. 2
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¬le=introduction.tex: LP
6 Chapter 1. A Short Introduction.

1·1. The Goal of Finance: Relative Valuation

Finance is such an important part of modern life that almost everyone can bene¬t from under-
standing it better. What you may ¬nd surprising is that the ¬nancial problems facing PepsiCo
or Microsoft are not really di¬erent from those facing an average investor, small business
owner, entrepreneur, or family. On the most basic level, these problems are about how to allo-
cate money. The choices are many: money can be borrowed or saved; money can be invested
into projects, undertaken with partners or with the aid of a lender; projects can be avoided
altogether if they do not appear valuable enough. Finance is about how best to decide among
these alternatives”and this textbook will explain how.
There is one principal theme that carries through all of ¬nance. It is value. It is the question
Theme Number One:
Value! Make Decisions “What is a project, a stock, or a house worth?” To make smart decisions, you must be able to
Based on Value.
assess value”and the better you can assess value, the smarter your decisions will be.
The goal of a good corporate manager should be to take all projects that add value, and avoid
Corporate managers
need to know how to those that would subtract value. Sounds easy? If it only were so. Valuation is often very
value”and so do you.
It is not the formulas that are di¬cult”even the most complex formulas in this book contain
The math is not hard.
just a few symbols, and the overwhelming majority of ¬nance formulas only use the four major
operations (addition, subtraction, multiplication, and division). Admittedly, even if the formu-
las are not sophisticated, there are a lot of them, and they have an intuitive economic meaning
that requires experience to grasp”which is not a trivial task. But if you managed to pass high-
school algebra, if you are motivated, and if you keep an open mind, you positively will be able
to handle the math. It is not the math that is the real di¬culty in valuation.
Instead, the di¬culty is the real world! It is deciding how you should judge the future”whether
The tough aspect about
valuation is the real your Gizmo will be a hit or a bust, whether the economy will enter a recession or not, where
you can ¬nd alternative markets, and how interest rates or the stock market will move. This
book will explain how to use your forecasts in the best way, but it will mostly remain up to you
to make smart forecasts. (The book will explain how solid economic intuition can often help,
but forecasting remains a di¬cult and often idiosyncratic task.) But there is also a ray of light
here: If valuation were easy, a computer could do your job of being a manager. This will never
happen. Valuation will always remain a matter of both art and science, that requires judgment
and common sense. The formulas and ¬nance in this book are only the necessary toolbox to
convert your estimates of the future into what you need today to make good decisions.
To whet your appetite, much in this book is based in some form or another on the law of one
The law of one price.
price. This is the fact that two identical items at the same venue should sell for the same price.
Otherwise, why would anyone buy the more expensive one? This law of one price is the logic
upon which almost all of valuation is based. If you can ¬nd other projects that are identical”
at least along all dimensions that matter”to the project that you are considering, then your
project should be worth the same and sell for the same price. If you put too low a value on
your project, you might pass up on a project that is worth more than your best alternative uses
of money. If you put too high a value on your project, you might take a project that you could
buy cheaper elsewhere.
Note how value is de¬ned in relative terms. This is because it is easier to determine whether
Value is easier relative.
your project is better, worse, or similar to its best alternatives than it is to put an absolute value
on your project. The closer the alternatives, the easier it is to put a value on your project. It is
easier to compare and therefore value a new Toyota Camry”because you have good alternatives
such as Honda Accords and one-year used Toyota Camry”than it is to compare the Camry
against a Plasma TV, a vacation, or pencils. It is against the best and closest alternatives that
you want to estimate your own project™s value. These alternatives create an “opportunity cost”
that you su¬er if you take your project instead of the alternatives.
¬le=introduction.tex: RP
Section 1·2. How do CFOs do It?.

Many corporate projects in the real world have close comparables that make such relative Relative value often
works well in the
valuation feasible. For example, say you want to put a value on a new factory that you would
corporate world.
build in Rhode Island. You have many alternatives: you could determine the value of a similar
factory in Massachusetts instead; or you could determine the value of a similar factory in
Mexico; or you could determine how much it would cost you to just purchase the net output of
the factory from another company; or you could determine how much money you could earn if
you invest your money instead into the stock market or deposit it into a savings account. If you
understand how to estimate your factory™s value relative to your other opportunities, you then
know whether you should build it or not. But not all projects are easy to value in relative terms.
For example, what would be the value of building a tunnel across the Atlantic, of controlling
global warming, or of terraforming Mars? There are no easy alternative projects to compare
these to, so any valuation would inevitably be haphazard.

1·2. How do CFOs do It?

Table 1.1. CFO Valuation Techniques

Method CFO Usage Recommended Explained
Internal Rate of Return (IRR) (76%) Often Chapter 8
Net Present Value (NPV) (75%) Almost Always Chapter 2
Payback Period (57%) Rarely Chapter 8
Earning Multiples (P/E Ratios) (39%) With Caution Chapter 10
Discounted Payback (30%) Rarely Chapter 8
Accounting Rate of Return (20%) Rarely Chapter 10
Pro¬tability Index (12%) Often Chapter 8
Re¬nements Useful in NPV and IRR
Sensitivity Analysis? (52%) Highly Chapter 7
Real Options Incorporated? (27%) Highly Chapter 7
Simulation Analysis (or VaR)? (14%) Highly Chapter 7
Adjusted Present Value (11%) Highly Chapter 22
Cost of Capital ” An Input Into NPV and Needed for IRR
Chapter ??
CAPM (73%) With Caution
Chapter ??
Historical Average Returns (39%) Rarely
Chapter ??
Modi¬ed CAPM (34%) With Caution
Backed out from Gorden Model (16%) Occasionally Chapter 3
Whatever Investors Tell Us (14%) Occasionally Chapter 2

Rarely means “usually no, and often used incorrectly.”

This book will explain the most important valuation techniques. But how important are these The Survey.
techniques in the real world? Fortunately, we have a good idea. In a survey in 2001, Graham and
Harvey (from Duke University) surveyed 392 managers, asking them what techniques they use
when deciding on projects or acquisition. The results are listed in Table 1.1. Naturally, these
are also the techniques that will consume most of this book. Until I explain them formally, let
me try to give you a brief, informal explanation of what these techniques are.
¬le=introduction.tex: LP
8 Chapter 1. A Short Introduction.

The main techniques.

• The gold standard of valuation is the “Net Present Value” (NPV) method. It tries to translate
all present and future project cash ¬‚ows into one equivalent value today. The project is
worth taking only if this value is positive. You will spend much of your time learning the
intricacies of NPV.

• The “Internal Rate of Return” (IRR) method and its variant, the “Pro¬tability Index,” try
to determine if the investment rate of return is higher or lower than the cost of capital.
For example, if a project would earn 30% and you can ¬nance this project with capital
obtained at a rate of return of 10%, IRR suggests that you take this project. For many
projects, IRR comes up with the same recommendation as NPV.

• The “Payback Period” method and its variant, “Discounted Payback,” ask how long it takes
before a project earns back its investment”and both are usually bad methods to judge
projects. (The survey also found that payback is especially popular among managers who
do not have an MBA and who are more advanced in years.)

• The “Earnings multiples” method tries to compare your project directly to others that
you already know about. If your project costs less and earns more than these alternative
opportunities, then the multiples approach usually suggests you take it. It can often be
used, but only with extreme caution.

• The “Accounting Rate of Return” method judges projects by their accounting performance.
This is rarely a good idea. (You will learn that ¬nancial accounting is not designed to
always accurately re¬‚ect ¬rm value.)

