally pays a fixed dividend. Thus, eight-dollar preferred refers to a share of preferred stock that
promises to pay a dividend of $8 once per year into the foreseeable future. Preferred stock is known as a hybrid security in that it combines features of both fixed claims
(bonds—i.e., debt) and residual claims (stocks—i.e., equity). Table 4.1 outlines the hybrid quali�es of preferred stock.

Table 4.1: Hybrid security features of preferred stock

Bond-like quali�es Stock-like quali�es

• Fixed dividend payments • Dividend payments not legally guaranteed

• Callable • Interest is not tax deduc�ble

• No maturity date

Preferred stock is fixed in the sense that the amount the issuing corpora�on is obligated to pay does not vary; it is fixed like the coupon rate on debt (in this case it is $8
once every year). Also, similar to some bond issues, preferred stock can be redeemable or callable (we will discuss this further in Sec�on 4.6).

On the other hand, preferred stock is residual because there is no legal obliga�on for the company to pay a dividend unless it has cash flows le� over a�er all other fixed
claims (such as interest on bonds) have been paid. Preferred claims have a lower priority than other fixed claims, but a higher priority than common stock. Therefore, no
dividends can be paid to common stockholders unless preferred dividends have been paid first. This contrasts to interest on debt, which must be paid or the company risks
legal ac�on by bondholders. Interest on debt is tax deduc�ble, but dividends on preferred stock (and common stock) is not, making $1.00 of dividends more expensive for
the company than $1.00 of interest. Also, like common stock, preferred stock has no maturity date.

The present value of a perpetuity formula is used to find the price of a share of eight-dollar preferred. The interest rate equals 16% in the example.

PV0 = CF/r is re-expressed as = D/r because today’s price (P0) is equal to the present value of future cash flows (PV0) and

preferred’s dividend (D) is the perpetuity’s cash flow (CF). The price is $50 per share.

Most preferred stock in the United States is issued by public u�li�es, financial ins�tu�ons, or REITs (real estate investment trusts).
For example, Pacific Gas & Electric (PG&E with �cker symbol PCG), a large California public u�lity, has eight different issues of
preferred stock. One issue is the 6.0% nonredeemable preferred with a par value of $25.00. Each year the stock pays a dividend of
6% of $25 or $1.50. Dividends are paid quarterly so each quarter an investor receives $0.375 per share in dividends. The 5%
preferred, also with a $25.00 par value, pays quarterly dividends of $0.3125 = 5% × $25/4.

On May 29, 2012, the 6% PG&E preferred stock sold for $29.35, down a bit from a price of $29.92 a few days before. At the
$29.35 price, investors are earning a return of 5.1107%. See the Web Resources at the end of the chapter for a link to the trades in
this preferred stock. Preferred stock issues, like PG&E, do not trade very o�en.

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Common stocks for companies such as Google are more familiar to
us than preferred stocks. What are the benefits and drawbacks to
common stocks?

Associated Press

4.3 Common Stock: A Growing Perpetuity

Unlike preferred stock, common stock does not pay dividends that are a constant amount through �me. On the
other hand, common dividends are equally spaced in �me and do con�nue indefinitely. Common stock,
therefore, sa�sfies all the criteria of a perpetuity except the changing amount of its dividend payment.

To find the price of common stock we might use the general formula for the present value of a stream of cash
flows. Again, recognizing that PV0 = P0 and CF1 = D1, CF2 = D2, and so on, we may re-express the formula in

terms of the price (P0) and dividends (D1, D2, . . .) of the common stock:

Clearly, it is impossible to solve this equa�on explicitly because the cash flows (dividends) go on forever.

A number of models have been developed to allow this formula to be solved. The simplest model requires the assump�on that successive dividends grow at a constant rate.
We call this a growing perpetuity.

Let that rate be termed gN, the long-run normal growth rate of dividends. The dividends may be expressed as (1 + gN) mul�plied by the preceding year’s dividend payment:

D0 = the current dividend

D1 = D0(1 + gN)

D2 = D1(1 + gN) = D0(1 + gN)
2

D3 = D2(1 + gN) = D0(1 + gN)
3

Subs�tu�ng into Equa�on (4.1) yields a geometric series:

Mathema�cians have shown that as long as g is less than r, this series can be summed fairly easily.