Both NPV and IRR are simple ideas, but they rely on inputs that can be di¬cult to obtain.
Input Methods.
Table 1.1 also describes CFOs™ use of some highly recommended techniques that try to help. A
“Sensitivity Analysis” asks what happens if you change your input estimates and/or forecasts.
If you are not 100% sure”and you will rarely be 100% sure”this is always a prudent exercise.
Spreadsheets were designed to facilitate such scenario analyses. “Real options” are embedded
projects that give you a lot of future possibilities. Their valuation is as important as it is di¬cult.
“Simulations” are a form of automated sensitivity anlysis. And “Adjusted Present Value” is a
way to take corporate income taxes into account.
One input that is of special interest to us ¬nance types is the cost of capital. It is an opportunity
The Cost of Capital
cost”where else could you invest money instead? The standard to ¬nd the cost of capital is
the “Capital-Asset Pricing Model,” more commonly abbreviated as CAPM. It tries to tell you
the appropriate expected rate of return of a project, given its contribution to most investors™
portfolio risk. It is a nice and consistent model, but not without problems. Still, the CAPM
and some of its generalizations are often the best methods we have. Interestingly, CAPM use
is more common in large ¬rms and ¬rms in which the CFO has an MBA.
¬le=introduction.tex: RP
Section 1·3. Learning How to Approach New Problems.

1·3. Learning How to Approach New Problems

This book is not just about teaching ¬nance. It also wants to teach you how to approach novel Theme Number Two:
Learn how to approach
problems. That is, it would rather not merely ¬ll your memory with a collection of formulas and
facts”which you could promptly forget after the ¬nal exam. Instead, you should understand
why it is that you are doing what you are doing, and how you can logically deduce it for yourself
when you do not have this book around. The goal is to eliminate the deus ex machina”the god
that was lowered onto the stage to magically and illogically solve all intractable problems in
Greek tragedies. You should understand where the formulas in this book come from, and how
you can approach new problems by developing your own formulas. Learning how to logically
progress when tackling tough problems is useful, not only in ¬nance, but also in many other
disciplines and in your life more generally.
The method of approaching new problems in this book is to think in terms of the simplest Always start simple and
possible example ¬rst, even if it may sometimes seem too banal a problem or a step that you
would rather brush aside. Some students may even be put o¬ by doing the basics, wanting
to move immediately on to the truly interesting, philosophical, or complex problems right
away. However, you should try to avoid the temptation of skipping the simpler problems, the
foundation. Indeed, arrogance about the basics is often more a sign of insecurity and poor
understanding than it is a sign of solid understanding”and even after many years of studying
the subject, I am always surprised about the many novel insights that I still get from pondering
even the most basic problems. I have studied ¬nance for almost two decades now, and this is
an introductory textbook”and yet I still learned a lot thinking about basic issues while writing
this textbook. There was plenty of “simple” material that I had thought I understood, which I
then realized I had not.
Now, working up from simple examples is done in this book by the method of numerical ex- Numerics work well.
ample. You should translate the numerics onto algebra only after you have understood the
simplest numerical form. Start simple even if you want to understand complex problems. This
will take the sting out of the many formulas that ¬nance will throw at you. Here is an example
of how this book will proceed. If you will receive $150 next year if you give me $100 today, you
probably already know that the rate of return is 50%. How did you get this? You subtracted
$100 from $150, and divided by your original investment of $100:

$150 ’ $100
= 50% .

The next step is to make an algebraic formula out of this. Name the two inputs, say, CFt=1 and
CFt=0 for cash ¬‚ow at time 1 and cash ¬‚ow at time 0. Call your result a rate of return and name
it r . To explain the correspondence between formulas and numerics, in this book, the formula
is placed under the numerics, so you will read
$150 ’ $100
50% =
CFt=1 ’ CFt=0

Looks silly? Perhaps”but this is how I ¬nd it easiest to learn. Now you can ask much more
interesting and complex questions, such as what you would end up with if you started with
$200 and earned 50% rate of return two years in a row, what the e¬ect of in¬‚ation and imperfect
competition would be on your rate of return, etc. There will be dozens of other complications
to this formula in this book. But, we are getting ahead of ourselves. So trust me. This book
will cover a lot of theory”but the theory will not be di¬cult when properly defanged.
¬le=introduction.tex: LP
10 Chapter 1. A Short Introduction.

1·4. The Main Parts of This Book

This book will now proceed as follows:
This book has four parts,
plus a synthesis pro
forma chapter.
1. The ¬rst part covers how your ¬rm should make investment decisions, one project at a
time. It covers the basics”rates of returns, the time value of money”and capital budget-
ing. It explains why we often rely on “perfect markets” when we estimate value.

2. The second part explains how corporate ¬nancial statements work, and how they relate
to ¬rm value.

3. The third part covers “investments.” The novel part here is the consideration of how one
investment in¬‚uences the risk of other investments. For example, a coin bet on heads is
risky. A coin bet on tails is risky. Half a coin bet on heads and half a coin bet on tails
has zero risk. This part explains how ordinary investors should look at your portfolio of
bets in overall terms. It then relates this investor problem to what the consequences are
in terms of the corporate cost of capital”that is, the opportunity cost of capital that your
investors incur if they give their money to your corporation rather to another one.

4. The fourth part covers how your projects should be ¬nanced. Should you ¬nd partners to
join you, or borrow money? The former is called equity ¬nancing, the latter is called debt
¬nancing. This part also describes how ¬rms have historically ¬nanced themselves and
how investment banking works. It closes with the subject of corporate governance”how
¬rm owners assure that their ¬rm and other owners will not steal all their money.

The book ends with a keystone chapter”a pro forma analysis of a real company, here PepsiCo.
The synthesis chapter is
not only the standard A pro forma is a projection of the future for the purpose of valuing the company today. In
way of business
virtually every corporation, new corporate propositions have to be put into a pro forma. This
communication, but it
is how new business ideas are pitched”to the CFO, the board, the venture capitalist, or the
also requires you
knowing everything!
investment bank. Pro formas bring together virtually everything that you learn in this book.
To do one well, you have to understand how to work with returns and net present values,
the subject of the ¬rst part of the book. You have to understand how to work with ¬nancial
statements, the next part of the book. You have to understand how to estimate the ¬rm™s cost
of capital, the next part of the book. You have to understand how capital structure, taxes and
other considerations in¬‚uence the cost of capital, the ¬nal part of the book. You will learn what
is easy and what is hard. You will learn what is science and what is art. And you will learn the
limits to ¬nancial analysis.

Let™s set sail.
The Time Value of Money

(Net) Present Values
last ¬le change: Feb 23, 2006 (14:08h)

last major edit: Mar 2004, Nov 2004

In this chapter, we assume that we live in an idealized world of no taxes, no in¬‚ation, no
transaction costs, no di¬erences of opinion, and in¬nitely many investors and ¬rms”which is
called a “perfect market.” Of course, this ¬nancial utopia is often unrealistic, but all the tools
you will be learning in this chapter will continue to work just as well in later chapters where
the world becomes more complex and more “real.”
After some de¬nitions, we begin with the concept of a rate of return”the cornerstone of ¬nance.
You can always earn interest by depositing your money today into the bank. This means that
money today is more valuable than the same amount of money next year. This concept is called
the time-value of money”the present value of $1 is above the future value of $1.
Now, the other side to investing money today in order to receive money in the future is a project,
company, stock or other investment that requires funding today to pay o¬ money in the future”
we want to invest, and companies want to borrow. The process by which ¬rms decide which
projects to undertake and which projects to pass up on is called capital budgeting. The idea
behind this term is that each ¬rm has a “capital budget,” and must allocate its capital to the
projects within its budgets. Capital budgeting is at the heart of corporate decision-making.
You will learn that, to determine the value of projects with given cash ¬‚ows in the future, the
¬rm should translate all future cash ¬‚ows”both in¬‚ows and out¬‚ows”into their equivalent
present values today, and then add them up to ¬nd the “net present value” (NPV). The ¬rm
should take all projects that have positive net present value and reject all projects that have
negative net present values.
This all sounds more complex than it is, so we™d better get started.

¬le=constantinterest.tex: LP
12 Chapter 2. The Time Value of Money.

2·1. Basic De¬nitions

Before we can begin, we have to agree on a common language”for example, what we mean by
a project, a bond, and a stock.

2·1.A. Investments, Projects, and Firms

As far as ¬nance is concerned, every project is a set of ¬‚ows of money (cash ¬‚ows). Most
To value projects, make
sure to use all costs and projects require an upfront cash out¬‚ow (an investment or expense or cost) and are followed
bene¬ts, including, e.g.,
by a series of later cash in¬‚ows (payo¬s or revenues or returns). It does not matter whether the
opportunity costs and
cash ¬‚ows come from garbage hauling or diamond sales. Cash is cash. However, it is important
pleasure bene¬ts.
that all costs and bene¬ts are included as cash values. If you would have to spend more time
to haul trash, or merely ¬nd it more distasteful than other projects, then you would have to
translate these project features into equivalent cash negatives. Similarly, if you want to do a
project “for the fun of it,” you must translate your “fun” into a cash positive. The discipline
of ¬nance takes over after all positives and negatives (in¬‚ows and out¬‚ows) from the project
“black box” have been translated into their appropriate monetary cash values.
This does not mean that the operations of the ¬rm are unimportant”things like revenues,
The black box is not
trivial. operations, inventory, marketing, payables, working capital, competition, etc. These business
factors are all of the utmost importance in making the cash ¬‚ows happen, and a good (¬nan-
cial) manager must understand these. After all, even if all you care about is cash ¬‚ows, it is
impossible to understand them well if you have no idea where they come from and how they
can change in the future.
Projects need not be physical. For example, a company may have a project called “customer
These examples show
that cash ¬‚ows must relations,” with real cash out¬‚ows today and uncertain future in¬‚ows. You (a student) are a
include (quantify)
project: you pay for education and will earn a salary in the future. In addition, some of the
non-¬nancial bene¬ts.
payo¬s from education are metaphysical rather than physical. If knowledge provides you with
pleasure, either today or in the future, education yields a value that should be regarded as a
positive cash ¬‚ow. Of course, for some students, the distaste of learning should be factored
in as a cost (equivalent cash out¬‚ow)”but I trust that you are not one of them. All such non-
¬nancial ¬‚ows must be appropriately translated into cash equivalents if you want to arrive at
a good project valuation!
A ¬rm can be viewed as just a collection of projects. Similarly, so can a family. Your family
In ¬nance, ¬rms are
basically collections of may own a house, a car, tuition payments, education investments, etc.,”a collection of projects.
This book assumes that the value of a ¬rm is the value of all its projects™ net cash ¬‚ows, and
nothing else. It is now your goal to learn how to determine these projects™ values, given cash
There are two important speci¬c kinds of projects that you may consider investing in”bonds
Stocks and Bonds are
just projects with and stocks, also called debt and equity. As you will learn later, in a sense, the stock is the
in¬‚ows and out¬‚ows.
equivalent of investing to become an owner who is exposed to a lot of risk, while the bond is
the equivalent of a lending money, an investment which is usually less risky. Together, if you
own all outstanding bonds (and loans) and stock in a company, you own the ¬rm:

Entire Firm = All Outstanding Stocks + All Outstanding Bonds and Loans .

Anecdote: The Joy of Cooking: Positive Prestige Flows and Restaurant Failures
In New York City, two out of every ¬ve new restaurants close within one year. Nationwide, the best estimates
suggest that about 90% of all restaurants close within two years. If successful, the average restaurant earns
a return of about 10% per year. One explanation for why so many entrepreneurs are continuing to open up
restaurants, despite seemingly low ¬nancial rates of return, is that restauranteurs so much enjoy owning a
restaurant that they are willing to buy the prestige of owning a restaurant. If this is the case, then to value
the restaurant, you must factor in how much the restauranteur is willing to pay for the prestige of owning a
restaurant, just as you would factor in the revenues that restaurant patrons generate. (But there is also an
alternative reason why so many restaurants fail, described on Page 175.)
¬le=constantinterest.tex: RP
Section 2·1. Basic De¬nitions.

This sum is sometimes called the enterprise value. Our book will spend a lot of time discussing
these two forms of ¬nancing”but for now, you can consider both of them just investment
projects: you put money in, and they pay money out. For many stock and bond investments
that you can buy and sell in the ¬nancial markets, we believe that most investors enjoy very
few, if any, non-cash based bene¬ts.
Solve Now!
Q 2.1 In computing the cost of your M.B.A., should you take into account the loss of salary while
going to school? Cite a few non-monetary bene¬ts, too, and try to attach monetary value to them.

Q 2.2 If you purchase a house and live in it, what are your in¬‚ows and out¬‚ows?

2·1.B. Loans and Bonds

Plain bonds are much simpler than stocks or corporate investment projects in general. You Why bonds ¬rst?
should view bonds as just another type of investment project”money goes in and money comes
out”except that bonds are relatively simple because you presumably know what the cash ¬‚ows
will be. For stocks and other projects the complications created by having to guess future cash
¬‚ows can quickly become daunting. Therefore, it makes sense to ¬rst understand the project
“plain bond” well before proceeding to other kinds of projects. Aside, much more capital in the
economy is tied up in bonds and loans than is tied up in stock, so understanding bonds well is
very useful in itself.
A loan is the commitment of a borrower to pay a predetermined amount of cash at one or Finance Jargon: Loans,
Bond, Fixed Income,
more predetermined times in the future (the ¬nal one being called maturity), usually for cash
upfront today. A bond is a particular kind of loan, named so because it binds the borrower
to pay money. Thus, “buying a bond” is the same as “extending a loan.” Bond buying is the
process of giving cash today and receiving a promise for money in the future. Similarly, instead
of “taking a loan,” you can just say that you are “giving a bond,” “issuing a bond,” or “selling
a bond.” Loans and bonds are also sometimes called ¬xed income instruments, because they
“promise” a ¬xed income to the holder of the bond.
Is there any di¬erence between buying a bond for $1,000 and putting $1,000 into a bank savings Bond: De¬ned by
payment next year.
account? Yes, a small one. The bond is de¬ned by its future promised payo¬s”say, $1,100
Savings: De¬ned by
next year”and the bond™s value and price today are based on these future payo¬s. But as the payment this year.
bond owner, you know exactly how much you will receive next year. An investment in a bank
savings account is de¬ned by its investment today. The interest rate can and will change every
day, and next year you will end up with an amount that depends on future interest rates, e.g.,
$1,080 (if interest rates will decrease) or $1,120 (if interest rates will increase).
If you want, you can think of a savings account as consecutive 1-day bonds: when you deposit A bank savings account
is like a sequence of
money, you buy a 1-day bond, for which you know the interest rate this one day in advance,
1-day bonds.
and the money automatically gets reinvested tomorrow into another bond with whatever the
interest rate will be tomorrow. Incidentally, retirement plans also come in two such forms:
de¬ned bene¬t plans are like bonds and de¬ned by how much you will get when you retire;
and de¬ned contribution plans are like bank deposit accounts and de¬ned by how much money
you are putting into your retirement account today”in the real world, you won™t know exactly
how much money you will have when you will retire.
You should already know that the net return on a loan is called interest, and that the rate of Interest and
Non-Interest. Limited
return on a loan is called the interest rate”though we will soon ¬rm up your knowledge about
interest rates. One di¬erence between interest payments and non-interest payments is that the
former usually has a maximum payment, while the latter can have unlimited upside potential.
Not every rate of return is an interest rate. For example, the rate of return on an investment in
a lottery ticket is not a loan, so it does not o¬er an interest rate, but just a rate of return. In real
life, its payo¬ is uncertain”it could be anything from zero to an unlimited amount. The same
applies to stocks and many corporate projects. Many of our examples use the phrase “interest
rate,” even though the examples almost always work for any other rates of return, too.
¬le=constantinterest.tex: LP
14 Chapter 2. The Time Value of Money.