A constant-growth stock may be valued using the constant-growth formula,

To illustrate the formula, let’s assume a stock has just paid a $5.00-per-share dividend. We believe that future dividends will grow at a 6% rate forever, and investors require
a 13% return on their investment in this stock. The stock’s price should be

The growth rate plays a very important role in determining the value of a share of stock (or any asset). In this example, suppose the dividend growth rate had been 2%
instead of 6%. Then the value of the stock today would be $46.36 = $5.00(1.02)/(0.13 – 0.02). Had the growth rate been zero (like a share of preferred stock), the value
today would be just $5.00/0.13 = $38.46. You can see that the addi�onal growth has a large effect on the value of the stock.

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Even a company as successful as McDonald’s experiences non-constant
growth and drops in stock prices. There are many other examples of non-
constant growth corpora�ons. How is non-constant growth significant to a
financial manager?

4.4 Common Stock: Nonconstant Dividend Growth

The constant-growth valua�on model works well for securi�es whose forecasted financial behavior corresponds to the model’s assump�on of dividends that grow at a
constant rate. Some companies, such as electric u�li�es, compete in mature markets that offer few prospects for rapid growth. Demand for their product is pre�y stable,
varying li�le with economic cycles. Such firms may be good candidates for valua�on using the constant-growth model.

For many corpora�ons, however, the constant-growth assump�on does not hold. O�en firms have new products that have compe��ve advantages over their compe�tors’
products. Patent protec�on, new technology, low-cost produc�on methods, and brand name recogni�on may enable a firm to experience rapid growth for a period of �me.
In the long run, though, this rapid growth is not sustainable as compe�tors’ technology, manufacturing efficiency, and so on, catches up with the industry leader’s, leveling
the playing field in the marketplace. A constant-growth valua�on model is clearly inappropriate for firms that experience a period of nonconstant growth.

McDonald’s Rise from Financial Downturn

Table 4.2 shows the annual dividends that Johnson & Johnson has paid since 2000. No�ce that the constant-growth model would probably not be appropriate for Johnson &
Johnson since the growth rates have been somewhat erra�c. Also no�ce that the firm experienced double-digit growth rates between 2000 and 2008, but then dividend
growth tapered off fairly drama�cally.

Table 4.2: Johnson & Johnson annual dividends since 2000

Year Dividend Annual % change

2000 $0.620 13.8%

2001 $0.700 12.9%

2002 $0.795 13.6%

2003 $0.925 16.4%

2004 $1.095 18.4%

2005 $1.275 16.4%

2006 $1.455 14.1%

2007 $1.620 11.3%

2008 $1.795 10.8%

2009 $1.930 7.5%

2010 $2.110 9.3%

2011 $2.250 6.6%

Note: Based on data from h�p://finance.yahoo.com (h�p://finance.yahoo.com)Processing math: 0%

http://finance.yahoo.com/

There are many reasons why a company might have nonconstant dividend growth. What might account for
abnormal growth periods?

Most corpora�ons, even large ones like
Johnson & Johnson, experience
nonconstant growth due to compe��on
with other companies. How is the ability
to calculate dividends helpful when
buying stock?

Ge�y Images News/Ge�y Images

One method for valuing firms in a nonconstant-growth cycle is presented here. Let’s assume that we are valuing a stock whose
dividends are expected to grow at an 18% rate for each of the next three years. A�er this abnormal growth period, normal growth
will con�nue at a 5% annual rate. The company’s last annual dividend was $2.00 per share. The discount rate for the stock is 16%.
The �meline in Figure 4.3 illustrates the growth assump�ons of this example.

Figure 4.3: Nonconstant dividend growth

As shown in Figure 4.3

gA = abnormal growth rate = 18%

A = length of abnormal growth period = 3

gN = normal or constant-growth rate = 5%

Because today’s price should equal the present value of future dividends, the first step is to find the size of these dividends.

D0 = $2.00 last dividend paid

D1 = $2.00(1.18) = $2.36 In year 1 dividends grow at 18%

D2 = $2.36(1.18) = $2.78 In year 2 dividends grow at 18%

D3 = $2.78(1.18) = $3.28 In year 3 dividends grow at 18%

D4 = $3.28(1.05) = $3.44 In year 4 dividends grow at 5%

Dividends growth at 5% from D4. . .

It is impossible to solve explicitly for the value of all future dividends, and, thus, it is also impossible to find explicitly the present value of all future dividends. But, note that
from point A forward, the growth rate is constant. This means that the assump�ons of the constant-growth valua�on model are met from period 3 onward. We can,
therefore, solve for P3, the stock’s price at �me 3, using the constant-growth model. This value, P3, incorporates the value of all the dividends from �me 3 onward. P3
includes the present value of D4, D5, D6, and so on. Recognizing this gives us a strategy for solving for P0, the current price.