2·1.C. U.S. Treasuries

Bonds may be relatively simple projects, but bonds issued by the U.S. government”called
Start with the simplest
and most important Treasuries”are perhaps the simplest of them all. This is because Treasuries cannot fail to
bonds: Treasuries.
pay. They promise to pay U.S. dollars, and the United States has the right to print more U.S.
dollars if it were ever to run out. So, for Treasuries, there is absolutely no uncertainty about
repayment. This is convenient because it makes it easier to learn ¬nance”but you should study
them not just because they are convenient tutorial examples. See, Treasuries are the single most
important type of ¬nancial security in the world today. As of October 2004, the United States
owed about $7.4 trillion, roughly $25,000 per citizen. After Treasuries are sold by the United
States government, they are then actively traded in what is one of the most important ¬nancial
markets in the world today. It would not be uncommon for dedicated bond traders to buy
a 5-year Treasury originally issued 10 years ago, and 10 seconds later sell a 3-year Treasury
issued 6 years ago”buyers and sellers in Treasuries are easily found, and transaction costs
are very low. In 2001, average trading volume in Treasuries was about $300 billion per trading
day (about 255 per year). Therefore, the annual trading volume in U.S. Treasuries of about $70
trillion totaled about ¬ve to ten times the U.S. economy™s gross domestic product (GDP) of $10
The abbreviation Treasury comes from the fact that the debt itself is issued by the U.S. Treasury
U.S. Treasury Bills,
Notes, and Bonds have Department. Treasury bills (often abbreviated as T-bills) with maturities of less than one year,
known and certain
Treasury notes with maturities between one and ten years, and Treasury bonds with maturities
greater than ten years. The 30-year bond was often called the long bond, at least before the
Treasury suspended its issuance in October 2001”now, even a ten-year bond is often called
the long bond. These three types of obligations are really conceptually the same, so they are
usually called U.S. Treasuries or just Treasuries.

2·2. Returns, Net Returns, and Rates of Return

The most basic ¬nancial concept is that of a return. The payo¬ or (dollar) return of an invest-
De¬ning: Return, Net
Return, and Rate of ment is simply the amount of cash it returns. The net payo¬ or net return is the di¬erence
between the return and the initial investment, which is positive if the project is pro¬table and
negative if it is unpro¬table. The rate of return is the net return expressed as a percentage
of the initial investment. (Yield is a synonym for rate of return.) For example, an investment
project that costs $10 today and returns $12 in period 1 has

Return at Time 1 $12

= ,
Returnt=1 CF1

= $12 ’ $10 =
Net Return from Time 0 to Time 1 $2
= CF1 ’ CF0 ,
Net Returnt=0,1

$12 ’ $10 $12
Rate of Return from Time 0 to Time 1 = = ’ 1 = 20%
$10 $10
CF1 ’ CF0 CF1
r1 = rt=0,1 = = ’1 .

Percent (the symbol %) is a unit of 1/100. So, 20% is the same as 0.20. Also, please note my way
to express time. Our most common investment scenario is a project that begins “right here
right now this moment” and pays o¬ at some moment(s) in time in the future. We shall use the
letter t to stand for an index in time, and zero (0) as the time index for “right now.” The length
of each time interval may or may not be speci¬ed: thus, time t = 1 could be tomorrow, next
month, or next year. A cash payout may occur at one instant in time, and thus needs only one
time index. But investments usually tie up cash over an interval of time, called a holding period.
¬le=constantinterest.tex: RP
Section 2·2. Returns, Net Returns, and Rates of Return.

We use a comma-separated pair of time indexes to describe intervals. Whenever possible, we
use subscripts to indicate time. When the meaning is clear, we abbreviate phrases such as the
interval “t = 0, 1” to simply 0, 1, or even just as 1. This sounds more complicated than it is.
Table 2.1 provides some examples.

Table 2.1. Sample Time Conventions

Cash Right Now (index time 0). The time index (“t =”)
is given explicitly.
CashMidnight, March 3, 2025 Cash on Midnight of March 3, 2025. We rely on the
subscript to tell the reader that the explicit subscript
t is omitted.
Cash1 Cash in the Future (at index time 1).
Investment0,Midnight March 3 2025 An Investment made right now to pay o¬ on March
3, 2025.
Investment0,1 A One Period Investment, From Right Now To Time
Returnt=1,2 A One Period Return, From Time 1 To Time 2.
Investment0,2 A Two Period Investment, From Right Now To Time
Return2 A Two Period Return, From Right Now To Time 2.

Returns can be decomposed into two parts: intermittent payments and ¬nal payments. For Capital Gains vs.
example, many stocks pay cash dividends, many bonds pay cash coupons, and many real estate
investments pay rent. Say, an investment costs $92, pays a dividend of $5 (at the end of the
period), and then is worth $110. What would its rate of return be?
$110 + $5 ’ $92 $110 ’ $92 $5
r0,1 = = + = 25%
$92 $92 $92
CF1 + Dividend0,1 ’ CF0 CF1 ’ CF0 Dividend0,1
r0,1 = = + .
Percent Price Change Dividend Yield

The capital gain is the di¬erence in the purchase price over the holding period, not counting
interim payments. Here, the capital gain is the di¬erence between $110 and $92, i.e., the $18
change in the price of the investment. The dividend or coupon divided by the original price
is called the dividend yield or coupon yield when stated in percentage terms. Of course, if
the dividend/coupon yield is high, you might earn a positive rate of return but experience a
negative capital gain. For example, a bond that costs $500, pays a coupon of $50, and then sells
for $490, has a capital loss of $10 (which comes to a ’2% capital yield), but a rate of return of
($490 + $50 ’ $500)/$500 = +8%. Also, when there are dividends, coupons, or rent, prices
follow a predictable pattern”this is because the price has to fall by about the amount of the
payment. For instance, if a stock for $20 were to pay a dividend for $2 and stay at $20, you
should immediately purchase this stock”you would get $2 for free. In fact, in a perfect market,
anything other than a price drop from $20 to $18 at the instant of the dividend payment would
not make sense. Such predictable price change patterns do not appear in rates of return. You
will almost always work with rates of return, not with capital gains”though sometimes you
have to draw the distinction, for example because the IRS treats capital gains di¬erently from
dividends. (We will talk about taxes in Section 6).
When interest rates are certain, they should logically always be positive. After all, you can (Nominal) interest rates
are usually non-negative.
always earn 0% if you keep your money under your mattress”you thereby end up with as
much money next period as you have this period. So why give your money to someone today
who will give you less than 0% (less money in the future)? Consequently, interest rates are
indeed almost always positive”the rare exceptions being both bizarre and usually trivial.
¬le=constantinterest.tex: LP
16 Chapter 2. The Time Value of Money.

Most of the time, people (incorrectly but harmlessly) abbreviate a rate of return or net return by
People often use
incorrect terms, but the calling it just a return. For example, if you say that the return on your $10,000 stock purchase
meaning is usually clear,
was 10%, you obviously do not mean you received 0.1. You really mean that your rate of return
so this is harmless.
was 10%. This is usually benign, because your listener will know what you mean. Potentially
more harmful is the use of the phrase yield, because it is often used as a shortcut for dividend
yield or coupon yield (the percent payout that a stock or a bond provide). So, if you say that
the yield on a bond is 5%, then some listeners may interpret this to mean that the overall rate
of return is 5%, while others may interpret this to mean the coupon yield to be 5%. And there
is yet another complication, because coupon yields are often not quoted relative to the current
price, but relative to the ¬nal payment. If in doubt, ask for a detailed explanation!
Here is a language problem. What does the statement “the interest rate has just increased by
Basis Points avoid an
ambiguity in the English 5%” mean? It could mean either that the previous interest rate, say 10%, has just increased
language: 100 basis
from 10% to 10% · (1 + 5%) = 10.05%, or that it has increased from 10% to 15%. Because this
points is 1 percent.
is unclear, the basis point unit was invented. A basis point is simply 1/100 of a percent. So,
if you state that your interest rate has increased by 50 basis points, you de¬nitely mean that
the interest rate has increased from 10% to 10.05%. If you state that your interest rate has
increased by 500 basis points, you mean that the interest rate has increased from 10% to 15%.

Important: 100 basis points constitute one percent.

Solve Now!
Q 2.3 A project o¬ers a return of $1,050 for an investment of $1,000. What is the rate of return?

Q 2.4 A project o¬ers a net return of $25 for an investment of $1,000. What is the rate of return?

Q 2.5 If the interest rate of 10% increases to 12%, how many basis points did it increase?

Q 2.6 If the interest rate of 10% decreased by 20 basis points, what is the new interest rate?

Anecdote: Interest Rates over the Millennia
Historical interest rates are fascinating, perhaps because they look so similar to today™s interest rates. In 2004,
typical interest rates may range between 2% and 20% (depending on other factors). Now, for over 2,500 years,
from about the thirtieth century B.C.E. to the sixth century B.C.E., normal interest rates in Sumer and Babylonia
hovered around 10“25% per annum, though 20% was the legal maximum. In ancient Greece, interest rates in the
sixth century were about 16“18%, dropping steadily to about 8% by the turn of the millennium. Interest rates
in ancient Egypt tended to be about 10“12%. In ancient Rome, interest rates started at about 8% in the ¬fth
century B.C.E., but began to increase to about 12% by the third century A.C.E. (a time of great upheaval). When
lending resumed in the late Middle Ages (12th century), personal loans in England fetched about 50% per annum
though they tended to hover between 10“20% in the rest of Europe. By the Renaissance, commercial loan rates
had fallen to 5“15% in Italy, the Netherlands, and France. By the 17th century, even English interest rates had
dropped to 6“10% in the ¬rst half, and even to 3“6% in the second half. Mortgage rates tended to be lower yet.
Most of the American Revolution was ¬nanced with French and Dutch loans at interest rates of 4“5%.
¬le=constantinterest.tex: RP
Section 2·3. The Time Value of Money.

2·3. The Time Value of Money

Now turn the rate of return formula 2.2 around to determine how money will grow over time,
given a rate of return.

2·3.A. The Future Value of Money

How much money will you receive in the future if the rate of return is 20% and you invest $100? Future Payoffs Given a
Rate of Return and an
The answer is
Initial Investment.
$120 ’ $100
20% = $100 · (1 + 20%) = $100 · 1.2 = $120
CF1 ’ CF0
r0,1 = CF0 · (1 + r0,1 ) = .


Because the interest rate is positive, a given amount of money today is worth more than the
same amount of money in the future”after all, you could always deposit your money today into
the bank and thereby get back more money in the future. This is an example of the time value
of money”a dollar today is worth more than a dollar tomorrow. This is one of the most basic
and important concepts in ¬nance. The $120 next year is therefore called the future value (FV)
of $100 today. It is the time-value of money that causes its future value to be a bigger number
than its present value (PV). Using these abbreviations, you could also have written the above
r0,1 = FV = PV · (1 + r0,1 ) .

Please note that the time value of money has nothing to do with the fact that the prices of
goods may change between today and tomorrow. (In Section 6, we will discuss in¬‚ation”the
fact that the purchasing power of money can change.) Instead, the time value of money, the
present value, and future value are based exclusively on the concept that your money today can
earn a positive interest, so the same amount today is better than the same amount tomorrow.

2·3.B. Compounding

Now, what if you can earn the same 20% year after year and reinvest all your money? What Interest on Interest (or
rate of return on rate of
would your two-year rate of return be? De¬nitely not 20% + 20% = 40%! You know that you
return) means rates
will have $120 in year 1, which you can reinvest at a 20% rate of return from year 1 to year 2. cannot be added.
Thus, you will end up with
$120 · (1 + 20%) = $144
CF1 · (1 + r1,2 ) = .

This $144”which is, of course, again a future value of $100 today”represents a total two-year
rate of return of
$144 ’ $100 $144
= ’ 1 = 44%
$100 $100
CF2 ’ CF0 CF2
= ’1 = r0,2 .
This is more than 40%, because the original net return of $20 in the ¬rst year earned an addi-
tional $4 in interest in the second year. You earn interest on interest! Similarly, what would be
your three-year rate of return? You would invest $144 at 20%, which would provide you with

$144 · (1 + 20%) = $172.80
CF2 · (1 + r2,3 ) = ,
¬le=constantinterest.tex: LP
18 Chapter 2. The Time Value of Money.

so your three-year rate of return would be
$172.80 ’ $100 $172.80
= ’ 1 = 72.8%
$100 $100
CF3 ’ CF0 CF3
r0,3 = = ’1 = r0,3 .

If you do not want to compute interim cash ¬‚ows, can you directly translate the three sequential
one-year rates of return into one three-year holding rate of return? Yes! The compounding
formula that does this is the “one-plus formula,”

(1 + 72.8%) = (1 + 20%) · (1 + 20%) · (1 + 20%)
(1 + r0,3 ) = (1 + r0,1 ) · (1 + r1,2 ) · (1 + r2,3 ) .

In this case, all three rates of return were the same, so you could also have written this as
r0,3 = (1 + 20%)3 . Figure 2.2 shows how your $100 would grow if you continued investing it
at a rate of return of 20% per annum. The function is exponential, that is, it grows faster and
faster, as interest earns more interest.

Table 2.2. Compounding Over 20 Years at 20% Per Annum


123456789 11 13 15 17 19


Start End Total Factor Total Rate
Period Value Rate Value on $100 of Return
0 to 1 $100 (1+20%) $120.00 1.2 20.0%
1.2 · 1.2 = 1.44
1 to 2 $120 (1+20%) $144.00 44.0%
1.2 · 1.2 · 1.2 = 1.728
2 to 3 $144 (1+20%) $172.80 72.8%
. . . . .
. . . . .
. . . . .

When money grows at a rate of 20% per annum, each dollar invested right now will be worth $38.34 in 20 years. The
money at ¬rst grows about linearly, but as more and more interest accumulates and itself earns more interest, the
graph accelerates upward.
¬le=constantinterest.tex: RP
Section 2·3. The Time Value of Money.

Important: The compounding formula translates sequential future rates of
return into an overall holding rate of return:

(1 + r0,T ) = (1 + r0,1 ) · (1 + r1,2 ) · ... · (1 + rT ’1,T ) . (2.11)
Current Spot Rate
Holding Rate Future 1-Period Rate Future 1-Period Rate

The ¬rst rate is called the spot rate because it starts now (on the spot). If all spot and
future interest rates are the same, the formula simpli¬es into (1+r0,T ) = (1+rt )T .

The compounding formula is so common, it is worth memorizing.

You can use the compounding formula to compute all sorts of future payo¬s. For example, Another example of a
payoff computation.
an investment project costing $212 today and earning 10% each year for 12 years will yield an
overall holding rate of return of

r0,12 = (1 + 10%)12 ’ 1 ≈ 213.8%
(1 + r ) ’ 1 = r0,12 .