But

Note that P3 is discounted for three periods because it is the price as of period 3 in Figure 4.3. We already know the value of D1, D2, D3, and r, so these values may be

subs�tuted into Equa�on (4.4).

To solve for P3, recall the constant-growth formula from the prior sec�on:

It solved for P0 using D1 because the constant-growth assump�on held from �me 0 onward. In this example, the constant growth holds from �me 3 onward, so we can

adjust the formula rela�ve to �me 3 and solve for P3.

We now have all the values we need to solve for P0, the current price of the stock.

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This price, $26.24, accounts for the present value of all future dividends. The present values of D0, D1 through D3 are solved for explicitly. The present values of D4, D5, D6,

and so on are solved for implicitly by finding the present value of P3. P3 is able to incorporate the values of all dividends a�er �me 3 because dividends grow at a constant

rate from �me 3 onward.

This method may be generalized in the following formula.

where

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Bonds have characteris�cs of annui�es and single cash flows.

A bond’s price is equal to the present value of the coupon stream, plus the present value of par value.

4.5 Bonds: An Annuity and Single Cash Flow

Bonds can be thought of as a combina�on of an annuity and a single cash flow. Bond investors receive from the corpora�on both a stream of coupon interest payments over
the life of the bond (annuity), and a payment of par value at maturity (single cash flow). Most bonds make coupon payments semiannually, and corporate bonds generally
carry a $1,000 par value. The cash flows for a typical bond are illustrated in Figure 4.4.

Figure 4.4: Cash flows for a typical bond

In the figure, m is the number of coupon payment periods un�l the bond matures. For bonds paying coupons semiannually, m is twice the number of years un�l maturity.
Every semiannual coupon payment equals one-half the coupon rate mul�plied by the bond’s par value, so a bond with an 8% coupon rate would make two $40 interest
payments every year (8% × $1,000)/2.

The price of a bond is the present value of the coupon stream plus the present value of par value. The coupon stream is an annuity and the repayment of par value is a
single cash flow. The �meline in Figure 4.5 may be used to find the present value (the price) of a bond.

Figure 4.5: Determining the present value of a bond

The formula for solving for a bond’s value is given here; keep in mind that r is the investors’ required return for the bond (the discount rate per payment period).

A bond that carries an annual coupon rate of 6.5%, makes coupon payments semiannually, has a $1,000 par value, and matures in 10 years would have a value of $684.58 if
the investors discount its cash flows at a 12% annual rate. Note that the 6.5% annual coupon rate is equal to 3.25% semiannually, yielding the $32.50 semiannual coupon
payment. The 12% annual required return is re-expressed as 6% semiannually to agree with the semiannual payment period, and the number of periods is (10)(2) = 20.

This problem is also prac�ced in the Applying Finance: The Price of a Bond feature.

Applying Finance: The Price of a Bond

Finding the Price of a Corporate Bond: If an investor wants to earn a 12% annual return, how much would she pay today for a bond that carries an annual coupon rate of
6.5%, makes coupon payments semiannually, has a $1,000 par value, and matures in 10 years?

To Solve Using TI Business Analyst

6 [I/Y], 20 [N], 1000 [FV], 32.50 [PMT], [CPT] [PV]

Note that both the FV and the PMT are signed posi�ve because they are both cash inflows for investors, and therefore the answer for PV will be nega�ve because it will
be the price investors are willing to pay. Of course, one could make both the FV and PMT nega�ve, and the answer would be posi�ve, taking the cash flows from the
firm’s perspec�ve.

To Solve Using Excel

Excel Solu�on: Use the PV func�ons with the following inputs:

RATE = 6% (This is the semiannual version of the 12% annual discount rate.)Processing math: 0%

Important informa�on can be obtained by examining the historical price changes of corporate bonds.

Many corpora�ons, like Caterpillar Incorporated, rely on various
types of bond issues. What do you think are the benefits of bonds
versus stocks?

Associated Press

NPER = 20 (There are 20 semiannual periods in 10 years.)

PMT = $32.50 (This is the semiannual interest payment to investors = $1,000×6.5%/2)

FV = $1000 (This is the par or face value of the bond that is repaid at maturity.)

TYPE = 0 (Interest payments are paid at the end of periods a�er interest has had �me to accrue.)

= PV(6%,20,32.50,1000,0)

Display shows ($684.577). This is signed nega�ve because this is the amount the investor will pay to purchase the future promised payments.

Note that this bond is selling below its par value ($684.58 < $1,000). It is said to be selling at a discount. Had the bond been valued at $1,000 so that the price was the par value, the bond would be selling at par. A bond whose price is above par is selling at a premium. These pricing differences can be a�ributed to the rela�onship between the bond's annual coupon rate and the investor's required return for the bond. In our example, the annual coupon rate was below the annual required return (6.5% < 12%). If investors paid full par value for the bond, it would only yield the coupon rate—below their requirements for making the investment. Bondholders could not sell this bond for $1,000 because there would be no demand. In order to market the bond, the bondholder must lower the price un�l the yield to the buyer equals the required return. Note that when purchasing a bond at a discount, investors will receive not only coupon payments but also a capital gain because they invest less than $1,000 yet receive the full par value when the bond matures. Buying a bond priced at a premium will lower the yield to investors because they will realize a capital loss over the life of the bond, offse�ng a por�on of their return from the coupon payments. A capital gain or loss becomes part of the bond's return to investors.

Caterpillar Incorporated (CAT) has a number of bonds outstanding. One bond issue has 9.375% coupon rate and
matures in 2021. In Figure 4.6, we show how the price of the bond (the red line) changes as the yield on BAA-
rated corporate bonds changes (the blue line) from 2003 through 2010. You can see the almost inverse
rela�onship between market yields and the price of the bond. As yields fall, as they did in early 2008, the price
of the bond rises. Since the bond has a high coupon rate (9.375%), investors place a high value on the bond when yields are low.

Figure 4.6: CAT bond price change in rela�on to BAA-rated corporate bonds

Based on data from www.bondsonline.com (h�p://www.bondsonline.com)

The CAT 9.375% bonds are not callable; that is, they cannot be redeemed before their maturity date in 2021. CAT has over a dozen different bonds issues. Some are callable,
so the company can redeem them (buy them back for a fixed price) before the maturity date if it wants. To see the en�re list of CAT bonds (as well as bonds issued by its
financing subsidiary), see the Web Resources sec�on at the end of the chapter.

Bonds are useful for illustra�ng the rela�onship between the �me value of cash flows and interest rates. Consider a 20-year bond that carries a 10% annual coupon rate, has
a $1,000 par value, and makes coupon payments semiannually. If investors require a 10% return on the date the bond is ini�ally sold to the public, then the bond’s price will
be $1,000. It will sell at par. On the following day, let’s assume that interest rates rise drama�cally and investors now require a 12% annual return on the bond. Those
investors who bought the bond on the previous day own a security that pays a series of fixed payments that yield 10% on their $1,000 outlay. In order to sell the bond, they

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Homepage

As interest rates rise, bond prices fall, and vice versa. Why is this?

must lower the price so the series of payments will yield a 12% return to the purchaser. Solving for the present value of the bond, given the new 12% annual discount rate,
yields a price of $849.54. If rates had dropped to 8%, for example, the bond would sell for $1,197.93. Thus, the price of a bond moves in the opposite direc�on from interest
rates. To see how to find the price for both these rates using Excel, see the Applying Finance: Solving for Price on a 20-year Bond feature.

Applying Finance: Solving for Price on a 20-year Bond

We can find the bond prices for the 20-year bond easily in Excel:

Excel formula: =PV(6%,40,50,1000,0) = $849.54

Excel formula: =PV(4%,40,50,1000,0) = $1197.93

Now let’s consider a bond iden�cal to the one just described, except it matures in 30 years rather than 20 years. Again, when the appropriate discount rate is 10%, the bond
will sell at par, $1,000. We solve for the price of the bond using a 12% discount rate and an 8% discount rate (see Applying Finance: Solving for Price on a 30-year Bond).

Applying Finance: Solving for Price on a 30-year Bond

We can find the bond prices for the 30-year bond in Excel:

Excel formula: =PV(12%,60,50,1000,0) = $838.39

Excel formula: =PV(8%,60,50,1000,0) = $1226.23

Compare the way that the 20-year and 30-year bond’s prices responded to changes in the interest rate. Note that the longer the maturity of the bond, the more sensi�ve it
is to interest rate changes. Investors, knowing this, generally require a higher return for longer maturity bonds because their prices will have greater responses to any
changes in interest rates. For this reason, yields on longer-term bonds tend to be higher than short-term bond yields, assuming they are issued by equally risky borrowers.
Table 4.3 shows how the prices of bonds vary with maturity and market interest rates. No�ce the longer the maturity is, the greater price change is due to shi�s in market
interest rates.