Your $212 investment today would therefore turn into a future value of

CF12 = $212 · (1 + 213.8%) ≈ $665.35
CF0 · (1 + r0,12 ) = .

Now, what constant two one-year interest rates (r ) would give you a two-year rate of return of Turn around the
formula to compute
r0,2 = 50%? It is not 25%, because (1 + 25%) · (1 + 25%) ’ 1 = 56.25%. Instead, you need to solve
individual holding rates.

(1 + r ) · (1 + r ) = (1 + r )2 = 1 + 50% . (2.14)

The correct answer is

r= 1 + 50% ’ 1 ≈ 22.47%
= 1 + r0,t ’ 1 = r .

(Appendix 2·3 reviews powers, exponents and logarithms.) Check your answer: (1 + 22.47%) ·
(1+22.47%) ≈ (1+50%). If the 12 month interest rate is 213.8%, what is the one-month interest
rate? By analogy,
(1 + r )12 ≈ 1 + 213.8%

1 + 213.8% ’ 1 = (1 + 213.8%)1/12 ≈ 10% ,

but you already knew this.
Interestingly, compounding works even over fractional time periods. So, if the overall interest You can determine
fractional interest rate
rate is 5% per year, to ¬nd out what the rate of return over half-a-year would be that would
via compounding, too.
compound to 5%, compute

(1 + r0,0.5 ) = (1 + r0,1 )0.5 = (1 + 5%)0.5 ≈ 1 + 2.4695% . (2.17)

Compounding 2.4695% over two (six-month) periods indeed yields 5%,

(1 + 2.4695%) · (1 + 2.4695%) ≈ (1 + 5%)
(1 + r0,0.5 ) = (1 + r0,1 ) .
¬le=constantinterest.tex: LP
20 Chapter 2. The Time Value of Money.

If you know how to use logarithms, you can also determine with the same formula how long it
You need logs to
determine time needed will take at the current interest to double or triple your money. For example, at an interest rate
to get x times your
of 3% per year, how long would it take you to double your money?

log(1 + 100%)
(1 + 3%)x = (1 + 100%) x= ≈ 23.5

log(1 + 3%)
log(1 + r0,t )
(1 + rt )T = (1 + r0,t ) T= .

log(1 + rt )

Solve Now!
Q 2.7 A project has a rate of return of 30%. What is the payo¬ if the initial investment is $250?

Q 2.8 If 1-year rates of return are 20% and interest rates are constant, what is the 5-year holding
rate of return?

Q 2.9 If the 5-year holding rate of return is 100% and interest rates are constant, what is the
annual interest rate?

Q 2.10 If you invest $2,000 today and it earns 25% per year, how much will you have in 15

Q 2.11 What is the holding rate of return for a 20 year investment which earns 5%/year each
year? What would a $200 investment grow to?

Q 2.12 What is the quarterly interest rate if the annual interest rate is 50%?

Q 2.13 If the per-year interest rate is 5%, what is the two-year total interest rate?

Q 2.14 If the per-year interest rate is 5%, what is the ten-year total interest rate?

Q 2.15 If the per-year interest rate is 5%, what is the hundred-year total interest rate? How does
this compare to 100 times 5%?

Q 2.16 At a constant rate of return of 5% per annum, how many years does it take you to triple
your money?

Q 2.17 A project lost one-third of its value each year for 5 years. What was its rate of return,
and how much is left from a $20,000 investment?

Q 2.18 From Fibonacci™s Liber Abaci, written in the year 1202: “A certain man gave one denaro
at interest so that in ¬ve years he must receive double the denari, and in another ¬ve, he must
have double two of the denari and thus forever. How many denari from this 1 denaro must he
have in 100 years?”

Q 2.19 (Advanced) In the text, you received the dividend at the end of the period. In the real
world, if you received the dividend at the beginning of the period instead of the end of the period,
could it change the rate of return? Why?
¬le=constantinterest.tex: RP
Section 2·3. The Time Value of Money.

2·3.C. Confusion: Interest Rates vs. Interest Quotes

Unfortunately, when it comes to interest rates, confusion and “sloppy talk” abounds. See, some Adding rather than
compounding can make
people mistakenly add interest rates instead of compounding them. When the investments, the
forgivably small
interest rates, and the time periods are small, the di¬erence between the correct and incorrect mistakes in certain
computation can be minor, so this practice can be acceptable, even if it is wrong. For example, situations”but don™t be
ignorant of what is
when interest rates are 1%, compounding yields

(1 + 1%) · (1 + 1%) ’1 = 2.01%

(1 + r0,1 ) · (1 + r1,2 ) ’1 = r0,2

1 + r0,1 + r1,2 + r0,1 · r1,2 ’1 = r0,2 ,

which is almost the same as the simple sum coming of r0,1 and r1,2 which comes to 2%. The
di¬erence between 2.01% and 2% is the “cross-term” r0,1 · r1,2 . When returns are small”here
if both returns are about 0.01”then the cross-product will be even smaller”here, it is 0.0001.
This is indeed small enough to be ignored in most situation, and therefore a forgivable approxi-
mation. However, when you compound over many periods, you will accumulate more and more
cross-terms, and eventually the quality of your approximation will deteriorate. It is also the
same approximation if you just work out an average interest rate instead of an annualized in-
terest rate. Doing so again ignores the interest on the interest. And again, this can be forgivable
if the number of time periods and the interest rates are small.

Table 2.3. How Banks Quote Interest Rates

Bank quotes annual rate of 10%
(sometimes confusingly called annual rate, compounded daily)

(should better be called annual quote)

Bank pays daily rate of 10%/365 = 0.0274%
(1 + 0.0274%)365 ’ 1 = 10.5%
Daily rate compounds over 365 days to
(sometimes called e¬ective annual rate, sometimes abbreviated EAR or just EFF)

(Even this is an oversimplication: banks can also compute interest rates based on 360/days per year. Fortunately,
this di¬erence between 360 and 365 days compounding is truly trivial.)

Even banks and many other lenders”who should know how to compound”have adopted a Banks add to the
confusion, quoting
convention of quoting interest rates that may surprise you. What they quote as the annual
interest rates in a
interest rate, is really lower than the actual annual interest rate your money will earn. The strange but traditional
banks will compute daily interest at a rate of their annual interest quote, divided by 365. So, way.
as Table 2.3 shows, in e¬ect, if the bank quotes you an annual interest rate of 10%, it is paying
you 10.5% per annum ($10.50 for every $100) if you leave your money in the bank for a year.
Similarly, many lenders who receive monthly payments”such as mortgage lenders”use the
same method to quote an “annual rate compounded monthly.” That is, if they quote 12% per
annum, they mean to collect 1.0112 ≈ 12.68% per year on the money lent to you. Trust me:
interest rates are not intrinsically di¬cult, but they can be tedious and de¬nitional confusions
often reign in their world.

Important: My best advice when money is at stake: If in doubt, ask how the
interest rate is computed! Even better, ask for a simple illustrative calculation.
¬le=constantinterest.tex: LP
22 Chapter 2. The Time Value of Money.

Digging Deeper: If you want to look up the rate of return on a Treasury bill, you may ¬nd that the Wall Street
Journal quotes a number like 95. What does this mean?
For example, in a Treasury auction, in which the government sells 180-day T-bills that will pay $10,000 in 180
days for $9,500, the discount quote would be

Quoted TB Price = $10, 000 · [1 ’ (180/360) · 10] = $9, 500
$10, 000 · [1 ’ (days to maturity/360) · discount rate] .

The Wall Street Journal then simply prints 95, because T-bills are quoted in units of 100. The real interest rate
at the 95 quote is (10, 000/9, 500) ’ 1 ≈ 5.26%. Therefore, even ignoring the extra 5 days, the 360 day interest
rate is 1.05262 ’ 1 ≈ 10.8%, not 10%. Be this as it may, a big advantage is that it is less confusing in that no one
will confuse 95 for an interest rate. (Incidentally, I have not memorized the meaning, either. If I need it, I read
this box.)

Solve Now!
Q 2.20 If you earn an (e¬ective) interest rate of 12% per annum, how many basis points do you
earn in interest on a typical day?

Q 2.21 If you earn an (e¬ective) interest rate of 12% per annum, and there are 52.15 weeks, how
much interest do you earn on a deposit of $100,000 over one week?

Q 2.22 If the bank quotes an interest rate of 12% per annum, how many basis points do you
earn in interest on a typical day?

Q 2.23 If the bank quotes an interest rate of 12% per annum, and there are 52 weeks, how much
interest do you earn on a deposit of $100,000 over one week?

Q 2.24 How much will your money grow to over the year?

Q 2.25 If the bank quotes an interest rate of 6% per year, what does a deposit of $100 in the
bank come to after one year?

Q 2.26 If the bank quotes a loan rate of 8% per year, what do you have to pay back in one year
if you borrow $100 from the bank?
¬le=constantinterest.tex: RP
Section 2·4. Capital Budgeting.

2·4. Capital Budgeting

Now turn to the ¬‚ip side of the investment problem: if you know how much money you will have Capital Budgeting:
should you budget
next year, what does this correspond to in value today? In a corporate context, your question
capital for a project?
is, “Given that Project X will return $1 million in 5 years, how much should you be willing to
pay to undertake this project today?”

2·4.A. Discount Factor and Present Value (PV)

Start again with the rate of return formula 2.2, The “Present Value
Formula” is nothing but
the rate of return
CF1 ’ CF0 CF1
r0,1 = = ’1 . (2.22) de¬nition”inverted to
CF0 CF0 translate future cash
¬‚ows into (equivalent)
You only need to turn this formula around to answer the following question: if you know the today™s dollars.
prevailing interest rate in the economy (r0,1 ) and the project™s future cash ¬‚ows (CF1 ), what is
the project™s value to you today? For example, if the interest rate is 10%, how much would
you have to save (invest) to receive $100 next year? Or, equivalently, if your project will return
$100 next year, what is the project worth to you today? The answer lies in the present value
formula, which translates future money into today™s money. You merely need to rearrange the
above formula to solve for CF0 ,
≈ $90.91
1 + 10%
= = PV( CF1 ) .
1 + r0,1

Check this: investing $90.91 at an interest rate of 10% will indeed return $100 next period:
$100 ’ $90.91 $100
10% ≈ = ’1 (1 + 10%) · $90.91 ≈ $100

$90.91 $90.91
CF1 ’ CF0 CF1
r0,1 = = ’1 (1 + r0,1 ) · CF0 = .


Thus, you can also state that the present value (PV) of $100 next year is $90.91”the value today
of future cash ¬‚ows. If you can borrow or lend at the interest rate of 10% elsewhere, you will

Anecdote: Fibonacci and the Invention of Net Present Value
William Goetzmann argues that Leonardo of Pisa, commonly called Fibonacci, may have invented not only the
famous “Fibonacci series,” but also the concept of net present value, which is the focus of our chapter.
Fibonacci™s family were merchants in the Mediterranean in the 13th century, with trade relations to Arab
merchants in Northern Africa. Fibonacci wrote about mathematics primarily as a tool to solve merchants™
problems”in e¬ect, to understand the pricing of goods and currencies relative to one another. Think about
how rich you could get if you could determine faster than your competition which goods were worth more in
relation to others! In fact, you should think of Fibonacci and other Pisan merchants as the “¬nancial engineers”
of the 13th century.
Fibonacci wrote his most famous treatise, Liber Abaci at age 30, and published it in 1202. We still are solving
the same kinds of problems today that Fibonacci explained. One of them”which you will solve at the end of
this chapter”is called “On a Soldier Receiving 300 Bezants for his Fief”:

A soldier is granted an annuity by the king of 300 bezants per year, paid in quarterly installments
of 75 bezants. The king alters the payment schedule to an annual year-end payment of 300. The
soldier is able to earn 2 bezants on 100 per month (over each quarter) on his investment. How
much is his e¬ective compensation after the terms of the annuity changed?

To solve this problem, you must know how to value payments at di¬erent points in the future”you must
understand the time value of money. What is the value of 75 bezants in one quarter, two quarters, etc.? What
is the value of 300 bezants in one year, two years, etc.? Yes, money sooner is usually worth more than money
later”but you need to determine by exactly how much in order to determine how good or bad the change is for
the king and the soldier. To answer, you must use the interest rate Fibonacci gives, and then compare the two
di¬erent cash ¬‚ow streams”the original payment schedule and the revised payment schedule”in terms of a
common denominator. This common denominator will be the two streams™ present values.
¬le=constantinterest.tex: LP
24 Chapter 2. The Time Value of Money.

be indi¬erent between receiving $100 next year and receiving $90.91 in your project today. In
contrast, if the standard rate of return in the economy were 12%, your speci¬c project would
not be a good deal. The project™s present value would be
≈ $89.29
1 + 12%
= ,
1 + r0,1

which would be less than its cost of $90.91. But if the standard economy-wide rate of return
were 8%, the project would be a great deal. Today™s present value of the project™s future payo¬s
would be
≈ $92.59
1 + 8%
= = PV( CF1 ) ,
1 + r0,1
which would exceed the project™s cost of $90.91. So, it is the present value of the project,
weighed against its cost, that should determine whether you should undertake the project
today, or whether you should avoid it. The present value is also the answer to the question,
“How much would you have to save at current interest rates today if you wanted to have a
speci¬c amount of money next year (CFt=1 )”?
Let™s extend the example. If the interest rate were 10% per period, what would $100 in two
The PV formula.
periods be worth today? In two periods, you could earn a rate of return of r0,2 = (1 + 10%) ·
(1 + 10%) ’ 1 = 21% elsewhere, so this is your appropriate comparable rate of return. The value
of the $100 is then
$100 $100
= ≈ $82.64
(1 + 10%)2 1 + 21%
CF0 = = = PV( CF2 ) .
(1 + r0,1 ) · (1 + r1,2 ) 1 + r0,2

In this context, the rate of return, r , with which the project can be ¬nanced, is often called
The interest rate can
now be called the “cost the cost of capital. It is the rate of return at which you can borrow money elsewhere. This
of capital.”
cost of capital is determined by the opportunity cost that you bear if you fund your speci¬c
project instead of the alternative next-best investment project elsewhere. Remember”you can
invest your money at this rate instead of investing it in the project. Now, the better these
alternative projects in the economy are, the higher will be your cost of capital, and the lower
will be the value of your speci¬c project with its speci¬c cash ¬‚ows. A project that promises
$1,000 next year is worth less today if you can earn 50% elsewhere than when you can earn only
5% elsewhere. (In this ¬rst valuation part of our book, I will just inform you of the economy-
wide rate of return, here 10%, at which you can borrow or invest. The investments part of the
book will explain how you can this rate of return is determined.)
Remember how bonds are di¬erent from savings accounts? The former is pinned down by
Bond present values and
prevailing interest rates its promised ¬xed future payment, while the latter pays whatever the daily interest rate is.
move in opposite
This induces an important relationship between the value of bonds and the prevailing interest
rates”they move in opposite directions. For example, if you have a bond that promises to
pay $1,000 in one year, and the prevailing interest rate is 5%, the bond has a present value of
$1, 000/(1 + 5%) ≈ $952.38. If the prevailing interest rate suddenly increases to 6%, the bond™s
present value becomes $1, 000/(1 + 6%) ≈ $943.40. You would have lost $8.98, which is about
0.9% of your original $952.38 investment. The value of your ¬xed bond payment in the future
has gone down, because investors now have relatively better opportunities elsewhere in the
economy. They can earn a rate of return of 6%, not just 5%, so if you wanted to sell your bond,
you would have to price it to leave the next buyer a rate of return of 6%. If you had waited
invested your money, the sudden change to 6% would have done nothing to your investment,
because you could now earn the 6%. This is a general implication and worthwhile noting:
¬le=constantinterest.tex: RP
Section 2·4. Capital Budgeting.