Table 4.3: Varia�ons in price for a 10% coupon (semiannual payments)

Years to maturity 6.00%
Market rate

8.00%
Market rate

10.00%
Market rate

12.00%
Market rate

14.00%
Market rate

5 $1,170.60 $1,081.11 $1,000.00 $926.40 $859.53

10 $1,297.55 $1,135.90 $1,000.00 $885.30 $788.12

15 $1,392.01 $1,172.92 $1,000.00 $862.35 $751.82

20 $1,462.30 $1,197.93 $1,000.00 $849.54 $733.37

25 $1,514.60 $1,214.82 $1,000.00 $842.38 $723.99

30 $1,553.51 $1,226.23 $1,000.00 $838.39 $719.22

There are two important lessons here. First, bond prices move in the opposite direc�on as movements in interest rates; and second, the longer the maturity of the bond is,
the greater the change in its price is for a given change in rates. This rela�onship is illustrated in Figure 4.7 using a teeter to�er. When the interest rate side goes down, the
price side goes up. The length of the right-hand side of the teeter-to�er may be thought of as the �me un�l the bond matures. The longer the right-hand side, the greater
the movement in price for a given movement in interest rates. Therefore, the longer the maturity, the more risk there is of a large adverse price change. This risk is termed
interest rate risk.

Figure 4.7: The interest rate-bond price teeter-to�er

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Sources like the Wall Street
Journal are useful when
calcula�ng expected returns.
What are some other helpful
resources?

Jerry Arcieri/Corbis

4.6 Solving for Expected Returns

In the preceding sec�ons, we solved for the value of preferred stock, common stock, and bonds. Many issues of these securi�es are ac�vely
traded in financial markets. It is o�en more useful for investors to solve for the returns they might expect to realize from an investment in such
securi�es than to solve for their value. A�er all, prices are generally known in the marketplace, so investors would be more interested in expected
returns on compe�ng securi�es, rather than prices. Similarly, corporate managers can compare expected returns from prospec�ve LHS projects
when deciding how to allocate the firm’s investment dollars among assets. Solving for expected returns is analogous to finding value because the
same formulas are used. Instead of knowing the discount rate and solving for price, however, now we know the price and are solving for the rate
of return.

Let’s consider a preferred stock with a price, as quoted in the Wall Street Journal, of $53.50. We note that this preferred stock pays a $4.50
dividend annually. Recognizing that this preferred stock is a perpetuity, we subs�tute the known quan��es into the perpetuity formula:

The return on this preferred stock is 8.41%. More precisely, 8.41% is the expected return because buyers cannot be certain that they will realize the 8.41% return (the firm
could go bankrupt).

The expected return for common stock is found using Equa�on (4.3), if we assume the stock’s dividends will grow at a constant rate.

Equa�on (4.3) is useful for two reasons. First, it may be used to find the expected return on a share of stock. For example, if a share is selling for $35, next year’s dividend is
expected to be $3 per share, and dividends are expected to grow at a 6% rate indefinitely, then the expected return on an investment in the stock is 14.57%:

The second use of Equa�on (4.3) is to illustrate the sources of the expected return. The first term to the right of the equal sign in Equa�on (4.3) is the dividend yield, D1/ P0.

The second term, gN, is equal to the capital gains rate. For our stock, investors expect an 8.57% return each year from dividends and a 6% return from price apprecia�on.

Let’s now turn to bonds. Because of the complexity of the bond formula, expected returns from bond investments must be solved using either trial and error or a good
financial calculator. To illustrate the trial-and-error method, let’s solve for the expected return on a bond that sells for $800, pays coupons semiannually, matures in 10 years,
carries a 9% coupon rate, and has a $1,000 par value.

Now we must take an educated guess at what r might be. We do have a clue about r: The bond is selling at a discount. Recall that a bond sells at a discount when its yield is
greater than the coupon rate. Therefore, we know that r > 4.5% (expressing rates on a semiannual basis to conform to the coupon payment period). Say that our first guess
for r is 5.5%:

Because $880 is above the actual price, we know we must raise the interest rate, lowering the value of the right-hand side of the equa�on. This �me let’s try 6%. Using 6%,
we get a value of $827.95, s�ll too high, but closer. Now let’s try 6.25%. This �me the answer is $803.29, close enough using trial and error. The approximate expected return
when buying this bond for $800 is 6.25% semiannually, or 12.5% per year. For a bond, the expected return is also called the bond’s yield to maturity, or YTM. This problem
is prac�ced in the Applying Finance: Corporate Bond YTM feature.

Applying Finance: Corporate Bond YTM

Solving for the rate that equates the price to the promised future cash flows: What is the YTM (yield to maturity) of a 10-year bond with a 9% coupon rate, $1,000 face
value, if its price today is $800, and it pays interest semiannually?

To Solve Using TI Business Analyst

20 [N], 45 [PMT], 800 [+/–] [PV], 1000 [FV], [CPT][I/Y]

= 6.284%

Note: The answer will be given as a percentage and will be a semiannual rate so it must be doubled to find the YTM. 12.568% = 6.284 x 2

To Solve Using Excel

=RATE(20,45,–800,1000,0,5%) = 6.284%

This is the semiannual rate, so the annual return (yield to maturity) is 12.568%.

GUESS can be le� out as in : =RATE(20,45,–800,1000,0)
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Many bonds are what is known as callable, meaning that the issuing corpora�on has the op�on to repurchase the bond at a fixed price above the bond’s par value at some
date prior to the bond’s maturity. These features are a�rac�ve to corpora�ons because if the firm issues bonds at a �me when interest rates are high, the call allows the
company to repurchase the bonds early and avoid high interest payments in the future. Corpora�ons that call a bond issue usually finance the repurchase by issuing new
bonds that carry lower yields. This process is known as refunding debt. For an investor, a callable bond carries the risk that the corpora�on may repurchase the bond prior to
maturity, and the bondholder, therefore, will not collect the high interest payment for the length of �me ini�ally an�cipated. Thus, the yield to maturity for a callable bond
may be misleading, and most investors also calculate the yield to call to see what return on their bond investment they are likely to realize. The yield to call is calculated in
the same fashion as the yield to maturity except the �me un�l call is subs�tuted for the �me un�l maturity (thus the number of coupon payments is reduced), and the call
price is subs�tuted for the call premium (thus the ending cash flow is greater than par).

Let’s demonstrate finding the yield to call by using the same bond we just used in the YTM example (see Applying Finance: Yield to Call). We assume that the call price is
$1,100, and the call date is five years from now. This changes the future value to $1,100 rather than the $1,000 used in the earlier example and changes the number of
semiannual coupon payment periods to 10 instead of 20. The discount rate that equates the $800 price with the future cash flows is 8.2% semiannually, for a yield to call of
16.4%. The yield to call is higher than the yield to maturity of 12.5% because of the higher ending cash flow that will be paid sooner if the bond is called. In this case,
investors should not expect that the bond will be called. The bond is selling at a discount, meaning that market rates are currently above the coupon rate offered by the
bond. Thus, the corpora�on would not choose to refund such an issue because it would have to issue bonds carrying a higher yield to replace these exis�ng bonds. It is
cheaper for the corpora�on to let the bonds mature rather than call them. This is the case whenever the yield to maturity is below the yield to call.

Applying Finance: Yield to Call

To Solve Using TI Business Analyst

800 [+/–] [PV] Today’s Price

1100 [FV] The Call Price

10 [N] Semiannual Periods Un�l the Call Date

45 [PMT] Semiannual Coupon

[CPT][ I/Y] = 8.19%, the Semiannual Yield

8.19% x 2 Yield to Call

Most callable bonds have a period during which they cannot be redeemed, usually the first three to seven years a�er issuance. This assures investors that they will receive
some of the interest payments before the bond is redeemed. Another standard feature of callable bonds is a call premium. If the bond is called before it reaches half of its
stated maturity, the company has to pay investors a bonus to repay the bond. Think of this as an early payment penalty. Usually the call premium is an extra year’s interest.

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Ch. 4 Conclusion

Chapter 4 has applied the �me value skills from Chapter 3 to the valua�on of corporate securi�es. Pricing preferred stock was shown to be an applica�on of the formula for
valuing perpetuity. Common stock, when dividends are expected to grow at a constant rate, was valued using the growing perpetuity formula. Bonds were priced using a
combina�on of the present value of a single cash flow (to value the return of par value at maturity) and the formula for finding present value of an annuity (to value the
coupon payments). Varia�ons of the formulas were also used to solve for the expected returns of traded securi�es.

The ability to express equivalent values of cash flows at different points in �me is a fundamental skill in finance. As with any skill, prac�ce increases proficiency and
understanding in solving �me value problems.

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Ch. 4 Learning Resources

Key Ideas

Zero-coupon bonds differ from typical bonds in that they make no payments to the bondholders un�l maturity.
Preferred stock is the most common type of perpetuity and is known as a hybrid security because it combines features of both fixed and residual claims.
Common stock does not pay dividends that are a constant amount through �me; rather, common dividends are equally spaced in �me and con�nue indefinitely.
The constant-growth assump�on does not hold for all companies, and these firms must use the nonconstant-growth dividend model.
Bonds can be thought of as a combina�on of an annuity and a single cash flow.
Callable bonds carry the risk that the corpora�on may repurchase the bond before maturity, meaning the bondholder would lose the high interest payment for the length
of �me ini�ally an�cipated.