Important: The price and implied rate of return on a bond with ¬xed payments
move in opposite directions. When the price of the bond goes up, its implied rate
of return goes down.

The quantity The discount factor is
closely related to the
Discount Factor0,t = (2.28) cost of capital.
1 + r0,t
is called the discount factor (or sometimes, less correctly, the discount rate, though you should
use discount rate as a name for r0,t in this context). When you multiply a cash ¬‚ow by its
appropriate discount factor, you end up with its present value. If you wish, you can also think
of discounting”the conversion of a future cash ¬‚ow amount into its equivalent present value
amount”as the reverse of compounding. In other words, the discount factor translates one
dollar in the future into the equivalent amount of dollars today. Because interest rates are
usually positive, discount factors are usually less than 1”a dollar in the future is worth less
than a dollar today. Figure 2.4 shows how the discount factor declines when the cost of capital
is 20% per annum. After about a decade, any dollar the project earns is worth less than 20
cents to you today.

Table 2.4. Discounting Over 20 Years at a Cost of Capital of 20% Per Annum
Present Value of $1, in dollars


123456789 11 13 15 17 19

in Year

Each bar is 1/(1 + 20%) = 83.3% of the size of the bar to its left. After 20 years, the last bar is 0.026 in height. This
means that $1 in 20 years is worth 2.6 cents in money today.
¬le=constantinterest.tex: LP
26 Chapter 2. The Time Value of Money.

2·4.B. Net Present Value (NPV)

An important advantage of present value is that all cash ¬‚ows are translated into the same unit:
Present Values are alike
and thus can be added, cash today. To see this, say that a project generates $10 in 1 year and $8 in 5 years. You cannot
subtracted, compared,
add up these di¬erent future values to come up with $18”it would be like adding apples and
oranges. However, if you translate both future cash ¬‚ows into their present values, you can
add them. For example, if the interest rate was 5% per annum (and (1 + 5%)5 = (1 + 27.6%) in
over 5 years), the present value of these two cash ¬‚ows together would be
PV( $10 in one year ) = ≈ $9.52 ,
1 + 5%
$8 (2.29)
PV( $8 in ¬ve years ) = ≈ $6.27 .
(1 + 5%)5
PV( CFt ) .
1 + r0,t

Therefore, the project™s total value today (at time 0) is $15.79.
The net present value (NPV) is really the same as present value, except that the word “net”
The de¬nition and use of
NPV. upfront reminds you to add and subtract all cash ¬‚ows, including the upfront investment
outlay today. The NPV calculation method is always the same:

1. Translate all future cash ¬‚ows into today™s dollars;
2. Add them all up.

So, if obtaining your project costs $12 today, then this is a positive NPV project, because
$10 $8
NPV = ’$12 + + ≈ $3.50
1 + 5% (1 + 5%)5
+ + = .
1 + r0,1 1 + r0,5

There are a number of ways to think about the NPV. One way is to think of the NPV of $3.50 as
Thinking what NPV
means, and how it can the di¬erence between the market value of the future cash ¬‚ows ($15.79) and the project™s cost
be justi¬ed.
($12)”the di¬erence being “value added.” You can also think of the equivalent of purchasing
bonds that exactly replicates the project payo¬s. Here, you would want to purchase one bond
that promises $10 next year. If you save $9.52”at a 5% interest rate”you will receive $10.
Similarly, you would have to save $6.27 in a bond that promises $8 in 5 years. Together, these
two bonds exactly replicate the project cash ¬‚ows. The law of one price tells you that your
project should be worth as much as this bond project”the cash ¬‚ows are identical. You would
have had to put away $15.79 today to buy these bonds, but your project can deliver these cash
¬‚ows at a cost of only $12”much cheaper and thus better than your bond alternative.
Still another way is to think of NPV as an indicator of how your project compares to the alterna-
Yet another way to
justify NPV: opportunity tive opportunity of investing at the capital markets, the rates of return being in the denominator
(the discount factors). What would you get if you took your $12 and invested it in the capital
markets instead of this project? You could earn a 5% rate of return from now to next year and
27.6% from now to ¬ve years. Your $12 would grow into $12.60 by next year. Like your project,
you could receive $10, and be left with $2.60 for reinvestment. Over the next 4 years, at the 5%
interest rate, this $2.60 would grow to $3.16. But your project would do better for you, giving
you $8. So, your project achieves a higher rate of return than the capital markets alternative
would achieve.
The conclusion of this argument is not only the simplest but also the best capital budgeting
Take all positive NPV
projects. rule: If the NPV is positive, as it is here, you should take the project. If it is negative, you should
reject the project. If it is zero, it does not matter.
¬le=constantinterest.tex: RP
Section 2·4. Capital Budgeting.


• The Net Present Value Formula is

NPV = CF0 + PV( CF1 ) + PV( CF2 ) + PV( CF3 ) + PV( CF4 ) + ...

= CF0 + + + + + ...
1 + r0,1 1 + r0,2 1 + r0,3 1 + r0,4

= .
1 + r0,t

The subscripts are time indexes, CFt is the net cash ¬‚ow at time t (positive
for in¬‚ows, negative for out¬‚ows), and r0,t is the relevant interest rate for
investments from today to time t. (The Greek sigma notation is just a shorter
way of writing the longer summation above. You will see it again, so if you
are not familiar with it, it is explained in Appendix Section 2·3.)

• The Net Present Value Capital Budgeting Rule states that you should accept
projects with a positive NPV and reject projects with a negative NPV.

• Taking positive NPV projects increases the value of the ¬rm. Taking negative
NPV projects decreases the value of the ¬rm.

• NPV is de¬nitively the best method for capital budgeting”the process by
which you should accept or reject projects.

The NPV formula is so important, it is worth memorizing.

Why is NPV the right rule to use? The reason is that in our perfect world, a positive NPV A “free money”
interpretation of NPV.
project is the equivalent of free money. For example, if you can borrow or lend money at 8%
anywhere today and you have an investment opportunity that costs $1 and yields $1.09, you
can immediately contract to receive $0.01 next year for free. (If you wish, discount it back to
today, so you can consume it today.) Rejecting this project would make no sense. Similarly,
if you can sell someone an investment opportunity for $1, which yields only $1.07 next year,
you can again earn $0.01 for free. Again, rejecting this project would make no sense. (In our
perfect world, you can buy or sell projects at will.) Only zero NPV projects ($1 cost for $1.08
payo¬) do not allow you to get free money. More interestingly, this allows you to conclude
how a perfect world must work: Either positive NPV projects are not easy to come by”they are
not available in abundant amounts, but can be available only in limited quantities to a limited
number of individuals”or the NPV rule must hold. Positive NPV projects must be scarce, or
everyone with access to these positive NPV projects would want to take an in¬nite amount of
these projects, which in turn would continue until the economy-wide appropriate rate of return
would go up to equal the project™s rate of return. Of course, this argument is not here to show
you how to get rich, but to convince you that the NPV rule makes sense and that any rule that
comes to other conclusions than NPV would not.
The translation between future values and present values”and its variant net present value” Recap: NPV may be the
most important building
is the most essential concept in ¬nance. Cash ¬‚ows at di¬erent points in time must ¬rst be


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