Key Equa�ons

Cri�cal Thinking Ques�ons

1. Corporate bonds can have lots of different features. To test your intui�on about how investors look at bonds consider how the presence of the following features would affect
the price an investor would be willing to pay for a bond.

Sinking Fund: A sinking fund requires a company to set aside money over �me to re�re its bonds. For example, if a bond has a 20-year maturity, the company might
begin se�ng aside funds in Year 11 so it will have a significant por�on of the principal accumulated before the bond matures. Some�mes the sinking fund amount is
used to purchase bonds early, but for this ques�on assume that the fund is invested and the bonds are repaid at maturity. How would the existence of a sinking fund
change the price investors are willing to pay for a bond?
Call Feature: This allows a company to repurchase bonds before maturity. Usually callable bonds will have a no-call period (say the first five years) during which the
company cannot repay the debt. For the next five years, the company may have to pay a premium to repay the bonds early. How would the existence of a call feature
change the price investors are willing to pay for a bond? Think about when companies are most likely to want to repay bonds early.
Collateral: Some loans are backed up with collateral; that is, the lenders have the first right to funds from specific corporate assets. We usually call such loans
mortgages. How would the existence of collateral �ed to a bond change the price investors are willing to pay for the bond?

2. Suppose you purchase a bond with an 8% coupon rate for $1,000, which is the bond’s face value. If you hold the bond to maturity, you will earn 8%. If you sell the bond
before it matures, why might you not earn 8%?

3. Preferred stock has fixed dividend payments, though they are somewhat discre�onary. The dividends are not tax-deduc�ble like interest on debt. The dividend payout rate
must be higher than the yield on a company’s debt since the preferred stock has lower priority in bankruptcy. Why would companies issue such a security?

4. We developed a model to value stock based on dividends and dividend growth. If a stock doesn’t pay any dividends does it have no value? Since there are many companies
with valuable stock that don’t pay dividends, we know the answer is “No.” Explain why.

5. What type of investors would be a�racted to zero-coupon bonds? Why would they give up the interim coupon payments in favor of a larger payout at maturity?
6. Common stock is o�en modeled as a growing perpetuity in which the growth rate is constant. If the overall economy is expected to grow at 6% per year over the long run,

would it be reasonable to expect a firm’s stock to grow at an average constant rate of 15% forever? Why or why not? (Hint: Imagine the economy as a pie that gets 6% bigger
each year; then imagine the firm as a piece of that pie that gets 15% bigger every year.)

7. Consider the constant-growth model for valuing a share of stock:

Can you develop a rule regarding the maximize value of gN that can be used in the constant-growth model?

8. Nega�ve values for gN are allowable. Under what circumstances might a firm’s growth be seen as nega�ve? How would stock price be affected as investors con�nue to lower

their expecta�ons about a company’s growth prospects? Do the results from the constant-growth model match your intui�on about how the price would change?
9. If the constant-growth formula is applied to a stock whose growth rate is zero (gN = 0), what will the formula resemble? Will D1 differ from D0 from for a zero-growth stock?

Key Terms

Click on each key term to see the defini�on.

call premium
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Amount in excess of par value that a company must pay when it calls a bond. It is the difference between the call price and the maturity value.

SLIDE 1 OF 7

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callable bond
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A bond that gives the issuing corpora�on the op�on to repurchase the bond at a price above the bond’s par value at some date prior to the bond’s maturity.

constant-growth formula
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A present value formula applied to stock valua�on where dividends are modeled as a growing perpetuity.

coupon rate
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The fixed interest paid by a bond, stated as a percentage of par value.

dividend yield
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The return due to dividends received equals the annual dividend divided by share price.

expected return
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The probability weighted value of an investment, computed by assigning a probability of occurrence to the various possible future values.

growing perpetuity
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An infinite cash flow stream that makes payments at regular intervals (e.g., monthly, annually, etc.), with each payment equaling its predecessor �mes a fixed growth factor.

hybrid security
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Investments that combines features of both fixed claims and residual claims.

interest rate risk
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The amount a bond will repay to the bondholder when it matures. Corporate bonds o�en have a face amount of $1.000.

preferred stock
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The most common type of perpetuity and generally pays a fixed dividend.

refunding debt
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Corpora�ons that call a bond issue usually finance the repurchase by issuing new bonds that carry lower yields.

selling at a discount
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Selling the bond below par value.

selling at par
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Selling the bond at the face amount.

selling at a premium
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Selling the bond at a price above par.

yield to call
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Calcula�on used to see what the return on bond investment is for callable bonds.

yield to maturity (YTM)
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The yield of a debt security computed by considering its price and the �ming of all cash flows, iden�cal to an IRR (internal rate of return).

zero coupon bond
(h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Bonds issued by corpora�ons and the government which make no payments to bondholders un�l maturity.
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Web Resources

PG&E 6% nonredeemable preferred price and volume data for May 2012: h�p://finance.yahoo.com/echarts?s=PCG-PA+Interac�ve#symbol=pcg-
pa;range=5d;compare=;indicator=volume;char�ype=area;crosshair=on;ohlcvalues=0;logscale=off;source= ; (h�p://finance .yahoo.com/echarts?s=PCG-
PA+Interac�ve#symbol5pcg-pa;range=5d;compare=;indicator=volume;char�ype=area;crosshair=on;ohlcvalues=0;logscale=off;source= 😉

List of CAT bonds:
h�p://quicktake.morningstar.com/StockNet/bonds.aspx?Symbol=CAT&Country=USA (h�p://quicktake.morningstar.com/StockNet/bonds.aspx?Symbol=CAT&Country=USA)

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http://finance.yahoo.com/echarts?s=PCG-PA+Interactive#symbol5pcg-pa;range=5d;compare=;indicator=volume;charttype=area;crosshair=on;ohlcvalues=0;logscale=off;source= ;

http://quicktake.morningstar.com/StockNet/bonds.aspx?Symbol=CAT&Country=USA

Discussion 1 Initial Investment

After reading Chapters 3 and 4 of your textbook, address each of the following questions:

· Think of something you want or need for which you currently do not have the funds. It could be a vehicle, boat, horse, jewelry, property, vacation, college fund, retirement money, etc. Select something which costs somewhere between $2,000 and $50,000. Use the “Present Value Formula”, which computes how much money you need to start with now to achieve the desired monetary goal. Assume you will find an investment that promises somewhere between 5% and 10% interest on your money (you choose the rate) and pretend you want to purchase your desired item in 12 years. (Remember that the higher the return, usually the riskier the investment, so think carefully before deciding on the interest rate.) How much do you need to invest today to reach that desired amount 12 years from now?

· You wish to leave an endowment for your heirs that goes into effect 50 years from today. You don’t want to be forgotten after you pass so you wish to leave an endowment that will pay for a grand soirée yearly and forever. What amount would you like spent yearly to fund this grand party? How much money do you have to leave to your heirs 50 years from now assuming that will compound at 6% interest? Assuming that you have not invested anything today, how much would you have to invest yearly to fully fund the annuity in 50 years, again assuming a 6% monthly compounding rate?

Discussion 2 Managing Earnings

Companies often try to keep accounting earnings growing at a relatively steady pace in an effort to avoid large swings in earnings from period to period. They also try to manage earnings targets. Reflect on these practices and discuss the following in your discussion post.

· Are these practices ethical?

· What are two tactics that a financial manager can use to manage earnings?

· What are the implications for cash flow and shareholder wealth?

· Using the financial balance sheet as displayed in the text, provide an example of how purchasing an asset or issuing stocks or bonds could potentially impact earnings targets.

Your post should be 200-250 words in length.

Required Resource

Text

Byrd, J., Hickman, K., & McPherson, M. (2013).

Managerial Finance

[Electronic version]. Retrieved from https://content.ashford.edu/

· Chapter 3: Time is Money

· Chapter 4: Time Value Applications – Security Valuations and Expected Returns

· Chapter 6: Capital Budgeting – Investing to Create Value

Week 2 – Assignment

Return on Investment – Education Funding

Develop a three- to five-page analysis (excluding the title and reference pages) on the projected return on investment for your college education and projected future employment. This analysis will consist of two parts.

Part 1: Describe how and why you made the decision to pursue an MBA. In the description, include calculations of expenses and opportunity costs related to that decision.

Part 2: Analyze your desired occupation. Determine how much compensation (return) you expect to earn and how long will it take to pay back the return on this investment. Use the financial formulas, Net Present Value (NPV), Internal Rate of Return (IRR), and Payback, provided in Chapter 6 of your text.

If you do not have any educational costs due to employee reimbursements or scholarships, you should estimate the cost of your education for your calculations.

The analysis should be comprehensive and reference specific examples from a minimum of two scholarly sources, in addition to your text. The paper must be formatted according to APA.

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